Advances in Imaging and Electron Physics, Volume 149: Electron Emission Physics

ADVANCES IN IMAGING ANDELECTRON PHYSICS

VOLUME 149

ELECTRON EMISSION PHYSICS

EDITOR-IN-CHIEF

PETER W. HAWKESCEMES-CNRS

Toulouse, France

HONORARY ASSOCIATE EDITORS

TOM MULVEY

BENJAMIN KAZAN

Advances in

Imaging andElectron Physics

Electron Emission Physics

BY

KEVIN L. JENSENElectronics Science and Technology Division

Naval Research Laboratory

Washington, DC

VOLUME 149

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PRINTED IN THE UNITED STATES OF AMERICA07 08 09 10 9 8 7 6 5 4 3 2 1

In memory of William D. Jensen (July 16, 1938 – July 4, 2007)

for his inspirational devotion to science

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CONTENTS

Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

Future Contributions . . . . . . . . . . . . . . . . . . . . . . . . . xi

Foreword . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii


Kevin L. Jensen

I. Field and Thermionic Emission Fundamentals . . . . . . . . . . . . . . 4

II. Thermal and Field Emission. . . . . . . . . . . . . . . . . . . . . . 47

III. Photoemission . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

IV. Low–Work Function Coatings and Enhanced Emission . . . . . . . . . . 280

V. Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304

VI. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325

vii

PREFACE

Electron emission physics is too vast a theme to be confined within a regular

review article. In this volume, Kevin Jensen examines numerous aspects of

the subject, in particular those of importance in recent generations of the

related devices. A first long section recapitulates the fundamentals and

serves as an introduction to the three succeeding sections. The second covers

the mechanisms of thermal and field emission; the various models

are described and expressions for current density and related quantities

are derived in the two extreme cases. A valuable feature of this chapter is the

meticulous examination of the approximations involved, always a source of

debate. All the steps in the relatively complicated derivations are shown.

Next comes a long section on photoemission with, as before, a presentation

of the models used and the associated physics, culminating in a study of the

emittance and brightness of photocathodes. A last section, very much

the physics of electron emission, discusses coatings with materials of low

work-function and the resulting increase in emission.

This monograph undoubtedly fills a gap in the literature, and I am

delighted that it should appear in these Advances. I shall not be alone in

appreciating the eVort made to present all this material so clearly.

Peter W. Hawkes

ix

FUTURE CONTRIBUTIONS

S. Ando

Gradient operators and edge and corner detection

P. Batson (special volume on aberration-corrected electron microscopy)

Some applications of aberration-corrected electron microscopy

C. Beeli

Structure and microscopy of quasicrystals

A. B. Bleloch (special volume on aberration-corrected electron microscopy)

Aberration correction and the SuperSTEM project

C. Bontus and T. Kohler (vol. 151)

Reconstruction algorithms for computed tomography

G. Borgefors

Distance transforms

Z. Bouchal

Non-diVracting optical beams

A. Buchau

Boundary element or integral equation methods for static and time-

dependent problems

B. Buchberger

Grobner bases

L. Busin, N. Vandenbroucke, and L. Macaire (vol. 151)

Color spaces and image segmentation

G. R. Easley and F. Colonna

Generalized discrete Radon transforms and applications to image

processing

T. Cremer

Neutron microscopy

I. Daubechies, G. Teschke, and L. Vese (vol. 150)

On some iterative concepts for image restoration

xi

A. X. Falcao

The image foresting transform

R. G. Forbes

Liquid metal ion sources

C. Fredembach

Eigenregions for image classification

A. Golzhauser

Recent advances in electron holography with point sources

D. Greenfield and M. Monastyrskii

Selected problems of computational charged particle optics

M. Haider (special volume on aberration-corrected electron microscopy)

Aberration correction in electron microscopy

M. I. Herrera

The development of electron microscopy in Spain

N. S. T. Hirata

Stack filter design

M. Hytch, E. Snoeck, and F. Houdellier (special volume on aberration-

corrected electron microscopy)

Aberration correction in practice

K. Ishizuka

Contrast transfer and crystal images

J. Isenberg

Imaging IR-techniques for the characterization of solar cells

A. Jacobo

Intracavity type II second-harmonic generation for image processing

B. Kabius (special volume on aberration-corrected electron microscopy)

Aberration-corrected electron microscopes and the TEAM project

L. Kipp

Photon sieves

A. Kirkland and P. D. Nellist (special volume on aberration-corrected

electron microscopy)

Aberration-corrected electron micrsocpy

xii FUTURE CONTRIBUTIONS

G. Kogel

Positron microscopy

T. Kohashi

Spin-polarized scanning electron microscopy

O. L. Krivanek (special volume on aberration-corrected electronmicroscopy)

Aberration correction and STEM

R. Leitgeb

Fourier domain and time domain optical coherence tomography

B. Lencova

Modern developments in electron optical calculations

H. Lichte

New developments in electron holography

M. Matsuya

Calculation of aberration coeYcients using Lie algebra

S. McVitie

Microscopy of magnetic specimens

S. Morfu and P. Marquie

Nonlinear systems for image processing

T. Nitta

Back-propagation and complex-valued neurons

M. A. O’Keefe

Electron image simulation

D. Oulton and H. Owens

Colorimetric imaging

N. Papamarkos and A. Kesidis

The inverse Hough transform

R. F. W. Pease (vol. 150)

Significant advances in scanning electron microscopy, 1965–2007

K. S. Pedersen, A. Lee, and M. Nielsen

The scale-space properties of natural images

S. J.Pennycook (special volumeon aberration-corrected electronmicroscopy)


FUTURE CONTRIBUTIONS xiii

E. Plies (special volume on aberration-corrected electron microscopy)

Electron monochromators

T. Radlicka (vol. 151)

Lie algebraic methods in charged particle optics

V. Randle (vol. 151)

Electron back-scatter diVraction

E. Rau

Energy analysers for electron microscopes

E. Recami

Superluminal solutions to wave equations

J. Rodenburg (vol. 150)

Ptychography and related diVractive imaging methods

H. Rose (special volume on aberration-corrected electron microscopy)

The history of aberration correction in electron microscopy

G. Schmahl

X-ray microscopy

J. Serra (vol. 150)

New aspects of mathematical morphology

R. Shimizu, T. Ikuta, and Y. Takai

Defocus image modulation processing in real time

S. Shirai

CRT gun design methods

T. Soma

Focus-deflection systems and their applications

J.-L. Starck

Independent component analysis: the sparsity revolution

I. Talmon

Study of complex fluids by transmission electron microscopy

N. Tanaka (special volume on aberration-corrected electron microscopy)

Aberration-corrected microscopy in Japan

M. E. Testorf and M. Fiddy

Imaging from scattered electromagnetic fields, investigations into anunsolved

problem

xiv FUTURE CONTRIBUTIONS

N. M. Towghi

Ip norm optimal filters

E. Twerdowski

Defocused acoustic transmission microscopy

Y. Uchikawa

Electron gun optics

K. Urban and J. Mayer (special volume on aberration-corrected electron

microscopy)

Aberration correction in practice

K. Vaeth and G. Rajeswaran

Organic light-emitting arrays

M. van Droogenbroeck and M. Buckley

Anchors in mathematical morphology

R. Withers

Disorder, structured diVuse scattering and local crystal chemistry

M. Yavor

Optics of charged particle analysers

Y. Zhu (special volume on aberration-corrected electron microscopy)


FUTURE CONTRIBUTIONS xv

FOREWORD

There is much to the observation of J. M. Ziman (2001), an exceptionally

clear translator of the whisperings of the tenth Muse, when he noted in his

preface to Electrons and Phonons that ‘‘Like a chemical compound, scientific

knowledge is purified by recrystallization,’’ followed by several more breath-

taking metaphors about the value of distilling hard‐won scientific insights

into texts. The debt of the present effort to him lurks behind many a page

written here, giving credence to his insight. After my having profited enor-

mously from the hard‐won nucleations of previous generations, it is time to

contribute in turn.

There is much merit in the international literature on electron emission

physics. To do justice to the field in a short work—or to even read what

is there, much less distill it—is daunting. Present aims perforce are much

more modest. Recognizing that a representation of what exists cannot be

adequately conveyed to those who wish to look, I shall instead try to

convey what I saw when I looked, along with travel notes of the journey

(which describes some features of the process—‘‘random walk’’ describing

the others). To the many whose work has been ignored by such an itinerary,

my intent is not to slight by omission of discussion or reference to

meritorious work.

The whole process of getting a simple electron from inside a material into a

vacuum cuts across many disciplines in physics, and it is therefore no surprise

that many renowned names appear, often repeatedly, from the early decades

of the twentieth century. If not for the ‘‘physicists’ war,’’ as World War II

has come to be called in some circles (see, for example, Chapter 20 of Kevles,

1987), perhaps some of the great names of physics that are reverently men-

tioned herein would not be so widely appreciated outside the high walls of

academia. But greatness is not something that is only born of conflict.

Indeed, progress in physics is largely due to international collegiality, open

discussion, much input from colleagues, and mentorship. I have had the

pleasure of association with many whom I hold in high regard. My experi-

ence, such as it is, is that physics only looks magisterial in the foundation

myths where goateed graybeards pontificate from podiums. Physics research

is a gritty, wonderful struggle, and the give and take, the clashing of ideas, the

absence of certainty, make for very powerful and compelling theater where

the boundary between actor and audience is gone. I am grateful for the honor

and pleasure of sharing the stage with many colleagues. I have tried to give

xvii

some of them their due here where possible, perhaps imperfectly. I would like

to thank some by name, although there are many more to whom I am

grateful (they know who they are). It is a sublime feature of physics that

the enterprise is far greater than its practitioners, traceable to progress being

a collective effort. Still, what defects exist herein are mine and do not reflect

on those whom I call colleagues and friends.

I have had the distinct pleasure of learning a great deal from my colleagues

at the Naval Research Laboratory over the years: F. A. Buot, J. Calame,

H. Freund, A. Ganguly, M. A. Kodis, Y. Y. Lau, B. Levush, K. Nguyen,

P. Phillips, T. Reinecke, J. L. Shaw, A. Shih, J. E. Yater, and E. G. Zaidman.

A research environment in which expertise is but a walk down the hall or

near a coffee pot has no equal.

In 2001, I had the distinct pleasure of spending a sabbatical at the Univer-

sity of Maryland and since then have enjoyed my weekly visits. My UMD

colleagues have been open, gregarious, stimulating, and beneficial: P. G.

O’Shea, D. W. and R. Feldman, N. A. Moody (now at Los Alamos National

Laboratory), D. Demske, and E. Montgomery. I remain deeply indebted to

P. O’Shea and D. Feldman for encouraging interesting problems at seren-

dipitous moments. I would like to thank the Feldmans in particular for

sharing their international friendships simply because of an idle dinner

conversation remark that has allowed me to pursue something I have long

dreamed of doing—namely, this.

There have been many whose camaraderie, insight, and/or guidance

have been invaluable, some of whom are T. Akinwande, S. Bandy, I. Ben‐Zvi,S. Biedron, V. T. Binh, C. A. Brau, I. Brodie, H. Busta, F. Charbonnier,

W. B. Colson, P. Cutler, D. H. Dowell, R. G. Forbes, B. E. Gilchrist,

M. C. Green, C. Holland, M. A. Hollis, C. Hunt, J. W. Lewellen,

L. G. Il’chenko, R. T. Longo, W. A. Mackie, C. Marrese‐Reading,

R. A. Murphy, R. Nemanich, G. Nolting, W. D. Palmer, J. K. Percus,

J. J. Petillo, T. Rao, Q. Saulter, P. R. Schwoebel, J. Severns, J. M. Smedley,

T. Smith, D. Temple, A. Todd, R. J. Umstattd, E. G.Wintuckey, W. Zhu, and

J. D. Zuber. I have particularly enjoyed the many occasions I have spent with

C. A. (Capp) Spindt, who has always been gracious, a good friend, and a

pleasurable colleague.

I wish to honor the memory of three people, each of whom has left

their unique mark on me during my tenure: H. F. Gray, R. K. Parker,

and C. Bohn. They shall always live on in their work, but so, too, in my

recollections of my time with them.

I owe considerable gratitude to the Naval Research Laboratory for its

many years of support, for the broad education I was able to pursue during

my tenure there, and for indulging my brand of basic research. What I

have to give was made possible through their investments in me, particularly

xviii FOREWORD

while R. K. Parker was at the helm of the Vacuum Electronics Branch. I also

thank the Office of Naval Research and the Joint Technology Office for their

support over the years.

I thank Peter Hawkes for his great patience, for making possible this

wonderful opportunity, and for his efforts to make its realization good,

hopefully as good as the dream; and Tracy Grace for the difficult task of

dampening stochastic thought into coherent narrative. They did so with

much humor and poise.

To my children, who keep me young, but who have first made me old—and

I hope much wiser. And to my parents, who raised me to hold the passions

and ethics I do. I’ve never regretted following their footsteps. And to my wife,

whose centrality especially in uncertain times was never in doubt.

I owe much to three generations of women in my life: grandmother,

mother, and wife, each of whom has bequeathed their own special gifts to

me. The Bard spoke truly: ‘‘From women’s eyes this doctrine I derive: / They

are the ground, the books, the academes, / From whence doth spring the

true Promethean fire’’ (Shakespeare, Love’s Labor’s Lost, Act 4, Scene III).

Thank you, thank you, thank you.

Kevin L. Jensen

FOREWORD xix


KEVIN L. JENSEN

I. Field and Thermionic Emission Fundamentals . . . . . . . . . . . . . 4

A. A Note on Units . . . . . . . . . . . . . . . . . . . . . . 4

B. Free Electron Gas . . . . . . . . . . . . . . . . . . . . . . 5

1. Quantum Statistical Mechanics. . . . . . . . . . . . . . . . . 5

2. The Fermi–Dirac Integral . . . . . . . . . . . . . . . . . . 8

3. The Chemical Potential . . . . . . . . . . . . . . . . . . . 9

4. A Phase Space Description . . . . . . . . . . . . . . . . . . 11

C. Nearly Free Electron Gas . . . . . . . . . . . . . . . . . . . 11

1. The Hydrogen Atom . . . . . . . . . . . . . . . . . . . . 11

2. Band Structure and the Kronig–Penney Model . . . . . . . . . . . 13

3. Semiconductors . . . . . . . . . . . . . . . . . . . . . . 20

4. Band Bending . . . . . . . . . . . . . . . . . . . . . . 20

D. The Surface Barrier to Electron Emission . . . . . . . . . . . . . . 22

1. Surface Effects and Origins of the Work Function . . . . . . . . . . 22

2. Ion Core Effects . . . . . . . . . . . . . . . . . . . . . . 31

3. Dipole Effects Due to Surface Barriers . . . . . . . . . . . . . . 33

E. The Image Charge Approximation . . . . . . . . . . . . . . . . 40

1. Classical Treatment . . . . . . . . . . . . . . . . . . . . 40

2. Quantum Mechanical Treatment . . . . . . . . . . . . . . . . 42

3. An ‘‘Analytical’’ Image Charge Potential . . . . . . . . . . . . . 43

II. Thermal and Field Emission . . . . . . . . . . . . . . . . . . . . 47

A. Current Density. . . . . . . . . . . . . . . . . . . . . . . 47

1. Current Density in the Classical Distribution Function Approach . . . . . 47

2. Current Density in the Schrodinger and Heisenberg Representations . . . . 49

3. Current Density in the Wigner Distribution Function Approach . . . . . 52

4. Current Density in the Bohm Approach . . . . . . . . . . . . . 62

B. Exactly Solvable Models . . . . . . . . . . . . . . . . . . . . 65

1. Wave Function Methodology for Constant Potential Segments. . . . . . 65

2. The Square Barrier . . . . . . . . . . . . . . . . . . . . . 67

3. Multiple Square Barriers . . . . . . . . . . . . . . . . . . . 69

4. The Airy Function Approach . . . . . . . . . . . . . . . . . 71

5. The Triangular Barrier . . . . . . . . . . . . . . . . . . . 80

C. Wentzel–Kramers–Brillouin WKB Area Under the Curve Models . . . . . 85

1. The Quadratic Barrier . . . . . . . . . . . . . . . . . . . . 85

2. The Image Charge Barrier . . . . . . . . . . . . . . . . . . 87

D. Numerical Methods . . . . . . . . . . . . . . . . . . . . . 94

1. Numerical Treatment of Quadratic Potential . . . . . . . . . . . . 95

2. Numerical Treatment of Image Charge Potential . . . . . . . . . . 95

3. Resonant Tunneling: A Numerical Example . . . . . . . . . . . . 99

1ISSN 1076-5670/07 Copyright 2007, Elsevier Inc.

DOI: 10.1016/S1076-5670(07)49001-2 All rights reserved.

ADVANCES IN IMAGING AND ELECTRON PHYSICS, VOL. 149

E. The Thermal and Field Emission Equation . . . . . . . . . . . . . 102

1. The Fowler–Nordheim and Richardson–Laue–Dushman Equations . . . . 104

2. The Emission Equation Integrals and Their Approximation . . . . . . . 106

3. The Revised FN and RLD . . . . . . . . . . . . . . . . . . 110

F. The Revised FN‐RLD Equation and the Inference of

Work Function From Experimental Data . . . . . . . . . . . . . . 118

1. Field Emission . . . . . . . . . . . . . . . . . . . . . . 118

2. Thermionic Emission . . . . . . . . . . . . . . . . . . . . 121

3. Mixed Thermal‐Field Conditions . . . . . . . . . . . . . . . . 123

4. Slope‐Intercept Methods Applied to Field Emission . . . . . . . . . 127

G. Recent Revisions of the Standard Thermal and Field Models . . . . . . . 131

1. The Forbes Approach to the Evaluation of the Elliptical Integrals . . . . 131

2. Emission in the Thermal‐Field Transition Region Revisited . . . . . . . 136

H. The General Thermal‐Field Equation . . . . . . . . . . . . . . . 139

I. Thermal Emittance. . . . . . . . . . . . . . . . . . . . . . 143

III. Photoemission . . . . . . . . . . . . . . . . . . . . . . . . 147

A. Background . . . . . . . . . . . . . . . . . . . . . . . . 147

B. Quantum Efficiency . . . . . . . . . . . . . . . . . . . . . 148

C. The Probability of Emission . . . . . . . . . . . . . . . . . . 151

1. The Escape Cone . . . . . . . . . . . . . . . . . . . . . 151

2. The Fowler–Dubridge Model . . . . . . . . . . . . . . . . . 152

D. Reflection and Penetration Depth . . . . . . . . . . . . . . . . 154

1. Dielectric Constant, Index of Refraction, and Reflectivity . . . . . . . 154

2. Drude Model: Classical Approach . . . . . . . . . . . . . . . 156

3. Drude Model: Distribution Function Approach . . . . . . . . . . . 158

4. Quantum Extension and Resonance Frequencies . . . . . . . . . . 162

E. Conductivity . . . . . . . . . . . . . . . . . . . . . . . . 165

1. Electrical Conductivity . . . . . . . . . . . . . . . . . . . 165

2. Thermal Conductivity . . . . . . . . . . . . . . . . . . . . 167

3. Wiedemann–Franz Law . . . . . . . . . . . . . . . . . . . 170

4. Specific Heat of Solids. . . . . . . . . . . . . . . . . . . . 171

F. Scattering Rates. . . . . . . . . . . . . . . . . . . . . . . 174

1. Fermi’s Golden Rule . . . . . . . . . . . . . . . . . . . . 174

2. Charged Impurity Relaxation Time . . . . . . . . . . . . . . . 177

3. Electron-Electron Scattering . . . . . . . . . . . . . . . . . 180

4. A Sinusoidal Potential. . . . . . . . . . . . . . . . . . . . 185

5. Monatomic Linear Chain of Atoms . . . . . . . . . . . . . . . 186

6. Electron-Phonon Scattering . . . . . . . . . . . . . . . . . . 194

7. Matthiesen’s Rule and the Specification of Scattering Terms . . . . . . 212

G. Scattering Factor . . . . . . . . . . . . . . . . . . . . . . 215

H. Temperature of a Laser-Illuminated Surface . . . . . . . . . . . . . 222

1. Photocathodes and Drive Lasers . . . . . . . . . . . . . . . . 222

2. A Simple Model of Temperature Increase Due to a Laser Pulse . . . . . 223

3. Diffusion of Heat and Corresponding Temperature Rise . . . . . . . . 225

4. Multiple Pulses and Temperature Rise . . . . . . . . . . . . . . 227

5. Temperature Rise in a Single Pulse: The Coupled Heat Equations. . . . . 234

6. The Electron-Phonon Coupling Factor g: A Simple Model . . . . . . . 236

2 KEVIN L. JENSEN

I. Numerical Solution of the Coupled Thermal Equations . . . . . . . . . 239

1. Nature of the Problem. . . . . . . . . . . . . . . . . . . . 239

2. Explicit and Implicit Solutions of Ordinary Differential Equations . . . . 240

3. Numerically Solving the Coupled Temperature Equations With

Temperature-Dependent Coefficients. . . . . . . . . . . . . . . 247

J. Revisions to the Modified Fowler–Dubridge Model: Quantum Effects . . . . 253

K. Quantum Efficiency Revisited: A Moments-Based Approach . . . . . . . 255

L. The Quantum Efficiency of Bare Metals . . . . . . . . . . . . . . 260

1. Variation of Work Function With Crystal Face . . . . . . . . . . . 261

2. The Density of States With Respect to the Nearly Free Electron

Gas Model. . . . . . . . . . . . . . . . . . . . . . . . 264

3. Surface Structure, Multiple Reflections, and Field Enhancement . . . . . 264

4. Contamination and Effective Emission Area . . . . . . . . . . . . 269

M. The Emittance and Brightness of Photocathodes . . . . . . . . . . . 274

IV. Low–Work‐Function Coatings and Enhanced Emission . . . . . . . . . . 280

A. Historical Perspective . . . . . . . . . . . . . . . . . . . . . 280

B. A Simple Model of a Low–Work‐Function Coating . . . . . . . . . . 281

C. A Less Simple Model of the Low–Work‐Function Coating . . . . . . . . 282

D. The (Modified) Gyftopoulos–Levine Model of Work

Function Reduction . . . . . . . . . . . . . . . . . . . . . 286

E. Comparison of the Modified Gyftopoulos–Levine Model to

Thermionic Data . . . . . . . . . . . . . . . . . . . . . . 292

F. Comparison of the Modified Gyftopoulos–Levine Model to

Photoemission Data . . . . . . . . . . . . . . . . . . . . . 296

V. Appendices. . . . . . . . . . . . . . . . . . . . . . . . . . 304

A. Integrals Related to Fermi–Dirac and Bose–Einstein Statistics . . . . . . . 304

B. The Riemann Zeta Function . . . . . . . . . . . . . . . . . . 305

VI. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 306

References . . . . . . . . . . . . . . . . . . . . . . . . . . 309

How can my Muse want subject to invent,

While thou dost breathe that pour’st into my verse

Thine own sweet argument, too excellent,

For every vulgar paper to rehearse?,

O give thy self the thanks, if aught in me

Worthy perusal stand against thy sight,

For who’s so dumb that cannot write to thee,

When thou thy self dost give invention light?

Be thou the tenth Muse, ten times more in worth

Than those old nine which rhymers invocate,

And he that calls on thee, let him bring forth

Eternal numbers to outlive long date.

If my slight Muse do please these curious days,

The pain be mine, but thine shall be the praise.

Sonnet 38, William Shakespeare

ELECTRON EMISSION PHYSICS 3

I. FIELD AND THERMIONIC EMISSION FUNDAMENTALS

A. A Note on Units

The widespread application of electron source technology as a subdiscipline

of physics and engineering disciplines is beholden to the use of SI (Inter-

national System of Units) (meter‐kilogram‐second‐ampere [MKSA]) in

formulas useful to experimenters. Despite its practical value, such a yoke is

not always easy. For electron emission from nanoscale sites, SI units necessi-

tate bookkeeping of inconveniently large exponents. The description of emis-

sion phenomena often finds units comparable to those of the Bohr atom

(the sine qua non of the physicist’s lexicon) to be in play, for which scales of

energy, distance, and charge are naturally introduced and described by elec-

tron volts, nanometers, femtoseconds, electron charge, andKelvin (eV‐nm‐fs‐q‐K) and are often used here alongside SI units. Thewaning unit ofAngstrom,

which occasionally appears, seems an odd choice, but it, along with the use of

electron volt for energy, is commonly used in surface physics and emission

phenomena. The indolent convention of q¼ h¼ c¼m¼ 1, adopted when the

relation of theory to experiment is not pressing or when obfuscation is useful,

is shunned. Tables 1 and 2 summarize common relationships and conversions.

Particularly important is how the electron charge is handled. The work

function and electron affinity of metals and semiconductors is generally

expressed in electron volts. Thus, rather than deal with electron charge, poten-

tials, and fields separately, it is inordinately convenient to combine the unit

charge with potential to get energy (eV) and with field to get force (eV/nm).

Moreover, equations concerning potential (e.g., Poisson’s equation) are easily

related to those concerning energy (e.g., Schrodinger’s equation) if the product

of unit charge and volt are combined; if the charge of the electron is the unit

used, then charge density and current are interchangeable with number density

TABLE 1

FUNDAMENTAL CONSTANTS

Quantity Symbol MKSA eV‐A‐fs‐q

Bohr radius ao 0.529177 10–10m 0.529177 A

Electron rest energy mc2 8.1871 10–31 J 510999 eV

Rydberg energy Ry 2.17987 10–18 J 13.6060 eV

Permittivity of free space o 8.85419 10–12 C/Vm 5.52635x10–3 q2/A eV

Planck’s constant h 1.05457 10–34 J s 0.658212 eV fs

Speed of light in vacuum c 2.997924 108 m/s 2997.924

Fine structure constant afs 1/137.036 1/137.036

MKSA, meter‐kilogram‐second‐ampere.

4 KEVIN L. JENSEN

and current. The convention used here is to combine potentials and fields with

unit charge q so they become potential energy V [eV] and force F [eV/nm],

respectively.Aparticularly useful related unit is the product of the fine structure

constant, Planck’s constant, and the speed of light, orQ¼ afshc=4¼ 0.359991

eV‐nm ¼ q2/16peo. Q appears frequently in the discussion of the image charge

contribution to the potential in vacuum, for which the classical image charge

potential energy is Q/x, x being the distance from the surface.

B. Free Electron Gas

1. Quantum Statistical Mechanics

The energy and the density of a gas of electrons permeates the discussion of

the physics of electron emission, and it is therefore only fitting to explore

them in the requisite detail. Consider a box ofN (spinless) particles with total

energy E. If the energy is parabolic in momentum, (which will be assumed

henceforth), then energy levels are characterized by Ek ¼ ðhkÞ2=2m, where k

is the vector corresponding to momentum. In a cubic box, the momentum is

quantized as per

k ¼ kxxþ kyyþ kzz;kx ¼ plx=L; ky ¼ ply=L; kz ¼ plz=L

ð1Þ

TABLE 2

RELATION OF NANO UNITS TO SI*

Quantity eAfq Conversion factor SI (MKSA)

Charge q 1.60218 1019 Coulomb

Length A 1010 meter

Time fs 1015 second

Energy eV 1.60218 1019 joule

Current q/fs 1.60218 104 amp

Current density q/fs A2 1.60218 1012 amp/cm2

Density q/A3 160218 Coulomb/cm3

Field eV/q A 1010 volt/meter

Energy eV 1.60218 1019 joule

Potential eV/q 1 volt

Resistance eV fs/q2 1.60218 1014 ohm

Permitivity q2/eV A 1.60218 109 Farad/meter

MKSA, meter‐kilogram‐second‐ampere.

*To obtain (eAfq) units in terms ofMKSA,multiplyMKSA by the conversion factor;MKSA

units in terms of (eAfq) is given by the inverse of the conversion factor; e.g., for current density,

(q/fs A2) ¼ 1.60218 1012 A/cm2. The units in the MKSA column are those often used in

practice, as in A/cm2 for current density.


where l is an integer and V ¼ L3. The subscript k on E is not bold, as the

energy depends only on the magnitude of the momentum. Particles of the

same energy are grouped into levels characterized by an energy Ei. A state

consists of ni particles distributed among gi levels. Consequently, the total

particle number and energy for the system are given by

N ¼Pknk ¼P

ini;E ¼PknkEk ¼

PiniEi

ð2Þ

where the first sum is a sum over quantum numbers (i.e., nk is an occupation

number) and the second a sum over levels (i.e., ni is the sum over all nkcharacterized by energy Ei). DefineW nif g as the number of states of the box

corresponding to the set of occupation numbers nif g. The entropy of the

system is given by

S ¼ kB ln ðW nif gÞ; ð3Þwhere kB is Boltzmann’s constant. Isolated systems in equilibrium are in a

state of maximum entropy; that is, fluctuations will cause a decrease in S

if the system is in equilibrium. The state variables are given by particle

number N, volume V (recall the definition of Ek), and entropy S for

systems in thermal and mechanical contact with the outside (Reichl, 1987).

Changes in energy are therefore related to the state variables by

dE ¼ TdS PdV þ mdN; ð4Þwhere m is the ‘‘chemical potential’’—which is therefore seen as the change in

energy when the number of particles is increased. If wi is the number of ways

in which ni particles can be allocated to the gi locations with a cell, then it

follows that

lnðWfnigÞ ¼X

iln wið Þ; ð5Þ

where wi is deduced from counting arguments. The entropy of a system is the

sum of the entropies of the subsystems, and so Si ¼ kB ln wið Þ. The ‘‘statistics’’of the particles is crucial in the understanding of emission current, for example,

and so it is profitable to concentrate on the meaning of the designation.

a. Maxwell–Boltzmann Statistics. For Maxwell–Boltzmann (MB) statis-

tics, there are N! ways to place N particles into different levels, but if the

particles are indistinguishable, then there are only N!=Pi ni!ð Þ distinct

arrangements. Within each level, each particle can be placed in gi locations,

so ni particles will each separately contribute a factor of gi to the combina-

torics. In order that wi so defined is the asymptotic limit of the Fermi–Dirac

6 KEVIN L. JENSEN

(FD) and Bose–Einstein (BE) distributions, wi is divided by N! (correct

Boltzmann counting) and so (Leonard and Martin, 1980)

wijMB ¼X

i

gnii

n!: ð6Þ

b. Fermi–Dirac Statistics. There are (gi) locations to place the first

particle within a level. The Pauli exclusion principle restricts the occupation

number of eachmomentum state to be 0 or 1, so there are but gi 1ð Þ locationsfor the next particle, and so on until the ni‐th particle. As with the MB case,

a factor of ni! accounts for indistinguishable permutations within a level, and

so (accounting for spin‐1/2 particles will square each term in the sum)

wijFD ¼X

i

1

ni!

Yni1

k ¼ 0gi kð Þ ¼

Xi

1

ni!

gi!

gi nið Þ!

: ð7Þ

c. Bose–Einstein Statistics. For bosons, there is no restriction on the

number of particles that can occupy a given momentum state. The number

of permutations of the ni particles and the gi 1ð Þ partitions must both be

accounted for, and so

wijBE ¼X

i

1

ni!

ni þ gi 1ð Þ!gi 1ð Þ!

ð8Þ

From Eqs. (6)–(8), both BE and FD statistics approximate MB statistics if

gi ni, that is, the number of particles in each level is small compared with the

number of available locations, a circumstance characteristic of high tempera-

ture. Invoking Stirling’s approximation ln n!ð Þ n ln nð Þ n and neglecting

terms < O(1/n), the subsystem entropies satisfy

@

@niSi ¼ kB

@

@niln wið Þ ¼ kB ln

gi

ni s

; ð9Þ

where s ¼ 1; 0;1f g for FD, MB, and BE statistics, respectively. Maximizing

the entropy S subject to the constraints of Eq. (2) is equivalent to finding the nifor which

0 ¼ @ni

PjSj þ a N Pjnj

þ b E PjnjEj

h i¼ @niSi a bEi

ð10Þ

where a and b are undetermined multipliers. From Eqs. (9) and (10), it follows

that for each level, ni Eið Þ ¼ gi= sþ exp aþ bEið Þ½ and therefore, for each

momentum vector k

n Ekð Þ ¼ sþ exp aþ bEkð Þ½ 1: ð11Þ


To find a and b, combine the derivative of Eq. (2) with Eq. (10) to obtain

dE ¼Pi nidEi þ Eidnið Þ

¼Pi ni@Ei

@V

0@

1AdV þ 1

kBb

XidSi a

b

Xidni

ð12Þ

where the sums over dSi and dni give dS and dN, respectively. Comparing

the coefficients of dS and dN in Eq. (12) with Eq. (4) identifies b ¼ 1=kBTand a ¼ m=kBT and therefore

n Ekð Þ ¼ sþ exp b Ek mð Þ½ f g1: ð13Þ

The sum of Eq. (13) over all momentum states, as per Eq. (2), gives the

total number of particles N. In the continuum limit for fermions

(s ¼ 1), and including a factor of 2 to account for the spin‐1/2 nature of

electrons,

N ¼X

k!n kð Þ ) 2

L

2p

3 ðdk

1þ exp b EðkÞ mð Þ½ ½ : ð14Þ

The chemical potential m was treated as an inauspicious parameter, but it

is of central significance and is the derivative of the free energy with respect

to the occupation number. In the free electron model for a box of volume L3,

the energy is given by

E kð Þ ¼ h2p2

2mL2l2x þ l2y þ l2z

¼ EðkÞ: ð15Þ

In the (zero temperature) ground state, electrons are added until each

level is filled to its maximum capacity; the momentum of the last electron

in is the Fermi momentum hkF . The chemical potential is identified with

the corresponding Fermi energy.

2. The Fermi–Dirac Integral

Introducing the number density r ¼ N/V. E(k) depends on the

magnitude of k, so that in spherical coordinates dk ¼ 4p k2 dk, Eq. (14)

becomes

r m,Tð Þ ¼ 4ffiffiffip

p m

2pbh2

3=2

F1=2 bmð Þ; ð16Þ

8 KEVIN L. JENSEN

where the FD integral of order p, denoted Fp(x), is defined by

Fp xð Þ ¼ð10

yp

1þ eyxdy: ð17Þ

Blakemore (1987) provides a general discussion and tables of Fermi–Dirac

integrals of order p. For negative argument and p ¼ 1/2

F1=2 x < 0ð Þ ¼ffiffiffip

p2

X1n ¼ 1

n3=2 1ð Þn þ 1enx

ffiffiffip

p2

ex 1þ ex

2ffiffiffi2

p þ 1

8

ffiffiffi3

p

9

0@

1Ae2x

24

351 ð18Þ

where the second line is good to better than 1% for x 0.2. For positive

argument (Jensen and Ganguly, 1993)

F1=2 x > 0ð Þ ¼ x3=22

3þð10

ffiffiffiffiffiffiffiffiffiffiffi1þ y

p ffiffiffiffiffiffiffiffiffiffiffi1 y

pexy þ 1

dyþð11

ffiffiffiffiffiffiffiffiffiffiffi1þ y

pexy þ 1

dy

8<:

9=;: ð19Þ

For x 1, the last integral can be ignored. Taylor expanding the radicals

in the middle integral and taking the upper limit to (þ1) results in terms pro-

portional to the Riemann zeta function z(2n) (see Appendix 1). A reasonable

approximation for x 2.5 is

F1=2 x 1ð Þ ¼ x3=22

3þX1n ¼ 0

4nð Þ!24n 2nð Þ! 1 22n1

z 2nþ 2ð Þx2nþ2

8<:

9=;

2

3x3=2 1 1

2

p2x

0@

1A

2

þ 3

40

p2x

0@

1A4

24

351 ð20Þ

For intermediate values of x, a quadratic approximation with an error of less

than 1% is

F1=2 0:2 x 2:5ð Þ 0:1897x2 þ 0:5362xþ 0:6705 ð21ÞThe performance of the approximations in Eqs. (18)–(21) is shown in

Figure 1.

3. The Chemical Potential

At room temperature, the coefficient of F1=2 bmð Þ in Eq. (16) is 2.832 1019

#/cm3. In the ‘‘free electron Fermi gas’’ model (Kittel, 1996), the electron

number density is approximately the same as the atomic number density and


on the order of 0.1 moles/cm3, or three orders of magnitude larger than the

coefficient. It is clear, therefore, that bm is generally large and positive for

metals, and Eq. (20) is a good approximation.

For semiconductors, the mass m is interpreted as the effective electron

mass m*. The coefficient of F1=2 bmð Þ is designated NC 2=ffiffiffip

pð Þ, where NC is

the ‘‘effective density of conduction band states’’ (an analogous equation exists

for the valence band). The number of conduction band electrons in an n‐typesemiconductor is a temperature‐dependent fraction of the dopant con-

centration. For silicon, generic doping concentrations are 1015 to 1018 #/cm3,

indicating that the chemical potential is negative and that Eq. (18) holds.

The number density r does not vary with regard to temperature; therefore,

the chemical potential is temperature dependent in such a way as to offset

the temperature‐dependence of NC. The ‘‘Fermi level’’ EF ¼ mo is taken

asm T ¼ 0Kð Þ mo ¼ hkFð Þ2=2m, where hkF is the Fermi momentum. In the

zero temperature limit

r mo; 0ð Þ ¼ limb!1

4ffiffiffip

p m

2pbh2

3=22

3bmoð Þ3=2 ¼ k3F

3p2ð22Þ

For metals, the temperature dependence of m(T) is obtained by setting

r(mo,0) ¼ r(m,T ) and using Eq. (20) to derive

m Tð Þ mo 1 1

3

pkBT2mo

2

1

5

pkBT2mo

4 !

ð23Þ

0.1

1

10

−1.0

−0.5

0.0

0.5

1.0

−2 −1 0 321 4

F1/

2(x)

% E

rror

x

[20]

[21]

[18]

FIGURE 1. The Fermi–Dirac integral (circles) compared to the approximations (lines) and

the associated error (dashed lines).

10 KEVIN L. JENSEN

For a generic metallic density of 0.1 mole/cm3, mo ¼ 5.6023 eV and

kF ¼ 1.2126 A–1; even at 3000 K, m(T) is within 99.8% of mo. Consequently,the temperature dependence of m is often neglected and m is taken as

interchangeable with EF for metals.

4. A Phase Space Description

The generalization of the Fermi distribution is a phase space distribution

f ðr,kÞ such that f ðr,kÞd3rd3k is the number of particles in the phase space

element d3rd3k. The Boltzmann transport equation describes the evolution of

the distribution function: from the conservation of the distribution along a

flow line, that is, f ðr; k; tÞ ¼ f ðrþ dr,kþ dk; tþ dtÞ, which implies

@

@tþ v r!r þ F

h r!k

f r,k,tð Þ ¼ @f

@t

coll

, ð24Þ

where v and F=m are the velocity and acceleration, respectively (recall that

F is the product of the electron charge and electric field,) and the right‐handside (RHS) represents the effects of collisions and scattering on the distribu-

tion. For the steady‐state case, and neglecting the collision operator, Eqs. (13)and (24) indicate that the electrochemical potential accounts for spatial

variations in electron density and is of the form m r;Tð Þ ¼ mo Tð Þ þ f rð Þ,where r!f ¼ F. Consequently, for slow variations in electron density, the

electrochemical potential m r;Tð Þ increases in regions where the density is

larger. The phase space description is too important to leave for long, and

therefore will be revisited often below.

C. Nearly Free Electron Gas

1. The Hydrogen Atom

Crystalline solids are aggregates of individual atoms brought together in

an orderly arrangement such that, in the case of metals, the outermost

electron(s)—originally bound in a Coulomb potential—become free to

move about the crystal. The free electron gas model obscures all traces of

the bound‐state energy levels and so is unable to, for example, explain the

optical spectra of solids or the transition from metallic to semiconducting

or insulating behavior. Heuristic models such as the hydrogen atom indi-

cate how such properties result from an arrangement of outermost elec-

trons loosely bound to an orderly array of ionic cores. It is therefore

considered in detail.


A quantum mechanical treatment of the hydrogen atom considers the

electron wave function c for a rotationally invariant potential as a product

of radial RE;l rð Þ and angular Yl;m y;fð Þ functions for which m is the spin

quantum number (not mass); for present purposes, there is no profit in

retaining it, and so m as a quantum number shall henceforth be ignored.

For the Coulomb potential Vðr!Þ ¼ q2=4peor Schrodinger’s equation for

RE;l rð Þ, where E specifies energy and l angular momentum, is

h2

2m

1

r2

@

@rr 2

@

@r

l l þ 1ð Þ

r2

q2

4peor

( )RE;lðrÞ ¼ EðkÞRE;lðrÞ: ð25Þ

The natural length scale is the Bohr radius ao ¼ 4peoh2= mq2ð Þ. Let

RE;lðrÞ ¼ GðrÞekr=r so that

@

@r

2

þ 2

rl þ 1 krð Þ @

@rþ 2

aor1 ðl þ 1Þkaoð Þ

( )GðrÞ ¼ 0: ð26Þ

Expressing G as a power series in r such as

G rð Þ ¼XN

j¼1Cjr

j ð27Þ

in Eq. (26) provides a recursion relation for the coefficients Cj

Cnl ¼ 2

ao

nkao 1

nþ 1ð Þn l l þ 1ð Þ

Cnl1, ð28Þ

where the principal quantum number n ¼ j þ l þ 1 has been introduced.

Consider the l ¼ 0 case for convenience: in the limit of large n,

Cn 2k=nð ÞCn1, that is, G(r) has an asymptotic series expansion character-

istic of e2kr, which will dominate the factor of e–kr in RE;l rð Þ unless the seriesterminates. Therefore, k ¼ 1/nao, implying that the energy E(k) is quantized

to the values of En ¼ afshc= 2aon2ð Þ where the fine structure constant

afs ¼ h= mcaoð Þ has been introduced. States with larger l are degenerate in

energy and such states are termed orbitals; in hydrogen atom parlance, they

are called s, p, d, f, and so on. These sharp (discrete) levels have their analogs

for heavier ions such that when these ions are brought together with their

attendant outermost electrons, the levels evolve into the band structure of

solids. Differences in energy between the various l‐orbitals due to spin‐orbitcoupling and relativistic effects are not considered here; they break the

energy degeneracy and cause the outermost electrons for the heavier atoms

of a metallic character to be s states. The first few s radial functions Rn0ðrÞ

12 KEVIN L. JENSEN

(where n rather than E is used to indicate the principal quantum number

in the subscript) are shown in Figure 2. The expectation values of

1=rh i1 ¼ n2ao are also shown, where the n ¼ 1 case corresponds to the

Bohr radius.

2. Band Structure and the Kronig–Penney Model

For multielectron atoms, the innermost electrons shield the nucleus from the

outermost electron, typically an s‐state electron of higher quantum number n

formetals. In fact, if both of the outermost s states are filled, as for barium, then

each of the s electrons partially shields the nucleus from the other, thus affecting

how that atom rests on a surface of other metal atoms, which in turn impacts,

for example, the Gyftopoulos–Levine theory for the work function of partially

covered surfaces (see Section IV). The interaction between atoms as they are

brought together alters the interatomic potentials such that some of the outer-

most electrons may be free to roam throughout the lattice. The origin of bands

and their characteristics is a staple of solid‐state physics texts (Ziman, 1985;

Jones and March, 1985; Kittel, 1996; Ibach and Luth, 1996; Quere, 1998);

herein it suffices to show that bands arise in a one‐dimensional (1D) model

with characteristics that extend to three‐dimensional (3D) crystals.

Consider an atom (taken to be a metal) relieved of its outermost electron

and immersed in distribution of electrons in a uniform background positive

charge otherwise known by the descriptive moniker jellium. Electrons are

attracted to and therefore cluster about the ionic core, shielding it and

screening the Coulomb potential of the core as experienced by other elec-

trons. The change in density dr causes a change in the electrochemical

00 2 4 6 8

0.2

0.4

0.6

0.8

1.0

1.2

dr

r [Angstroms]

n=1

n=2n=3

4p r2Rn0(r)

9ao4aoao

FIGURE 2. Probability density for the radial hydrogen atom function for l = 0.


potential dm ¼ df, and so

drdf

¼ 2

2pð Þ3ðd3k

dfFD EðkÞð Þdf

¼ bð10

DðEÞebðEmÞ

1þ eb Emð Þð Þ2dE; ð29Þ

where the density of states per unit volume of the crystal, defined as the

number of states between E(k) and E(k) þ dE, is given by

D Eð ÞdE ¼ 2pð Þ34pk2dk ¼ m

2p2h3ffiffiffiffiffiffiffiffiffiffi2mE

pdE; ð30Þ

where the second expression is a consequence of the parabolic relation

between E(k) and k (the factor of 2 for spin has not as yet been included).

Embedded in the integral are terms that can be rewritten as

1= 1þ exð Þ 1þ exð Þ½ 1=4ð Þexp x2=4ð Þ, implying that the integrand is

sharply peaked about the Fermi level for general temperatures characteristic

of electron sources. By comparison, D(E) does not vary appreciably com-

pared to the remainder of the integrand and may be replaced by D(m) and

removed from the integral. For bm 1, the remaining integral is unity, so

dr rð ÞD mð Þdf rð Þ: ð31ÞTherefore, the terms other than D in Eq. (29) tend to conspire and act very

much like a Dirac delta function, a feature that becomes uncommonly useful

in the following text.

For rotationally symmetric potentials, Poisson’s equation is r2@r r2@rdfð Þ ¼q2=e0ð Þdrrecall that f is an energy and r a number density so that the

traditional minus (–) sign is absent—and therefore it follows that

dfðrÞ ¼ q2

4peorexp k

TFrð Þ

kTF jbm 1 ¼q2D mð Þ

eo

0@

1A

1=2

¼ffiffiffiffiffiffiffiffi4kF

pao

s ð32Þ

the Thomas–Fermi screening length is given by 1/kTF. A metallic‐like electrondensity of 0.1 moles/cm3 implies the screening length is 1/kTF ¼ 0.5854 A. For

pedagogical reasons, however, consider a smaller density associated with a

simple lattice of spheres of radius n2ao where n ¼ 3, for which the screening

length is 1.131 A. A cross‐section of such potentials is shown in Figure 3

where the n ¼ 2 and 3 energy levels of the hydrogen atom are shown for

comparison (though the energy levels of the potential given in Eq. (32) will be

higher).

14 KEVIN L. JENSEN

The derivation of Eq. (32) presumed that bm 0, but this need not be

so for semiconductors, where, because the carrier density is orders of mag-

nitude smaller, the chemical potential can be negative. When bm 1, then

dfFD /df bfFD df, in which case kTF becomes

kTF jbm 1 ¼q2r

eokBT

1=2

: ð33Þ

Bare charges are therefore screened by a redistribution of the electron gas

surrounding them. If the charge is inside a material with a dielectric constant

of Ks, then eo ) Kseo in kTF.

The small resistance of metals implies that some fraction of the available

electrons are relatively free to move about; such a ‘‘free electron’’ model

was developed by W. Pauli and A. Sommerfeld to treat metals, in which a

weakly bound valence electron propagates in a lattice of nuclei with

their tightly bound core electrons. What, then, is the consequence of these

periodic disturbances on the free electrons’ motion? The 1D Kronig–Penney

model gives a qualitative sense of what arises (Kronig and Penney, 1931).

Consider a square barrier periodic potential V(x) of well width a, barrier

width b, such that V(xþaþb) ¼ V(x), and barrier height Vo. According

to Bloch’s theorem, the wave function is then given by cðxÞ ¼ uðxÞexp ikxð Þ,where k ¼ 2pj=L, L is the (macroscopic) region defining the crystal, and

u(x) is a periodic function in x with period (aþb). If barrier

regions are designated by cI and the well regions by cII, Schrodinger’s

equation is

20151050−5−10−15−20−4

−3

−2

−1

0

Pot

ential

[a.

u.]

Position [a.u.]

n=3

q2 exp(−kTFr)/4peora

n=2

FIGURE 3. Screened Coulomb potential (red) and multiple adjacent potentials (black).


@2

@x2þ 2ik

@

@xþ k2 k2 k2o 8<

:9=;uI ðxÞ ¼ 0

@2

@x2þ 2ik

@

@xþ k2 k2 8<

:9=;uIIðxÞ ¼ 0

ð34Þ

where EðkÞ ¼ hkð Þ2=2m andVo ¼ hkoð Þ2=2m. Solutions are

uI ðxÞ ¼ Aexp kv ikð Þx½ þ Bexp kv þ ikð Þx½ uIIðxÞ ¼ Cexp i kþ kð Þx½ þDexp i k kð Þx½

ð35Þ

where k2v k2o k2. Continuity of the wave function implies that

uI ð0Þ ¼ uIIð0Þ and periodicity implies that uI ðaÞ ¼ uIIðbÞ; these two equa-

tions, along with twomore relating the first derivatives, provide four equations

for four unknown coefficients. In matrix notation,

1 1 1 1

kv ik kvþ ikð Þ i kþkð Þ i kkð ÞeðkvikÞa eðkvþikÞa eiðkþkÞb eiðkkÞb

kv ikð ÞeðkvikÞa kvþ ikð ÞeðkvþikÞa kþkð ÞeiðkþkÞb kkð ÞeiðkkÞb

0BB@

1CCA

A

B

C

D

0BB@

1CCA¼0

ð36Þ

The determinant of the matrix of coefficients must vanish for a solution

to exist, which specifies the relation between momentum k and energy

via k(E):

cos k aþbð Þ½ ¼ k2vk2 2kkv

sin kbð Þsinh kvað Þþcos kbð Þcosh kvað Þ ð37Þ

For k > ko, then kv ) ijkv|, and the RHS develops an oscillatory nature. The

magnitude of the left‐hand side is constrained to be 1, whereas the magni-

tude of RHS can vary substantially depending on parameters, and for k < kois generally in excess of unity. Therefore, allowable solutions of k(E) occur

only in certain ranges, or bands, the widths of which are determined by

how quickly the RHS varies with kva. Consider two limits: first, in the limit

ko ) 0, the RHS becomes cos[k(aþb)], indicating that E¼ hkð Þ2=2m, or

16 KEVIN L. JENSEN

the free electron result, as expected. In the opposite limit, when ko ) 1,

solutions exist only when tan bkð Þ2k=ko!0, or k jp for integer j,

which is the square well limit. For intermediate values of ko, the discrete

energy levels of the square well merge into the continuum levels of the free

electron, as shown in Figures 4 and 5.

0.2 0.4 0.6 0.8 1

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

k (Barrier)

k (E

nerg

y)

FIGURE 4. Transition from discrete levels to bands as the barrier k value increases.

10

5

0

−5

0.5

0 1

0.5

0−10

1

FIGURE 5. Surface plot and contour map based on Eq. (37).


The consequences of the previous treatment indicate that the wave func-

tion of electrons above the potential barriers more or less mimics free

electron wave functions and that the extent of the band gap is dependent

on the magnitude of the potential barrier. That this is not merely an artifact

of the square barrier potentials considered is seen by investigating a smooth

sinusoidal potential. In bra‐ket notation, consider a 1D region of width L

with (unperturbed) basis states defined by

xjnh i ¼ L1=2exp iknxð Þ ¼ L1=2exp i2pnx=Lð Þ

1 ¼ 1

L

ðL=2L=2

xj i xh jdx ¼ 1

N

Xn

jnihnj ð38Þ

such that the distance between adjacent sites (e.g., atoms) is L/N. Introduce

creation and annihilation operators a and a such that a nj i ¼ nþ 1j iand a nj i ¼ n 1j i, and a potential operator

_

V ¼ VlfðaÞl þ alg so that

xjV j0 ¼ 2Vl cosðklxÞ. We have

Hjni ¼ E nj i ) H0 þV

n0j i þ n1j ið Þ ¼ E0 þ E1ð Þ n0j i þ n1j ið Þ; ð39Þwhere the subscript indicates the order of the approximation for basis states

defined by Eq. (38). It follows from the orthogonality relation nh jmi ¼ dnm,where dmn is the Kronecker delta function, that

E1 ¼ n0h jV jn0i ¼ Vl n0h jðnþ lÞ0 þ n0h jðn lÞ0ig ¼ 0: ð40Þ

that is, the presence of the perturbation potential does not alter the free electron

relation E0ðnÞ ¼ hknð Þ2=2m to first order (i.e., there is no first‐order change inenergy). However, the density becomes

jhxjnij2 ¼ jhxjn0ij2 þ

Xj 6¼n

hxj j0ih j

0jV jn

0ihn

0jxi þ c:c:

E0ðnÞ E

0ð jÞ

þXj 6¼n

Xj0 6¼n

hxj j00ih j0

0jV jn

0i n

0jV j j

0

j0jxh i

E0ðnÞ E

0ð jÞ½ E

0ðnÞ E

0ð j0Þ½

ð41Þ

where c.c. indicates complex conjugate. In the first summation, as a conse-

quence of the creation/annihilation operators comprisingV, it follows that only

those terms for which j ¼ n 1 survive, and these can be combined to yield

Xj 6¼n

xj j0

h i j0jV jn

0

D En

0jxh i þ c:c:

E0ðnÞ E

0ð jÞ ¼ 4Vlcos klxð Þ

E0ðlÞ E

0ð2nÞ : ð42Þ

With a commensurately greater effort, the last double summation can be

combined to give

18 KEVIN L. JENSEN

Xj 6¼n

Xj0 6¼n

xj j00

D Ej00j _

V jn0

D EDn

0j _

V j j0

EDj0jxE

E0ðnÞ E

0ð jÞ½ E

0ðnÞ E

0ð j0Þ½ ¼

2V 2l

E0ðlÞ E0ðlÞ E0ð2nÞ½ cosð2klxÞ þ E0ðlÞ þ E0ð2nÞE0ðlÞ E0ð2nÞ

8<:

9=;

ð43Þ

With the introduction of v ¼ 2mVl=h2, Eqs. (41)–(43) become

jhxjnij2 ¼ 1þ v4cos k

lxð Þ

l2 4n2

þ v2

l2 l2 4n2ð Þ cos 2klxð Þ þ l2 þ 4n2

l2 4n2

: ð44Þ

The integers l and n are generally large, so that j xjnh ij2 is generally constantand close to unity except when 2n l (the pedagogical case of v ¼ 1 and

n ¼ 51 is shown in Figure 6). Depending on whether l approaches 2n from

below or above, the sign of l – 2n changes from negative to positive, and the

density at the ‘‘atomic’’ sites is reduced or increased accordingly. Conse-

quently, a substantially different behavior results for a small change in a

parameter characterizing the wave function; it can be shown that to second

order, the change in density profile is associated with a change in energy.

In other words, a band gap has developed and a forbidden region has occurred

for momenta near k(l) k(2n) as a consequence of the sinusoidal

0.98

0.99

1

1.01

1.02

−6 −4 −2 0 2 4 6

−27−8−11827

|y|2

k(n)x

FIGURE 6. Eq. (44) for the values of v = 1 and n = 51 for values of l approaching n from

above and below.


perturbation—but away from that region, the wave function behaves, to a

good approximation, as a free electron (plane wave basis states with

energy parabolic in momentum). Near the band gap, of course, the situation

is different, but—as shall be seen—emission is generally dominated by

momentum states where the ‘‘free electron’’ approximation is good.

3. Semiconductors

For intrinsic semiconductors, the Fermi level lies in the band gap between the

conduction and valence band levels. Excitations of electrons into the con-

duction band are accompanied by the creation of ‘‘holes’’ in the valence

band. Conditions can be arranged (e.g., by doping) so that a preponderance

of electrons or holes occurs. As the distribution of electrons is given byDe(E)

f(E), the distribution of holes will be given by Dh(E)[1 – f(E)], where the

e and h subscripts denote electron and hole, respectively, and f(E) is the dis-

tribution in energy of the particles (i.e., the FD distribution). The distinction

is required as the ‘‘mass’’ of holes need not equal the electron mass. When

charge transport is predominantly carried by electrons, the Fermi level lies

closer to the conduction band, and the semiconductor is designated ‘‘n‐type.’’Conversely, when charge transport is predominantly carried by holes, the

Fermi level lies closer to the valence band, and the semiconductor is desig-

nated ‘‘p‐type.’’ Moreover, if the Fermi level lies within the band gap and

more than 3kBT below the conduction band or above the valence band, the

semiconductor is termed nondegenerate. When the Fermi level lies within

3kBT of either band, or falls within either band, the semiconductor is degen-

erate. Much has been written on the equilibrium carrier concentrations of

electrons and holes in doped semiconductors, thereby obviating the need to

write more here. For the present, rather, interest lies in the behavior of the

semiconductor subject to an applied external field so that carriers migrate to

shield out the field in the bulk of the semiconductor.

4. Band Bending

Unbound electrons in a material migrate in response to an electric field,

thereby shielding the interior of a conductive material from an externally

applied electric field. Poisson’s equation relates the unbalanced charge to

spatial variations in the potential energy; in one dimension, it is

@2

@x2f xð Þ ¼ q2

Kseor xð Þ roð Þ; ð45Þ

where the traditional negative sign on the RHS is absent due to r being a

number density and f being a potential energy, courtesy of the hidden

20 KEVIN L. JENSEN

multiplicative factor of electron charge. Ks ¼ e/eo is the dielectric constant ofthe material, large for metals and of O(10) for semiconductors. The relation-

ship F(x) ¼ @xf(x) allows for the substitution

@2

@x2f ¼ @

@xf

@

@fF ¼ 1

2

@

@fF2; ð46Þ

therefore

@

@fF2 ¼ 2q2

kse0fro þ

4q2Ncffiffiffip

pkseo

ð10

dy

ðf0

dy

1þ exbð yþmoÞ

ffiffiffix

pdx; ð47Þ

where m(f ¼ 0) ¼ mo and x ¼ bE. Performing the integration over y yields

F2 ¼ 2q2

Kse0fro þ

4q2Ncffiffiffip

pKse0b

ð10

ln1þ ebmx

1þ ebmox

ffiffiffix

pdx: ð48Þ

For metals, bmo 1 so that to leading order in f,

F ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3q2ro2moKseo

s1þ f

12mo

f 1þ f

12mo

fl; ð49Þ

where the length parameter l(mo) for the canonical metal (ro¼ 0.1 moles/cm3)

is 58.5 nm. Eq. (49) implies that in the limit f mo, the potential energy

exponentially decays into the bulk with a length factor l. At the surface of

a metal, the field F is related to an externally applied (vacuum) field Fvac by

F ¼ Fvac /Ks; the largeness of Ks indicates that for metals even under high

fields, f remains small, and the potential in the interior remains, to a good

approximation, flat (e.g., for Fvac ¼ 10 eV/nm and Ks ¼5000, f< 0.0083 eV).

For semiconductors, however, the situation is different by virtue of the

relative smallness of Ks and ro: the former is of order O(10), and the latter is

of such a magnitude that mo is generally negative. Two limits then exist,

depending on whether the electron density is degenerate or nondegenerate

as a consequence of band bending. For the more familiar nondegenerate case

(bm 1),

F ¼ 2q2robKseo

ebf 1 bf 1=2

Fo

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiebf 1 bf

p; ð50Þ


whereas for the degenerate case (bm 1)

F ¼ 2Fo

bmp

1=4

ebm=22

15bmð Þ2 þ 1

1=2

; ð51Þ

where, for T ¼ 300K, Ks ¼ 12, and ro ¼ 1 1017 cm–3, Fo ¼ 2.7922 10–3

eV/nm. A comparison of Eqs. (50) and (51) with Eq. (48) is shown in Figure 7.

D. The Surface Barrier to Electron Emission

The origins of the work function are complex and, indeed, depend very much

on surface conditions, material parameters, and many‐body physics. A num-

ber of intensive treatments exist in books (Modinos, 1984; Jones and

March, 1985; Monch, 1995), and the periodical literature (aside from articles

cited in the following text in context, an excellent recent review may be

found in Yamamoto, 2006). Such in‐depth treatments are recommended to

compliment the treatment here.

1. Surface Effects and Origins of the Work Function

Having shown that to a good approximation, electrons in a conducting

material move about in a quasi‐free fashion, and therefore that electron

motion is well described by plane‐wave basis states, the origin of the barrier

to electron emission at the surface of a material, that is, the ‘‘work function,’’

10−6

10−5

10−4

10−3

10−2

10−1

100

0.01 0.1 1 10 100

Exact

Fie

ld [eV

/Å]

Parameters T=300 K Ks=12.0

ro=1017 cm−3

bmo= −5.5238

bf

bm 1

bm −1

FIGURE 7. Comparison of Eq. (48) to its asymptotic approximations Eqs. (49) and (50).

22 KEVIN L. JENSEN

becomes readily explicable. It requires a consideration of how the potential

and kinetic energy terms become operators in a basis dictated by particle

number (Reichl, 1987; Quere, 1998; Feynman, 1972).

The Hamiltonian of Schrodinger’s equation for many electrons is the sum

of several terms: their kinetic energy and the interaction of the electrons

among themselves (Hel), their interaction with the background (Vel–B), and

finally, the self‐interaction of the background (VB), or

HN ¼ HNel þ VelB þ VB

HNel ¼

PNi¼1

ðhkÞ22m

þ q2

4pe0

XNi<j¼1

eajrr0 j

jr r0 j

VelB ¼ q2

4pe0

PNi¼1

Ðdr

eaj rrij

jrr0ij rþðrÞ

VB ¼ þ q2

8pe0

PNi<j¼1

Ðdrdr0 e

ajrr0 j

jrr0 j rþðrÞrþðr0Þ

ð52Þ

where the factor ear in the Coulomb potential is inserted to enforce conver-

gence—at the end of the evaluations, the limit of a ! 0 is taken. In the

language of creation/annihilation operators, a quantity O as a function of

position r, momentum k, and spin s, becomes, in field operator notation,

O r, k

¼Xs1s2

ðdr

1

ðdr

2r1s

1jO r, k

jr2s

2

D Ecðr

1s

1Þcðr

2s

2Þ, ð53Þ

where for convenience, O is presumed to be spin independent. The notation

becomes burdensome quickly, and it is common to introduce a bra‐ketnotation that hides the vector nature and includes spin, that is, jri jr;siand jki jk;si. Analogously, interpret integrals over dr to indicate integra-

tion over dr and summation over s, and summations over k to be over k and

s, that is, letP

s

Ðdr ) Ð

dr, and likewise for momentum (though k is

discrete due to finite volume). The following relations hold

1 ¼ Pk jkihkj ¼

Ðdrjrihrj

rjkh i ¼ V1=2exp ik rð Þrjr0h i ¼ ds;s0 dðr r

0 Þ; kjk0h i ¼ ds;s0dk;k0

ð54Þ

The field operators are represented by

cðr ,sÞ )Xk

rjkh iak; cðr ,sÞ )Xk

kjrh iak, ð55Þ


and so Eq. (53) becomes

O ¼Xk1k2

k1jOjk

2h ia

1

a2; ð56Þ

where an indicates an annihilation operator for a number state characterized

by momentum kn, and similarly for an. By way of example, the number

operator N becomes

N ¼X

k1k2

k1j1jk

2h ia

1

a2¼X

ka

k

ak; ð57Þ

where, in the absence of a numerical subscript on k, the numerical subscript

on a reverts to the k notation. Likewise, the kinetic energy operator (the first

term of Hel) becomes

T ¼X

k1k2

k1j h2k22mjk2

* +a

1

a2¼X

k

h2k2

2ma

k

ak: ð58Þ

The potential terms are more involved, although the self‐interaction of the

background is straightforward, as it does not involve the electrons; for a

uniform background positive charge, rþðrÞ ¼ Nh i=V (a consequence of

global charge neutrality, as the average electron and background densities

must be equal) and so

VB¼ q2

8peo

ðdr1

ðdr2

exp ajr1 r2jð Þjr1 r2j

Nh i2V 2

¼ q2

8peo

Nh i2V 2

ðdr

ð10

4pR2 eaR

R

0@

1AdR

¼ q2

8peo

Nh i2V

4pa2

ð59Þ

Likewise, the electron‐background contribution is

VelB¼ q2

4peo

Nh iV

Xk1k2

ðdrhk1j exp ajri rjð Þ

jri rj jk2ia1

a2

¼ q2

4peo

hNiV

Xk1k2

4pa2

0@

1Adk1;k2a1

a2

¼ q2

4peo

hNiV

4pa2

0@

1AN

ð60Þ

24 KEVIN L. JENSEN

and the electron‐electron contribution (the second term of Hel) is

Vee¼ q2

8peo

Xk1k2k3k4

k1k

2j exp ajr

1 r

2jð Þ

jr1 r2j jk3k

4

a

1

a2

a3a

4: ð61Þ

Eq. (61) hides a rather subtle sleight of hand: wave functions for fermions

must be antisymmetric (i.e., the sign must change) when particles are

exchanged. Consequently, |k1k2i is not as simple as |k1i|k2i. Rather, |k1k2imust be interpreted as the combination of all |k1i and |k2i arranged such that

the antisymmetry is manifest and results in the introduction of the 2 2

Slater determinant

jk1k

2iðÞ ¼ 1ffiffiffi

2p det

k1

j i k2

j ik

1j i k

2j i

; ð62Þ

where det indicates that the determinant of the matrix is to be taken, and the

superscript minus (–), which shall be ignored as soon as convenient, indicates

‘‘antisymmetric.’’ The generalization of Eq. (62) to more than two particles

should be evident. With Eq. (56), |k1k2i ¼ – |k2k1i, as switching particles is

tantamount to switching columns in the matrix, resulting in a sign change in

the determinant. It then follows

k1k2j exp ajr1 r

2jð Þ

jr1 r

2j jk3k4

* +¼ 1

Vdk1þk2;k3þk4

ðdR

eaR

Rexp i k1 k3ð Þ R½

¼ 1

V

4pdk1þk2;k3þk4

a2 þ jk1 k3j2ð63Þ

where the V in the denominator is volume, not potential—an unfortunate

convergence in notation—and the influence of spin has largely been ignored.

Proceeding further requires greater attention to the properties of the creation

and annihilation operators.

The action of the a0s on the number‐representation kets is

aj jn1. . .nj . . .n1i ¼ ffiffiffiffinj

p jn1. . .ðnj 1Þ. . .n1iaj jn1. . .nj . . .n1i ¼ ffiffiffiffiffiffiffiffiffiffiffiffi

nj þ 1p jn1. . .ðnj þ 1Þ. . .n1i ð64Þ


so that

aj aj

D E¼ nj

ai, aj

¼ a

i , a

j

h i

D E¼ 0

ai, aj

h iþ

¼ nj þ 1

dnj, 0 þ njdnj, 1

di, j ¼ di, j

ð65Þ

where the last line is a consequence of nj ¼ 0 or 1 for fermions, and [A,B] ¼AB BA. Thus, Vee for zero momentum transfer (i.e., k1 ¼ k3) becomes

Veeð0Þ ¼ q2

8peo

Xk1k2

4pVa2

a1

a2

a1a

2

¼ q2

2Veoa2X

k1k2

a1

a1; a

2

þ a1a

2

n o

a2

¼ q2

2Veoa2N2 N

ð66Þ

Combining VB, Vel‐B, and the zero‐momentum transfer component of Vee

gives

VB þ VelB þ Veeð0Þ ¼ q2

2V eoa2Nh i2 2 Nh iNþN2N

ð67Þ

In the thermodynamic limit, hNni ¼ N n

and so hNi2 2 Nh iN þ N2 ¼N N

2 ) 0. Moreover, in the same limit, hNi/V remains finite, but

hTi becomes increasingly large, so that hNi/V is negligible by comparison.

The remaining terms are therefore

H ¼Pk

h2k2

2makak

1

2

X0

k1k2k3k4k1k2jVee k1 k3ð Þjk3k4

a1

a2

a3a4

¼Pk

h2k2

2makak

1

4

X0

k1k2k3k4k1k2jVee k1 k3ð Þjk3k4 ðÞ

a1

a2

a3a4

ð68Þ

where the prime on the summation indicates that the no‐momentum

transfer term (k1 ¼ k3) has been excluded; the overall negative sign for

the potential term is a consequence of the ordering of the annihilation opera-

tors, and the superscript minus (–) indicates that hk1k2jVeejk3k4i is replaced by

hk1k2|Veejk3k4i – hk1k2jVeejk4k3i, which is antisymmetric to a switch in k3 and

k4 and therefore balances the sign change when a3a4 are switched.

26 KEVIN L. JENSEN

The various components of hHi ¼ hn1. . .n1jHjn1. . .n1i can now be eval-

uated. At zero temperature, the states nj are filled until hkj 2

=2m > m,beyond which they are empty. Consequently, hakaki functions as the

Fermi–Dirac distribution function, the finite temperature extension of

which was encountered in Eq. (11) for s ¼ 1, and is the probability that the

kth state is occupied, or

akak

D E/ 1þ exp b EðkÞ mð Þ½ f g1: ð69Þ

The kinetic energy per unit volume is therefore

1

V

Xk

h2k2

2makak

* +) 2

V

ð10

h2k2

2my m EðkÞð Þ L

2p

3

4pk2dk ¼ h2k5F10p2m

;

ð70Þwhere y(m – E(k)) is the Heaviside step function and the factor of 2 in the

coefficient of the integral is from a summation over spin. The exchange term,

as the second component of Eq. (68) is called, only gives the contribution

1

4V 2

X0

k1k2hk1k2jV ee k1 k2ð Þjk1k2iðÞ

a1a1a

2a2

D E;

) q2

2pð Þ6eo

ðdk1

ðdk2

y m Eðk1Þð Þy m Eðk2Þð Þa2 þ jk1 k2j2

ð71Þ

where commutation of the creation/annihilation operators has been used,

and the zero temperature limit has been taken to ease the evaluation of the

integrals. As the Fermi–Dirac distribution function changes appreciably only

near the Fermi momentum, such an approximation is, in fact, rather reason-

able. Recall that the prime on the summation indicates the zero‐momentum

transfer component has already been removed. Introduce k ¼ k1 k2and k0 ¼ 1

2k1þ k2

. The integral over k0 is then

ðdk

0y m E k1ð Þ½ y m E k2ð Þ½ )

ðdk

0y kF k

0 þ 1

2k!

y kF k

0 1

2k

ð72Þ

The integralÐy r jx yjð Þdx is the volume of a sphere of radius r offset

from the origin by y. Consequently, the RHS of Eq. (72) is interpreted as the

volume of intersection of two spheres of radius kF, the origins of which are

separated by k kF. Thus


Ðdk0 y½kF jk0 þ 1

2kjy½kF jk0 1

2kj

¼ 2pk3Fy 2kF kð Þ Ð arccosðk=2kF Þ0

sin3ðxÞdx

¼ p12

4kF þ kð Þ 2kF kð Þ2y 2kF kð Þ

ð73Þ

which (as it should) reduces to (4p/3)kF3 when the spheres overlap (k ¼ 0).

The integration over k is then trivial, and results in (where the limit a ! 0 has

been taken)

q2

2pð Þ6eo

ð2kF0

4pdkp12

4kF þ kð Þ 2kF kð Þ2 ¼ q2k4F

2pð Þ4eo: ð74Þ

To this order, the total energy per unit volume U is the sum of Eqs. (70)

and (74), and is

U r;T ¼ 0ð Þ ¼ h2k5F10p2m

q2k4F16p4eo

¼ 3

5mro

m

p3h2aom2

ð75Þ

where, in the second line, ro ¼ r(mo,0) from Eq. (22) and the definition of the

Bohr radius ao have been used. In the literature, Eq. (75) is not the most

common representation; rather, a dimensionless parameter rs (Table 3) is

introduced such that

TABLE 3

CORRELATION ENERGY TERMS*

rs eke þ eex þ ecor

1.0 1.174(1)

2.0 0.0041(4)

5.0 0.1512(1)

10.0 0.10675(5)

20.0 0.06329(3)

50.0 0.02884(1)

100.0 0.015321(5)

*Correlation energy values as a function of rs from

Ceperley and Alder (1980).

Parentheses represent the error bar in the last digit.

28 KEVIN L. JENSEN

Nh iV

ro ¼4p3

rsaoð Þ3 1

ð76Þ

in terms of which the energy per unit volume is

U0 r;T ¼ 0ð Þ ¼ Ryro3p2

2

0@

1A

1=3

9

10r2s 3

2p

0@

1A

2=3

3

2rs

8><>:

9>=>;

¼ Ryro2:2099

r2s 0:91633

rs

0@

1A

ð77Þ

where Ry ¼ 13.6063 eV is the Rydberg energy.

The next term, generally called the correlation energy ecor (alternately, thestupidity energy, as sardonically suggested by Feynman, 1972) in the rsexpansion is an arduous exercise that is fortunately well treated elsewhere.

It accounts for the difference between the total energy and the sum of

the kinetic energy and exchange term. An indication of what is entailed can

be inferred from the following. In the language of Feynman diagrams,

the second term in Eq. (68) can be diagrammatically expressed as

V

k3

k1 k2

k4

k1k2 k3k4Vee ⇒ ð78Þ

Consequently, the analogous potential interaction term in Eq. (71) generates

a diagram of the form (where the line has been compacted to a point ( ) forconvenience)

k1k2 k2k1Vee ⇒ ð79Þ

Eq. (79) is the lowest‐order Feynman diagram to contribute. The higher‐order ‘‘polarization’’ diagrams give a contribution DU composed of the

higher‐order Feynman diagrams

+ …++∆U = ð80Þ

where, for sake of convenience, labeling and arrows are suppressed. All such

polarization diagrams must be summed to remove the divergence that occurs


for low momentum transfer. A tedious calculation (Feynman, 1972;

Quere, 1998) shows that including these diagrams results in the small rsexpansion

DU r;T ¼ 0ð Þjrs!0 ¼ Ryro2

p21 ln ð2Þð Þ ln rsð Þ 0:096þO rsð Þ

0@

1A

¼ Ryro 0:06218 ln rsð Þ 0:096þO rsð Þð Þð81Þ

where the term in parentheses is identified as ecor. The terms eex and ecor ofEqs. (77) and (81) represent the low rs, or high electron density, limit of the

exchange‐correlation energy term. In the low‐density, or large rs limit, as shown

byWigner, the electron gas ‘‘crystallizes’’ into a lattice.Wigner suggested ecor0.878Ry/(rs þ 7.79) (Haas and Thomas, 1968), although the form due to

Ceperley and Adler (Kiejna and Wojciechowski, 1996)

DU r;T ¼ 0ð ÞRyro 0:862849

rs þ 3:22016ffiffiffiffirs

p þ 3:03546

ð82Þ

is perhaps better. The various contributions are shown in Figure 8.

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

1 10 100

Corr (C&A)

Corr (rs<<1)

Corr (wigner)

TotalMetals

Ene

rgy

[Ry]

rs

Al

Mg Cu

AuAg

BaNa

Cs

Si@ 1E19

FIGURE 8. Exchange and correlation energy and the position of various metals on the total

curve as a function of the (dimensionless) radius parameter rs.

30 KEVIN L. JENSEN

The inference from Eqs. (22) and (75) that the energy of the system can be

expressed in terms of the density is correct: the ground‐state energy of an

interacting electron gas is given as a functional of the density (Hohenberg

and Kohn, 1964). Minimization of the energy with respect to the constraint

thatÐrðrÞdr ¼ N ¼ constant, analogous to the procedures leading to

Eq. (12), serves to relate ro to the effective one‐body potential under which

an electron in the material is considered to move (Jones and March, 1985).

The variation of Vxc from deep in the bulk of the material to the vacuum

outside the surface allows for the determination of the largest component of

the work function. The exchange‐correlation potential is determined by the

functional derivative of the exchange‐correlation energy Exc (i.e., U0 þ DUwithout the kinetic energy term), or

Vxc rð Þ ¼ dExc r rð Þ½ dr rð Þ : ð83Þ

Technically, Eq. (83) is valid for a uniform electron density—its application

to a non‐uniform density makes use of the local‐density approximation

(LDA) in which Vxc is calculated for a small‐volume element for which the

local density is ro. Surprisingly, however, the procedure continues to

work well even when the electron density is rapidly varying, as near an

ionic core—but more to the point here, near the surface of a metal—and

therefore, the LDA is made in almost all density functional calculations

(Sutton, 1993).

Consider the case of sodium, for which rs ¼ 3.93 in bulk (Kittel, 1996).

Therefore, Vxc(rs ¼ 3.93) – Vxc(rs ¼ 1) is 5.266 eV and m ¼ 3.245 eV. Their

difference corresponds to a potential change from bulk to vacuum of 2.021 eV

(Ceperley and Adler) or 2.073 eV (Wigner), values surprisingly close to the

work function of sodium (F¼ 2.29 eV; Haas and Thomas, 1968). The success

of sodium is quickly tempered by the divergence of the method for other

metals, inviting the justifiable suspicion that the physics of other effects is

being neglected. These effects are discussed next.

2. Ion Core Effects

As alluded to in the discussion of the Kronig–Penney model, the ion cores

associated with the metal atoms cannot be neglected. Crudely, the ionic cores

can be thought of as residing in spheres of radius rsao surrounded by an

electron charge cloud, so that the overall sphere is neutral and the spheres

in the crystal, if nonoverlapping, therefore noninteracting (except for, per-

haps, van derWaals and repulsive forces; Herring and Nichols, 1949—effects

that are small for metals and therefore judiciously ignored). Inside the


rs sphere, the electron density is relatively constant except near the core.

Two contributions to the energy exist: the electron cloud interacting with

itself, and with the ion core. Within the spherical approximation, the electron

self‐ and core interactions are easily determined: the cloud–ion core

interaction is given by

eei ¼ ðrsao0

ncq

4pe0r

qro4pr

2dr ¼ 3Ry

rsnc; ð84Þ

where the term in parentheses is the Coulomb potential of a ion core with

charge ncq. Similarly, the cloud self‐interaction term is

eee ¼ðrsao0

nsq

4pe0rnsq

r

rsao

3" #

ro4pr2dr ¼ 3Ry

5rsn2s ; ð85Þ

where ns is the number of electrons in the sphere (henceforth, for ease of

discussion, ns and nc, will be taken as unity—that is, ignored—to avoid

the discussion becoming needlessly complex), and the terms in the integrand

are the Coulomb potential for a charge nsq, the charge within a uniform

charge density sphere of radius r (note that r is a length, but rs is dimension-

less) and the charge in the shell comprised of the product of the charge

density with the differential volume element of the shell. Consequently, a

term ei ¼ 9Ry=5rs

may be added to the exchange‐correlation energy to

account for the effect of the ion cores of the metal.

The inclusion of ei contains a sleight of hand: the ion core is not a bare

charge, but rather is surrounded by an inner cloud of electrons that shield it

from the valence electrons. Therefore, the ability of the valence electron to

penetrate the core is circumscribed. A simple approximation is to assume

that up to a radius ai (a notation evocative of the Bohr radius ao), the core

region excludes the outer electron completely, but for r > ai riao,

the potential of the core is a simple Coulomb potential. While such a

‘‘pseudopotential’’ approximation appears to be draconian, in fact it per-

forms rather well (Ashcroft, 1966; Quere, 1998). Eqs. (84) and (85) must

therefore be modified to exclude the core region, both in the e terms, but also

the density ro term, giving

ei ) 6Ry

r3s r3i ðrs

ri

rr3 r3ir3s r3i

1

0@

1Adr

¼ 3Ry

5

3r5s 5r2i r3s þ 2r5i

r3s r3i 2

0@

1A

ð86Þ

32 KEVIN L. JENSEN

The ground state of the system may be approximated by the following

argument (Jones and March, 1985). Within the rs sphere, the electron has a

wave function of the form ckðrÞ ¼ eikr j0ðkrÞ þ Kan0ðkrÞ½ , where j0(x) and

n0(x) are spherical Bessel functions of order zero, and a is a scattering length

from the zero‐momentum limit of the scattering amplitude f(y), but whichfor simplicity will be evaluated using the Born approximation, for which

a ¼ 2m

h2limk!0

ðai0

sinðkrÞkr

q2

4peor

r2dr ¼ a2i

a0; ð87Þ

where ai ¼ riao. The factor of K is found by requiring that the first derivative

of the wave function vanish at rs, giving K2 ¼ 3a= rsaoð Þ3. A constant

energy term

eo ¼ hKð Þ22m

¼ 3Ry

r2ir3s

ð88Þ

is then added to the overall energy expression. The total energy is the sum

of the kinetic, exchange‐correlation, ion core, and ground‐state energies, orE ¼ Uo þ DU þ Ei þ Eo. With the removal of the kinetic energy term in Uo,

the variation of the remainder with respect to the electron density r(rs), as perEq. (83), gives rise to the largest component of the potential barrier that an

electron experiences at the surface of a material.

3. Dipole Effects Due to Surface Barriers

To the potential resulting from exchange‐correlation energy and ion core

terms must be added any effects due to self‐consistency. At the surface, the

quantum mechanical, or wave, nature of the electron allows the electron to

be found in the classically forbidden region of the surface barrier, which

serves to prevent electron escape into the vacuum. Amodel for the estimation

of the magnitude of the dipole effect is obtained by considering Schrodinger’s

equation for a potential barrier in the form of a wall of heightVo and width L,

or VðxÞ ¼ VoðxÞðL xÞ, where is the Heaviside step function (Jensen,

2003a). Wave functions approaching the barrier (from the left) and leaving

(to the right) have the form

ck x < 0ð Þ ¼ 1ffiffiffi2

p eikx þ rðkÞeikx

ck x Lð Þ ¼ tðkÞeikxð89Þ

where hk is the momentum of the electron with corresponding

energyE ¼ h2k2=2m. Momentum‐like variables prove convenient, so

introduce


ko 2mVo=h2

1=2k2 k2o k2

ð90Þ

The requirement to match the wave function and its first derivative at each

(abrupt) change in potential is concisely expressed in the matrix equation for

k < ko (for k > ko, k ) ik)

1 1

ik ik

1

rðkÞ

¼ 1 1

k k

ekL ekL

kekL kekL

1eikL eikL

ikeikL ikeikL

tðkÞ0

ð91Þ

The solution of Eq. (91) for r(k) and t(k) is straightforward albeit monot-

onous. It suffices to quote the results, the general methodology of the

calculation being deferred to the development of the emission equations.

The solutions are

tðkÞ ¼ 2kkeikL

2kkcosh Lkð Þ þ i k2 k2ð Þsinh Lkð Þ

rðkÞ ¼ i k2 þ k2ð Þsinh Lkð Þ2kkcosh Lkð Þ þ i k2 k2ð Þsinh Lkð Þ

ð92Þ

in the limit kL 1, then jr k < koð Þj 1, indicating that for electrons

with energies below the barrier maximum, total reflection occurs when the

barrier is tall, wide, or both, and a reflection (assuming such a pun is

permissible) of the exponential decay of the electron density within the

classically forbidden region under the barrier. To find ck(x) to the left of

the barrier, let r be given as

rðkÞ RðkÞexp 2i’ðkÞð Þ

RðkÞ ¼ 1þ 2kkk2osinh Lkð Þ

24

35

8<:

9=;

1

tan 2’ðkÞð Þ ¼ 2kk cosh Lkð Þk2 k2ð Þ sinh Lkð Þ

ð93Þ

34 KEVIN L. JENSEN

For tall, wide (or both) barriers, the reflection coefficient is, to a good

approximation, independent of barrier width, a consequence of cosh(z) sinh(z) for large z so that

jckðxÞj2 ¼1

21þ R2 R cos 2 kxþ ’ðkÞð Þ½

1 cos 2 kxþ ’ðkÞð Þ½ ð94Þ

When k2o k2 in particular, Eq. (93) indicates that ’ðkÞ k=ko kxo,

so that to leading order the effect of a tall barrier, regardless of width, is to

simply shift the density to the left by an amount inversely proportional

to V1=2o .

The impact of the shift by xo is most readily seen in the zero‐temperature

limit for density, which (as for all things quantummechanical) is altered from

Eq. (14): the density matrix (Shankar, 1980) is defined as r ¼Pinijii ijh in the

number representation, where ni ¼ 0 or 1 for fermions. In the momentum

representation, then

r ¼X

knkjkihkj ) 2

2pð Þ3ðfFD EðkÞð Þjki kjdkh ð95Þ

which contains some more sleight of hand: the k‐kets to the left contain three

momentum components and spin, as in Eq. (54), whereas to the right, the

spin states have been summed over, the vector nature of k is made explicit,

and the transition to the continuum limit has been made, where nk is replaced

by the FD distribution function. Consequently, the k of Eq. (94) is but one

momentum component of the k in Eq. (95), say kx. The density of interest

here is the diagonal of the density matrix, or xjrjxh i r xð Þ, which becomes

r xð Þ ¼ 2

2pð Þ3ð11

jckðxÞj2dkð10

2pk⊥dk⊥ f FD EðkÞð Þ;

1

2p

ð11

f ðkÞjckðxÞj2dkð96Þ

where kx ) k, and the 1D ‘‘supply function’’ f(k) has been introduced. f(k)

is seen to be the FD with the transverse momentum components integrated

over; its sly introduction via Eq. (96) belies its significance, as its usage

in electron emission theory is ubiquitous. With the assumption of a para-

bolic relationship between energy and momentum, it is straightforward to

show that

f ðkÞ ¼ m

pbh2ln 1þ exp b m EðkÞð ½ g;f ð97Þ


where hk is understood in Eq. (97) to be the 1D momentum into (or

away from) the barrier. In the zero‐temperature limit b ) 1 so that

f ðkÞ ) 2pð Þ1k2F k2

kF kð Þ, for which r(x) becomes

limb!1

rðxÞ ¼ 2

2pð Þ2ðkF0

k2F k2

1 cos 2k x xoð Þð Þ½ dk

¼ k3F3p2

1þ 3cos zð Þz2

3sin zð Þz3

8<:

9=;

ð98Þ

where z ¼ 2kF x xoð Þ and the coefficient, equal to ro, is familiar from

Eq. (22). The behavior of r(x) is shown in Figure 9, along with a hyperbolic

tangent fit ra(x) and a step‐function ri xð Þ ¼ ro xi xð Þ, where xi is the

location of the origin of the background positive charge. The oscillations

visible, due to the trigonometric functions in Eq. (98), are known as

Friedel oscillations and are a consequence of the wave nature of the electron.

The value of xi is found by demanding global charge neutrality, orð11

re xð Þ ri xð Þ½ dx ¼ 0 ð99Þ

A bit of manipulation shows that Eq. (99) is equivalent to

3

2kFlimd!0

Re

ð10

1

s2 þ d2þ i

s s2 þ d2

( )eisds

( )þ xi xoð Þ ¼ 0 ð100Þ

0

0.2

0.4

0.6

0.8

1.0

1.2

−6 −4 −2 0

r(x)/ro

ra(x)/ro 1.244

Ion

r(x)

/ro

2kF(x–xo)/p

Excess (+) chargeExcess (−) charge

ra(x)/ro 2.554

FIGURE 9. Electron density compared to the bulk value and the nature of Friedel Oscilla-

tions at the surface.

36 KEVIN L. JENSEN

The first term contains integrals familiar from the calculus of residues and is

straightforwardly shown to be

Re

ð10

1

s2 þ d2þ i

s s2 þ d2

( )eisds ¼ p

2ded p

2d21 ed

: ð101Þ

It follows

xi ¼ xo 3p8kF

ð102Þ

For example, if ro ¼ 0.1 moles/cm3 and F ¼ 4.5 eV, then xo ¼ 0.61415 A and

xi ¼ –0.35749 A. Evaluating the dipole resulting from the charge distribution

in Figure 9 requires a bit more finesse. Integration over the field F ¼ @xfgives the dipole contribution Df:

Ð11 @xfð Þdx ¼ Ð11 x @2

xf

dx

¼ q2

eo

ð11

x xoð Þ reðxÞ riðxÞð Þdxð103Þ

where integration by parts, the vanishing of the field at the boundaries,

Poisson’s equation, and global charge neutrality have been used. Let

s ¼ 2kF (x – xo) and D ¼ 2kF(xi – xo) ¼ –3p/4. The integrand for the electrons

is then proportional to s cosðsÞ=s2 sinðsÞ=s3f g ¼ @s sinðsÞ=s½ so

Df ¼ q2ro4eok2F

3

ð01

d

ds

sinðsÞs

0@

1Ads

ð0Dsds

8<:

9=;

¼ QkF

16p32 3p2 ð104Þ

where the definitions of Q and ro have been used. For the canonical example

of a 0.1 moles/cm3 density for which kF ¼ 1.213 A1, Df ¼ 0.4153 eV.

Two of the approximations undermine the clean simplicity of Eq. (104):

the approximation’(k) kxo has been used, and thermal effects (which were

neglected) come into play. The former causes the Friedel oscillations to be

more pronounced than would be evident from a numerical evalua-

tion of Eq. (96). The latter introduces electrons with energy greater than

the Fermi level that penetrate more deeply into the barrier, causing the

dipole term to be larger. The effects also trickle down to modify the definition

of xi. Amore phenomenological theory (Jones andMarch, 1985; Smith, 1969)


is to consider the approximate electron density profile to be defined by

raðxÞ ¼ro

1þ exp 2lkF x xið Þ½ : ð105Þ

If l is chosen such that @xre(x) ¼ @xra(x) at x ¼ xi along with Eq. (98) and

D ¼ –3p/4, then

l ¼ 36

D3cosðDÞ þ 12

D4D2 3

sinðDÞ ¼ 1:24356: ð106Þ

The integral for Df is then trivially evaluated by

Dfa ¼q2ro

2e0 lkFð Þ2ð10

s

1þ esds ¼ 2p

9l2QkF : ð107Þ

Use of Eq. (106) makes Eq. (107) substantially larger than Eq. (104) (the

value l0 that would give equality between Df and Dfa is l0 ¼ ð4p=3Þ

32 3p2½ 1=2 ¼ 2.5546). The discrepancy lies with the fact that the fields

generated by the charge further away from the interface contribute in the

tanh model, whereas for the Friedel‐model, the fields generated by the charge

between(2nþ 1)p 2kF(x – xo)(2n 1)p (i.e., zeros of sin(2kF(x – xo))

vanish, and no net charge exists past x > xo, a relic of the infinite barrier

approximation). How much do small excesses of charge inconveniently

located elsewhere change the presumed potential profile? In fact, a great

deal. For the generic metal example of re ¼ 0.1 moles/cm3, a thin sheet of

charge of widthDx produces a field of reDx=2eo; in other words, ifDx¼ 0.2 A,

then a field of 0.11 eV/nm results, indicating that over Angstrom‐scaledistances, electron volt–scale potential differences are created. The ‘‘wings’’

that accompany the tanhmodel can be clipped by setting the integration limits

in Eq. (103) to xi xo. With re replaced by ri, Eq. (107) becomes

Dfa )2p

9l2QkF 1M 2lkFxoð Þ½

M zð Þ ¼ 12

p2X1k¼1

1ð Þkþ1 1þ kzð Þk2

ekz

ð108Þ

For example, M(2) 0.468, showing that the hyperbolic tangent approxi-

mation by itself gives estimates to Df that are a bit too large (see Figure 14).

Nevertheless, insofar as Eq. (108) is a reasonable model if a good choice of lis found, it demonstrates that the magnitude of the dipole potential

38 KEVIN L. JENSEN

contribution is dependent on both the electron density and the height of the

barrier, and while crude, it provides a qualitatively satisfactory account of

the variation of the dipole term with electron density for simple metals.

An estimate of the work function is then obtained from the relation,

suggested by Eqs. (83) and (104) (the signs reflect that the vacuum level is at 0)

mþ Fð Þ ¼ @

@rr eex þ ecor eo þ eið Þ½ Df ð109Þ

Performing the derivative is cumbersome, but the resulting equation yields a

rough estimate of F if rc is known. Given the qualitative treatment of Df and

the ion core terms here, however, ri is not known and is therefore only qualita-

tively similar to the actual ionic core radii.Approaches that pay better attention

to the ionic core terms, such as by Ashcroft and Langreth (1967), or to the

dipole terms, such as by Lang and Kohn (1970), fare better. Nevertheless, if

actual work function values are used, then Eq. (109) can be used to predict ri,

and the values of ri so found compared to, for example, the Pauling radii. The

results of that analysis are shown inFigure 10 for two values of l. Overall, given

the approximations, the relationship between ri and the Pauling radius is good,

particularly for ‘‘good’’ metals such as cesium, barium, sodium, and potassium

(columns 1 and 2 on the periodic table)—but not in all cases, especially those

where the impact of d‐shell electron contributions are nontrivial, such as

tungsten, iron, and others. Moreover, the qualitative behavior of the core

radius in the simple model here correlates well with trends of, for example,

Ashcroft and Langreth (1967) (also shown in Figure 10).

1

2

3

Cs Rb K Ba Na Li Zr Ag Mg Au Co Cu Pb Ni Zn Al Fe W

Core (2.71)

Core (1.24)

Pauling

A&LC

ore

radi

us [an

gstr

oms]

Element

FIGURE 10. The core radius ri compared to the Pauling radius and radii from Ashcroft and

Langreth for two values of l (2.71 and 1.24).


The 1D model does not account for expressly 3D complications associated

with real surfaces; the representation of a surface by a uniform and feature-

less plane is a substantial idealization. Real crystal surfaces appear

corrugated and complicate the work function. Smoluchowski (1941)

proposed a surface of regular pyramids and pyramidal depressions mimick-

ing a real surface such that the apexes and valleys become equally and

oppositely charged (Figure 11). The base of the pyramid is of length scale

L and the charge at the apex of the period is fq (where f is presumed to be1)

over an area L2, equivalent to a surface charge density of fq/L2. Crudely, the

potential energy drop across the dipole formed by the pyramids, on average,

is then dfD( fq2/L2)/eo¼ 18.095 eV‐nm f D/L2, whereD is the approximate

separation of the dipole layers after the charge has partially smoothed out.

Considering the pedagogical numbers of D ¼ 2 A, L ¼ 24 A, and f ¼ 0.3,

then df 0.189 eV. Since different crystal orientations have different

corrugated shapes at the surface, the model makes plausible the small differ-

ences in work functions that occur for different faces, albeit that it is defective

as an actual description of surface atomic structure. A similar argument may

be used to explain the tendency of adsorbates that charge either positively or

negatively to lower or raise the work function accordingly, such as barium

and oxygen on tungsten in a B‐type dispenser cathode, which has a work

function several tenths of an electron volt lower than bulk barium (this topic

is discussed in the section on the Gyftopoulos–Levine theory). Contours of

electron density for surfaces such as tungsten, with or without a barium

overlayer, hint at such a triangular structure (see, for example, Figure 3 of

Hemstreet, Chubb, and Pickett, 1989). Other effects not treated here also

leave their mark (e.g., effective mass differences, field enhancement). Never-

theless, Eq. (109) accounts for the dominant influences and is adequate to

anticipate the behavior of the image charge potential near the surface.

E. The Image Charge Approximation

1. Classical Treatment

Classically, a charged particle outside a conductor causes a redistribution of

charge on the surface of the conductor that serves to screen out the external

field. For a charge –q a distance x outside a conducting surface, it is a common

+

− −

+ +

L

fq

FIGURE 11. Smoluchowski model of corrugation of a real surface, showing a migration of

electron charge and the subsequent creation of a dipole. The “net charge” at the apex is fq.

40 KEVIN L. JENSEN

result, familiar from electrostatics, that to ensure that the surface remains at

zero potential, an image charge þq lies at –x from the surface inside of the

conductor (Eyges, 1972). The potential energy of the external charge is

obtained from the integral over the field between the charged particle and its

image, or

Vimage xð Þ ¼ð1x

q2

4peo 2xð Þ2 !

dx ¼ q2

16peoxð110Þ

or Vimage(x)¼ –Q/x. For a metal surface subject to an external field, and using

the bottom of the conduction band as the reference level in energy, the image

charge potential is then given by

Vimage xð Þ ¼ mþ F FxQ

xð111Þ

whereF is the product of electron charge and electric field. Far from the surface,

Vimage must be correct at macroscopic distances—if one is flexible about the

definition of F—and a more subtle analysis using the exchange‐correlationpotential and the jellium model is required to reveal the 1/x dependence (Lang

and Kohn, 1973). Near the surface, Eq. (111) is unphysical, plummeting to –1at x ¼ 0þ.

The hyperbolic tangent approximation to the density in Eq. (105)

with the exchange‐correlation analysis does serve to qualitatively validate

Eq. (111) near the surface. For the generic metal with re ¼ 0.1 moles/cm3, the

exchange‐correlation potential associated with the hyperbolic density plus the

dipole is compared to the image charge potential in Figure 12. The agreement

between the exchange‐correlation potential and the image charge potential is

not optimal—a consequence of the approximations behind the hyperbolic‐tangent density approximation—but the comparison clearly does show that

image‐charge–like variations in the potential exist near the surface simply as a

consequence of variations in density and the exchange‐correlation potential

prescription of Eq. (109). The more careful the theoretical analysis, the better

the agreement, further supporting the utility of the image charge potential

model of the surface (Kiejna, 1991, 1993, 1999).

The shifting of the density by an amount xo as a consequence of Eqs. (94) and

(98) creates an expectation that the positionx inEq. (111) should be replaced by

xþ xo, so thatVimage(x)¼ mþF – F(xþ xo) –Q/(xþ xo). Some (e.g., Lang and

Kohn. 1973) give xo with a negative sign—the choice reflects convention—and

in their parlance, xo represents a ‘‘center of mass’’ given by

xo ¼Ð10

xre xð ÞdxÐ10

re xð Þdx ð112Þ


The shifting of the potential by an amount inversely proportional to the

square root of the barrier height, as well as variations in potential arising

from fluctuations in density, are expectations borne out in more rigorous

treatments. It is important, however, to bear in mind that the model consid-

ered here is idealized: the jellium charge distribution is slightly different than

a realistic charge distribution of actual metals, which serves to complicate the

specification of the image plane position (see, for example, Forbes, 1998).

2. Quantum Mechanical Treatment

A Green’s function approach to the determination of the metal‐vacuum inter-

face potential accounts for potential variation near the surface due to quantum

effects and correctly accounts for the asymptotic image charge behavior

(Il’chenko and Kryuchenko, 1995; Qian and Sahni, 2002). When the interface

is subject to an external field, a Thomas–Fermi approximation (TFA) can quite

elegantly provide useful approximations to xo in the image charge potential;

accounting for the quantum mechanical nature of the conduction band elec-

trons via the random‐phase approximation (RPA) illuminates the nature of the

Friedel oscillations under more general conditions than considered above. The

TFA and RPA approaches differ in their approximations to the dielectric

function in a metal (Il’chenko and Goraychuk, 2001), and therefore in their

predictions, for example, in barrier height, but both validate the qualitative

0

0.2

0.4

0.6

0.8

1.0

1.2

0

2

4

6

8

10

−8 −4 40 8

Friedel

Tanh model

Tanh+dipole

Image potential

Den

sity

/bul

k de

nsity

Potential [eV

]

Position [Å]

F=0.4 eV/Å

FIGURE 12. Density (thin red line) showing Friedel oscillations and the related exchange-

correlation potential (thick red line). Also shown are the hyperbolic tangent approximation to

the density (thin blue dashed line), and the image charge potential (green dashed line).

42 KEVIN L. JENSEN

behavior of the classical image charge potential andmodifications to it. RPA is

preferred as it can, for example, deal with complications related to the shape of

the Fermi surface such as occur for realmetals as a consequence of the behavior

of the dielectric function.

The TFA, however, is easier to express concisely and analytically; there-

fore it is of greater utility in illuminating modifications to the image charge

potential. Approximate expressions to V(x) based on the TFA are then

given by (Il’chenko and Kryuchenko, 1995)

V xð Þ F

kTFekx Q

x 2xoe2kTFx x < 0ð Þ

2kTFQ F xþ 4

3xo

0@

1A Q

xþ xox 0ð Þ

8>>>>><>>>>>:

ð113Þ

where kTF is from Eq. (32), and xo has been redefined as xo ¼ 3/(4kTF). For

semiconductors, the image charge term Q is modified for a material with

dielectric constant Ks according to

Q ) Ks 1

Ks þ 1

afshc4

ð114Þ

The work function therefore becomes F ¼ 2kTFQ – m. In this sense, xo does

scale as 1/(barrier height)1/2, as expected in the discussion surrounding

Eq. (112), and the intuition that x should be (more or less) replaced by

x þ xo in the classical image charge potential is vindicated. The behavior

of Eq. (113) is shown in Figure 13 for an applied field of F ¼ 0.4 eV/A and

kF ¼ 1.3 A–1. Appealing though it is, the barrier predicted by Eq. (113) is

smaller than that predicted using RPA, and the TFA approach lacks the

conspicuous Friedel oscillations in Eq. (98) and Figure 12 that the RPA

approach replicates; however, it provides a justification for the development

of an analytical image charge potential by showing that quantum and physi-

cal effects can be accommodated (however crudely) through modifications

to the work function, field, and image charge terms.

3. An ‘‘Analytical’’ Image Charge Potential

The analytic image charge potential is a simplistic approximation compared

with the RPA approach or treatments where the wave function and

the exchange‐correlation and dipole terms are evaluated self‐consistently(Figure 14). The justifications for considering an ‘‘analytical’’ image charge

model are as follows: (1) the development of emission equations using

approximation formulas for the potential are the most expedient, (2) Friedel


oscillations and variation in the bulk are secondary in the development of the

emission equations; and (3) the impact of both field and temperature are

easily accommodated. The effects to be included are dealing with the con-

sequences of the origin of the background positive charge in the jellium

model not being coincident with the origin of the electron distribution and

the temperature of and field dependence of barrier height. The shift in the x

parameter by xo has been treated previously. Regarding the temperature and

field dependence of the barrier height, the electron gas temperature and

applied field have an impact on the magnitude of the work function for

analogous reasons. Both allow greater penetration of the emission barrier

0

2

4

6

8

10

0 5 10 15

TFAImage

Pot

ential

[eV

]

Position [angstroms]

FIGURE 13. Comparison of the Thomas–Fermi approximation [Eq. (113)] to the image

charge approximation.

0.0

0.5

1.0

1.5

0.001 0.01 0.1

tanh approx.Friedel Approx

∆f[

eV]

Density [1024 #/cm3]

FIGURE 14. Comparison of the dipole term calculated from the hyperbolic tangent approxi-

mation compared to the density using the Friedel approximation.

44 KEVIN L. JENSEN

by either a larger population of more energetic electrons (for temperature) or

the ability of the electrons to penetrate (for field). Coupled with the depen-

dence of the exchange‐correlation potential on density, as well as the behav-

ior of the dipole contribution, the height of the barrier above the Fermi level

changes. It is therefore expected that terms accounting for changes in the

work function that depend on temperature, field, and extent of the dipole will

need to be introduced, presumably resulting in a change in the definition of

F: the image charge potential should therefore resemble

V xð Þ ) m Tð Þ þ f F ;Tð Þ Fx Q

xþ xo; ð115Þ

where the use of f accommodates field‐ and temperature‐dependent effects.The origin of the background positive charge and the electron gas have been

treated as equivalent, but the shifting of the density in response to the barrier

height shows that the assumption is not strictly valid. Let the separation

between the electron gas origin and the background positive charge origin be

dxi. The density of the background positive charge is ro, and, due to global

charge neutrality, only the potential dropDfi across the slab needs to be added

to the work function estimate. It follows from Poisson’s equation that

Dfi ¼roe0

ðdxi0

ðx00

dx00

dx0 ¼ 8

3pQk3Fdx

2i: ð116Þ

While dxi can be found for simple models, in general it must be found

by enforcing global charge neutrality in a numerical simulation (an example

of which is found in Jensen [1999, 2000], although the present treatment

has changed) as it depends on electron temperature, applied field at

the surface, and the barrier height. Assuming that has been done, what remains

of the image charge approximation? A rearrangement of terms results in

Vanalytic xð Þ ¼ m Tð Þ þ Feff F xþ xoð Þ Q

xþ xo, ð117Þ

where an ‘‘effective’’ work function Feff is introduced. Eq. (117) does not

appear to confer any benefit, as it appears to substitute one set of difficult

parameters for another, but its advantage lies in the fact that in estimates of

emission current, the classical image charge figures prominently in analytical

formulae, particularly the Richardson–Laue–Dushman (RLD) equation for

thermionic emission and the Fowler–Nordheim (FN) equation for field emis-

sion. The introduction of an effective work function allows for subtle effects

related to temperature and field on the barrier height and the transmission

probability to be ‘‘smuggled’’ into the RLD and FN formulas without necessi-

tating grueling numerical or analytical effort provided the behavior of Feff

can be ascertained. What might that behavior be like? Experimentally,


the work‐function variation with temperature is linear, with the coefficient of

the temperature material dependent (see (Haas and Thomas, 1968) for a listing

of coefficients for various elements). Similarly, from the expression

xo ¼ k1o ¼ h=

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2m mþ fð Þp

, where f is the height of the barrier above the

Fermi level, the lowering of the image charge barrier by the application of

field (known as Schottky barrier lowering) will relate to the shifting of the

density profile, and therefore to the value of dxi. Numerical findings suggest

that the relation between the barrier height and Fxo is also linear to a good

approximation. Therefore, the effective work function should resemble

Feff T ;Fð ÞFo þ aokBT þ 8

3pQk3Fdx

2i þ 2Fxo ; ð118Þ

where parameters such as the (dimensionless) parameter ao are from

the thermal dependence of the work function, known from the literature

(see examples in Table 3) and the others chosen so that Feff is equal to

the experimental work function at zero field and a known temperature.

The temperature and field variation of dxi is slight, and as a pragmatic

matter, it is adequate to absorb the term in which it resides into Fo, or to

parameterize it based on numerical simulations. Regardless of the method

used to determine the temperature and/or field dependence of the work

function, the determination of emission current from the classical image

charge potential is convenient as long as the fact that the work function in

it depends on temperature and field in the manner suggested by the discus-

sion surrounding Eq. (103) is kept in mind. With that caveat, any equation in

the following text dependent on the image charge potential can be trivially

modified by the usage of Eq. (118) for the replacement of the work function

with an ‘‘effective’’ work function (see Table 4).

TABLE 4

PARAMETERS OF WORK FUNCTION*

Atomic number Element Fo ao

47 Ag 4.31 0.12

79 Au 4.25 0.17

56 Ba 2.3 5.80

72 Hf 3.6 1.62

25 Mn 3.83 1.28

42 Mo 4.33 0.12

41 Nb(100) 3.95 0.35

14 Si 3.59 2.67

74 W 4.52 0.70

*F(T) ¼ Fo þ ao kBT; adapted from Haas and Thomas, 1968.

46 KEVIN L. JENSEN

II. THERMAL AND FIELD EMISSION

A. Current Density

In the discussion of the hydrogen atom, the Kronig–Penney model, and the

dipole contribution to the work function, Schrodinger’s equation was

‘‘smuggled’’ into the discussion; when analyzing the surface barrier, a meth-

odology was introduced by fiat to yield transmission and reflection coeffi-

cients in Eqs . (89) and (92) , which are ne eded to determ ine the extent to

which the wave function penetrated the barrier. A return to the central issue

in the development of the emission equations, namely, the barrier problem—

but now with the proper formalism—is desirable. In contrast with perhaps

standard treatments, the approach here is to start with a distribution

function and move to the Schrodinger representation to allow for the

introduction of the classical approach followed by its quantum extension.

1. Current Density in the Classical Distribution Function Approach

The Boltzm ann equ ation in Eq. (24) is fami liar from the classical stat istical

mechanics of an ideal gas (Reichl, 1987; Reif, 1965), for which the number of

particles dn ¼ f(x,k,t) dx dk in a small region at time t is conserved. After a

time t0 ¼ tþdt, all the particles must be accounted for, or

f ðx; k; tÞdxdk ¼ f ðx0; k0; t0Þdx0dk0: ð119ÞTo order O(dt), x0 ¼ xþ hk=mð Þdt and hk0 ¼ hkþ F=mð Þdt, so that

the Jacobian is (the 1D case is shown, but the 3D case represents a

straightforward generalization)

]xx0 ]xk0]kx0 ]kk0

¼ 1 0

hdt 1

¼ 1: ð120Þ

Therefore dx0dt0 ¼ dxdt, from which the conclusion f ðx; k; tÞ ¼ f ðx0; k0; t0Þfollows. A Taylor expansion then shows

0 ¼ f xþ dx; kþ dk; tþ dtð Þ f ðx; k; tÞdt

¼ ]

]tþ hk

m

]

]xþ F

h

]

]k

8<:

9=;f ðx; k; tÞ;

ð121Þwhere F ¼ ]x V ð xÞ (compa re Eq. (24) ). ‘‘Mom ents’’ of a general dist ribution

function are defined by


hO r;kð Þi

ðdrdkO r; kð Þf r; kð Þð

drdkf r; kð Þ: ð122Þ

They are important because the first momentum moments are proportional

to the density and the average velocity.When the distribution is the equilibrium

FD distribution, then they have been encountered before, as Eq . ( 14 ), for

example, can be written in the language of hknxi /ðknx fFD EðkÞð Þdk for n ¼ 0.

For one dimension the transverse momentum components in the numerator

and denominator of Eq. (122) can be performed, resulting in the supply

function encountered in Eq . ( 97 ), and allowing for the suppression of the

subscript on the remaining component via kx ) k. For one dimension, then,

the number density is proportional to hk0i and current density to hk1i.The generalization to present circumstances for a distribution function,

which is spatially dependent, follows analogously:

r xð Þ ¼ 1

2p

ð11

f x; kð Þdk

J xð Þ ¼ 1

2p

ð11

hk

mf x; kð Þdk

ð123Þ

A cautionary note about Eq. (123): as written, r and J are number density

and number current density, respectively, as the charge of the carrier is not

present. In the Poisson and continuity equations, for example, the charge q

reappears in their coefficients.

Consider three well‐known examples. First, the equipartition theorem

follows from the evaluation of the second moment when the distribution

function is an MB distribution via

hEðkÞi ¼

ð11

EðkÞebEðkÞdkð11

ebEðkÞdk¼ h2

8m

ffiffiffiffiffiffiffiffiffiffip=a3

pffiffiffiffiffiffiffiffip=a

p ¼ 1

2b; ð124Þ

where the parabolic relationship between E and k is used, b ¼ 1=kBT ,

and a ¼ bh2=2m. Note the (1/2) coefficient on the RHS, reflecting that

Eq. (124) is a 1D evaluation: when all three dimensions are considered,

hExi þ hEyi þ hEzi ¼ 3kBT=2. Parenthetically, Eq. (124) reappears in the

discussion of thermal emittance.

Second, consider the case when f(x,k) is a Gaussian distribution in k only,

that is,

48 KEVIN L. JENSEN

f kð ÞjGaussian ¼ro

Dkffiffiffip

p exp k ko

Dk

2( )

: ð125Þ

It then follows from Eq. (123) that J ¼ hko=mð Þr, or current density is the

product of the mean (or center) velocity and number density.

Third, and extending the second example, evaluate ]tr using Eq. (123) and

insert it into Eq. (121) to obtain

]

]tr x; tð Þ ¼ 1

2p

ð11

hk

m

0@

1A ]

]xf ðx; k; tÞ þ F

m

]

]kf ðx; k; tÞ

24

35dk

¼ ]

]xJ x; tð Þ

ð126Þ

where use has been made of f x;1ð Þ ¼ 0, allowing the ]kf term to be

integrated and summarily dispensed with. Eq. (124), known as the continuity

equation, is the classical distribution function version whose quantum

mechanical counterpart is superficially similar but in detail a great deal

more subtle, a task to which we now turn.

2. Current Density in the Schrodinger and Heisenberg Representations

To find the quantum extensions of expressions in Eqs. (123) and (126),

the bra‐ket notation requires additional formalism (Rammer, 2004).

In the continuum limit and in an arbitrary number of dimensions, the

kets jxi and jki satisfy

hxjki ¼ 2pð Þd=2exp ik xð Þ

hxjx0i ¼ d x x0ð Þhkjk0i ¼ d k k0ð Þ

ð127Þ

The vector notation significantly complicates the formulas without a com-

mensurate pedagogical benefit, so the 1D (d ¼ 1) case shall be considered

henceforth, leaving the higher‐dimensional analogs to be intuited (or

obtained from hardier treatments) (Reichl, 1987). An operator of particular

importance is the identity operator: generalizing from when x and k are

discrete, it is

I^¼ 2pð Þ1=2

ð11

jxihxjdx ¼ 2pð Þ1=2

ð11

jkihkjdk: ð128Þ


Evolution of the wave function is governed by the evolution (alternately,

propagation) operator UðtÞ such that jcðtÞi ¼ UðtÞjcð0Þi. It follows

from the definition of jcðtÞiand the constancy of the energy E that for a

Hamiltonian H, then UðtÞ must satisfy

E ¼ hcðtÞjHjcðtÞi ¼ hcð0ÞjHjcð0Þi ð129Þ

jcðt1 þ t2Þi ¼ Uðt1ÞUðt2Þjcð0Þi ¼ Uðt1 þ t2Þjcð0Þi ð130Þ

hcðtÞjcðtÞi ¼ hcð0ÞjUðtÞUðtÞjcð0Þi ¼ hcð0Þjcð0Þi: ð131Þ

Equation (129) indicates that H; UðtÞ ¼ 0, where A; Bh i

AB BA

(the commutator relation). Eq. (130) indicates that UðtÞ ¼ exp f ðHÞt (where f is a function), and Eq. (131) demonstrates that U is unitary

(i.e., U1 ¼ U , or the inverse of U is its adjoint). Taken together, the

simplest function to satisfy these requirements is UðtÞ ¼ exp iHt=h

.

The treatment of jcðtÞi so far has used time‐independent operators andtime‐dependent wave functions, collectively known as the Schrodinger picture

or representation. Operators in this representation are designated with an S

subscript (S). An alternate approach, referred to as the Heisenberg picture or

representation, designated by an H subscript (H), makes the wave function

time independent and transfers the time dependence to the operators O via

OHðtÞ ¼ UðtÞOSUðtÞ: ð132ÞThe variation in time of OHðtÞ is then

]

]tOHðtÞ ¼ ]

]tUðtÞ

0@

1AOSUðtÞ þ UðtÞOS

]

]tUðtÞ

0@

1A

¼ i

hHS; OHðtÞh i ð133Þ

where Eqs. (131) and (132) have been used. Thus, in the Heisenberg repre-

sentation the time variation of an operator is given by its commutation with

the Hamiltonian.

Let A, B, and C be operators. It is trivial to show A;BC½ ¼A;B½ C þ B A;C½ . Coupled with the Heisenberg relation xn; k

h i¼ i, it follows

that

xn; kh i

¼ inxn1 ¼ i]x xnð Þ: ð134Þ

50 KEVIN L. JENSEN

Expressin g f ðx Þ as a power serie s in x, it foll ows that

f ðxÞ ; kh i

¼ i]

] x f ðxÞ : ð135Þ

Now co nsider the density operator defin ed by [comp are Eq. (96) ]

nH ð t Þ ¼X

Efo Eð Þjc ðt Þih cð t Þj; ð136Þ

from which it follows that nð x; t Þ ¼ hxjnð t Þjxi an d therefo re

h xj V ðx Þ ; nðt Þ½ jx i ¼ 0. Anothe r way to express this is by obs erving that the

diagonal elem ents h x jO j xi vanish if O is the commut ator of a position ‐depend ent ope rator with the den sity ope rator. It follows that for the diago nal

elemen ts (the ‘‘off ‐ diagonal ’’ elem ents of the de nsity operator are con sidered

in the discus sion of the Wigner fun ction)

HS ; n H ð t Þ ¼ h 2

2mk 2 ; nH ð t Þ

h i¼ i h2

2 m

]

] xk ; nH ð tÞn o

; ð137Þ

where the antic ommut ator is defined by A; Bf g ¼ AB þ BA . Identify the

current operator as

jS ¼1

2mnS ; h kn o

jH ð t Þ ¼1

2mU ð tÞ nS ; h k

n oU ð tÞ :

ð138Þ

Coupled with Eq. (137) an d Eq. (133) , Eq. (138) implies that

]

] t nH tð Þ þ ]

] xjH tð Þ ¼ 0: ð139Þ

Eq. (139) is the quan tum analog to Eq. (126) . Using h xjk jc ðt Þi ¼i ]x cð x; t Þ , it foll ows that

j ð x; t Þ ¼ hco jj H ðx; t Þjc o i ¼h

2m h c ð tÞj nS ; k

n ojcðtÞi

¼ h

2micðx; tÞ]xcðx; tÞ cðx; tÞ]xcðx; tÞ ð140Þ

which is the conventional relation for the current density in the Schrodinger

picture. Consider the special case when jcoi ¼ jki, that is, the wave functionis a moment um eigens tate, as sho wn in Figure 15. The incide nt wave is

cinc ¼ ð2pÞ1=2exp ikxð Þ, for which the incident current is jinc ¼ hk=m; as

n(x,t) is a number density, j(x,t) will be the flux of particles. Current density,

in the conventional sense of charge per unit area per unit time, corresponds to


qj(x,t). After interacting with a potential V jxj 1ð Þ ¼ 0, the transmitted

and reflected waves are given by ctrans ¼ ð2pÞ1=2tðkÞexp ikxð Þ and

cref ¼ ð2pÞ1=2rðkÞexp ikxð Þ, respectively. Thus,

jtrans ¼ jtðkÞj2hk=mjref ¼ jrðkÞj2hk=m ð141Þ

Conservation of particles demands that jtrans þ jref ¼ jinc or

jtðkÞj2 þ jrðkÞj2 ¼ 1; ð142Þwhich is obtained by taking the ratio of both sides with the incident current.

The transmission probability for a given momentum is therefore taken as

the ratio between the transmitted and incident current densities, but

T kð Þ ¼ jtðkÞj2 only for V jxj 1ð Þ ¼ F jxj 1ð Þ ¼ 0; when the RHS and

the LHS differ in reference energy, or the fields are different, then changes

occur in the expression for transmission probability.

3. Current Density in the Wigner Distribution Function Approach

The common usage of Eq. (140) lacks resemblance to the distribution func-

tion approach of Eq. (126). In a related vein, regardless of the (visual)

similarity, a difference exists between f x; k; tð Þ and foðEÞjckðxÞj2, and so

the latter is not the quantum analog of the distribution function sought.

Something else is required. Wigner suggested the following function as a

candidate (Hillery et al., 1984):

r(k) e−ikx

e ikx

t(k) e ik'x

FIGURE 15. Schematic representation of incident, transmitted, and reflected waves incident

on a general barrier for which V( 1) ¼ 0.

52 KEVIN L. JENSEN

f x; k; tð Þ ¼ 2

ð11

e2ikyhxþ yjnðtÞjx yidy: ð143Þ

Combining Eqs. (137) and (139), it follows

]

]tf x; k; tð Þ ¼ 2

ð11

e2ikydy hxþ yj h

2m]xfnðtÞ; kgjx

8<: yi

þhxþ yj ih½V ðxÞ; nðtÞjx yi

9=;

ð144Þ

Using Eq. (136), first term of the integrand is

hxþ yj]xfnðtÞ; kgjx yi ¼ kð]xþy þ ]xyÞhxþ yjnðtÞjx yi¼ 2k]xhxþ yjnðtÞjx yi ð145Þ

The second term is a bit more involved:

hxþ yj V xð Þ; nðtÞ½ jx yi ¼ V ðxþ yÞ V ðx yÞð Þhxþ yjnðtÞjx yi

¼ V ðxþ yÞ V ðx yÞð Þ 1

2p

ð11

e2ik0yf x; k0; tð Þdk0

ð146ÞIntroducing the concise notation

V x; k k0ð Þ ¼ i

ph

ð11

e2iðkk0Þy

V ðxþ yÞ V ðx yÞf gdy ð147Þ

and combining Eqs. (144)–(147) results in the time‐evolution equation for the

Wigner distribution function (WDF):

]

]tf x; k; tð Þ ¼ hk

m

]

]xf x; k; tð Þ þ

ð11

V x; k k0ð Þf x; k0; tð Þdk0: ð148Þ

The immediate impact of quantum effects is the dependence of

V x; k k0ð Þ on the potential at locations other than x; that is, the integrand

in Eq. (147), is nonlocal, meaning the behavior of f at x depends on the

behavior of V(x0) for x0 away from x. The impact of quantum effects can be

related to Boltzmann’s transport equation (BTE) by expanding the integrand

of Eq. (147)


V xþ yð Þ V x yð Þ ¼X1n¼0

2y2nþ1

2nþ 1ð Þ!]

]x

2nþ1

V xð Þ ð149Þ

and using the substitution

ð11

yne2ikydy ¼ i

2

]

]k

n

2pd kð Þ: ð150Þ

The generalization to the BTE is then

]

]tf ðx; k; tÞ ¼ hk

m

]

]xf ðx; k; tÞ þ 1

h

X1n¼0

1ð Þn22n 2nþ 1ð Þ! ]

2nþ1x VðxÞ

]2nþ1k f ðx; k; tÞ

:

ð151ÞFor potentials that are at most quadratic, all but the first term in the sum in

Eq. (151) vanish. Only the field term survives, and the classical form of the

BTE is satisfied by f(x,k,t). It follows that under such circumstances, f(x,k,t)

¼ f(xcl(t),kcl(t),0), where xcl(t) and kcl(t) are classical trajectories—but it does

not follow that f(x,k,t) is a classical distribution function as there are regions

for which f(x,k,t) can be negative, and probability distributions do not

behave in such a manner (Hillery et al., 1984; Kim and Noz, 1991; Rammer,

2004; Reichl, 1987). Nevertheless, the first two moments of the WDF provide

the particle number density and current density, respectively, or

nðx; tÞ ¼ hxjnðtÞjxi ¼ 1

2p

ð11

f x; k; tð Þdk

Jðx; tÞ ¼ 1

2p

ð11

hk

mf x; k; tð Þdk

ð152Þ

as can be shown using Eq. (143) andðeizdz ¼ 2pdðzÞ, which mirrors

the behavior of the classical distribution function. Moreover, Eq. (139)

follo ws from Eq. (152) and Eq. (148) and the an tisymm etry of V(x,k) .

Examples highlighting the quantum behavior and its differences from

classical distributions are given next.

a. Wave Packet Spreading (No Potential). Consider a wave packet con-

structed by summing over plane wave states with a Gaussian weighting factor

in momentum centered about k¼ 0: it is trivial to shift the center momentum

to a non‐zero value for a traveling wave packet. The normalized wave

function is

54 KEVIN L. JENSEN

cðx; tÞ ¼ 2

pDk2

0@

1A

1=4ð11

exp k

Dk

0@

1A

2

þ ikx iot

8<:

9=;dk

¼ 2

pDk2

0@

1A

1=4

1

Dk2þ iht

2m

0@

1A

1=2

exp Dk2x2

41þ ihDk

2mt

0@

1A

18<:

9=;

ð153Þ

where hoðkÞ ¼ h2k2=2m. The probability density then is a Gaussian given by

r x; tð Þ ¼ 2

pDk2

0@

1A

1=2

1

Dk4þ ht

2m

0@

1A

20@

1A

1=2

exp Dk2x2

2

1

Dk4þ ht

2m

0@

1A

20@

1A

18><>:

9>=>; ð154Þ

The wave packet therefore spreads as time increases, as shown in

Figure 16. Compared to Eq. (154), the Wigner formulation of the same

problem is elegant. At time t ¼ 0, f(x,k,0) is given by

f ðx; k; 0Þ ¼ 1

2p

ð11

e2ikyc xþ y; 0ð Þc x y; 0ð Þdy

¼ 2

Dk2exp 1

2Dk2x2 1

2

k

Dk

0@

1A

28<:

9=;

ð155Þ

The integrations that lead to Eq. (155) are readily employed when using

Eq. (153). A regrouping of terms then results in

−2

−1

0

1

2

−3 −2 0 1 2 3−1

k

t = 0.0 fs

t = 0.2 fs

[x1(t), k1(t)]

[x2(t), k2(t)]

x

FIGURE 16. Spreading of the Gaussian wave packet in the Wigner distribution function

approach. Two trajectories are shown that demonstrate the classical trajectory behavior.


f x; k; tð Þ ¼ 2

D k2 exp 1

2 Dk2 x hk

m t

2

1

2

k

D k

2( )

; ð 156 Þ

from which (with much effor t) an integ ration ov er k reproduc es Eq. (153) .

Exa mination sho ws that Eq. (156) can be rew ritten as

f x; k ; tð ÞjV ð xÞ¼0 ¼ f x hk

m t ; k; 0

; ð 157 Þ

wher e the subscri pt reinforces that force ‐ free evo lution is occurri ng. Thespreadi ng of the wave pa cket is theref ore tanta mount to a sheari ng of a n

ellipse in pha se space such that while the area of the ellip se bounde d by a

given contour line remain s constant , it is pro gressivel y elong ated (conser va-

tion of area in pha se space is a not ion reappea ring in the discussion of

emittance). Analogous to the spreading of a wave packet for an electron

shown in Figure 16 ( Dk ¼ 0.1 nm 1 an d t ¼ 0.0 and 0.2 fs for a region 1 nm

across and 1 nm1 wide), Eqs. (153) and (156) are shown schematically in

Figure 17.

More can be evoked from this example. First, observe that for potentials

that are at worst quadratic, Eq. (151) can be written as

]

]tf x; k; tð Þ ¼ hk

m

]

]xþ 1

h]xV ðxÞ]k

f x; k; tð Þ; ð158Þ

which is seen as the BTE for one dimension. The term in curly brackets

on the RHS can be treated as an operator O. For systems in which

the energy is constant, O E ¼ 0 (a restatement of Hamilton’s equations),

0

0.2

0.4

0.6

0.8

−3 −2 −1 0 1 2 3

t = 0.0 fst = 0.2 fst = 0.4 fs

Den

sity

[a.

u.]

Distance [0.1 nm]

FIGURE 17. Spreading of the Gaussian distribution density (0th moment of the distribution

function) as a function of time.

56 KEVIN L. JENSEN

then f(x,k,t) is written as f(E) and Eq. (158) is automatically satisfied along

trajectories, designated by the pair of phase‐space coordinates (xp(t),kp(t)),

which satisfy

h2k2p2m

þ V xp ¼ E: ð159Þ

For a free particle, V(x) ¼ 0 and E ¼ ðhk2pÞ=2m, and Eq. (158) demon-

strates xpðtÞ ¼ x hkmt, in agreement with Eq. (157), which is what is meant

when it is stated that Wigner trajectories are equivalent to classical trajec-

tories for up to quadratic potentials. The equivalence allows for the conclu-

sion that if V(x) is a linear function of x, or V(x) ¼ gx (the notation

reflecting the common gravitational example of a linear potential), then it

follows that

f x; k; tð Þ ¼ f x hk

mt g

2mt2; kþ g

ht; 0

; ð160Þ

which explicitly satisfies Eq. (158) and satisfies f(x,k,t) ¼ f(E(x,k)). The

treatment here is more improvised than precise; Rammer (2004) and Kim

and Noz (1991) provide a careful demonstration of Eq. (160). Before Wigner

trajectories are dismissed as idle curiosities of a slothful imagination, they do

in fact have some value: numerical simulations of particle transport are

beholden to trajectories of pointlike creatures, and so introducing a classical

notion into a system with quantum behavior, a ‘‘quantum trajectory’’ con-

cept, has merit (Hsu and Wu, 1992; Jensen and Buot, 1989, 1990, 1991;

Martin et al., 1999). Another quantum trajectory concept, the Bohm trajec-

tories (Bohm and Staver, 1951; Dewdney and Hiley, 1982; Vigier et al., 1987)

similarly makes a clever attempt to introduce classical trajectory concepts via

Schrodinger’s equation itself.

b. The Harmonic Oscillator. The simplicity of Eq. (156) erroneously

implies that the Wigner function is a probability distribution function and

therefore is positive for all values of x and k, not only because the contours

act as trajectories but because f(x,k,t) acts like a classical phase‐spaceprobability distribution function by giving momentum and current density

as moments of the distribution. The Wigner function, regardless of its other

virtues, is not a probability distribution; the simplest system to see how it is

not (but also how quantum mechanics is intriguingly different) is the

harmonic oscillator. Classically, the energy of an oscillator can be written as

E ¼ h2k2

2mþ h2k2o

2m

x

L

2; ð161Þ


where a characteristic length (L) and momentum hkoð Þ have been introduced.

Trajectories correspond to contours of E, and therefore,

xpðtÞ ¼ xo cos otþ fð ÞkpðtÞ ¼ xoo sin otþ fð Þ ð162Þ

where f is a phase and ho ¼ h2ko=2mL h2a=2m. As is generally true, it is

pragmatic to know one’s final destination before embarking; to that end, a

concise account of the quantum treatment of the oscillator is given to show that

f(x,k,t) ¼ f(E(x,k)) in a manner foretold by Eq. (155). The unconventional

representation of the energy in Eq. (161) and the introduction of a is, as

expected, to simplify Schrodinger’s equation in operator parlance, which

becomes

k2 þ a2x2

jcni ¼ k2njcni; ð163Þ

where the n subscript (n) distinguishes the energy levels and anticipates

the conclusion that the energy of the oscillator will be quantized. The

observation that

ikþ ax

ikþ ax

¼ k2 þ x2 þ ia x; k

h i¼ k

2 þ x2 að164Þ

where the commutator of position and momentum has been used, suggests

the introduction of ‘‘creation’’ and ‘‘annihilation’’ operators (the nomenclature

to become clear) defined by

a ¼ a1=2 ikþ ax

a ¼ a1=2 þikþ ax ð165Þ

and satisfying

a; a½ ¼ 2i x; kh i

¼ 2

Enjcni ¼h2a2m

aaþ 1ð Þjcnið166Þ

The similarities to the creation and annihilation operators introduced

earlier in Eq. (39) are intent ional, but there are impor tant differences .

Using the A;BC½ commutation relations

58 KEVIN L. JENSEN

Hajcni ¼h2a2m

aaþ 1ð Þajcni

¼ h2a2m

a a; a þ aaþ 1 jcni

¼ ah2a2m

aaþ 3ð Þ8<:

9=;jcni ¼ En þ hoð Þajcni

ð167Þ

Therefore, the effect of a onjcni has been to raise the energy eigenvalue by

h2a=m ¼ ho; that is, En ¼ nhoþ E0. Normalization is resolved by starting

with the ground state

hxjajc0i ¼ 0 ¼ ]

]xþ ax

c0ðxÞ; ð168Þ

for which the solution is

c0ðxÞ ¼ap

1=4exp 1

2ax2

: ð169Þ

The action of the Hamiltonian on Eq. (169) identifies E0 ¼ h2a=2m ¼ ho=2.Let jcni ¼ Nn að Þnjc0i, where Nn is the normalization found by insisting that

1 ¼ hcnjcni ¼ Nn=Nn1ð Þ2hcn1jaajcn1i

¼ Nn=Nn1ð Þ2hcn1jaaþ 2jcn1i

¼ Nn=Nn1ð Þ2hcn1j2 n 1ð Þ þ 2jcn1i ¼ Nn=Nn1ð Þ22n

ð170Þ

It follows thatNn ¼ 2nn!ð Þ1=2 a=pð Þ1=4. The solution to the harmonic oscilla-

tor is now complete, and the representation in the jxi basis can be obtained

from the definition of the ground state

hxjajc0i ¼ a1=2 ]x þ axð Þc0ðxÞ ¼ 0; ð171Þthe normalized solution is

c0 xð Þ ¼ ap

1=4exp 1

2ax2

: ð172Þ

The use of Eq. (172) to find the ground‐state Wigner function, while

certainly a candidate for consideration, is not a particularly compelling one

because it does not differ in appearance much from the wave packet example.

Higher n wave functions are of greater pedagogical interest, but they require

further work. Let


cnðxÞ ¼ NnhðxÞexp 1

2ax2

; ð173Þ

where h(x) is to be determined. From ajcn1i,

hnðxÞexp 1

2ax2

¼ a1=2 ]x þ axð Þ hn1ðxÞexp 1

2ax2

: ð174Þ

It follows that

hn xð Þ ¼ a1=2 ]xhn1ðxÞ þ 2axhn1ðxÞð Þ; ð175Þwhere h0(x) ¼ 1. Eq. (175) is a variation of the recurrence relation

for Hermite polynomials (Abramowitz and Stegun, 1965), namely,

Hnþ1ðyÞ ¼ ]yHnðyÞ þ 2yHnðyÞ and so for the nth level,

cnðxÞ ¼ 2nn!ð Þ1=2 ap

1=4Hn a1=2x

exp 1

2ax2

: ð176Þ

Eq. (176) is what was sought to find the nth‐level Wigner function. Straight-

forward evaluation of Eq. (155) but with Eq. (173) shows that for n¼ 0 and 1

f0ðx; k; 0Þ ¼ exp 1

ak2 þ a2x2 2

435

f1ðx; k; 0Þ ¼ 2

ak2 þ a2x2 1

24

35exp 1

ak2 þ a2x2 2

435

ð177Þ

The higher n Wigner functions are evaluated analogously. The

classical orbits for the harmonic oscillator are xðtÞ ¼ e=að Þ1=2cos otþ ’ð Þ andkðtÞ ¼ aeð Þ1=2sin otþ ’ð Þ, where ’ is specified by initial conditions

and e ¼ 2E=ðhoÞ. It is easily shown that

f0 xðtÞ; kðtÞð Þ ¼ ee

f1 xðtÞ; kðtÞð Þ ¼ 2e 1ð Þeeð178Þ

In other words, as suggested, the classical trajectories correspond to the

contour lines of the Wigner function. The exploitation of the trajectory

concept is of more than just pedagogical interest; it may, in fact, serve as a

bridge between quantum (Wigner function) and classical (BTE) simulations

(Hsu andWu, 1992; Jensen and Buot, 1989, 1991; Jensen and Ganguly, 1993;

Martin et al., 1999; Vigier et al., 1987). Some care is required, however,

as Eq. (178) contains an additional feature: unlike a classical distribution

function, the Wigner function can assume negative values, as is apparent

when e < 1 for f1(x,k) (Figure 18), a feature that prevents its interpretation

60 KEVIN L. JENSEN

as a true probability distribution function, regardless of the utility of its

moments for the evaluation of the number and current densities. Less

timorous spirits have tackled the concept of negative probability head

on (Feynman, 1987), but here, such a feature is a quixotic artifact of an

otherwise useful approach.

c. The Gaussian Potential Barrier. A final pedagogically valuable

example is the case where V(x) is of the form VðxÞ ¼ Voexp x=Dð Þ2h i

.

A blessedly short derivation then shows that

V x; kð Þ¼ i

ph

ð11

e2iky V ðxþ yÞ V ðx yÞ½ dy

¼ 2Dffiffiffip

phVoexp D2k2

sin 2kxð Þ

ð179Þ

Eq. (179) is curious and useful not because real potentials are Gaussian (they

are not) nor because the Wigner function is easily evaluated (it is not) but

because consideration of the Gaussian potential provides a relatively clear

distinction between the regimes where through‐the‐barrier (tunneling) versusover‐the‐barrier (thermionic emission) dominate without the necessity of

indulging gradient expansions characteristic of Eq. (151). The argument is

simple: if D is large, indicating that the barrier is wide, then the exponential in

Eq. (179) is sharply peaked, so that only values for which k k0survive in

Eq. (148) so that because of the sine term, all the even terms of a Taylor

expansion including the 0th‐order term vanish, leaving the classical equation

of Eq. (121). Conversely, if D is small, then the exponential term is broad and

momentum values far from k0 contribute. Quite generally, then, quantum

effects are unimportant when kx oscillates many times over a length scale

characteristic of the barrier width at E(k), but are important when kx

f1(x,k) < 0

f1(x,k) > 0

FIGURE 18. The Wigner distribution function f1(x,k) for the n = 1 harmonic oscillator.


wiggles only a few times over the characteristic length scale. The value of k is

for the largest value appreciably present, which, for field emission from

metals is the Fermi value kF, whereas x is of order D. Transplanting this

newly acquired intuition to the field emission barrier, but ignoring the image

charge term, the potential barrier for an applied field F has a thickness of F/Fat an incident energy equal to the chemical potential. Thus, tunneling is

important when

kFFF 2p ) F F

2ph

ffiffiffiffiffiffiffiffiffi2mm

p: ð180Þ

For example, using values characteristic of copper (m ¼ 7 eV, F ¼ 4.6 eV),

then F is on the order of 10 eV/nm, or equivalent to 10 GV/m. In practice,

fields between 4 GV/m and 8 GV/m produce appreciable tunneling current

from metals when the work function is several electron volts.

4. Current Density in the Bohm Approach

An approach to the evaluation of current density and the transmission

probability due to Bohm and Hiley (1985) is a natural introduction to the

Wentzel–Kramers–Brillouin (WKB) methods used below [Dicke and Wittke

(1960); although, as emphasized by Forbes (1968), given modern usage

epitomized by Murphy and Good (1956) the designation Jeffreys‐Wentzel‐Kramers‐Brillouin (JWKB) is perhaps preferred]. The wave function is

represented at cðxÞ ¼ RðxÞexp iSðxÞ½ , where R and S are real functions.

When inserted into the time‐dependent Schrodinger equation

ih]tc ¼ h2

2m]2x þ VðxÞ

!c; ð181Þ

then Eq. (181) becomes, on separating the real and imaginary parts to the

LHS, respectively,

h2

2m]xSð Þ2 R1]2xRþ k2v

h iþ h]tS ¼ i hR1]tRþ h2

2mR2 ]xRð Þ ]xSð Þ þ R]2xS 8<

:9=; ð182Þ

whereVðxÞ ¼ hkvðxÞð Þ2=2m. Because R and S are real, each side of Eq. (182)

separately is zero. The LHS gives the Hamilton–Jacobi equation (Goldstein,

1980) if h]xS is identified as a momentum (i.e., KE ¼ h]xSð Þ2=2m is the

kinetic energy) and a quantum potential (distinct from the classical potential

V(x)) is defined by

62 KEVIN L. JENSEN

OðxÞ ¼ h2

2mR1]2xR; ð183Þ

in which case the LHS of Eq. (182) becomes

h]tS þ h2

2m]xSð Þ2 R1]2xRþ k2v

h i¼ h]tS þ KE þ OðxÞ þ V ðxÞ ¼ 0: ð184Þ

The quantum potential and the potential V(x) can be used to chart the

dynamics of particles following Bohm trajectories (Dewdney and Hiley,

1982; Vigier et al., 1987; analogous to the Wigner trajectories), and has

been used to provide a trajectory interpretation to the interaction of wave

packets with general barriers and resonant tunneling diode (RTD) barriers.

The current density is obtained by inserting the wave function into Eq. (140)

to yield

JðxÞ ¼ RðxÞ2 h

m]xSðxÞ

; ð185Þ

which, when compared to jðxÞ ¼ hk=mð ÞrðxÞ, matches the interpretation of

RðxÞ2 as the number density and ðh=mÞ]xS as the velocity. The RHS of

Eq. (182) reproduces the continuity equation

i hR1]tRþ h2

2mR2 ]xRð Þ ]xSð Þ þ R]2xS 8<

:9=; ¼ ih

2r]trþ ]xJð Þ ¼ 0: ð186Þ

Equations (184) and (186) are beautiful, but unlike the Wigner trajectory

case, evaluating the Bohm trajectories for situations even as appealing as the

harmonic oscillator serves no further pedagogical value. Rather, in the

development of the emission equations, the Bohm approach is useful as a

backdoor approach to defining the most widely used approach to the deriva-

tion of current density for potentials encountered in electron emission,

namely, the area under the potential approach to evaluating the transmission

coefficient. In this situation, the time‐independent Schrodinger equation for

momentum eigenstates is used

]2x þ kvðxÞ2

ckðxÞ ¼ k2ckðxÞ; ð187Þ

where EðkÞ ¼ ðhkÞ2=2m, which, upon incorporating Eq. (185) to render in

terms of J(x), becomes

]xJ ¼ i2rh

E V Oð Þ m

hrJ2

: ð188Þ


Because of the continuity equation and presumed time independence, the

LHS is zero. From the definition of J, it follows that

]xS ¼ i

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2m

h2V ðxÞ þ OðxÞ Eð Þ

r: ð189Þ

The motive for extracting the ‘‘i’’ explicitly is because of the particular

interest in the case where the electron energy is below the potential maximum

so that tunneling occurs. The integral of Eq. (189) is between the zeros of the

integrand. Of particular interest is the case when the density is slowly vary-

ing, which is generally (but not always) associated with a slow variation in the

potential. If the density is exponentially decaying over some length scale l,then the quantum potential is of the order O h2=2ml2, and therefore

small if l is large. With the designationðhkðxÞÞ2=2m ¼ VðxÞ E, a good

approximation to c is then

cðxÞ / kðxÞ1=2exp

ðxkðx0Þdx0

; ð190Þ

where the coefficient follows from Eq. (185). As introduced previously, the

transmission coefficient T(E) is the ratio between the current transmitted

through the barrier and the incident current. If the transmitted wave function

is a plane wave whose magnitude is decreased as per Eq. (190), it then follows

that

TðEÞ exp 2

ðxþx

kðx0Þdx0

; ð191Þ

where the limits of the integral are the zeros of the integrand. Eq. (191) is

the form most commonly invoked in the determination of tunneling current

(e.g., field emission), even though its most serious defect is a neglect of

the momentum dependence of the coefficient of the exponential term,

approximating it rather by unity. Still, the exponential term in Eq. (191)

captures the dominant features and, coupled with other approximations,

enables tractable analytical solutions for classes of potentials of particular

importance here. Another problem is the behavior of Eq. (191) near

the barrier maximum, where the neglect of O(x) is an issue. There is

therefore pedagogical value to examine exactly solvable cases to find how

well Eq. (191) holds—and conversely, where it fails and how it should be

modified.

64 KEVIN L. JENSEN

B. Exactly Solvable Models

Classes of potentials whose simplicity or particular features facilitate meth-

ods for which numerical evaluation of Schrodinger’s equation is not needed

are considered next. The first class is those in which Schrodinger’s equation is

exactly solvable (the square barrier and triangular barrier potentials) for

which the general methodology is to find basis states that are analytically

tractable. The second class is those for which the integral in the area under

the curve (AUC) method suggested by the Bohm analysis is analytic.

1. Wave Function Methodology for Constant Potential Segments

The general technique of the wave function methodology was encountered in

the discus sion of Eq. (35) , but in the present an alysis, consider ably mo re

attention to its detail is useful. Reconsider the 1D Schrodinger equation in

position space, that is, hxjcEi ¼ cEðxÞ for which

h2

2m

]

]x

2

þ V ðxÞ( )

cE xð Þ ¼ EcE xð Þ ð192Þ

for potentials that are at worst piece‐wise discontinuous for a finite number

of regions, that is, for n ¼ 1. . .N,

limd!0

jV ðxn þ dÞ Vðxn dÞj < 1: ð193Þ

As the energy is also finite, Eqs. (192) and (193) therefore requ ire that the

second derivative also be at most piece‐wise discontinuous, and therefore,

that the wave function, as well as its first derivative, is continuous, conditions

formally expressed as

limd!0

jcEðxn þ dÞ cEðxn dÞj ¼ 0

limd!0

jc0E ð xn þ dÞ c

0 E ð xn dÞj ¼ 0

ð194Þ

where prime (0) indicates derivative with respect to argument. Momentum

eigens tates are co nvenient for the evaluat ion of cu rrent, and they are he nce-

forth exclusively used. The wave functions in Eq. (194) are superpositions of

positive and negative momentum states, both of which have the same energy

eigenvalue (a consequence of the parabolic relationship between E and k).

For the special case where the potential for xn x xnþ1 is constant and

equal to Vn,


cnðxÞ ¼ tnexp iknxð Þ þ rnexp iknxð Þhkn ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2mðE VnÞ

p ð195Þ

where t and r are complex coefficients for waves moving to the right and left,

respectively. A matrix representation of Eq. (194) therefore suggests itself

(anticipated by the Kronig–Penney treatment)

cnðxÞ]xcnðxÞ

¼ exp iknxð Þ exp iknxð Þ

iknexp iknxð Þ iknexp iknxð Þ

tnðkÞrnðkÞ

ð196Þ

The coefficients’ vector shall be designated byzn xð Þ, and the 2 2 matrices

byMn xð Þ. Matching wave function and first derivative entails

Mn1 xnð Þzn1ðxnÞ ¼ Mn xnð ÞznðxnÞ: ð197Þ

Solving introduces matrices given by (Brennan and Summers, 1987; Jensen

2003b; Tsu and Esaki, 1973; Vassell et al., 1983):

SðnÞ Mn1ðxnÞ1MnðxnÞ

¼ 1

2kn1

kn þ kn1ð Þexp i kn kn1ð Þxn½ kn þ kn1ð Þexp i kn þ kn1ð Þxn½ kn þ kn1ð Þexp i kn þ kn1ð Þxn½ kn þ kn1ð Þexp i kn kn1ð Þxn½

( )

ð198ÞFor a potential separated into N regions, subject to t0 ¼ 1, r0 ¼ r, tN ¼ t, and

rN¼ 0, that is, the incident wave on the left is normalized to unity and there is

no wave incident from the RHS, then

1

r

¼

YNn¼1

SðnÞ( ) t

0

: ð199Þ

By virtue of the fact that no wave is incident from the right, Eq. (199)

therefore indicates

tðkÞ ¼YNn¼1

SðnÞ" #

1;1

8<:

9=;

1

; ð200Þ

where the (1,1) subscript (1,1) indicates that the first‐row, first‐column entry

of the matrix is given by the product of S(1) through S(N). From the

definition of current density given by Eq. (140), it follows that the incident,

reflected, and transmitted currents are, respectively,

66 KEVIN L. JENSEN

jincðkÞ ¼ hk

m

jref ðkÞ ¼ hk

mjrðkÞj2

jtransðkÞ ¼ hk

mjtðkÞj2

ð201Þ

when k0 ¼ kN ¼ k the transmission coefficient T(k), representing the ratio

of the transmitted current with the incident current, is as beforejtðkÞj2. When

kN 6¼ k, thenTðkÞ ¼ kN=kð ÞjtðkÞj2. The simplest case of a (N ¼ 1) step

function V(x) ¼ Vo for x 0 and 0 otherwise, for example, results in

tðkÞ ¼ 2k

kþ k1)

TðkÞ ¼ 4k1k

kþ k1ð Þ2 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiE E Voð Þp

ffiffiffiffiE

p þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiE Voð Þp 2

ð202Þ

where the form of T(k) is valid only for hk > hk1 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2m E Voð Þp

.

2. The Square Barrier

The next level of complexity is a simple square barrier of heightVo ¼ h2k2v=2m,

for N ¼ 2, such that k0 ¼ k2 ¼ k, x0 ¼ 0, and x1 ¼ L. Consequently,

k1 ¼ k2 k2v 1=2

is real or imaginary, depending on whether the E > Vo or

E < Vo, respectively. It is advantageous to introduce the strictly real

k ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffijk2 k2v jp

. A straightforward but possibly tedious exercise shows

t kð Þ ¼ 4ikkeikL

ik kð Þ2ekL ikþ kð Þ2ekLk < kvð Þ

4kkeikL

k kð Þ2eikL kþ kð Þ2eikLk > kvð Þ

8>>>><>>>>:

ð203Þ

T kð Þ ¼

4k2k2

4k2k2 þ k2 þ k2ð ÞsinhðLkÞ½ 2 k < kvð Þ

4k2k2

4k2k2 þ k2 k2ð ÞsinðLkÞ½ 2 k > kvð Þ

8>>>><>>>>:

ð204Þ

A representative case of Eq. (204) is shown in Figure 19 for L ¼ 5 A and

Vo ¼ 10 eV. Also shown are asymptotic (‘‘approximate’’) limits given by


Tappr ox kð Þ

2k

k

0@

1A

2

exp 2kLð Þ k < kvð Þ

2k kk2 þ k 2

0@

1A

2

k > kvð Þ

8>>>>>>><>>>>>>>:

ð 205 Þ

wher e, for k > kv, the lower limit line replaces sin( Lk ) by 1 (the uppe r limit is

self ‐ evident ly unity) . Several observation s are fort hcoming. First , the a p-

proxim ate solution , remi niscent of the AUC WKB ap proach, is reasonabl y

goo d for values of mo mentum below the barrier value kv —reas onably go od,

0

0.2

0.4

0.6

0.8

1.0(a)

1050 15 20 25

T(E)

Approx (E<Vo)

Approx (E>Vo)

Tra

nsm

ission

coe

ffic

ient

Energy [eV]

10−6

10−5

10−4

10−3

10−2

10−1

100(b)

1050 15 20 25

T(E)

Approx (E < Vo)

Approx (E > Vo)

Tra

nsm

ission

coe

ffic

ient

Energy [eV]

F IGURE 19. Transmission probability (thick black line) for a rectangular barrier of height

10 eV and of width 0.5 nm. The thin dashed and solid lines are for the two limiting cases shown

in Eq. (205). (b) Same as (a), but on a log scale.

68 KEVIN L. JENSEN

that is, when E is well below Vo, in contrast to near the barrier maximum,

where the approximation degrades as expected from the behavior of

Eq. (204). Second, ln TðkÞð Þis approximately linear with respect to E(k) for

narrow ranges in the vicinity of E¼ m. Third, at the barrier maximum (as well

as particular momentum values above it), the transmission coefficient is

not unity. These observations will have bearing on the emission equations

developed for general potentials in what follows.

3. Multiple Square Barriers

The ‘‘area‐under‐the‐potential’’ method of evaluating the transmission coef-

ficient can be approximated by using the trapezoidal approximation to

evaluating integrals as in

ln TðEÞf g¼ðxmax

xmin

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2m

h2V ðxÞ Eð Þ

vuut dx

1

h

ffiffiffiffiffiffiffi2m

p XNn¼1

DxnffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiVðxnÞ E

p ð206Þ

Not surprisingly, Eq. (206) looks very much like a sequence of square

barriers whose cumulative effect is the product of their respective trans-

mission coefficients Tn, whereln TnðEÞf g ¼ Dxffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2m=h2

V ðxnÞ Eð Þq

. Super-

ficially, it appears that T(E) would not be different if the barriers were

adjacent or separated by a distance. The wave nature of the electron, how-

ever, renders that conclusion inaccurate. When the barriers are far enough

apart to allow a resonant level for values of the energy below the barrier

maximum, then T(E) can approach unity for particular energy levels.

When there are many barriers the Kronig–Penney model is approached.

The opposite limit of but two barriers gives the case considered by Esaki

and Tsu (Tsu and Esaki, 1973) in their analysis of the RTD. The methodo-

logy of Eq. (200) reveals the subtlety nicely: in Figure 20 the effect of

repeatedly doubling the number of barriers on T(E) clearly shows

the development of ‘‘bands’’ for energies above the barrier maximum of

Vo ¼ 10 eV. Similarly, in Figure 21 for T(E), for which the energy range is

generally below the barrier maximum, the intuition motivated by Eq. (206)

accounts for much, but not all, of the behavior of T(E), in that if TnðEÞ is thetransmission probability for n barriers, then T2nðEÞ TnðEÞ½ 2 T1ðEÞ½ 2nfor 2n barriers; that is, doubling the number of barriers tends to square the

transmission probability (except near resonances).


0

0.2

0.4

0.6

0.8

1.0(c)

252015105

2 Barriers4 Barriers

Tra

nsm

ission

coe

ff.

Energy [eV]5 10 15 20 25

0

0.2

0.4

0.6

0.8

1.0(d)

4 Barriers8 Barriers

Tra

nsm

ission

coe

ff.

Energy [eV]

0 10 15 20 25

0

0.2

0.4

0.6

0.8

1.0(a)

Tra

nsm

ission

coe

ff.

Energy [eV]

Step function

Single barrier

5 10 15 20 25

0

0.2

0.4

0.6

0.8

1.0(b)

Tra

nsm

ission

coe

ff.

Energy [eV]

1 Barrier

2 Barriers

FIGURE 20. (a) Step function versus single barrier transmission probability (barrier height = 10 eV). (b) Same as

(a) but for single and double barriers. (c) Same as (b) but for double (two) and four barriers. (d) Same as (c), but for

four and eight barriers. Evidence of bandlike formation is becoming discernible.

70

KEVIN

L.JE

NSEN

4. The Airy Function Approach

Return ing to the step functi on barrier, consider the case wher e, inste ad of

being constant , the barri er is of the form V(x) ¼ Vo – Fx, wher e F is

the produc t of the elect ric fie ld and the elect ron charge. Retaini ng the

notatio n Vo ¼ hk oð Þ2 =2m and introd ucing F ¼ s h2 f =2m (note that f ha s

units of [1/A 2] and is assum ed posit ive), Sch ro dinger’s eq uation becomes

] 2x ck ð x Þ þ k 2o k2 þ sfx

ck ð xÞ ¼ 0; ð207Þ

10−20

10−18

10−16

10−14

10−12

10−10

10−8

10−6

10−4

10−2

100(a)

105 6 7 8 9 11 12

1 Barrier2 Barriers4 Barriers8 Barriers

Tra

nsm

ission

coe

ff.

Energy [eV]

10−5

10−4

10−3

10−2

10−1

100(b)

1098765 11 12

1 Barrier2 Barriers4 Barriers8 Barriers

T(E

)1/

n

Energy [eV]

FIGURE 21. (a) Summary of Figure 20 on a log scale, but showing the existence of a

previously indiscernible resonance level at 8 eV. (b) Same as (a), but with the transmission

probability take to the root of the number of barriers n—as expected, the area under the curve

exponential factor is largely seen to govern the behavior away from resonance.


where s indicates the inclination of the field, or 1 for descending, þ1 for

ascending: for a triangular barrier, s is therefore (1). Such an awkward

notation may appear at best to be feigned madness, but there is method to it:

extra work now will be well worth the investment later. Introduce

zðxÞ f 2=3jk2o k2 þ sfxjc ¼ sign k2o k2 þ sfx

ð208Þ

for which Eq. (207) becomes Airy’s differential equation

]2zc c2zc ¼ 0

ckðxÞ ¼ aAi c2zðxÞð Þ þ bBi c2zðxÞð Þ ð209Þ

where c ¼ 1,i, ]xz ¼ c2sf 1=3, and a and b are arbitrary constants deter-

mined by boundary conditions (c should not be confused with the speed of

light, and a and b are not to be identified with an and bn below). Note that s is

apparently hidden, as when it does appear, it does so as s2 ¼ 1, but s will

return below. Although Eq. (209) is correct, its utility is compromised by

hiding behind the Airy functions and thereby obscuring the smooth transi-

tion to the field‐free case where the wave function become plane waves. An

approach that explicitly calls out the asymptotic behavior of the Airy func-

tions is numerically advantageous. Recalling the Bohm analysis leading to

Eq. (190), a reasonable ansatz to the wave function is

Ziðc; zÞ ¼ FcðzÞ2ffiffiffip

p z1=4exp2

3cz3=2

8<:

9=;

1

4c 1ð Þ c2 þ 2cþ 3

Ai c2z þ 1

4cþ 1ð Þ 3c2 2cþ 1

Bi c2z ð210Þ

where the new function Fc(z) (not to be confused with the field F) reflects the

desire to move beyond the estimates leading to Eq. (190), and the second line

(sans the peculiar coefficients) corresponds to a more traditional method of

representing the wave function for a linear potential. Eq. (210) introduces

and defines the Zi functions—so named to emphasize their connection to

Ai(z) and Bi(z). In fact, ck(x) is linear combinations of the Zi functions

for c2 ¼ 1, but it is easer to treat one case at a time, an approach that

presents no difficulty provided at the end one is mindful of the shortcut.

Inserting Eq. (210) into the Airy differential equation gives for F(z)

16z2]

]z

2

þ 8z 4cz3=2 1 ]

]zþ 5

( )FcðzÞ ¼ 0: ð211Þ

72 KEVIN L. JENSEN

Introducing a change of variables given by zðzÞ ¼ z3=2, Eq. (211) becomes

36z2]

]z

2

þ 24 3z 2cð Þ ]

]zþ 5

( )Fc zðzÞð Þ ¼ 0 ð212Þ

Inspection of Eq. (212) suggests that Fc(z) is a polynomial in z and

that its adjacent coefficients (which depend on c) are related. Inserting

Fc zðzÞð Þ ¼X1n¼0

anzn into Eq. (212) and setting the coefficients of different

powers of z to 0 gives for the an

an

an1

¼ s36n 1ð Þ 6n 5ð Þ

48n; ð213Þ

or

an ¼ sð Þnffiffiffip

p 1

32n 2nð Þ!G 3nþ 1

2

0@

1Aa0

a0 ¼ 1

4sþ 3ð Þ s2 þ 1

þ 2ffiffiffi2

ps 1ð Þ s2 1

n o ð214Þ

where the values of a0 are determined from the asymptotic expansions of the

Airy functions compared to Eq. (210). As pleasing (and well‐known; Hoch-

stadt, 1986) as the expansion entailed by Eq. (214) is, it is numerically

unusable because the coefficients eventually dominate z3n/2 and the terms

fail to converge. Eq. (210), though appearing to be useful, instead hides its

computational limitations behind the allure of its simplicity. Tunneling

calculations routinely encounter exponentials of large terms, and their com-

putation therefore results in machine precision limitations even when using

widespread and useful numerical packages such as IMSL or LAPACK

(Linear Algebra PACKage), unless care is taken to partition the calculation

to appropriate regions.

To appreciate what works numerically, there is value in showing what

does not work. At first glance, a naıve approach is to adopt a polynomial fit,

with z ¼ z3/2 of the form

Fc zðzÞð Þ ¼X4

n¼0bnðcÞzn ð215Þ

where the bn are the nth row of a b vector determined from solving the matrix

equation


Cb ¼ Fc

Ch i

j;k¼ zj k1

Fc½ j ¼ Fc z zj ð216Þ

and zj ¼ (j1)/4 for a fourth‐order polynomial, 0 j 4, and where Fc(zj)

can be determined from a table of Airy functions (Abramowitz and Stegun,

1965) and the relation

FcðzÞ ¼ 2ffiffiffip

pz1=4exp 2

3cz3=2

0@

1A

1

4c 1ð Þ c2 þ 2cþ 3

Ai c2z þ 1

4cþ 1ð Þ 3c2 2cþ 1

Bi c2z 2

435

ð217Þ

In the case of a fourth‐order polynomial fit

bð1Þ ¼

2:000000:100471:12626

1:791340:76213

0BBBB@

1CCCCA; bð1Þ ¼

1:000000:102320:06402

0:039330:01176

0BBBB@

1CCCCA; bðiÞ ¼

1:4142 1:4142i0:15439 0:15237i0:06408 0:17965i0:11339 0:10189i

0:04509 0:02511i

0BBBB@

1CCCCA ð218Þ

where the argument of the b vector is the value of c. For c ¼ i, the relation

Fi (z) ¼ F‐i (z) is used. Tunneling calculations using the approximation of

Eq. (217) for arbitrary barriers are generally good to 1%, depending on the

potential examined; a consequence is that the numerical estimation of

the transmission probability can exceed unity by a small amount for energies

above the barrier maximum. Eq. (215) sacrifices more than aesthetic beauty

for computational simplicity, it sacrifices accuracy: for large arguments,

small discrepancies give rise to cumulative errors.

This leads (begrudgingly) to the final, and workable, approach based on

interpolating between known values (Tables 5 and 6). The form of Eq. (218)

and the dependence of Eq. (212) on cmotivate defining Fc by the real functions

X and X0 as per

Fc zðzÞð Þ ¼ Xc zð Þ cX 0c zð Þ: ð219Þ

The behavior of X and X0 are shown in Figure 22 as well as tabulated in

Table 6. Closely related to the Fc functions are the Hc functions, needed for

derivatives of the Zi functions, defined according to

]zZiðc; zÞ ¼ffiffiffiz

pHc zðzÞð ÞZiðc; zÞ ð220Þ

From Eqs. (212)–(214), it follows that Hc(0) ¼ c. Analogous to Eq. (219),

introduce the real functions Wc and W0c defined according to

74 KEVIN L. JENSEN

TABLE 5

AIRY POLYNOMIAL FUNCTIONS I: VALUES OF XC z AND X0C (z) FOR c ¼ 1 AND i

z X1 Xi X01 X0

i

0.000 1.50000 1.41421 0.50000 1.41421

0.025 1.50138 1.41047 0.50394 1.41781

0.050 1.50293 1.40658 0.50794 1.42127

0.075 1.50465 1.40259 0.51204 1.42453

0.100 1.50658 1.39851 0.51627 1.42763

0.125 1.50875 1.39436 0.52066 1.43054

0.150 1.51121 1.39017 0.52528 1.43328

0.175 1.51400 1.38594 0.53017 1.43585

0.200 1.51712 1.38170 0.53533 1.43825

0.225 1.52053 1.37745 0.54074 1.44049

0.250 1.52418 1.37321 0.54632 1.44257

0.275 1.52797 1.36898 0.55201 1.44451

0.300 1.53183 1.36477 0.55772 1.44631

0.325 1.53568 1.36058 0.56337 1.44799

0.350 1.53945 1.35643 0.56891 1.44954

0.375 1.54308 1.35231 0.57427 1.45098

0.400 1.54654 1.34823 0.57942 1.45230

0.425 1.54979 1.34420 0.58433 1.45353

0.450 1.55280 1.34021 0.58897 1.45466

0.475 1.55557 1.33626 0.59334 1.45570

0.500 1.55809 1.33236 0.59743 1.45665

0.525 1.56036 1.32850 0.60123 1.45753

0.550 1.56237 1.32469 0.60475 1.45832

0.575 1.56414 1.32093 0.60800 1.45905

0.600 1.56566 1.31722 0.61099 1.45971

0.625 1.56696 1.31356 0.61372 1.46031

0.650 1.56804 1.30994 0.61621 1.46085

0.675 1.56890 1.30637 0.61846 1.46133

0.700 1.56957 1.30285 0.62050 1.46176

0.725 1.57005 1.29938 0.62232 1.46214

0.750 1.57035 1.29595 0.62395 1.46248

0.775 1.57049 1.29256 0.62539 1.46276

0.800 1.57047 1.28923 0.62666 1.46301

0.825 1.57031 1.28593 0.62776 1.46322

0.850 1.57001 1.28268 0.62871 1.46339

0.875 1.56958 1.27947 0.62952 1.46352

0.900 1.56903 1.27631 0.63019 1.46362

0.925 1.56838 1.27318 0.63074 1.46368

0.950 1.56762 1.27010 0.63117 1.46372

0.975 1.56677 1.26705 0.63148 1.46373

1.000 1.56583 1.26405 0.63170 1.46371


TABLE 6

AIRY POLYNOMIAL FUNCTIONS II: VALUES OF WC z AND W 0C (z) FOR c ¼ 1 AND i

z W1 Wi W01 W0

i

0.000 0.00000 0.00000 1.00000 1.00000

0.025 0.00625 0.00625 0.99990 1.00010

0.050 0.01253 0.01247 0.99961 1.00039

0.075 0.01885 0.01865 0.99910 1.00086

0.100 0.02526 0.02478 0.99837 1.00151

0.125 0.03178 0.03083 0.99739 1.00233

0.150 0.03848 0.03680 0.99610 1.00329

0.175 0.04540 0.04268 0.99448 1.00439

0.200 0.05255 0.04847 0.99253 1.00562

0.225 0.05987 0.05415 0.99029 1.00696

0.250 0.06732 0.05974 0.98783 1.00839

0.275 0.07482 0.06522 0.98523 1.00992

0.300 0.08229 0.07061 0.98258 1.01153

0.325 0.08965 0.07589 0.97995 1.01321

0.350 0.09686 0.08107 0.97740 1.01495

0.375 0.10386 0.08616 0.97499 1.01676

0.400 0.11062 0.09116 0.97274 1.01861

0.425 0.11712 0.09606 0.97069 1.02051

0.450 0.12334 0.10087 0.96885 1.02245

0.475 0.12928 0.10559 0.96723 1.02442

0.500 0.13493 0.11023 0.96584 1.02643

0.525 0.14030 0.11479 0.96467 1.02846

0.550 0.14540 0.11926 0.96373 1.03052

0.575 0.15023 0.12366 0.96299 1.03261

0.600 0.15481 0.12799 0.96247 1.03471

0.625 0.15914 0.13224 0.96213 1.03682

0.650 0.16323 0.13642 0.96199 1.03895

0.675 0.16711 0.14054 0.96202 1.04110

0.700 0.17077 0.14459 0.96222 1.04325

0.725 0.17424 0.14857 0.96257 1.04541

0.750 0.17751 0.15249 0.96307 1.04758

0.775 0.18061 0.15636 0.96371 1.04976

0.800 0.18354 0.16016 0.96447 1.05194

0.825 0.18632 0.16391 0.96536 1.05412

0.850 0.18895 0.16760 0.96635 1.05631

0.875 0.19143 0.17124 0.96746 1.05850

0.900 0.19379 0.17483 0.96865 1.06069

0.925 0.19602 0.17837 0.96994 1.06288

0.950 0.19813 0.18186 0.97131 1.06507

0.975 0.20013 0.18530 0.97276 1.06726

1.000 0.20203 0.18870 0.97429 1.06944

76 KEVIN L. JENSEN

1.50

1.52

1.54

1.56

1.58(a)

(c)

0.52

0.56

0.60

0.64

0.68

0 0.2 0.4 0.6 0.8 1

X1

X1'

X1(z) X

1 '(z)

c = 1

z0 0.2 0.4 0.6 0.8 1

1.28

1.32

1.36

1.40

(b)

−1.47

−1.46

−1.45

−1.44

−1.43

−1.42

−1.41

Xi

X'i

Xi(z

) X'i (z)

c = i

z

−0.20

−0.16

−0.12

−0.08

−0.04

0.00

0.96

0.97

0.98

0.99

1.00

W'1

W1

W1(z)

z

c=1

0 0.2 0.4 0.6 0.8 1

W'i (z)

−0.20

−0.16

−0.12

−0.08

−0.04

0.00(d)

1.00

1.02

1.04

1.06

1.08

Wi'

Wi

Wi(z

) W'i (z)

c = i

z0 0.2 0.4 0.6 0.8 1

F IGURE 2 2 . (a) B ehavi or of the Airy c oe ffici ents X1 and X0 1 introduced in Eq. (219). (b) Be havior of t he Airy c oefficients Xi and X

0 i

introduced in Eq. (219). (c) Be havior of the A iry c oef ficients W1 and W0 1 introduced in Eq. (220). (d) Behavior of the A iry c oefficient s Wi

and W0i introduced in Eq. (221).

ELECTRON

EMISSIO

NPHYSIC

S77

H c zð Þ ¼ W c zð Þ cW 0c zð Þ: ð 221 ÞThe be havior of W an d W 0 are shown in Figure 22 and tabula ted in

Table 6. Although the X and W appear to be polynomials, they are not. In

practice, then, the usage of tabulated values of X, X0, W, and W0 and

interpolation to intermediate values of z via

Xc zn1 < z < znþ1ð Þ ¼ 1

2y 1 yð ÞXc zn1ð Þ þ 1 y2

Xc znð Þ þ 1

2y 1þ yð ÞXc znþ1ð Þ

ð222Þwhere yðzÞ ¼ 2 z znð Þ= znþ1 zn1ð Þ, zn are uniformly spaced, and similar

equations hold for X0, W, andW0 is found to provide the accuracy needed for

tunneling calculations.

a. Large Argument Case. On the face of it, Zi functions appear to offer

no advantages over the Airy functions. The reason for introducing them is

that ratios of the Zi functions (as shall appear during the matrix evaluations

below) asymptotically approach the plane wave or exponential functions as

the field vanishes, and so the methodology meshes well with the approach

based on Eq. (196). Consider the case of c ¼ 1, and examine the ratio

Zið1; zÞ=Zið1; zoÞ where zo ¼ z(x ¼ 0). From the definition of Zi, to

leading order

Ziði; zÞZiði; zoÞ exp

2i

3fk2 k2o sfx 3=2 k2 k2o

3=2h i8<:

9=;

exp is k2 k2o 1=2

xh i ð223Þ

when k2o k2 fx. The wave functions for non‐zero f are combinations of

the Zi functions, or

ckðxÞ ¼ tðkÞZiðc; zðxÞÞ þ rðkÞZiðc; zðxÞÞ ð224Þfor c ¼ 1 (over the barrier) or i (under the barrier), and the choice of t and r

for coefficients reflecting which of the terms (t in particular) correspond to an

outgoing wave. Continuity of the wave function also requires the evaluation

of the gradient, or]

]xckðxÞ ¼ 3

2f 1=3sc2z5=3

]

]zckðxÞ: ð225Þ

78 KEVIN L. JENSEN

The quantity of interest embedded in Eq. (225) is the relation

]

]xZiðc; zÞ ¼ f

64z

24

351=3

sc2

Fc zðzÞð Þ 4c zð ÞFc zðzÞð Þ 6z2]zFc zðzÞð Þ Ziðc; zÞ

Diðc; zÞZiðc; zÞð226Þ

It follows from Eq. (220) that

Di c; zð Þ ¼ sf

zðzÞ

0@

1A

1=3

Wc zðzÞð Þ þ cW 0c zðzÞð Þ

¼ s

c

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2o k2 þ fsx

pWc zðzÞð Þ þ cW 0

c zðzÞð Þ ð227Þ

where, in the second line, z has been expressed in terms of the parameters

introduced by Schrodinger’s equation. Eq. (227) is well behaved for the range

of conditions characteristic of tunneling and therefore is of considerable use

below.

b. Small Argument Case. When z is small, numerical work benefits from

expansions if library routines for the evaluation of the Airy functions are not

available (Abramowitz and Stegun, 1965). In practice, truncating the series

expansions such that six‐digit accuracy is preserved is sufficient for accurate

numerical tunneling calculations. Therefore

AiðzÞ ¼ PðzÞ

32=3G2

3

0@1A

SðzÞ

31=3G1

3

0@1A

BiðzÞ ¼ PðzÞ

31=6G2

3

0@1A

þ SðzÞ

31=6G1

3

0@1A

ð228Þ

The truncated polynomials P(z) and S(z) are defined by

PðzÞ ¼ 1þ z3

1710720285120þ z3 9504þ z3 132þ z3

SðzÞ ¼ z

70761607076160þ z3 589680þ z3 14040þ z3 156þ z3

ð229Þ


Gradients of the Airy functions are then evaluated by

P0ðzÞ ¼ z2

14256071280þ z3 4752þ z3 99þ z3

S0ðzÞ ¼ 1þ z

544320181440þ z3 7560þ z3 120þ z3

ð230Þ

Using Eqs. (228) – (230), small argument values of Di(c,z) can likewise be

found.

c. Wronskians of the Airy Functions. The Airy functions satisfy the

relation (Watson, 1995)

AiðzÞ ]zBiðzÞð Þ ]xAiðzÞð ÞBiðzÞ ¼ 1

pð231Þ

for arbitrary z, a relationship useful due to Wronskians appearing for the

matrix inverses below. In terms of the Zi functions, this becomes (where the

gradient with respect to position is explicitly shown to avoid confusing primes

to be gradients with respect to argument)

Ziðc; zÞ ]xZiðc; zÞð Þ Ziðc; zÞ ]xZiðc; zÞð Þ ¼ 1

2pf 1=3sc 1 3c2

ð232Þ

5. The Triangular Barrier

The triangular barrier represents an example of the Airy function approach

and is the basis for the derivation of the FN equation as originally given

(Fowler and Nordheim, 1928) in the treatment of field emission. Consider the

potential V(x) ¼ Vo – Fx. In the parlance introduced above, it becomes

k2o k2 þ sfx with s¼ 1. Matching the wave function and first derivative at

x ¼ 0, corresponding to z(x¼0) ¼ zo, gives

1 1

ik ik

1

r

¼ Ziðc; zoÞ Ziðc; zoÞ

Zi0ðc; zoÞ Zi0ðc; zoÞ

t0r0

: ð233Þ

For transmission over the barrier (c ¼ i), it follows that t0 ¼ t and r0 ¼ 0, but

for under the barrier (c ¼ 1), the value of c changes to i when the wave

emerges at the location defined by z ¼ 0. Consequently, a transition matrix

must be introduced whenever c changes value. Therefore, for the case of

under the barrier to over

Zið1; 0Þ Zið1; 0ÞZi0ð1; 0Þ Zi0ð1; 0Þ

t0r0

¼ Ziði; 0Þ Ziði; 0Þ

Zi0ði; 0Þ Zi0ði; 0Þ

t

0

; ð234Þ

80 KEVIN L. JENSEN

and for over the barrier to under

Ziði; 0Þ Ziði; 0ÞZi0ði; 0Þ Zi0ði; 0Þ

t0r0

¼ Zið1; 0Þ Zið1; 0Þ

Zi0ð1; 0Þ Zi0ð1; 0Þ

t

0

: ð235Þ

Use of the Wronskians shows that

t0r0

¼

i i

1 1

t

0

k2o > k2

1

2

i 1

i 1

t

0

k2o < k2

8>>><>>>:

ð236Þ

Now let us restrict attention to electron energies below the barrier maximum

(under to over). Then the matrix equation to be solved is

1 1

ik ik

1

r

¼ Zið1; zoÞ Zið1; zoÞ

Zi0ð1; zoÞ Zi0ð1; zoÞ i i

1 1

t

0

: ð237Þ

The solution for t(k) is revealed by expanding the matrices and finding

tðkÞ ¼ 2k

k iDið1; zoÞð ÞZið1; zoÞ i k iDið1; zoÞð ÞZið1; zoÞ : ð238Þ

The wave function becomes (outside the barrier to the right)

ckðxÞ ¼2kZiði; zÞ

k iDið1; zoÞð ÞZið1; zoÞ i k iDið1; zoÞð ÞZið1; zoÞ : ð239Þ

Inside the barrier to the right, the Zi in the numerator would be replaced

with Zi(1,z), but that case is ancillary to our present focus on the emitted

current. If the electron energy is well below the barrier height, then

ckðxÞ 2ik

k iDið1; zoÞð ÞZiði; zÞZið1; zoÞ

: ð240Þ

The employment of the Zi functions, argued to be useful when their

ratios are taken, therefore reveals their utility. From the relation

Ziði; zÞ ¼ Ziði; zÞ, it follows that the transmission coefficient T(k) is

(where the smaller terms neglected in Eq. 240 are kept)

Tðk < koÞ ¼ 4k2

k2 þDið1; zoÞ2

Zið1; zoÞ2 þ 2p f

1=3kþ k2 þDið1; zoÞ2

Zið1; zoÞ2f 1=3

pk

ð241ÞAn analogous equation follows for emission over the barrier. It is a good

pedagogical (if slightly pedantic and definitely tedious) exercise to examine

the limit of a step function potential and demonstrate that, as expected, the


wave function and transmission coefficient are as described previously in the

derivation of Eq. (202). It is an exercise to show that moving from the Zi

functions to the traditional Airy functions results in expression of Eq. (241) as

Tðk < koÞ ¼ 4k2

k2 AiðzoÞ2 þ BiðzoÞ2

þ 2p f

1=3kþ f 2=3 Ai0ðzoÞ2 þ Bi0ðzoÞ2 f 1=3

pk

;

ð242Þwhere the center term in the denominator is a consequence of the Wronskian.

Performing the same analysis for k > ko replaces zo in Eq. (242) by –zo,

in contrast to Eq. (241), which instead becomes

Tðk > koÞ ¼ 4k2

k2 þDiði; zoÞDiði; zoÞð ÞZiði; zoÞZiði; zoÞ þ 2p f

1=3k

f 1=3

pk

:

ð243ÞThe FN equation, developed for field emission from metals, was princi-

pally concerned with electron energies below the barrier maximum for large

work functions. Therefore, the asymptotic limit of Eq. (241) is desired

for zo ¼ jk2o k2j3=2=f 2=3 k3=f 2=3 1. Let twice the AUC term be

designated by 4k3=3f 2=3 ¼ y, and use the asymptotic expansions for the

under‐the‐barrier Zi

Zi 1; zoð Þ f 1=6

4ffiffiffiffiffiffipk

p 3 1ð Þexp y2

8<:

9=;

k2 þDi 1; zoð Þ2

Zi 1; zoð Þ2 k2oZi 1; zoð Þ2ð244Þ

The inclusion of a factor of (2) in the definition of y is a slight departure fromother analyses (Jensen, 2001) where y is identified with the AUC term

directly. Eq. (241) becomes

Tðk < koÞ ¼ 16kkk2o 4ey þ eyf g þ 8kk

: ð245Þ

The FN approximation to the transmission coefficient is then

TFNðkÞ ¼ limko!1

TðkÞ ¼ 4k

koexp 4

3fk3

: ð246Þ

Even though the potential is sharply peaked, it is clear that the coefficient

is field independent and the argument of the exponential is the AUC term,

both keeping in line with Eq. (190).

82 KEVIN L. JENSEN

For k > ko, an analysis analogous to the one leading to Eq. (245) in turn

gives rise to the asymptotic approximation

Tðk > koÞ ¼ 4kk

kþ kð Þ2 : ð247Þ

The two limits of Eq . ( 24 5) and (247) a re s ug ge st iv el y s im il ar , but no t

quite the same. They suffer from the problem that both vanish when k ¼ ko(i.e., k ¼ 0), whereas neither Eq. (241) no r Eq. (243) vanishes. Pursuing an

expansion that is correct through the point k¼ komay appear churlish, but the

effort belies a subtlety that is useful for the analysis of other barriers, in

particular the quadratic barrier considered below. A careful analysis shows

that the problems at k¼ko arise from the presence of z1/4 in the denominator of

the asymptotic (large z) expansion ofZi(c,z) in Eq. (210). The simplest approx-

imation is to remove the singularity by appending a small, finite term, as in

z1=4o ! z2o þ p2

1=8. The same analysis that yielded Eq. (247) then gives

Tðk > koÞ ¼ 4k k4 þ f 4=3p2ð Þ1=4

k2 þ k2 þ 2k k4 þ f 4=3p2ð Þ1=4: ð248Þ

The value of p is found by demanding that Eq. (248) be valid at k¼ ko, using

Eq. (242), and the zero‐argument terms given in Eq. (228). The resulting

expression depends on both ko and f and it can be shown that

p ¼ 9k2o4p

0@

1A

2

34=3f 2=3

G 1=3ð Þ2 þ32=3k2o

G 2=3ð Þ2

24

352

¼ 0:398593k4o

k2o þ 0:531457f 2=3 2

ð249Þ

for vanishing field, p approaches a barrier‐independent constant. How does

this help? First, note that the term ey is negligible in Eq. (245) except near

y ¼ 0, and so neglecting it in general is useful. Second, as T(k) is a continu-

ous function of k, then y(k) should be likewise continuous. As y depends on

k2, this amounts to continuity in E. We therefore take y to be the AUC factor

for energies below the barrier maximum but to be the linear continuation in

E of that function for energies above the barrier maximum. Consequently, the

procedure is to replace y for energies above the barrier maximum with

the linear extension y0ðmþ FÞ E m Fð Þ, where the prime on y denotes

derivative with respect to argument, when the energy exceeds the barrier

maximum. For the FN triangular barrier, such a procedure is trivial:

y0ðEÞ ¼ ð2=hFÞ 2m mþ F Eð Þ½ 1=2 vanishes at the barrier maximum, and

so y vanishes for energies above the barrier. In contrast, for barriers


whe re y0 does not vanis h at the b arrier maxi mum, the prescripti on is to linearly

extend the below ‐ba rrier resul ts to abo ve the ba rrier. For the trian gularbarri er, then, the form of T(k) valid for all k is

T ð kÞ ¼ 16 k k4 þ f 4 =3 p2ð Þ1 =4

4 k 2 þ k2ð Þexp yðk Þ½ þ 8k k4 þ f 4 =3 p2ð Þ1= 4 ; ð 250 Þ

wher e yð kÞ ¼ 4k3 =3f½ , for k < ko and 0 for k k o, and wher e k2 ¼ jk2o k 2 j .Obser ve that in the limit of vanishi ng field, Eq. (202) is recover ed.

Consid er the perfor mance of Eq. (250) for copp erlike pa rameters, that is,

m ¼ 7.0 eV and F ¼ 4.6 eV and an app lied field of 0.4 eV/A , for which

ko ¼ 1.7636 6 1/A and f ¼ 0.1049 87 1/A 3. Figure 2 3 compares the exact resul t

with the FN approx imation [ Eq. (246) ] and Eq. (250) . The p prescr iption of

Eq. (246) works quite well. In a ddition, the exact solution is shown for severa l

fields in Figure 24 . The pe dagogica l value of Eq. (250) is suffici ent to justify

the effor t invested in its de rivation reveal ing the nature of the denominat or;

loosel y, the trans mission coeffici ent is ap proxim ately of the form

Taprx ð kÞ C ð k Þ1 þ exp yð kÞð Þ ; ð251Þ

0

0.2

0.4

0.6

0.8

1.0

7 8 9 10 11 12 13 14

ExactAnalyticFNm & m + Φ

Tra

nsm

ission

coe

ffic

ient

Energy [eV]

m = 7 eVΦ = 4.6 eVF = 0.4 eV/Angko = 1.745 Ang−1m m + Φ

FIGURE 23. Comparison of the numerically evaluated transmission probability using the Zi

functions [exact Eq. (241)] with the traditional Fowler Nordheim equation [FN Eq. (246)] and

the analytical approximation Eq. (250).

84 KEVIN L. JENSEN

where C(k) for large k ap proaches unity. The form of Eq. (251) is a general

form that we wish to retai n be low. Above the ba rrier in general , y k > koð Þcan be approxim ated by

y k > koð Þ ¼ yo E ð k o Þð Þ þ y0o E ð ko Þð Þ E ð kÞ E ð ko Þð Þ; ð252Þ

where the prim e ind icates deriva tive with respect to energy, even though

in the case of the triangular ba rrier, both y and its first deriva tive vanish atE ¼ m þ F (that is, saying the AUC fact or vanishe s above the barri er is a

consequ ence of the special de penden ce yð E Þ / m þ F Eð Þ3= 2 for E < m þ Ffor the triangu lar barri er). In gen eral, y(E) and its first de rivative do not sovanish (such as for qua dratic barri ers), and theref ore the form of Eq. (252) is

useful in a relation such as Eq. (251) .

C. Wentze l–Kram ers–Bri llouin WKB Are a Under the Curve Models

1. The Quadr atic Barrier

The quadrati c barrier c an be gen erally writt en for | x| < L as

Vquad ð xÞ ¼ V o 1 x

L

2 : ð253Þ

5 10 15 200.0

0.2

0.4

0.6

0.8

1.0

2 eV/nm4 eV/nm6 eV/nm8 eV/nm10 eV/nmµm + ΦT

rans

mission

coe

ffic

ient

Energy [eV]

m = 7 eVΦ = 4.6 eV

F IGURE 24. The triangular barrier emission probability calculated according to Eq. (250)

for copper-like parameters for various fields.


The AUC express ion for y then is sim ple to evaluat e and yields

yquad ð E Þ ¼ 2


h2

s ðxo xo

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiVo 1 x

L

0@1A

28<:

9=; Edx

vuuut

¼ pko L 1 E

Vo

0@

1A

ð 254 Þ

wher e hkoð Þ2 =2m ¼ V o and xo ð E Þ ¼ L 1 E =V oð Þ½ 1 =2. The ex tension ofyquad E > Voð Þ is then trivial as yquad E < V oð Þ is alrea dy linea r. Conse quen tly,the approxim ation to the qua dratic barri er us ing the form suggested by

Eq. (251) is then

Tquad ð E Þ 1 þ exp yquad ð E Þ 1

: ð 255 ÞThe perfor mance of Eq. (255) is shown in Figu re 25 for co pperlik e para-

mete rs ( m ¼ 7 .0 eV, F ¼ 4.6 eV) and a barrie r wid th of 2L ¼ 1 nm . Clearl y,

the perfor mance near the ba rrier maxi mum (11.6 eV) is quite good; less

clear ly visible is that nea r E ¼ m, Eq. (255) is ap proxim ately 23% large r

than the Airy functio n solut ion. Before much is made of the latter discr epan-

cy, recal l that in light of the generally unknown surfa ce co nditions, there are

substa ntial differences between real surfa ces an d mod els that purpo rt to

descri be them. But before much is mad e of that , the ab sence of a perfec t

model is not lice nse to use a malad apted one . Mode ls such as Eq. (255) pr ove

their utility when the emission current contai ns contri butions near the barri er

maxi mum, as in therm ionic an d photoemi ssion, a poin t retur ned to in the

following text.

0.0

0.2

0.4

0.6

0.8

1.0

1098 11 12 13 14

Exactexp(−θ)1/[1 + exp(θ)]

Tra

nsm

ission

coe

ff.

Energy [eV]

2L = 1 nmCopper-like:

m = 7.0 eVΦ = 4.6 eV

FIGURE 25. Comparison of the exact quadratic barrier transmission probability with the

standard area under the curve approximation exp(y) and Eq. (255).

86 KEVIN L. JENSEN

2. The Image Charge Barrier

The last of the analytic models to be considered is arguably the most influen-

tial one, as it is the basis for the thermionic (Richardson) and field emission

Fowler Nordheim (FN) equations treated below. Consider, therefore,

the potenti al given by the image c harge potenti al Eq. (110) , for which the

associated AUC expression is

yimageðEÞ ¼ 2

ðxþðEÞxðEÞ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2m

h2mþ F FxQ

x E

0@

1A

vuuut dx

xðEÞ ¼ 1

2Fmþ F Eð Þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffimþ F Eð Þ2 4FQ

q ð256Þ

Introducing a change of variables governed by the length L(E) ¼ xþ – x,

Eq. (256) becomes

y Eð Þ ¼ 4L

h

ffiffiffiffiffiffiffiffiffiffiffiffiffi2mLF

pR0

xL

; ð257Þ

where

R0ðxÞ ¼ðp=20

cos2ðsÞsin2ðsÞxþ sin2ðsÞ 1=2 ds: ð258Þ

The form of Eq. (257) has a certain utility to it: it is in the form of a product

of a length term (L) with a wave number term related to the height of the

barrier above the Fermi level (F ¼ FL) with a dimensionless correction term

(R0(x)) accounting for the difference between the image charge barrier and

the triangular barrier, a feature repeated for other potentials. Two limits of

R0(x) are easily found to be R0ð0Þ ¼ 1=3ð Þ and R0 x 1ð Þ p= 16x1=2ð Þ. TheFN triangular barrier result is obtained by setting Q ¼ 0 and using R0(0). A

more detailed analysis based on a partial summation of the series involved

provides better approximations (Jensen, 2001), namely,

R0ðxÞ

ffiffiffiffiffiffiffiffiffiffiffi1þ x

p

42241408þ px 336ln

x

1þ x

0@

1A 151

24

35

0@

1A x < 0:125ð Þ

0:35657 0:28052ffiffiffix

p þ 0:086441x 0:125 x 1:0ð Þpffiffiffix

p4 4xþ 1ð Þ x > 1:0ð Þ

;

8>>>>>>>>>><>>>>>>>>>>:

ð259Þ


wher e a crit ical feature, namel y the logari thmic depend ence on the small x

beh avior, is shown to exist. The performan ce of Eq. (259) compared to

Eq. (258) is shown in Figure 26 : before much is made of it and its accuracy ,

the utilit y of equ ations such as Eq. (259) ha ve been permanent ly eclipsed by

an approx imation due to For bes (2006) , discus sed in great er de tail be low,

renderi ng furt her discour se on Eq. (259) a bit anach ronistic and only of

hist orical inter est.

a. Expansio n of y near E ¼ m . W hat passes for the ‘‘tradi tional treat-

ments ’’ of the FN equ ation often is based on the form ulatio n of Murphy an d

Good (1956) to accoun t for imag e charge modificat ions on the current

den sity form ulas through the intr oduction of functi ons v(y) and t(y) (see

For bes an d Jensen, 2001, for tabula ted values ), which aris e when Eq. (257) is

rendered linear in energy E about the expan sion point m. They are relat ed tothe R0(x ) for 0 y 1 functi ons by

v cos ð xÞð Þ ¼ 3

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffisin 3 ð xÞ

qR0

1 sin ð xÞ2sin ð xÞ

0@

1A

t cos ðx Þð Þ ¼ 1 þ 2

3 co tð xÞ ] x

24

35v co s ðx Þð Þ

ð 260 Þ

The traditional form t ð yÞ ¼ 1 ð2=3Þ y]y

v ð yÞ is more often encountered

(Modinos, 1984). The literature is replete with clever approximations to accom-

plish various ends, although the most common end sought is to approximate

effective emission area, work function, or both from current versus voltage data

rendered on an FN plot (Forbes, 1999a). Although the Forbes approximation

to v(y) is deferred to later, in the literature much effort is often devoted to the

0 0.5 1 1.5 20

0.1

0.2

0.3 R0(x)Small xLarge xMid xDomains

R0(

x)

Sqrt(x)

FIGURE 26. Comparison of the exact Eq. (258) with its approximation Eq. (259).

88 KEVIN L. JENSEN

form vðyÞ vo y2 so that a plot of current density versus field on an FN plot

is explicitly linear, and so there is historical interest in describing such efforts.

Expanding v(y) about yo to order y2 results in

vquadðyÞ vðyoÞ þ ðy2 y2oÞ3

4

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2zo þ 1

p3R zoð Þ þ 2zo þ 1ð ÞR 0

o zoð Þ ;

ð261Þwhere zo is the argument of R0 evaluated at y ¼ yo. If yo is chosen such that

the coefficient of y2 is identically unity, then yo ¼ 0.599161, and

vquadðyÞ 0:936814 y2: ð262ÞA widely used form vðyÞ vo y2, with vo ¼ 0.95, was introduced by

Spindt et al. (1976). It is a challenging but ultimately pointless exercise to

aspire to an analog of Eq. (262) for t(y); often it is merely approximated by a

constant. The fact is that low‐order Taylor expansions perform poorly as a

consequence of the embedded logarithmic dependence hinted at in Eq. (259).

A crude three‐point fit is

tðyÞ 1 yð Þ 1 2yð Þtð0Þ þ 4y 1 yð Þt 1

2

0@1A y 1 2yð Þt 1ð Þ

¼ 1þ 0:06489yþ 0:0458308y2

ð263Þ

where t(0) ¼ 1, t(1/2) ¼ 1.0439, and t(1) ¼p=ffiffiffi8

pand is compared to exact

values (Figure 27). This, however, is only a temporary mathematical

‘‘fix’’; better approximations are described in the discussion of the Forbes

approximation; fortunately, that approximation is worth the wait.

Regardless of how v(y) and t(y) are obtained, the linearized y(E) is givenby (where the subscript ‘‘fn’’ refers to Fowler–Nordheim, as this form is

needed in the derivation of the FN equation, as given byMurphy and Good):

yfn E mð Þ ¼ bfn

Fþ cfn m Eð Þ

bfn Fð Þ ¼ 4

3h

ffiffiffiffiffiffiffiffiffiffiffiffi2mF3

pv

ffiffiffiffiffiffiffiffiffiffi4QF

pF

0@

1A

cfn Fð Þ ¼ 2

hF

ffiffiffiffiffiffiffiffiffiffi2mF

pt


pF

0@

1A

ð264Þ


wher e the unusua l ch oice of bfn/F is made so that when the qua dratic form of

v(y) is used, the resul ting interce pt is linea r in F—a useful feat ure in the

repres entat ion of current den sity on an FN plot of ln J =F 2ð Þ versus 1/ F.

In the limit that Q approach es 0 (i.e., as the image charge is neglect ed an d

the potenti al ba rrier becomes trian gular), the origina l FN rep resentati on,

whi ch woul d be obt ained from the linea rization in en ergy about m of theargume nt of the exp onent in Eq. (246) [and as sugge sted in Eq. (252) ] is

recover ed.

b. Expansio n of y near E ¼ m þ f : The Quadr atic Barrie r. W hen the

work functio n is low, or when the tempe rature is high, the trans ition from

tunneli ng (und er the barri er) to therm al (over the ba rrier) emission occurs

(Gadz uk and Plu mmer, 1971) an d there the expansi on point needs to be

taken, not at the Fermi level , but closer to the potenti al maxi mum. Near the

barri er maxi mum, the image charge pot ential resem bles an invert ed parabo-

la, that is, a qua dratic potential, in whic h case the linear expansi on is

sim ply Eq. (254) but with Vo and symm etry axis of the quadrati c poten tial

dicta ted by the image charge parame ters Vo ¼ V image ( xo) and

] 2x Vimage ð xo Þ ¼ ] 2x V quad ð xo Þ , (the corresp onden ce is sho wn for several fieldsin Figure 28 ), resul ting in

yquad ðE Þ ¼ p2h


F

rF


pþ m Eð Þ

n o: ð 265 Þ

The behavior of Eq. (265) is shown for copper pa rameters at fields charac-

teristic of tunneling in Figure 29, labeled by the acronym SICT (standard

image charge theta) and compared to AICT (approximate image charge

0 0.2 0.4 0.6 0.8 1

0.6

0.0

0.2

0.4

0.8

1.0

1.00

1.02

1.04

1.06

1.08

1.10

1.12

v(y)vquad(y)

t(y)t quad(y)

v(y)

t(y)

y

FIGURE 27. Performance of the “crude” quadratic approximations to v(y) and t(y)

[Eqs. (262) and (263), respectively] compared to exact (numerically evaluated) results.

90 KEVIN L. JENSEN

theta) designating Eq. (264) and Quad designating Eq. (265). The figure

showing the ratios of the approximations with the WKB y shows that

AICT performs well near the Fermi level, but Eq. (265) is accurate near the

barrier maximum, where the image charge potential is better represented by a

quadratic, and also better for high fields, where the triangular nature of the

barrier is suppressed.

c. Reflection Above the Barrier Maximum. Use of Eq. (265) for energies

larger than the barrier maximum as per the prescription of Eq. (252) worked

well for the quadratic barrier. For the image charge potential, however, the

correspondence is not quite as cozy: evaluations of T(E) using numerical

methods (such as those described below) show that T(E) does not approach

unity for E > m þ F nearly as rapidly as the linear extension of y model

suggests. Another factor contributes, as suggested by the differences between

the triangular barrier and quadratic barrier models: C EðkÞð Þ for the former,

as inferred from a comparison of Eqs. (250) and (251), is a nontrivial

creature, whereas for the latter, as inferred from the success of Eq. (255), it

is unity, an effect therefore inferred to be related to the abruptness of the

triangular barrier compared to the far more composed rise of the quadratic

barrier. The image charge barrier has elements of both—an abrupt rise near

the origin due to the image charge term, and a leisurely decline far from the

origin due to the field term. A good analytic model does not present itself, but

the Bohm analysis suggests a reasonable kludge (Jensen, 2003b).

Consider an incident plane wave to the left of the first zero of the image

charge potential barrier. To the right, let the wave function be approximated

by ckðxÞ ¼ tðkÞRkðxÞexp iSkðxÞð Þ, but after x ¼ xþ assume that the potential

6 9 12 1537

8

9

10

11

Pot

ential

[eV

]


1 eV/nm

5 eV/nm

9 eV/nm

ImageQuadratic

Cuµ = 7.0 eVΦ = 4.6 eV

FIGURE 28. Comparison of the image charge potential (thick line) to the quadratic barrier

potential (thin line) accurate near the apex for increasing fields.


0.6

0.7

0.8

0.9

1.0

11109876

Rat

io q a

prx(E

)/q(

E)

Energy [eV]

(c)

9 eV

/nm

5 eV

/nm

1 eV

/nm

m

5 6 7 8 9 10 11−2

0

2

4

6

8

10(b)

SICT 9AICT 9Quad 9SICT 5AICT 5Quad 5

q(E

)

Energy [eV]

5 V/nm

9 V/nm

5 6 7 8 9 10 110

10

20

30

40

50

60(a)

SICT1AICT1

Quad1

q(E

)

Energy [eV]

1 V/nm

FIGURE 29. (a) Comparison of the linear expansions for the y function evaluated using the

approximate image charge y (AICT) and the quadratic approximation (Quad), compared to the

exact result (SICT) for a field of 1 GV/m. (b) Same as (a), but for the higher fields of 5 GV/m and

9 GV/m. (c) The approximations (AICT and Quad) to y compared to the numerical evaluation.

As expected, the approximations are good only near the expansion points of the chemical potential

(i.e., Fermi level).

92 KEVIN L. JENSEN

is flat such that the wave function resumes its plane wave behavior

(an example for copperlike parameters being given in Figure 30). Equating

wave function and first derivative at x ¼ x– for energies in excess of the

barrier maximum suggests that t(k) is given by

tðkÞ ¼ 2ikexp ikx iSð ÞikRþ ]xRþ iR]xS

jtðkÞj2 ¼ 4k2

]xRð Þ2 þ R2 kþ ]xSð Þ2n o ð266Þ

where x is evaluated at x– (for simplicity the k subscript (k)on R and S is

suppressed) and EðkÞ ¼ h2k2=2m. Neglecting the quantum potential indicates

that ]xS ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2 kvðxÞ2

qand R E= E VðxÞð Þ½ 1=4, where kv is defined by

VðxÞ ¼ h2kvðxÞ2=2m. It follows that S(x–) ¼ 0 and R(x–) ¼ 1 because V(x–)

is identically 0 by definition, and, in this approximation,

]xRðxÞ ¼ F

4EðkÞxo

x

2

1

" #: ð267Þ

Joining components suggests that C(k) for the image charge potential is

given by |t(k)|2, or

Cimage EðkÞð Þ ¼ E3

E3 þ h2F2

128mxox

2 1

2 : ð268Þ

0

3

6

9

12

0 2 4 6 8 10

Pot

ential

[eV

]


Copper-likem = 7.0 eVΦ = 4.6 eVL = 1.0 nm

FIGURE 30. Quadratic potential (inverted parabola) for copper-like parameters.


The impact of Eq . ( 26 8) on the tanh‐WKB model is shown for copperlike

parameters under an applied field of 8 eV/nm for the image charge potential in

Figure 31. At the barrier maximum (E ¼ m þ F ¼ 8.21 eV), C(k) is approxi-

mately 88.4% and slowly increases to 95% at E ¼ m þ F ¼ 11.6 eV.

The improvement is evident. The dependence of C on energy is nevertheless

comparatively weak compared to e y ðE Þ , and therefore, it is often enough toreplace C(E) by C( Em), where Em is the location of the integrand maximum

(which, for thermionic emission, is approximately at m þ f) when the currentdensity is being evaluated.

D. Numeri cal Method s

The numerical evaluation of T(k) uses the m odified A iry f unction

approach, r eplacing the plane waves previously considered in Eq. (197) .

Analogou sly, at each region where a change in slope or a discontinuity in

height occurs (or both), the relation between the c oefficients to the l eft and

right i s

tn 1

rn 1

!¼ 2 p

f 1 = 3n 1 sn 1 cn 1 3 c

2n 1 1

]x Zi c n 1 ; zn 1ð Þ Zi cn 1 ; zn 1ð Þ]x Zi c n 1 ; zn 1ð Þ Zi cn 1 ; z n 1ð Þ

!

Zi cn ; znð Þ Zi cn ; z nð Þ]x Zi cn ; znð Þ ]x Zi c n ; z nð Þ

!tn

rn

! ð 269 Þ

wher e the coeffici ent is a consequ ence of the W ronskia n of the Zi functi on.

As in the treat ment of the triangu lar barrier, if the nth region includes a

transition from above the barrier to below (or vice versa), then the transition

0

0.2

0.4

0.6

0.8

1.0

6 9 12 15

T(E)hyp-SICTAICTAnalyticT

rans

mission

pro

babi

lity

Energy [eV]

F IGURE 31. Product (thick line) of the energy-dependent coefficient in Eq. (268) to

the hyperbolic tangent approximation (thin line) to exact results. Also shown is the AICT

approximation for comparison, which performs disastrously near the barrier maximum.

94 KEVIN L. JENSEN

matrices of Eq. (236) are require d. In all other respect s, the methodol ogy is

analogous to the tri angular barri er and square barrier exampl es—albe it wi th

more segme nts, necessi tating great er atte ntion to when the trans ition matri-

ces must be invo ked. (The methodol ogy here is analogous to, but simpler,

than that found in Jen sen, 20 03b.) Consi der the quad ratic and image ch arge

potenti als an d their numeric al solut ion as ca se example s.

1. Numeri cal Treatment of Quadr atic Pot ential

A discretization of the quadratic potential using 24 linear segments is shown in

Figure 30 where copperlike parameters are used. From this potential,

the transmission coefficient was calculated for 200 values of energy

(Figure 32). The tanh‐WKB approximation 1= 1 þ e y ðE Þð Þ is compared to the

nu me ri ca l T( E) , and the more familiar WKB approximation e y ðE Þ , where y isas given in Eq . ( 25 4) . On this scale, the tanh‐WKB approximation works

well. The numerical calculation, from the generation of the potential, the

initialization of the Zi functions, the evaluation of T(E) for 200 cases, and

the output of the data is rapid, taking less than a second on a contemporary

de skt op c om pu te r.

2. Numeri cal Treatment of Imag e Char ge Potent ial

Repre senting the image charge potenti al as a sequ ence of piece ‐ wise linea rregions is more art than scienc e: wher e the poten tial varies rapidl y and

nonlinear ly (near the origin ), many smal l potential regions are requ ired,

wherea s with a predomi nantly linea r poten tial (far away wher e the image

charge term is negligible ), the lengt h of the segme nts can be substan tially

0

0.2

0.4

0.6

0.8

1.0

6 8 10 12 14

Numericaltanh–WKB

WKB

Tra

nsm

ission

pro

babi

lity

Energy [eV]

Copper-likem = 7.0 eVΦ = 4.6 eVL = 1.0 nm

Quadratic potential

FIGURE 32. Numerical solution of the quadratic barrier of Figure 30 compared to the

tanh-approximation and the exp(-y) approximation.


longer. Generally, precision of a method does not necessarily guarantee the

accuracy of its result. The relation between art and precision takes some

quantification. Consider the representation of image charge potential, start-

ing with a small number of segments and increasing their number. A crude

measure of ‘‘accuracy’’ is whether doubling the number of segments results in

a negligible change in the variation of the transmission probability.

In particular, consider the following schemes referred to as ‘‘linear,’’

‘‘quadratic,’’ and ‘‘proper,’’ in which x(i) is evenly spaced, the length of

the segments increases quadratically, and the length of the segments reflects

the importance of the region, respectively. They are

xlinearðiÞ ¼ x þ xþ xð Þ i 1

N 1

0@

1A

xquadðiÞ ¼ x þ xþ xð Þ i 1

N 1

0@

1A

2

xproperðiÞ ¼x þ xo xð Þ N 2i

N 2

0@

1A

3=2

2i N

xo þ xþ xoð Þ 2i N

N

0@

1A

2

2i > N

8>>>>>>><>>>>>>>:

ð270Þ

where xþ, x–, and xo are the larger and smaller zeros ofV(x) and the location

of the barrier maximum, respectively. The schemes can be characterized as

follows. Linear takes no account of details of the potential, and is therefore

expected to perform poorly. Quad accounts for the steep variation near the

origin and minimizes the variation far away, but it does not take notice of

the actual barrier maximum location and value, and in particular, has coarser

discretization there than near x–. Proper takes pains to discretize finely near

the barrier maximum and less so as the points move further from xo (it is

proper only in the sense that it respects details of the potential, not that

it is the optimal choice, which it is not), and—significantly—the maximum of

the potential is one of the grid points.

If the calculation of T(E) by numerical means is accurate, then doubling

the number of linear segments in the modified Airy function approach has

minimal impact. The change in doubling the number of points chosen for the

potential as per Eq. (270) is shown in Figure 33. As a consequence of the

increase in the number of segments, the effects on the numerically determined

T(E) are shown in Figure 34. Clearly, the linear method is pathetic: a large

portion of the potential is shaved off in the N ¼ 8 case, which, by intuition

96 KEVIN L. JENSEN

0

2

4

6

8

10(a)

0 4 8 12 16 20 24

Image potentialN = 8N = 16

Pot

ential

ene

rgy

[eV

]


Copper-likem = 7.0 eVΦ = 4.6 eVF = 5 eV/nm

Linear x(i)

0

2

4

6

8

10(b)

0 4 8 12 16 20 24


0

2

4

6

8

10(c)

0 4 8 12 16 20 24


Pot

ential

ene

rgy

[eV

]



Proper x(i)

FIGURE 33. Discrete representation of the image charge potential (a) linearly spaced

regions; (b) quadratically spaced regions, (c) fine spacing near maximum, coarse in linear regions.


0

0.2

0.4

0.6

0.8

1.0

7 8 9 10 11

N = 8 N = 16

Tra

nsm

ission

pro

babi

lity

Energy [eV]


Linear x(i)

(a)

0

0.2

0.4

0.6

0.8

1.0

7 8 9 10 11

N = 8N = 16

Tra

nsm

ission

pro

babi

lity

Energy [eV]


Quadratic x(i)

(b)

0

0.2

0.4

0.6

0.8

1.0

7 8 9 10 11

N = 8N = 16

Tra

nsm

ission

pro

babi

lity

Energy [eV]


Proper x(i)

(c)

FIGURE 34. (a) Comparison of the N = 8 and N = 16 linear schemes: agreement is poor,

therefore accuracy is poor. (b) Comparison of the N = 8 and N = 16 quadratic schemes:

agreement is moderate, therefore accuracy is moderate. (c) Comparison of the N = 8 and

N = 16 proper schemes: agreement is close, therefore accuracy is good.

98 KEVIN L. JENSEN

born of the WKB AUC method, will have predictable consequences—name-

ly, T(E) is shifted to lower energies. The quadratic method fares better, but

changes in the AUC factors near the barrier maximum have a noticeable

impact. Finally, in the propermethod, the scheme was designed with a goal of

minimizing the discrepancies between the AUC factors; indeed, N ¼ 24 does

not result in a readily discernible change in the behavior of T(E) compared

to N ¼ 16. Accuracy, in a numerically cognizant interpretation, therefore

implies rapid convergence of TN(E) as the parameter N increases.

3. Resonant Tunneling: A Numerical Example

The investment behind the Airy function approach was considerably more

significant than the AUC approaches based on theWKBmethod, and yet the

methods generally yield comparable results for tunneling sufficiently below

the barrier maximum. Given that surface conditions are extraordinarily

complex (Haas and Thomas, 1968; Monch, 1995; Prutton, 1994), or that

surface roughness itself (much less a deliberately pointed cathode geometry,

such as Spindt‐type or carbon nanotube field emitters) introduces complica-

tions that cause the macroscopic applied field to differ substantially from the

field at the emission site, effort directed toward the accurate calculation of

the transmission probability seems to be the obsession of the aesthete. There

are two responses. As a general matter, the presence of unknowns or impen-

etrable complexity is not license for indolence. As in philosophy, knowing

what is not the case bounds what is, thus making even simple models inordi-

nately useful. As a matter of practical importance, the AUC fails spectacu-

larly if resonance contributes to the tunneling current; consider the treatment

of an adsorbate on the surface of a metal as discussed in the magisterial tome

by Gadzuk and Plummer (1973) and its more recent incarnations (Binh

et al., 1992), which motivate (but are not identical to) an example considered

below. A systematic treatment of resonance has been dealt with elsewhere

(Jensen, 2003a,b) and is characterized by numerical gymnastics. Instead, a

pleasantly straightforward numerical model that captures the main points

will dominate the present focus.

Returning to the FN triangular barrier, consider the excision of a small

rectangular region from the potential barrier as shown in Figure 35. The

potential is characterized by four grid points and one field, making for a

particularly simple application of the matrix method. In form, the structure is

similar to that of an RTD (Tsu and Esaki, 1973), (an account of the develop-

ment of the idea behind resonant tunneling and its relationship to the FN

equation is in the Nobel lecture of Esaki, 1973), the theoretical analysis and

simulation of which has received considerable attention (Frensley, 1990;

Price, 1988), except that electrons are not incident from the left, and metallic


parameters shall be considered (which sidesteps effective mass variation

problems). Such a model is suggestive of (albeit in a simplistic fashion), for

example, field emission from a single atom tip (Binh et al., 1992), the

potential introduced by a barium atom on tungsten in a dispenser cathode

(Hemstreet, Chubb, and Pickett, 1989), and defects at metal‐semiconductor

interfaces (Monch, 1995). Therefore, the blue line marked ‘‘ion’’ shows the

Coulomb potential associated with a screened charge outside the surface or

VionðxÞ ¼ Voionexp ajx xionjð Þ=jx xionj. The parameters of the example

potential are again copperlike (Vo ¼ mþ F ¼ 11.6 eV, F¼ 4 eV/nm), the well

region is 0.4 nm wide and 13.6 eV deep (the ‘‘ion’’ curve is obtained from

Voion ¼ 13.6 eV, xion ¼ 0.7 nm, and a ¼ 2 nm1).

The numerically calculated transmission probability is shown in Figure 36,

along with the transmission probability for the FN triangular barrier, which

is orders of magnitude smaller. Three resonant levels manifest themselves

as peaks in ln(T(E)), their locations given coarsely by the infinite well energy

levels. The triangular barrier with a square well excised is a convenient

choice, because the transmission probability can be envisioned as a conse-

quence of a big triangular barrier of height Vo with a smaller triangular

barrier of height Vo – FL (where L is the location of the LHS of

the well) excised from it, and an even smaller triangular barrier of height

Vo – F(LþW) (where W is the width of the well) reinserted, inasmuch as the

AUC formulation is only concerned about the barrier characteristics for

the area above the energy E. Thus, it appears as though the transmission

probability T(E) can be approximated as

−5

0

5

10

0 5 10 15 20 25

V(x)Grid Pointsion

Pot

ential

[eV

]


Barrier = 11.6 eVField = 4 eV/nm

Well = −13.6 eVWidth = 0.4 nm

FIGURE 35. A linear segment potential (linear black lines with gray dots at segment origins)

containing a well for which a resonant state will occur, constructed to mimic the potential of an

ion (shown in dark gray) but with minimum complexity.

100 KEVIN L. JENSEN

T ð E Þ TFN Vo ; Eð Þ TFN Vo F ð L þ W Þ; Eð ÞTFN Vo FL ; Eð Þ

; ð271Þ

where the term in the curly brackets . . .f g is referred to as Tbarr in Figure 36.

Dividing T(E) by the RHS of Eq. (271) (i.e., we assume

T ð E Þ TFN T barr T re s and isolate Tres) therefore reveals the resonances

Tres in stark contrast, as shown in Figure 37. Resonances are often represented

as Lorentzians of the form (Price, 1988)

Tres ð E Þ To

pE Eres

d

2

þ 1

( ) 1

; ð272Þ

where the facto r of p antic ipates that, for small d, a Loren tzian behav esanalogous ly to a Dir ac delta functi on when integrate d with other smoot hly

varyin g function s. Asid e from the reson ances, the trans mission pro bability is

handled quite well by products and ratio s of AUC terms in that away from

the resonances , the ratio is of ord er unity. If the resonances are indexe d from

n ¼ 1 to 3 for low est to highest , then d(n) ¼ 0.006115 , 0.0611 56, 0.4701 28;

Eres (n) ¼ 0.8729 1 eV, 6.8533 eV, an d 11.954 e V; and To(n) ¼ 384,08 5.00,

2520.5 8, and 6.3249 5, respect ively . The wid th of the energy spread coup led

with the magni tude of the coefficie nt guarant ee that the pre sence of reso-

nances wi ll cause a substan tial increa se in the transmit ted current . Ther efore,

when resonances are pos sible, ne ar the energy levels of the wel l, AUC ‐ likeformali sms miss the physics : great er diligenc e is demand ed, and the Airy

function approach provides it.

−40

−30

−20

−10

0

2 4 6 8 10 12

T(E)TFN

Tbarr

ln

T(E

)

Energy [eV]

FIGURE 36. The transmission probability of Figure 35: TFN is the Fowler–Nordheim trans-

mission probability without the excised well region; see Eq. (271) for the definition of Tbarr.


E. The Thermal and Field Emission Equation

In addition to the transmission probability, the distribution of electrons in

energy is needed for the estimation of current density. The emitted distri-

bution depends on the particulars of the barrier, whether the majority of

electrons tunnel through or are emitted over the barrier—or some combina-

tion thereof. To evaluate the total current density, a naive approach to

evaluate the current density for a given momentum k, shown previously to

be jtrans ¼ jtðkÞj2hk=m, is simply to integrate jtrans(k) with its distribution

function fo(k) (the problems with this approach have been examined in

the discussion of the Wigner function representation for current density).

The Tsu–Esaki formula (Tsu and Esaki, 1973) for current differences

between left‐ and right‐flowing electrons from opposite boundaries uses

such a method; letting dE=h ¼ ðhkdk=mÞ and T(E) ¼ |t(k)|2,

J ¼ q

2ph

ðTðEÞ f lefto Eð Þ f righto Eð Þ

dE: ð273Þ

For supply functions based on the FD distribution (as for RTD simula-

tions) it follows that

J ¼ qm

pbh2

ðTðEÞln 1þ ebðmEÞ

1þ ebðmE’Þ

dE; ð274Þ

−4

−2

0

2

4

6

8

10

12

2 4 6 8 10 12

T(E)/(Tbarr x TFN)

Σ lorentzians

ln

Tra

nsm

ission

pro

b.

Energy [eV]

n = 1

n = 2

n = 3

FIGURE 37. The extraction of Tres from T(E), compared to the sum over Lorentians

modeling the resonances.

102 KEVIN L. JENSEN

where’ is the bias drop across theRTDstructure. For the equations of electron

emission, however, frighto ¼ 0; coupledwith the general transmission probability

given by Eq. (251) and taking y(E) to be linear in E suggests that

JðF ;TÞ ¼ qm

pbTh2

ð10

CðEÞln 1þ ebF ðmEÞ 1þ ebT EoEð Þ 1

dE; ð275Þ

where bT ¼ 1/kBT and bF are the slope factors of the supply function and

transmission coefficient, respectively, in units of inverse energy, the notation

serving to emphasize their analogous role. Eq. (275) is the general form from

which limiting cases yield thermal, field, or photoemission equations. The

quantity bF(Eo – E) is the equation of the tangent line to y(E) at E¼ Em. It is

convenient to recast Eq. (275) in terms of a dimensionless integral (Jensen,

O’Shea, and Feldman, 2002)

JðF ;TÞ ¼ C Emð ÞARLD kBbTð Þ2N

bTbF

; bF ðEo mÞ; bFEc

; ð276Þ

where Em is the maximum of the integrand, and ARLD ¼ mqk2B=2p2h3 ¼

120.173 amp/k2cm2 is the Richardson constant (Richardson and Young,

1925; augmented by Fowler, 1928 by a factor of 2 to account for electron

spin) evaluated using contemporary values of the fundamental constants

(e.g., see http://physics.nist.gov/constants). The coefficient C(E) is presumed

to be slowly varying (an intuition shown to be reasonable from the analytical

models for which an exact evaluation is possible, as well as Airy function

approach numerical studies) and of order unity; few have patience for such

things given the uncertainty in quantities such as emission area and local

work function, so that simply approximating it by unity is an irresistible

temptation. Nevertheless, in low field thermionic emission studies, the wave

nature of the electron induces ripples captured by C(E(k)) that can be

measured (Haas and Thomas, 1968). However, below we succumb to temp-

tation and approximate C(E(k)) by unity. The introduced function N(n,s,x)

is represented by dimensionless integral defined by

N n; s; uð Þ ¼ n

ðu1

ln 1þ exp nðz sð Þ½ 1þ expðzÞ dz; ð277Þ

where n ¼ bT=bF and s ¼ bF (Eo – m). Eq. (271) is general; it is applied to

either field or thermal emission by specifying whether n > 1 or n < 1,

respectively. The value of s and Eo in each case is different.

A general expression for Eq. (277) will be found in due course, but first, it is

pedagogically valuable to investigate two of the three historical current

density antecedents in the canonical equations of field, thermionic, and

photoemission emission that are based on the image charge potential, namely,


the FN (field) and RLD (thermionic) equations. Near the barrier maximum

V(xo) ¼ m þ F, where xo ¼ffiffiffiffiffiffiffiffiffiffiQ=F

pand f ¼ F ffiffiffiffiffiffiffiffiffiffi

4QFp

, V(x) is well

approximated by a quadratic; therefore, y(E) is linear and given by

Eq. (265), the slope factor bF for which is smaller than for energies closer to

m. What is said about the quadratic barrier bF therefore can be used as a guide

to the image charge bF. For example, using the quadratic bF, n is given by

nquad ¼ 2

ph2

2m

!1=2F3

Q

1=41

kBT: ð278Þ

In particular, n ¼ 1 for T ¼ 1000 K and F ¼ 1.1 eV/nm. Higher tempera-

tures or lower fields are therefore indicative of the thermal regime, whereas

lower temperatures or higher fields are indicative of the field regime. The

slope factor bF evaluated from the FN terms is larger, but the qualitative

behavior is similar. It is important to note that bF is not beholden to either

the FN or quadratic parameterization. It varies depending on where the

tangent line to y(E) is taken. Two limits are considered in turn.

1. The Fowler–Nordheim and Richardson–Laue–Dushman Equations

Strong fields and low temperatures make n large and signal that tunneling

dominates thermal emission. Conversely, high temperatures and weak fields

render n small and signal that thermal emission dominates tunneling. The

asymptotic limits of Eq. (277) under the assumption that u s 1 are

readily represented as

N n; s; uð Þ ) ens n ! 0ð Þn2es n ! 1ð Þ :

ð279Þ

It is then a question of determining the Eo component of s. For n asymptoti-

cally small because bF is large, the transmission probability approximates a

Heaviside step function in energy with the step occurring at the barrier maxi-

mum mþF; the integrandmaximummust therefore occur at (rather, very near

to) that energy. Conversely, for n asymptotically large because bT is large, thenthe supply function vanishes at the Fermi level and the integrand maximum

occurs near m and the FN approximation to y is warranted. Thus,

asymptotically,

Eo mþ F n ! 0ð Þ

mþ bfn

Fcfn

0@

1A n ! 1ð Þ

8>><>>: ð280Þ

104 KEVIN L. JENSEN

The ratio bfn= FFcfn ¼ 2vðyÞ= 3ð1 yÞtðyÞð Þ, where y ¼ ffiffiffiffiffiffiffiffiffiffi

4QFp

=F, variesfrom 2/3 to 1 as y varies from 0 to 1. Thus, the value of Eo from the thermal

(small n) side is comparable to the field (large n) side. Use of Eq. (280) in

Eq. (276) with the approximation for N given by Eq. (279) results in the

following asymptotic limits of the general emission equation

JðF ;TÞ ) JRLD ¼ ARLDT2exp f=kBT½ ðn ! 0ÞJFN ¼ ARLD kBcfn

2exp bfn=F ðn ! 1Þ

ð281Þ

These are not the forms encountered in the literature (although they follow

naturally from the present analysis). The most commonly given forms are

JRLD Tð Þ ¼ qm

2p2h3kBTð Þ2exp F


p =kBT

h i

JFN Fð Þ ¼ q

16p2hFtðyÞ2 F2exp 4


p

3hFvðyÞ

0@

1A ð282Þ

The roles of F and T are interchanged in the asymptotic limits, apart from

changes in the work‐function–dependent coefficients.A sense of the magnitude of the various terms is useful. For typical

thermionic emission conditions of a barium dispenser cathode operating at

1 A/cm2, an extraction grid is held at kilovolt potentials fractions of a

millimeter above the emitter surface, which in turn is heated. Using values

of F¼ 1 eV/mm (corresponding to an electric field of 1 MV/m),F¼ 2 eV, and

T ¼ 1300 K implies n b=cfn ¼ 1=1500. Conversely, field emission from a

sharpened Spindt‐type field emitter cone (Spindt et al., 1976) emitting 25 mAfrom an emission area about (5 nm)2 with F ¼ 7.86 eV/nm, F ¼ 4.41 eV, and

T ¼ 300 K, to give n ¼ 13.13. Under such conditions, an array of Spindt‐type emitters with a packing density of 108 #/cm2 likewise produces a current

of 1 A/cm2 (these are ad hoc numbers; arrays driven harder need not be so

tightly packed—the hardest‐driven Spindt‐type emitters have achieved more

than 1 mA per tip (Schwoebel, Spindt, and Holland, 2003)). The peculiar

middle ground near n ¼ 1 can occur when dispenser cathode temperatures

get cold or field emitters get (very) hot, circumstances that are not generally

encountered (an exception being Schottky emission cathodes; Fransen, Over-

wijk, and Kruit, 1999), or heated (via high‐intensity lasers) metallic needles

subject to high fields (Garcia and Brau, 2001, 2002; Jensen, et al., 2006b).

Aside from its historical significance (which is great) and its usage (which is

widespread), further discussion of Eq. (282) from a pedagogical view pro-

vides diminishing returns; such treatments are replete in the literature and

cleaved to with steadfast (and on occasionally unthinking) tenacity, but they

are in fact incomplete by virtue of ignoring tunneling in the RLD equation


and thermal emission in the FN equation, although the analyses of Murphy

and Good (1956) and Gadzuk and Plummer (1971) are notable counter-

examples in efforts to consider thermal‐field emission and provide a thermal

correction to the most often used form of the FN equation. Nevertheless,

treatments of the middle ground characterized by n 1 are rare. Technolog-

ical advances rarely leave stones unturned for long, and therefore, the tun-

neling modifications to thermal emission or the thermal modifications to field

emission have a utility apart from the symmetrical beauty of a more general

analysis that we shall now develop.

2. The Emission Equation Integrals and Their Approximation

In an age of breathtaking desktop computational power, the numerical evalua-

tion Eq. (277) so effortlessly reproduces the FN and RLD equations in the

appropriate limits (Hare,Hill, andBudd, 1993;Xu,Chen, andDeng, 2000) that

the pursuit of analytic formulas to augment Eq. (281) seems either academic,

anachronistic, or obsessive. That is mistaken; numerical methods do not reveal

the underlying connection between the two equations hinted at by the formal

dependence on F andT evinced in Eq. (281). Early in the twentieth century, the

striking similarity of the RLD and FN equations suggested to Millikan and

Lauritsen (1928, 1929) that a general form of the current density is

J ¼ A T þ cFð Þ2exp B=ðT þ cFÞð Þ. The actual relation, as shall be seen,

bears a subtle beauty well beyond Millikan’s erroneous conjecture.

The function N(n,s,u) can be separated into four integration regions that

admit of series expansions such that each term in the expansion can be

analytically integrated. Therefore

Nðn; s; uÞ ¼X4i¼1

Niðn; s; uÞ: ð283Þ

The integrals corresponding to N1 and N2 are field emission dominant

and are

N1ðn; s; uÞ ¼ n

ðus

n z sð Þez þ 1

dz

8<:

9=;

N2ðn; s; uÞ ¼ n

ðus

ln 1þ enðzsÞ½ ez þ 1

dz

8<:

9=;

ð284Þ

106 KEVIN L. JENSEN

whereas N3 and N4 are thermal emission dominant and are

N3ðn; s; uÞ ¼ n

ð01


dz

8<:

9=;

N4ðn; s; uÞ ¼ n

ðs0


dz

8<:

9=;

ð285Þ

The evaluation of N1 is trivial and gives

N1ðn; s; uÞ ¼ n UðsÞ UðuÞf g nðu sÞln 1þ euð Þ; ð286Þ

where the Fowler–Dubridge function (Bechtel, Lee, and Bloembergen,

1977; DuBridge, 1933; Fowler, 1931; Girardeau‐Montaut and Girardeau‐Montaut, 1995) has been introduced and is defined by

UðxÞ ¼ p2

12þðx0

ln 1þ ezð Þdz

UðxÞ ¼ 1

2x2 þ p2

6UðxÞ

ð287Þ

although a convenient analytical approximation of reasonable accuracy is

given by Jensen et al. (2003b) as

UðxÞ ex 1 cbexpðcaxÞ½ x 0x2

2þ p2

6 ex 1 cbexpðcaxÞ½ x > 0

8><>: ð288Þ

where cb ¼ 1 p2=12ð Þ ¼ 0.17753 and ca ¼ 1 cb ln2ð Þ=cb ¼ 0.72843.

Considering typical values across the range of thermal, field, and photoemis-

sion processes, the size of u is such that all terms containing eu are generally

negligible.

The rationale for partitioning the sums in this manner is that convergent

expansions for ln 1þ z½ and 1þ z½ 1for z 1 can be used in theNi such that

the remaining integrals become summations. In each of the integrals, the

following replacements are made


1þ ezð Þ1 ¼X1k¼1

ð1Þkþ1ekz

ln 1þ enðzsÞð Þ ¼X1j¼1

ð1Þjþ1

jejnðzsÞ

ð289Þ

Term‐by‐term integration over the elements of the summations is now

possible. The integral for N1 is straightforward and gives

N1 n; s; uð Þ ¼ n2 UðsÞ UðuÞf g þ n2ðs uÞln 1þ euð Þ) n2UðsÞ ð290Þ

where the second line results from neglecting u‐dependent terms. The series

expansions of Eq. (289) and term‐by‐term integration results in

N2 n; s; uð Þ ¼X1k¼1

X1j¼1

1ð Þkþjeks

j nj þ kð Þ 1 eðsuÞðnjþkÞ

)X1k¼1

1ð Þkþ1eksZ

k

n

0@1A ð291Þ

where the k and j terms reflect the series of Eq. (289) with the same index; the

second line is obtained by neglecting the u‐dependent terms. For large s, only

the k¼ 1 term survives. The leading order terms of interest for N1 andN2 are

independent of u: such will also be the case with N3 andN4, and in all cases it

is the same; when u appears, it appears in an exponent with a negative

coefficient, and its size indicates that such terms are negligible, and so

exponential terms containing (‐u) are summarily neglected. The Z function

has been introduced and is defined by

Z xð Þ ¼X1j¼1

1ð Þjþ1

j j þ xð Þ: ð292Þ

Special cases are Zð0Þ ¼ zð2Þ=2 ¼ p2=12, where z(x) is the Riemann zeta

function, and Z(1) ¼ 2ln(2) – 1. Asymptotic expansions for large and small x

are, for large x

108 KEVIN L. JENSEN

Z xð Þ¼ 1

xlnð 2Þ

X1j ¼ 1

1

2j þ xð Þ 2j þ x 1ð Þ

8<:

9=;

) 1

2xln 4

x þ 1

x þ 2

0@

1Aþ 1

x þ 2ð Þ x þ 1ð Þ

8<:

9=;

ð293Þ

where the secon d line follo ws from the integ ral approxim ation to the seri es

summ ation, an d for small x, an expansi on of 1/( jþ x) gives

Z ð xÞ ¼ 1

x

X1j ¼ 1

1 2 j ð1Þ j z 1 þ jð Þxj : ð294Þ

The asympt otic limit s are therefore

limx!0

Z xð Þ ¼ x þ 3

2 x þ 1ð Þ x þ 2ð Þ þ2 z ð2Þ 3ð Þ2

2 6z ð 3Þ 7½ x þ 2 2z ð 2Þ 3½ f g

limx!1 Z xð Þ ¼ 1

2xln 4

x þ 1

x þ 2

0@

1Aþ 1

x þ 2ð Þ x þ 1ð Þ

8<:

9=;

ð295Þ

Figure 38 shows the exact value of Z(x) compared to its asymptotically large

and small (Eqs. (293) and (294), respectively) approximations. The awkward

asymptotic expressions are purposefully constructed to reveal the singular

0.01

0.1

1

0 0.8 1.6 2.4 3.2 4

Z(x)

Large x

Small x

Z(x

)

ln (1 + x)

FIGURE 38. Comparison of Z(x) to its asymptotic limit formulae [Eqs. (293) and (294)].


behavior at x ¼ 1. Continuing, the integral for N3 can be recast using

ez þ 1ð Þ1 ¼ 1 ez þ 1ð Þ1and j kþ jnð Þ½ 1 ¼ jkð Þ1 n k kþ jnð Þ½ 1

to

obtain

N3ðn; s; uÞ ¼ UðnsÞ lnð2Þnlnð1þ ensÞ þ n2X1k1

1ð Þkþ1eknsZðknÞ: ð296Þ

The final integral is

N4ðn; s; uÞ ¼ lnð2Þln 1þ ensð Þ þ nX1k¼1

ð1Þkþ1 eks

nZ k

n

0@

1Aþ neknsZ knð Þ

8<:

9=; ð297Þ

Different terms survive depending on whether ns or s is larger, thereby

leading to the RLD and FN equations.

3. The Revised FN and RLD Equations

Consider now the RLD and FN‐like limits, which correspond to s ns 1

and ns s 1, respectively. In the RLD limit (small n), N1 and N2

are negligible, only the k ¼ 1 terms survive in the series expansions,

and the U and log functions can be replaced by their leading order terms

(e.g., ln(1þx) x). In the FN limit (large n), N3 is negligible, N1 is replaced

by its leading order terms, and the k¼ 1 terms survive in the series expansion of

N3 andN4. When going through the mechanics of finding the dominant terms,

it becomes apparent, after a bit of regrouping, that to leading order

N n; s; uð Þ ! N n; sð Þ and

N n; sð Þ ¼ S1

n

0@1Aes þ S nð Þens

S xð Þ 1þ x2 Z xð Þ þ Z xð Þf gð298Þ

In other words,N naturally separates into two parts: the part containing ens

is the thermal‐like term, and the part containing es is the field‐like term, so

called because their asymptotic limits give rise to the canonical RLD and FN

equations, respectively. Explicitly, the revised FN‐RLD equation can be

written

J F ;Tð Þ ¼ JF=n2ð Þ þ JT n < 1ð ÞJF þ n2JT n > 1ð Þ

JT ARLD kBbTð Þ2S nð Þens

JF ARLD kBbFð Þ2S1

n

0@1Aes

ð299Þ

110 KEVIN L. JENSEN

The symmetry between field emission and thermionic emission is made a bit

more manifest by using the series expansion form of S(x) to show

NRLDðn; s; uÞ ¼ 1þ zð2Þn2 þ 7

4zð4Þn4 þ 31

16zð6Þn6 þ . . .

8<:

9=;ens

NFNðn; s; uÞ ¼ 1þ zð2Þn2 þ 7

4zð4Þn4 þ 31

16zð6Þn6 þ . . .

8<:

9=;n2es

ð300Þ

of which the leading terms have been anticipated by Eq. (279). Using the

explicit forms for the Riemann zeta functions and utilizing Eq. (276), it

follows that the revised FN and RLD equations become (remember the

temptation to approximate the coefficient C by unity)

JRLDðF ;TÞ ¼ ARLD kBbTð Þ21þ p2

6

bTbF

0@

1A

2

þ 7p4

360

bTbF

0@

1A

4

þ . . .

0@

1Aexp bT ðEo mÞf g

JFNðF ;TÞ ¼ ARLD kBbFð Þ21þ p2

6

bFbT

0@

1A

2

þ 7p4

360

bFbT

0@

1A

4

þ . . .

0@

1Aexp bF ðEo mÞf g

ð301ÞAs a historical note, the first correction term in parentheses for JFN is, using

the FN representation for bF, the same as an expansion of the thermal correc-

tion term found by Murphy and Good (1956) in their Eq. (77). The term

Eo changes from thermal to field emission conditions. It is known from the

thermal and field regimes that Em is at the barrier maximum or the chemical

potential, respectively, that is

Em n 1ð Þ ¼ mþ fEm n 1ð Þ ¼ m

ð302Þ

from which Eo is given by (as seen in the FN and RLD equations)

Eoðn 1Þ ¼ mþ 2vðyÞ3tðyÞ F

Eoðn 1Þ ¼ mþ f

ð303Þ

Restricting n to very much larger or smaller than 1, as done here, is slight

overkill, as they generally work reasonably well under less stringent

demands—but it is precisely the region where n is near 1 that difficulties

arise; these are explored next.


The symmetry of Eq. (301) is appealing, although it should be observed

that (1) the term Eo differs between the FN and RLD limits, and (2) while the

two expressions converge for n¼ 1, neither is correct at that point. The point

where n ¼ 1 constitutes the transition region, in which the current integrand

peak shifts from near the barrier maximum to near the chemical potential as

n advances from below unity to above it. Near the n ¼ 1 transition region, all

the integrals entailed by N1 through N4 contribute, and N4 in particular

contains a term whose denominator goes as (n – 1)1. To leading order,

Ntransðn; sÞ ¼ n es ensð Þn 1

þ es þ n 1ð Þ2lnð2Þes: ð304Þ

The vanishing denominator is therefore offset by a vanishing numerator,

so that L0 Hopital’s rule may be used. Therefore, when n¼ 1, the ‘‘transition’’

current density is

JtransðF ;TÞ ¼ ARLD kBbTð Þ2 bF Eo mð Þ þ 1½ exp bF Eo mð Þf g; ð305Þwhich is larger than Eq. (301) when bF ¼ bT for a given Eo. Therein lies a

difficulty in the implementation of Eqs. (301) and (305). From Equations

(302) and (303), the value of Eo changes depending on the asymptotic limit of

n. To use Eqs. (301) and (305), two questions must be addressed. First, how

shall Eo be calculated when n is of order unity? And second, how is bF to be

determined under general conditions?

It is numerically evident that the optimal tangent line to y(E) should be

taken at the maximum of the current integrand, Em; errors in the integrand

away from this energy are exponentially damped by either the transmission

probability or the supply function. Thus,

n ¼ bT ]Ey E ¼ Emð Þf g1

bTEo ¼ bTEm þ ny Emð Þ ð306Þ

For thermionic emission conditions, it is clear that Em lies close to mþF. Forenergies above the barrier maximum, the linear extension approximation to

y(E) [see the discussion following Eq. (249)] ensures that n in Eq. (306) is

trivially evaluated using Eq. (275), and it therefore follows that

ntherm ¼ 2

ph2

2m

!1=2F3

Q

1=41

kBT; ð307Þ

for example, n ¼ 0.01 for F ¼ 103 eV/nm and T ¼1047 K. Observe that n

scales as n / F3=4 for n < 1. It is natural to inquire if n for n > 1 follows a

similar power law behavior—as shown by direct numerical evaluation

(Jensen and Cahay, 2006) and as shall be demonstrated in the following

112 KEVIN L. JENSEN

text, it does. How ever, the optimal evaluat ion of the power for nfield be nefits

most from good ap proxima tions to the elliptical integ ral function s v(y) and

t(y) an d therefore must await the intr oduction of the For bes approxim ation

to v(y) below .

In the meant ime, numeri cal means suffice to consider the performan ce of

the revised FN ‐ RLD equati on. It is a stra ightforw ard matter [usin g the form

of T(E) pro vided by Eq. (251) with C(k) 1 and y evaluated using theWKB AUC method of Eq. (256) ] to find the locat ion of the current density

integran d maxi mum, Em , by bra cketing and bisection. Havi ng foun d E mnumeri cally, n is evaluated using Eqs. (306) and (256) . The beh avior of

n(F) for copper pa rameters at 800 K is shown in Figu res 39 and 40 along

with the therm al and fie ld power ‐ law relations

ntherm ¼ 1: 164 F 3= 4

nfield ¼ 0: 661 F 0: 948 ð308Þ

where the coeffici ents and pFN ¼ 0.948 are determ ined from the F ¼ 0.02 eV/

nm (ther mal) and F ¼ 10 eV/n m (field) da ta poi nts.

Clearly, therefore, the current integ rand maximum migrates from ne ar

the barrier maxi mum (m þ F ) to near the Fermi level (m ) as the field increa ses(Dolan and Dyke, 1954; Gadzu k and Plu mmer, 1971; Jensen, O’She a, and

Feldman, 200 2; Murphy and Good, 1956). The next que stion is: How

does the shape of the integrand change during the same evolution?

In Figure 41 the location of the integrand maximum is bracketed by the

two values where the integrand is 1% of its maximum (designated Emax and

Emin for the larger and smaller energy, respectively) for the same conditions

as in Figures 39 a nd 40. Several feat ures are notice able. First, in the therm al

regime, Em remains fairly close to the barrier maximum (m þ F). Second, inthe field regime, Em is close to, but generally not at the Fermi level, and at

high fields, Em can be below the Fermi level: when the tunneling electrons are

replaced by electrons from higher energies, the excess energy appears as heat

in a process called Nottingham heating (Ancona, 1995). Conversely, at low

fields, tunneling electrons primarily come from above the Fermi level and

cooling occurs. Third, as shown in Figure 42, the energy full width at half

maximum (EFWHM) increases substantially near the n ¼ 1 region; the loca-

tions where n differs from 1 by less than 2% are shown by the open circles.

For almost all fields of technological interest (F 10 eV/nm), EFWHM is largest

in the n 1 transition region. For copperlike parameters, compared to stan-

dard field or thermionic conditions of 4 eV/nm and 300 K (field) or 0.05 eV/

nm (i.e., 5 MV/m) and 1500 K (thermionic), the integrand for the n 1

transition region (1.36 eV/nm and 800 K) is substantially broader than either

the field or thermionic cases (the total current density in each case is


substantially different; the parameters are chosen for pedagogical rather than

pragmatic reasons). The transition is revealed more readily by holding the field

fixed at 1.36 eV/nm and raising the temperature as shown in Figure 43a (the

temperature variation of the chemical potential is ignored) for the temperatures

300 K, 700 K, 800 K, 900 K, and 1500 K f or c opp er pa ra me te rs ; t he

900 K case shows a broad distribution in particular. Figure 43b shows

the behavior of the (normalized) integrand as the temperature is adjusted

0.1

1

10

0.1 1 10

nntherm

nfield

n = 1n

= b

T/b

F

Field [eV/nm]

1.164 F3/4

0.661 F0.948

Thermal regime

Field regime

Φ = 4.6 eV m = 7.0 eV T = 800 K F = 1 eV/nm

FIGURE 39. Behavior of the slope factor ratio ( n) as a function of field for copper-like

parameters and moderate temperatures. The n = 1 line demarcates the transition region between

thermal ( n < 1) and field (n > 1) conditions.

0.4

0.8

1.2

1.6

0.5 1 1.5 2 2.5

nexact

ntherm

nfield

n = 1

n =

bT/b

F

Field [eV/nm]

1.164 F3/4

0.661 F0.948

F IGURE 40. Close-up of the n = 1 region of Figure 39.

114 KEVIN L. JENSEN

upward for constant field. The nature of the broadening of the integrand as

the peak shifts through the transition region bounded by m < Em < m þ F isreadily apparent.

It remai ns to compare the performan ce of the revised FN ‐ RLD equ ation

where n is found num erically (an analyt ical method will be present ed after the

Forbes approxim ation to v(y) is intr oduced ). First , the compari sons are

made for regim es in which the FN and RLD equati ons are known to perfor m

well. Figure 4 4a compares the revised FN ‐ RLD and the standar d FN and

2

4

6

8

10

12

0.1 1 10

Emax

Em

Emin

µ + φ

µEne

rgy

[eV

]

Field [eV/nm]

Thermal regime

Fieldregime

n = 1RegimeT = 800K

FIGURE 41. Behavior of the full-width-at-half-max (FWHM) separation as a function of

field for the same parameters as Figure 39.

6

8

10

12

0.0

0.4

0.8

1.2

1.6

2.0

0.1 1 10

Ene

rgy

[eV

] EF

WH

M [eV

]

Field [eV/nm]

|n(F)-1| ≤ 0.02

FIGURE 42. Location of the integrand maximum Em and the full-width-half-maximum

energy width as a function of field for the parameters of Figure 39.


RLD equ ations. In the standar d FN equatio n, the Spindt quadrati c approxi-

matio n v(y) mentio ned afte r Eq. (262 ) and t(y) ¼ 1.0566 is used (the reason

for the designa tion ‘‘standar d’’ is be cause of the wide use of this form in

infer ring work function from slope on an FN plot). Not surpri singly, the

agreem ent is excell ent. Second , for the fic titious case wher e a low work

functi on coating is present wi th work func tion of F ¼ 1.8 eV (but otherwis e

the co pper pa rameters are retained), as sh own in Figu re 44b, the standard

RLD eq uation is general ly adequ ate to within 1 5%, whereas the revised

FN ‐ RLD is goo d to be tter than 1%. The inter est comes, howeve r, for

moderat e tempe ratur e and field regim es, a s in Figure 44c, wher e the tempe r-

ature is high (but not as high as for therm ionic cathod es) and the work

0

0.2

0.4

0.6

0.8

1

1.2(a)

(b)

6 8 10 11 12

dJ(E

)/dJ

max

Energy [eV]

T = 300KF = 4 eV/nm

T = 800KF = 1.36 eV/nm

T = 1500KF = 0.05 eV/nm

Cu

7 9

0

0.2

0.4

0.6

0.8

1

1.2

7 8 9 10

dJ (

T)/dJ

max

Energy [eV]

300K700K

800K900K

1500K

Cu

FIGURE 43. (a) Behavior of the current density integrand for thermal (right), field (left) and

mixed (middle) conditions. (b) Same as (a) but showing the intermediate cases. Note the width of

the curve labeled “900 K.”

116 KEVIN L. JENSEN

10−9

10−7

10−5

10−3

10−1

101

103

105

107

109

1011

1

(a)

10

Numerical Revised FN-RLD Standard FN (Spindt aprox)

Cur

rent

den

sity

[A

/cm

2 ]

Field [eV/nm]

Cu under field conditions(T=300K, Φ = 4.6 eV)

(b)

102

103

104

105

106

107

0.1 1

Cur

rent

den

sity

[A

/cm

2 ]

Field [eV/nm]

Cu w/coating under thermal conditions(T = 1400 K, Φ = 1.8 eV)

Numerical

Revised FN-RLD

Standard RLD

10−1

101

103

105

107

109

1011

0.1

(c)

1 10

Cur

rent

den

sity

[A

/cm

2 ]

Field [eV/nm]

Cu under mixed conditions(T = 800 K, Φ = 1.8 eV)

NumericalRevised FN-RLDStandard FNStandard RLD

FIGURE 44. Performance of the revised Fowler-Nordheim–Richardson-Laue-Dushmann

equation [Eq. (301)] (a) compared to the most commonly used forms of the Fowler–Nordheim

equation for copperlike parameters; (b) compared to the most commonly used forms of the

Richardson-Laue-Dushmann equation for cesium on copperlike parameters; (c) compared to

mixed conditions challenging the Fowler–Nordheim and Richardson-Laue-Dushmann equa-

tions for cesium on copperlike parameters. Note the high-field behavior.


function is low (by comparison to field emitters). Here the superiority of the

revised FN‐RLD equation is manifest where it is expected to be better, but

also in the high field region where the superiority of the FN‐RLD equation is

likewise evident. Before considering a general thermal‐field equation it is

profitable to determine the performance of the FN and RLD equations

(through the eyes of the FN‐RLD equation) in practice.

F. The Revised FN‐RLD Equation and the Inference of Work

Function From Experimental Data

1. Field Emission

The common motivation for representing v(y) as a linear function in y2 and

t(y) as a constant is that the FN coordinate ln J=F 2ð Þ is linear in 1/F, or

ln J=F2 A B

F: ð309Þ

Using the approximations vðyÞ ¼ vo y2 and tðyÞ ¼ to, where vo and to are

constants independent of field, then

B 4vo

3h


p

A 16

3hQ


F

sþ ln

q

16p2ht2oF

0@

1A ð310Þ

For example, consider the ‘‘prediction’’ of the work function using as data

points the evaluation of current density using the revised FN‐RLD equation

and copperlike parameters in the range 2 eV/nm < F < 10 eV/nm. Inferring

the work function using Eq. (310) (with vo ¼ 0.93685 and to ¼ 1.0566) from

the calculated slope B ¼ 63.472 eV/nm indicates F ¼ 4.6162 eV, very close to

the value of 4.60 eV used in the simulation of J.

Backing out theoretical parameters from ad hoc simulations is scholasti-

cism, even though it indicates the accuracy of an approach. Of greater

interest is to what accuracy the work function can be determined from actual

field emission data based on the FN equation. Of the several existing meth-

ods to measure work function (Haas and Thomas, 1968), estimating F from

the value of B is widely used (a good example is Swanson and Strayer, 1968),

and therefore, showing how it fares is useful. In early studies of field emission

from tungsten wires, data with and without barium adsorbed onto the apex

of the emitter were taken by Barbour et al., (1953; Figure 3 of Barbour et al.

is represented in Figure 45; the straight line fits are explained below). The

tungsten needle geometry allowed for fields greater than 1 GV/m to be

118 KEVIN L. JENSEN

generated at the apex. An immediate complication is that current is measured

as a function of potential differences between cathode and anode, whereas

the FN equation relates current density to field at the emission site. Why the

naive presumption that current is proportional to current density by an area

factor (and voltage to field by a ‘‘beta’’ factor) is addled has been subject of

much work (Forbes, 1999a; Forbes and Jensen, 2001; Jensen and Zaidman,

1995; Nicolaescu, 1993; Nicolaescu et al., 2001, 2004; Zuber et al., 2002).

The problem is twofold: the field varies considerably over the sharpened

structures required to obtain significant field enhancement, and the rapid

variation of field over the surface means that the emission area changes

depending on field strength on‐axis and its variation off‐axis. Barbour et al.(1953) note this in their analysis of the emission data, but assume that the

emission area is constant in order to facilitate the analysis.

A simple model of the impact of both field enhancement and emission area

can be used to obtain a refined analysis compared to that of Barbour et al.

(1953). Consider emission from a hemisphere of radius a. It is a simple

problem in electrostatics to show that the field along the surface of such a

hemisphere on a grounded plane is given by F yð Þ ¼ 3ðV=DÞcos yð Þ for a

sufficiently distant anode held at a potential V a distance D away. The field

enhancement factor at the apex of the hemisphere is therefore (3/D), that is,

Ftip ¼ 3=Dð ÞV ¼ boV (reflecting the nomenclature beta factor: the prolifera-

tion of quantities referred to by the b symbol induces the ‘‘o’’ subscript (o) to

avoid confusion with the temperature and field slope factors).

−36

−32

−28

−24

−20

1 2 3

ln

I/V

2 [A

mp/

Vol

t2]

104/V [Volt]

1 2 3 4

Direct currentMeasurements

Pulsed currentMeasurements

FIGURE 45. Fit to the data of figure 3 of Barbour et al., (1953) for a clean tungsten emitter

(1) and the same emitter with increasing amounts of surface coverage by barium.


It follows that the current from the hemisphere can be written as the product

of an ‘‘effective’’ emission area and apex current density, or

I Vð Þ ¼ barea Ftip

J Ftip

where

barea Ftip

¼ 2pa2

J Ftip

ðp=20

J F yð Þ½ sin yð Þdy: ð311Þ

The approximation that t(y) is constant and v(y) ¼ vo – y2 is very

convenient, and from that approximation, it follows

barea Ftip

¼ 2pa2ð10

exp B

Ftip

x 1ð Þ8<:

9=;x2dx 2pa2

BFtip

Bþ 2Ftip

2 : ð312Þ

An area factor for a hemisphere is but the simplest approximation possible,

but considering either an ellipsoidal or hyperbolic emitter geometry does not

change the overall conclusion that the area factor is linearly dependent on

field; using F ¼ 4.5 eV and Eq. (262), then Ftip/B ¼ 0.082 for Ftip ¼ 5 eV/nm.

The weak field dependence of the denominator changes in the RHS of

Eq. (312) for the geometries characteristic of wires and field emitter arrays,

respectively, but it is still found that for field emission from metals in general,

B is sufficiently larger than Ftip that to a good approximation, barea scales

linearly with apex field.

A field‐dependent area factor undercuts the common practice to plot

current‐voltage data in FN‐like coordinates of (1/V) versus ln(I/V2) and to

infer from linearity of the resulting plot that field emission is occurring and

the coefficient of 1/V gives information about the work function. Given

experimental uncertainty, ln(I) versus 1/V is likewise fairly linear—and

given the variation in emission area with apex field, there is no reason to

expect an FN‐like coordinate ln I=V 2ð Þ to occupy any privileged role. In fact,

given the behavior of barea, the proper coordinates are ln I=V 3ð Þ versus 1/V(if the statistics of dissimilar emitters is considered, then another power of

V appears in the denominator of the log function; Cahay, Jensen, and

vonAllmen, 2002; Jensen and Marrese‐Reading, 2003) Nevertheless, in the

literature, when the slope factor B is referenced (typically to infer work

function), it is under the approximation that ln I=V 2ð Þ is linear in 1/V. An

example in the case of carbon nanotubes, which have small emission areas

and sharp apexes indeed, is the work of Fransen et al. (1999). It is possible to

reconcile the standard approach with the physics, and such is the approach

taken here.

Let Vo be a particular voltage for which the current is Io and the apex field

is Fo ¼ boVo, so that IðVÞ ¼ bareaðFoÞ F=Foð ÞJFN Fð Þ. Then it follows

120 KEVIN L. JENSEN

ln I Vð Þ=V 2f g A0 B0

V

B0 ¼ B

boþ Vo

A0 ¼ 1þ Aþ ln b2obarea boVoð Þ ð313Þ

Therefore, values of B0 extrapolated from experimental data are related to

the work function (assuming it is unique and not a compilation of averaged

values over different crystal planes) by

F ¼ 9b2oh2

32mv2oB0 Vo½ 2

( )1=3

: ð314Þ

If work function changes are occurring [as when barium is being deposited

on the tungsten needle, as Barbour et al. (1953) did], then it follows that

F1

F2

¼ B10 Vo

B20 Vo

2=3

: ð315Þ

Reconsider now the data of Barbour et al. in FN coordinates. The B0values of the linear fits of lines 1–4 are 145.5, 89.91, 72.23, and 55.02 kV,

respectively. Using as the reference point for line 1 values of Io ¼ 0.6457 mAat Vo ¼ 7981 V, an assumed F for tungsten of 4.5 eV, and A0 ¼ 14.22, it is

inferred that bo ¼ 4441 q/cm and barea(boVo) ¼ 3.930 1010 cm2, values

comparable but not equal to those of Barbour et al. The other lines corre-

spond to progressively greater amounts of barium deposited on the surface.

A partial coverage of alkali and alkali earth metals on other metals tends to

lower the work function, so that the lines 2–4 reflect decreasing values of

the effective work function as the surface coverage of barium increases.

We infer from the linear fits and the work function of clean tungsten

that the value of F for lines 2–4 are 3.19 eV, 2.71 eV, and 2.20 eV, respective-

ly, which are comparable to (but smaller than) the values obtained by

Barbour et al. because of the present approximation of a field‐dependentarea factor.

2. Thermionic Emission

For thermionic emission, the variation of current density with temperature

allows for the estimation of work function, as the Richardson coordinate

ln J=T2ð Þ is linear in 1/T. Here, complications such as area factors do not

arise. Wh ere F ! T in Eq. (309 ), we ha ve


B 1

kBF


p

A lnqmk2B

2p2h3

0@

1A ð316Þ

Compare the work function evaluated from the slope of current density

on an RLD plot using Eq. (316) when the current is given by the revised

FN‐RLD equation for parameters somewhat at the edge of generic

parameters (e.g., an applied field of 20 MV/m and a work function of

1.8 eV, as suggested by Figure 46). From a slope of 1.641/kB, the work

function is inferred by Eq. (316) to be 1.811 eV, quite close to the input

value. Similarly, the numerical intercept is close to the theoretical value of

ln(120.17 A/cm2K2) ¼ 4.790.

Inferring work function from experimental data, however, is the challenge.

One difficulty is that the work function is temperature dependent

(i.e., F(T) ¼ Fo þ aT) in addition to its dependence on crystal face. Since

many experimental metal emitters are polycrystalline and require high tem-

peratures to achieve significant current, the inference of a single or

well‐defined work function from experimental data is problematic. Current

density is inferred from current and a presumed emission area, but even

correcting for area results in A values that differ from theoretical predictions.

However, when such factors are corrected for by carefully designed experi-

ments, an estimate of F from the slope of a Richardson plot can indicate

−20

−16

−12

−8

81 01 21 14

ln

J / T

2

1/kB T

y(x) = 4.734 – 1.641 x

Field = 20 MV/mWork function = 1.6 eV

FIGURE 46. A hypothetical data set created using the Revised Fowler-Nordheim–Richardson-

Laue-Dushmann equation so as to compare the accuracy of inferring work function from a

Richardson plot.

122 KEVIN L. JENSEN

work function with some accuracy. An effort to extract work function

estimates, allowing for such complications (and others), was undertaken by

Shelton (1957). Shelton’s data (shown in Figure 47) give from the slope a

naive estimate for the work function for tantalum to be 4.55 eV, but correct-

ing for the work function variation with temperature reduces the value to

4.35 eV, close to the accepted polycrystalline value of 4.25 eV (Weast, 1988).

3. Mixed Thermal‐Field Conditions

Under mixed thermal‐field conditions, estimating work function from

slopes on RLD and FN plots becomes problematic. At low fields, thermal

emission compromises the slope on an FN plot in a manner suggested by

Figure 48 so that, apart from the disagreement introduced by a low work

function, changes are introduced by an increasingly nontrivial thermal com-

ponent as the field declines. In such cases, a comparison to numerical

evaluations of current density is preferable. It is then a question of what

complications can arise, and as expected, complications do arise with the

procedure of comparison. Gadzuk and Plummer (1973) describe how total

energy distribution (TED) measurements are affected by the finite energy

resolution of energy analyzers, and therefore, the energy distribution appears

to be smeared out. For example, even a zero‐temperature energy distribution,

which would in principle have a sharp edge, nevertheless has a broadening

of the distribution near the Fermi level that appears similar to the effects

of a raised temperature. We therefore cannot expect a priori an exact

−30

−27

−24

6.4 6.8 7.2 7.6

ln

J/T

2 [A

/cm

2 K2 ]

1/(kBT [eV])

yA = 4.69 − 4.55/ kBT

yB = 6.52 − 4.70/ kBT

FIGURE 47. Data considered by Shelton (1957) in the determination of work function from

Richardson plots.


correspondence between theoretical models predicated on simple emission

calculations with difficult‐to‐obtain energy distribution measurements (even

though the total current measurements can be good, representing as they do

an integration over the energy distribution). An overall agreement in the

qualitative features is satisfactory.

A word of caution is necessary; we have not made a distinction between the

TED and the normal energy distribution; the latter represents the distribu-

tion of our designation Ez. The difference is both nontrivial and important

(Young, 1959; Young andMuller, 1959). A measured field emission distribu-

tion measures the TED corresponding to E ¼ h2k2=2m, whereas the normal

energy distribution is the outcome of looking at the passage of the longitudi-

nal momentum component through the 1D barrier and is therefore

Ez ¼ h2k2z=2m. The distinction has been hidden heretofore because of the

focus on 1D emission equations—the TED benefits from a moments‐basedanalysis (which is not treated here but is discussed later). The discussion here

blurs the distinction between Ez and E, although comments about the normal

energy distribution will have analogs for the TED (and so the z subscript will

not be used on E); a correct analysis is well summarized by Groning et al.

(1999, 2000).

Let the energy analyzer have a distribution of energies S(EEo) that it

accepts when measuring the particle count at an energy Eo; for example,

S may be Gaussian of the form S Eð Þ ¼ 2pgð Þ1=2exp E2=2gð Þ. Thus, the

particle count per unit area per unit time (proportional to the current density

integrand) for the energy Eo is

0

5

10

15

20

25

30

1 10 100

NumericalFNRLDFN

lnJ

/F2

1/(F [eV/Å])

FIGURE 48. Departures from the Fowler–Nordheim relation for simulated data based upon

the Fowler-Nordheim–Richardson-Laue-Dushmann equation.

124 KEVIN L. JENSEN

d

dtNðEoÞ ¼ 1

2ph

ð11

S E Eoð ÞT Eð Þfo Eð ÞdE: ð317Þ

By the normalization of Eo, the integration of dN/dt over Eo reproduces the

current density (to within a factor of the electron charge). The more sharply S

is peaked (the sharper the energy resolution), the more the experimental

results resemble the theoretical energy distribution of emitted electrons.

Consider two examples: first, the case of simple Richardson‐like (thermionic)

emission, and second, a more Gaussian‐like distribution characteristic of the

situation n ¼ 1.

For the thermionic case, TðEÞfoðEÞ / y E m Fð Þexp bT ðE mÞð Þ,and so the integration over the Gaussian form of S in Eq. (317) is readily

performed and is

d

dtNRLDðEoÞ ¼ 1

2pð Þ3=2h ffiffiffig

pð1

mþf

exp E Eoð Þ22g

bT E mð Þ8<:

9=;dE

¼ 1

4ph1 Erf

gbT þ mþ f Eoffiffiffiffiffi2g

p24

35

8<:

9=;exp

1

2gb2T bT Eo mð Þ

24

35

ð318Þwhere Erf(x) is the error function. The presence of the error function

complicates matters, but the effect is a progressive blunting of the sharpness

of the emitted electron distribution to a more Gaussian‐like shape; as gbecomes larger, the large argument approximation to the error function

can be invoked, and it can be shown that

d

dtNRLDðEoÞ

ffiffiffiffiffiffig2p

rbTgþ mþ F Eoð Þ1

exp bTFþ Eo m Fð Þ22g

" #;

ð319Þwhich demonstrates that as the resolution of the detector becomes

progressively worse (g1/2 becomes progressively larger), the measured distri-

bution becomes more Gaussian rather than the characteristic MB‐likebehavior. Figure 49 shows the effect on a theoretical distribution for dispens-

er cathode conditions for various values of g. In Figure 50, the impact of

Eq. (317) with g¼ 0.1 eV2 on a (normal) energy distribution suggested by the

experimental conditions of Gadzuk and Plummer (1973) are shown. Note

that one of the effects is to make the ‘‘thermal tail’’ appear to be at a higher

temperature than is the case. Therefore, when comparing the energy distri-

butions with experimental data, the impact of the resolution of the energy

analyzer must be considered in the analysis. Here, a measure of the success of

the theory is whether the qualitative dependence is captured—as it is.


Consid er, then, co nditions such that n trans itions through unity, and

compare the theoret ical current densit y integrand to the measur ement s of

Gadzuk and Plumme r (1973). The criteri a for a good co mparison are that

fields taken in the same pro portion as the volta ges co nsider ed experimen tally

give rise to simila r qua litative relations for the e nergy dist ributions. As shown

in Figu re 51 , the theoret ical model bears a relation to the experi menta l trends

and large ly accounts for the e volution from field to therma l con ditions.

0.0

0.5

1.0

1.5

9.2 9.4 9.6 9.8 10 10.2 10.4

0.000.050.100.20

Cur

rent

int

egra

nd [a.

u.]

Energy [eV]

m = 8.0 eVΦ = 1.8 eVF = 10 MV/mT = 1300 K

γ1/2

FIGURE 49. Smoothing out of the normal emission distribution with a Gaussian energy

analyzer [Eq. (317)].

10−15

10−14

10−13

10−12

10−11

10−10

10−9

5 6 7 8 9 10 11 12 13

Cur

rent

int

egra

nd [a.

u.]

Energy [eV]

F = 3.50 eV/nm

F = 2.19 eV/nm

Φ = 4.8 eVm = 8.0 eVT = 1570 K γ = 0.1 eV2

FIGURE 50. Effect of Gaussian energy analyzer on emitted distribution (lines constitute no

energy analyzer, symbols are effects of an energy analyzer—both are generated from theoretical

models).

126 KEVIN L. JENSEN

4. Slope‐Intercept Methods Applied to Field Emission

In the previous section on inferring work function from the FN relation, it

was explicitly assumed that the field was related to the applied voltage by a

scale factor (the ‘‘beta’’ factor), but it was also implicitly assumed that the

current is related to the current density by an emission area. A general

argument was presented to show that the emission area must be field depen-

dent—but the usage of the slope and intercept parameters is more useful than

simply that; it can illuminate the nature of changes that occur on the emitter

during conditioning, especially if a model of the field enhancement and area

factors can be developed as done by Mackie et al. (2003) and Charbonnier

et al. (2005). The present treatment is similar in intent but differs in detail.

Complications associated with simultaneously saying something intelligent

about work function and field enhancement have been capably treated by

Groning et al. (1997, 1999, 2000).

Let us reconsider the field‐dependent area factor analysis behind Eq. (313)

and use it to express current versus voltage using current density versus applied

field relations. For Spindt‐type field emitters, where the voltage in question

comes from a close‐proximity gate, let the relationship between apex fieldF and

gate voltageV be given by F ¼ bgV , where the g subscript (g) identifies that it is

the gate that is primarily responsible for the apex field rather than a distant

anode. Next, explicitly separate out the field dependence of the area factor.

Combining the large B limit of Eq. (312), using Eq. (310) it follows that

1

2

3

4

5

6

7

0

1

2

3

4

5

6

7

−2 −1 0 1 2 3 4 5

Mea

sure

d ch

arge

(a.

u.) C

urrent integrand (a.u)

E-EF [eV]

1600/3.501200/2.631000/2.19800/1.75600/1.31500/1.09

V [V]/F [eV/nm]

FIGURE 51. A comparison of the theoretical energy distributions (lines demarcated by field)

with the experimental distributions from Gadzuk and Plummer (1971, 1973) (symbols

demarcated by voltage). Relative ordering of numbers reflect relative position of lines.


IFNðV Þ 3qa2

64pffiffiffiffiffiffiffiffiffiffiffiffi2mF5

pvot2o

bgV 3

exp 16

3hQ


F

s 4vo

3hbgV


p8<:

9=;:

ð320ÞIt is clear, therefore, that Eq. (320) can be transformed into a linear

relationship where the slope (s) and intercept (zo) factors are defined

according to the relation

lnIFNðVÞV 3

¼ s

Vþ zo ð321Þ

and are determined from experimental data. Thus, a comparison between

Eqs. (320) and (321) uniquely determines the apex radius and the work

function if the slope and intercept are known from the relations

s 4voffiffiffiffiffiffiffiffiffiffiffiffi2mF3

p

3hbg

zo 16Q

3h


F

sþ ln

3qa2b3g64pvot2o


p24

35 ð322Þ

In contrast to claims (implicit or otherwise) occasionally made in the

literature, Eq. (322) does not show that the slope on an FN plot gives work

function but rather that the slope is proportional to F3=2=bg and the

enhancement factor must be considered, the methods of Groning et al.

being a case study in point (Groning et al., 1999, 2000).

In practice, the application of Eq. (322) is fraught with complications.

At the apex of a field emitter, more than one crystal face can be exposed, and

crystal faces have different work functions; contamination and oxides can

impart features of their own or even reduce the effective emission area;

emission can come off‐axis, whereas the theoretical model above presumes

on‐axis emission from a rotationally symmetric surface; migration of materi-

al can occur; and so on. An extensive study of several of these effects was

done by Dyke et al. (1953) in greater detail than allowed by the present

treatment. Nevertheless, it is of pedagogical value to see if expectations borne

of Eq. (322) are manifest in experimental data.

The variation of work function F can be addressed by considering an

effective, or averaged, work function over the apex. The geometry factor bgis a bit more difficult as it depends on the particulars of the emitting surface.

An approximation, based on the hyperbolic model of a Spindt‐type emitter,

suggests that to leading order, the field enhancement factor is inversely

proportional to tip radius (simple models, such as needles, also give an

128 KEVIN L. JENSEN

inverse relationship to tip radius to leading order). Other factors, such as

cone angle and gate radius, contribute additional factors beyond our scope

here. The simplest approximation is to use a polynomial (quadratic) fit of

abg; based on the hyperbolic model and expanded about a ¼ 10 nm, we use

bg 11:654þ að Þ 59:224 að Þ

2429:1a; ð323Þ

where a is in units of nanometers and bg in units of 1/nanometer (parenthetical-

ly, note that the gate radius and cone angle are implicitly assumed to be 0.5 mmand 15, respectively, in the evaluation of the numerical parameters in

Eq. (323)). Of course, there are higher‐order effects, the neglect of which will

affect, for example, the value of the work function converged on, but these

considerations are ancillary to the present treatment.

Now consider the data from Figure 2 of Schwoebel, Spindt, and Holland

(2003) showing the changes wrought on single‐tip Spindt‐type field emitters

subject to conditioning (reproduced in Figure 52 and recast in Figure 53) in

FN coordinates from which the slope and intercept factors are ascertained.

The ‘‘conditioning’’ entailed controllably heating the field emitter tips by

drawing intense currents; the heat was sufficient to smooth and recrystallize

the apex by surface diffusion, as well as to drive off contaminants by thermal

desorption. Surface self‐diffusion tends to come into equilibrium with

applied field for a particular apex configuration (Barbour et al., 1960;

10−7

10−6

10−5

60 80 100 120 140 160 180

I1FN I1I2FN I2F1FN F1F2FN F2

Cur

rent

[A

mp]

Voltage [V]

FIGURE 52. Preconditioning and post-conditioning current-voltage plots of the emitter tips

examined by Schwoebel et al. (2003). Symbols are experimental data; lines are based on the

theoretical models examined in the text.


Char bonni er, 1998), so that tips cond itioned in such a manne r can be made

more like each other, thereby impr oving emission unifor mity from an array

of su ch emitter s.

Our purpose here, howeve r, is to determ ine whet her such changes are

cap tured in the varia tion of the theo retical model of the slope and intercept

fact ors. The curves labeled ‘‘I1’’ and ‘‘I 2’’ are as fabricat ed, wher eas the

cu rves labeled ‘‘F1’ ’ and ‘‘F2’’ (follow ing the notation of Sch woebel et al. )

are afte r condit ioning . Table 7 shows the slope and fie ld values , from whi ch

the effective radius is determined for a presum ed average work functio n of

4.5 eV. As opposed to inferring both field enh ancement fact or and work

functi on from slope ‐ inter cept data, the work functi on here is he ld at apresu med value, the slope is used to infer the apex radius based on a field

en hancement model, and the intercep t is pred icted. For aforem entione d

reason s, the intercep t shou ld not be exp ected to be ex actly predict ed

(see, for exampl e, For bes, 1999b for a gen eral discussion on the prob lems

associated with inferring emission area and work function from FN

−34

−32

−30

−28

−26

6 8 10 12 14 16 18

FNI1Fit FNI1FNI2Fit FNI2FNF1Fit FNF1FNF2Fit FNF2

ln

I/V

3 [A

/V3 ]

1000/V [Volts]

FIGURE 53. The data of Figure 52 represented on a Fowler–Nordheim Plot.

TABLE 7

FOWLER‐NORDHEIM FACTORS

Curve Slope (Exp) Intercept (Exp) a [A] Slope (Theory) Intercept (Theory)

I1 728.73 17.385 42.9 728.44 17.85

I2 1331.0 18.340 93.8 1331.8 18.10

F1 1580.0 17.659 118.6 1580.0 18.14

F2 1623.7 17.560 123.1 1623.3 18.14

130 KEVIN L. JENSEN

slope‐intercept data). As seen in the table, the general trend is captured and

the data shown to be commensurate with the hypothesis that the tips are in

fact blunting through conditioning to about the same magnitude.

G. Recent Revisions of the Standard Thermal and Field Models

1. The Forbes Approach to the Evaluation of the Elliptical Integrals

Up to this point, the methodologies used to tackle thermal and field emission

in the pursuit of emission equations have not departed significantly in the

reliance on common representations of the elliptical functions v(y) and t(y)

to determine n(F,T) and the onset of the transition region. Recent improve-

ments have enabled a truly general thermal‐field equation beyond the simple

addition of correction terms used by the revised FN and RLD equations that

moreover does not rely on numerical searches for the integrand maximum in

the transition region; they make use of recent advances by Forbes (2006) in

creating extraordinarily convenient and elegant analytical forms of v(y) and

t(y) over the entire range of y. We first explore the development of the

analytical forms and then apply them to the physics of the transition region

of the thermal‐field model. Although the goal is the form given by Forbes, the

path is different and based on series expansion methods.

Recall the defining equation for v(y) introduced in Eq. (260), which

facilitates the development of a series summation that will be particularly

useful and rewrite it as

vðyÞ ¼ 3

8

ffiffiffi2

p1 y2

Sffiffiffiffiffiffiffiffiffiffiffiffiffi1 y2

pn o

S xf g ¼ðp0

sin2yffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ xcosy

p dyð324Þ

The slight rewriting has immediate payoff, as S can be series expanded

to give

S xð Þ ¼ðp=20

1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ xcosy

p þ 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 xcosy

p8<:

9=;sin2ydy

¼X1

n¼01ð Þn 2nð Þ!

22n n!ð Þ2xn

ðp=20

1ð Þnsin2ycosnyþ sin2ycosny

¼ 2X1

n¼0

4nð Þ!24n 2nð Þ!ð Þ2x

2n

ðp=20

sin2ycos2nydy

¼ 1

2pX1

n¼0

4nð Þ!26n 2nð Þ! n!ð Þ nþ 1ð Þ!x

2n

ð325Þ


Initially, this appears to be of little benefit, but a great simplification arises if the

lowest‐order approximation to n! (Stirling’s approximation, or n! nnenffiffiffiffiffiffiffiffi2pn

p)

is used, for which S(x) becomes approximated by So(x) where (note the

changes in the summation limits)

So xð Þ p2þ 1

4

ffiffiffi2

p X1n¼1

x2n

n nþ 1ð Þ

¼ p2þ 1

4

ffiffiffi2

pþ

ffiffiffi2

p1 x2ð Þln 1 x2ð Þ

4x2

ð326Þ

where the integration of the commonly known series expansion for ln(1–s)

with s ¼ x2 has been exploited to convert the summation into a closed

formula. Using So in place of S, it follows that v(y) is approximated to

leading order by

v yð Þ 3

161þ p

ffiffiffi2

p 1 y2 þ 3

8y2ln yð Þ: ð327Þ

As simple as the result appears, it cannot be correct; while v(1) ¼ 0 (as it

should), v(0) does not equal 1, but rather 1.0205, and the problem is traceable

to the ungainly coefficient of 1 y2ð Þ. The consequences of using Stirling’s

approximation have made themselves felt. It was Forbes’ insight, based on

examining the tabulated function and then experimenting on expansions and

summations using the Maple mathematical package (Maplesoft, Waterloo,

Ontario, Canada), that the better approximation is

v yð Þ 1 y2 þ 1

3y2ln yð Þ; ð328Þ

where the coefficient (1/3) was ascertained to be a fairly close fractional

representation of the actual numerical coefficient. The elegant simplicity of

Eq. (328) is breathtaking for those who have squandered countless hours

searching for a good analytical representation: it contains the proper end

points and at its worst is still good to within 0.332% of the numerically

evaluated value (occurring at y ¼ 0.54). Moreover, the form lends itself to

the ready evaluation of t(y) and the interpretation of FN slope factors.

A little effort shows why the (1/3) coefficient is in fact a reasonably good

approximation, how good ‘‘good’’ is, and how close Eq. (328) is to a proper

account of the summation terms. Compared to the ‘‘derivation’’ of Eq. (328),

it is not as appealing and relies on some patience with series expansion

methods (Jensen, 2007).

Introduce a difference function D(x) that varies from 0 to 1 and represents

the difference between S(x) and its approximation So(x). Using the series

form of each, it follows that

132 KEVIN L. JENSEN

DðxÞ SðxÞ SoðxÞð Þ Sð0Þ Soð0Þð ÞSð1Þ Soð1Þð Þ Sð0Þ Soð0Þð Þ ¼ 12

SðxÞ SoðxÞ13

ffiffiffi2

p 6p; ð329Þ

so that

vffiffiffiffiffiffiffiffiffiffiffiffiffi1 x2

p ¼ 3

8

ffiffiffi2

px2 SoðxÞ þ 1

1213

ffiffiffi2

p 6p

DðxÞ

: ð330Þ

It is seen that Eq. (327) results if the term containing D(x) is neglected. Theterm So(x) has explicitly selected the singular term for vanishing x. Therefore,

whatever remains is a rapidly convergent power series in x2, or equivalently,

in y2, that is


p ¼X1n¼1

Anx

2n þ 3

161 x2

ln 1 x2

vðyÞ ¼ ð1 y2Þ 1þX1

n¼1An y2 nn o

þ 3

8y2ln yð Þ

ð331Þ

where the first summation in the top representation starts at n ¼ 1 because

v(1) ¼ 0, and the bottom term in the curly brackets is a consequence of both

regrouping (the reason for the asterisk in the x power series being dropped in

the y power series) and the demand that v(0) ¼ 1; it is precisely that observa-

tion that allows us to bypass the problematic boundary condition at y ¼ 0 if

we choose to truncate the series expansion after a few terms, which is our

intention to obtain the An coefficients of the y‐power series. Therefore,

consider the first handful of terms in the series expansions of v(y) and

So(x) from Eqs. (324)–(326), that is


p 3

16pffiffiffi2

px2 1þ 3

32x2 þ 35

1024x4

0@

1A

So xð Þ p2þ 1

8

ffiffiffi2

px2 1þ 1

3x2 þ 1

6x4

0@

1A

ð332Þ

and insert these into Eq. (330) to determine D(x):

DðxÞ ffiffiffi2

px2 96 3

ffiffiffi2

pp 16

þ 105ffiffiffi2

pp 512

x2 þO x4ð Þ

1024 13ffiffiffi2

p 6p : ð333Þ


Clearly, truncating the series early incurs an error that is increasingly large

when x approaches unity. The affront to a finite series representation of D(x)is minimized, therefore, by appending a correction term only to the coeffi-

cient of the highest power kept. That is, if f(x) is an infinite series and fa(x) is

a finite series approximation to it, or

f xð Þ ¼X1

k¼1akx

k

fa xð Þ ¼Xn

k¼1akx

k þ anþ1xnþ1

ð334Þ

and where both vanish at x ¼ 0 and are unity at x ¼ 1, then

anþ1 ¼X1

j¼nþ1aj ¼ 1

Xn

j¼1aj: ð335Þ

The error of the approximation vanishes at the boundaries and is

f ðxÞ faðxÞ ¼ xnþ1X1

j¼1anþjþ1 1 xj

ð336Þ

in the middle. Demanding that D(1) ¼ 1 determines the correction to the last

coefficient, and so, using Eqs. (334) and (335)

DðxÞ )ffiffiffi2

px2 96 3

ffiffiffi2

pp 16

þ 105ffiffiffi2

pp 512

x2 þ 15 512

ffiffiffi2

pp 512

x4

1024 13

ffiffiffi2

p 6p

ð337Þ

Putting Eq. (337) plus the closed form of Eq. (326) into Eq. (330) and then

collecting terms shows that if the series is truncated at n¼ 3 in Eq. (331), then

A1 ¼ 9897

16384pffiffiffi2

p 85

32 0:02754

A2 ¼ 5145

8192pffiffiffi2

p 89

32 0:009114

A3 ¼ 15

16384pffiffiffi2

p 15

16 0:0021112

ð338Þ

Using these values, it follows that truncating the series at the third term gives

rise to the approximations

134 KEVIN L. JENSEN

vðyÞ 3

8y2lnðyÞ þ 1 y2

1þ y2 A1 A2y

2 þ A3y4

tðyÞ 1

8y2lnðyÞ þ 1þ y2 B1 þ B2y

2 B3y4 þ 13

3A3y

6

8<:

9=;

ð339Þ

where

B1 ¼ 3299

16384pffiffiffi2

pþ 31

32 0:074153

B2 ¼ 33645

16384pffiffiffi2

p 145

16 0:061084

B3 ¼ 41265

16384pffiffiffi2

p 357

32 0:033665

ð340Þ

and where the maximum error of Eq. (339) is 0.029% for v(y) and 0.039% for

t(y). Using this representation, which is designed to be accurate at the

boundaries, it can be shown that the boundaries are (correctly) given by

vð0Þ ¼ tð0Þ ¼ 1

limy!1

vðyÞ1 y2

0@

1A ¼ 3

16pffiffiffi2

p

tð1Þ ¼ 1

4pffiffiffi2

pð341Þ

It must be emphasized that the use of a truncated series in An* to find a

truncated series in An is a procedure that does not provide the exact values of

An (even though for all but AN*, theAn* coefficients are specified exactly) but

rather approximations to these terms. However, approximations are all we

are seeking; namely, we are striving to find a method that steadily improves

commensurate with the level of effort corresponding to the number of terms

retained.

How does this relate to Forbes’ beautiful result? We seek to confirm the

value of C in

v yð Þ 1 y2 þ Cy2ln yð Þ ð342Þ

such that the y¼ 1 limit of Eq. (331) is reproduced in Eq. (342) (field emission

conditions are such that y is generally closer to that boundary). We find


C ¼ limy!1

3

8 1 y2ð Þ

lnðyÞ A1 A2y2 þ A3y

4 8<

:9=;

¼ 3

8pffiffiffi2

pþ 2 0:33392

ð343Þ

where, from L’Hopital’s rule, the y ¼ 1 limit of 1 y2ð Þ=lnðyÞ ¼ 2. There-

fore, the finding that C be approximated by 1/3 is well supported. It follows

that

t yð Þ ¼ v yð Þ 2

3y]yv yð Þ

1

8y2ln yð Þ þ 1þ 1

121 4A1ð Þy2 þ 5

3A1 þ A2ð Þy4

3 A2 þ A3ð Þy6 þ 13

3A3y

8

1

3Cy2ln yð Þ þ 1þ 1

31 2Cð Þy2

ð344Þ

where the second line uses the power series (An) approximation and the third

uses the Forbes‐like (C) approximation. Finally, the choice of C defined by

Eq. (343) entails that the boundaries in Eq. (341) are respected.

Figure 54 shows the performance of both the An and C forms of v(y)

and t(y). Finally, note that what is designated C is actually C(y¼1).

The Forbes‐like equations can be retained with their evident simplicity and

utility by replacing C(1) with C(yo), where yo ¼ffiffiffiffiffiffiffiffiffiffiffi4QFo

p=F and Fo is a

characteristic or midpoint temperature for the experiments under consider-

ation. The behavior of C(y) is shown in Figure 55. However, given that the

Forbes approximation is better than 0.4% for all y, the incentive to do so is

rather weak.

2. Emission in the Thermal‐Field Transition Region Revisited

The punishing analysis to obtain the deceptively simple Forbes‐like represen-tation of the elliptical integral functions v(y) and t(y) serves a purpose: it is

precisely in the y ¼ 1 limit that the transition region from field to thermal

emission occurs. We may now profitably revisit the n ¼ bT/bF approxima-

tions to the thermal‐field equation and deal with the transition region explic-

itly. The behavior of the transition region was the subject of a detailed

investigation by Dolan and Dyke (1954, in their Figure 2), which chronicles

the evolution of the transition region as a function of both temperature and

136 KEVIN L. JENSEN

field, albei t us ing the FN fact ors; the intent here is to impr ove upon the

analys es. Recall the compelling beha vior in Figure 40: the transit ion from

thermal to field emi ssion ‐ like regimes is demarc ated by a regim e in which n is

unity —thou gh not exactly, still to a very goo d app roximati on. Thi s suggest s

the FN slope factor bF (which dep ends on v(y) ) can be used up to thetransiti on region, and the quad ratic barrier slope factor can be used afte r

the trans ition region. Wh at is needed now is an an alytic method to find the

slope fact or withi n the transitio n region for whi ch n ¼ 1 (recall that the

numerical method is to simply crudely sum the cu rrent density integ ral

using any appropri ate num erical algorithm after finding the slope fact or bF

0

0.1

0.2

0.3

0.4

0 0.2 0.4 0.6 0.8 1

% Error v(A)% Error v(C)% Error t(A)% Error t(C)

% e

rror

y

FIGURE 54. Comparison of the Forbes approximation ( C = 1/3) with the polynomial

approximation to C(y) in Eq. (342)

0.33

0.34

0.35

0.36

0.37

0 0.2 0.4 0.6 0.8 1

C(y

)

y

Cquad(y) = 0.368 − 0.049y + 0.015y2

C(y)QuadC(1)

FIGURE 55. Comparison of the numerical, polynomial fit, and constant (1/3)

representations of C(y).


and expansion point Eo using numerical search algorithms). What is happen-

ing while n ¼ 1 is that the peak of the current density integrand as a function

of energy is migrating from the Fermi level to the barrier maximum level (see,

for example, Figure 8 of Murphy and Good, 1956), and in that simple

observation, a solution is suggested.

The expansion of AUC term y(E) constitutes the starting point, where it

was observed that in the transition region between the FN and quadratic

barrier forms, the slope factor bF appeared to be well approximated by a

quadratic in energy. Using a standardized notation, a polynomial approxi-

mation in the difference between energy and Fermi level for y(mþxf)of the form

ya mþ xfð Þ Ba þ CaxþDax2 þ Eax

3 ð345Þis sought in such a way that the two linear approximations of primary

interest, namely, the linear FN given by

yfn mþ xfð Þ ¼ BFN CFNfx

BFN ¼ 4

3hF


pv yð Þ

CFN ¼ 4fhF


pt yð Þ

ð346Þ

and quadratic barrier approximations given by

yq mþ xfð Þ ¼ Bq Cqx

Bq ¼ Cq ¼ phfffiffiffiffiffiffiffi2m

p Q

F 3

0@

1A1=4

ð347Þ

where y ¼ ffiffiffiffiffiffiffiffiffiffi4QF

p=F and f ¼ 1 yð ÞF, shall determine the coefficients in

Eq. (345). Note explicitly that x is dimensionless, its value at 0 designating an

energy at the chemical potential, and its value at 1 designating the barrier

maximum. Demand that y(E) be approximated by the fn‐form for E < m, bythe q‐form for E > m þ F, and by ya in the intermediate region, where the

value of ya and its first derivative are continuous at the boundaries with

the linear forms. We find for x ¼ ðE mÞ=f that

ya mþ xfð Þ ¼ BFN CFNxþ x2 CFN Cq

2 xð Þ BFN Bq

3 2xð Þ

ð348Þ

138 KEVIN L. JENSEN

and for the slope factor

bF Eð Þ ]

] E y Eð Þ

1

fBq x þ C FN 1 xð Þ þ 3 2BFN B q CFN

x 1 xð Þ

ð349ÞThe trans ition region app roximati on shall be invo ked when the integ rand

maximu m expansi on point lies between m and m þ f , at which poin t n will betaken as identi cally 1 and the integ rand exp ansion point x ¼ xc determined

the cond ition bF E ð xc Þð Þ ¼ bT (anothe r way of saying n is identi cally equ al tounity) and for whi ch

Eo ¼ m þ 2F Co

Bo þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiB2o þ 4A o C o

p0@

1A

Ao ¼ 6B FN 3B q 3C FN

Bo ¼ 6BFN þ 2Bq þ 4CFN

Co ¼ bt F þ C FN

ð350Þ

As an exampl e, consider copperli ke pa rameters (m ¼ 7.0 eV, F ¼ 4 .5 eV)

under condition s of a field of 1 GV/m and a temperatur e of 723 K,

and assuming the Forb es approxim ation for v(y) and t(y) : then Ec –

m ¼ 2.27 eV, or xc ¼ 0.687.

A key limit ation of the revis ed FN and RLD equati ons ha s now been

overcome , namel y, the specifica tion of the Eo pa rameter by other than

numeri cal means. We are now in a posit ion of evaluating bF an d E o wi thoutrelying solely on the FN linear approximation which, for mixed thermal‐fieldconditions, was unsatisfactory. We therefore turn to the development of a

truly general thermal‐field equation.

H. The General Thermal‐Field Equation

As shown by Figure 40, to a g ood approx imation, n ¼ 1 in the trans ition

region between the thermal and field regimes. A reasonable approximation

can then be made by taking n to be equal to 1 when the temperature falls

within a critical region that occurs when T is larger than the FN‐like temper-

ature TFN yet smaller than the RLD‐like temperature TRLD, both of which

are obtained by finding the equivalent temperatures corresponding to the

slope factors. In other words, n ¼ 1 when


TFN 1

kBbF mð Þ T 1

kBbF mþ fð Þ TRLD: ð351Þ

For completeness, when temperatures are above TRLD or below TFN, then

Eo T > TRLDð Þ ¼ mþ f

Eo T < TFNð Þ ¼ mþ 2vðyÞ3tðyÞ F

ð352Þ

as before, but when bF Emð Þ bT (a relation that serves to define Em in the

transition region), then

Eo Em þ y Emð ÞbF Emð Þ ; ð353Þ

where the boundary cases for field (Em ¼ m) and thermal (Em ¼ m þ f) inEq. (353) agree with Eq. (352). In turn, s(Em) is always given by

s F ;Tð Þ bF Emð Þ Eo Emð Þ mf g ð354Þfrom which it can be shown that s(TRLD) ¼ Bq and s(TFN) ¼ BFN. The

remaining terms still require a tractable form in order to obtain a truly

general thermal‐field equation. Starting from the form of N(n,s)

N n; sð Þ ¼ S1

n

es þ S nð Þens ð355Þ

[where the expressions of J(F,T), JT, and JF have not changed from their

forms given in Eq. (299)], recall that S(x) can be written as

S xð Þ ¼ 1þX1

j¼11 212j

z 2jð Þx2j: ð356Þ

Such a form can be cumbersome; a reasonable approximation is given by

S xð Þ 1

1 x x 1þ xð Þ þ 1

4x3 7x 3ð Þ þ zð2Þx2 1 x2

; ð357Þ

where, as done above on other infinite series, the highest‐order term is

amended by the next‐order term to respect boundary conditions (in the

present case, making the nonsingular part of S(1) unity). The singular parts

of Eq. (355) at n ¼ 1 cancels, and the remaining terms are well behaved.

Finally, the Forbes approximation is used to revisit the power law depen-

dence of n on F, which can be rewritten as

n Fð Þ ¼ n Foð Þ F

Fo

p

; ð358Þ

140 KEVIN L. JENSEN

where Fo is a refer ence field and p is a power . As shown in Eq. (307) , ptherm ¼3/4. It is the form of pfield that was num erically found in Eq. (308) but for

which the For bes approxim ation now allow s an a nalytical e xpression to be

ascertained. Introduc e y 2o ¼ 4QF o =F2 . It then foll ows from Eq. (358) ,

the field ‐ regime term bF( m), constant temperatur e, and the Forb es e xpression

for t(y ) that for F close to Fo we discove r

pfield ¼ 18 þ y2o18 t yoð Þ ¼

18 þ y2o18 þ 2y2o 1 ln yoð Þð Þ : ð359Þ

Unfortuna tely, Eq. (359) con tinues to rely on a refer ence fie ld Fo.

Howev er, a graph of pfield exh ibits a minimum at yo ¼ 0.9740 for which pfield¼ 0.950, close to the num erical value discove red previou sly. In practi ce,

pfield ¼ 0.950 remai ns a goo d app roximati on to p for n > 1.To summ arize the app roximati on, the be havior implied by increa sing the

field mono tonically from low values to high is that initially, the location of

the current den sity integ rand maxi mum is at the barri er maximu m as F

increa ses a nd n increa ses. When n ! 1 (the tempe rature is equal to T RLD),

the locat ion of the integ rand maxi mum be gins to migrate from the barri er

maximu m to the chemica l poten tial, caused by the smoo th change of the fie ld

slope factor from qua dratic barri er–like to FN ‐ like. Whe n n ! 1þ , then thelocation of the current density integ rand maxi mum takes root at the chemi cal

potenti al and n increa ses to large r values . Figure 5 6a sho ws the pe rformance

for copperli ke parame ters with the tempe rature held at cold (300 K) or hot

(1500 K) con ditions, indicating that the RLD and FN appro ximations work

rather well in their respect ive regimes. The perfor mance in the inter media te

regime is obfuscat ed on a log ‐ log plot, so Figu re 56b shows the ratio of theFN and RLD current s with the therma l‐ field model, showing how wel l each

equati on performs in the transitio n regim e. Switchi ng conditi ons to cesium

on tungsten cathode ‐ like co ndition s but for intermedi ate tempe ratur es and

fields (and in pa rticular , for a work func tion of 2.0 eV, slightl y high er than

the 1.8 eV suggested earlier, simply for effect), the ratio comparison in

Figure 56c sh ows the de gree to which the FN an d RLD mod els de part.

The resultant general thermal‐field emission equation for which the two

equations FN and RLD are shown to be limiting cases has been constructed

and works for arbitrary n, even in the transition region specified by n ¼1. By

formulating the theory in this manner, the present formulation allows for the

determination of the effects of temperature on field emission as well as fields

on thermal emission. The ability to unify the equations was a consequence of

developing a good approximation to y(E) that smoothly transitioned from

below the barrier to above it. We shall see below that the example of

photoemission benefits by extending the analysis begun here.


−8

−4

0

4

8(a)

(b)

(c)

−3 −2.5 −2 −1.5 −1 −0.5 0

JTF(300 K)

JTF(1500 K)

JRLD(1500 K)

JFNlog 1

0(J

[A/c

m2 ])

log10(field [eV/Å])

Cu-likem = 7 eV, Φ = 4.5 eV

0.1

1

10

0.001 0.01 0.1 1

Rat

io o

f cu

rren

t

Field [eV/nm]

1500 K

300 K

Cu-likem = 7 eV, Φ = 4.5 eV

JRLD/JTF

JFN/JTF

R = 1

0.1

1

10

100

0.02 0.04 0.06 0.08 0.1

JTF/JRLD

JTF/JFN

R = 1

Rat

io o

f cu

rren

t

Field [eV/nm]

1500 K

300 K

Cs-on-Cu-likem = 7 eV, Φ= 2 eV

FIGURE 56. (a) Comparison of the thermal-field equation (JTF) with both the Richardson–

Laue–Dushman equation (JRLD) and the Fowler–Nordheim equation (JFN) for copperlike con-

ditions for 300K and 1500K.. (b) Ratio of the thermal-Field equation (JTF) with both the

142 KEVIN L. JENSEN

I. Thermal Emittance

Consider a symmetrical beam of electrons and let the symmetry axis be z. If

all electrons had their velocities wholly in the direction of the symmetry axis

(that is, if k ¼ kz), then the beam would not diverge as it propagates.

Electrons, however, are always emitted on average with some perpendicular

velocity component kr, and as the beam moves in the forward direction,

these electrons find themselves farther and farther away from the symmetry

axis. If no forces complicate matters, then the ratio of the spread of the beam

to the distance the beam has propagated goes as x0 ¼ dx=dz dkx=dkz,

where the last relationship assumes that the axial velocity dominates the

radial velocity. If every particle was tagged by a pair of coordinates x; x0ð Þ

and these points plotted on the axes x and x0, then the area that encompassed

all of the points—that is, the trace space defined by

Ax ðð

dxdx0 ð360Þ

provides a measure of the quality of the beam. Problems inherent with a

trace‐space definition of ‘‘emittance’’ are discussed more fully by Reiser

(1994), even though the quantity is commonplace in the literature, but for

ideal beams with linear focusing fields, the relationship between the rms

emittance ~ex (rms ¼ root‐mean‐squared and relates to the statistics of the

distribution of points; see below) and the trace space of Eq. (361) is

Ax ¼ 4pex: ð361Þ

As seen in Eq. (360), the units of emittance are a bit odd on first encounter;

while x has units of length, x0 does not—rather, it has units of radians. In the

community of electron sources, a commonly used unit is 106 meter‐radians,or, as it is more often encountered, mm‐mrad. Although ‘‘microns’’ are

also used, such a designation obscures the angular nature inherent in the

definition of emittance.

Another measure of the quality of a beam is the total beam current for a

given emittance, which can be shown to be related to the current density

for a given solid angle. Brightness is therefore defined as

Richardson–Laue–Dushman equation (JRLD) and the Fowler–Nordheim equation (JFN) for

copperlike conditions. (c) Ratio of the thermal-field equation (JTF) with both the Richardson–

Laue–Dushman equation (JRLD) and the Fowler–Nordheim equation (JFN) for cesium on

copperlike conditions.


B ¼ J=dO: ð362ÞFor idealized particle distributions whose trace‐space is confined by a

hyperellipsoid (an ellipsoid in four dimensions (x,y,x0,y0)), it can be shown

that the average brightness is given by

hBi ¼ 2I

p2exeyð363Þ

and therefore has the units of A/(mm‐mrad)2.

As seen previously, the distribution function approach leads naturally to a

continuity equation, ]trþrJ ¼ 0. Generalizing, if the current is repre-

sented as the product of a (six‐dimensional) density in phase space and a

velocity, then

]trþ vrr ¼ drdt

¼ 0 ð364Þ

if the number of particles dN in a small region dV is not changing, then

dN ¼ rdV (again, it is emphasized that dV is a small volume in phase space

and therefore six‐dimensional). Thus,

d

dtdN ¼ dr

dt

0@

1AdV þ r

d

dtdV

0@

1A

¼ rd

dtdV

0@

1A ¼ 0

ð365Þ

where the second line follows as a consequence of Eq. (364). We conclude

d

dtdV ¼ d

dt

ððdxdk ¼ 0: ð366Þ

That is, the volume of a given number of particles in phase space is invariant,

a conclusion known as Liouville’s theorem (see Reiser, 1994, for a discussion).

Insofar as coupling does not occur between motion in the various directions,

the finding of the invariance of the phase space volume is equivalent

to invariance of its projections on various pairs of axes such as dxdkx, and

so it is found that trace‐space area is conserved. By extension, this

has bearing on the behavior of the emittance as per the relationship between

Ax and ex.Cathodes for advanced accelerators and advanced linear accelerator

(LINAC)‐based light sources, vacuum electronic devices, high‐energy phys-

ics, and the like are responsible for generating well‐collimated beams as the

consequences of errant electrons outside the intended path lead to very

144 KEVIN L. JENSEN

undesirable results (Bohn and Sideris, 2003; O’Shea, 1995)—stray electrons

from a high‐energy beam still have a strong negative impact on whatever

they strike. Intrinsic emittance, that is, emittance originating with the

photocathode, is important because what is generated there cannot be

compensated for by subsequent beam optics.

Emittance e appears in the envelope equation (Reiser, 1994; Serafini and

Rosenzweig, 1997) as a parameter governing the evolution of the beam

radius (r)

r00 þ k2or2

r

I

Io

1

bgð Þ3 e2

r3¼ 0; ð367Þ

where ko is the betatron wave number of the focusing fields, b ¼ vz=c, andg ¼ 1 b2

1=2are the dimensionless velocity and relativistic correction

factor, Io ¼ 4pe0mc3=q ¼ 17045 A is a characteristic current. Emittance is

related to the ‘‘moments’’ hx2i and hx02i, where x0 ¼ dkx=dkz where hk=m is

the velocity of the particle, kx being the conjugate variable to x. The related

rms emittance is defined by erms ¼ e/4 for a uniform beam. A beam without

emittance may propagate with pencil‐like straightness, whereas when emit-

tance is present, the beam can diverge and the extent to which it diverges over

a given propagation distance is a measure of the transverse velocity compo-

nents. Brightness is also affected, and so a normalized brightness is also

defined in terms of the normalized emittance as

Bn B

bgð Þ2 ¼2I

p2e2n: ð368Þ

When the particle velocity is small or when the transverse and perpendicular

components are comparable (i.e., near the cathode), then using x0 posesproblems so the definition used here is (O’Shea, 1998)

en;rmsðzÞ ¼ h

mc

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffihx2ihk2xi hxkxi2

q: ð369Þ

Moments are defined according to

hOi

ðdrdkO r; kð Þf r; kð Þð

drdk f r; kð Þ;ð370Þ

where f is the distribution of emitted particles. Attention shall be restricted to

axisymmetric beams for which hxkxi ¼ 0. In the case of thermionic emission,

only those electrons whose energy exceeds the barrier height (m þ f) may be

emitted and so


f r; kð Þ ¼ y r rcð Þy EðkzÞ m Fð ÞfFD EðkÞð Þ; ð371Þ

where EðkÞ ¼ h2jkj2=2m, EðkzÞ ¼ h2k2z=2m, rc is the radius of the cathode

(cylindrical coordinates), y is the Heaviside step function, and fFD is

the FD distribution. The symmetry of the distribution results in

hx2i ¼ hr2i=2 ¼ r2c=2. For typical work functions, the distribution function

for energies above the barrier height is well approximated by a MB distri-

bution, and so the moment for momentum is equally straightforward.

Minimal effort shows that

hk2xi ¼ hk2ri=2

ð10

exp bTh2k2r=2m

n ok3rdkr

2

ð10

exp bTh2k2r=2m

n okrdkr

¼ m

bTh2; ð372Þ

yielding the oft‐quoted result that the emittance of a thermionic cathode is

en;rmsðthermalÞ ¼ rc4bTmc2ð Þ1=2

: ð373Þ

A numerical example is to consider a cathode 0.5 cm in radius and at

1300 K. Eq. (373) then indicates that the emittance is 1.171 mm‐mrad.

Two points merit attention, as the question of emittance is considered

afresh in the treatment of photoemission in the effort to derive an equation

of comparable simplicity to Eq. (373). First, if ‘‘moments’’ of the distribution

function are defined by

Mn /ðknrf EðkÞð Þdk; ð374Þ

then Eq. (372) proportional to M2=2M0. Second, the replacement of the FD

by the MB distribution is crucial to facilitate the stunning ease by which

Eq. (372) is obtained. In a more general circumstance, such as in photoemis-

sion, the convenience entailed by the MB distribution will be of no avail.

Conversely, emittance for field emission is so significantly complicated by

questions of field variation over sharpened emitter structures and the change

of field lines with emitted charge that the evaluation of emittance for such

structures is a question of considerable complexity (Jensen et al., 1996, 1997)

and is not considered further here.

146 KEVIN L. JENSEN

III. PHOTOEMISSION

A. Background

The explanation of the photoelectric effect in terms of quanta liberating

electrons from the surface of a metal earned Albert Einstein the Nobel

Prize in 1921. As interesting as the liberation of a few electrons is, the

liberation of many electrons complicates the physics significantly and affects

the transition of that physics into technology. The approach in this section

continues its focus on electron emission and current density; thus the treat-

ment of photoemission is considered in that light.

Photocathodes are excellent sources for the production of short bunches

of electron beams for injection into radiofrequency (RF) LINACs, free elec-

tron lasers, and related devices (Nation et al., 1999; Rao et al., 2006). While

requirements vary, the European Organization for Nuclear Research

(CERN) (linear collider) test accelerator is a measure of the state of the art.

It uses a Cs2Te photocathode illuminated with a 262‐nm yttrium‐lithium‐fluoride (YLF) laser to generate electron bunches postaccelerated to 50 MeV

containing 30 nC per bunch at a modulation frequency of 3 GHz. As such, its

nominal characteristics are in interesting contrast to field and thermionic

technologies; the fields at the surface are an order of magnitude greater than

thermionic sources but two orders smaller than field emission sources, yet its

average current is 10 A and the peak current substantially higher. Other

photocathodes in use at, for example, at the Stanford Linear Accelerator

(SLAC), Thomas Jefferson Lab National Accelerator Facility (JLAB), the

ELETTRA Synchrotron Light Source in Trieste, and the German Electron

Synchrotron (DESY), make related demands, although the details differ

depending on the circumstance (mostly in charge per bunch and repetition

rate). What is demanded of photocathodes modifies what merits discussion:

demand much, and interesting physics is thereby revealed. Representative

numbers that drive much of the following text (but do not represent a

realized achievement) are 1 nC per bunch produced in 10 ps every 1 ns

from a 1‐cm2 area corresponds to peak current density of 1 kA/cm2, and an

average current density of 1 A/cm2. If such numbers were realized, then a

megawatt (MW) class free electron laser (FEL) would be potentially brought

to realization, so these are, in fact, numbers of interest.

Intense current densities from sub–square‐centimeter regions are not

uncommon, so much so that space charge effects within the bunch can affect

its dynamics in nontrivial ways (Dowell et al., 1997; Harris, Neumann, and

O’Shea, 2006). For comparison, using typical numbers suggested by Dowell

et al. for a current and current density of 77 A and 530 A/cm2, respectively, a


pancake bunch containing 2 nC from an area 0.145 cm2 produces a local field

of approximately (2 nC)/(0.145 cm2)2e0 ¼ 7.8 MV/m, which is sufficient to

affect internal structure and adjacent bunches. At extraordinarily high laser

intensities, multiphoton effects are revealed, wherein the quantum efficiency

depends on higher powers of laser intensity than simply a linear relation;

further, the electron gas can be brought to such temperatures so quickly that

thermionic emission results even as the electron gas temperature decouples

from that of the lattice (Girardeau‐Montaut et al., 1993, 1994, 1996;

Girardeau‐Montaut and Girardeau‐Montaut, 1995; Logothetis and Hartman,

1969; Papadogiannis and Moustaizis, 2001; Papadogiannis, Moustaizis,

and Girardeau‐Montaut, 1997; Papadogiannis, Moustaizis, Loukakos, and

Kalpouzos, 1997; Papadogiannis et al., 2002; Riffe, Wertheim, and Citrin,

1990; Riffe et al., 1993; Tomas, Vinet, and Girardeau‐Montaut, 1999).

B. Quantum Efficiency

The ability to liberate electrons for a given laser intensity is measured by

quantum efficiency (QE). Various materials commend themselves for differ-

ent reasons. Metal photocathodes are rugged and prompt emitters and can

produce very short bunches but require higher‐intensity lasers to do so as

their QEs are on the order of 0.001–0.01% (Papadogiannis, Moustaizis, and

Girardeau‐Montaut, 1997; Srinivasan‐Rao, Fischer, and Tsang, 1991, 1995).

Semiconductor photocathodes such as GaAs require much lower‐intensitydrive lasers and can produce polarized electron bunches, but they generally

require better vacuum conditions since they are more fragile (Aleksandrov

et al., 1995; Maruyama et al., 1989). Direct bandgap p‐type semiconductors

(alkali antimonides and alkali tellurides; Michelato, 1997; Spicer, 1958;

Spicer and Herrera‐Gomez, 1993), and bulk III‐V with cesium and oxidant

(Maruyama et al., 1989) have high QEs on the order of 30% but are chemically

reactive and easily poisoned, damaged by back ion bombardment (Sinclair,

1999), and for Negative Electron Affinity (NEA) III‐V photocathodes, which

have excellent QE, have a long response time of tens of picoseconds (Table 8).

The required drive laser intensity is related to the number of electrons that

can be liberated for a given number of incident photons on the surface of a

material. (The speed with which lasers can be turned on and off coupled with

a fast‐response photocathode enables the generation of bunches of electrons

with a short spatial extent that is unavailable by other means, thereby

explaining the strong interest of the technology, for example, in the accelera-

tor community when RF photoinjectors are used; Michelato, 1997; O’Shea

et al., 1993; Travier, 1994). An electron absorbing a photon will be raised

in energy by an amount ho. If ho > F, then the electron has a nontrivial

148 KEVIN L. JENSEN

TABLE 8

PHOTOCATHODE‐DRIVE LASER COMBINATIONS*

Material n l (nm) Efficiency [%] QE (%) Lifetime

Time

response

Vacuum

tolerance

Power at

photo‐cathode(W/cm2)

1.06‐mm drive

laser power

(W/cm2)

K2CsSb 2 532 50 8 4 hours Prompt Poor 44 88

Cs2Te 4 266 10 5 >100 hours Prompt Very good 140 1401

GaAs 2 532 50 5 58 hours <40 ps Poor 70 140

Cu 4 266 10 1.4 102 >1 year Prompt Excellent 50043 500430

Mg 4 266 10 6.2 102 >1 year Prompt Excellent 11300 113000

Goal 3 355 30 1 kHr 0 Prompt Excellent 525 1752

*Power on the photocathode at a specified drive laser frequency required to produce 1 nC from a photocathode area of 0.125 cm2 in 50 ps.

ELECTRON

EMISSIO

NPHYSIC

S149

probability of escape. The number of photons is the ratio of the amount of

incident energy with the photon energy, or No ¼ DE=ho, whereas the num-

ber of electrons is the ratio of the emitted charge with the electron unit

charge, or Ne ¼ DQ=q. The QE is then the ratio of the number of emitted

electrons with the number of incident photons, or

QE Ne

No¼ ho

q

DQDE

: ð375Þ

Metal photocathodes, being prompt, produce electron pulses that follow

the light pulse, and the emission area is equal (or nearly so) to the illumination

area. It follows that DE ¼ IoADt and DQ ¼ JeADt, where Io and Je are the

illumination intensity of the incident light and the resulting current density,

respectively, and A and Dt are the characteristic areas and pulse times. If the

incident intensity is measured in watts per square centimeter and the current

density in amps per square centimeter, then a convenient relation is

QE ) hoq

Je

Io

¼ 1:2398

Je½A=cm2l½mmIo½W=cm2 ð376Þ

The photon energy must exceed the work function, implying that ultravio-

let (UV) light is needed (e.g., the fourth harmonic of a neodymium‐yttrium‐aluminum‐garnet [Nd:YAG] laser, for which l ¼ 266 nm, corresponds

to ho ¼ 4:661eVÞ. The laser intensity required to obtain 1 A/cm2 from a

photocathode with QE ¼ 0.01% is then 46.61 kW/cm2. A variety of issues

are associated with such intensities. The ‘‘interesting physics’’ suggested by

such conditions is twofold: (1) can the QE be predicted (and how), and

(2) what impact do large laser intensities have?

Regarding the prediction of QE, Spicer suggested a three‐step model

(Berglund and Spicer, 1964b; Spicer, 1960; Spicer and Herrera‐Gomez,

1993) based on three events: (1) photon penetration and absorption, (2) elec-

tron excitation and transport to the surface, and (3) electron emission over

the surface barrier. Spicer’s focus was principally on semiconductors (later

compression of the electron beam can generate short pulses so that the higher

QE of semiconductors can be profitably exploited), whereas the present dis-

cussion is on metals; the paradigm nevertheless is still useful. Until the

discussion of the moments‐based formulation, the QE is therefore related

to a product of factors—one accounting for absorption of light (leading to

the treatment of reflectivity and penetration depth); one accounting for the

probability of emission (leading to the notion of an escape cone, which will be

replaced subsequently by a combination of the Fowler–Dubridge model

and transport models); and one accounting for losses due to scattering during

transport to the surface (leading to a model of a scattering loss factor).

150 KEVIN L. JENSEN

C. The Probability of Emission

1. The Escape Cone

Under the assumption that photoexcited electrons should be isotropically

distributed, the QE then depends on what fraction of photoexcited electrons

are optimally directed so as to surmount the surface barrier, thereby intro-

ducing the concept of an ‘‘escape cone’’ (mentioned here as such parlance

appears in the literature), though other methods not beholden to the concept

are adopted below. Fields that exist on the surface of a photocathode are

typically on the order of 10–100 MV/m (for RF photoinjectors, which can

support higher fields). As seen in the discussion of thermionic emission, fields

of such magnitude preclude a tunneling contribution to the emitted current;

therefore, it is sufficient to assume that the transmission probability is

governed by the Richardson approximation, in which the electrons escape

only if their momentum component directed at the surface (kx) is higher than

the momentum corresponding to the barrier height, which, for a quadratic

relationship between energy and momentum results in

kx >

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2m

h2mþ fð Þ

r ko: ð377Þ

From the relation kx ¼ k cos(y), where y is the polar angle coordinate, the

fraction fe of electrons that escape from the surface is given by (where Y is

the Heaviside step function)

fe Eð Þ ¼ÐOT kxð Þk2dOÐ

Ok2dO

¼ 1

4p

ðp0

sinðyÞdyð2p0

djY k cosðyÞ koð Þ

¼ 1

21 ko

k

0@

1A ¼ 1

21

ffiffiffiffiffiffiffiffiffiffiffiffimþ fE

s0@

1A ð378Þ

The integral over fe(E) (not to be confused with the supply function) is

proportional to the QE, and for a zero‐temperature electron gas

QE /ðmþho

mþf

fe Eð ÞdE ¼ mþ 1

2hoþ fð Þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffimþ fð Þ mþ hoð Þ

p: ð379Þ

An expansion of Eq. (379) shows QE / ho fð Þ2. The fact that not all

photoexcited electrons make it to the surface because their mean free path

(distance between collision events) is less than their distance to the surface

is considered separately when the impact of scattering is analyzed in greater

detail.


2. The Fowler–Dubridge Model

The dependence of QE on photon energy and barrier height was uncovered by

Fowler (1931), augmented by Dubridge (1933), and enjoys wide use (Bechtel,

Smith, and Bloembergen, 1977; Girardeau‐Montaut and Girardeau‐Montaut,

1995; Jensen et al., 2003b; Jensen, Feldman, and O’Shea, 2005; Papadogiannis,

Moustaizis, Loukakos, andKalpouzos, 1997; Riffe et al., 1993). It can be easily

understood in the context of the 1D supply function–transmission coefficient

model familiar from the escape cone analysis, and as indicated in that model, it

relies on the approximation that the effect of the photon energy is to raiseEx by

an amount ho (all the photon energy is directed at the surface). Such a

conjecture, on the face of it, is overreaching, but as Fowler noted it is surpris-

ingly effective in explaining experimental data and capturing its qualitative

dependence, as follows from the limit of Eq. (379); for photon energies near the

barrier height, unlessEx is augmented by themajority of the photon energy, the

electron is unlikely to be emitted.

The principal effect of augmenting Ex, then, is to make transmission more

likely, so thatT(E) in the current density integral is replaced withT E þ hoð Þ.The probability of emission is then a ratio of the current density emitted with

the incident current density on the surface barrier. Electrons with an energy

ho below the Fermi level are unlikely to find their final state unoccupied and

hence cannot make the transition. It follows that the probability of emission

should then resemble

P hoð Þ Ð1mho T E þ hoð Þ f Eð ÞdEÐ1

0f Eð ÞdE ¼ U b ho fð Þ½

U bm½ ; ð380Þ

where the Richardson (thermionic) approximation to T(E) is used, and where

the Fowler–Dubridge function U(x) has been introduced and is defined by

U xð Þ ¼ Ð x1 ln 1þ eyð Þdy¼ 1

2x2 þ 2U 0ð Þ U xð Þ ð381Þ

Aspecial case isU(0)¼ z(2)¼ p2/12, where z is theRiemann zeta function.

For negative argument, the log function can be series expanded to give

U xð Þ ¼X1j¼1

1ð Þ jþ1

j2exp jxð Þ; ð382Þ

which is useful for large |x|. As observed in the treatment of the General

Thermal Field Equation, for small |x| an approximate form good to better

than 1% is

U xð Þ ex 1 beaxð Þ; ð383Þ

152 KEVIN L. JENSEN

where a and b are found by demanding that Eqs. (381) and (383) agree for

U(x) and dU(x)/dx at x ¼ 0, or

a ¼ 1 lnð2Þð Þ= 1Uð0Þð Þ ¼ 1:7284b ¼ Uð0Þ 1ð Þ ¼ 0:17753

ð384Þ

The approximation given by Eq. (383) is shown in Figure 57, the relation

for positive argument being trivially obtained by Eq. (381). To leading

order, then, when the photon energy is in excess of the barrier height, the

probability of escape becomes

U ½b ho fð ÞU bmð Þ 6 ho fð Þ2 þ pkBTð Þ2

6m2 þ pkBTð Þ2 : ð385Þ

As is often the case for metals under UV illumination, the difference

between the photon energy and the barrier height term in Eq. (385) signifi-

cantly exceeds the thermal term, and so the common observation that

QE / ho fð Þ2 results. When the photon energy, however, is comparable

to the barrier height, then the thermal term makes its presence known

(Figure 58), but clearly, the analytical approximation based on Eq. (385) is

good for photon energies almost to the barrier height for moderate (e.g.,

room) temperatures and lower. For a metal like copper subject to a field of

10 MV/m and with incident 266‐nm laser light, the probability of emission

suggested by Eq. (380) is 0.0714%, which is larger than reported values of QE

for copper (Dowell et al., 2006; Srinivasan‐Rao, Fischer, and Tsang, 1991)—

there is more physics in play, and we now turn to the other contributions.

100

10–1

10–2

0 1 2 3 4 5

U(x)Approx

U(−

x)

x

FIGURE 57. Comparison of the numerically calculated Fowler–Dubridge function with its

analytical appoximation for negative argument [see Eq. (381) for positive argument].


D. Reflection and Penetration Depth

1. Dielectric Constant, Index of Refraction, and Reflectivity

The optical properties of solids are thoroughly discussed elsewhere (e.g.,

chapter 8 in Ziman, 1985, and chapter 6 in Marion and Heald, 1980). The

present concern is with the degree to which light is reflected from a surface

and the extent to which it penetrates into a metal. The electric field compo-

nent E of an electromagnetic wave satisfies the propagating wave equation

with dissipation derivable from Maxwell’s equations:

= E ¼ mo@tH=H ¼ eo@tEþ J

) =2 1

c2

@

@t

2( )

E ¼ 1

eoc2@

@tJ; ð386Þ

where it is assumed that there is no spatial variation in electron density, eoand mo are the electric permittivity and magnetic permeability, moeo ¼ c2, and

some vector identities and the other Maxwell equations have been surrepti-

tiously used. If the material exhibits magnetic or polar characteristics, the

situation is slightly more complicated, but such complications are ignored in

the present analysis. The relation between current J and electric field E is

given by J ¼ sE so that

=2 1

c2

@

@t

2( )

E ¼ seoc2

@E

@t: ð387Þ

10–2

10–3

10–4

10–5

260 270 280 290

NumericalAnalyticQuadratic

Pro

babi

lity

of e

mission

Wavelength [nm]

Copper @ 500 K and 50 MV/m

FIGURE 58. Comparison of the numerical evaluation of the Fowler–Dubrdige function with

the analytic and quadratic approximations.

154 KEVIN L. JENSEN

Taking E to be given by Eo exp i K r otð Þf g, where K is the propagation

constant and o is the frequency, then

K2 þ oc

2¼ i

soeoc2

: ð388Þ

The complex refractive index n (the caret denoting a complex quantity) is

then defined by

K ¼ 1þ isoeo

1=2 oc

n

oc

: ð389Þ

In free space (s¼ 0), the familiar relation c¼o/K follows, but the presence

of resistance (inverse conductivity) implies a dampening due to the imaginary

part of the complex refractive index n nþ ik.

An electron accelerated by an electric field over a distance L ¼ v dt

increases its energy by (qEv)dt. For a density r of electrons, the power

absorbed from the electromagnetic wave heating the conductor is given by

qrvE ¼ JE ¼ sE2 for normal incidence and electron motion parallel to the

field. It therefore follows that the length scale d characteristic of power

absorption is

@z ln EðzÞj j2h in o1

¼ c=2ko ¼ l=4pk d; ð390Þ

a quantity known as the penetration depth. (A word of notational caution: in

contrast to past notation, k is the imaginary part of n, and not a momentum

term, here.)

Regarding how much power actually enters the metal, for simplicity,

consider normal incidence (off‐angle incidence is a staple of textbooks and

readily found elsewhere). For the electromagnetic wave, the amplitudes of

the electric and magnetic components must be equal at the interface. If the

electric field is E ¼ iEo exp i Kz otð Þf g (where i is the unit vector along

the x‐axis), then from Maxwell’s equations H ¼ j coK

Eo exp i Kz otð Þf g.The equations relating the amplitude of E andH become, in matrix notation

(recall the quantum tunneling problems for which the present problem bears

a passing similarity),

1 1

Kinc Kinc

Einco

Ereflo

¼ 1 1

Ktrans Ktrans

Etranso

0

; ð391Þ

where the superscript denotes whether the wave is incident, transmitted, or

reflected. Because the incident (or LHS) medium is assumed to be vacuum,

Kinc is real; similarly, because the RHS is a metal, Ktrans is complex, as is


indicated by the caret. These expressions allow Eorefl to be expressed in terms

of Eoinc, and it is found

Ereflo ¼ Ktrans Kinc

Ktrans þ KincEinco : ð392Þ

The reflectivity R is the ratio of the absolute square of the magnitudes of

the reflected with the incident wave, and so

R ¼ jK trans Kincj2jK trans þ Kincj2 ¼

jn 1j2jnþ 1j2 ¼

n 1ð Þ2 þ k2

n 1ð Þ2 þ k2:ð393Þ

Thus, the two parameters that govern how much light is absorbed by a

material, the penetration depth d and the reflectivity R, can be ascertained

from the complex index of refraction Eq. (393). The question then becomes

how the values of n and k are ascertained, which in turn is related to the

question of how s is determined.

2. Drude Model: Classical Approach

When an electric field is maintained across a material, such as when the gap

between a capacitor is filled with a dielectric, the field within the dielectric is

less than that which would exist if the gap were a vacuum. If the material is a

metal, then electrons would flow to the surface in such numbers as to

completely screen out the field within the metal. In dielectrics, the electrons

are bound, such that the electron‐ion units deform into dipoles (Figure 59),

whose cumulative effect is to partially shield out the external field. The degree

to which the electron‐ion unit deforms (i.e., the strength of the dipole)

is related to the magnitude of the electric field, and so the polarization P is

related to the electric field by P ¼ eowE, where w is the susceptibility and

static conditions are assumed. For dielectric materials, therefore, Maxwell’s

equations can be retained in form by introducing the D field given by

E = 0

r

E ≠ 0

FIGURE 59. Deformation of the ion‐electric cloud by the application of a (vertical) electric field.

156 KEVIN L. JENSEN

D ¼ e0Eþ P ¼ e0 1þ wð ÞE; ð394Þso that = D ¼ qr, where r is still the number charge density of free (not

bound) electrons.

In the presence of an electric field that is time varying, the polarization of

charge within the dielectric proceeds after changes in the E field occur.

Consequently, w acts as a response function and the polarization satisfies

PðtÞ ¼ e0

ð11

w t t 0ð ÞEðt 0Þdt 0; ð395Þ

where w(t) is a real function. The Fourier transform of equations of the form

of Eq. (395) into frequency space results in P(o) being simply given by the

product of the Fourier transforms of each of the integrand functions, or

P oð Þ ¼ e0w oð ÞE oð Þ ð396Þ(albeit that wðoÞ is not defined with the customary 2p of Fourier transforms

so as to retain the form of Eq. (396)) but now, and as indicated by the

caret atop w, the susceptibility is no longer necessarily real and will have

an imaginary component. The relationship between the susceptibility and

the previously considered index of refraction is then shown to be (by the

consideration of the wave equation in terms of the polarizability)

n oð Þ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ wðoÞ

p: ð397Þ

So, the index of refraction has been expressed in terms of the susceptibility—

another quantity that requires a model. Consider first a classical argument for

its evaluation, after which a quantum‐based argument is made. In the former,

as a result of a time‐dependent electric field, a bound electron oscillates about

an atom according to the equation

m@2t rðtÞ þ ðm=tÞ@trðtÞ þmo2

orðtÞ ¼ qEðtÞ; ð398Þwhere the term mo2

o corresponds to a restoring force, m=t corresponds to a

dissipation or dampening term (moving electrons both radiate and scatter)—

and t therefore a relaxation time, and the bound electron is treated as a

harmonic oscillator. The Fourier transform of r(t) is

r oð Þ ¼ qEðoÞm

o2 o2o þ i o=tð Þ 1

: ð399Þ

The induced (atomic) dipole moment is the product of the electron charge

with r(o), and it is also equal to the product of the atomic polarizability with

the electric field. The macroscopic polarizability P is the sum over all such

atomic ones, of which the number density is ro, and so (the sign change being

due to the negative electron charge)


wðoÞ ¼ q2roeom

o2o o2 iðo=tÞ 1

o2p

o2o o2 iðo=tÞ

ð400Þ

where the plasma frequency is defined by

op ¼ q2ro=eom 1=2

: ð401ÞFor ro characteristic of the number density of metals, or 1023 atoms/cm3,

the plasma frequency is on the order of 1.784 1016 rad/s (UV regime). For

a metal, there is no restoring force, meaning the electrons are free to move

about so that oo ¼ 0. Consequently,

n2 ¼ 1þRe wð Þ þ i Im wð Þ; ð402Þor, in terms of the real and imaginary parts,

n2 k2 ¼ 1 o2pt2

1þ o2t2ð Þ

2nk ¼ o2pt

o 1þ o2t2ð Þð403Þ

Defining n2 k2 ¼ N1, 2nk ¼ N2, it can be shown

n ¼ 1

2N2

1 þN22

1=2 þN1

h i8<:

9=;

1=2

k ¼ 1

2N2

1 þN22

1=2 N1

h i8<:

9=;

1=2 ð404Þ

A representative case loosely based on copperlike parameters is instruc-

tive. A conductivity of s ¼ 5.95 105 (O cm)1 plus a number density of

ro ¼ 8.411 1022 cm3 (corresponding to m ¼ 7 eV) entails t ¼ 25 fs and

op ¼ 1.64 016 1/s.

3. Drude Model: Distribution Function Approach

The distribution function approach to the evaluation of number and current

density provides another avenue. Consider smoothly varying electric fields

that change over length scales that are comparatively long, so that the

Wigner function is a solution to the Boltzmann equation. Although there is

158 KEVIN L. JENSEN

no need to restrict attention to one dimension, as the 3D case is straight-

forwardly the same, it is simply a matter of convenience. Thus, recall

J ¼ q

2p

ð11

hk

mf ðkÞdk; ð405Þ

where f(k) satisfies

@t f þ _x@x f þ _k@k f ¼ @c f ; ð406Þwhere the dots indicate time derivatives and @c f is the scattering term. With

_x ¼ hk=m and h _k ¼ F (the velocity and acceleration, respectively) and

invoking the relaxation time approximation, then

@t f þ hk

m@x f F

h@k f ¼ 1

tð f f0Þ; ð407Þ

where f0 is the equilibrium distribution in the absence of fields and tempera-

ture gradients. Consider turning on the field F gradually using a parameter l(not a wavelength), which changes from 0 to 1, that is, let F ) lF . It followsthat the distribution function will likewise ‘‘turn on’’ to the full distribution

via a series of progressively smaller terms characterized by the power of l.In other words,

f ) f0 þ lf1 þ l2f2 þ . . . : ð408ÞInserting Eq. (408) into Eq. (407) and equating like powers of l results

in for l0

@t f0 þ hk

m@x f0 ¼ 0; ð409Þ

which is a restatement of equilibrium: the distribution does not vary spatially

in the absence of forces or with time and depends at most on k. The time

independence of f0 implies, as per the continuity equation @trþ @xJ ¼ 0 that

f0 is a symmetric function in k: it is tantamount to the supply function in the

derivation of the Richardson and FN equations. The next power is l1, forwhich

@t f1 þ hk

m@x f1 F

h@k f0 ¼ 1

tf1: ð410Þ

Recall that along the trajectories of the equilibrium distribution, the

energy does not change (e.g., the harmonic oscillator treated in the Wigner

distribution approach surrounding Eq. (160)). It is therefore reasonable that

the effect of the field will be to change the energy, and therefore f1 will

be proportional to an energy‐like term. In fact, viewing f1 as the first term

in a Taylor expansion implies


f1 ¼ f f0 ¼ Fðx; k; tÞ @E f0ð Þ; ð411Þwhere F is to be determined (and is not to be confused with work function).

Inspection of Eq. (410) suggests that F has the same spatial and temporal

dependence as F ¼ Fo exp iðKx otÞf g, but with a k‐dependent coefficient, or

Fðx; k; tÞ ¼ FoðkÞexp iðKx otÞf g: ð412ÞCoupled with the relation @k f ¼ ð@kEÞð@E f Þ, where E is the energy,

Eqs. (411) and (412) allow Eq. (410) to be written

ioþ ihk

mK

FoðkÞ Fo

hð@kEÞ

ð@E f0Þ ¼ 1

tFoðkÞð@E f0Þ: ð413Þ

With the assumption that the energy is parabolic in k and that, first,

K ¼ no=c and second, that hk=m c (i.e., the electron velocity is much

smaller than the speed of light), Eq. (413) entails

FoðkÞ ¼ tFo

1 iothk

m

: ð414Þ

Finally, using Eq. (405), the definition J ¼ s F=qð Þ (a rather peculiar way

of stating Ohm’s law given that F is a force) and the zero‐temperature limit of

the supply function entails

s oð Þ ¼ qJ

F¼ q2

2p

ðkFkF

tFo

1 iothk

m

0@

1A

2

m

ph2

0@

1Adk

¼ tq2k3F3p2m

1 iotð Þ1

ð415Þ

Using the relationship between density ro and kF, the expression for direct

current (DC) conductivity, and Eq. (389) it follows

sðoÞ ¼ sð0Þ 1þ iot1þ o2t2

; ð416Þ

from which Eq. (403) follows. For metals, the scattering time t is on the order

of 10 fs; therefore, for optical frequencies (e.g., a wavelength of 532 nm),

ot is on the order of 35.

The distribution function approach points out that the Drude relations

break down when higher‐order l terms are nonnegligible (large fields), the

relation between k and E is more complex than assumed, or temperature

gradients add complications. If these complications can be ignored (for

the present, they can), the consequences of Eq. (403) are twofold. First,

in the limit of vanishing frequency, n and k become approximately equal,

and the reflectivity R approaches

160 KEVIN L. JENSEN

limot!0

R ¼ 1 2

ffiffiffiffiffiffiffiffiffiffi2e0osð0Þ

s; ð417Þ

which is known as the Hagen–Rubens equation; that is, at low frequencies

(wavelengths longer than infrared), the metal is nearly 100% reflective

(R ¼ 1). In the opposite limit, n approaches unity and k approaches 0, so

that R ¼ 0, indicating that metals are transparent in the UV limit (omust be

larger than the plasma frequency—for copper, the plasma frequency corre-

sponds to a wavelength of 100 nm, or smaller than the lower limit of the

visible spectrum at 400 nm). In the o ! 0 limit, the static dielectric constant

results and is defined by

eð0Þ ¼ e0 1þ op

oo

2( )

: ð418Þ

In the case of metals, where oo [encountered in Eq. (398)] is vanishingly

small, Eq. (418) suggests inordinately large static dielectric constants.

For semiconductors, where the physics is a bit different, the same equation

suggests more reasonable values. By way of example, consider a semiconduc-

tor‐like material (for which number of electrons per atom, effective mass

variation depending on crystal orientation, and similar complications are

suppressed) nominally modeled after silicon. For a number density (the ratio

of the density with the atomic mass) of 5 1022 atoms/cm3, each of which

contributes one electron,op is 12.6 1016 rad/s. Ifoo corresponds to an optical

wavelength, such as 500 nm, then oo ¼ 3.77 1015 rad/s, implying that

e0¼ 12.2, typical of semiconductors—like silicon, not surprisingly. The energy

hoo ¼ 2.48 eV is similar in energy to where sharp changes in the absorption

coefficient k occur (Philipp and Ehrenreich, 1963) in the dielectric constant of

silicon that are indicative of a contribution such as Eq. (400).

The extension of the Drude model to semiconductors is possible, but

complications arise from the lower free‐carrier density of electrons compared

to metals and the presence of a band gap (Jensen, 1985; Jensen and Jensen,

1991). Importantly, the behavior of the dielectric components is related to the

validity of the quasiclassical Boltzmann transport equation widely used to

model solids. In the quantum mechanical theory (using either the density

matrix approach, or second order perturbation theory approach which gives

the same result), the relaxation time is constant at low frequencies but is

frequency dependent at high frequencies so that absorption coefficient varies

not as the second power of the wavelength as in the Drude model, but as the

third or fourth (depending on the scatteringmechanisms), though a treatment

revealing the dependence is outside the scope of this monograph.


Reflectivity is complicated by the presence of resonances in the visible

regime of the spectrum, as such resonances occur for optical frequencies,

which is precisely where present interests lie. The drive lasers for photo-

cathodes use harmonics of the fundamental frequency of an Nd:YAG laser

(wavelengths of (1064 nm)/n so that n ¼ 3, 4, and 5 correspond to 355 nm,

266 nm, and 213 nm, respectively) or titanium‐sapphire (Ti:Saph) laser (witha fundamental of 800 nm so that n ¼ 2 corresponds to 400 nm). Thus, the

reflectivity in the visible regime is of interest.

4. Quantum Extension and Resonance Frequencies

Let the charge electric field in the vicinity of the atom be given by Fo cos(ot)and the ground state of the electron in the absence of the field be desig-

nated fo. As a result of the perturbation FðtÞ r (where the caret indicates

operator), the wave function becomes

jcðtÞi ¼ jfoiexp iootð Þ þX

cjðtÞjfjiexp ioj t

; ð419Þwhere hoj is the energy difference between the j th level and the ground state

(oth) level. The coefficients satisfy the relation (a consequence of the time‐dependent Schrodinger equation and the orthogonality of the basis states)

ih@tcjðtÞ ¼ hfjjF tð Þ rjfoiexp i oj oo

t

; ð420Þ

the solution is (for sake of argument the electric field is taken to be along the

x‐axis)

cjðtÞ ¼ 1

2ih

ðt0

Fxhfj jxjfoicos otð Þexp iðoj ooÞt

: ð421Þ

Consequently, the expectation value of the dipole term is

hcðtÞjqxðtÞjcðtÞi ¼ 2qFx

h

Xj

jhfjjxjfoij2 oj o 1 þ oj þ o

1n o

cos otð Þ:

ð422Þwhere the terms that oscillate rapidly and out of phase with the field have

been ignored. The dipole moment of the atom is therefore proportional

to the electric field (which was assumed previously). Use the relation

oj o 1 þ oj þ o

1 ¼ 2oj=ðo2j o2Þ and introduce the oscillator

strength term fj ¼ 2m=h2

hojjhfjjxjfoij2: The atomic polarizability is then

q2=mð ÞPj fj=ðo2j o2Þ: Summing over all such terms in a unit area then

shows that the quantum extension of Eq. (340) is (where the dampening

term has been reinserted by hand)

162 KEVIN L. JENSEN

wðoÞ ¼ q2roeom

Xj

fj

o2j o2 iðo=tÞ

: ð423Þ

Some subtleties have been glossed over between the local and the macro-

scopic fields, for which the reader is referred to Ziman (1985, 2001).

How much of a consequence does Eq. (423) make? Consider a simple

pedagogical example based on copperlike parameters, for which hop ¼10:769eV. If the conductivity is taken to be s ¼ 5.959 105 (O‐cm)1, then

t ¼ 25.144 fs. Values of n and k (for example, from the CRC tables; Weast,

1988) at normal incidence, from which R can be obtained, are shown in

Figure 60 on the line designated ‘‘Exp.’’ Similarly, the behavior of n and k

as calculated using Eqs. (403) and (404) is shown by the line designated

‘‘Drude.’’ Clearly, while the Drude line captures the changeover from reflec-

tive to transparent, some physics is missing. An ad hoc Lorentz term using

the parameters f1 ¼ 1, ho1 ¼ 5:9239eV and t1 ¼ 0.25 fs (being a resonance

line, the value of t1 will not necessarily equal t) is shown in the line f1, from

which it is seen much of the difference is captured. Likewise, choos-

ing the somewhat arbitrary parameters f1 ¼ 4/5, f2 ¼ 1/5, o2 ¼ 2o1, and

t2 ¼ t1 improves the correspondence by addressing the tail. Clearly, an

optimized fitting procedure using Lorentzian components will have good

success capturing the behavior of R(o). Such an exercise is of pedagogical,

but for purposes herein not practical, interest—in assessing the role of

reflectance in diminishing QE, it is simpler to extract the values of R from

actual optical constant data available from the literature. In the event that

0.0

0.2

0.4

0.6

0.8

1.0

1 10

Exp.

Drudef1f1 & f2

Ref

lect

ance

Energy [eV]

FIGURE 60. Comparison of experimental reflectance to the Drude model and two

Lorentzian terms.


the incident light is not normal to the surface, then the off‐angle formulas

allow the reflectance to be obtained (Gray, 1972). The relations

Rs ¼ a2 þ b2 2a cosyþ cos2ya2 þ b2 þ 2a cosyþ cos2y

Rp ¼ Rs

a2 þ b2 2a siny tanyþ sin2y tan2y

a2 þ b2 þ 2a siny tanyþ sin2y tan2y

R ¼ 1

2Rs þ Rp

d ¼ l

4pk

ð424Þ

where y is the angle from normal, and the terms a and b are given by

2a2 ¼ ðn2 k2 sin2yÞ2 þ ð2nkÞ2h i1=2

þ ðn2 k2 sin2yÞ

2b2 ¼ ðn2 k2 sin2yÞ2 þ ð2nkÞ2h i1=2

ðn2 k2 sin2yÞð425Þ

allow the value of R for several common metals to be obtained (Figure 61).

It is observed that the maximum absorption can occur off normal, as with

tungsten and lead.

40

60

80

100

0 30 60 90

Ref

lect

ivity

[%]

Angle [deg]

CuAg

AuPb

W

FIGURE 61. The reflectivity of various metals as a function of incidence angle using Eq. (424).

164 KEVIN L. JENSEN

E. Conductivity

The scattering rate factor appearing in the Drude model and alluded to in

the discussion of the mean free path is referred to as the relaxation time

in the discussion of the Wigner and Boltzmann equations. Electrons moving

through a solid may interact among themselves or scatter off of phonons and

thermalize with the lattice. Such a bland observation in fact entails a great

deal of physics relevant to photoemission models. The probability that a

photoexcited electron can transport to the surface depends on its scattering

possibility and the temperature of the electron gas, as scattering rates are

temperature dependent, which in turn depends on the amount of laser energy

absorbed by the material. Photocathodes of the variety considered herein

must produce a great deal of current on demand, unlike their photodetector

brethren. In fact, the peak current densities demanded of photocathodes for

particle accelerators is enormous (on the order of kiloamps, albeit over very

short times) and if the QE is poor, then a considerable incident laser intensity

and therefore significant heating occur. Scattering affects how the laser

energy is distributed into the lattice, which in turn affects the electron

temperature, which in turn affects the scattering rate. The scattering factors

are therefore not simply parameters to be inferred from, say, electrical

conductivity, but merit consideration in their own right. The evolution of

the distribution model represents a convenient starting point.

In keeping with the approach so far, the concern is when electron flow

is in the direction of applied fields (if any), and insofar as a surface exists,

it is normal to the direction of electron transport. This bland assumption

indicates that the problem at hand can be treated as a 1D problem, but what

is stated here can easily be expressed in full 3D parlance. The only motive

for one dimension is purely the argument of ease, but this rationale has much

to commend it. Many excellent sources consider the problem in its full

3D glory but arrive at the same conclusions (Hummel, 1992; Ibach and

Luth, 1996; Kittel, 1962), albeit via a more rigorous and therefore arduous

route.

1. Electrical Conductivity

Reconsider the linearized Boltzmann equation for a time‐independent equi-librium distribution, for which the scattering term vanishes (as many elec-

trons scatter into a state as scatter out). The equilibrium distribution would

then nominally appear to satisfy

hk

m@x f ðx; kÞ þ F

h@k f ðx; kÞ ¼ 0: ð426Þ


A naive solution to Eq. (426) would seem to be

f ðx; kÞ fSðx; kÞ m

pbh2ln

1þ exp

b mðxÞ EðkÞð Þ ; ð427Þ

where mðxÞ mþ jðxÞ, F ¼ @xj; and the subscript on f reinforces that

the result is symmetric in k. But this cannot be correct; only the antisym-

metric component ( fA) of a distribution in k gives rise to an overall current.

Consider, however, how many electrons are actually required to produce an

appreciable current. In the Richardson equation—the number of electrons

per unit time and area passing over the barrier compared to the number

hitting the barrier is phenomenally small:ð1ko

hk

m

foðkÞdkð1

0

hk

m

foðkÞdk

4p2h3

qmm2

!JRLD JRLD

Jmax

: ð428Þ

For m ¼ 7 eV and even for JRLD ¼ 100 A/cm2, the ratio in Eq. (428) is

approximately 2.52 1010 because Jmax ¼ 3.96 1011 A/cm2 for copper.

A similar analysis, using the FN equation, yields a larger ratio due to the

higher current densities, but the conclusion essentially remains the same:

the ratio is negligible. The antisymmetric part is missing in Eq. (427), which

prevents it from being used to ascertain currents. Nevertheless, the naive

approach lends credence to the concept of a ‘‘local’’ chemical potential m(x),elsewhere termed the electrochemical potential, which may be used as long as

the potential variation is weak in some sense. In the following discussion,

this can be implicitly assumed to have been done.

From the definition of current density and Eq. (407) then

JF ¼ q

2p

ð11

hk

mf ðx; kÞ foðx; kÞð Þdk

¼ qt2p

ð11

hk

m

hk

m@x f1 þ F

h@k f0

0@

1Adk

ð429Þ

where the F subscript (F) reinforces that it is current due to electric field,

the first line is a consequence of the symmetry of fo, and the approximation

that the relaxation time is independent of k (while not strictly true) is used.

In the second line, and recalling Eq. (410), the second term in the brackets is

dominant. Neglecting the first term and integrating by parts yields

J ¼ qtF2pm

ð11f0ðkÞdk ¼ qtF

mr0

qtF3p2m

k3F ; ð430Þ

166 KEVIN L. JENSEN

where the zero‐temperature approximation to the bulk number density has

been used. This identifies the DC conductivity s ¼ qJ/F as

sð0Þ ¼ q2t3p2m

k3F ; ð431Þ

where the scattering rate is now identified as the relaxation time. The approach

leading toEq. (431) is now recapitulated for gradients in temperature to reveal a

fundamental relation between electrical and thermal conductivity.

2. Thermal Conductivity

The energy carried by the electrons during their migration is transferred to the

lattice when they scatter. The question arises: What is the thermal conductivity

of an electron gas, that is, the proportionality between current and temperature

(as opposed to potential) gradient? A related concept is specific heat, or the

change in energy with temperature. Consider by way of introduction

the statistical mechanics of an ideal gas. In a cubic box, a third of the particles

are moving along each of the coordinates, and half of those are in the plus (þ)

direction with the other half in the minus (–) direction. If the particles share the

same velocity v, then the number per unit area that impact the face of the cube in

a time t is rvt=6, where r is the number density. After impact, the change

in momentum is 2 mv. Therefore, the pressure P is the product of density and

momentum transfer per unit time, and so P ¼ rmv2=3. The quantity mv2=2 is

the kinetic energy. The ideal gas equation plus the expression for kinetic energy

combine to show that the energy of a particle in terms of temperature is 32kBT ,

and it follows that the energy density of the ideal gas of particles is

E ¼ 3

2rkBT : ð432Þ

The coefficient of T in Eq. (432) is the specific heat capacity Cv ¼ ð3=2ÞrkB:If there is a temperature gradient, in the short distance v dt that the particles

travel between thermalizing collisions, the flow of energy is the difference in

the number per unit area of the particles from the left with their unit energy,

and the particles from the right

JT ¼ vt6

dE

dt¼ vt

6

dx

dt

0@

1A 3

2rkB

dTðxÞdx

¼ 1

2rv2tkB

dT

dx

¼ 1

3Cvv

2tdT

dx K

dT

dx

ð433Þ

where vt is the length between collisions and the (1/6) comes from the

arguments above relating to how many particles pass through a given face.


An alternate method of writing the relationship, assuming that only the

electrons at the Fermi level matter in heat flow—and this remains to be

shown—is given by K ¼ 2m=3mð ÞCvt.Regrettably for the derivation just presented, electrons do not have a

uniform energy and therefore do not travel at a uniform velocity. However,

Eq. (433) provides an indication of what must occur, and so the problem is

reconsidered afresh from the distribution function approach now that there

is some confidence in the destination. (Remember that the transport of heat is

the issue.) When it was charge that was transported, the current density was

the product of the charge, its velocity, and the density of charge. Now,

however, not charge but energy is flowing. From thermodynamics a quantity

of heat dQ obeys

dQ ¼ dU mdr: ð434ÞIn a zero‐temperature equilibrium distribution, any small quantity of

energy added to a particle is added at the Fermi level. The flow of heat is

therefore the net imbalance of a particle traveling in one direction with

an energy E compared to its matching particle traveling in the opposite

direction at the Fermi level. In equilibrium, there is no net flow (dQ ¼ 0),

and the disturbance from equilibrium implied by Eq. (434) is small and

affects but one particle. Using the relationship dU ¼ EðkÞfFD EðkÞð Þdkthen dJE ¼ EðkÞ hkx=mð ÞfFD EðkÞð Þdk, where, out of necessity, the full 3D

approach reappears and the shorthand dk ¼ dkxdky dkz ¼ 4pk2dk is useful.

The number current density is, as before, dJe ¼ hkx=mð Þ fFD EðkÞð Þdk: Con-sequently, the current of heat JQ is then related to the current of energy by

JQ ¼ JE mJe

¼ 2pð Þ3

ðEðkÞ m hkx

m

0@

1AfFD EðkÞð Þdk

¼ 2pð Þ3

ðEðkÞ mð Þt EðkÞð Þ hkx

m

0@

1A2

@x fFD EðkÞð Þ½ dk

ð435Þ

In the last line, the term containing the equilibrium distribution vanishes

by the asymmetry of the integrand in kx. Recalling the density of statesD(E)

defined by Eq. (30), Eq. (435) may be written in a more general form that

allows for nonspherical Fermi surfaces in terms of which

JQ ¼ð10

ðE mÞ hkxm

0@

1A2

tðEÞDðEÞ@x fFDðEÞ

dE

¼ 2

3m

ð10

ðE mÞEtðEÞDðEÞ @TfFDðEÞ½ dE8<:

9=;ð@xTÞ

ð436Þ

168 KEVIN L. JENSEN

where Ex ¼ E/3 by spherical symmetry. The coefficient of @xT is the thermal

conductivity. To proceed further [i.e., reclaim Eq. (433)], the nature of

@TfFD must be investigated.

The gradient with respect to temperature of the FD distribution, for

any reasonable temperature encountered in practice, is a sharply peaked

function. Letting u ¼ bðE mÞ in the FD distribution, then

@TfFDðEÞ ¼ kBbu

ð1þ euÞð1þ euÞ : ð437Þ

For a smoothly varying function g(du), a Taylor expansion and Appen-

dices 1 and 2 can be used to show that for small d

ð11

gðduÞ1þ euð Þ 1þ euð Þ du gð0Þ þ 2d2

2!g00ð0ÞWþð2;1Þ þ 2d4

4!g0000ð0ÞWþð4;1Þ

¼ gð0Þ þ d2g00ð0Þzð2Þ þ 7

4d4g0000ð0Þzð4Þ ð438Þ

where primes indicate derivatives with respect to argument, the function

W(n,x) is discussed at length in Appendix A, and z is the Riemann zeta

function discussed in Appendix B. The leading‐order term justifies the stan-

dard approximation that @TfFDðEÞ mimics a Dirac delta function when

mixed with slowly varying functions over the range jE mj kBT . Any

smoothly varying function in E can have its argument replaced by m and

pulled from the integral, letting Eq. (436) be approximated by

JQ ¼ 2m3m

t mð Þ@Tð10

E mð ÞfFD Eð ÞD Eð ÞdE

@xTð Þ

2m3m

t mð ÞCe Tð Þ @xTð Þ k @xTð Þð439Þ

Eq. (439) is formally equivalent to Eq. (433) if v2 ¼ 2m=m and t ¼ tðmÞ, thatis, the velocity and relaxation time are evaluated at the Fermi level. The

specific heat can be further approximated by

CeðTÞ @

@T

ð10

E mð Þ fFD Eð ÞD Eð ÞdE

k2BT

ðbm0

u2 D mþ u

b

0@

1AþD m u

b

0@

1A

8<:

9=;

1þ euð Þ 1þ euð Þ du

k2BT 2D mð ÞWþ 2; bmð Þ D mð Þ2bmð Þ2 Wþ 4; bmð Þ

8<:

9=;

ð440Þ


where Eq. (A4) is invoked and bm 1 has been used. For temperatures of

present concern, only the leading‐order term inWþ(2,1)¼ z(2)matters, and so

Ce Tð Þ ¼ 1

3p2k2BTD mð Þ 1 7

40

pbm

0@

1A

28<:

9=;

p2

3k2BD mð ÞT gT

ð441Þ

where k2F ¼ 2mm=h2. The theoretical value of g differs, in general, from the

experimental value; given the linear dependence of g on m, it is common to

define a ‘‘thermal’’ mass by

mth

m¼ gexp

gtheory: ð442Þ

Several examples are (Cu) ¼ 1.375, (Ag) ¼ 1.0136, and (Al) ¼ 1.4855,

indicating that the simple model works reasonably well. The difference

between the thermal mass and the electron mass is attributed to the influence

of the periodic potential of the atoms on electron motion, as well as their

vibration modes (phonons), which is discussed in more detail below. For

other metals, it is found that (Fe) ¼ 7.931, and other transition metals are

comparably high. In these cases, the partially filled d shells contribute to the

density of states (DOS) and thereby undercut the simple model put forward

here (Ibach and Luth, 1996).

3. Wiedemann–Franz Law

Let us now compare the electrical conductivity to the thermal conductivity

under the assumption that the thermal and electrical relaxation times are the

same—an assumption that is not a priori obvious. We find

ks¼

2

3m

h2k2F2m

!t mð Þ m

3h2k2BkFT

q2t mð Þ3p2m

k3F

¼ p2

3

kB

q

2

T LT ; ð443Þ

where L ¼ 2.44301 108 ohm‐watt/Kelvin2 is the Lorentz number. The

empirical Lorentz number Lexp kexp=sexp for various metals compared to

the theoretical value L is shown in Figure 62.

Implicit in Eq. (443) is the assumption that the relaxation time for

thermal processes is equivalent to that for electrical conduction. Such a

circumstance is not a priori true, as electrical currents resemble a displacement

170 KEVIN L. JENSEN

of the Fermi sphere, but thermal effects affect the distribution in electron

energy near the Fermi level. The point, however, is that even though L for a

variety of metals is not the Lorentz number, it is remarkably close.

A final comment concerning the electrical and thermal conductivities is

that, as generally observed, good thermal conductors make good electrical

conductors because energy is transported by free electrons. For insulators,

where free electrons are scarce, the absence of electrical current is related to

the fact that heat is transported phonons, so that poor conductors are related

to thermal insulators as well.

4. Specific Heat of Solids

Thermal energy is also related to motion of the lattice; atoms in the lattice

vibrate and the degree of their vibration relates to the temperature of the

solid. Vibrations constitute harmonic oscillators, for which the average

kinetic energy is the same as the average potential energy. A simple model

is that the total energy held by the harmonic oscillators representing the

lattice is E ¼ 2 3=2ð ÞkBT ¼ 3kBT . For N atoms per mole, then, the total

internal energy is 3NkBT . The coefficient of temperature, identified as heat

capacity per unit volume, is then Ci ¼ 3 N=Vð ÞkB ¼ 3rikB for solids—an

i subscript (i) is used to distinguish the contribution to specific heat from

phonons from that due to electrons. At low temperatures the relation fails,

and a quantum mechanical treatment must be considered.

Treat the atoms in the lattice as a set of coupled quantized harmonic

oscillators. If they are in contact with a heat bath of temperature T,

2.0

2.2

2.4

2.6

2.8

3.0

Cu Au Ag Al W Mo Na Pb

ExperimentTheoretical

L [10

−8 W

-Ω/K

2 ]

Element

FIGURE 62. Theoretical (line) versus experimental Lorentz numbers for various metals.


then the probability that the harmonic oscillator will be at a level

En ¼ nþ 1=2ð Þho hon is proportional to exp hon=kBTð Þ ¼ exp bhonð Þ.The constant of proportionality is obtained by observing that the sum over

all probabilities must be 1, orX10Pn ¼ No

P10 ebhon ¼ Noebho

P10 enbho

¼ Noebho=2

1 ebho¼ 1

ð444Þ

which defines No. The probability that the oscillator at the energy level n is

Pn ¼ enbho ebho 1

: ð445ÞThe mean energy of the system is the sum over the products of the energy

levels En with their probability Pn, or

E oð Þ ¼X1

n¼0EnPn ¼ ebho 1

hoX1

n¼0nþ 1

2

enbho: ð446Þ

The sums are readily evaluated by

1 ¼X1

n¼0xn

X1n¼1

xn

¼X1

n¼0xn x

X1n¼0

xn ¼ 1 xð ÞX1

n¼0xn

ð447Þ

and tricks such as X1n¼1

nxn ¼ @xX1

n¼0xn ¼ x

1 xð Þ2 : ð448Þ

Both terms on the RHS of Eq. (446) may then be evaluated, giving

E oð Þ ¼ ho1

2þ 1

ebho 1

: ð449Þ

The recasting of E(o) as hni þ 1=2ð Þho identifies

hni ¼ 1

ebho 1; ð450Þ

that is, the oscillators obey BE statistics.

A complication is the realization that a complex arrangement of atoms in

a crystal does not oscillate with only one fundamental frequency o; rather,N unit cells with r atoms per unit cell oscillating along three axes implies 3r N

modes, all of which can be excited. For an isotropic crystal of volume V, the

number of modes per unit volume in frequency space is constant. Given that

the number of atoms present in even a paltry bit of matter is enormous,

172 KEVIN L. JENSEN

the summations can be converted integrals without difficulty. The DOS

Dp(o) (p designating phonon) in reciprocal space q will then be

DpðoÞdo ¼ 1

2pð Þ3 d3q ¼ 1

2pð Þ3 4pq2dq ¼ 1

2p2o2

c3i

do; ð451Þ

where o=q ¼ ci is the sound velocity for the ith branch. If the transverse T

sound velocities are assumed to be the same (but different than the longitu-

dinal L), then

DpðoÞdo ¼ V

2p21

c3Lþ 2

c3T

o2do: ð452Þ

The total energy of the crystal is then a summation (now integration) over

all frequencies of the energies and the DOS, or

E ¼ðoD

0

DpðoÞEðoÞdo; ð453Þ

where oD is the maximum, or Debye cutoff, frequency determined from the

requirement that the number of modes is equal to the number of atoms

(concepts discussed in greater detail in the treatment of the electron‐latticerelaxation time), or

3rN ¼ðoD

0

Dp oð Þdo ¼ V

6p21

c3Tþ 2

c3L

o3

D: ð454Þ

It is more common to speak of a Debye temperature TD defined by the

relation hoD ¼ kBTD and estimated from

TD ¼ hvskB

6p2Nr 1=3

; ð455Þ

where be ¼ 1/kBTe is the electron temperature thermal factor, N [#/cm3] is

the number density of the crystal, r is the number of atoms per unit cell, and

vs is the velocity of sound. Values among metals vary; examples are 165 Kfor gold, 343 K for copper, and up to 400 K for tungsten. Introducing the

lattice density ri¼N/V and combining the components (note that o=2, beingtemperature independent, does not contribute) gives

CiðTÞ ¼ 9rrio3

D

d

dT

ðoD

0

o2 hoebho 1

do

¼ 9rkBriT

yD

0@

1A

3 ðyD=T0

x4

1 exð Þ 1 exð Þ dx

¼ 9rkBriT

yD

0@

1A

3

W 4;yDT

0@

1A

ð456Þ


For T yD, then Wð4; xÞ x3=3: the classical result is regained (for

simple metals, r ¼ 1). However, for T yD then Wð4; xÞ 4p4=15, andCi T yDð Þ 3rkBri 4p4=5ð Þ T=yDð Þ3. The ratio of Ci(T) with the classical

3 rrikB is shown in Figure 63. The total specific heat for a solid is then the

sum of the electron and lattice components.

F. Scattering Rates

The resistivity r of a metal is known to be temperature dependent, a common

relation being the linear form of

rðTÞ ¼ ro þ a T Toð Þ; ð457Þwhere T is temperature, To is a reference temperature (commonly room

temperature), and a is the temperature coefficient. As resistivity is the inverse

of conductivity, Eq. (457) implies that the relaxation time is temperature

dependent. To determine the dependence, evaluating the relaxation times

must be done using quantum transport theory (Jensen et al., 2007; Rammer,

2004; Ridley, 1999; Wagner and Bowers, 1978). The strategy here shall be to

provide descriptions of sufficient plausibility that when complex relations are

provided (deus ex machina), they are plausible.

1. Fermi’s Golden Rule

Let the Hamiltonian of a scattering problem be given by H ¼ H0 þ U , where

U represents the scattering potential and H0 the unperturbed Hamiltonian.

Define the unperturbed wave functions by

0.01

0.1

1.00

−3 −2 −1 0 3

Exact

Small approx

Large approx

Ci(T

)/C

i(∞)

In(T/qD)1 2

FIGURE 63. Comparison of Ci (T) calculated numerically with its asymptotic limits.

174 KEVIN L. JENSEN

jcðtÞi ¼PkckðtÞjckðtÞi ¼P

kckðtÞjkieiok t

H0jki ¼ E kð Þjki ð458Þ

Consider the scattering off of a weak potential in that scattering to other k

states is not large. Then it follows that Schrodinger’s equation

ih@tjcðtÞi ¼ H0 þ U jcðtÞi ð459Þ

that

ih@t ck0 ¼X

kckhk0jUðtÞjkiexp i ok0 okð Þtf g; ð460Þ

where k0 denotes the state after scattering from the state k, which is the object

of our attention. Although the potential may have a complex dependence

on time, it is easier to consider its Fourier components, and so assume

UðtÞ ¼ Uo e

iot: the sign on o will (shortly) indicate if a particle is emitted

or absorbed (when discussing phonons). If the initial state k0 is empty, then

an integration of Eq. (460) gives

ck0 ðtÞ ¼ i

h

Xkckhk0jU

o jkiexp i ok0 ok oð Þtf g 1

ok0 ok oð Þ

: ð461Þ

In theBorn approximation, one initial state ko dominates all others; therefore

all but one of the ck vanish—that is, ck dk;ko . In deference to keeping termi-

nology manageable, the ‘‘o’’ subscript will be ignored (i.e., ko ¼ k). It follows

ck0 ðtÞ hk0jUo jki

exp i ok0 ok oð Þtf g 1

ih ok0 ok oð Þ

: ð462Þ

The probability Pk(t) that a scattered electron ends up in the state k is

thenjckðtÞj2, for which Eq. (461) is less than computationally elegant.

As when evaluating density, current, or other moments of a distribution

function, an integration over the final states is performed, entailing an

examination of how Eq. (461) behaves. Let O ¼ ok0 ok oð Þ andeiOt 1

iOt¼ 2eiOt=2

sin Ot=2ð ÞOt

: ð463Þ

For large times t, the absolute square of the RHS of Eq. (462) is sharply

peaked, indicating that integrations with it are concentrated about O ¼ 0.

UsingÐ11 x2sin2 xð Þdx ¼ p, then

sin2ðOtÞðOtÞ2 2p

tdðOÞ; ð464Þ

where d(x) is the Dirac delta function, and so


jck0 ðtÞj2 jhk0jUo jkij2

2pth

d Eðk0Þ EðkÞ hoð Þ: ð465Þ

The notation ho in the far RHS represents either the absorption or

emission of something, and so the sum of each case should be considered in

Eq. (465). The probability that a transition has taken place at time t is less

interesting in the present circumstance than the rate at which transitions take

place (that is, scattering), and it is common to define the latter as the large

time limit of jck0 ðtÞj2=t S k; k0ð Þ; using Eq. (465) this yields

S k; k0ð Þ ¼ 2ph

0@

1A jhk0jUþ

o jkij2d Eðk0Þ EðkÞ þ hoð Þþn

jhk0jUo jkij2d Eðk0Þ EðkÞ hoð Þ

o ð466Þ

As noted by Ridley (Chapter 3, 1999), the delta function is an approximation

that applies provided the time between collisions is much longer than the

duration of a collision. The relaxation time is then

1

t kð Þ ¼X

k0S k; k0ð Þ: ð467Þ

Equation (467) works well for nondegenerate semiconductors where the

concentration of carriers is not great. However, for metals and degenerate

semiconductors, the occupation of the final state factors in, as does the

occupancy of the initial state. These complications are considered explicitly

when treating metals.

While Eq. (467) is an estimation of how often collisions occur, what is

additionally important is the rate at which both momentum and energy are

randomized through collisions. If scattering is not isotropic, the memory of

the initial direction (say in the z‐direction) of an initial electron beam takes

longer to dissipate, and an additional multiplicative factor equal to the

fractional change in the forward, or z, momentum is required, or

z hk hk0ð Þz hk ¼ 1 k k0ð Þ

k2¼ 1 k0

kcosy; ð468Þ

where k ¼ kz. If scattering is elastic and energy parabolic in momentum, then

(k0/k) is unity. Finally, the fractional change in energy is

1 Eðk0 ÞEðkÞ ¼ 1 k0

k

2

: ð469Þ

176 KEVIN L. JENSEN

The momentum tm and energy tE relaxation times (Lundstrom, 2000;

Ridley, 1999) are defined through the inclusion of the factors given by

Eqs. (468) and (469) in Eq. (467), respectively.

2. Charged Impurity Relaxation Time

A time‐independent shielded Coulomb potential of the form encountered in

Eq. (32) gives rise to

hrjUojr0i ¼ Zq2

4peorexp kTFrð Þd r r0ð Þ; ð470Þ

where Zq is the charge of the potential. For degenerate statistics,

kTF ¼ 4kF=paoð Þ1=2 [the nondegenerate statistics case is given in Eq. (33)].

Elementary evaluation shows that as kTF becomes large, exp kTFrð Þ=r actslike a delta function itself for a smoothly and weakly varying function g(r);

that is. to leading order

ðO

ekTF r

rgðrÞdr ¼ 4p

ð10

gðrÞekTF rrdr

¼ 4pk2TF

ð10

gx

kTF

0@

1Aexxdx 4p

k2TFgð0Þ

ð471Þ

Therefore, before considering the shielded potential, consider first the

Dirac delta function approximation, as it illustrates several features of relax-

ation times without undue burden. It follows

hk0jUojki ¼ Zq2

4pKseoV1

k2TF

0@

1Aðhk0jrid rð Þhrjkidr

¼ Zq2

4pKseoV2pk2TF

ð11

dx

ð10

drd rð Þexp iprxf g

¼ Zq2

Kseok2TFV

ð472Þ

where x ¼ cosy, V is the volume, and p ¼ jk k0j. It is only slightly more

difficult to show that (recall Eq. (63), the difference here being the mass of

the ionized scattering site)


hk0jUojki ¼ Zq2

4pKseoV

ðhk0jri exp kTFrð Þ

rhrjkidr

¼ Zq2

4pKseoV

ð10

ekTF rsin prð Þdr

¼ Zq2

KseoVk2TF þ p2 1

ð473Þ

Eq. (473) reduces to Eq. (472) for kTF » p. Conservation of momentum

and energy indicates that the collision is elastic, for which the magnitude

of the momentum in the initial and final states are the same; the

expression for p is obtained from (where y is now taken as the angle between

k and k0)

p ¼ jk k0j ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2 þ k02 2kk0 cosy

p¼ 2k sin y=2ð Þ ð474Þ

and so

hk0jUojki ¼ Zq2

KseoVk2TF þ 4k2sin2

y2

1

: ð475Þ

The scattering rate is then

1

tðkÞ ¼Pk0S k; k0ð Þ

) 2V

2pð Þ3ð2phjhk0jUojkij2d EðkÞ Eðk0Þð Þdk0 ð476Þ

¼ 2

phZq2

Kseo

0@

1A

2

1

V

ð10

k02dk0d EðkÞ Eðk0Þð Þðp0

sinydy

k2TF þ 4k2sin y=2ð Þ 2

The delta‐function integration is readily dispatched; introducing

BðkÞ2 ¼ k2TF=4k2 gives

1

tðEÞ ¼1

32p2

mE3

1=2Zq2

Kseo

2 ðp0

sinydy

BðkÞ2 þ sin y=2ð Þn o2

; ð477Þ

178 KEVIN L. JENSEN

whereas the momentum relaxation time, as per Eq. (468), is

1

tmðkÞ ¼1

32p2

mE3

1=2Zq2

Kseo

2 ðp0

siny 1 cosyð ÞdyBðkÞ2 þ sin y=2ð Þn o2

: ð478Þ

The integration can be performed—but note that so far, only one scatter-

ing center has been considered, whereas the number of scattering sites

(impurity centers) per unit volume ri must be included. Performing the

integration and including the scattering sites finally yields

1

tmðEÞ ¼ri32p

2

mE3

1=2Zq2

Kseo

2

ln 1þ B2 B2

1þ B2

: ð479Þ

The term in brackets is large for semiconductors, of order O(10) to O(100)

depending on the effective mass and the ionized impurity concentration

(generically about 1017 #/cm3). The coefficient provides a measure of the

size of the relaxation time in general:

32pri

mE3

2

1=2Zq2

Kseo

2

¼ 5:1767psKs

Z

2E½eV

r

3=2

; ð480Þ

where r ¼ m/mo is the effective mass ratio. Therefore, picosecond‐scalerelaxation times are to be expected.

On occasion it is written 1=tmð Þ ¼ vgsm=V , where sm is a cross section; the

inverse scattering rate can therefore be thought of as the ratio of the volume

swept out by an area traveling at a group velocity for a scattering time with

the cell volume V. The sm calculated via Eq. (480) is often denoted as the

Brooks—Herring (BH) approximation—it avoids problems in infinities that

arise in the evaluation of a Coulomb potential cross section by appealing to

the notion of screening. The BH approach is in contrast to removing the

offending infinities by truncating relevant integrals at a lower scattering angle

(the Conwell–Weisskopf approximation; Ridley, 1977, 1999).

The description of scattering given here relies on the Born approximation,

known to overestimate the amount of electron‐electron scattering. It is a

poor defense of an approach to claim it is not so bad (or could be worse), but

that is, in fact, true here. A study of the electron‐electron scattering in metals

along the lines herein accounts for screening provided by other electrons but

neglects the interaction of the screening electrons with each other. Including

that interaction gives rise to the random‐phase approximation in which a bare

Coulomb interaction is screened by the Lindhard dielectric function—but

that is also an approximation as the dielectric function of the electron gas is

unknown (Kukkonen and Wilkins, 1979). Kukkonen and Smith (1973) find

that rather than being a factor of 5 in error (as was believed at the time of


their work), the Born approximation instead overestimates by about a factor

of 2 the scattering cross section and the electron‐electron contribution to the

thermal resistivity. Regardless, for the present purposes, the Born approxi-

mation more than suffices to infer temperature and energy dependence of the

scattering terms.

3. Electron‐Electron Scattering

Any ionized impurity in a metal is quickly surrounded by the copious carriers

in the conduction band. So why the interest in the ionized impurity calcula-

tion? For two reasons: first, because electron‐electron scattering dominates

the other relaxation times for photoemission in metals (Tergiman et al.,

1997), and second, because electron‐electron scattering can be viewed as a

kind of ionized impurity scattering, albeit that the players are of equal mass

and identically charged. It is in fact a difficult calculation, but the ionized

impurity calculation provides guidance. The changes for electron‐electronscattering, though, are important. First, the electrons, being identical parti-

cles, both scatter into, and out of, a given state. Second, the occupation of the

initial state matters, as does the final state, as an electron cannot scatter into

an occupied state (a rather wordy way of saying the exclusion principle

holds). The language of distribution functions is useful. The modifications

for the two‐body scattering are needed, and they are to assign weighting

factors of f ðkÞ ¼ probability that an electron is scattering from the state

k, and 1 f ðkÞð Þ ¼ probability that an electron is scattering into the state k.

The scattering event is governed by the two‐body term S k1; k2; k3; k4ð Þ,in which k3 ! k1 (read as electron in state ‘‘3’’ scatters off the potential

into state ‘‘1’’) and k4 ! k2, as is symbolically indicated in

Vk4k3

k2k1

k3k4 Vee k1k2 ⇒

The formalism for calculating the collision term then consists of associat-

ing an f factor for every arrow entering the interaction region, a factor (1 – f )

for every factor leaving the interaction region, and then integrating over

all momenta. However, electrons scatter into states as well as from states.

Therefore, the companion diagram in which the indices are shuffled also con-

tributes, but with the opposite sign. The collision operator, corresponding to

the diagrams suggested by the above discussion (see Wagner and Bowers,

1978, for the formal treatment) results in the interpretation of the above

Feynman diagram as (Ridley, 1999; Tergiman, 1997)

180 KEVIN L. JENSEN

@cf k3ð Þ ¼ 2pð Þ9

ðdk1dk2dk4 S k1; k2; k3; k4ð Þf3f4 1 f1ð Þ 1 f2ð Þf

S k3; k4; k1; k2ð Þf1f2 1 f3ð Þ 1 f4ð Þgð481Þ

where the notation

fj f kj

;Ej E kj ð482Þ

has been introduced. The principle of detailed balance indicates that when

the system is in equilibrium, then the collision operator vanishes; it sets

restrictions both on S and the form of the fs. First, S is symmetrical, that

is, S k1; k2; k3; k4ð Þ ¼ S k3; k4;k1; k2ð Þ, or, alternately, the expression does not

change with k1 $ k3 and k2 $ k4. Second, when the fj’ s are replaced by their

equilibrium (FD) distributions then, using the shorthand

xn b EðknÞ mð Þ ð483Þ

plus the symmetry of S, Eq. (482) vanishes when

0¼ f3 f4 1 f1ð Þ 1 f2ð Þ f1 f2 1 f3ð Þ 1 f4ð Þf1 f2 f3 f4

¼ ex2þx1 ex3þx4

ð484Þ

when fj ¼ 1= 1þ exjð Þ. The second line of Eq. (485) is simply a restatement of

conservation of energy (the energy entering the vertexV is equal to the energy

leaving). The evaluation of the relaxation time therefore makes use of depar-

tures from the FD distribution in Eq. (482) in the linearized Boltzmann’s

equation.

If the potential interaction is independent of spin s, and introducing the

convenient but slightly obfuscating notation jki ¼ j k; si, then (recalling that

jkikji represents a Slater determinant)

S k1;k2; k3; k4ð Þ ¼ 2phjhk1k2jVeejk3k4ij2d k1 þ k2 k3 k4ð Þd E1 þ E2 E3 E4ð Þ

¼ 4ph

V 231 þ V 2

32 V31V32

d k1 þ k2 k3 k4ð Þd E1 þ E2 E3 E4ð Þ

ð485Þ

where a sum over spin coordinates has been performed and

Vij Vee jki kjj

. Conservation of energy and momentum are enforced


in the delta functions. Analogous to Eq. (473), we have

Vee jki kjj ¼ q2

eoV1

k2TF þ jki kjj2 : ð486Þ

To say that the solution of Eq. (481) using Eqs. (482)–(486) is difficult is

not an understatement, and the approach taken in most treatments is the one

used here, namely, defer to the archival literature (Lawrence and Wilkins,

1973; Lugovskoy and Bray, 1998, 1999; Morel and Nozieres, 1962; Wagner

and Bowers, 1978)—there are many components—at a convenient opportu-

nity, quote the result, and move on. However, a fair amount may be said that

renders the final result plausible, and this dominates the treatment here.

The focus is on the relaxation time for distributions distorted from the

equilibrium distribution such that f ðkÞ foðkÞ @EfoðEÞð ÞcðkÞ (which

defines c), where the fo is understood to be FD distribution. Observe that

d

dEfoðEÞ ¼ d

dE½1þ eb Emð Þ1 ¼ bfo 1 fof g; ð487Þ

a function that is sharply peaked about E ¼ m [recall the discussion following

Eq. (30)] indicating the magnitude of k3 approximates kF. It ‘‘follows’’ that

1

tee k3ð Þ ¼1

1 f3

0@

1A 1

2 2pð Þ9ððð

d k1d k2d k42phjhk1k2jVeejk3k4ij2

f4 1 f1ð Þ 1 f2ð Þ þ f1 f2 1 f4ð Þf g d k1 þ k2 k3 k4

d E1 þ E2 E3 E4ð Þ

ð488Þ

where ‘‘follows’’ means that as plausible as the result appears, there is

considerable effort showing it that has been passed over but can be found

elsewhere (Wagner and Bowers, 1978; see also Eq. 2 in Lugovskoy and Bray,

2002, which bears greater similarity to Eq. (488)). The interpretation is that

both scattering into state k1, as well as scattering from state k1, must be

considered. The solution of Eq. (489) is nontrivial, but the energy delta

function can be exploited to discern the behavior of interest to the present

treatment insofar as the temperature and E3 dependence of tee is to be

ascertained.

For metals, energies a few kBT below Fermi level are filled, and so final

scattering states there are precluded. The scattering electron is most probably

within a kBT of the Fermi momentum, and by energy and momentum

conservation, the final state is similarly constrained. That is, both jk1j andjk3j are comparable to kF and their difference is small—a conclusion pertain-

ing to jk2j and jk4j as well, indicating that the momentum delta function

182 KEVIN L. JENSEN

becomes a relation between the angular components rather than their mag-

nitudes. When screening is strong (degenerate statistics), then the term

Vee jki kj j

in Eq. (487) is approximately constant. Taken together, these

two observations imply the angular integrations may be handled separately

from the energy integration. It is therefore sufficient for present purposes to

consider the energy integral to ascertain the leading‐order temperature

and energy dependence of tee(E), as the angular integrations are the source

of the vexing dependence of tee on kTF—and so its evaluation is deferred to

the literature. To exploit the energy delta function, switch to an energy

integration via

dkn ¼ k2ndkn sinydydf 1

2b2m

h2

3=2

m1=2dxn sinydydf; ð489Þ

where the smallness of xn (by comparison to bm over the region where the

integrand is significant) has been exploited. Elsewhere [as Ziman (1985, 2001)

does], it is common to write Eq. (490) in the form d kn ¼ dEndOk

dk ¼ d3k dEdOk

@E=@kð Þ ¼ dEdOk

h2k=m ; ð490Þ

where the k in the denominator is evaluated at kF; such is the origin of factors

of vF ¼ hkF=m in the denominator of the coefficient of tee in the final result.

The integration over E4 leaves (where the largeness of bm has been used to

extend the lower limit to 1)

1

teeðk!3Þ/ððð

dE1dE2dE4 f4 1 f1ð Þ 1 f2ð Þ þ f1 f2 1 f4ð Þf gd E1 þ E2 E3 E4ð Þ

¼ 1

b2

ð11dx1

ð11dx2 ½ ex1þx2x3 þ 1ð Þ ex1 þ 1ð Þ ex2 þ 1ð Þ1n

þ½ ex1x2þx3 þ 1ð Þ ex1 þ 1ð Þ ex2 þ 1ð Þ1o

ð491Þ


The integral may be simplified by using the relationð11

dy

exþc þ 1ð Þ ex þ 1ð Þ ¼c

ec 1; ð492Þ

from which Eq. (492) can be shown to be

1

teeðk!3Þ/ 1

b2

ð11

dx1x3 x1ð Þ

½ ex1x3 þ 1ð Þ ex1 þ 1ð Þ x3 x1ð Þ

½ ex1þx3 þ 1ð Þ ex1 þ 1ð Þ

0@

1A

¼ ex3 þ 1ð Þb2

ð11

dx1x1

½ ex1x3 þ 1ð Þ 1 ex1ð Þ

¼ 1

2p2 þ x23 ð493Þ

Combining the components and hiding the angular integrations behind the

newly introduced function g yields

1

tee E3ð Þ ¼ C1

2pkBT

E3

2

1þ E3 mpkBT

2" #

g2jk3jkTF

; ð494Þ

where the collection of constants making up C and the behavior of the

function g remain to be determined. Because of jhk1k2jVeejk3k4ij2, it followsC will resemble q2=eoð Þ2 plus a smattering of factors of p and other numbers

for good measure. The function g is another matter; while its derivation is no

more odious than what has transpired so far, we cite Wagner and Bowers

(1978) and move on; they show

tee EðkÞð Þ ¼ 8hK2s

a2fspmc2

EðkÞkBT

0@

1A

2

1þ EðkÞ mpkBT

0@

1A

20@

1Ag

2k

qo

0@

1A

24

351

gðxÞ ¼ x3

4tan1xþ x

1þ x2 tan1 x

ffiffiffiffiffiffiffiffiffiffiffiffiffi2þ x2

p ffiffiffiffiffiffiffiffiffiffiffiffiffi2þ x2

p0@

1A

ð495Þ

where a notation invoking the fine‐structure constant instead of the permit-

tivity of free space is deferred to in order to make a unit analysis transparent.

The sudden introduction of Ks is a not‐so‐subtle sleight‐of‐hand, as it cer-tainly was not part of the original discussion of electron‐electron collisions.

It arises as a result of additional screening by d‐band electrons at zero laser

frequency, and values between 1 and 10 have been suggested for various

metals (as discussed later). That implies, however, that kTF might require

some changes, and so the qo factor (in a notation following Ridley) is used in

184 KEVIN L. JENSEN

its stead in g, where

k2TF ! q2o ¼ 4kF=pKsao: ð496Þ

The behavior of g(x) is shown in Figure 64, along with its asymptotic limits

given by

gðxÞ x3

4

x7x2 þ 12ð Þ15x2 þ 12ð Þ x 1ð Þ

p2

1 1

x

0@

1A x 1ð Þ

:

8>>>>><>>>>>:

ð497Þ

The important features of Eq. (496) are the energy and temperature

dependence. They are, first, that the electron‐electron relaxation time is

proportional to the inverse‐square of the temperature (T2), and second,

that the energy dependence is proportional to the square of the difference of

the electron energy with the Fermi level (E – m)2 when the difference is larger

than the thermal energy kBT. Both follow as a natural consequence of FD

statistics applied to a degenerate gas of electrons.

In some treatments (Papadogiannis, Moustaizis, and Girardeau‐Montaut,

1997) and elsewhere (Jensen, 2003b), tee is parametrically represented by an

equation of the form

0

0.4

0.8

1.2

1.6

−2 −1 0

Exact

x « 1

x » 1

4g(x

)/x3

In(x)1 2 3 4 5

FIGURE 64. Behavior of the angular function appearing in the electron‐electron relaxation

time compared to its asymtotes.


tee ¼ hmA

1

kBT

2

; ð498Þ

where A is a dimensionless parameter of order unity. A comparison of

Eqs. (495) and (498) suggests that the parameter A is approximately

A pR1=4m; where R1 is the Rydberg energy (13.6 eV). For m ¼ 7 eV,

such a comparison suggests A ¼ 1.53 if Ks ¼ 1.

4. A Sinusoidal Potential

While electron‐electron scattering dominates the scattering processes that

affect photoemission in metals, acoustic phonon scattering often dominates

the relaxation time in thermal transport. We now turn attention to its

evaluation. To prepare the way, consider the (far) simpler problem of when

the perturbing potential is simply a sinusoidal (in the x direction) of the form

hrjUepjr0i ¼ aqeiðqxotÞhrjr0i; ð499Þ

where the subscript ep (ep) reinforces that this exercise is in preparation for

treating electron‐phonon scattering. It follows that

Sðk; k0Þ ¼ 2ph

0@

1Ajaqj2 d Eðk0Þ EðkÞ þ hoð Þd k0 kþ qð Þþf

d Eðk0Þ EðkÞ hoð Þd k0 k qð Þgð500Þ

Two types of lattice vibrations occur. When the position of one vibrating

lattice atom is not so far off from its neighbor, then the values of |q| are small

and the dispersion relation isoðqÞ ¼ sq, where s is the sound velocity—such a

mode is termed acoustic.However, if the oscillation of one atom is out of phase

with its neighbors (it is ‘‘up’’ while its neighbors are ‘‘down’’), then oðqÞ o0

¼ a constant; it is generally observed that such vibrations can interact strongly

with light, and so the mode is termed optical. That these branches occur can be

intuited by considering a simple linear model (considered next).

5. Monatomic Linear Chain of Atoms

Consider a 1D chain of atoms of type B (dark) and A (light) schema-

tically illustrated below where they are joined by ‘‘springs’’ with spring

constant g.

un−1 un un+1 un+2

186 KEVIN L. JENSEN

where the deviation from the equilibrium value is indicated by u and the atom

by the subscript (e.g., n). When the atoms are displaced from their equilibri-

um positions, the restoring force they feel is the sum of two springs, as in the

force on atom n being

Fn ¼ g ðunþ1 unÞ ðun un1Þf g ¼ gðunþ1 þ un1 2unÞ; ð501Þwhere g is the spring constant (the use of g in the notation—not a wonderful

choice—rather than the customary k or K, is required as the latter symbols

are reserved for use below). Periodicity is assumed so that the nþj atom

executes the same motion as the jth atom. If the restoring forces between the

atoms depend only on the magnitude of their displacement, then the atoms

act as coupled harmonic oscillators, where M is the mass of the atom; in the

monatomic case, the masses are the same (MB ¼ MA); in the diatomic case,

they are different (and that case shall be handled separately in a subsequent

section). The energy of the linear chain of atoms is therefore

E ¼ h2

2M

XN

n¼1k2n þ

1

2gXN

n¼1ðunþ1 unÞ2: ð502Þ

Quantizing Eq. (502) straight away involves replacing the kn and un with

operators kn and un, respectively, which (as introduced in Section II.C.2)

satisfy the commutation relations ½ul ; kj ¼ idl;j . The odd notation u is per-

haps now appreciated; u is a displacement and is a function of x—and while

the commutation relations are reminiscent of those for x and k, it remains

the case that u is a function of x. The basis states of the atoms are simply the

product of the individual basis states, or

jci ¼ ju1iju2i . . . juNi ju1u2 . . . uNi; ð503Þand the Hamiltonian is

H ¼ h2

2M

XN

n¼1k2n þ

1

2gXN

n¼1unþ1 unð Þ2: ð504Þ

Periodicity dictates that for some integer j,

Hju1þj . . . uNþji ¼ Eju1þj . . . uNþji; ð505Þmeaning that the state with j can differ from the state with j ¼ 0 by at most a

phase factor

ju1þj . . . uNþji ¼ eijjn ju1 . . . uNi; ð506Þwhere jn ¼ 2pn/N, with n an integer.

The coupling of adjacent coordinates in Eq. (502) is somewhat awkward,

as it does not allow for the pleasing definition of creation and annihilation

operators that were gainfully used in the treatment of the harmonic oscillator


when only one oscillator was present. It is therefore profitable to introduce

‘‘normalized’’ coordinates Xn and Kn defined by

Xn ¼ 1ffiffiffiffiffiN

pXN

j¼1uje

2pinj=N ; Kn ¼ 1ffiffiffiffiffiN

pXN

j¼1kje

2pinj=N ; ð507Þ

which are immediately recognized as discrete Fourier transforms of the

position and momentum operators. In an exercise of incomparable pedagog-

ical value, it can be shown

Xj; Kj0 ¼ 1

N

XN

n¼1

XN

n0¼1un; kn

h ie2pij0n0=Ne2pijn=N

¼ i

N

1 e2piðjj0Þð Þ1 e2piðjj0Þ=Nð Þ ¼ idj; j0

ð508Þ

which shows that the normalized coordinates satisfy the requisite commuta-

tion relations sought in the treatment of the harmonic oscillator. Inserting

the normalized coordinates Xn and Kn into the Hamiltonian, the relevant

part of the kinetic energy is transformed to

XN

j¼1k2j ¼

1

N

XN

j¼1

XN

l¼1

XN

l0¼1KlKl0exp 2piðlj þ lj0Þ½

¼XN

j¼1

XN

l¼1KlKl0dll0

¼XN

l¼1KlKl

ð509Þ

where Kl ¼ K l (the * denotes complex conjugation). As shown by direct

substitution, XNn ¼ Xn, and so it is common (and shall be done below) to

take the range of n to be from N=2 to þN/2 instead of 1 to N. The terms

arising in the potential energy require more effort (and where the usage of the

condition of periodicity in the indices is a bit more subtle):

XN

j¼1ujþ1 uj 2 ¼XN

j¼12u2j uj ujþ1 uj uj1

¼XN

j¼12XjXj XjXje

2pij=N XjXje2pij=N

¼ 2

XN

j¼1XjXj 1 cos 2pj=Nð Þ½

ð510Þ

188 KEVIN L. JENSEN

0

(a)

0 10 20 30 40 50

10

20

30

40

Position [a.u.]

wt

n = 4, N = 64

00

(b)

10 20 30 40 50

10

20

30

40

Position [a.u.]

wt

n = 8, N = 64

0 10 20 30 40 500

10

20

30

40

Position [a.u.]

(c)

wt

n = 17, N = 64

FIGURE 65. (Continues)


Using complex conjugation rather than the negative index notation, the

Hamiltonian becomes

H ¼XN=2

n¼N=2

h2

2Mknk

n þ

1

2Mo2

nXnX n

!; ð512Þ

where the frequency on has been introduced and is defined by

o2n ¼ 2

g

M1 cosð2pn=NÞð Þ ¼ g

Msin2

qna

2

ð513Þ

and where the wave number qn ¼ 2pn=Na and the lattice spacing a have

been introduced. For the simple monatomic and isotropic system under con-

sideration in the large N limit, the small‐angle approximation to the sine

function shows oq ¼ vsq, where vs ¼ ga2=4Mð Þ1=2 is a sound velocity. For

example, for iron (M ¼ 0.055845 kg/mole; a ¼ 2.87 A; vs ¼ 4910 m/s) implies

g 117 J/m2. The question of dispersion, that is, the variation of on with qnbeyond the small‐angle approximation, is considered below in the treatment of

the two‐mass chain.

It is instructive to look at the ‘‘modes’’—that is, the Hamiltonian where all

of the normalized coordinates, save for one, are zero. Examples for a chain of

64 atoms are shown in Figure 65 for n ¼ 4, 8, 17, and 32. Time progresses up

the vertical axis, and only a portion of the chain is shown. The gray band

0 10 20 30 40 500

10

20

30

40

Position [a.u.]

wt

n = 32, N = 64(d)

FIGURE 65. (a) Time slices of a chain of equal masses for a low frequency; the gray area

shows a representative time. Clustering of atoms results in a changing density per unit length.

The line joins the regions of highest density. (b) Same as (a) but for twice the frequency; the

density clusterings are closer together. (c) Same as (b) but for a high frequency; the density

clusterings are very close together. The slope of the line (related to sound velocity) is larger.

(d) Same as (c) but for the highest frequency; the density clustering gives way to the masses

oscillating 180 out of phase with their nearest neighbors.

190 KEVIN L. JENSEN

represents a particular time slice, or snapshot, of the atoms’ position, and the

dashed lines represent the locations of the centers of the atoms. Inspection

reveals that clusters of atoms tend to form and that the center of those

clusters migrates as time increases. For small n, the clusters moving in a

particular direction contain a large number of atoms, wherein the direction

an atom is moving is likely to be the same as its neighbors. For larger n, the

clusters contain fewer atoms until n¼ N/2, at which point the atoms oscillate

around their zero point and out of phase with their immediate neighbors.

The advantage of Eq. (513) is that in the normalized coordinates the

Hamiltonian does not have cross terms among the Xn, so that the interpreta-

tion in terms of creation and annihilation operators, using the formalism

described in Section II.A.3.b , for each index n, is possible. What is being

annihilated and created are phonons, and the Hamiltonian in Eq. (513)

simply counts the number of phonons with each index n and sums their ener-

gies. This can be seen explicitly by [in complete analogy with Eq. (165)]

defining the operators

a ¼ ikn þ anX n

=ffiffiffiffiffiffiffi2an

p

an ¼ ik n þ anXn

=ffiffiffiffiffiffiffi2an

p ð514Þ

where an ¼ mon=h Using the commutation relations in Eq. (508), it follows

that

H ¼ 1

2hon

XN=2

n¼N=2ana

n þ anan

¼ hon

XN=2

n¼N=2anan þ

1

2

: ð515Þ

Switching to a number representation basis, where (compare Eq. (38) and

the following discussion)

anj1 . . . n . . .Ni ¼ ffiffiffin

p j1 . . . ðn 1Þ . . .Nianj1 . . . n . . .Ni ¼ ffiffiffiffiffiffiffiffiffiffiffi

nþ 1p j1 . . . nþ 1ð Þ . . .Ni ð516Þ

This allows for the identification of the mean occupation number for

the oscillators connected to a heat bath of temperature T to be calculated.

Switch from the n to the q notation suggested by Eq. (513) in preparation

for the ultimate shift to integrations over wave number (where q becomes the

relevant wave vector). Then, using hnqjaqaqjnqi ¼ nq; the average energy is

given by


U ¼X

qhoq

Xnq

nq þ 1

2

8<:

9=;ebðnqþ1=2Þhoq

Xnqeb nqþ1=2ð Þhoq

0BBBBBB@

1CCCCCCA

¼X

qhoq

1

eboq 1þ 1

2

0@

1A

Xqhoq hnðoqÞi þ 1

2

0@

1A

ð517Þ

The extension from a 1D chain to the 3D continuum limit then takes

Eq. (517) into Eq. (452). Such a cavalier transition from a chain of atoms

to a lattice of atoms should provoke unease [for example, there may be more

than one atom per unit cell, motivating the factor of r in Eq. (455)], but a

systematic analysis puts such arguments on better footing (the point here,

rather, is to suggest the plausibility of the transition). The point is to arrive at

the investigation of a lattice specific heat, having justified the notion of the

harmonic oscillator approximation and indicating why the Debye frequency

is limited by the number of modes (i.e., number of atoms).

To reiterate a central feature of phonons that affects the calculation of the

scattering terms below: no limit as to the number of phonons in a particular

state exists (that is, nj is not restricted to 0 or 1, as for fermions). The

occupation factors associated with electron‐phonon scattering calculations

below must account for the BE statistics of phonons. The actual mechanics

are a bit involved and will be considered after the modifications associated

with a linear chain with two types of atom have been more fully discussed.

The use of the terminology sound velocity (how fast disturbances move in

the lattice) is naturally related to the description of phonons as ‘‘acoustic’’

phonons. The picture is more complicated, however, and the subtle changes

introduced by allowing adjacent atoms to be of different mass allow for a

derivation of the dispersion relation in Eq. (513) by other means that help

illuminate the physics. Rather than using the quantum‐mechanical model

that was used previously (treated in full 3D glory by Ziman, 1985, 2001),

reconsider the problem from a classical perspective, where a harmonic force

exists between adjacent atoms. The acceleration of the nth atom €xn ¼d2xn=dt2 is related to the forces acting on it; Eq. (501) then becomes two

equations for each of the different masses, taken to be the n and (n þ 1)

atoms:

MA€un ¼ g unþ1 þ un1 2unð ÞMB€unþ1 ¼ g unþ2 þ un 2unþ1ð Þ ð518Þ

192 KEVIN L. JENSEN

with similar equations holding for other adjacent pairs of atoms. The con-

sequences of Eq. (507) suggest that solutions of the form

un ¼ CAexp i ot nqað Þf gunþ1 ¼ CBexp i ot ðnþ 1Þqað Þf g ð519Þ

are applicable—not surprising for harmonic oscillators. With the substitu-

tion of Eq. (519) into Eq. (518), the following equations follow

o2CA ¼ 2g

MA

CA CBcosðqaÞð Þ

o2CB ¼ 2g

MB

CB CAcosðqaÞð Þð520Þ

For solutions to exist, the determinant of the matrix of coefficients must

vanish, and so

0¼o2 2

g

MA

2g

MA

cos qað Þ

2g

MA

cos qað Þ o2 2g

MB

¼ o4 2g

MA þMB

MAMB

0@

1Ao2 þ 4g2

MAMB

sin2 qað Þ

ð521Þ

0.0

0.5

1.0

1.5

0 30 60 90

+ (r = 1)

− (r = 1)

+ (r = 0.5)

− (r = 0.5)

+ (r = 0.25)

− (r = 0.25)

(w±/w

g)

qa [degrees]

Optical

Acoustic

FIGURE 66. Origin of the acoustic and optical branches of a linear chain composed of two

types of atoms whose masses are related by the value of r.


Solutions are as follows:

o2 ¼ g

MA þMB

MAMB

1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 4MAMB

sin qað ÞMA þMB

2s8<

:9=;: ð522Þ

For the special caseMA¼MB, then o2 corresponds to o2n in Eq. (513), but

now another solution exists that did not occur for metals in the earlier

derivation. The behavior of o=og

2, where o2

g ¼ g=M and the ‘‘reduced’’

massM1 ¼ M1A þM1

B (or half of the mass of one of the atoms when their

masses are equal) is sought. Let the mass ratio MA=MB r, then Figure 66

shows for the linear chain the dimensionless relation

oog

2

¼ 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 4r

sin qað Þ1þ r

2s

: ð523Þ

The ‘‘–’’ branch is designated acoustic and the ‘‘þ’’ is designated optical (so

named because in ionic crystals, the oppositely moving atoms create a

polarization that can interact with light). In monovalent metals or crystals

with but one atom, the optical mode does not appear. Finally, note the near‐linearity for small qa for the acoustic mode. In three dimensions, a similar

Figure results, albeit that the acoustic mode for a 3D lattice is not as linear as

implied in Figure 66.

Optical phonons, by virtue of their higher frequency, are of higher ener-

gy—and their satisfying BE statistics implies that the average number of

optical phonons is well below the number of acoustic phonons at generic

temperatures. The particle representation of phonons allows the interaction

of electrons and phonons to be treated as a particle‐particle collision, in

which an electron gains or loses energy in the absorption or creation, respec-

tively, of a phonon. If the energy of the acoustic phonon is small, then to

leading order from the BE distribution in Eq. (517), it follows

hnðoÞijhokBT kBT

hoð524Þ

(where the q subscript (q) on o has been suppressed), indicating, as physical

intuition suggests, that as the temperature increases, so too does the number

of acoustic phonons. With more phonons, more scattering occurs during

current flow, and the expectation is therefore that the resistivity of the metal

will likewise increase. Eq. (524) therefore suggests that the acoustic phonon

relaxation time tep (which is inversely proportional to the resistivity of the

metal) should scale inversely with temperature. This holds for sufficiently

high temperatures, and the effort to quantify what is meant by ‘‘sufficiently’’

is a matter of some effort, as detailed below.

194 KEVIN L. JENSEN

6. Electron‐Phonon Scattering

In terms of the normalized coordinates introduced in Eq. (507), the oscilla-

tions described by the normal modes entail variations in the local density.

Consider a group of atoms in a volume Vo in their unperturbed state that,

through their motion, occupy a volume V when one of their modes is

stimulated. If the total mass of the crystal is M, then the change in density

is given by

dr ¼ M

V M

Vo

¼ MV Voð ÞVVo

MDVV 2

o

¼ riDVVo

; ð525Þ

where ri is the number density of the crystal lattice and the term

DV V Vo has been introduced. The fractional change in volume

DV=Vo is therefore related to the fractional change in density dr=ri. Atoms

moving across a surface dA, that is, x dA, cause the density to change and so

the integrated fractional change in volume can be recast asðdrri

dV ¼ ðx dA ¼

ð= xð ÞdV ; ð526Þ

where the last step exploits the relation between integration over surfaces to

integrations over volume involving vectors (Gauss’ theorem). Eq. (526) as

much as identifies dr ¼ ri= x. This is important because the ions are

slowly lumbering behemoths compared with the agile electrons, so that at

all times, the electrons can be assumed to be exactly following the lattice

dynamics much like flies about the plodding animals they torment; this

means that whatever changes in the lattice density occur, those changes are

reflected in analogous changes in the electron density. But changes in electron

density can be related to changes in the electrochemical potential, and

therefore, changes in the potential f under which the electrons move. As-

suming that the distribution is well approximated by the FD distribution

with an electrochemical potential given by mðxÞ ¼ mo þ fðxÞ, it follows

(where now r refers to electron, not lattice, number density)

dr¼ rðxÞ ro ¼ 2 2pð Þ3 Ðdk fFD EðkÞð Þ f oFD EðkÞð Þ

2 2pð Þ3 Ðdk fðxÞ@Ef oFD EðkÞð Þ

bfp2

ðk2dkd

bh2

2mk2F k2 2

435¼ mkF

p2h2fðxÞ

ð527Þ

where the replacement of the gradient of the FD distribution with a delta

function relied on the peakiness of @E f and the presumed smallness of f.


On the other hand, from Poisson’s equation

@2xfðxÞ ¼

q2

eodr ¼ q2

eo

mkF

p2h2

fðxÞ k2TFfðxÞ; ð528Þ

where the Thomas–Fermi parameter k2TF ¼ q2=e0ð Þ@mr is familiar from

Eq. (32). Thus, variations in the electron density caused by oscillations of

the normal modes give rise to a potential in which the electrons move, and it

follows that electron‐phonon scattering is defined by the relation by

Vep xð Þ ¼ riq2

eok2TF= u: ð529Þ

The coefficient may have been anticipated—when electron densities are

high, then a shielded Coulomb potential such as exists about the ions has

more of the appearance of a Dirac delta function:

q2

4peojx x0j ekTF jxx0 j ) q2

eok2TFd x x0ð Þ ð530Þ

in the strong shielding limit (recall Eqs. (470) and (471)). Although such a

relation is simple, it is not the one most commonly used; more often, the

sound velocity appears in the coefficient of Eq. (529) instead. Developing an

expression for the sound velocity requires a detour to develop the Bohm–

Staver relation. A few different methods can be used to accomplish that

result.

The first method is to work from the plasma frequency and observe that

the electrons are tracking the motion of the ions. The plasma frequency is

given by

op ¼ q2ri=eðkÞM 1=2

; ð531Þ

where M and ri are the mass of an ion and density of the ions, respectively,

and where eo has been replaced by eðkÞ; the trick is then finding eðkÞ. A cloud

of electrons around the ion potential gives rise to a ‘‘screened’’ Coulomb

potential [recall the form of Eq. (470)]. From the relation UðkÞ ¼ UextðkÞ=eðkÞ, where U is the Fourier transform of the Coulomb potential and Uext is

the unscreened potential (kTF¼ 0), for which the Fourier transform (ignoring

factors of 2p) is ðO

q2

4pe0rekTF rei k rd r ¼ q2

e0k2TF þ k2 1

: ð532Þ

196 KEVIN L. JENSEN

The dispersion relation is then given by

dopðkÞdk

2

¼ riq2

2Me0k2TF1þ k2

k2TF

1

: ð533Þ

The velocity is the k ¼ 0 limit. Charge neutrality relates the ion number

density to the electron number density via a factor for the valence Z, set here

to 1. It follows

vs ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiri

q2

2Me0k2TF

s¼ 1ffiffiffi

3p hkF

M

¼

ffiffiffiffiffiffiffiffi2m3M

r: ð534Þ

The second approach is from statistical mechanical arguments. The sound

velocity is related to the compressibility w of a gas via

v2s ¼ wmrð Þ1; ð535Þwhere

w ¼ @P lnðV Þ ð536Þand where P, V, and r¼ N/V are the pressure, volume, and number density,

respectively. Ions can be imagined as acting like massive electrons, and what

is said about the latter therefore has bearing on the former. From the

distribution function approach of Section I, it is known that the total energy

of a gas is related by

EN ¼ V 2pð Þ3

ðV

dkfFD EðkÞð Þ Vh rð Þ; ð537Þ

where the function h(r) is (temporarily) defined by this relation and the

relation between chemical potential and number density. Pressure, which is

defined by P ¼ @VEN , can be expressed therefore as

P ¼ h rð Þ þ rh0 rð Þ; ð538Þwhere prime indicates derivative with respect to argument. Combining

Eqs. (536) and (538), it follows that [usingV@VP ¼ r2h00ðrÞ]

1

w¼ r2h00 rð Þ ¼ Vr2@N h0ðrÞjV¼const

: ð539Þ

Recalling that m is the change in energy when an additional particle is

added, and from Eq. (537) that m ¼ @NEN ¼ @rhðrÞjV¼const, it immediately

follows that


1

w¼ Nr @Nmð ÞjV¼const ð540Þ

and so

v2s ¼N

m@Nmð ÞjV¼const ¼

rm

@mr 1 ¼ 2m

3m; ð541Þ

where the last relation follows from the zero‐temperature limit of Eq. (16).

This is the sound velocity of an electron gas. The ions, however, are behaving

like massive positive particles, and so the last step is to replace m withM and

recover the previous result of Eq. (534); alternately, one may consider the ion

background as jellium in its own right and the derivation of Eq. (541) is

unchanged but for the usage of M instead of m. Consequently, the Bohm–

Staver result, that the sound velocity can be expressed in terms of the

chemical potential, follows.

The Bohm–Staver relation allows Eq. (529) to be written

Vep xð Þ ¼ riq2

eok2TF= u ¼ 4

9

m2

Mv2s= u ¼ 1

3m= u: ð542Þ

In other words, if the Bohm–Staver relation holds, then the deformation

potential X is simply one‐third of the Fermi energy. However, this cannot be

quite correct—beyond the ion valency Z (which has been ruthlessly ignored),

there is the variation that must be expected in the lattice coupling constant g

from material to material. Something like the Bohm–Staver relation, how-

ever, should apply, modifying the RHS of Eq. (542) by a

TABLE 9

SOUND VELOCITIES OF VARIOUS METALS*

Element Atomic number vs (exp) [m/s] vs (BS) [m/s] [vs(BS)/vs(exp)]2

Na 11 3200 3006 0.882

Mg 12 4602 4344 0.891

Al 13 6420 5265 0.673

Fe 26 5950 3576 0.361

Cu 29 4760 2662 0.313

Ag 47 3650 1808 0.245

Ba 56 1620 1308 0.651

W 74 5174 2515 0.236

Au 79 3240 1341 0.171

*The Bohm–Staver relationship for sound velocity is vs ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2m=3M

p, where m is the chemical

potential and M is the mass of a lattice atom.

198 KEVIN L. JENSEN

dimensionless constant called, say, l. Since the sound velocity is the likely

culprit of differences between X and m, then l should be a ratio of energies—

one being the kinetic energy of the lattice and the remaining energy being the

Fermi energy, as deduced from the coefficient of Eq. (542). Therefore, let

Vep xð Þ ) 4

3lm= u

l Z

3

mMv2s

ð543Þ

where l ¼ 1/2 if the Bohm–Staver relation holds. Table 9 compares the

commonly accepted sound velocity with the sound velocity predicted by the

Bohm–Staver relation for several metals, along with the values of lexp, where‘‘exp’’ indicates the empirical value. A ‘‘good’’ metal like sodium has reason-

ably good agreement, whereas the agreement with the transition metals is

spottier.

The time is now ripe to switch to the canonical variables X and K that

describe the phonons, but difficulties immediately present themselves. First,

u is a real quantity, and so the urge to make a trivial multidimensional gener-

alization of Eq. (507) is ill advised. Second, the potential is the divergence of

u, but that is unlike the construction leading to the creation and annihilation

operators as done for the 1D case in Eq. (514). More care is indicated, and in

fact, much more care is required. It can be found elsewhere (e.g., Wagner and

Bowers, 1978); here, a brief sketch must suffice.

If u describes a wave phenomenon, then it vanishes when acted on by

the D’Alambertian operator =2 c2d2t ; that is, u ¼ 0, where the

velocity c is replaced here by vs (see Chapter 10 of Goldstein, 1980).

Equation (543) does not entail =2u but rather it would give rise to = = uð Þ,and if u ¼ 0 is to hold, then several consequences result. First, using the

relation

=2uðx; tÞ ¼ = = xð Þ = = xð Þ ð544Þindicates the second term must vanish if Eq. (543) is to be exploited. Second,

the relation

uðx; tÞ ¼ 1

V

Xk>0

Xkei kxo k

!tð Þ ð545Þ

subsequently entails that k k Xk. Both observations entail that shear and

vorticity do not occur, and the k > 0 condition on the summation in

Eq. (545) implies that an overall movement (translation) of the medium is

foresworn. In practice, these conditions mean that the proper generalization

of the 1D problem to 3D gives


Xk ¼ffiffiffiffiffiffiffiffiffiffiffiffiffi

h

2Mok

sake

iokt þ ake

ioktn o

; ð546Þ

from which it follows

u x; tð Þ ¼Xk>0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffih

2MokriV

sake

i kxoktð Þ þ ake

i kxoktð Þn o

; ð547Þ

where the direction of u is understood to be in the direction of k; the notation

could be better (but is not). Finally, note that ri is the ion number density, so

that riV ¼ N ¼ the number of ions, which harkens back to the definitions

introduced in Eq. (507).

With Eq. (547), much follows in short order. First and foremost is the

electron‐phonon potential interaction from Eq. (543), or

Vep xð Þ ¼ 4

3lm= u

¼ 1ffiffiffiffiV

pXk>0

akakei kxoktð Þ þ a kake

i kxoktð Þn o

ak i4

3lm

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffihok

2Mv2sri

s ; ð548Þ

where jakj2 ¼ mhok=3r when l ¼ 1/2. The coefficient of the divergence of u

has units of energy; the form here of Vep ¼ ð4=3Þlm= u is not the one

typically encountered in practice. Elsewhere (e.g., Ridley, 1999) it is instead

written Vep ¼ w= u, and X is known as the deformation potential. This shall

be returned to in due time, but it is worth noting here that semiconductors,

unlike the metals so far considered, have different properties along different

crystal axes, one example being effective mass and another being that the

deformation potential exhibits elastic anisotropy.

The matter of the creation and annihilation operators must now be con-

sidered, which in turn means greater attention to the initial and final states.

The approach here will be to consider one of them, then using Feynman

diagram–like arguments, to infer the remainder. The relaxation time involves

the evaluation of a term resembling jh f jUepjiij2, but what constitutes the

initial and final states needs examination. Clearly, there are momentum

states jki in the initial state jii, but by virtue of the harmonic oscillator

creation and annihilation operators in Eq. (548), there are also phonon

states jni. This leads to identifying jii ) jkijni, where the first ket refers to

the momentum states of the electrons and the second to the occupation

number of the phonons. Consider now what is meant by the first term in

200 KEVIN L. JENSEN

Eq. (548) sandwiched between the initial and final states in the circumstance

when the initial state is acted on by an annihilation operator. It follows

h f jak eikxakjii ¼ hk2jakeikxjk1ihn2jakjn1i¼ ÐÐdx1dx2hk2jx2ihx2jakei kxjx1ihx1jk1i ffiffiffiffiffiffiffiffiffiffiffiffi

n okð Þp¼ Ð dx1akei kþk1k2ð Þx ffiffiffiffiffiffiffiffiffiffiffiffi

n okð Þp¼ akd k2 k1 kð Þ ffiffiffiffiffiffiffiffiffiffiffiffi

n okð Þpð549Þ

The delta function ensures that jakj ) jak2k1 j. In particular,

S k2; k1ð Þ ¼ 2phjak2k1 j2d E2 E1 ho k2k1ð Þ

: ð550Þ

In finding the collision integral, the occupation of the initial state (given by

f1) and the vacancy of the final state (given by 1 – f2) figure in a form that

resembles ðdk1S k2; k1ð Þf1 1 f2ð Þn oq

; ð551Þ

where q k2 k1 for the subscript on o and the sign subscript on S refers to

the sign of k in the delta function embedded within S. There is a diagram-

matic interpretation of integrals of the form in Eq. (551): straight lines refer

to electrons and carry factors of f going into a vertex, or factors of 1 – f when

leaving a vertex; wavy lines refer to phonons and carry factors of n oq

when

going into a vertex or factors of n oq

þ 1 when leaving a vertex; and the

vertex itself carries a factor of S. An overall conservation of momentum

delta function is buried in S and originates in Eq. (551), whereas an overall

conservation of energy delta function from Fermi’s golden rule is appended

at the end. In Feynman‐diagram parlance, this amounts to the Feynman

diagrams represented in Figure 67: q is the difference between the entering

and exiting momenta, the electrons are straight lines, the phonons are wiggly

1 − f2 1 − f2

f1

S(k2,k1)

S(k2,k1)n(w)

S(k2,k1)f1(1−f2)n(wq) S(k2,k1)f1(1−f2)n(wq) + 1

f1

n(w ) + 1

FIGURE 67. Feynman diagrams for the phonon interaction terms.


lines, and the interaction is the center circle. There are a total of four such

diagrams: two with entering phonons, two with exiting, and a shuffling of

the Fermi lines. In the example shown in Figure 67, the absorption case is

the Figure on the left; the emission case is the Figure on the right. A careful

collection of terms then shows that the collision integral is of the form

@c f k2ð Þ ¼ 2pð Þ3

ðdk1 Sþ k1; k2ð Þ½ f2 1 f1ð Þ 1þ n21ð Þ f1 1 f2ð Þn21f

þS k1; k2ð Þ½ f2 1 f1ð Þn12 f1 1 f2ð Þ 1þ n12ð Þgð552Þ

where the shorthand of Eq. (482) is used, a new shorthand nk1k2 n12 is

introduced, and all other gremlins of notation can be inferred. Reducing this

equation to a simpler form takes additional work.

In writing Eq. (552), it was tacitly assumed that the S factors did not

depend on the order of the vector arguments. As natural as that may naıvely

seem, it is not generally true, but rather the case only for potentials of a

general type—namely, those unaffected by time reversal and inversion—but

showing why that means that S is insensitive to the order of its arguments is

rather complicated. Intuitively, it is known that time reversal does not affect

position but reverses the sign on momentum, as the velocity hk=m ¼ dx=dtchanges sign because dt changes sign, and that observing an interaction with

the clock running backward switches the entering and exiting momenta.

Neither intuition nor belief is proof: proof demands more. Invariance under

spatial inversion (‘‘parity’’) focuses on the consequences of an operator upthat behaves according to

Up

xi ¼ xihkupxi ¼ hk xi ¼ hk

xi ð553Þ

where the second line is a consequence of the 3D generalization of Eq. (38).

Similarly, it is obvious by repeated application that u2p ¼ I , where I is the

identity operator. Recall that an operator that is a constant of the motion

commutes with the Hamiltonian, implying that

UpHU1p ¼ H: ð554Þ

By inspection the kinetic energy operator is unaffected by up because

k

h2k22m

k0* +

¼ h2k2

2md k k0ð Þ ¼ k

h2k22m

k0* +

: ð555Þ

202 KEVIN L. JENSEN

Therefore the Hamiltonian’s invariance with respect to inversions requires

an examination of the potential term Vep. Using Eq. (549), it follows that

hxVep

x0i ¼ hxUpVepUp

x0i: ð556ÞThe LHS is

hxVep

x0i ¼ ðdkdk0hxkihkVep

k0ihk0x0i/ðdkdk0hxkihkq

k0ihk0x0i ð557Þ

From Eq. (557), a factor ofakk0

2 appears. The RHS of Eq. (556) is

slightly more work and yields

hxUpVepup

x0i / Ð dkdk0hxkihkq

k0ihk0 x0i/ Ð dkdk0hxkihk

q

k0ihk0x0i ð558Þ

and therefore yields a factor of jakþk0 j2. Invariance with respect to inversion

therefore allows us to conclude

S k; k0ð Þ ¼ S k;k0ð Þ: ð559ÞTime reversal is slightly more involved. If a wave function exists (assume

c is an energy eigenstate) such that the actions of time reversal result in

cðx; tÞ ) cðx;tÞ, then the operator formalism contains some interesting

differences. Observe that c ðx;tÞ satisfies the same (Schrodinger) equation

that cðx; tÞ does, and therefore the time‐reversal operator entails complex

conjugation as per Tcðx; tÞ ¼ c ðx;tÞ. In bra‐ket notation, given thatcðtÞi ¼ exp iHt=h cð0Þi; ð560Þ

it follows that

hxTcðtÞi ¼ hxt cðtÞi ¼ hxcðtÞi ¼ hcðtÞxi; ð561Þwhere the action of the lower‐case t operator is to simply change t to –t. Time

reversal T therefore entails the actions of t and complex conjugation; the

latter observation makes incoming particles appear to be outgoing when time

is reversed. We have

hcf ðtÞciðtÞi) hcf

T2cii ¼

ðdxhcf ðtÞ

TxihxTciðtÞi

¼ðdxhcf ðtÞxi hxciðtÞi ¼ hciðtÞcf ðtÞi

ð562Þ

where the facts that T is its own inverse andÐdxxihx ¼ I (apart from

factors of 2p) have been used. Reversal of time therefore entails that the


initial and final configurations switch, as intuitively expected but now proven.

The analog of Eq. (556) must now be checked (the invariance of the kinetic

energy operator under the action of T being trivially shown):

hcf ðtÞTVepT

1ciðtÞi ¼ hciðtÞVep

cf ðtÞi: ð563ÞTo determine the impact of time reversal on S, then, the action of T on the

momentum kets requires consideration. A short analysis shows

hkTcðtÞi ¼ ð dxhkxihxTcðtÞi¼ðdxhx kihcðtÞxi ¼ hcðtÞ ki

ð564Þ

after which it is straightforward to demonstrate that

hcf ðtÞTVepT

1ciðtÞi ¼

ðdkdk0hciðtÞk0ihk0

q

kihkcf ðtÞi: ð565Þ

Invariance to time reversal therefore entails

S k; k0ð Þ ¼ S k0;kð Þ ð566Þ(a related consequence is that oq ¼ oq). Consequently, the combined action

of Eqs. (559) and (566) shows that if the potential is both time and inversion

invariant, then

S k; k0ð Þ ¼ S k0; kð Þ; ð567Þa result that, however intuitive, did require some effort to show. Less obvious

but soon to be appreciated is that the outcome allows substantial simplifica-

tion of the equation for electron‐phonon scattering, and so we now return to

the consideration of Eq. (552).

It is seen immediately that Eq. (567) entails that @c f k2ð Þ ¼ 0 when the

distributions are replaced by equilibrium FD distributions—as it should.

After all, this is what is meant by equilibrium—that as many particles scatter

into as out of a particular state (and why Eq. (567) was intuitively expected).

Now consider deviations from equilibrium, so that

f k2ð Þ ¼ f Eðk2Þð Þ þ dk1;k2df k1ð Þ ¼ f2 þ dk1;k2df ; ð568Þfor which @c f k2ð Þ ) df =t k2ð Þ and the RHS of Eq. (552) becomes more

complicated. The following observation(s) help: if fj are FD, and n BE

distributions, then using the notation xj b EðkjÞ m

(for a moment) and

fj ¼ 1þ e xjð Þ1; nij ¼ exp½bhokikj 1 1

; ð569Þit can be shown that (it is emphasized that subscripts in the compact notation

204 KEVIN L. JENSEN

nij have a different meaning than they do for fj)

f1 f2 ¼ 1 f2ð Þ f1n12

¼ þ 1 f1ð Þ f2n21

ð570Þ

where either the top or bottom relation holds. The top will be shown

explicitly (and the proof of the bottom left to independent confirmation):

f1 f2 ¼ ex2 ex1

1þ ex1ð Þ 1þ ex2ð Þ ¼ 1 ex1x2ð Þ1þ ex1ð Þ 1þ ex2ð Þ ; ð571Þ

where the final transition, from Eq. (571) to Eq. (570), involves resubstituting

Eq. (569). It is straightforward, albeit requiring care, to show that with the

substitution f1 ) f1 þ df and all fj are then taken to be FD distributions,

it follows

f2 1 f1 þ dfð Þð Þ 1þ n21ð Þ f1 þ dfð Þ 1 f2ð Þn21 ¼ df f2 þ n21f gf2 1 f1 þ dfð Þð Þn12 f1 þ dfð Þ 1 f2ð Þ 1þ n12ð Þ ¼ df f2 n12 1f g

ð572Þ

which is as it should be; the terms independent of df cancel for FD statistics.

Insertion of Eq. (572) into Eq. (552) produces

1

tep k2ð Þ ¼2ph

1

2pð Þ3ðdk1 ak2k1j j2 f1 þ n21ð Þd E2 E1 þ hok2k1ð Þf

þ 1þ n21 f1ð Þd E2 E1 hok2k1ð Þg ð573Þwhere use has been made of oq ¼ oq and n2–1 ¼ n1–2. It is profitable to

switch from k1 as the integration term to q ¼ k1 k2, for which (where the

expanded notation has been reverted to)

1

tep k2ð Þ ¼2ph

1

2pð Þ3ðdqjaqj2 fo E2 þ hoq

þ n hoq

d E2 E1ðqÞ þ hoq

þ 1þ n hoq

fo E2 hoq

d E2 E1ðqÞ hoq

g ð574Þwhere the substitutions made in the FD functions are allowed by the

arguments of the Dirac delta functions. The arguments of the delta functions

appear to obfuscate matters, but on closer examination, we see that


E2 E1ðqÞ hoq ¼ h2

2mq2 þ 2k q hoq

h2k

mvsoqcosy hoq

¼ hovk

vscosy 1

0@

1A

ð575Þ

here vk ¼ hk=m is the electron velocity and q2 is taken to be negligible. The

argument of the delta functions, therefore, concerns the angle between k and q,

rather than the magnitude of q. The angular integration can therefore be

done separately from the q integration. Using d axð Þdx ¼ d yð Þdy=jaj (wherex, y, and a are dummy terms), Eq. (548) to find jaqj2, using dq ¼ 2pq2dqdx,recalling that vs ¼ o=q, suppressing the q subscript on o (because oq ¼ ojqj,there is no further utility in retaining it), collecting terms, and using Eq. (548),

then Eq. (574) becomes

1

tep kð Þ ¼1

4p2h2pð Þv3

s lp2h2

kFm

0@

1Að o2do hoð Þ1 vs

vk

ð11

dx d x vs

vk

0@

1A½1þ n f o þ d xþ vs

vk

0@

1A½nþ f þo

8<:

9=;

ð576Þ

where E ¼ E(k), n ¼ n hoð Þ, and f o ¼ fo E hoð Þ. Collecting terms and

noting that the second integral is simply the product of the factor in square

brackets withY vk vsð Þ results in1

t kð Þ ¼2lp

2kFvsð Þ2ðo2J E;oð ÞdoY vk vsð Þ

J E;oð Þ ½1þ 2n hoð Þ fo E hoð Þ þ fo E þ hoð Þð577Þ

which is fundamentally the same as the form given in Wagner and Bowers

(1978). We restrict attention to those cases in which the electron velocity

exceeds the sound velocity to dispense with the Heaviside step function.

Using the definitions of the BE and FD distributions, it is a straightforward

exercise to show that

J E;oð Þ ¼ exo þ 1ð Þ exE þ 1ð Þ2exEþxo þ 1ð Þ exExo þ 1ð Þ exo 1ð Þ

¼ cosh xEð Þ þ 1f g cosh xoð Þ þ 1f gcosh xEð Þ þ cosh xoð Þf gsinh xoð Þ

ð578Þ

206 KEVIN L. JENSEN

where the shorthand xE ¼ b E mð Þ and xo ¼ bho has been used, and

where the second line is intentionally reminiscent of Eq. (15) in Gasparov and

Huguenin (1993), whereas the formulation in the third line makes better use of

the inherent symmetry and asymptotes.

Several consequences immediately follow that are of interest here, and

they relate to the temperature dependence of the scattering rate at the

Fermi level and the energy dependence for low temperatures. We deal with

the temperature dependence at the Fermi level first. For E kð Þ m, thenJ m;oð Þ ¼ 2=sinh bhoð Þ. It follows that

1

tep kFð Þ ¼lp

kFvsð Þ2ðoD

0

o2

sinh bhoð Þ do

¼ lp

h hkFvsð Þ2 kBTð Þ3ðxD0

x2

sinh xð Þ dxð579Þ

where xD ¼ bhoD ¼ TD=T and where oD and TD ¼ hoD=kB are the Debye

frequency and temperature, respectively. The upper limit of the integral

assumes that oD /vs is smaller than kF; if this were not the case, then the

upper limit of the integral would have to be xF ¼ bhvskF . For large xD, theintegral in the second line is 7zð3Þ=2, whereas for small xD, it becomes x2D=2,which indicates that for low temperatures, tep xD 1ð Þ / T3, but for high

temperatures, tep xD 1ð Þ / T1. At room temperature for rather generic

parameters, the relaxation time is tens of femtoseconds. Finally, the units of

(kFvs) are [fs1], which offsets the units of o in the integrand— an observa-

tion useful when considering higher‐order powers of o to be encountered

in the momentum relaxation time below.

The high temperature relation is an oft‐used and well‐known result

(Jensen, Feldman, Moody, and O’Shea, 2006a; Papadogiannis, Moustaizis,

and Girardeau‐Montaut, 1997), a commonly used form being given by

tep k ¼ kFð Þ h

2pl0

1

kBT

; ð580Þ

where the ‘‘0’’ on l reiterates that it is different than the l used in Eq. (579)

by a matter of a few constant factors—in particular, for xD « 1, then lo¼ l/4.The low‐temperature limit implied by Eq. (579) needs modification for

the momentum relaxation time (which has bearing on the resistivity),

however, where a factor of 1 cosy must be inserted into the integrand.

The relationship

1 cosy ¼ 2sin2 y=2ð Þ; ð581Þ


where y is the angle between the initial and final momentum states, plus the

relation between the initial and final states at the Fermi level given by

q ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik1 k2 2q

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2k2F 1 cosyð Þ

q¼ 2kF sin

y2

ð582Þ

allows the factor in Eq. (581), when put into the relaxation time as per

Eq. (468), to result in an additional factor of o2 in the integrand, and

therefore, an additional factor of T2 in Eq. (579). The resulting low‐temperature dependence of the relaxation time as tep k ¼ kFð Þ / T5 is

additionally well known and referred to as Bloch–Gruneisen behavior.

The proper limits have now been ascertained. As a historical matter, a

detailed exposition of the Bloch–Gruneisen formulation can be found in

Wilson (1938), who observes with unflappable British understatement, that

‘‘(t)he formula we have just derived is so complicated that we must make

some approximations before going any further.’’ Wilson, after making the

promised and reasonable approximations, arrives at the conclusion that

the relaxation time is (a variation of his Eq. (18))

1

tep kFð Þ 8pl0kBTD

h

T

TD

0@

1A5 ðTD=T

0

x5

ex 1ð Þ 1 exð Þ dxþOT

TD

0@

1A32

435

¼ 8pl0kBTD

h

T

TD

0@

1A5

W 5;TD

T

0@

1A ð583Þ

The second term of the first line (the one we labor to ignore) corresponds to

the temperature equivalent of the minimum energy necessary to excite s‐dtransitions and is not considered further. The coefficient containing l0 is

made to converge with the common high‐temperature representation of te‐p.In the literature, W–(n,x) is designated as Jn(x) and referred to as the

Bloch–Gruneisen function (Fuller, 1974; Westlake and Alfred, 1968; White

and Woods, 1959). Note also that the idealized l of the Bohm–Staver

relation has been replaced with l0. Although a relation between the two

was suggested previously (specifically, in Eq. (580) and the discussion follow-

ing it), any connection in the future is casually ignored, as l0, used to

generalize from the Bohm–Staver relation but not otherwise specified, must

be pressured to take on whatever constants that additionally accrue in the

transition from Eq. (579) to (583); for our present purposes, it is sufficient to

know that its value is assigned below using comparisons to experimental data.

208 KEVIN L. JENSEN

100

10−1

10−2

101

100

(a)

1000

CuCu (BG)AgAg (BG)AuAu (BG)

PbPb (BG)MgMg (BG)WW (BG)

Res

istivi

ty [mΩ

-cm

]

Temperature [K]

−8

−6

−4

−2

0

2

0.1

(b)

101

BG

Ag

Au

Cu

Mg

W

Na

Fe

Al

Inr

(T)/r(

TD)

T/TD

0.0

100

200

300

400

500

Na

(c)

Mg Al Fe Ni Cu Mo Ag W Pt Au Pb

Kittel

Least squares

Deb

ye t

empe

ratu

re [K

]

Element

Increasing atomic number



The performance of Eq. (583), caveats and all, is rather spectacular.

If acoustic phonon scattering dominates (as it does at the Fermi level), then

the ratio of the scattering rate at temperature T to that at temperature TDwill

be the same as the ratio of the resistivities for the same temperatures.

Consequently, metals should exhibit a universal curve when t(T)/t(TD) is

plotted as a function of T/TD—given that the resistivity for metals is domi-

nated by acoustic phonon scattering, it follows that the ratio of the resistiv-

ities as a function of T/TD should likewise exhibit a universal characteristic of

r T ;TDð Þ ¼ r To;TDð Þ T

To

5W 5;TD=Tð ÞW 5;TD=Toð Þ ; ð584Þ

where To is a suitably chosen reference temperature. An example of the

performance of Eq. (584) for a variety of metals is shown in Figure 68,

where To is the highest temperature for which a resistivity value is available,

the ‘‘hybrid’’ approximation toW–(5,x) is used (see Eq. (A7) and Figure 69),

the resistivity values have been taken from CRC tables (Weast, 1988), and

the Debye temperatures are from Table 1 of Kittel (1996). The agreement

is astounding and can be repeated for other metals not shown. The values of

FIGURE 68. (a) Resistivity of select metals compared to the predictions of Eq. (584). (b)

Demonstration that a wide variety of metal resistivities are well described by a Bloch–Gruneisen

function dependence [Eq. (584)] using the least‐squares estimate for the Debye temperature.

(c) Debye temperatures evaluated from a least‐squares estimate on the presumption that metals

follow a Bloch–Gruneisen behavior as shown in (b). A comparison to the Debye temperatures

given by Kittel (1962) is also shown.

100

10−1

10−2

10−3

10−4

10−5−3 −2 −1 0

Numerical

120 z(5) x5

(x/4)/1 + (18x2)−1

Hybrid

W-(

5,x)

In(x)

1

FIGURE 69. Behavior of the Block–Gruneisen function W–(5,x) (calculated numerically)

compared to its asymptotic limits and the ‘‘hybrid’’ approximation.

210 KEVIN L. JENSEN

the Debye temperature for various solids depend on temperature range and

sample purity (White andWoods, 1959). Therefore, if the Debye temperature

is treated as an adjustable parameter in fitting Eq. (584) to actual resistivity

data, the universal feature of the behavior of metals follows the relation

implied by Eq. (584) quite well: if the reference temperature To is taken

to be the Debye temperature TD, then TD becomes a parameter that can

be extrapolated by performing a least‐squares comparison of tabulated

resisitivity with a Bloch–Gruneisen function such that

@

@To

Xn

lnr Tnð Þr Toð Þ

ln x5W 5; x1ð ÞW 5; 1ð Þ

2To¼TD

¼ 0 ð585Þ

is minimized when To ¼ TD, where n is an index of the tabulated values of the

resistivity as a function of temperature and W(n,x) is defined in Eq. (A2)

with approximations in Eq. (A7). Note that r(T)/r(TD) as a function of T/TD

is implied by Eq. (584) to be a universal relation for all metals. Performing

the minimization using CRC resistivity data (for which the lowest tempera-

ture values, typically 1 K and 10 K, are excluded) results in Figure 68b

(which shows greater correlation than Figure 11 of [White and Woods, 1959]

and thereby indirectly points out the sensitivity on the estimates of the Debye

temperature). Finally, the fitted values of the Debye temperature implied by a

TABLE 10

DEBYE TEMPERATURES*

Atomic number Element TD (Kittel) r(TD) LS‐Fit r(TD)‐fit

11 Na 158.00 4.9300 200.66 2.9018

12 Mg 400.00 14.400 313.77 4.7178

13 Al 428.00 10.180 399.54 3.8648

26 Fe 462.45 20.662 470.00 57.100

28 Ni 450.00 38.600 434.91 13.652

29 Cu 343.00 6.0410 274.61 1.5538

42 Mo 450.00 21.200 370.24 7.2660

47 Ag 225.00 5.6400 217.40 1.1334

74 W 400.00 21.500 361.57 6.9304

78 Pt 240.00 32.000 187.93 6.3008

79 Au 165.00 7.8600 123.52 0.84522

82 Pb 105.00 38.300 82.180 6.7215

*‘‘Kittel’’ refers to Table 1 of Kittel (1996), whereas ‘‘LS‐Fit’’ refers to a least‐squares fit ofthe ratio of the resistivity to its value at the Debye temperature as predicted by the Bloch–

Grunesien theory in which the Debye temperature is treated as an adjustable parameter.

Resistivity values were from the CRC tables (Weast, 1988); the fitting excluded the 1 K and

10 K resistivity values.


fit to Eq. (585) are in fact often quite close to tabulated values obtained via

other means (Figure 68c): comparison of the fitted Debye temperatures with

values quoted in Kittel is given explicitly in Table 10.

For electron energies in excess of the Fermi level, refer to Eqs. (577) and

(578) to consider the frequency dependence of J E;oð Þ. The presence of an

o2 in the integrand of Eq. (579) (or the higher powers ofo for the momentum

relaxation time) offsets the sinh xoð Þ in the denominator when the argument

is small, and so the small‐case o can be ignored. At low temperatures, xo and

xE are large because of b, and so

J E;oð Þjb1 cosh xEð Þ

cosh xEð Þ þ cosh xoð Þf g ð586Þ

in the regions for which the integrand is significant. It is seen, therefore,

that at low temperatures, J E;oð Þ behaves analogously to a Heaviside step

function, and so

1

tep kFð Þ ¼lp

kFvsð Þ2ðoD

0

o2J E;oð Þdo

lp

kFvsð Þ2ðoD

0

o2

exp½b jE mj hoð Þ þ 1f g doð587Þ

where the power of o is increased by 2 if the momentum relaxation time is

considered as per Eq. (582). The absolute value of E – m is a consequence of

the behavior of coshðxÞ ejxj=2 when the magnitude of x is large. To leading

order, then, it follows that if oD > jE mj, then the integral will be propor-

tional to E mð Þ3, whereas if oD < jE mj, then the integral is a constant,

that is, for large b

3h3ðoD

0

o2

exp½b E m hoð Þ þ 1f g do jE mj3 jE mj < hoD

hoDð Þ3 jE mj > hoD

ð588Þ

whereas for the momentum relaxation time, the power is increased by 2. For

general conditions and a generic metal with a Debye temperature of 350 K(similar to copper), then hoD 0:03 eV and for all practical purposes for

typical photoemission wavelengths that are to be considered below, the

acoustic relaxation time can be taken as approximately constant. That con-

clusion is not a priori so for semiconductors, but that is another, and much

longer, story.

It was suggested previously that the value of l would be specified by

empirical data rather than just taken as 1/2. That is not quite accurate;

what shall be adjusted instead is the value of the deformation potential.

The high‐temperature asymptote of the acoustic phonon relaxation time

212 KEVIN L. JENSEN

entailed by Eq. (580) can be written in the form (Ridley, 1999)

tep ¼ 2h

pl0kBT¼ pMrih

3v2smkF 2kBT

; ð589Þ

where the two equivalent formulations are given. This implies that

l0jBohmStaver ¼92

2m2ð590Þ

as long as the ion number density and electron number density are the

same, and the sound velocity is given by the Bohm–Staver relation

1=2ð ÞMv2s ¼ m=3. If so, l ¼ 1/2 implies X ¼ m/3, as before. However, Mri isthe mass density, the sound velocity is an empirical quantity, and so specify-

ing l by empirical relations is tantamount to finding appropriate values of X.

7. Matthiesen’s Rule and the Specification of Scattering Terms

The low‐temperature behavior of the resistivity was not shown in Figure 68.

According to the Bloch–Gruneisen relation entailed in Eq. (584), the resistiv-

ity should approach 0 as T5—but in fact, as shown in Figure 70 for another

sampling of metals, what is seen, rather, is that the resistivity tapers off to

what appears to be a finite value as the temperature becomes cryogenic. This

residual resistivity is due to defects in the lattice that serve as scattering

centers to electron transport (and the reason why the low‐temperature resi-

sitivity values were truncated in the least‐squares estimation of the Debye

temperature). If scattering mechanisms are independent, then the probability

of scattering should approximately follow the sum of the individual

10−1

10−2

10−3

0 20 40 60

Cu

Ag

Au

Ni

Mg

Res

istivi

ty [mΩ

-cm

]

Temperature [kelvin]

FIGURE 70. Low‐temperature behavior of the resistivity shows that the impact of the

scattering of defects persists at low temperatures.


scattering probabilities; that is, the sum of the inverse relaxation times, and

therefore, the total resistivity should simply be the sum of the partial resisi-

tivities, a relation that is empirically supported and known as Matthiesen’s

rule, and given by

1

ttotal¼ 1

tdlfþ 1

tee

þ 1

tep

ð591Þ

Experimental relations (Kanter, 1970) tend to consider the closely related

mean free path that follows the same relation, and in addition pay attention

to complications not considered here (DOS and interaction with d electrons,

for example), but these complications are outside the present scope. Relaxa-

tion times, being related to thermal conductivities via Eq. (439), allow for an

indirect method to determine how well a prescription like Eq. (591) applies in

practice. By knowing the tabulated values of thermal conductivity as a

function of temperature coupled with the specific heat evaluated at room

temperature and the relation entailed in Eq. (441), it is possible to recast the

thermal conductivity data as relaxation time data (Jensen, Feldman, Moody,

and O’Shea, 2006a) by using Eq. (439), or

t mð Þ 3m

2mkexp Tð ÞgexpT

; ð592Þ

where the exp subscript (exp) reinforces that values for these quantities are

obtained from tabulated data in the literature [e.g., Gray (1972) or CRC

tables]. Figure 71 shows such application for a variety of metals. Several

0

4

8

12

0 2

Cu

Ag

Au

Pb

Wln

(tex

p [fs])

ln(T [K])

4 6

FIGURE 71. The total relaxation time as inferred from thermal conductivity data.

214 KEVIN L. JENSEN

features are evident. For the metals shown (and for metals in general), the

relaxation time tends to go to a constant value, presumably dictated by tdef asthe temperature drops—note that lead (Pb) does not quite match this,

because lead becomes a superconductor at temperatures below 7.2 K.

In addition, there is a change in slope for the higher temperatures, primarily

corresponding to the change in behavior of the e‐p relaxation time as a

function of temperature: at the higher temperatures, the slope approaches

(1) as expected. Generally, the e‐e relaxation time is dominated by the

others; thus, not much can be said regarding it for now—but that changes

when the energy of the electron is higher than the Fermi level as a consequence

of photoexcitation.

The matching of thermal conductivity data (or rather, to the relaxation

time inferred from thermal conductivity data) requires that values of three

quantities that have not been heretofore given be provided as follows:

The value of tdef, which can be obtained from low‐temperature thermal

conductivity data

The value of Ks in the electron‐electron scattering rate, which shall be

obtained by comparisons to more comprehensive theory and simulation

than the treatment given here—where available; and finally

The value of X, which shall be inferred by a best‐fit model to the total

relaxation time inferred from thermal conductivity.

For a number of metals that are of interest, the requisite detailed studies

and simulations may not be available, and in those cases, a combination of

arguing by analogy and a minimization of least‐squares differences is used.While it is true that a number of approximations have been folded seamlessly

into the exposition, perhaps the most consequential ones are the parabolic

energy‐momentum relation and a general ignoring of the shape of the Fermi

surface and the DOS (which are not trivial). Nevertheless, for present pur-

poses, the specification of the three bulleted quantities above provides more

than adequate agreement that can be used to account for the role of scatter-

ing in the photoemission process and to allow for the usage of an idealized

model of the electron distribution.

Consider next the electron‐electron relaxation time, which has been the

subject of sustained interest in both theory and simulation (Campillo et al.,

1999; Krolikowski and Spicer, 1969; Ladstadter et al., 2004; Lugovskoy

and Bray, 1998; Quinn, 1962; Wertheim et al., 1992). By comparing the

theoretical formula entailed in Eq. (495) to the Monte Carlo simulations

and detailed calculations, it is possible to form estimates of Ks. Consider the

examples shown in Figure 72, where Eq. (495) is compared to theoretical

calculations of electron scattering for gold and copper by Ladstadter et al.


1

10

100

0

(a)

5

Ladstadter

Lugovskoy Ks = 1

Lugovskoy Ks= 5.2

Theory

tee

[fs]

E - m [eV]

CuKs(eff) = 1.684

1 2 3 4

1

10

100

Kanter

Krolikowski

Theory

AgKs(eff) = 14.417

tee

[fs]

E-m [eV]0

(b)

1 2 3 4

Ladstadter

Theory

1

10

100Au

Ks(eff) = 6.4546

tee

[fs]

E-m [eV]0

(c)

1 2 3 4

FIGURE 72. (a) Comparison of Monte Carlo simulations with Eq. (495) with Ks ¼ 1.684 for

copper. (b) Comparison of other theory and measurements with Eq. (495) with Ks ¼ 14.417

for silver. (c) Comparison of Monte Carlo simulations with Eq. (495) with Ks ¼ 6.4546 for gold.

216 KEVIN L. JENSEN

Cu

−2

0

2

4

6

8

10

12

14(a)

70 1 2 3 4 5 6

In(t )In(t ep)

In(t ee)

In(matt.rule)Liq. nitrogenRoom temp

In(r

elax

atio

n tim

e [fs])

In(temperature [kelvin])

Ag

0

2

4

6

8

10

12

14(b)

70 1 2 3 4 5 6

In(t )In(t ep)In(t ee)


In(r

elax

atio

n tim

e [fs])



0.1

1

10

100

8 10

KanterKrolikowskiKs = 14.417Ks = 2.060

Ag

t ee [fs]

E-m [eV]0 2 4 6

FIGURE 73. Same as Figure 72(b) but for a larger E range, showing impact of the d electrons.


Au

−2

0

2

4

6

8

10

12

14(c)

70 1 2 3 4 5 6

In(t )In(t ep)

In(t ee)


In(r

elax

atio

n tim

e [fs])


Pb

−2

0

2

4

6

8

10(d)

70 1 2 3 4 5 6

In(t )In(t ep)

In(t ee)

In(TOTL)Liq. nitrogenRoom temp

In(r

elax

atio

n tim

e [fs])


W

−2

0

2

4

6

8

10(e)

70 1 2 3 4 5 6

In(t )In(t ep)

In(t ee)

In(TOTL)Liq. nitrogenRoom temp

In(r

elax

atio

n tim

e [fs])


FIGURE 74. (a) Determination of the deformation potential and the defect scattering term

from thermal conductivity data from which the relaxation time is taken for copper. Room

temperature and liquid nitrogen temperature (300 K and 77 K, respectively) are also shown.

(b) Same as (a) but for silver. (c) Same as (a) but for gold. (d) Same as (a) but for lead. (e) Same as

(a) but for tungsten.

218 KEVIN L. JENSEN

(2004), the findings of Lugovskoy and Bray (1998) for copper, and the theory

and measurements of Kanter (1970) and Krolikowski and Spicer (1969) for

gold. A first take suggests that the Ks modification makes practical sense

given the ability to account for the data, but such an assessment is premature.

The first hint of complications is the manner in which Ks changes from metal

to metal in the cases considered. As noted separately by Lugovskoy and Bray

(1998) and Krolikowski and Spicer (1969) and alluded to previously, com-

plications occur as a consequence of the d electrons that change the effective

value of Ks, as can be seen by examining a greater range of energies as for

silver (Figure 73). A transition occurs in the scattering rate that can be

modeled by a change in the value of Ks as the higher‐energy photons probe

more deeply into a DOS that is at variance with the nearly free electron

model. Nevertheless, the theoretical description can be adapted to the regime

of interest through a modification of Ks.

If tdef and Ks can be independently ascertained, then the final specification

of X using thermal conductivity data is straightforward. If that option is not

viable, then a least‐squares minimization procedure can be used to form

adequate estimates from the thermal conductivity data, although such a

procedure is less satisfactory because of the domination of te‐e by te‐p for

scattering at the Fermi level. The results of such a calculation are shown in

Figure 74 for copper, silver, gold, lead, and tungsten, where the open circles

represent t as calculated from thermal conductivity data via Eq. (592) and

‘‘Matt.rule’’ represents Eq. (591).

z

x Surf

ace

Vacuum

q

Cathode

hω−

FIGURE 75. Relation of the parameters z, x, and y for the path of a photoexcited electron

inside a metal.


G. Scattering Factor

With an estimate for the relaxation time available, an estimate of the impact

of scattering on the QE of a metal can be made. The correct approach should

be to consider the scattering rate as a function of electron energy after it

absorbs a photon and then sum over the contributions of all electrons that

are photoexcited and satisfy conditions for emission, a method to be consid-

ered below and designated the moments approach. It is possible, however,

to form an estimate of the impact of scattering on QE in the context of the

Fowler–Dubridge model if it is assumed that (1) the electron energy prior to

photoexcitation is concentrated around the Fermi level, and (2) the emission

probability and the transport to the surface are unrelated. Both approxima-

tions are not strictly true, but they allow for the decomposition of the QE into

the product of an emission probability P and a scattering factor Fl, of which

the former was considered previously and the latter the topic of the present

inquiry.

The presence of scattering introduces a factor that accounts for the proba-

bility an electron will migrate to the surface with a kinetic energy component

normal to the surface sufficient to be emitted given by a weighted average of

the product of the probability that an electron will absorb a photon at a

depth x in the material and the probability that the electron will not scatter

before reaching the surface. Two modifications are that, first, only those

photoexcited electrons with energy sufficient for emission are considered,

and second, only those electrons with a velocity component toward the

surface are considered (Figure 75). Such a demarcation of the photoemission

process into absorption, transport, and emission is familiar from the success-

ful three‐step model of photoemission introduced by Spicer (Spicer, 1960;

Sommer and Spicer, 1965; Spicer and Herrera‐Gomez, 1993). Consider the

simpler question of the number of electrons, expressed as a fraction of the

number photoexcited, that reach the surface without suffering a scattering

event under the approximation that any scattering event is fatal to emission;

while not strictly correct and perhaps draconian, as electrons can scatter into

states that can be emitted in addition to being scattered out of such states,

nevertheless when the energy of the electron is sufficiently above the Fermi

level, electron‐electron scattering dominates in metals, and collisions with

other electrons generally divide the energy. The approximation therefore has

more in its favor than against it—the existence of a surface barrier and the

restriction of attention to incident photons with energies not much larger

than the barrier height make the approximation quite reasonable. Let the

incident photon be absorbed at a depth x with a probability proportional

to exp(–x/d), where d is the penetration depth. Let the mean distance the

electron travels in any direction before suffering a collision be

220 KEVIN L. JENSEN

l kð Þ ¼ hjkjm

t EðkÞð Þ; ð593Þ

where if scattering is isotropic, then only the magnitude of k need be con-

sidered. The probability that an electron will suffer a collision after traveling

l(k) is then proportional to exp(–z(y)/l(k)). Consequently, the fraction of the

electrons reaching the surface is approximately

Fl ¼ÐdkfFD EðkÞð ÞY EðkÞ þ ho m fð Þ

ð10

exp x

d zðyÞ

l kð Þ

dx

ÐdkfFD EðkÞð ÞY EðkÞ þ ho m fð Þ

ð10

exp x

d

dx

:

ð594ÞA number of subtle features exist. The first is that because only electrons

with an x‐component directed at the barrier are considered, the angular

integration is to p/2 instead of p, whereas in the denominator any direction

can result after photoexcitation. The second feature entailed by the Heaviside

step function is not just any photoexcited electron but only those with an

energy that would allow them to surmount the barrier if they were favorably

directed, contribute to the QE (at present, the possibility of a tunneling

component to the emission is ignored). Recall that the probability of

emission factor P containing the Fowler–Dubridge functions has already

dispensed with those electrons that cannot pass the surface barrier,

and the emphasis here is only on the remaining electrons and what addi-

tional processes they endure. With the identification that z ¼ x=cosðyÞ, thex‐integrations in both numerator and denominator are trivial, and result

in a multiplicative factor ofð10

exp x1

dþ 1

lðkÞcosðyÞ

dxð10

exp x

d

dx

¼ cos yð Þcos yð Þ þ d=lðkÞð Þ : ð595Þ

The angular integration is likewise analytic, giving

1

2

ðp=20

sinydycosy

cosyþ d=lð Þ

¼ 1

21þ d

l

ln

ddþ l

: ð596Þ

The final momentum integration calls for an understanding of the mom-

entum dependence of the relaxation time, which is complicated beyond

the needs of a simple approximation. Brief reflection indicates that the

majority of the photoemitted electrons have energies not much larger than

the barrier height, and so a reasonable approximation is to replace k by


kv ¼ ½2m mþ fð Þ1=2=h in l(k), thus allowing the final k integration to be the

same in the numerator and denominator and factor out. It follows

Fl 1

21þ d

lðkvÞ lnd

dþ lðkvÞ

0@

1A

8<:

9=;

kv ¼ 1

h

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2m mþ fð Þ

p ð597Þ

A numerical example is profitable. Assume that the dominant part of the

relaxation time is due to electron‐electron scattering, and use the low‐temperature leading‐order limit, for which

teeðkvÞ 32hm

f2

2Ks mþ fð ÞafshkFcp3

1=2

: ð598Þ

Using copper parameters and neglecting field, it is found that l is approxi-

mately 3.6 nm, whereas d is 12.9 nm. This suggests that Fl ¼ 0.059. With

the crude estimation provided by Eqs. (593)–(598), a first‐pass estimate of the

QE may be made using the low‐temperature expansion of the Fowler–

Dubridge functions

QE 1 Rð ÞFlho F

m

2

: ð599Þ

10−5

10−4

10−3

3.9 4.1 4.3 4.5

QE (theory)Rosenzweig et al.Rao et al.Dowell et al.

Qua

ntum

eff

icie

ncy

Work function Φ [eV]

Cu

FIGURE 76. Quantum efficiency as predicted by the modified Fowler–Dubridge model of

Eq. (599) compared to experiment for copper.

222 KEVIN L. JENSEN

Again use copper parameters of m ¼ 7.0 eV and R ¼ 34% at l ¼ 266 nm.

Work function varies with crystal face, a common value of F ¼ 4.48 eV for

the 110 face (Weast, 1988). The predictions of the approximation to QE given

in Eq. (599) are shown in Figure 76 for a range of work function values.

Compare this to three reported values in the literature (i.e., Table I of

Srinivasan‐Rao, Fischer, and Tsang, 1991; Figure 6 of Rosenzweig et al.,

1994; and Figure 8 of Dowell et al. 2006). These values are shown on the

labeled points at the work function of 4.3 eV (the value taken for the photo-

electric work function of copper by Srinivasan‐Rao and Dowell—though

the field for Rosenzweig (1994) was large enough to cause a consequential

Schottky barrier–lowering factor). As discussed by Dowell et al. (2006),

much variation can result as a consequence of surface preparation, so the

seemingly dissimilar experimental values are not surprising. What is notable,

instead, is how closely the crude estimate Eq. (599) for QE actually ap-

proaches measured values. The estimate is improved in the moments‐basedapproach discussed below.

H. Temperature of a Laser‐Illuminated Surface

1. Photocathodes and Drive Lasers

The QE of many materials is generally no better than a few percent (although

some, such as cesiated GaAs, can be high as 40%), and in the case of metals,

the QE is often on the order of 0.001% (Rao et al., 2006). As expected, the

application dictates the photocathode, although the applications that moti-

vate the present treatment principally are those that demand high peak and

average current densities, such as particle accelerators (Schmerge et al., 2006)

and high‐power free‐electron lasers (O’Shea and Freund, 2001; O’Shea et al.,

1993). While the QE of the semiconductor photocathodes is without parallel,

such photocathodes have response times that are longer than picoseconds

and so, if ultrashort bunches (sub‐picosecond or femtosecond; Riffe et al.,

1993) and crisply defined laser pulse profiles (sub‐picosecond rise and fall

times) are demanded to produce bunches from rugged photocathodes, metal

photocathodes are required. Alternately, the generation of polarized electron

beams (Maruyama et al., 1989) requires cesiated GaAs photocathodes char-

acteristic of, for example, the JLAB DC photoinjector. RF injectors, owing

to their generally more hostile environment, tend to rely on metal photo-

cathodes such as copper (Rosenzweig et al., 1994).

The consequences of attempting to extract a charge bunch require use of

a laser pulse. As shown in Table 10, such an ability is constrained as much

by the drive laser as by the QE. The wavelength of a drive laser is obtained

by nonlinear conversion crystals that reduce (for sake of argument) the


wavelength of a 1064‐nm laser by doubled (512 nm), tripled (355 nm), or

quadrupled (266 nm) Nd:YAG conversion, with conversion efficiencies of

approximately 50%, 30%, or 10%, respectively (Jensen et al., 2003a,b).

For the UV case, therefore, a substantial amount of waste heat is dumped

into the crystals, altering their operation and leading to nonlinear perfor-

mance. This effect is generally undesirable as the nonlinear conversion pro-

cess introduces fluctuations that scale as (laser intensity)n, where n is the

harmonic number (4 for 266 nm), thereby causing such fluctuations to ap-

pear in the resulting electron pulses. Noise of that character results in

degraded FEL operation. Even if the electron bunches are so separated in

time that heat generated at the cathode can be dissipated between bunches, as

for accelerators, it is still important to question what impact a local, short‐duration laser pulse will have on the temperature of the electron gas, as

the underlying processes, from escape probability to scattering factors, are

all dependent on the temperature. In fact, extreme laser intensities for very

short durations reveal nonlinear effects of considerable interest and

much else (Agranat, Anisimov, and Makshantsev, 1988, 1992; Fujimoto

et al., 1984; Girardeau‐Montaut, Girardeau‐Montaut, Moustaizis, and

Fotakis, 1994; Girardeau‐Montaut and Girardeau‐Montaut, 1995; Imamura

et al., 1968; Papadogiannis and Moustaizis, 2001; Papadogiannis et al., 2002;

Yibas and Arif, 2006).

2. A Simple Model of Temperature Increase Due to a Laser Pulse

For a crude approximation of the temperature rise from extracting 1 nC from

a surface area of (1/8) cm2 under the assumption that all the energy deposited

on the surface to extract that amount of charge is uniformly distributed over

0

500

1000

1500

2000

2500

0 1 2 3 4 5 6 7

Electrons

Lattice

Tem

pera

ture

[K

]

Peak intensity [GW/cm2]

FIGURE 77. Calculated peak temperature for copper subjected to 248‐nm wavelength 450‐pslaser pulse (after Figure 3 of Papadogiannis, Moustaizis, and Girardeau‐Montaut, 1997).

224 KEVIN L. JENSEN

a slab of thickness equal to the laser penetration depth 12.93 nm for copper,

consider the following model. The temperature rise is related to the specific

heat, the amount being heated, and the energy deposited, or

DT ¼ DEVCiðTÞ ; ð600Þ

where V ¼ Ad is the volume and Ci(T) is defined in Eq. (456) (often, the

replacement Ci ¼ ci di, where ci is the specific heat in units of joules per gram

Kelvin [J/gK] and di is the mass density, results in the more familiar

DT ¼ DE=MciðTÞ, where M is the total mass involved). The Debye tem-

perature of copper is larger than room temperature (343 K), and so an

approximation to Ci of

Ci Tð Þ 3kBri

1þ 1

20

TD

T

2ð601Þ

is useful. It follows

DT ¼ hoq

DQAdkBri

1

QE

1þ 1

20

TD

T

2" #

; ð602Þ

which, for a wavelength of 266 nm and DQ ¼ 1 nC is approximately 62.4 K.

If the laser pulse is of 50 ps in duration, the intensity of the laser is

5 MW/cm2. While 60 K temperature rises indicate some interesting physics

is in store, clearly if the laser pulse illuminates a smaller area and is of signi-

ficantly greater intensity (see, for example, Papadogiannis and Moustaizis,

2001, in which >1 GW/cm2 intensities are used), then the temperature excur-

sion can be significant, and the metal brought to high temperatures that give

rise to thermionic emission (Riffe et al., 1993) and show evidence of a

decoupling between the temperature of the electron gas and the lattice.

Such a state of affairs is shown in Figure 77 (which summarizes Figure 3 of

Papadogiannis, Moustaizis, and Girardeau‐Montaut, 1997).

Although Eq. (602) is pedagogically appealing, it is clearly incorrect in its

assumptions that the laser energy is uniformly distributed over a depth d andthat such a depth is independent of the duration of the laser pulse. In what

follows, methods to model the temperature rise and the impact on photo-

emission are considered that allow for the impact of sudden pulses on the

rise in temperature near the surface and how that relates to both energy

transfer to the lattice and thermal diffusion into the bulk, always with an


eye toward returning the discussion to the treatment of QE and laser heating

of a photocathode (Jensen, Feldman, Moody, and O’Shea, 2006a) at the

appropriate time.

3. Diffusion of Heat and Corresponding Temperature Rise

Thermal current density harkens back to the defining relations in Eq. (435),

and like charge current density, it obeys a continuity equation analogous

to Eq. (126) that relates the time rate of change of a density to the spatial

variation in the current. The ‘‘density’’ is the energy density, the time deriva-

tive of which is related to the specific heat, whereas the spatial derivative of

the current is ascertained from Eq. (433), a form of Fick’s law relating a

‘‘current’’ to a spatial variation in a density that dominates the treatment of

diffusion and transport phenomena: the current across a surface is related to

its gradient analogous to Eq. (526). Therefore, a combination of Fick’s law

and the continuity equation entails that a region where the thermal energy

is concentrated diffuses into regions where it is not, according to

Cv Tð Þ @@t

T ¼ @

@xkðTÞ @

@xT

; ð603Þ

where T is the temperature of the electron gas,Cv is the specific heat, and k(T)is the thermal conductivity (e.g., Cv ¼ Ce(T) þ Ci(T) ¼ 3.45 J/Kcm3 and

k ¼ 401 W/mK for copper at room temperature). Although Eq. (603) some-

what dominates the discussion, at the outset it is apparent that it cannot be

quite right for several reasons. First, the equation needs a source term

representing the drive laser. Second, since the electron and lattice tem-

perature can decouple, and the fact that the relaxation time depends on

both e‐e and e‐p scattering rates, the thermal conductivity actually should

be k Te;Tið Þ, where e indicates the electron temperature and i the temperature

of the phonon bath (reflecting a slavish obedience to conventions established

in the literature). Finally, if the electron and lattice temperature differ, then

a term accounting for the bleeding off of electron energy to the lattice as

electrons and phonons interact must be included. These complications will

appear after the simple form of Eq. (603) is examined.

The first and simplest approximation is to assume that the thermal con-

ductivity is at best weakly dependent on temperature so that @x k@xTð Þ k@2

xT , and second, that the temperature excursions are small, so that

k=Ce Do is approximately constant, where Do has units of square centi-

meters per second (cm2/s). Solutions exist of the form

Tðx; tÞ ¼ To þ coDTuðx; tÞ@tu x; tð Þ ¼ Do@2

xu x; tð Þ ð604Þ

226 KEVIN L. JENSEN

where DT and To are a temperature rise and the baseline (or bulk) tempera-

ture, respectively, and where the parameter co is a constant to act as a

placeholder for future factors that will invariably arise but which are incon-

venient to specify now. Let w be the spatial Fourier transform of u such that

wðk; tÞ ¼ 1ffiffiffiffiffiffi2p

pð11

uðx; tÞeikxdx; ð605Þ

then

@twðk; tÞ ¼ Dok2wðk; tÞ ) wðk; tÞ ¼ woexp Dok

2t

: ð606ÞInverting the Fourier transform and normalizing u so that its integral over

all space is unity gives

u x; tð Þ ¼ 4pDotð Þ1=2exp x2= 4Dotð Þ

: ð607ÞA feature exploited below is that for large Dot, the u function acts remark-

ably like a Dirac delta function (in point of fact, the derivation of Eq. (607)

is a useful approach to ‘‘deriving’’ the properties of the delta function;

see, for example, Butkov, 1968). Hence, Eq. (607) shall be referred to as a

delta‐function–like pulse, not because it is so sharp but because for small

times, when integrated with other x‐dependent functions, it behaves in a way

that mimics a delta function, even though it is Gaussian when the time

parameter is large. On a related note, the solution entailed by Eq. (604) is

analogous to the path integral formalism of quantum mechanics (Rammer,

2004), as the heat diffusion equation and Schrodinger’s equation are formally

analogous, but where the real temperature in the former is the imaginary time

in the latter.

From a macroscopic viewpoint in which the laser penetration depth is as

good as infinitesimally thin, the dumping of a quantity of energy in an

infinitesimally short pulse creates a temperature spike that proceeds to diffuse

into the solid.Near the surface, though, a complication arises in that heat does

not diffuse from the solid into the vacuum (it radiates—but that is ignored for

now), or equivalently, the boundary condition that the gradient of tempera-

ture at the surface vanishes is imposed. If the pulse is absorbed some distance

xo in the surface, then a method to ensure the boundary condition is to add an

image pulse a distance xo outside the surface, so that

uðx; tÞ ) 1ffiffiffiffiffiffiffiffiffiffiffiffiffi4pDot

p exp ðxþ xoÞ24Dot

!þ exp ðx xoÞ2

4Dot

!( ); ð608Þ

where the gradient at x ¼ 0 vanishes. In the limit that xo approaches


0 (the pulses are absorbed at the surface), then the modification is to insert a

factor of 2, one of the many small factors that are absorbed into the definition

of co at the appropriate time.

4. Multiple Pulses and Temperature Rise

Insofar as a pulse of arbitrary duration may be considered the sum of many

infinitesimal pulses, it is relevant to ask how such pulses sum. If our view-

point is enlarged to even longer times, then the pulse again appears to be like

Eq. (607), and perhaps that would suffice, but it glosses over an important

feature: rather than being an academic exercise for obscurantist theorists, a

train of equivalent pulses is what a photocathode endures in the operation of

an FEL or accelerator, and so the question of the cumulative rise in temper-

ature becomes related to the time separation between pulses and the energy

content of each pulse, apart from what happens in a particular finite duration

pulse, though that is critical as well.

Consider a train of Dirac delta‐function–like pulses, where each individual

pulse gives rise to a term like u(x,tn) for the nth pulse. The temperature as a

function of position and time is then the sum over such pulses, and it matters

whether the time of interest is during or after the period when the train of

pulses is incident on a surface. It is an initial assumption that the coefficients

co and DT are the same for each pulse; that this cannot be strictly true is

evident because as the temperature rises, the relaxation rates change and

therefore the conductivity changes, but to leading order and especially if the

energy content of each pulse is small, the approximation is quite reasonable.

Therefore, the temperature can be written

Tðx; tÞ To þ coDT SnðaðxÞ; sðtÞÞ SnðaþðxÞ; sðtÞÞf g

Snða; sÞ Xn

j¼1

1

j þ sð Þ1=2exp a

nþ s

24

35 ð609Þ

where the difference in S functions arises because the back boundary of the

cathode of finite thickness is to be held to the boundary condition that

the temperature there is To. New terms a and s have been introduced. They

are defined as follows. Time is a function of a characteristic time Dt (thepulse‐to‐pulse separation), a pulse number index n, and an offset parameter

s that will be (1/2) for times in between adjacent pulses or odd multiples of

(1/2) for times after the last pulse in a pulse train, or t ¼ tnðsÞ ¼ nþ sð ÞDt.It follows that for a total number of pulses N, if n < N, then s ¼ 1/2 and the

time period corresponds to heating due to absorbed laser pulses, but if nN,

then s ¼ nN þ 1=2ð Þ and the time period is one of cooling after the last

pulse has been absorbed and time elapses. Next, let the width of the cathode

228 KEVIN L. JENSEN

be L and the position x be a function of a dimensionless term y such that

xðyÞ ¼ 1 yð ÞL so that y ¼ 0 corresponds to the back contact and y ¼ 1

corresponds to the surface. A fictitious image pulse is needed equidistant

from the back contact, corresponding to x¼2L, so that the boundary con-

ditions of holding the back contact at fixed temperature can be maintained

[hence the aþ term in Eq. (609)]. Thus,

aðyÞ ¼ 1 yð Þ2L2

4DoDt ao 1 yð Þ2: ð610Þ

In the limit of large N, converting the summations to integrals shows that

SN a; sð Þ 2ffiffiffiffiffiffipa

pErf

ffiffiffiffiffiffiffiffiffiffiffiffia

sþN

s2435 Erf

ffiffiffias

s2435

8<:

9=;

þ 4ðsþNÞ þ 1ð Þ2ffiffiffiffiffiffiffiffiffiffiffiffisþN

p ea=ðsþNÞ 4s 1ð Þ2ffiffis

p ea=s

ð611Þ

where the error function is defined by

Erf zð Þ 2ffiffiffip

pðz0

exp x2

dx ð612Þ

and where the extra terms in Eq. (611) arise from the application of the

trapezoidal rule endpoints, which cannot be ignored when converting the

summation to an integral.

Two cases are of particular interest—first, early in the pulse train or when

the pulse train is short; and second, when the pulse train is so long that a

disturbance has propagated to the back of the slab. Treating the first case

first, using copperlike parameters at room temperature

DoðCuÞ ¼ kðTÞCvðTÞ

4:01W=cmK

3:45 J=cm3K¼ 1:16

cm2

s: ð613Þ

Consequently, a copperlike slab roughly half of a millimeter thick subject

to pulses roughly 1 ns in duration entails ao ¼ 500,000—assuredly a big

number, but one that pales in comparison to the number of pulses (109)

that make up a 1‐second engagement. Early in the train, however, when the

number of pulses is small compared to a, meaning the ratio a= N þ sð Þ is

large, and using the approximation to the error function for large argument

Erf x 1ð Þ 1 exp x2ð Þxffiffiffip

p 1 1

2x2

; ð614Þ


then noting that if a/(Nþs) is large, then a/s is far larger, it can be shown that

Eq. (611) is well approximated by

SN a; sð Þ 1

2ffiffiffiffiffiffiffiffiffiffiffiffiN þ s

p exp aN þ s

1þ 2

N þ sð Þ2a

( ); ð615Þ

which confirms the intuitive judgment that a train of pulses continues to look

like an expanding delta‐function–like pulse governed by a relation that

closely resembles Eq. (607), albeit that the coefficient has acquired a few

numerical constants; that is, the sum of N pulses has a form that resembles

one of its summation terms with n replaced by N. Figure 78a shows an

example of such an expanding pulse, although the Figure is equally valid if

0.0

0.5

0.1

1.5(a)

0.20 0.4 0.6 0.8 1

k = 0k = 1k = 4k = 8k = 64

y

a o = 100

s = 1/2

kth

term

of

S N(a

−) -

SN(a

+)

0

10

20

30

40

50

0

(b)

0.2 0.4 0.6 0.8 1

N = 1N = 4N = 16N = 64N = 256N = 1024Equilibrium

S N(a

o(1−

y)2 ,s)

-SN(a

o(1+

y)2 ,s)

y

ao = 100

s = 1/2

FIGURE 78. (a) Components of Dirac delta function–like thermal pulses at different times.

(b) The sum over the pulses shown in (a) for various total times.

230 KEVIN L. JENSEN

the time coordinate is scaled by a factor ls and the spatial coordinate byl1=2s ,

and where the time axis is begun away from the origin at 0.5 so as to not have

the Figure dominated by the sharpness of the pulse for earlier times.

The second case for consideration is when so many pulses have occurred

that heat is being lost to the back fixed‐temperature boundary and equilibri-

um ensues. Such a condition defines a maximum temperature parameter.

In this case, N is asymptotically large and ao/(Nþs) small. For small y,

neglecting s by comparison to N, and to order N1/2, a bit of work shows

that the small y limit is

SN ½a; s SN ½aþ; s ¼XN

k¼0

2ffiffiffiffiffiffiffiffiffiffiffikþ s

p sinh2aoykþ s

0@

1Aexp ao 1þ y2ð Þ

kþ s

24

35

4ffiffiffiffiffiffiffipao

p1 2

ffiffiffiffiffiffiffiaopN

s0@

1Ay

ð616Þ

In other words, a linear behavior with respect to x occurs at the back

boundary asN becomes large. Equilibrium entails time independence, and so

a linear function in x is what is expected from Eq. (604) after a long time.

The temperature declines linearly from the hot to the cold boundary,

as shown in Figure 78b for the example parameters of ao ¼ 100. Setting

y ¼ 1 in Eq. (616) defines a characteristic maximum temperature above

background given by

Tmax ¼ coDT 2L

ffiffiffiffiffiffiffiffiffiffiffip

DoDt

r ¼ coDT 2

ffiffiffiffiffiffi3p

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

L

vFDt

L

vFt

s( ); ð617Þ

where vF is the Fermi velocity hkF=m and the odd way of writing the RHS

shows that three length scales are involved: the width of the cathode, the

mean free path, and how far an electron at the Fermi level travels during the

duration of the pulse. Note, however, that we have been rather cavalier with

the parameter Dt: it has been treated as a differential element analogous to dt,

but the conclusions do not change if it is treated as the FWHM width of one

laser pulse or 1/100 of such a pulse—in fact, it could even be much larger than

t and the conclusions drawn by Eqs. (616) and (617) would not change.

Moreover, nothing has been said about whether adjacent pulses share a

common boundary (merge into a larger pulse) or are separated by a time

increment that can be much larger than the pulse length itself, so that

questions of heating due to a finite train of short‐duration pulses can be

investigated, an advantage of the manner in which the problem was

formulated.


The final factor needed to estimate Tmax is an expression for DT. If a total

amount of energy DE is deposited on the surface of a material, then

DE2

ð0L

dx

ð0Dt=2

dt CvðTÞ@tT Do@2xT

¼ð0L

dx

ð0Dt=2

dt @t1

2gT2 þ CiT

24

35

8<:

9=;

ð618Þ

where the assumption is that the energy deposited on the surface is done so

symmetrically in time (e.g., a Gaussian laser pulse), the disappearance of the

term containing Do is a consequence of @xT ¼ 0 at the boundaries, and

the approximation CeðTÞ gT has been used. The time integration is

straightforward, and so

DE 2

ð0L

dx1

2g T2 T2

o

þ Ci T Toð Þ24

35

2

ð0L

dx T Toð Þ gTo þ Cif g

¼ 2Cv Toð Þð0L

dx T Toð Þ

ð619Þ

01

10

100010010

ao = 50

aopao

20

30

40

50

104

Numerical

S(N

,0,1

/2)

- S(

N,4a

o,1/

2)

N

4 1–2pN

4N + 2

FIGURE 79. Comparison of the S functions with the asymptotic values for the evaluation of

surface heating.

232 KEVIN L. JENSEN

Using the relation Eq. (604), defining matters such that T(0,0) ¼ To þ DT,and taking L to be so large compared to other length scales that the lower

limit can be taken to infinity, it follows

DT ¼ DE

Cv Toð Þ ffiffiffiffiffiffiffiffiffiffiffiDoDt

p ; ð620Þ

TABLE 11

COPPERLIKE PARAMETERS

Parameter Value Units

R 33.7 %

Ce(300K) 0.0291 J/K cm3

Ci(300K) 3.297 J/K cm3

g 9.7105 J/K2cm3

TD 343 K

F/q 10 MV/m

Io 1 MW/cm2

k 4.007 W/K cm2

Lo 1 cm

Do 1.20 cm2/s

l 266 nm

Pulse‐to‐pulse 15 ns

Pulse width (FWHM) 10 ps

t 16.78 fs

ao 2075400 —

FWHM, full width at half maximum.

TABLE 12

GOLD AND COPPER PARAMETERS

Parameter Units Copper* Gold

Sound velocity vs m/s 4760 3240

Atomic mass M gram/mole 63.546 196.9665

Chemical potential m eV 7 5.51

Lattice temperature TiKelvin 1000 1000

Relaxation time [Eq. (638)] fs 20 36

gexp GW/K–cm3 60 40

g [Eq. (627)]/gexp — 2.38 0.64

g [Eq. (637)]/gexp — 0.44 6.8

From Wright and Gusev, 1995.From Fann et al., 1992.


which allows Tmax to be identified as

Tmax To ¼ 2Lffiffiffip

pCv Toð ÞDo

DEDt

2L

ffiffiffip

pk Toð Þ Il: ð621Þ

Consider the canonical copper example used in Eq. (613) for a 0.5‐cm thick

sample subject to a laser intensity of Il ¼ 100 W/cm2: Tmax – To under these

conditions is equal to 44 K.

Finally, there is the question of how fast the metal heats and how fast it

cools once the pulses stop arriving on the surface. Explicit use of the fact that

adjacent pulses can be separated in time can now be made: the sum of a train

of pulses separated in time by an increment even larger than the pulse width

itself is allowed by the formalism leading to Eq. (616). Two asymptotic

conditions are of interest for heating: the initial heating and the approach

to equilibrium at the surface (y ¼ 1). As shown in Figure 79 for the ad hoc

parameters ao ¼ 50, the behavior of heating (s ¼ 1/2) at the surface follows

the asymptotic expressions

SN 0; sð Þ SN 4ao; sð Þ 2ffiffiffiffiffiffiffiffiffiffiffiffiN þ s

pN aoð Þ

4ffiffiffiffiffiffiffipao

p1

ffiffiffiffiffiffiffi4aopN

s0@

1A N aoð Þ

8>><>>: ð622Þ

300

340

380

−8 −6 −4 −2 0

HeatingCoolingTo

Tmax

Tem

pera

ture

[K

]

log10t [s]

∆t = 15 ns

d tFWHM = 10 ps

Cu

FIGURE 80. Calculation of the temperature rise during laser pulse heating and cooling rate

after the last pulse for copperlike parameters.

234 KEVIN L. JENSEN

rather well. Easily evaluated models aside, actual parameters are of greater

pedagogical value. Consider again the canonical case of copper using

a photocathode simulation algorithm (Jensen, Feldman, Moody, and

O’Shea, 2006a; Moody et al., 2007), to be discussed in greater detail in the

modeling of a single pulse, to model the temperature rise and cooling of an

illuminated copper surface. Assume that the individual pulses are Gaussian

with a FWHM value of 10 ps (corresponding to a Gaussian time parameter

of 6 ps) and that the pulses are separated in time by 15 ns. Assume a QE of

0.0056%. Such values correspond to a peak and average current of 12 A/cm2

and 8 mA/cm2, respectively. Finally, for copper, the relevant values of the

various needed parameters are given in Tables 11 and 12. Under such

conditions, the heating and cooling profiles are shown in Figure 80.

5. Temperature Rise in a Single Pulse: The Coupled Heat Equations

Returning to Eq. (603) which, to accommodate the energy that a laser pulse

deposits on the surface, must now be written as noted by Papadogiannis,

Moustaizis, and Girardeau‐Montaut (1997) as

Cv Tð Þ @@t

T ¼ @

@xkðTÞ @

@xT

þ G z; tð Þ; ð623Þ

where the integral of G(z,t) over all time and space is DE, or the energy

dumped into the surface per unit area, and is given by

Gðx; tÞ ¼ 1 Rð ÞIlðtÞ ex=d

d

1U ½b ho fð Þ

U ½bm

; ð624Þ

where the reflectivity R is a function of incidence angle and Il is the laser

intensity per unit area incident on the photocathode. The overly pessimistic

term containing the Fowler–Dubridge U functions nominally accounts for

energy loss from direct photoemission (i.e., energy not absorbed and trans-

ferred to the lattice from scattering); to leading order it is ½ ho fð Þ=m2, sothat for photon energies at or near the barrier height, the term is negligible.

This equation is correct, however, only if the electrons and the lattice

are in thermal equilibrium, and it is quite possible (and widely done for

varied reasons; see Girardeau‐Montaut et al., 1996; Kaganov, Lifshitz, and

Tanatarov, 1957; Logothetis and Hartman, 1969; Lugovskoy and Bray,

1998, 1999 1999; Lugovskoy, Usmanov, and Zinoviev, 1994; Mcmillan,

1968; Papadogiannis and Moustaizis, 2001; Papadogiannis, Moustaizis,

and Girardeau‐Montaut, 1997; Papadogiannis et al., 1997; Riffe et al.,

1993; Rosenzweig et al., 1994; Wright and Gusev, 1995; Zhukov et al.,

2006) to make laser pulses of sufficient brevity that the electrons heat to

temperatures higher than the lattice without the lattice having time to catch


up. In that case, Eq. (624) becomes not one, but two coupled differential

equations for the electron and lattice temperature separately, or

Ce Teð Þ @@t

Te ¼ @

@xk Te;Tið Þ @

@xTe

0@

1AU Te;Tið Þ þ G x; tð Þ

Ci Tið Þ @@t

Ti ¼ U Te;Tið Þð625Þ

where U is the transfer in energy from the electrons to the lattice. To cleanly

solve Eq. (625), then U would have to be linear in the difference between the

electron and lattice temperatures Te – Ti (where the i subscript nominally

denotes ‘‘ions’’) and that is an approximation often made, in which the

electron‐phonon coupling constant—called g or some variant—is defined by

U Te;Tið Þ g Te Tið Þ: ð626ÞOne can do better than taking g as a constant. In fact, its determination

requires careful attention to competing effects and is important beyond our

interest in it here; see Corkum et al. (1988) and Kaganov, Lifshitz, and

Tanatarov (1957)—who, in articles often cited and possibly rarely seen—

obtained the relation

U Te;Tið Þ ¼ p2

6mv2sr

1

tep Teð Þ 1

tep Tið Þ

8<:

9=;

p2

6

mv2srtep Tið Þ

0@

1A Te Ti

Ti

0@

1A

ð627Þ

wherem, vs, and r are the electron mass, sound velocity, and electron number

density, respectively. Theoretical estimates of U(Te,Ti) have achieved

some sophistication (Girardeau‐Montaut and Girardeau‐Montaut, 1995;

Mcmillan, 1968). Still, the preference is to cleave to a simpler model, and

therefore a method based on a refinement of the approach developed by

Kaganov, Lifshitz, and Tanatarov suffices.

6. The Electron‐Phonon Coupling Factor g: A Simple Model

Because photoexcited electrons interact in metals via a fast electron‐electronscattering mechanism, an equilibrium temperature among the electrons is

achieved rapidly. Electron collisions with the lattice occur with much less

frequency, and so the lattice temperature trails the electron temperature. If

the electrons and the lattice are in thermal equilibrium, then the scattering

236 KEVIN L. JENSEN

operator

@c f ¼ 1

2pð Þ3ðdk2S k1; k2ð Þ n12 þ 1ð Þf1 1 f2ð Þ n12f2 1 f1ð Þf g; ð628Þ

where the FD f and BE n functions have been defined in Eqs. (482), (569), and

the S term originated in Eq. (500) but we shall use Eq. (550) preferentially.

The delta function in S, namely, d E1 þ ho E2ð Þ, entails that if the electronand lattice temperatures are equal, then the collision term is identically 0.

To show this, modify past notation slightly so that

f1 ! ex þ 1ð Þ1

f2 ! ex0 þ 1ð Þ1

n12 ! ey 1ð Þ1

ð629Þ

where x be m E1ð Þ, x0 be m E2ð Þ, and y ¼ biho, and where bs ¼1=kBTs and s designates either e or i. It is readily shown that

n12 þ 1ð Þ f1 1 f2ð Þ n12 f2 1 f1ð Þ ¼ ex0þyx 1

ex þ 1ð Þ ex0 þ 1ð Þ ey þ 1ð Þ ; ð630Þ

where the indices on either o or y are superfluous and ignored. The delta

function indicates that x0 þ y x ¼ 0 if be ¼ bi, and so Eq. (630) becomes

identically 0 (recall that the x’s have opposite signs than the E’s). The change

in the electron distribution that occurs when the electron and lattice temper-

ature become separated is mirrored in the change in the phonon distribution.

Consider, then, what occurs when, as a consequence of a temperature change

in the lattice so that n ! nþ dn. Eq. (628) becomes

@c f ) _n ¼ 1

2pð Þ3ðdk2S k1; k2ð ÞDn f1 1 f2ð Þ f2 1 f1ð Þf g

¼ 1

2pð Þ3ðdk2

2ph

0@

1Aa12

2d E1 þ ho E2ð Þ24

35DnDf ð631Þ

where

Df ¼ ex0x 1

ex þ 1ð Þ ex0 þ 1ð Þ : ð632Þ


The term Dn arises from a change in temperature in the BE distribution,

and so

Dn¼ 1

ebiho 1 1

ebeho 1¼ e bebið Þho 1

ebeho 1ð Þ 1 ebihoð Þ

be bið Þhoebiho 1ð Þ 1 ebihoð Þ

ð633Þ

where the second line is the leading‐order change (the subscripts require

particularly careful attention); the approximation is reasonable, as o oD,

and so be bið ÞhoD 1 for generic parameters. For scattering near

the Fermi level (that is, x ¼ 0) and using the delta function in Eq. (631), it

follows

Df ¼ ebeho 1ð Þ2 ebeho þ 1ð Þ ð634Þ

and so, using the definition of a in Eq. (548),

_n oð Þ ¼ 1

2pð Þ32ph

0@

1Ajaj22p 2m

h2

0@

1A

3=2

mþ hoð Þ1=2DnDf

¼ 2

9

l2m2

hprM2m

h2

0@

1A

3=2

hovs

0@

1A

2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffimþ ho

pebiho þ 1ð Þ 1 ebihoð Þ be bið Þ

ð635Þ

to leading order in be bi, which explains the flexible attitude toward the

e and i subscripts on the b’s. Note that r is the electron number density,

so that if it is assumed that one atom donates one electron, the product Mris the same as the mass density of the crystal. With _n oð Þ in hand, then

the approximate change in energy per unit time and volume is an integral

over the product of the change in the number of phonons at a given fre-

quency with the energy of the phonon at that frequency for all available

frequencies, or

U Te;Tið Þ ¼ 2pð Þ3

ðoD

0

_n oð Þho 4po2

v3sdo; ð636Þ

where o ¼ vsk for phonons. Inserting Eq. (635) into Eq. (636) for the case

ho m then after a bit of algebraic effort, and recalling that kBTD ¼ hoD,

it follows

U Te;Tið Þ 24=3pr lmð Þ2hTi

m

M

Te Tið Þ Ti

TD

5

W 6;TD

Ti

; ð637Þ

238 KEVIN L. JENSEN

which shows the sought‐for linear dependence on the temperature differ-

ence between the electrons and the lattice. In computation, l should be

evaluated via its definition in Eq. (543) rather than using the Bohm–Staver

value of (1/2). The dependence on theW– function and its (T/TD)5 coefficient

is hauntingly familiar and appears very similar to the electron‐phononrelaxation time, albeit that there the function W–(5,x) appears. That is,

1

tep Tð Þ ¼24=3plkBT

h

T

TD

4

W 5;TD

T

: ð638Þ

To leading order for small x, the series expansion solutions of W–(n,x)

show that

W 6; xð ÞW 5; xð Þ

4

5x 1 4

189x2

; ð639Þ

which, to leading order in TD/T, allows Eq. (637) to be cast as

U Te;Tið Þ 12

5l2 bimð Þ mv2sr

tepðTiÞ

Te Ti

Ti

; ð640Þ

a form similar to that found by Kaganov, Lifshitz, and Tanatarov (1957),

albeit it differs in having a different temperature dependence in the

TABLE 13

LASER HEATING OF TUNGSTEN PARAMETERS

Parameter Units Simulation Bechtel*

Wavelength nm 1064 1060

Reflectivity % 60.3184 60

Thermal conductivity at 300 K W/K‐cm 1.19715 1.78

Density g/cm3 19.3 19.3

Laser penetration depth nm 22.3654 25.0

Sound velocity m/s 5174 —

Ks — 18.0396 —

Debye temperature Kelvin 400.020 —

Chemical potential eV 18.08 —

Thermal mass ratio — 1.2036 —

Electron specific heat at 300 K J/K–cm3 0.04094 —

Lattice specific heat at 300 K J/K–cm3 2.39981 —

Relaxation time at 300 K fs 1.37942 —

G GW/K–cm3 33832.4 —

Laser penetration depth nm 22.3654 25.0

*Bechtel, 1975.


coefficients because of different approximations for the electron‐phononrelaxation time, but as a pedagogical exercise, the rederivation of the Kaga-

nov form has accomplished its objective of revealing the underlying behavior

of the thermal coupling between the electrons and the lattice.

Consider, as examples, gold and copper for the parameters given in

Tables 12 and 13, where Eq. (637) [rather than the Procrustean Eq. (640)]

is compared to Eq. (627)—the comparisons are pedagogical, given the nature

of the model and the wide variety of g (and sound velocity) values in the

literature—but the agreement is reasonable enough to conclude, first, that

the transfer of energy from the electron gas to the lattice is linear in the

temperature difference, and second, that the temperature dependence of

the coefficient g that governs the transfer follows the temperature depen-

dence of the electron‐phonon relaxation time as found by Kaganov Lifshitz,

and Tanatarov and therefore, the widespread use of the Kaganov form

(e.g., see Jensen, Feldman, Moody, and O’Shea, 2006a; Papadogiannis,

Moustaizis, and Girardeau‐Montaut, 1997; Yilbas, 2006) has merit.

I. Numerical Solution of the Coupled Thermal Equations

1. Nature of the Problem

The methods used to solve Eq. (625) are rather sophisticated and, through

the use of some simplifying approximations about the length of the laser

pulse, the temperature variation of the thermal conductivity, and the

temperature of the background lattice, analytical solutions are possible,

although to make use of them, numerical means are needed to evaluate the

terms of the series (Smith, Hostetler, and Norris, 1999). Our goals here are to

explore the regime in which the lattice and electron temperatures can diverge,

and so numerical methods are sought. A review of methodology is helpful.

Solving Eq. (625) using fashionable finite difference methods is a bit

premature because a simple finite difference numerical scheme (Smith,

1985) to solve the heat equation @tu ¼ Do@2xu ) @yu ¼ @2

z u sets limits on

the discretization spacing Dy tolerated in the time domain given a dis-

cretization spacing Dz in the position domain, where Dy and Dz are

normalized variables such that 0 zj 1 with a similar equation for yk,

with k and j being index coordinates. Stable and convergent solutions to

these parabolic equations for explicit schemes (i.e., ones where the j þ 1 time

step is straightforwardly calculated from the j time step solution) are only

possible if

r DyDz2

1

2, Dt Dx2

Do

; ð641Þ

240 KEVIN L. JENSEN

where the RHS is the largest Dt that can be considered. Taking as example

parameters Dx d=80, where d is the laser penetration depth (on the order

of 12 nm) and Do ¼ 1.2 cm2/s, then the largest tolerable time increment is

on the order of 0.1 fs. A simulation spanning 50 ps for a 3‐mm thick simula-

tion region would imply Nt ¼ 500,000 and Nx ¼ 20,000, or NtNx2 ¼ 2 1014,

and such investments of computer processing time are impractical (apart

from the fact that 50 ps is a short time and 3 mm is irrelevantly thin)—

even if Do ¼ k=Cv was more or less constant rather than dependent on the

evaluation of temperature‐dependent electron‐electron and electron‐phononrelaxation times. In that case, techniques analogous to the multipulse

treatment can be brought to bear and much accomplished via analytical

means (an excellent example being the analysis of Bechtel, 1975).

That, however, is not the situation here, and something more inventive is

required.

2. Explicit and Implicit Solutions of Ordinary Differential Equations

The resources available that cover numerical issues in the computational

solution of partial differential equation cousins, of which the heat equation

is a well‐examined representative, are legion (Anderson, Tannehill, and

Pletcher, 1984; Smith, 1985). The present interest is in solving such equations

when extraordinarily dissimilar time scales and conditions are involved. Over

sufficiently small scales, most functions are well approximated by polyno-

mials. The value of functions at various regularly spaced intervals provides a

useful estimate of the coefficients of those polynomials, and consequently

derivatives of those polynomials, albeit with a greater loss of accuracy the

higher the derivative. Let a polynomial P(x) take on the values yj at the

discrete points xj jDx, that is,

yj P jDxð Þ: ð642ÞIntroduce the notation

fn xj Dxn

d

dx

n

P xð Þx¼xj

: ð643Þ

A Taylor expansion of P(xj) about x ¼ 0 can then be written

yj ¼XN

n¼1

jn

n!fn 0ð Þ; ð644Þ

whereN is the order of the expansion. A little thought shows that Eq. (644) can

be elegantly expressed as a matrix equation when a multitude of yj’s are

available, such that the number of columns corresponds to N and the number

of rows to the quantity of yj’s available.When the number of rows and columns


are the same, there areN equations forN unknown coefficients of the f ’s so that

numerical estimates of the higher‐order derivatives can be made. Define

y ¼ M f

Mjk ¼ jk

k!

ð645Þ

For example, choosing points symmetrically about j ¼ 0, then for N ¼ 5

y2

y1

y0y1y2

0BBBB@

1CCCCA ¼ 1

24

24 48 48 32 16

24 24 12 4 1

24 0 0 0 0

24 24 12 4 1

24 48 48 32 16

0BBBB@

1CCCCA

f0f1f2f3f4

0BBBB@

1CCCCA ð646Þ

where, for notational simplicity, the ‘‘(0)’’ has been omitted from the f ’s, and

a common divisor has been extracted from the matrix. An obvious symmetry

about the center row is evident. The inverse of Eq. (646) gives the 5‐pointfinite difference approximation, or

f0f1f2f3f4

0BBBB@

1CCCCA ¼ 1

24

0 0 12 0 0

1 8 0 8 1

1 16 30 16 1

6 12 0 12 6

12 48 72 48 12

0BBBB@

1CCCCA

y2

y1

y0y1y2

0BBBB@

1CCCCA ð647Þ

There is nothing special about equispaced points. Values of the polynomial

on the half‐index (i.e., corresponding to xn þ 1/2) or nonuniformly spaced xjare equally subject to the same formalism, although slightly more cleverness

is involved. What is less evident, but of greater importance, is that the

matrices and vectors of Eq. (646) can be pared to obtain second‐order (i.e.,three‐point) approximations by crossing out the nth row and column

to eliminate yn to obtain convenient approximations that are useful. For

example, the much‐vaunted central difference scheme (CDS) is obtained by

eliminating the first and fifth rows and columns, and solving

y1

y0y1

0@

1A ¼ 1

2

2 2 1

2 0 0

2 2 1

0@

1A

f0f1f2

0@

1A)

f0f1f2

0@

1A ¼

y01

2y1 y1ð Þ

y1 2y0 þ y1

0BB@

1CCA ð648Þ

Often, at a boundary (that is, for j ¼ 1 or N), the forward ( j > N) or

backward ( j < 1) values are not available (this occurs, for example, if the

boundary is absorbing; Jensen and Ganguly, 1993). In this case, upwind

and downwind difference schemes are available. Consider explicitly the

second‐order upwind difference scheme (SUDS) that follows from eliminating

242 KEVIN L. JENSEN

the first and second rows and columns and solving for

y0y1y2

0@

1A ¼ 1

2

2 0 0

2 2 1

2 4 4

0@

1A

f0f1f2

0@

1A)

f0f1f2

0@

1A ¼

y01

23y0 þ 4y1 y2ð Þy2 2y1 þ y0

0B@

1CA:

ð649ÞObserve that to second order, the approximation to f2 has the same

structure of subtracting twice the central point from the sum of the end-

points—an indication that a second‐order polynomial has a constant second

derivative. The second‐order downwind scheme is trivially obtained by

changing the sign of the indices and the second row of Eq. (649).

Consider now the usage of finite differencing schemes to solve ordinary

differential equations. As a trivial case, consider how to solve the equation

@xuðxÞ ¼ vðxÞ with the boundary conditions of u0 and uNþ1. Using the CDS

scheme to approximate the first derivative, thematrix version of the equation is

1

2Dx

0 1 0 0 0

..

. ... ..

. ... ..

.

. . . 1 0 1 . . .... ..

. ... ..

. ...

0 0 0 2 2

0BBBBB@

1CCCCCA

u1

..

.

uj

..

.

uN

0BBBBBB@

1CCCCCCA

u0

..

.

0

..

.

0

0BBBBB@

1CCCCCA

8>>>>>><>>>>>>:

9>>>>>>=>>>>>>;

¼

v1

..

.

vj

..

.

vN

0BBBBBB@

1CCCCCCA; ð650Þ

where only the 1st, jth, and Nth rows are shown. In concise notation,

0

0.16

0.32

0.48

0.64

0 0.2 0.4 0.6 0.8 1

u(x)DDSSDDS & CDS

u(x)

N = 5

X

u(x) = 1− e−x

xu = e−x

x2u = −e−x

FIGURE 81. Comparison of the central differencing scheme with a first‐order differencingscheme used at the boundaries downward difference scheme (DDS) with a scheme using second‐order upwind and downwind differencing schemes (SDDS and central differences scheme (CDS).


1

2DxMuþ ubc ¼ v; ð651Þ

where Mj,j1 ¼ 1 (except for the last row) and Dx ¼ N þ 1ð Þ1. The second

vector ubc on the LHS is the vector of boundary conditions. Two important

but subtle features are noteworthy. First, because it is a first‐order differentialequation, Eq. (650) uses only one boundary (u0), whichmeans the other (uN þ 1)

must not be included (or vice versa). If thematrix equation is set up so that both

of these boundaries are specified, by which theNth row of the coefficient matrix

of u uses the second‐order scheme that the jth row uses, then it is quickly

discovered that the coefficient matrix does not have an inverse and a solution

is not possible. The second feature is that the solution of the matrix equation is

only as good as the worst differencing scheme used. In Eq. (650), the simple, or

Euler, downwind difference scheme is used for the Nth row of M, and so the

accuracy of the solution is to order Dx, even though the accuracy of the CDS

formula is to order Dx2. This is shown in Figure 81 for vðxÞ ¼ ex and

uðxÞ ¼ 1 ex andN¼ 5, for which the first‐order downward difference scheme

(DDS) used in the last row of the difference operator matrix in Eq. (650) is

responsible for the jagged appearance. If the last row is instead replaced with

the second‐order DDS of Eq. (649), that is, instead of 0 0 0 2 2ð Þ theNth row ofM resembles 0 0 1 4 3ð Þ, then the second‐order downwinddifference scheme (SDDS) line in Figure 81, accurate toDx2, results. It is worthemphasizing that the order of the solution is dictated by the order, in this case,

of the Nth‐row coefficients.

If, instead, the equation @2xuðxÞ ¼ vðxÞ (e.g., Poisson’s equation) were

being solved, the matrix version is

1

Dx2

2 1 0 0 0

..

. ... ..

. ... ..

.

. . . 1 2 1 . . .... ..

. ... ..

. ...

0 0 0 1 2

0BBBBB@

1CCCCCA

u1

..

.

uj

..

.

uN

0BBBBBB@

1CCCCCCA

þ

u0

..

.

0

..

.

uNþ1

0BBBBB@

1CCCCCA

8>>>>>><>>>>>>:

9>>>>>>=>>>>>>;

¼

v1

..

.

vj

..

.

vN

0BBBBBB@

1CCCCCCA: ð652Þ

Note the following: the coefficient matrix is tridiagonal throughout;

the boundary vector contains two (not one) boundaries. In the numerical

literature, solutions of tridiagonal matrix equations hold a special place, and

algorithms to rapidly solve them using a minimum of storage space are

widespread and are common in LAPACK1 or IMSL2 software. An example

1 http://www.netlib.org/lapack/2 http://www.absoft.com/Products/Libraries/imsl.html

244 KEVIN L. JENSEN

of the solution of the CDS equation of Eq. (652) for the same u(x) considered

previously results in the CDS line of Figure 81, which is indistinguishable

from the SSDS line.

A particularly expeditious solution to @2xuðxÞ ¼ vðxÞ is possible (Jensen

and Buot, 1991) without even numerically defining (i.e., creating) the matrix

M. If, on input, the boundary conditions vector is added to the vector

Dx2v ubc ) v, then the solution in a programming‐like notation becomes

For j ¼ 2 to N

vj ( j

j þ 12vj þ vj1

Next j

For j ¼ N 1ð Þ to 1

vj ( vj þ j

j þ 1vjþ1

Next j

u ¼ v=2

ð653Þ

Figure 82 contains an example for v(x) ¼ sin(2px) and N ¼ 24, where v(x)

compares very well to the numerical solution of –u(x).

Let @2xuðxÞ ) M u as considered above. The final step in preparing for the

heat equation is to consider solutions to

@tuðtÞ ¼ M uðtÞ: ð654Þ

−1

−0.5

0

0.5

1

0 105 15 20 25

v(x)−u(x)/2

v(x)

and

u(x

)

j

v(x) = sin(2px)

FIGURE 82. Numerical solution of v(x) compared to its exact representation using the

low‐memory solution.


Symbolically, the solution to this equation is

u tþ dtð Þ ¼ exp dtM u tð Þ; ð655Þ

where, as familiar in quantum mechanics (e.g., Eq. (130), and as expected

given the formal similarity between the heat equation and Schrodinger’s

equation), the exponential operator is understood to be replaced by its

power series expansion

exp dtM

X1n¼1

dtn

n!M n

: ð656Þ

That Eq. (654) in Eq. (655) solves Eq. (654) can be verified by substitution.

To order dt2, Eq. (656) can be approximated by the Cayley representation

(Press, 1992) because

exp dtM ¼ 1þ dtM þ 1

2dt2M2 þO dt3

¼ 1 dt2M

0@

1A

1

1þ dt2M

0@

1A ð657Þ

which implies, with Eq. (655), that

u tþ dtð Þ ¼ 1 dt2M

1

1þ dt2M

u tð Þ: ð658Þ

An alternate method to reach the same result is to use the simple Euler

scheme for the time derivative, but using an implicit scheme for the spatial

derivative term, where the average of the future and past solutions are joined.

Such a scheme is sometimes referred to as the Crank–Nicolson method. Why

the average? The spatial derivative should (one would think) be evaluated

at the midpoint between the future u(x,tþdt) and past u(x,t) solutions, that is,

u(x,tþdt/2), but the midpoint need not be available but is presumably near

the average. ‘‘Implicit’’ here is taken to mean that the future value of u is

acted on by a nontrivial matrix rather than the identity matrix, and so a

matrix inversion is required to solve for u (whereas an ‘‘explicit’’ scheme

would have only the identity matrix acting on the future value and therefore

require no such inversion). That is,

@tu x; tð Þ ) 1

dtu x; tþ dtð Þ u x; tð Þf g

@2xu x; tð Þ ) 1

2M u x; tþ dtð Þ þ u x; tð Þf g

ð659Þ

246 KEVIN L. JENSEN

which results in Eq. (658) after rearrangement. Implicit schemes can use

much larger time steps and still maintain stability, so the added cost of

inverting a matrix is often well worth the effort—particularly if an exponen-

tially decaying solution is sought, as exponentially growing solutions fre-

quently satisfy the same differential equation and are otherwise difficult to

suppress. If, as shall occur in the heat diffusion case, a source term v(x,t) is

added, then, like the spatial derivative term, the average of its future and past

values is used rather than simply its past value.

A final and widely‐used methodology is based on the ‘‘predictor‐corrector’’ methods such as that of Runge and Kutta (Press, 1992). A simple

Euler scheme might have us conclude, if trying to solve an equation such as

@tu ¼ f ðuÞ, where the RHS is a function of the function we are trying to find,

that a solution would be

u tþ dtð Þ u tð Þ þ dtf uðtÞð Þ )unþ1 ¼ un þ dtf unð Þ ð660Þ

where the second line defers to a simpler notation in which the index refers

to the time step. This scheme is accurate only to order O(dt), and that

is generally inadequate. A better approach is to take a ‘‘guess’’ as to what

unþ1/2 would be and use that in the evaluation of unþ1, or

unþ1=2 un þ dt2f unð Þ )

unþ1¼ un þ dtf unþ1=2

ð661Þ

The accuracy of this approach is substantially better, but one need not

stop there, and use a guess to get to the one‐quarter point, use that to guess

the half point, and so on, leading up to the ‘‘fourth‐order’’ Runge–Kutta

method, which is quite reliable.

Thus, of the numerical methods in which we are interested, second‐orderdifferencing schemes for spatial derivatives (CDS away from the boundaries

but SDDS and SUDS at the boundaries), coupled with some combination of

implicit and predictor‐corrector schemes, may be what is required to avoid

the limitations otherwise obstructing our ability to circumvent time scales

of widely different magnitude in the solution of the laser‐heated surface.

That, as determined below, is in fact a useful approach.


3. Numerically Solving the Coupled Temperature Equations With

Temperature‐Dependent Coefficients

In the parlance of the previous section, we shall solve for the electron

and lattice temperature using both implicit and predictor‐corrector schemes

(Jensen, Feldman, Moody, and O’Shea, 2006a). The discrete temporal

and spatial coordinates are defined by tj ¼ j 1ð ÞDt and xj ¼ j 1ð ÞDxfor 1 j Nt or Nx, respectively, where surface lies at x ¼ 0, and negative

x corresponds to the region of space occupied by the photocathode material.

For accuracy, the coefficients are temporally averaged as well. The transition

from continuum to discrete for a coefficient C(t) and a parameter T(t)

proceeds according to (where dependence on x is hidden)

CðtÞ@tTðtÞ ) 1

2Dt½C tþ Dtð Þ þ C tð Þ½T tþ Dtð Þ T tð Þ: ð662Þ

For the spatial derivatives, the dependence of k on temperature results in

@x½kðxÞ@xTðxÞ ¼ 1

2Dx2kjþ1 þ kj

Tjþ1 Tj

kj þ kj1

Tj Tj1

:

ð663Þ

If k were constant, the CDS approximation to the second derivative

results. The temperatures T are represented as vectors whose jth component

corresponds to the spatial coordinate xj; similarly, the coefficients become

matrices defined by (where g is the factor from U(Te,Ti) and is approxi-

mately constant for high(er) temperature as shown previously; alternately,

see Papadogiannis, Moustaizis, and Girardeau‐Montaut, 1997)

½Cel;j ¼1

2DtCe½Te þ Ce½Tef gdlj

½Cil;j ¼1

2DtCi½Ti þ Ci½Tif gdlj

½Hl;j ¼1

2gdlj

½Jl;j ¼g½Cil;j

2½Cil;j þ gdlj

ð664Þ

where dlj is the Kronecker delta function and the temperatures in k are

evaluated at the x location at a particular time t. Define

248 KEVIN L. JENSEN

½DðtÞl;j ¼1

4Dx2½ kjþ1 þ kj

dl;jþ1 2 kjþ1 þ 2kj þ kj1

dl;j þ kj þ kj1

dl;j1ð665Þ

0

(a)

10

20

30

40

−40 −20 0 20 40 60

Te(t)-Tbulk

Laser (scaled)Bechtel (fig. 5)

Ele

ctro

n te

mpe

ratu

re [K

]

Time [ns]

W @1 MW/cm2

30 ns (FWHM)

0

(b)

200

400

600

800

−40 −20 0 20 40 60

Te(t)-Tbulk

Laser (scaled)Bechtel (fig. 7)

Ele

ctro

n te

mpe

ratu

re [K

]

Time [ps]

W @ 1 GW/cm2

30 ps (FWHM)

FIGURE 83. (a) Calculation of temperature rise for illuminated tungsten surfaces showing

the impact of a temperature‐dependent thermal conductivity term (‘‘laser’’) compared to a

constant thermal conductivity as done by Bechtel (1975). (b) Same as (a) but for a higher laser

intensity over a shorter time.


700600500400300200100

Tem

pera

ture

[K

]

6040

200

−20−40

−0.25−0.20

−0.15−0.10

−0.05

Time [ps]

(a)

Distance

[micro

n]

−0.25

−0.5

0.5

1.0

1.5

2.0

(b)

−40−20

20

Tem

pera

ture

[K

]

Time [ps]

4060

0.0

0 −0.20−0.15

Distance

[micro

n]−0.10

−0.05


250 KEVIN L. JENSEN

Cu: Electrons(c)

800

600

400

200

Tem

pera

ture

[K

]

Time [ps]

Distance

[micro

n]

5

0

−5−0.25

−0.20−0.15

−0.10−0.05

Cu: Lattice

800

(d)

600

400

200

Tem

pera

ture

[K

]

Time [ps]Dista

nce [m

icron]

5

0

−5−0.25

−0.20−0.15

−0.10−0.05



The matrix form of the coupled temperature equations is then represented

as

Ce þ J Dð ÞjtþDt Te tþ Dtð Þ ¼ Ce J þDð Þjt Te tð Þ þ 2J TeðtÞþ 1

2

ðGðtÞdtþ Tbc ð666Þ

DþHð ÞjtþDt Ti tþ Dtð Þ ¼ DþHð Þjt Ti tð Þ þH Teðtþ DtÞ þ TeðtÞð Þ

whereÐGdt is the integral of the laser term over the time increment, and Tbc

accounts for the boundary conditions; far into the bulk, the temperature

is held fixed, and at the surface, the gradient of the temperature vanishes.

A complex wrinkle to the ‘‘implicit’’ nature of the problem is now evident

because the coefficients on the LHS of Eq. (666) must be evaluated at

the future time tþ Dt, whereas the temperatures at that time are being solved

for and therefore a priori unknown. This is handled by approximating Te and

Ti by their values at time t (the guess), solving Eq. (666), and using the

Cu: Difference(e)

800

600

400

200

0

Tem

pera

ture

[K

]

Time [ps]Dista

nce [m

icron]

5

0

−5−0.25

−0.20−0.15

−0.10−0.05

FIGURE 84. (a) Temperature profile into bulk tungsten for electrons for Bechtel‐like condi-tions as a function of distance from the surface and time compared to the center of the Gaussian

laser pulse. (b) Same as (a) but for difference between electron and lattice temperature in bulk

for tungsten. (c) Laser heating of copper for Papadogiannis conditions: electron temperature.

(d) Same as (c) but for lattice temperature using same scale; note the differences in peak

temperature. (e) Difference between (c) and (d). Note the much greater temperature differences.

252 KEVIN L. JENSEN

predicted values of the temperatures at time t þ Dt to create a new guess to

the coefficients (the refinement). The process is iterated several times, the

number of iterations being determined by when subsequent refinements have

negligible effect. By this ruse, it is possible to choose time steps that are much

larger than those tolerated by criteria such as Eq. (641).

The numerical solution is decidedly nontrivial to implement, as all manner

of terms are dependent on the temperature and particulars of the material

parameters that Figure in the scattering terms evaluated at the Fermi level and

other quantities discussed throughout this section. The temperature depen-

dence of the thermal conductivity (and other quantities) results in differences

in the temperature evolution as compared to solutions where such terms are

held fixed, as done by Bechtel (1975). Bechtel considered short laser pulses

incident on tungsten for laser intensities of 1 MW/cm2 and 1 GW/cm2 for

pulse widths 30 ns and 30 ps in his Figures 5 and 7, respectively. Figure 83

shows the comparison of the numerical solution of Eq. (666) with Bechtel’s

findings for the temperature at the surface for the parameters shown in

Table 13. To be sure, Bechtel used quantities from the literature (e.g., the

AIP Handbook; Gray, 1972), whereas here the same quantities (e.g., relaxa-

tion times, specific heat and thermal conductivities, reflectivity, penetration

depth) are calculated from the underlying models developed in preceding

sections; a comparison of parameters is given in Table 13. Initially, the

solutions track reasonably well, but as time progresses, the impact of

temperature‐dependent terms becomes evident. The important feature of

either Bechtel’s results or the present simulation is that compared to the

laser pulse, the temperature maximum occurs after the laser pulse maximum,

and the temperature profile is asymmetrical in contrast to the symmetrical

laser profile, modeled as a Gaussian with a center at t ¼ 0, as the heat

dissipates into the bulk material.

Next, consider how heat propagates into the bulk material. The numerical

solution of the electron temperature profile for the parameters considered in

the 1 GW/cm2 case above results in a temperature profile for the electrons

given in Figure 84a, where only a subregion of the entire simulation near the

surface is shown. For such parameters, the lattice temperature tracks

the electron temperature closely, a consequence of the rapidity of the scatter-

ing rates in comparison to the duration of the laser pulse. The difference

between the electron temperature and the lattice temperature in such a case is

more instructive (as shown in Figure 84b). Here, even though the difference

in temperature is never more than a few degrees, the electrons heat up in

comparison to the lattice as the laser pulse rises, but after the pulse begins to

fade, the electron thermalization causes the temperature to drop below the

lattice, at which point the lattice transferring energy back to the electrons

prevents their rapid decline in temperature.


Effects are perhaps more evident in the extreme, so consider conditions

reminiscent of the high‐intensity studies of Papadogiannis and Moustaizis

(2001), in which various metals were subject to GW/cm2‐intensity lasers for

very short durations. In such cases, the decoupling between the lattice

temperature and the electron temperature is far more pronounced. Here, a

Gaussian laser pulse of the form IlðtÞ ¼ Ioexp½ t=dtð Þ2with dt¼ 2 ps and an

intensity of 3 GW/cm2 under a field of 1 MV/m is incident on a bare copper

surface for a wavelength of 266 nm (the reflectivity of copper in IR is quite

high, so that copper photocathodes are generally subject to the fourth

harmonic of an Nd:YAG laser for which the wavelength is 1064 nm/n,

where n is the harmonic number). Now, and in support of similar findings

by Papadogiannis et al. (2001), the temperature rise of the electron gas is

rather substantial—on the order of 1000 K. However, unlike the case for

tungsten where the pulse was both longer and far weaker, now the heating of

the lattice follows the electron gas with a lag so that it remains hotter than the

electrons past the pulse even though it does not experience nearly as large a

temperature rise. Consequently, the temperature of the electron gas is kept

high by the lattice returning energy to the electrons after the laser pulse is

over. The difference between the electron and lattice temperature in

Figures 84c and d, respectively, is shown in Figure 84e. Under such circum-

stances, metals can be raised to a high enough temperature that thermionic

emission can result and complicate the interpretation of whether the electron

emission is photoemission or thermionic emission in nature (Bechtel, Smith,

and Bloembergen, 1977)—or, for that matter, when coupled with very high

fields, to what extent field emission contributes (Brau, 1997; Jensen, Feldman,

and O’Shea, 2005). Determining which is which shall be taken up below.

J. Revisions to the Modified Fowler–Dubridge Model: Quantum Effects

The methodology introduced in the general thermal field equation can

be extended to the modified Fowler–Dubridge model to assess the impact

of the transmission probability not being a step function on the QE.

With the development of the moments‐based approach in the next section,

such treatment is perhaps ancillary but is given for aesthetic completeness.

For photoemission, the N function introduced in the GTF equation is

N n;s; uð Þ n

ðu1

ln½1þ en xþsð Þ1þ ex

dx; ð667Þ

where u ¼ bF Emð Þ mþ f hoð Þ, s ¼ bF Emð Þ ho fð Þ. In particular, and in

contrast to the GTF equation, Em ¼ mþ f under all conditions as the

emission is dominated by electrons passing over the barrier. Therefore,

254 KEVIN L. JENSEN

since nothing is added by retaining the argument of bF in the case of

photoemission, it is neglected here so that bF without an argument refers to

the quadratic approximation, that is, bF ¼ bF mþ fð Þ. As before,N separates

into regions:

N n;s; uð Þ ¼ N1 n;s; uð Þ N2 n;s; uð Þ þN3 n;s; uð Þ þN4 n;s; uð Þ;ð668Þ

where the sign on N2 deserves note. Observe that these are not the same

integrals obtained by simply changing the sign of s; rather, they are regions

defined according to whether a closed‐form series representation of the

integrand components is allowed. N1 and N2 can be done exactly

N1ðn;s; uÞ ¼ n

ðs

1ln 1þ enðzþsÞ

dz

8<:

9=; ¼ U 0ð Þ

N2ðn;s; uÞ ¼ n

ðux

ln 1þ enðzþsÞ½ ez þ 1

dz

8<:

9=;

¼ n2 U sð Þ U uð Þ uþ sð Þ ln½1þ euf g n2U sð Þ

ð669Þ

N3 is given by (the perfunctory treatment being a direct consequence of

having explained the series methodology in the context of the GTF equation)

N3ðn;s; uÞ ¼ n

ðus

n zþ sð Þez þ 1

dz

8<:

9=;

X1k¼1

1ð Þkþ1eksZ

k

n

0@1A

Z xð Þ X1j¼1

1ð Þjþ1

j j þ xð Þ

ð670Þ

The fourth term N4 is difficult but can be shown to be

N4ðn;s; uÞ ¼ n

ðus

ln½1þ enðzþsÞez þ 1

dz

8<:

9=; ð671Þ

1

2z 2ð Þ þ

X1k¼1

1ð Þk eksZ k

n

0@

1Aþ ekns

k21þ kn2 kZ knð Þ þ Z knð Þð Þ 8<

:9=;

where throughout, by virtue of the largeness of u, terms such as e–u have been

neglected. For large s, only the k ¼ 1 terms and terms of order e–s at most

need consideration. Combining and keeping dominant terms shows that the

photoemission extension to the N function is


N n;s; uð Þ 1

2n2s2 þ z 2ð Þ½n2 þ 1 esn2S

1

n

þ ensS nð Þ

; ð672Þ

where for standard technologically accessible photoemission conditions, n is

in general smaller than unity. Consider as an example, for cesium on copper

(m ¼ 7 eV, F ¼ 1.8 eV) for room‐temperature conditions and a field of

100 MV/m (n ¼ 0.552, f ¼ 1.42 eV) and a photon wavelength of 800 nm

(ho¼1.55 eV). The three separate groupings in Eq. (672) are then 12.5, 2.15,

and 0.0114, respectively, the sum of which is 14.6.

The MFD equation does not consider currents per se as does the GTF

equation, but rather probabilities, and it is therefore the ratio P hoð Þ ¼J F ;T ; hoð Þ=Jmax hoð Þ that is of interest, where the numerator is obtained

from Eq. (672) by appending the requisite coefficients on the N function as

done for the thermal‐field equation. Following the arguments familiar from

the evaluation of the scattering rates, an electron will not be photoexcited

unless its final state above the Fermi level is unoccupied, after which only

those electrons with a momentum component toward the surface are of

importance. Therefore

Jmax ¼ 2ARLD

k2B

ð10

EfFD Eð Þ½1 fFD E þ hoð ÞdEðp=20

sinydy

ARLD

k2Bho 2m hoð Þ

ð673Þ

10−3

10−2

10−1

100

200 400 600 800

FD (Cu)GP (Cu)FD (CsCu)GP(CsCu)

Rat

io w

ith

ref. (

o) v

alue

Wavelength [nm]

Field = 100 MV/m; lo = 200 nm

T = 600 K

Φ = 1.8 eV

T = 300 K

Φ = 4.5 eV

FIGURE 85. Comparison of the modified Fowler–Dubridge formulation (FD) with the

general photoemission formulation of Eq. (674)

256 KEVIN L. JENSEN

The general photoemission equation that updates the MFD equation, as

the GTF did with the FN and RLD equations, is obtained by replacing the

probability of photoemission in expressions for QE with

U bT ho fð Þð ÞU bTmð Þ ) P hoð Þ ¼ ho fð Þ2 þ 2b2

T zð2Þ 1þ n2ð Þ2ho 2m hoð Þ : ð674Þ

If the LHS of Eq. (674) is designated as PFD hoð Þ and the RHF PGP hoð Þ(where FD and GP indicate Fowler—Dubridge and general photoemission,

respectively), then a measure of the impact of Eq. (674) is obtained by

considering PFD hoð Þ=PFD hooð Þ compared toPGP hoð Þ=PGP hooð Þ, where oo

is an ad hoc reference frequency (here chosen to correspond to a wavelength

of 200 nm); the results are shown in Figure 85. For metallic‐like parameters,

the Fowler–Dubridge model is adequate; for a low work function surface,

however, the quantum effects distinctively make their impact known.

K. Quantum Efficiency Revisited: A Moments‐Based Approach

The elements of the three‐step model introduced in the discussions of

Eq. (376) and combined in Eq. (599) are now reconsidered. As before, the

problem of photoemission naturally partitions itself into absorption, trans-

port, and escape. Recall that the modified Fowler–Dubridge approach to

estimate QE focused on an independent estimation of each from which QE

was obtained by considering their product. Reflection suggests that treating

such processes as distinct may be lacking for several reasons. First, treating

the transmission probability (leading to the Fowler–Dubridge functions) as

a strictly 1D model is at odds with the consideration of the scattering factor

Fl as a 3D construct. Second, it would be superior to account for the

emission probability in light of the fact that only part of its total momentum

is directed at the surface, rather than all. Finally, the formulation of the

modified Fowler–Dubridge formula, for reasons intimately connected with

these issues, is not adaptable to the evaluation of beam emittance at the

cathode in the discussion of thermal emittance; the transverse momentum

components are ignored when they are of central importance.

Recall that in the development of the 1D current‐density equations, the

concept of a supply function was invoked, in which the integrations over

the momentum components parallel to the plane of the surface were per-

formed on the FD function characterizing the electron distribution. This was

a consequence of the transmission probability being dependent only on the

momentum normal to the surface. With photoemission—as noted in the

derivation of the Fowler–Dubridge model—the electron energy is augmented

by the photon energy. However, the approximation in Fowler–Dubridge that

the entire energy of the photon was manifested in the forward direction of the


electron toward the surface was far too optimistic, even though it worked

well when the photon energy was comparable to the barrier height above the

Fermi level. Rather, the transmission probability (expressed in terms of an

energy argument) should be

T Ex þ hoð Þ ) T E þ hoð Þcos2y ; ð675Þ

where y is the angle of the electron trajectory with respect to normal. Another

approximation used in the development of the thermal and field emission

equations that must be modified in the case of photoemission is that the final

state of the electron after absorption Figures into the analysis. Consequently,

for the distribution of electrons in the current density, the replacement

fFDðEÞ ) fFDðEÞ 1 fFD E þ hoð Þf g ð676Þis made, in which the occupancy of the final state of the electron matters as

to what electrons can be excited. Last (but not least), an electron of a given

final‐state energy must transport to the surface without suffering a debilitat-

ing collision, and so the scattering factor as a function of energy must be

present. That is, the factor

fl cosy;E þ hoð Þ ¼ cosycos yþ p E þ hoð Þ

p Eð Þ mdhk Eð Þt Eð Þ

ð677Þ

is included in the integrand, where d is the laser penetration depth,

k Eð Þ ¼ ffiffiffiffiffiffiffiffiffiffi2mE

p=h, and the relaxation time t(E) has by now become all too

familiar. Define

Mn ksð Þ ¼ 2pð Þ3 2m

h2

0@

1A

3=2 ð10

E1=2dE

ðp=20

sinydy ksð Þn

T ðE þ hoÞcos2yf gflðcosy;E þ hoÞfFDðEÞ 1 fFDðE þ hoÞf gð678Þ

where particles traveling away from the surface (y > p/2) have been excluded

and the following definitions for parallel ks ! kz and transverse ks ! krmomentum components are defined by

h2k2z2m

¼ Ecos2y

h2k2r2m

¼ Esin2y

ð679Þ

258 KEVIN L. JENSEN

Thus, in Eq. (678), a sin2y term in the integrand raised to the power n/2 is

recognized as the longitudinal momentum to the nth power. By way of

contrast, if emittance were the focus, then the cos2y would be replaced by

sin2y in the integrand to examine the transverse momentum moments.

It is readily seen that Eq. (678) is far different than the modified Fowler–

Dubridge approach, but it is also seen how the Fowler–Dubridge approxi-

mation is a consequence for photon energies not much in excess of the barrier

height; the transmission probability in such cases only admits electrons fairly

well pointed at the surface at the outset, and the integrand tapers off quickly

for larger values of y. The solving of Eq. (678), however, is a rather pro-

tracted problem for which the energy, field, and temperature dependence,

especially of the relaxation time embedded in p(E), the transmission proba-

bility T(E,) and the FD distribution function, make a numerical approach

all but inevitable. Nevertheless, the leading‐order behavior is instructive

to ascertain, and it is obtained by making the zero‐temperature, small‐fieldapproximation. The former turns the FD distributions into step functions;

the latter does the same with the transmission probability.

The moments approach to the evaluation of current density (for reasons to

be seen over time, the current‐density calculation is easier to consider than

emittance) is obtained from the first moment of the distribution function for

kz, for which

Jo ¼ 2ARLD

k2B

ðmmþfo

EdE

ð1jðEÞ

x2

xþ p E þ hoð Þ dx; ð680Þ

where the energy ratio j(E) has been introduced and is defined by

j Eð Þ mþ fE þ ho

1=2

: ð681Þ

Clearly, photoemission does not occur unless j(E) < 1; that is, the final

electron energy exceeds the barrier maximum. Note that barrier factor f is

used rather than the work function F as the low‐field approximation man-

ifests itself as rendering the transmission probability to be a step function

independent of whether the Schottky barrier–lowering factor is included or

not. Using the approximation


ð11d

x2

xþ pdx ¼ p2 ln 1 d

1þ p

8<:

9=;þ 1 pð Þd 1

2d2

d1þ p

þO d2 ð682Þ

then Eq. (680) becomes

Jo 2ARLD

k2B

ðmmþfho

1 j Eð Þ3n o1þ p E þ hoð ÞEdE: ð683Þ

In turn, the leading‐order approximation to Eq. (683) is given by

Jo 2ARLD

k2B

ho fð Þ2 3mþ f hoð Þ12 mþ fð Þ½1þ p mþ hoð Þ

( ): ð684Þ

Not unexpectedly, the dependence on the factor ho fð Þ2, anticipatedfrom the modified Fowler–Dubridge approach, is prominent, but even more

can be said. Compare Jo with a ‘‘current’’ that accounts for all the excited

electrons and directed at the surface defined by the relation

Jmax ¼ 2ARLD

k2B

ðmmho

EdE

ð10

dx ¼ ARLD

k2Bho 2m hoð Þ: ð685Þ

It immediately follows that for f < ho < m

P ho; bF ; bTð Þ Jo

Jmax

¼ ho fð Þ2 3mþ f hoð Þ6ho 2m hoð Þ mþ fð Þ½1þ p mþ hoð Þ : ð686Þ

Eq. (686) seems rather far from the Fowler–Dubridge model. However,

if the photon energy is approximately equal to the barrier height above

the Fermi level, then

P ho f; bF ; bTð Þ ho fð Þ24m2½1þ p mþ fð Þ ; ð687Þ

an expression which, apart from a factor comparable to 2 to 4, very closely

resembles the product of the scattering factor Fl and Fowler–Dubridge

probability ratio U b ho fð Þð Þ=U bmð Þf g. Rather than succumb to such a

temptation, however, the moments‐based Eq. (686) is used in preference to

the modified Fowler–Dubridge‐like Eq. (687) in calculations of QE below in

cases where an analytical approximation is used, and so the moments‐basedapproach identifies the total photo‐field‐thermal current as

260 KEVIN L. JENSEN

Je ho;F ;Tð Þ ¼ q

ho1 RðyÞð ÞP ho; bF ; bTð ÞIl þ JGTF F ;Tð Þ; ð688Þ

where JGTF(F,T) is the general thermal‐field contribution if tunneling and/or

thermal emission are appreciably present.

The photocurrent expression in Eq. (688) has required extensive calcula-

tion. Its historical development and performance has been cataloged in

the literature (Jensen, 2003a; Jensen, Feldman, and O’Shea, 2003; Jensen,

Feldman, Virgo, and O’Shea, 2003a,b; Jensen, Feldman, and O’Shea, 2004,

2005; Jensen, O’Shea, Feldman, and Moody, 2006; Jensen and Cahay, 2006;

Jensen, Feldman, Moody, and O’Shea, 2006a,b; Jensen, Lau, and Jordan,

2006) as it was systematically tested in the treatment of bare metals, coated

surfaces, and progressively more complex systems. Its evaluation requires a

full‐fledged numerical solution to account for time dependence, temperature,

scattering factors, reflectivity, and other explicit and/or implicit quantities

that are otherwise carefully hidden in the folds of such a deceptively unas-

suming equation. In the following text, QE shall be the numerically evaluated

ratio between total emitted charge and total incident energy as per Eq. (375),

where the emitted charge is the time integral over a current density [Eq. (688)]

for a uniformly illuminated area, and the total incident energy is the time

integral over laser intensity over the same area wherein the laser profile is

presumed to be Gaussian in time and uniform in space.

10−3

10−2

10−1

200 220 240 260 280

ExperimentalΦ = 4.31Φ = 5.1060−40

Qua

ntum

effic

ienc

y [%

]

Wavelength (nm)

Cu (SLAC)

FIGURE 86. Comparison of experimental data (circles) with theory (all parameters from

literature sources). Assuming that the surface is composed of two crystal faces in 60/40 propor-

tion, the solid blue line results (weighted average of the 4.31‐eV and 5.10‐eV lines). (Experimental

data courtesy of D. Dowell, SLAC.)


L. The Quantum Efficiency of Bare Metals

Metal photocathodes are common photocathodes: being relatively simple by

comparison to photocathodes using low–work function coatings (necessary

to significantly enhance the QE) in addition to being desirably rugged (al-

though they require cleaning; Schmerge et al., 2006), metal photocathodes

are natural testing grounds for the quality of the theoretical models that have

so far been developed. The prerequisite factors to evaluate the photoemission

(a)

(b)

FIGURE 87. (a) Surface of a sintered tungsten dispenser cathdode, showing evidence of crystal

face variation, pore and profilimetry, and surface roughness as a consequence of machining.

(b) Same as (a) but at a lower magnification. (Photographs courtesy of N.Moody (UMD/LANL).

262 KEVIN L. JENSEN

current from simple metals have been described, from the reflectivity and

laser penetration depth to the dependence of the scattering factors on tem-

perature and finally to the probability of emission. Several sources are

available for comparisons.

As a first comparison, consider the measured QE of copper (keeping in

mind the caveats about DOS), as a function of wavelength before and after

cleaning with a hydrogen ion beam, shown by Dowell et al. (2006) in their

4

4.4

4.8

5.2

100 110 111 112 113 114 116 310 332

Cu

W

Nb

Mo

Wor

k fu

nction

[eV

]

Crystal plane

FIGURE 88. The work function of various crystal faces for several metals topical to photo-

cathodes for the accelerator community.

10−6

10−5

10−4

10−3

10−2

10−1

100

210 240 270 300

110 Face only

100 + 110 + 111 Face

QE

(l)/Q

E (

190

nm)

Wavelength [nm]

Copper

FIGURE 89. Difference in quantum efficiency for a pure (110) face compared to a surface

equally composed of the 100, 110, and 111 faces of copper.


Figure 2 for the line designated ‘‘10.23 mC’’; in this line, it was argued that

the contamination that had collected on the surface was removed—albeit not

entirely, as a residual 8% of the surface was claimed to be covered with

carbon (a high–work function material as a contaminant). Analogous results

were obtained by Moody et al. (Moody, 2006; Moody et al., 2007) for the

cleaning of tungsten with an argon ion beam. A comparison of the Dowell

et al. data with a simulation based on Eq. (688) with all quantities such as

relaxation time, reflectivity, and other embedded factors calculated using the

models of Sections II and III is shown in Figure 86, where the intensity of the

incident light is so low as to not make demands on the numerical calculation

of a temperature rise.

Various sources of differences and errors make a comparison to experi-

mental data somewhat of an art. In assessing the performance of the theory,

it is important to estimate these effects and the comparative magnitude of

change that they would entail. They are variations of work function with

crystal face, differences in the DOS compared to the nearly free electron gas

approximation, effects of surface structure and/or reflections and/or field

enhancements, and the impact of contamination. The existence of so many

seemingly suggests that agreement between the theory herein and actual data

bears a serendipitous relationship, but that would be an overly cynical

insinuation. The various complications create changes that can be at odds

(in the direction for which they modify the model above) with each other and,

moreover, do not result in large multiplicative factors. In actuality, the

success of the moments‐based emission model is notable. Moreover, there

are factors that, if not unknown, are unknowable and therefore must be

accounted for by other means (e.g., surface profilimetry; Jensen, Feldman,

Virgo, and O’Shea, 2003b; Jensen, Lau, and Jordan, 2006), which will tend to

result in an ‘‘effective’’ field enhancement factor to account for surface

roughness, and geometrical features complicate transport near the surface in

complex ways (Mayer and Vigneron, 1997). Most important, surface condi-

tions are not static; they are affected by themigration of coatings across metal

surfaces, evaporation rates, degradation effects, and performance character-

istics (Jensen et al., 2007). Complications to the simple metals are considered

here, deferring until the next section the significantly more complicated

impact of surface coatings on all manner of electron emission effects.

1. Variation of Work Function With Crystal Face

Particularly in studies of thermionic emission from dispenser cathodes, where

sintered metals such as tungsten expose a number of crystal faces (an example

is shown in Figure 87) on which low–work function coatings rest, it has long

been appreciated that the consequences of the presence of different exposed

faces translated into different emission current densities (Adler and Longo,

1986; Haas and Thomas, 1968;). Figure 88 shows the variation of work

264 KEVIN L. JENSEN

function for typical faces for a few thermionic, field emission, and photo-

emission metals. The cause of the different work functions for various faces,

according to Brodie (1995), relates the crystal face work function to dimen-

sions of the underlying atoms and the effective mass with respect to Fermi

energy of the electron along different crystal planes in bulk. Consider the

expected photoemission differences if the first three crystal faces of copper

(100, 110, and 111 with F ¼ 5.1, 4.48, and 4.94, respectively) are present in

equal proportion on a surface, as compared to a monocrystalline surface of

the 110 face, the results of which are shown in Figure 89 under the assumption

that impact of reflectivity and scattering are more or less equal, and by using

the Fowler–Dubridge representation for the escape probability, as it is easier

to implement and is approximately correct. By taking the ratio with a refer-

ence QE at l ¼ 190 nm, factors common to all three faces cease to figure into

the estimate. There are reductions in the QE that vary as a function of

wavelength as the photon energy drops below the various crystal face work

functions as the wavelength increases; consequently, the agreement is better

for shorter wavelengths (the QE for 3 faces is approximately 70% of the QE

for the 110 face at 190 nm) than it is for longer wavelengths (the QE for 3 faces

is 1/3 of theQE for the 110 face for wavelengths longer than 270 nm,where the

factor of 3 represents the assumption that the crystal faces are equally

represented).

1

1.5

2

2.5

3

200 220 240 260 280

Rat

io

Wavelength [nm]

s = 3 T = 300 K

F = 1 MV/m

U bT 4QsFhw−Φ +

U bT 4QFhw−Φ +

FIGURE 90. Behavior of the Fowler–Dubridge function as a consequence of field enhancement

for a hemisphere.


40

50

60

70

80

90

100

0 20 40 60 80

Copper

Lead

Ref

lect

ivity

[%]

Angle [deg]

FIGURE 91. Reflectivity of copper and lead as a function of incidence angle.

FIGURE 92. The surface of solid lead. The white square is 4 mm on a side. (Photograph

courtesy of J. Smedley, Brookhaven National Laboratory.)

266 KEVIN L. JENSEN

FIGURE 93. The surface of magnetron‐sputtered lead. The white square is 5 mm on a side.

(Photograph courtesy of J. Smedley, Brookhaven National Laboratory.)

r1 r2 1-r1-r2

FIGURE 94. Surface roughness and its relation to increased absorption, as multiple hits on

the surface increase the absorption probability.


2. The Density of States With Respect to the Nearly Free Electron

Gas Model

The nearly free electron gas model has formed the basis of most models

described herein due to its simplicity and easy explanatory value. In contrast,

proper calculations should use the correct DOS in three dimensions for the

actual metals under question. There are substantial differences between the

simple metals and the transition metals for which the narrow d band fills and

for which the noble metals (a surreal title given the passions they arouse),

such as gold, silver, and copper, have completely filled d bands (Sutton,

1993). A proper account of electron emission (Modinos, 1984) and in partic-

ular photoemission (Berglund and Spicer, 1964a,b; Dowell et al., 1997;

Ishida, 1990; Janak, 1969; Krolikowski and Spicer, 1969) pays attention to

the empirical DOS or uses sophisticated theoretical methods to estimate the

DOS, especially for the transition metals. Alternately, dedicated sites for the

calculation of the DOS for various elements exist (‘‘NRL Electronic Struc-

tures Database.’’ http://cst-www.nrl.navy.mil/ElectronicStructureDatabase).

Calculating the DOS, however, requires an understanding of the underlying

crystal structure, is nontrivial, and requires that fairly substantial theoretical

methods be brought to bear; the case for copper is a particularly complicated

one (Campillo et al., 2000)—the repeated use of copper as a case study herein

is therefore not without a bit of irony. It induces complexity far beyond the

nearly free electron gas model that provides a useful simple model for

dr

Perspective Top-down

z

ρ

q

FIGURE 95. Relation of the incidence angle to the differential surface area for the model of a

hemisphere.

268 KEVIN L. JENSEN

example cases we have considered. This most important of modifications to

the photoemission models is therefore relegated to the in‐depth treatments of

the literature.

3. Surface Structure, Multiple Reflections, and Field Enhancement

In the moments‐based model behind Eq. (678), changes induced by surface

structure are not explicitly accommodated. These manifest themselves in two

ways: as a field enhancement changing of the emission barrier, and as a

change in the reflection of the incident light due to a crystal face being at

an angle to the incident light.

Consider field enhancement first, and as a pedagogical example, consider a

hemispherical bump (a ‘‘boss’’). Field enhancement tends to change over a

bump; recall that Schottky barrier–lowering for a field of 1 MV/m is on the

order of 0.04 eV, which, while not great, can affect estimates of QE. Letting

s represent the field enhancement factor compared to a flat surface, then

the approximate increase in QE may be estimated from the modified

Fowler–Dubridge formula as

Jo sFð ÞJo Fð Þ U ½bT ho Fþ ffiffiffiffiffiffiffiffiffiffiffiffi

4QsFpð Þ

U ½bT ho Fþ ffiffiffiffiffiffiffiffiffiffi4QF

pð Þ : ð689Þ

The behavior of Eq. (689) for copperlike parameters is shown in Figure 90.

The impact of field enhancement is offset by the areas involved over which

the enhancement factor is significant. Using the example of the hemisphere,

while the enhancement on‐axis is a factor of 3, the effective area over which

this occurs is dA ¼ 2pr2sinydy, and therefore, smaller areas contribute near

the axis where the enhancement is strong (similar arguments are at work in

the definition of the emission area of a field emitter (Forbes and Jensen, 2001,

for example).

Next consider the change in reflectivity, again for the boss example as the

angle the incident light makes with the normal to the surface is equivalent to

the polar angle measured from the apex. The reflectivity is dependent on the

particular material; the cases for copper and lead are shown in Figure 91. The

reflectivity does not change appreciably until past 60 degrees, at which point

it begins to climb to unity. Such an effect offsets the greater emission area

associated with the rings of larger diameter over which the reflectivity is

constant. Thus, the centermost parts of the hemisphere contribute the most

to the QE of a rough surface, although the effective emission area is less than

suggested by the dimensions of the illumination area.

To gain an appreciation of the complexity of the physical surfaces, consid-

er the cases of solid and magnetron sputtered lead surfaces, images of which

(taken by J. Smedley, BNL) are shown in Figures 92 and 93, respectively.


Both appear smooth on a macroscopic scale, but micron‐scale resolution

shows just how complex the surfaces are, particularly the magnetron sput-

tered example, which evinces greater QE than the solid lead surface (in the

latter, the sharp eye will notice sandlike grains pressed into the lead surface,

which are residual grains from diamond polishing). The canyonlike complexity

of these surfaces suggests yet another possible effect—the probability that light

reflected from the side of a protrusion, rather than being sent on its way from

the surface, is rather sent to strike another region on the surface, as suggested in

Figure 94. With high reflectivity a disproportionately greater impact results.

QE depends on the amount of light absorbed, so the question arises as to how

much more light is absorbed when multiple reflections are present. Let the

proportion of the surface accounting for one reflection be r1, that for two

reflections be r2, and assume that all photons experiencing more than two

reflections are in fact absorbed. The increase in QE will then be, to a first

approximation, the ratio of photons absorbed on a rough surface on regions

where one, two, andmore than two reflections occur compared to the condition

where only one reflection occurs. The number of photons absorbed from

those incident on region 3 is unity by assumption; the number absorbed

incident on region 1 is (1 – R); and the number absorbed incident on region

2 is R(1 – R). Therefore, the ratio of the number of absorbed photons for the

rough surface compared to the smooth should approximately behave as

QErough

QEsmooth

1 r1 r2ð Þ þ 1 Rð Þr1 þ R 1 Rð Þr21 R

¼ 1 Rr1 R2r2

1 R: ð690Þ

For a rather stylized example, if the three regions are equally represented

and the reflectivity is 75%, then the improvement is 9/4 ¼ 2.25.

A complication is the fact that the reflectivity generally depends on inci-

dence angle and Eq. (690) presumes the reflectivity to be more or less

constant. The question arises, then, as to how much the variation in reflec-

tivity will affect matters; its impact will be to reduce the effective absorbing

area as surfaces faceted away from the normal to the macroscopic surface

will subject incoming photons to more oblique incidence angles, as suggested

in Figure 95. For uniform intensity light incident from the top, the boss will

be ‘‘seen’’ as a circle (the ‘‘top‐down’’ perspective) so that the intensity of

light Io illuminating each ribbon defined by 2prdr will be the same, even

though the intensity Iocosy on the actual surface ribbon 2pa2sinydy, wherea is the radius of the boss and r ¼ asiny, diminishes as y increases. The

product of the reduced intensity and the increased ribbon area

2prffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffidr2 þ dz2

p ¼ 2prdr=cosy offset each other, resulting in an integrand

that puts the work of the y‐variation only on the reflectivity factor. There-

fore, in the modified Fowler–Dubridge model, the ratio of the QE for an

270 KEVIN L. JENSEN

600

550

500

450

400

350

350 400 450 500 550 600x pixel

y pi

xel

1

0.95

0.85

0.75

0.65

0.55

0.80

0.70

0.60

0.90

October 31, 2001(a)

November 04, 2001

600

(b)

600

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

550

550

500

500

450

450

400

400

350

350x pixel

y pi

xel



December 04, 2001

600

(c)

600

0.3

0.2

0.1

0.4

0.5

0.6

0.7

0.8

0.9

1.0

550

550

500

500

450

450

400

400

350

350x pixel

y pi

xel

x pixel650

December 10, 2001

600

(d)

600

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

550

550

500

500

450

450

400

400

350

350

y pi

xel

FIGURE 96. (a) Quantum efficiency plots of the APS Mg LEUTL Photocathode: pixels are

approximately 10 mm on a side. This image is before any cleaning. (b) Same as (a) but after first

cleaning. (c) Same as (b) but 1 month later, showing degradation from operation. (d) Same as (c)

but after second cleaning. Uniformity has been improved and contamination reduced, but the

pattern has evolved from (b). (Data for all images courtesy of J. Lewellen, Argonne National

Laboratory.)

272 KEVIN L. JENSEN

illuminated boss compared to a uniformly illuminated disk of the same

radius is given by

QEboss

QEdisk

ða0

1 R½y rð Þf g2prdrða0

1 R 0ð Þð Þ2prdr

¼

ðp=20

1 R yð Þf gsin 2yð Þdy1 Rð0Þð Þ

ð691Þ

For copper, where R behaves (to a good approximation) as

RCu yð Þ Ro þ 1 Roð Þexp mo

p2 y

h i; ð692Þ

with Ro ¼ 0.49775 and mo ¼ 6.2329, the evaluation of Eq. (691) is analytic

and results inQEboss 0:95338QEdisk, a difference that is not eye‐catching fora hemisphere but which, when applied to the multifaceted structures perhaps

analogous to the magnetron sputtered surfaces, might have consequences

of greater significance.

4. Contamination and Effective Emission Area

The problem of cleaning contamination from metal surfaces to expose clean

70

80

90

100

200 220 240 260 280 300

QE

patc

h/Q

Eba

re [%

]

Wavelength [nm]

7%

15%

30%

FIGURE 97. Changes in effective quantum efficiency for a surface partially covered with a

higher–work function material (percentages indicate degree of coverage of said material).


crystal faces is well known (Haas and Thomas, 1968). In the case of thermal‐field emission from refractory metals such as tungsten, a grueling heating of

the needles can be performed to drive off all manner of contaminations (and

also allow for deformation of the emitter tip as a consequence of the balanc-

ing of surface tension and field; Barbour et al., 1960). Such techniques cannot

be used with microfabricated field emitters (field emitter arrays) or the metals

generally used as photocathodes because the temperatures required are far

too high for the materials used or close to the melting point of the favored

metals (such as copper, which has a melting point of 1358 K), but contami-

nation is still problematic. Although Spindt et al. discussed molybdenum

field emitters, their description of the condition of emitter tips (from chapter

4 of Zhu, 2001) is an elegant summary for metal surfaces in general:

‘‘. . .microfabricated emitter arrays are rarely heated for cleaning at more than

450 C, and this is not sufficient to produce an atomically clean surface. As a

result, we find ourselves working with an ill‐defined emitting surface that can

probably be best described as a combination of several microcrystalline surfaces,

grain boundaries, and adsorbates. In addition, it is a dynamic situation as adsor-

bates diffuse about the surface and the surface evolves toward equilibrium with its

environment.’’ [Spindt et al., in chapter 4 of Zhu, 2001)]

For photoemitters, methods other than, or in addition to, heating must be

used to reduce the impact of adsorbates and contamination/degradation.

One such method is to subject the metal surface to a laser beam focused to

an intensity just below the damage threshold of the metal and then scan the

surface, a process that alters the surface as revealed by changes in the emission

pattern (Girardeau‐Montaut, Tomas, and Girardeau‐Montaut, 1997;

Smedley, 2001 Srinivasan‐Rao et al., 1998; Tomas, Vinet, and Girardeau‐Montaut, 1999). More recently, methods of cleaning the surface using hydro-

gen (Dowell et al., 2006) and argon (Moody, 2006; Moody et al., 2007) have

proven quite successful at cleaning and restoring a metal surface to initial

QE values. Even so, in the hydrogen ion cleaning of copper, as Dowell et al.

point out, the data suggest that 7% of the surface retains some carbon

coverage; since the work function of carbon is high, this suggests that a

fraction of a cleaned surface nevertheless does not contribute and therefore

gives the appearance of a lower overall QE than would otherwise be the case,

a problem conceptually very similar to the impact of nonuniformity (see

Chapter 2 of Herring and Nichols, 1949) or of poisoning of a low–work

function coating on the surface of a thermionic emitter of particular concern

to the dispenser cathode community (Marrian and Shih, 1989).

An example of the changing of the QE associated with an actual metal

photocathode is shown in a series of QE measurements of the magnesium

photocathode used for the advanced photon source (APS) low‐energy undu-

lator test line (LEUTL) at Argonne National Laboratory (Lewellen and

274 KEVIN L. JENSEN

Borland, 2001; Lewellen et al., 2002) taken by J. Lewellen (from ANL) over a

few months. In the sequence of images shown in Figure 96, two trends are

apparent. First, the ‘‘cleaning’’ of the photocathode by a laser significantly

improves the QE, both in magnitude and uniformity (the images were scaled

to theirmaximumvalue in each case and therefore show relative, not absolute,

performance so as to accentuate contrast—therefore the apparent QE of one

image does not correspond to the QE of another, although the scale band to

the right does indicate the relative magnitude within the image). Second,

recleaning the surface does not return the cathode to its initial state, although

it does improve matters; changes to the surface accompany the cleaning

process. One such change, not apparent, is the increase in dark current after

the cleaning process, an indication of changes in surface structure and geom-

etry—that is, cleaning ‘‘roughens up’’ the cathode. This affects emittance,

field emission, and the like—apart from the impact of changes in work

function due to removal and redeposition of contamination and adsorbates.

If a region of a photoemitting surface experiences conditions such that it

exhibits a higher work function than surrounding areas (due to contamina-

tion, crystal face, or another effect), then the overall QE is reduced. If an area

dA of a total area A exhibits a work function of F þ dF, then, compared to a

bare (or uniform) surface, the QE becomes, as estimated by the modified

Fowler–Dubridge model,

QEpatch

QEbare

dAA

U ½bT ho f dFð ÞU ½bT ho fð Þ

þ A dAð Þ

A: ð693Þ

For photon energies well above the barrier, the ratio is close to unity, but

for energies closer to the barrier, the reduction in QE is approximately

proportional to the uncontaminated proportion of the surface. As an exam-

ple, consider the contamination to have a work function comparable to

polycrystalline carbon of approximately 5.0 eV. The impact on QE as a

function of wavelength is shown in Figure 97 for coverages of 7%, 15%,

and 30%.

M. The Emittance and Brightness of Photocathodes

The moments‐based formalism used to determine the QE of bare metals

can now be used to determine the emittance associated with photoemission.

The need for such descriptions arises, for example, in the effort to provide

physics‐based emission models needed by advanced simulation codes

(Lewellen, 2001; Petillo et al., 2005; Travier et al., 1997; Zhou et al., 2002).

The tacit assumption underlying the present description is that the electron

beam is used to convert spontaneous electromagnetic radiation to coherent


radiation from a beam‐wave interaction characteristic of a broad class of

vacuum electronic devices (Abrams et al., 2001; Gilmour, 1986; Parker et al.,

2002). There are other uses of electron sources, but it is the VE applications

in particular that have provided the fundamental paradigm for demands on

the electron source that drives the present discussion. Traditionally, therm-

ionic cathodes are used in or sought for microwave and power amplifiers

(cold cathodes have also been considered and used (Makishima et al., 1999;

Whaley et al., 2000), whereas photocathodes are the source of choice for the

‘‘big dogs’’ of advanced RF photoinjectors for high‐power free‐electronlasers—and interesting combinations of photo and field emission may enable

‘‘little dogs’’ based on photostimulated needle cathodes (Brau, 1997, 1998;

Jensen, Feldman, Moody, and O’Shea, 2006b; Lewellen and Brau, 2003).

The two concepts that determine the quality of an electron beam intro-

duced in Section II.I (Thermal Emittance) are emittance and brightness.

They are of such paramount importance for the accelerator and vacuum

electronics communities that even an extended description in the confines of the

present treatment would only scantily cover the literature (the canonical texts

of Reiser, 1994, and Humphries, 1986 and 1990, are general treatments, but

see also Abrams et al., 2001; Anderson et al., 2002; Brau, 1997; 1998; Carlsten

et al., Fraser and Sheffield, 1987; 1988; Fraser et al., 1985; Humphries, 1990;

Jensen, O’Shea, Feldman, and Moody, 2006; O’Shea, 1995, 1998; O’Shea

et al., 1993; Parker et al., 2002; Rao et al., 2006; Reiser, 1994; Rosenzweig

et al., 1994; Travier, 1991; Tsang, Srinivasanrao, and Fischer, 1992). Therm-

ionic emittance was treated before (Eq. (373)); extending that treatment

to photoemission is the present objective. FELs represent ‘‘tunable’’ sources

of narrow‐band light—wavelengths in the hundreds (UV) to the tens

(XUV) of nanometers—and megawatt‐class devices may be possible if the

brightness of the electron source can be improved, and RF photocathodes

appear to be the most likely source capable of doing so (O’Shea et al., 1993,

1995). The average power of the FEL is limited by the electron beam average

power and beam brightness, for which the improvements entailed by photo-

cathodes literally outshine the thermionic cathode competition (Dowell et al.,

1993) in terms of beam brightness. In an FEL, a pulse train of electron

bunches is created, each containing a substantial amount of charge (on the

order of 0.1 to 2.0 nC). Overlap of the lightwave field with the electron bunch

is critical for gain, and that entails a tolerable upper limit on the transverse

emittance that can be endured (Fraser and Sheffield, 1987) (longitudinal

emittance, another concern, is not discussed here). Beams with higher current

and smaller emittance enable shorter wavelength and more powerful FELs.

Electrons outside the laser beam do not contribute much to the coherent

radiation and are wasted, or—what may be worse—electrons outside the

core beam (generally called halo) cause some damage elsewhere where it is ill

276 KEVIN L. JENSEN

tolerated (Bohn and Sideris, 2003). The extension of the derivation of the

emittance of a thermionic source to a photocathode requires more effort

(Jensen, O’Shea, Feldman, and Moody, 2006) than the pleasingly (and

relatively) uncomplicated derivation for thermal emittance.

0

1

2

3

4(a)

200 300 400 500 600 700

e n,rm

s [m

m-m

rad]

Wavelength [nm]

Cs on Cu Φ = 1.8 eV

CuΦ = 4.5 eV

Thermal

NumericalEq. 699

0

1 3 109

2 3 109

3 3 109

4 3 109(b)

200 300 400 500 600 700

Cu

Cs on Cu

Bn

[A/c

m2 ]

Wavelength [nm]

FIGURE 98. (a) Comparison of the analytical model of emittance [Eq. (699)] with

its numerical evaluation. (b) Estimates of brightness based on Eq. (700) for copper and

cesium‐coated copper (work functions of 4.5 and 1.8 eV, respectively).


The low‐temperature limit of Eq. (678) and the Richardson (step function)

approximation to the transmission probability result in the approximation

Mn ¼ 1

2pð Þ22m

h2ho fð Þ

24

35ðnþ3Þ=2 ð1

0

mðh fÞ þ x 1

24

35ðnþ1Þ=2

G p½ mþ fð Þ 1þ Dxð Þ; 1

1þ Dx

0@

1A

1=2

;n

2

264

375dx

ð694Þ

where the dimensionless quantity D has been introduced and defined by

D ¼ ho fmþ f

: ð695Þ

For photoemission conditions such that the photon energy is not much

larger than the barrier height, then D can be small for metals. The new

function G is defined by

G a; b; sð Þ ð1b

x 1 x2ð Þsxþ að Þ dx 1 b2ð Þsþ1

2 sþ 1ð Þ 1þ að Þ ; ð696Þ

where the RHS is an approximation rather than an exact result. In fact, when

0.5

1.0

1.5

2.0

2.5

3.0(a)

1 10 100

e n,rm

s [m

m-m

rad]

Field [MV/m]

Cs on Cul = 400 nmΦ = 1.8 eV

Cu

Φ = 4.5 eVl = 266 nm

Numerical Eq. (699)


278 KEVIN L. JENSEN

s is an integer, an exact result can be found, namely,

1 109

2 109

3 109

4 109

5 109

6 109

1 10

(b)

100

Bn

[A/c

m2 ]

Field [MV/m]

Cs on Cul = 400 nmΦ = 1.8 eV

Cul = 266 nmΦ = 4.5 eV

1

2

3

(c)

400 600 800 1000 1200 1400

e n,rm

s [m

m-m

rad]

Temperature [K]

Cs on Cu: l = 400 nm, Φ = 1.8 eV

Cu: l = 266 nm, Φ = 4.5 eV

NumericalEq. (699)

Eq. (373)

FIGURE 99. (a) Comparison of the numerical evaluation of emittance using the moments

to the analytical formula of Eq. (699) for copper and cesium on copper. (b) Brightness as

evaluated using Eq. (700) The brightness for bare copper has been multipled by a factor of

10 so as to allow for a visual comparison. (c) Numerically evaluated photoemittance compared

to analytical model: the latter is temperature‐independent, causing the numerical evaluation

(which is dependent on T) to diverge at larger temperatures. Also shown is the thermal emittance,

Eq. (699).


IV. LOW–WORK FUNCTION COATINGS AND ENHANCED EMISSION

A. Historical Perspective

In the presumptively halcyon days of the 1920s and 1930s, when the equations

of electron emission physics were born from the marriage of quantum me-

chanics and statistical mechanics, much effort was devoted to understanding

emission, characterizing work function, ferreting out the impact of different

crystal faces, and assessing the consequences of the absorption of materials

such as cesium and thorium on metals with the tendency to increase emission

current. Many of the great names of physics left their mark in disparate fields

from which the literature on electron beams trace their origins. At about the

same time, following the pioneering research of both Heinrich Hertz and

Nikola Tesla, Albert Taylor and Leo Young at the U.S. Naval Research

Laboratory (NRL)3 demonstrated (both by accident and intent) the first

continuous‐wave (CW) radar system that another NRL scientist, Robert W.

Page, succeeded in transforming into a pulsed radar system in the early 1930s

( Alliso n, 1981; Kevles, 198 7). The onset of war accele rated matt ers consider -

ably. In response to the urgent need of theUnitedKingdom for radar systems,

a magnetron developed by the U.K. scientists John Randall and Harry Boot

demon strated enough power to make radar practical (Osepc huk and Ruden,

2005 ; de cades late r, magn etrons filled another, albei t more be nign need for

microwave ovens). Klystrons were developed by the Varian brothers in 1937

( Tallerico, 2005 ), and the trave ling wave tube was inven ted by the U.K .

scientist Rudolph Kompfner and later refined by him and John R. Pierce at

Bell Labs in the United State s (Ler ner and Trigg, 1 991 ). Were it not for the

political events of that time that held history in thrall, the intellectual ferment

was likely exhilarating.

By the 1940s, the technologiesmade possible by harnessing electron emission

for vacuum tubes came to be recognized by those who understood how the

capabilities could be used to advantage in the pressing global conflicts of the

time.4 Nothing spawns innovation and investment quite like a convergence of

military and commercial interests. The needs of radar, communications, elec-

tronic warfare (Granatstein and Armstrong, 1999), and directed‐energy devices

3 Coincidentally, thermionic emission (referring to the emission of ‘‘thermions’’) was origi-

nally designated the ‘‘Edison effect’’ after Thomas Edison, who went on to champion the

creation of the U.S. Naval Research Laboratory and who is that laboratory’s patron saint.

The history of NRL and radar development there is detailed in Allison (1981).4 Power tubes refer to early magnetrons, klystrons, traveling wave tubes, and later gyrotrons

and free electron lasers; this is a peculiar appellation as there is nothing glass tube–like about

them.

280 KEVIN L. JENSEN

(Bennett and Dowell, 1999; O’Shea and Bennett, 1997) f or t he c ap ab ili ti es

of vacuum devices led to rapid advances on several fronts. Radar and

vacuum electronic research at NRL itself and many other institutions in

the United States and worldwide became vigorous for several decades and

remains an area of active research. While solid‐state technology applied

to radar has made astounding advancements in a moderately shorter time,

for high‐power applications the playing field still belongs to the ‘‘tubes’’

(Abram s et al., 2001; Freund and Neil, 1999; Granat stein, Parker, and

Armstr ong, 1999; O’She a and Freund , 2001 ).

Of the five technologies necessary for the maturation of RF vacuum

technol ogy ( Parker et al., 2002 )—nam ely, the linea r beam, pe riodic pe r-

manent magnet focusing, the depressed collector, the dispenser cathode,

and the metal/ceramic packaging—the innovation that is of present concern

is the fourth: the dispenser cathode. Its ubiquitous presence in all manner of

devices, such as cathode ray tubes in displays, advanced radar systems,

particle accelerators, satellite communications, electronic warfare systems,

microwave generators, attests to its sweeping importance. In time, other

cathodes offering other capabilities came to the fore, but the idea of lowering

the work function of a material through the selective application of materials

has captured the attention not only of the dispenser cathode community,

but also the field emission and photoemission communities as well as attested

by more than 80 years of research. What happens on the surface of a metal

when elements like cesium and barium come to roost is a protracted problem

in surface science for whi ch extens ive treatment s are to be foun d ( Haas

and Thoma s, 1968; Mod inos, 1984; Mo nch, 1995 ; Prutton, 1994; Sommer ,

1968 ). Here we provide a n accou nt of the phy sics and its app lication to the

interpretation of photoemission data from partially covered surfaces in a

manner that uses what has come before and a theory of work function

reducti on de veloped by Gyft opoulos and Levine ( Gyfto poulos an d Lev ine,

1962; Jen sen, Feldman , Mo ody, an d O’Shea, 2006a ,b; Jen sen, Feld man,

Virgo, and O’She a, 2003b; Levine an d Gyfto poulos, 1964a ,b; M oody

et al., 2007 ).

B. A Simple Model of a Low–Work Function Coating

When an atom of cesium sits on the surface of a metal such as tungsten, its

weakly bound outer electron easily transfers to the bulk material. The ion—

or, perhaps more correctly, the polarized atom—left behind induces an image

charge. A very trivial model of work function reduction is to then envision

that a sheet of charged ions opposite of their image charges exists, looking

very much like a capacitor. The surface charge density s and the distance of


the partial ion to its image charge d allow for an estimation of the potential

drop that can be interpreted as a reduction in the work function of the

surface. Assume for sake of argument that the charge density is a fraction

s of a unit charge for one atom over an area equal to the atomic diameter

squared, or s ¼ sq=ð2rCÞ2, where rC is the radius of the atom (for cesium,

rC ¼ 0.52 nm), which suggests that d is larger than, but near, 2rC. It then

follows from elementary considerations that

DF qseo

d ¼ 2pafshc

rCs ð701Þ

For cesium‐like parameters, the work function reduction is from 4.5 eV to

1.6 eV, or 2.9 eV, so that s is approximately 1/6, reinforcing the notion that

the cesium atoms are more like polarized atoms, or dipoles, than ions, as

suggest ed by sim ulation s (Hems treet an d Chubb, 1993; Hems treet, Chubb,

and Pickett, 1989 ).

If the surface coverage is not a monolayer (y ¼ 1) but rather exhibits

fractional coverage (y < 1), then s qy=rC and the work function decreases

with reduced coverage. This would suggest that the overall work function

decreases from bulk values to the monolayer coverage value as y increases

from 0 to 1. What is observed is that for very low coverage values, the

reduction is in fact roughly linear, but as monolayer coverage is approached,

changes in y do not change the work function appreciably from its mono-

layer values. In other words, rather than DF being linear in y, it resembles

a more complex function. The determination of that function is the goal of

Gyftopoulos–Levine theory.

C. A Less Simple Model of the Low–Work Function Coating

From the late 1970s and thereafter, considerable industrial effort was

devoted to understanding the operation of the dispenser cathode. Much

was to be gained from a longer‐life, lower–work function cathode for mili-

tary, space, and commercial applications, and a commensurate effort was

devoted by industry and government to characterizing them, finding new

candidates, and understanding the operation of these complex constructs.

A small and pragmatic literature base aimed at studying the operational

characteristics of these cathodes was published in the journal literature but

also in the Technical Digest of the International Electron Devices Meeting

(IEDM ) and the Tri ‐Se rvice/N ASA Cat hode Workshop (see Adle r and

Longo, 1 986; Chubun an d Suda kov a, 1997; Cort enraad et al., 1999; Falce

and Longo, 2004; Gartn er et al., 1999; Green, 1980; Haas an d Thomas, 1968;

Haas, Shih, and M arrian, 1 983; Haas, Thomas, M arrian, and Shi h, 1989;

282 KEVIN L. JENSEN

Haas, Thom as, Shih, an d M arrian, 1989; Jensen et al., 2003b; Je nsen, Lau,

and Lev ush, 2000; Jones and Grant, 1983; Longo, 1978, 1980, 2000, 2003;

Longo, Adler, and Falce, 1984; Longo, Tighe, and Harr ison, 2002; Marri an,

Haas, and Shih, 1983; Ma rrian and Shi h, 1989 ; Marri an, Shih, and Haas,

1983; Schmidt an d Gomer , 1965; Shih, Yater, and Hor, 2005; Thom as, 1985;

Vancil and Wintuc ky, 2006 , for a represen tative cross sectio n). A mo del by

Longo, Adle r, and Fal ce (1984) provides a con cise accou nt of the work

functio n varia tion.

Dispenser ca thodes are de veloped by pressing small grains of tungsten

togeth er under he ating (si ntering). The joined grains are porou s; the spac es

between are filled with mate rial that, when heated, liberates barium, which

then migr ates to the surfa ce. In the ope ration of a catho de, barium diffuses to

the surfa ce an d exud es from pores that are randoml y sp aced but general ly

such that the por e ‐to ‐ pore separat ion is on the order of the grain size:6–10 mm. Early in the life of the cathode, the barium arrival rate at the

surface can exceed what is requir ed for mon olayer co verage. At most

a mon olayer of barium a toms builds on the surfa ce as bulk evap oration

rates are ord ers of magn itude faster than the monolay er evaporat ion rates ,

a con sequence of the different bond stre ngth between barium and itself

compared to barium an d tun gsten ( Forman, 1984 ). Never thele ss, given the

pores, specula tion that smal l islands of barium form ed aroun d them was a

hypothesi s worth invest igating and so Longo, Adle r, and Falc e (1984) set out

to asses s what the average work function might be and the possibl e co n-

sequence( s) of isl and formati on on the operati onal lifeti me of dispen ser

cathodes . While more phen omenologi cal than the Gyftop oulos–Levi ne

theory, it cap tures some features rather easily.

It is assumed that the work function of a surface is a weighted average

between the work function of the bulk material (in this example, tungsten—W)

and the work function of the coating (designated by a C; the coatings can

change from barium to barium oxide to cesium to whatever, and so a generic

designation is used). Measurements of the work function of partially coated

surfaces exhibit a minimum, sometimes at values under a monolayer, as

shown in Figure 100 for data adapted from fig ur e 2 2 of Schmidt and Gomer

(1965) for the metals cesium, potassium, barium, and strontium. Assuming for

the moment that the crystal plane on which the coating rests is uniform (it need

not be; unless a single crystal is used, coverage and work function will be

affected by crystal face; the photographs in Schmidt and Gomer provide a

rich catalog of images of differing coverage on different planes of a needle),

then a fictitious ‘‘picture’’ of such a surface near a pore might well resemble

Figure 101, which suggests regions about which there may be no coating

(‘‘bare’’), a monolayer coating (‘‘monolayer’’), or many layers (‘‘multiple

layers’’) for which the work function of that region looks like the bulk work


function of C. This suggests that the ‘‘macroscopic’’ work function, as might

be obtained from a Richardson plot, is a sum of differing terms of the form

hFðyÞi ¼ fwAwðyÞ þ fcAcðyÞ; ð702Þwhere the urge to interpret the A factors as areas is strong—but should be

resisted as they are instead weights of a distribution. They should, however,

have some relation to the actual areas of coverage, and Longo et al. (1984)

suggests the appellation weighted areas. If they act like areas, then small

2

3

4

0 2 4 6

Cs

K

Ba

StW

ork

func

tion

[eV

]

n [1014 atoms/cm2]

FIGURE 100. Variation of work function with surface coverage for various coverings

(surface density), based on figure 22 of Schmidt and Gomer (1965).

Bare

Monolayer

Multiple layer

FIGURE 101. Representation of how the coverage near a pore on a dispenser cathode surface

may appear in the Longo model.

284 KEVIN L. JENSEN

changes in the coverage will cause small changes in the average work function

andwill depend on the amount of each area so covered. Thus, onemight expect

@yAcðyÞ ¼ aAcðyÞ@yAwðyÞ ¼ b Awð0Þ AwðyÞf g ð703Þ

where a and b are rate constants,and y is the fractional monolayer coverage:

y > 1 means more than a monolayer present, and Awð0Þ is a bare surface.

Solutions to Eq. (703), when inserted into Eq. (702) and normalized to unit

area, suggest that the average work function is then

hFðyÞi ¼ eayfw þ ð1 ebyÞfc ; ð704Þan equation that properly expresses an intuitive feeling: when the coverage is

low, then changes in the average work function appear to be linear in y but,

depending on the values of the rate constants a and b, thenEq. (704) can exhibita minimum at submonolayer coverage. Define ym to be the value of y that

minimizes hFðyÞi, that is,

limy!ym

@yhFðyÞi ¼ 0 ) ab

fw

fc

¼ exp ðb aÞymf g; ð705Þ

where FðymÞ ¼ Fmin can be less than the bulk work function of the covering

material. For example, barium on tungsten has a minimum work function

of 2.0 eV, whereas the work function of bulk barium is 2.55 eV. For low

coverage, the work function variation with coverage is almost linear, and so

limy!0

@yhFðyÞi Sf ¼ afw bfc: ð706Þ

From experimental data the variation of work function (for example, as

shown in Figu re 102 for ba rium) on the assum ption that 4 :3 10 14 # =cm 2

atoms co nstitute s a monolay er (as suggest ed by the scali ng of Schmidt ,

1967 ) and letting a=b 2, then the Longo approxim ation of Eq. (704) com-

pares with the data of Schmidt in Figure 102; the Longo approximation

provides quite reasonable agreement for a simple model using generic

parameters.

Longo’s concern was obtaining a model of the degradation rate of the

barium dispenser cathode, and so a simple model that captured the essential

features of work function variation as a function of coverage was useful.

However, it does not illuminate why the work function is reduced in the first

place. For that, models that address how the covering atoms create dipoles,

and how those dipoles interact, are required. An oft‐used model is the

Toppin g form ula ( Topping, 1927 ; Sc hmidt and Gomer , 1965 ). How ever, as

it shares elements with the Gyftopoulos–Levine model, which has a good


correspondence with data (even if the interpretation is a bit ambiguous;

see the discus sion in Haas and Thom as, 1968 ), its discus sion is left to the

literature.

D. The (Modified) Gyftopoulos–Levine Model of Work

Function Reduction

The Gyftopoulos–Levine (GL) theory is a hard‐sphere model of the coverage

atoms atop the bulk metal atoms, and it accounts for dipole and dipole‐dipole proximity effects on the magnitude of F. It performs quite well, if

one is not too persnickety in insisting that what the parameters purport

to describe are physically realizable for hard spheres or whether the work

function reduction is due to two sources (electronegativity differences and

dipole effects) or just dipole effects. Leaving such questions of interpretation

aside, the GL theory gives rather breathtaking agreement with experimental

data. The account here is directed toward comparing theory to recent experi-

mental data from photoemission studies—and to make use of small changes

in atomic parameters on which the GL theory relies that have occurred in the

decades since the theory made its debut.

The GL theory postulates that the work function variation with coverage

owes its existence to differences in electronegativity (W) and a dipole effect (d ).

Electronegativity is the tendency of an atom to attract electrons to itself.

Pairs of atoms have differing ability to do this, so one component of the work

function represents the differing abilities of atoms to attract and retain

2

3

4

0 0.5 1.0 1.5

SchmidtApprox

Wor

k fu

nction

[eV

]

Coverage q

qm = 1.15

b /a = 1/2

a = 2.1542

F IGURE 102. The data of Schmidt (L. D. Schmidt, 1967) compared to the model of Longo

et al. (1984) using generic values.

286 KEVIN L. JENSEN

electrons, and the other to a dipole effect resulting from a charge redistribution.

This can be written as

FðyÞ ¼ WðyÞ þ dðyÞ: ð707ÞMullikan suggested that the atomic electronegativity be taken as the mean

between the ioniza tion potenti al and the elect ron affinit y of an atom ( Gray,

1964 ). The effective work functi on Fe of a surface was related to the electro-

negativ ity X by Go rdy and Thom as (1956) , who synthes ized a rough form ula

relating the two given by

Fe eV½ ¼ 2:27X PU½ þ 0:341 eV; ð708Þwhere the work function is measured in electron volts and the electronega-

tivity X in Pauling units. The quality of this approximation is shown in

Figure 103 using values of electronegativities and work functions from the

CRC table s (W east, 1988 ); the app roximati on retai ns its ap peal and so is

adopted for historical continuity. Pauling units are such that the electroneg-

ativity of fluorine (100.45 kJ/mole ¼ 10.411 eV) is 3.98 Pauling units (PU).

Therefore, a Pauling unit is 2.616 eV. The odd factor of 0.341 eV is

attributed by Gyftopoulos and Levine as the energy to overcome image

charge forces and therefore is the same for all metals.

With the relationship between electronegativity and work function estab-

lished, W(y) is taken to be the simplest polynomial that will give rise to the

correct boundary conditions. These boundary conditions are as follows. For

no coverage, the work function of the bulk material should arise, and the

addition of a few coverage atoms should not change that. Thus,

Wð0Þ ¼ fw

limy!0

dWdy

¼ 0 ð709Þ

An analogous relationship holds for the monolayer coverage case:

Wð1Þ ¼ fc

limy!1

dWdy

¼ 0 ð710Þ

Unlike the Longo case, here fc refers to the work function of the monolayer,

not the bulk material. The simplest polynomial that satisfies two boundary

conditions and two derivatives at the boundaries is a cubic. It is easily shown


WðyÞ ¼ fc þ ðfw fcÞð1þ 2yÞð1 yÞ2 fc þ ðfw fcÞHðyÞ ð711Þ

and , a s per Eqs. (377) and (378) , H (0) ¼ 1 and H (1) ¼ H0 (0) ¼ H 0 (1) ¼ 0,

where prime indicates differentiation with respect to argument.

The dipole term d(y) is more difficult. Returning to Pauling, the dipole

moment between two atoms, A and B, is proportional to the difference in

their electronegativities. The assumption of the GL theory is that the same

2

3

4

5

6

0.5 1.0 1.5 2.0 2.5

Work function

Gordy and Thomasf = 2.27 (x + 0.15)

Wor

k fu

nction

[eV

]

Electronegativity [ pauling units]

(a)

Data from CRC handbook ofchemistry and physics

2

3

4

5

6 Work functionGordy and Thomas

Wor

k fu

nction

[eV

]

Atomic #

(b)

200 40 60 80

FIGURE 103. (a) Work function variation with electronegativity compared to the linear fit of

Gordy and Thomas (1956). (b) Comparison of work function trend compared to the Gordy and

Thomas model.

288 KEVIN L. JENSEN

holds true for a site composed of four substrate atoms, represented by hard

spheres, in a square array with an absorbed atom at the apex of the pyramid

(Figure 104). By Eq. (708), a difference in electronegativities is tantamount

to a difference in work function values apart from a constant coefficient.

The distance from the center of atom ‘‘c’’ to atom ‘‘w’’ is designated R. For

the four dipoles that result, only the components parallel to the vertical axis

survive; the others have equal and opposite contributions. Let Mwc be the

dipole moment between a c‐atom and a w‐atom. The dipole for the group of

four is then Mo 4MwccosðbÞ, where b is the angle the line joining the atom

centers makes with the vertical. GL theory suggests that Mwc is given by

kðfw fcÞ=2:27, where k ¼ 43.256 eo is a composite of factors deduced from

the relationship between electronegativities and molecular dipole moments.

The dipole term is then

MðyÞ ¼ MoHðyÞMo ¼ 4eor 2o cosðbÞðfw fcÞ ð712Þ

where b and R ¼ rw þ rc are as illustrated in Figure 104. A constant radius

parameter ro ¼ (k/2.27eo)1/2 ¼ 4.3653 A has been introduced, and the factor

of 2.27 is the previously encountered factor relating electronegativity and

Top Side

C

W

Rb

Perspective

W

C

FIGURE 104. Schematic of coverage atom (e.g., cesium) atop a layer of bulk (e.g., tungsten)

atoms in the Gyftopoulos‐Levine model.


work function. The cos(b) term is deduced from geometrical arguments

regarding the pyramid in Figure 104 to be

sin2ðbÞ ¼ 2

w

rW

R

2ð713Þ

where w/(2rw)2 is the number of substrate atoms per unit area, and w the

number of atoms per unit cell, where the cell size is dictated by the hard‐sphere radius. This notation slightly departs from the path chosen by GL in

terms of symbols and their meaning, but the arguments, being the same,

produce the same final conclusions.

Adjacent dipoles introduce a depolarizing effect such that the effective

dipole moment Me(y) is the difference between the dipole moment M(y) andthe depolarizing field E(y), the latter of which is proportional toMe(y) as per

EðyÞ ¼ 9

4pe0

f

ð2rCÞ2y

!3=2

MeðyÞ ð714Þ

where, analogous to w, the dimensionless factor f is the number of adsorbate

atoms per unit cell at monolayer coverage. The effective dipole moment is

found by solving

MeðyÞ ¼ MðyÞ aEðyÞa ¼ 4peonr3c

ð715Þ

where a is the polarizability and the form is as given by GL. The term rcis taken as the covalent radius of the adsorbate. The factor n is slightly more

tricky; it accounts for the electronic shell structure of the atom on the

polarizability. Alkali metals (column 1 on the periodic table) have but one

electron in the outermost shell, so n ¼ 1. Alkaline earth metals (column

2 on the periodic table) have two valence electrons, and these electrons

tend to shield each other from the nucleus; thus, to account for that shielding,

n ¼ 1.65 for alkaline earth elements. The dipole term d(y) is then the product

of the effective dipole moment, the surface density of coating atoms, and the

coverage factor, or

dðyÞ ¼ MeðyÞ f

ð2rcÞ2

0@

1A y

eo

¼ MðyÞ

1þ 9a4pe0

f

ð2rCÞ2y

0@

1A

3=2

f

eoð2rCÞ2y

0@

1A ð716Þ

290 KEVIN L. JENSEN

Combining all the factors gives the work function in terms of the coverage

factor y

FðyÞ ¼ fC ðfC fW Þð1 yÞ2ð1þ 2yÞ 1 GðyÞf g

GðyÞ ¼

ro

rC

0@

1A

2

1 2

w

rW

R

0@

1A

20@

1A

1þ nrC

R

0@

1A

30@

1A 1þ 9n

8ð f yÞ3=2

0@

1A

f yð717Þ

To reiterate, the values of rw and rc are the covalent radii of the substrate and

adsorbate atoms, respectively; R is the sum of them; ro is a constant radius

parameter, and n depends on whether the covering is alkali or alkali earth.

In Eq. (385) , two parame ters f and w remain to be determined by empir ical

data and the specifics of the system under consideration. They are not inde-

pendent, as the coverage atoms reside on a surface dictated by the substrate

atoms. The ratio of the substrate and adsorbate values for the number of

atoms per unit cell depends on crystal face and whether the adsorbate is alkali

metal or alkaline–earth metal. The nature of the surface is further dictated by

which crystal plane is exposed, for example, the [100] in a body‐centered cubic(bbc) crystal. Knowledge of one crystal plane can be related to the others and

therefore relates the values of f and w. Let No represent the crystal face.

GL argue

½100 ) No ¼ 1

½110 ) No ¼ 2

½B ) No ¼ 3

ð718Þ

The first two cases appear straightforward enough, but tossing in a B

demands explanation; after a certain point, the crystal face simply looks

Bumpy. On a sintered tungsten surface, the best representation is to use the

B value—but there are cases where crystalline surfaces are considered, and

then greater care is demanded. GL then argue that the quantities ff = ffiffiffiffiffiffiNo

p gand fw= ffiffiffiffiffiffi

No

p g are approximately constant fromone face to another. Values for

a variety of coverings and substrates are given in Table 14, which updates an

equivalent table in Gyftopoulos and Levine (1962). The constraint between

f and w is given by the ratio of surface densities and takes the form

w

f

rC

rW

2

¼ 4 for Cs on W;Mo;Ta . . .2 for Ba on Sr;Th;W . . .

ð719Þ


Eq. (719) suggest s that the ratio is an integ er, but in fact , it need not be.

How ever, it shall be treated as such and the unc ertainty in the actual ratio is

absorbed by defi ning ‘‘ef fective’’ values of w (or f ) such that Eq. (719)

is correct .

The GL theory is not without complications. Since the time of the GL article,

the value of the covalent radii of various metals has changed slightly; values

are taken from Winter (see WebElements; http://www.webelements.com/

webelements/). Also, there is some ambiguity surrounding how f (and hence

w) is defined. Values in the literature for the surface number density of cesium

on tungsten or barium on tungsten from various sources (e.g., Gyftopoulos and

Levine, 1962; Haas and Thomas, 1968; Haas, Thomas, Shih, and Marrian,

1983; Schmidt, 1967; Taylor and Langmuir, 1933; Wang, 1977) tends to evolve

over time. What, then, should be made of f—and more important, how should

it be evaluated? The answer to such a question is intimately related to the

question of how to compare different experimental data sets.

E. Compari son of the Modified Gyftopo ulos– Levine Mo del to

Thermi onic Data

In rep orting the variation of work function with co verage, the latter is often

express ed in terms of fractions of a monolay er. However, this is not how

coverage is measur ed ; rather , expe rimental da ta infer ‘‘cov erage’ ’ eithe r by

assum ing a linear relation between coverage and deposition time ( Wang,

1977 ), dep osited mass measur ed using a quartz crystal ba lance ( Moo dy et al.,

2007 ), or other means. Therefor e, experi menta l error, inco rrect scali ng fac-

tors, or both can a lter the estimat e of y that is quoted —and if ‘‘coverage’ ’ is

the only parame ter shown without referen ce to the scaling facto r used , there

TABLE 14

COVERAGE F ACTOR PARAMETERS *

Cover rc [nm] n Substrate rw [nm] f /ffiffiffiffiffiffiNo

pw /

ffiffiffiffiffiffiNo

pRatio

Cs 0.230 1 W 0.146 0.5060 0.8530 4

Cs 0.225 1 Mo 0.145 0.5161 0.8574 4

Cs 0.225 1 Ta 0.138 0.4666 0.7012 4

Sr 0.192 1.65 W 0.146 0.7377 0.8530 2

Ba 0.198 1.65 W 0.146 0.7840 0.8530 2

Th 0.165 1.65 W 0.146 0.5440 0.8530 2

*Coverage factor parameters (after Jensen, Feldman, Moody, and O’Shea, 2006a). Values of

f are constructed to replicate the values of the surface densities for the adsorbate and substrate

metals and other values tabulated by Gyftopoulos and Levine (1962). ‘‘Ratio’’ refers to Eq. (719).

Radii are in nanometers.

292 KEVIN L. JENSEN

is no apparent ‘‘good way’’ to compare differing measurements. That this

occurs can be ascertained from comparing differing data sets from the

literature for barium on tungsten; for example, see Figure 105, where a

compilation of several experimental measurements is compared directly to

GL theory (Longo 1 and 2 refer to Longo, Adler, and Falce, 1984 ; Haas

refers to Haas, Shih , and Ma rrian, 1983 ; Sch midt refer s to Sch midt, 1967 ).

A similar plot can be made of, for example, cesium on tungsten. If a measure

of science is reproducibility via independent measurements, then this is not

reassuring.

In a procedure perhaps more art than science, theoretical predictions and

experimental data can be brought into line satisfactorily. A few parameters

remain for which there is some ambiguity; these are the work function of the

monolayer, the exact value of f, and the scale factor that must multiply a

slightly off coverage estimate or which is the coefficient of the experimentally

measured term (such as deposition time, etc.). The first two are tightly

constrained, if they are taken to vary at all. What parameters remain to

vary can then be pinned down by demanding the minimization of the least‐squares error between GL theory and the experimental relations. When

done, disparate findings coalesce satisfactorily along the GL relation for

scale factors that are reasonable, as in Figure 106 for the case of barium on

tungsten for a surface assumed to be bumpy (B) in deference to the polycrys-

talline form presumed to exist at the surface. The impact of crystal face is

shown in Figure 107, which changes the value of f (that is, changing the value

2

3

4

5

0 0.2 0.4 0.6 0.8 1.0 1.2

Longo-1Longo-2HaasSchmidtGyfto Lev

Wor

k fu

nction

[eV

]

Coverage q

FIGURE 105. Comparison of barium on tungsten as reported by several sources available in

the literature using their estimates of the relationship between the experimental parameter and

coverage, as compared to Gyftopoulos‐Levine theory.


of f so as to keep f =ffiffiffiffiffiffiNo

pconstant) but otherwise uses the same parameters

as in Figure 106. A similar result obtains for cesium on tungsten and the

analys is of the data of Wang (1977) and Taylor and Lang muir (1933) , as

shown in Figure 108 (the Wang data used correspond to the ‘‘no‐oxygen’’data, as oxygen tends to result in an even lower work function), again on the

presumption of a bumpy surface. As with Figure 107, Figure 109 examines

2.5

3.5

4.5

0 0.2 0.4 0.6 0.8 1.0

B110100

Wor

k fu

nction

[eV

]

Coverage q

Ba on W

FIGURE 107. Effects of changing the f value by considering the different crystal faces for

barium on tungsten.

2

3

4

5

0 0.2 0.4 0.6 0.8 1.0

Longo-1Longo-2HaasSchmidtGyfto Lev

Wor

k fu

nction

[eV

]

Coverage q

FIGURE 106. A re‐analysis of the coverage factor of Figure 105 using a least‐squares analysisfor the determination of the surface density parameter f.

294 KEVIN L. JENSEN

the changes wroug ht by diff ering crystal face, but for cesium on tungst en.

Overall , the experi mental data are brough t into remar kably consis tent agree-

ment for the values used in the Table 12. Note that the same value of f is used

for Longo, Schmidt , and Haas (bari um on tungsten ), as wel l a s for Taylor and

Wang (cesium on tungst en). It is instru ctive to co mpare the surfa ce numb er

densities to values found in the literat ure. Gyftop oulos and Lev ine (1962)

give the surfa ce densit y of a mono layer of cesium on tungst en [purpo rted to

1.5

2.5

3.5

4.5

0 0.2 0.4 0.6 0.8 1.0

B110100

Wor

k fu

nction

[eV

]

Coverage q

Cs on W

FIGURE 109. Same as Figure 107 but for the parameters of cesium on tungsten.

1.5

2.0

2.5

3.0

3.5

4.0

4.5

0 0.2 0.4 0.6 0.8 1.0

WangTaylorGyfto Lev

Wor

k fu

nction

[eV

]

Coverage [q ]

FIGURE 108. Same as Figure 106 but for the least‐squares analysis for determining f applied

to cesium on tungsten, compared to the data of Wang (1977) and Taylor and Langmuir (1933).


be due to Tay lor and Lang muir (1933) ] to be sCs ¼ 4: 8 10 14 # =cm2 ,

wher eas Wan g (1977) gives sCs ¼ 5 10 14 # =cm 2 . The v alues in Table 12

suggest sCs ¼ 4: 404 10 14 # =cm2 , whi ch is in reasona ble agreem ent.

Conver sely, Gyftopou los and Lev ine give sBa ¼ 8: 65 10 14 # =cm 2 for a

bump y surfa ce [ Schmidt (1967) suggest s sBa ¼ 6 10 14 # =cm 2 for the square

close ‐ packing density, a numb er between the [100] and [110] plan e

surfa ce de nsities], wher eas the table suggest s s ¼ 8: 78 1014 #=cm2 : again,

satisfa ctory agreem ent.

F. Com parison of the Modified Gyftopo ulos–Le vine Model to

Photoem ission Dat a

Of the varie ty of photocat hode s that exist, the focus here is on a pa rticular

cand idate intende d for FELs. FELs are argua bly one of the most deman ding

of the applic ations in term s of phot ocathode pe rformance an d charact eris-

tics, as well as the hostilit y of the operatio nal environm ent ( Col son, 2001;

Neil an d Mermin ga, 2002 ; O’She a and Fr eund, 2001 ). W hile meta l photo-

cathodes are a pprecia ted for their rugged behavior , their relative ly low QE

has alway s rank led. Kno wledge of how coatin gs lowered the work functio n

of meta ls in the dispens er cathode, and more impor tantly, how the dispens er

cathode ‘‘hea led’’ itself , spoke to a knowl edge base that eventual ly foun d its

way into speculation about how to substantially improve the QE of photo-

cathodes. The advantage was appreciated early in the history of the FEL

program ( Lee and Oet tinger, 1985 ), exami ned in the co ntext of photoinj ec-

tors (Tr avier e t al., 1995, 19 97 ), and then various off‐ the ‐ shelf dispens ercathodes syst ematical ly investiga ted (Fel dman et al., 2003; Jen sen,

Feldman , Virgo, and O’She a, 2003a ,b; Jensen, Feldman, and O’She a, 2004 )

for their utility in high‐power devices as a prelude to the development of a

con trolled porosity dispens er photocat hode (Jen sen, Feldman, Moo dy, an d

O’She a, 2006a ; Moo dy et al., 2007 ). The low work function was , as antici-

pated, a boon, but in an effort to characterize and baseline the impact of

cesiation on metal surfaces, a systematic study was performed by Moody

et al. ( Mood y, 2006; Moo dy et al., 2007 ) to ch aracterize the QE as a functio n

of surface coverage of cesium on tungsten and other metals. Those experi-

ments became a useful testing ground for the photoemission models that

have been discussed in previous sections. Here, rather than survey all such

investigations, the more relevant portion focusing on the QE of cesiated

surfaces as a function of wavelength will suffice.

Not all cesiated surfaces for photocathodes rely on a dispenser cathode

architecture—quite the contrary: for example, the present record‐holder inthe pursuit of a high‐power FEL presently resides at the Thomas Jefferson

296 KEVIN L. JENSEN

National Accelerator Facility (more commonly referred to in the FEL com-

munity as JLab) in Newport News, Virginia. Its cathode is a cesiated gallium

arsenid e (Cs ‐ GaA s) crystal ( Gubeli et al., 2001 ). GaAs phot ocathodes sport

high QEs of better than 10% ( Neil et al., 20 06; Sincla ir, 2006 ) in addition to

being a unique source of polarized electrons. The type of injector gun that

uses it applies constant fields (a DC gun in the parlance) and performs

bunching of the electron beam elsewhere. It has been argued, however, that

if beam brightn ess is c rucial, then RF gun s ( Lewel len and Br au, 2003;

O’She a, 199 5) are the injec tor of choice, in whi ch very high elect ric fie ld

gradients (on the order of 50 to 150 MV/m) rapidly accelerate short charge

bunch es from the photocat hode ( Todd, 2006 ). The choice of injector depend s

on the particular application and materials, and so great variety exists

worldw ide (see Colson, 2001 , for a summary).

One problem, however, is that photocathodes that can be used in an RF

gun environment, which tends to not be as pristine as for a DC gun, are not

of comparable QE. As seen from the discussion of scattering and transport to

the surface, higher‐QE cathodes such as GaAs tend to have longer response

times of tens of picoseconds (compared to metal photocathodes, which are

essential ly insta ntaneous; Spicer and Herr era ‐ Gomez, 1993 ), and in the

generation of short bunches at the cathode, this detail can be problematic.

To understand such an issue, let the incident laser pulse be Fourier trans-

formed into a representation given by

IlðtÞ ¼ IoyðtÞyðT tÞXN

n¼0cn cosðontÞ, ð720Þ

where l refers to the laser wavelength, but on refers to the Fourier fre-

quencies. If there is an emission delay time characterized by t, which has a

connection to the scattering relaxation time and the depth to which the laser

penetrates, then the emission current Ie (t ) can be obtaine d from ( Lewel len,

2007 )

IeðtÞ ¼ QE

t

Z t

1IlðsÞ exp t s

t

h i: ð721Þ

It is a straightforward problem in integration to show that

IeðtÞ /XNn¼0

cn

1þ ðontÞ2cos ontð Þ þ ont sin ontð Þð ÞeT=t 1½ et=t t < T

ðeT=t 1Þet=t t T

ð722ÞThe important conclusion is that all the components have tacked upon

them a term of the form expðt=tÞ, indicating that the electron beam will

continue to ebb out of the photocathode with an exponential tail. If the

bunch lengths of the electron pulse are desired to be on the order of 10 ps


(a peculiar nomenclature: the ‘‘length’’ T of a pulse is measured by its

duration, usually the FWHM duration), then delay times on the order of

1 ps may be somewhat beneficial, but longer delay times destroy the pulse

shape. To observe the benefit of a mild delay time, consider Figure 110;

assume that the photocathode surface is at the left y‐axis. The top‐hat profilenext to it is a laser pulse traveling to the left. After a time, the electron pulse

(the right pulse) is moving to the right, away from the photocathode.

In Figure 110a (for instantaneous emission), the electron pulse is simply the

mirror image profile of the incident laser pulse. The structure on top of the

pulse represents fluctuations in the laser (noise), which are a consequence of

its generation by frequency‐doubling crystals and become more pronounced

the shorter the wavelengt h of the light is made (Jen sen, Feldman, Virgo, an d

O’She a, 2003b ), an d such fluc tuations are unde sirab le. Eve n a modest delay

time inserted into Eq. (388) greatly curbs such fluctuations, as in Figure 110b

for t¼ 0.4 ps. Circumstances quickly degrade after that, though, and the 3.2‐ps delay time has already morphed the electron bunch into something quite

different than the top‐hat‐like laser pulse (shown in Figure 110c). The 12.8‐pspulse of Figure 110d bears little resemblance to what was desirable in the

original top‐hat distribution.Meeting the needs of pulse shaping and QE is difficult to achieve simulta-

neously given that metals favor the former but semiconductors the latter.

A self–re‐cesiating surface based on the dispenser cathode model that is

projected to have greater ruggedness in an RF environment but far higher

QE than a metal photocathode has been the candidate investigated at

the University of Marylan d ( Moo dy, 2006; Moody et al. , 2007 )—hence the

interest in knowing how cesium evolves on the surface of metals. The effects

on QE have been studied in a companion program at the Naval Research

Labor atory ( Jensen et al., 2006a ). As QE can be theo retically inferred from

the work function, and the work function inferred fromGL theory, consider-

ing QE as a function of coverage naturally follows. Using the QE moments‐based model and the GL work function model, such measurements can be

compared to a theoretical prediction.

A certain amount of latitude is offered for explorations under laboratory

conditions, where the laser intensities are low and fields not appreciable.

Consequently, complications that would otherwise be due to laser heating of

the surface do not manifest themselves and thus temperature excursions can

be neglected (though the full‐fledged temperature code is used below), as

can temperature‐induced thermal desorption and migration of low–work

functi on coati ngs (Go mer, 1990; Husm ann, 1965; Swans on, Straye r, and

Char bonnier, 196 4; Taylor and Langmuir , 1933 ). W hat cann ot be neglect ed

is the quality of the surface, which can contaminate rather easily and is

difficul t to clean ( Dowe ll et al., 2 006; Moody et al., 2007; Sommer , 1983 ).

298 KEVIN L. JENSEN

0

0.2

0.4

0.6

0.8

1.0

1.2

−5 0 5 10 15 20 25 30 35

Las

er/E

lect

rons

am

plitud

e

Time [ps]

(a)

0.0

(b)

−5 0 5 10 15 20 25 30 35

0.2

0.4

0.6

0.8

1.0

1.2

Time [ps]

0.4 ps

Las

er/E

lect

rons

am

plitud

e

0.0

0.2

0.4

0.6

0.8

1.0

1.2

−5 0 5 10 15 20 25 30 35

Las

er/E

lect

rons

am

plitud

e

Time [ps]

3.2 ps

(c)



As noted in the discussion of the thermionic cathodes, diffusion through

and from pores raises questions of variable coverage. Consequently, for the

comparisons here, evaporating cesium on the surface of an argon‐cleanedpolycrystalline tungsten sample is used to avoid possible pooling of the

cesium. Values for field, temperature, laser intensity, and other parameters

for which the comparisons are made are given in Table 15.

Some discussion of the parameters is needed. Although the time‐dependent QE code is used, experimentally the pulse duration is long (on

the order of tenths of seconds), which is outside the allowable bounds of the

simulation model. However, the laser intensity is also very low, and since

TABLE 15

PARAMETERS FOR QUANTUM EFFICIENTY FROM A CESIATED

TUNGSTEN SURFACE

Parameter Value Unit

Field 1.7 MV/m

Temperature 300 KLaser intensity 0.1 MW/cm2

f 1.4 —

Cesium atomic radius 0.5309 nm

12.8 ps

0.0

0.2

0.4

0.6

0.8

1.0

1.2

−5 0 5 10 15 20 25 30 35

Las

er/E

lect

rons

am

plitud

e

Time [ps]

(d)

FIGURE 110. (a) Laser pulse (left structure) traveling to surface (left boundary) and the

resulting electron pulse profile (right structure) traveling to the right if the delay time is 0 ps. (b)

Same as (a) but for a delay time of 0.4 ps. (c) Same as (a) but for a delay time of 3.2 ps: note how

even this short time degrades the top‐hat–like structure. (d) Same as (a) but for a delay time of

12.8 ps: the electron pulse bears little resemblance to the incident laser pulse.

300 KEVIN L. JENSEN

no temperature excursion occurs, the QE of a 1‐s laser pulse is close (if not

identical) to the QE of a 10‐ns pulse, and so no distinction is drawn between

the two. With regard to coverage, the amount of cesium deposited is

measured with a quartz crystal balance which, given the density of cesium,

returns a thickness l rather than a measurement of coverage directly via

l ¼ r=mA, where r is density and m and A are the mass of the deposition

and the area, respectively. Given surface structure, the width of a monolayer

has some ambiguity: the cesium‐to‐cesium separation distance (distance

between nuclei) in bulk is 5.309 A, the empirical atomic diameter of a cesium

atom is 5.2 A, and the cube root of the atomic volume of cesium is 4.9 A. We

have chosen the scale factor to be the inverse cesium‐to‐cesium separation

distance, or 5.309 A. Finally, as is evident from Figure 87, the tungsten on

which the cesium is deposited is not pretty nor is it particularly flat. Argon

cleaning of the surface roughens it. Therefore, the value of f in GL theory is

not a priori unambiguous. The experimental data for QE as a function of

coverage tend to show a stronger hump than GL theory with f ¼ 1. It is

plausible that a rougher surface typical of sintered materials would therefore

have a larger f va lue (as discussed by Haas and Thom as, 1968 ), an d so the

value f ¼ 1 .4 was chosen. All oth er pa rameters not exp licitly specified are

taken from common values available in the literatu re ( Mood y, 2006 , contain s

a complete description of the experimental arrangement and description of

the surface and its treatment). The comparison between the modified GL þtime‐dependent simulation theory is shown alongside experimental data

from cesium deposited onto tungsten, and the results are shown in

Figure 111. Notably, even though every part of the theory is based on

countless subordinate models of underlying processes and conditions, as a

whole, the theory performs quite well in accounting for the qualitative and

quantitative behavior of the experimental data with only one parameter ( f )

subject to some uncertainty. Similar agreement is found in comparisons of

cesium on sil ver (Jen sen et al., 2006; Mo ody et al., 20 06 ).

The simulation code used to generate Figure 111 can be put to other uses,

as it provides a full account of the effects of temperature rise on scattering,

thermal‐field emission, and photoemission in a time‐dependent framework

utilizing the numerical temperature‐rise algorithms (in other words, it embo-

dies everything that has come before). Investigating the impact of higher

intensity can therefore be done theoretically. A higher temperature is corre-

lated with a reduction in the scattering time, and therefore should correlate

with a reduction in QE; alternately, an increase in applied field is correlated

with a reduction in the Schottky barrier, and therefore correlated with an

increase in QE. These hypotheses are tested in Figure 112 where fields of 100

MV/m and an intensity of 1 GW/cm2 are considered. The impact of a high

laser intensity, even for a short 10‐ps FWHM pulse, is such that the temper-

ature rise can climb to 895 K from a starting temperature of 300 K;


375

405

532

655

8080.00

0.04

0.08

0.12

Qua

ntum

eff

icie

ncy

[%]

Coverage %0

(a)

20 40 60 80 100

10−3

10−2

10−1

Qua

ntum

eff

icie

ncy

[%]

Coverage %

375

405

532

655

808

0

(b)

20 40 60 80 100

10−3

10−2

10−1

1.5 2.0 2.5 3.0 3.5

Theory max

Set A max

Set B max

Set C max

Max

imum

QE

[%

]

(c)

Photon energy [eV]

375405

532

655

808

FIGURE 111. (a) Comparison of theoretical quantum efficiency model (lines) with experi-

mental data for cesium on tungsten at five different laser wavelengths (in nanometers). (b) Same

as (a) but on log scale. (c) The maxima of the data in (a) as a function of photon energy.

302 KEVIN L. JENSEN

the resulting degradation of QE from heightened scattering more than

offsets the increase in QE from Schottky barrier lowering.

Finally, it is only fitting to consider as a last example the case of cesium on

copper, where the expectation is that the effects will be similarly pronounced

given the good conduction characteristics of copper. In Figure 113, for

standard parameters from the literature and f ¼ 1, the case of 1 GW/cm2 is

compared to 1 MW/cm2 for a 10‐ps FWHM laser pulse at 355 nm, a field of

0.0

0.10

0.20

0.30

0.40

0.50

0 20 40 60 80 100

1 GW/cm^21 MW/cm^2Q

uant

um e

ffic

ienc

y [%

]

Coverage [%]

Cs on Cu

FIGURE 113. Cesium on copper for differing laser intensities, in which a temperature rise in

the copper is induced at the higher intensity.

0.00

0.05

0.10

0.15

0

1.7 MV/m; 0.1 MW/cm^2

100 MV/m; 0.1 MW/cm^2

100 MV/m; 1.0 GW/cm^2

Qua

ntum

eff

icie

ncy

[%]

Coverage [%]0 20 40 60 80 100

FIGURE 112. The 375‐nm line of Figure 111 for different fields and laser intensities. The

former contributes to Schottky barrier lowering; the latter induces a temperature rise as the

intensity increases.


50 MV/m, and where the monolayer work function for cesium on copper is

taken to be 1.8 eV. Copper is a better conductor of heat, so a longer pulse

length of 0.1 ns (to elevate the temperature up to 1490 K) was chosen.

The maximum QE of the higher‐intensity 1 GW/cm2 case is but 68% of the

1 MW/cm2 case. These cases emphasize that in the simulation of systems

under extreme conditions, the various dependencies conspire in nontrivial

ways to render the outcome not an a priori certainty.

V. APPENDICES

A. Integrals Related to Fermi–Dirac and Bose–Einstein Statistics

Integrals that appear frequently in the evaluation of energy, specific heat, and

the distribution functions of fermions (s ¼ þ1) and bosons (s ¼ –1) are

ðx0

yn1

ey þ sdx ¼ 1 ð1þ sÞ

2n

0@

1AGðnÞzðnÞ

ð1x

yn1

ey þ sdx

¼ 1

n

xn

ex þ sþWsðn; xÞ

8<:

9=;

ðA1Þ

where G(n) is the gamma function, z(n) is the Riemann zeta function, and

Ws is ( Jensen, Feldman, Vi rgo, and O’She a, 2003b )

Wsðn; xÞ ¼ðx0

yn

ðey þ sÞð1þ seyÞ dy: ðA2Þ

Adequate approximations to W(n,x) are therefore sought. For large x,

use ( Grads hteæin, Ryzhi k, and Jeff rey, 1980 )

ð10

½lnðxÞn1þ 2xcosðtÞ þ x2

dx ¼ n!

sinðtÞX1

k¼1ð1Þnþk1 sinðktÞ

knþ1, ðA3Þ

from which it can be shown

Wsðn; xÞ ¼ 1 ð1þ sÞ2n

n!zðnÞ n!xn

X1j¼1

ðsÞjþ1ejx

Xn

k¼0

xk

jkðn kÞ! :

ðA4Þ

304 KEVIN L. JENSEN

Conversely, for small x,

Wsðn; x 1Þ 2xnþs

ð3sþ 5Þðnþ sÞ xnþsþ2

2ðsþ 7Þðnþ sþ 2Þ xnþsþ4

24ð3s 7Þðnþ sþ 4ÞðA5Þ

Finally, the special case s ¼ –1, n ¼ 5 is known as the Bloch–Gruneisen

function, for which it can be shown that useful limits, accurate to four

significant digits, are

Wð5; xÞ 6x2 72 ln 1þ 1

12x2

0@

1A ðx < 0:5Þ

120zð5Þ ðx5 þ 5x4 þ 20x3 þ 60x2 þ 120xþ 120Þex ðx > 8Þ

8>><>>:

ðA6ÞA reasonable estimate of W–(5,x) may be formed from the asymptotic

limits, a ‘‘hybr id’’ polynomi al (shown in Figure 69 ), via

Wð5; xÞ W> ð5; xÞ1 þW< ð5; xÞ1n o1

120zð5Þx580

3zð5Þx2ð18x2 þ 1Þ þ 1

,ðA7Þ

which has a maximum error of 21% occurring at x ¼ 0.135.

B. The Riemann Zeta Function

The function z(n) is defined according to

zðnÞ ¼ 1

ð1 21nÞGðnÞ

ð10

xn1

ex þ 1dx

¼ 1

GðnÞð10

xn1

ex 1dx

ðB1Þ

Alternately, a series definition is

zðnÞ ¼P1k¼1

1

kn

¼ 1

ð1 21nÞX1

k¼1

ð1Þkþ1

kn

ðB2Þ


Several special cases often encountered are for n ¼ 2, 3, and 4:

zð2Þ ¼ p2

6

zð3Þ ¼ 1:202057

zð4Þ ¼ p4

90

ðB3Þ

In particular, z(3) is on occasion referred to as Apery’s constant. For large

values of n, a convenient relation is

zðnþ 2Þ 1

4ðzðnÞ þ 3Þ: ðB4Þ

VI. CONCLUSION

Apart from either the simple pleasure brought about by understanding why

physical processes behave as they do or the slightly more complex thrill

associated with using that understanding to dragoon natural phenomena to

enable technological marvels, it is natural to enquire as to the utility of

models of electron emission. In the discussion of the various material and

operational parameters that affect emission, it has been clear that emission

characteristics (by which is meant current density, emission non‐uniformity,

emittance, and the other characterizations) are all affected by a host of

complications. A number of them, particularly field enhancement effects

due to surface structure, work function variation due to crystal faces and

monolayer coatings, temperature, and other complications, have been the

primary focus. Such conditions matter when thermal effects complicate field

emission, dark current intrudes on photoemission, and joint thermal‐fieldeffects are rife. The pristine and tightly constrained world of experimental

characterization is then in stark contrast to either the inherent complexity of

a surface or the complicated architecture of devices which exploit electron

beams. Whatever utility of simple models exists therefore seems at the outset

to be remarkably circumscribed. The justification for the ones considered

here couples well with musings authors traditionally offer in summaries of

their tomes and so such musings will be the final questions to consider: why

do simple models matter, to what purposes can their improvements be

directed and in what way are the models lacking or incomplete? The answers

necessarily point to research underway or under consideration.

306 KEVIN L. JENSEN

Why simple models matter is a consequence of the complexity of modern

electron beam devices: modeling and simulation are often the only clairvoy-

ant that can describe what is happening to and on account of the electron

beam as it propagates, particularly as dimensions shrink in pursuit of ever

higher frequency where imperfections are of greater consequence. Compres-

sion of the electron beam after it leaves the cathode region produces undesir-

able scalloping and halo in rf devices. High brightness sources can disrupt the

electron beam by making it dependent on the variation existing at the emitter

surface. The predictions are no better than the models that go into them, and

such a bland observation points to the concerns here. In earlier times when

the computational power brought to bear in simulating devices was far more

limited, the emission models were rudimentary. As computational power

increases, the impact of simulation on ‘‘first pass design success’’ is far

more critical to the costly effort of designing amplifiers and rf injectors for

accelerators: an account of beam evolution and spread and the impact of

imperfections in the beam on the performance is essential in the design

of high power devices. To accentuate the point, systems of higher operational

frequency entail reduced dimensions, meaning that the quality of the electron

beam in all its varied metrics has disproportionately greater impact, and the

passably adequate simple models of a previous time are increasingly limited,

or worse, maladapted. PIC codes such as MICHELLE are presently able to

consider variation in emission over micron length scales in the modeling of

macroscopic devices, so the question of variation takes on a pressing nature

and is an area of active research. The more comprehensive emission models

have found in the power of modern simulation codes a strong argument for

their utility.

The improved models are needed to address the operation of electron

sources in mixed conditions when the canonical equations are inoperative,

subject to conditions which vary from one regime to another, or which

involve parameters that are not static throughout the emission process.

Some examples suffice to convey what is envisioned.

A simple but by no means trivial complication is how much current

comes from how small of an area: for field emission from sources such as

Spindt cathodes, the transconductance (that is, the variation in current with

applied voltage, the measure of which bears on the Class of an amplifier)

depends on whether small amounts of current come from a great many points

or a great deal of current from a few points; such considerations are in

addition related to the scalability of the cathode (that is, whether 100 times

as many emitters will produce 100 times as much current or – as is in fact

more often the case – a smaller amount). A small number of emitters driven


hard have a different signature and therefore impact on modulation of the

resultant beam than a small amount of current per site from a great number

of sites.

The nature of the surface chemistry on advanced thermionic (e.g.,

scandate) cathodes affects emission because it affects the mechanism (dipole

versus a semiconductor model) by which the barrier to emission is lowered.

In addition, the manner in which dispenser cathodes are ‘‘rejuvenated’’ by

the flowing of the coating materials like barium introduces variation as a

consequence of simultaneous diffusion and evaporation of the coating,

producing work function variation more complex than the uniform

sub‐monolayer coatings that were the focus of Gyftopoulos‐Levine theory

as treated here.

For photoemitters, the aforementioned surface effects are in addition to

whatever properties and their dependencies on temperature and photon

frequency that exist which affect electron transport in bulk. Semiconductor

photoemitters are subject to a host of complications that our focus on metals

allowed us to side‐step, such as band gaps, band bending, additional scatter-

ing mechanisms, and effective mass variation. Investigations of such comp-

lications on emission are predicated on models that are more cognizant of

material specific properties or behavior.

The analytical models of emittance for thermal and photoemitters is in

contrast to the absence of a useful one for field emitters. In addition to the

rapid decline of field away from the apex of the field emitter, the addition of a

close‐proximity gate that is used to create the high fields necessary in turn

significantly complicates the electron trajectories as they emerge from the

gate region. The emitters are not identical, meaning that the electron trajec-

tories are further buffeted by asymmetries in the extraction field caused by

emission ‘‘hot’’ spots. In a related note, if the current density is high, then the

impact of space charge in general is quite complex. Addressing such issues is

the province of simulation, but given the critical nature of initial conditions,

the emission theories that must be brought to bear must be more detailed

than the canonical equations, or the simulations are hobbled at the outset.

Finally, there is the question of the impact of additional physics that has

not been considered here. A simultaneous solution of Poisson’s equation and

the equations of emission are called for to investigate ‘‘quantum space charge

effects’’ particularly as the dimensions of the emitters shrink into the nanoe-

lectronics regime. A looming problem is the question of what impact nano-

scale dimensions have on the emission characteristics: recall that all of the

expressions for current density and emittance herein presupposed bulk‐likeand nearly free electron model conditions. In the case of Spindt‐typeemitters, such an approximation is perhaps adequate, but for carbon

308 KEVIN L. JENSEN

nanotubes where the diameter of the tube is only several nanometers, assum-

ing a bulk emission model strains credulity (apart from the question of

transport between the multiwalled layers). Ab initio studies provide a

means for determining what can be retained of the models considered

here that will account for their Fowler‐Nordheim‐like behavior, and better

models are required for when the that behavior departs from the same idyllic

FN characteristics.

Gratifyingly, there is much left to do.

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Index

A

Absorption probability, 267

Accelerator community, 263

Acoustic phonon, 193

scattering, 208

time, 212

Ad hoc simulations, 118

Adjacent dipoles, 290

Adjacent pulses, 231

Advanced photon source (APS), 270

After conditioning, 130

Airy coeYcients, 77

Airy functions, 74

approach, 71–80, 101

polynomial, 75, 76

Wronskians of, 80

Alkaline-earth metals, 290, 291

Analytical formula, 279

Angstrom-scale distances, 38

Angular integrations, 183, 205, 221

Annihilation operators, 25, 200

Anticommutator, 51

Apex configuration, 129

Approximation, 179

born, 33, 175

crue quadratic, 90

emission equation, 106–110

finite diVerence, 241

Forbes, 137

Fowler-Dubridge, 257

Friedel, 44

harmonic oscillator, 191

hyperbolic tangent, 41, 44, 94

image charge, 40–46

local-density, 31

Longo, 285

polynomial, 137

pseudopotential, 32

random-phase, 42, 179

Spindt quadratic, 116

Stirling’s, 132

Thomas-Fermi, 42

APS. See Advanced photon source

Area under the curve (AUC), 65

Area-under-the-potential, 69

Argon-cleaned polycrystalline

tungsten, 300

Asymptotic expressions, 109, 232

Asymptotic limits, 109, 174

Atomic polarizability, 157, 162

Atoms

chain to lattice transition of, 191

coverage, 289

monatomic linear chain of,

186–194

multielectron, 13

polarized, 281

radial hydrogen, 13

AUC. See Area under the curve

Awkward asymptotic

expressions, 109

B

Balance

detailed, 181

quartz crystal, 292

Band bending, 20–22

Band structure, 13–20

band bending, 20–22

semiconductors, 20

Bare metals, 148–150

density of states of, with respect to

nearly free electron gas

model, 264

quantum eYciency of, 260–273

Barium, 283, 293

325

Barrier

exact quadratic, transmission

probability, 86

Gaussian potential, 61–62

height, 276

image charge, 87–94

k value, 17

maximum, 83, 91–94

multiple square, 69–71

quadratic, 85–86, 90–91

Schottky, lowering, 222, 303

single, 70

square, 67–69

surface, 22–40

triangular, 80–85

BE. See Bose-Einstein

Bechtel-like conditions, 253

Behavior

Bloch-Grneisen, 207

of electron-electron relaxation

time, 185

low-temperature, 213

Bessel functions, 33

Beta dactor, 119

BH. See Brooks-Herring

Bloch-Grneisen behavior, 207

Bloch-Grneisen function, 208, 210

Blue line results, 261

Bohm approach, 62–64

Bohm trajectories, 57

Bohm-Staver relation, 195,

198, 208

Bohm-Staver result, 197

Boltzmann’s constant, 6

Boltzmann’s equation, 158

Boltzmann’s transport equation

(BTE), 11, 53

Born approximation, 32, 175

Bose-Einstein (BE), 7

Bose-Einstein statistics,

304–305

Bra-ket notation, 17–18

Brooks-Herring (BH), 179

BTE. See Boltzmann’s transport

equation

C

Calculated peak temperature, 225

Canonical copper, 232

Canyonlike complexity, 266

Carbon nanotubes, 120

Cathodes. See also Photocathodes

dispenser, 283, 284, 296

lower-work function, 282

Schottky emission, 105

sintered tungsten dispenser, 262

CDS. See Central diVerence schemes

Central diVerence scheme (CDS), 241

Cesium, 293, 296

Charged impurity relaxation

time, 177–179

Chemical potential, 6, 9–11, 92

Classical distribution function

approach, 47–49

Classical image charge, 5

Classically forbidden region, 34

Clean tungsten emitter, 119

CoeYcients

airy, 77

matrix, 244

requisite, 254

temperature-dependent, 247–253

Collision integral, 200

Collision operator, 180

Complex conjugation, 203

Complexity

canyonlike, 266

considerable, 146

Conductivity, 165–174

DC, 167

electrical, 165–167

photoemission, 165–174

thermal, 167–170

thermal, data, 214

Conjugation, 203

Considerable complexity, 146

Constants

Boltzmann’s, 6

dielectric, 15, 154–156

electron-phonon coupling, 236

326 INDEX

fundamental, 4

potential segment, 65–67

representations, 138

Richardson, 103

Contamination, 269–273

Continuity equation, 49

Contour map, 17

Copper

canonical, 232

parameters, 93, 131, 233, 238

reflectivity of, 266

Core radius, 39

Correlation energy, 29

Coulomb potential, 13, 195

Coupled heat equations, 234–235

Coupled temperature equations, 248

Coupled thermal equations

numerical solution of, 239–253

nature of problem, 239–240

ordinary diVerential

equations, 240–246

numerically solving, 247–253

Coverage atom, 289

Coverage factor parameters, 239

Crank-Nicolson method, 245

Creation/annihilation

operators, 27, 58

Crue quadratic approximations, 90

Crystal faces

diVerent, 295

work function, 261–263

Current of energy, 168

Current of heat, 168

D

D electrons, 217

D’Alambertian operator, 198

DC. See Direct current

DC conductivity, 167

Debye cutoV, 173

Debye frequency, 191, 206

Debye temperature, 208

Deformation potential, 199, 218

Delta-function-like pulse, 227

Denominator, 221

Detailed balance, 181

Diatomic case, 186

Dielectric constant, 15, 154–156

DiVerent crystal faces, 295

Dimensionless parameter, 28

Dipoles

adjacent, 290

contribution, 37

eVective, moment, 289

eVects, 33–40, 286

molecular, moments, 289

term, 44

Dirac delta function, 14, 101, 169,

176, 205

Direct current (DC), 160

Direct numerical evaluation, 112

Discrete representation, 97

Dispenser cathodes, 283, 284, 296

Distribution

classical, function approach, 47–49

electron, 214

emitted, 126

energy, 127

FD, 27

function, 48, 257

Gaussian, 48, 56

general, 48

normal emission, 126

theoretical energy, 127

Wigner Distribution Function

Approach, 52–62

Downwind diVerence schemes, 242

Drive lasers, 222–223, 250

Drude model, 156–162

Drude relations, 160

E

Edison eVect, 279

EVective dipole moment, 290

EVective emission area, 269–273

EVective quantum eYciency, 273

EYciency, 273. See also Quantum

eYciency

INDEX 327

Electrical conductivity, 165–167

Electrochemical potential, 166

Electromagnetic wave, 155

Electron(s)

background contribution, 24

collisions, 236

D, 217

density, 36

distribution, 214

electron collisions, 184

electron relaxation time, 185

behavior of, 185

electron scattering, 180–185,

220, 221

emission, 22–40

eVects, 261

physics, 279

surface eVects/origins of work

function, 22–31

energies, 210

fast electron-electron scattering

mechanism, 236

FEL, 147

free electron gas, 5–11, 264

free, gas model, 264

free, model, 8

gas, 197, 238

high-power free, lasers, 222

IEDM, 282

multi, 13

nearly, electron gas, 11–22

number density, 237

photoemitted, 221

photoexcited, 219, 236, 278

uniform, density, 31

zero-temperature, gas, 151

Electronegativity, 286, 288, 289

Electron-phonon

coupling constant, 236

coupling factor, 236–239

relaxation time, 238

scattering, 194–212

calculations, 191

Elliptical integrals, 131–136

Emission. See also Field emission;

Photoemission

eVective area, 270–274

electron, 22–40

enhanced, 279–304

equation integrals, 106–110

general, equation, 105

low field thermionic, studies, 103

normal, distribution, 126

in thermal-field transition region

revisited, 136–139

thermionic, 279

triangular barrier,

probability, 85

Emittance, 273, 277

Emitted distribution, 126

Emitter

clean tungsten, 119

field, 265

microfabricated, 270

Spindt-type, 128

thermionic, 270

tungsten, 119

Energy

correlation, 29

current of, 168

distribution, 127

exchange/correlation, 30

Gaussian, analyzer, 126

high, photons, 215

kinetic, 27, 34, 188, 219

LEUTL, 270

photoexcited electron, 278

photon, 303

Rydberg, 29, 185

stupidity, 29

theoretical, distribution, 127

total, per unit volume, 29

Enhanced emission, 279–304

less simple model of, 282–286

simple model of, 281–282

Equations

approximation, emission,

106–110

Boltzmann, 158

Boltzmann transport, 11, 53

continuity, 49

coupled heat, 234–235

coupled temperature, 248

coupled thermal, 239–253

328 INDEX

Fowler-Nordheim, 104–106, 143

Fowler-Richardson-Laue-

Dushmann, 117

general emission, 105

general thermal-field, 131, 139

linearized Boltzmann, 165

ordinary diVerential, 240–246

parabolic, 240

Poisson’s, 14, 45, 195

revised FN-RLD, 118–130

Richardson, 166

RLD, 104–106, 110–118

Schrodinger’s, 12, 15–16, 162

thermal-field, 139–146

Equivalent formulations, 21

Error function, 228

Escape cone, 151–154

Exact quadratic barrier transmission

probability, 86

Exactly solvable models, 65–85

airy functions approach, 71–80

multiple square barriers, 69–71

square barrier, 67–69

triangular barrier, 80–85

wave function methodology for

constant potential

segment, 65–67

Exchange term, 27

Exchange/correlation energy, 30

Exchange-correlation potential, 31

Experimental reflectance, 163

Experimental relations, 213

F

Fast electron-electron scattering

mechanism, 236

FD. See Fermi-Dirac; Fowler-

Dubridge formulation

FEL. See Free electron laser

Femtoseconds, 206

Fermi level, 138, 168, 220, 237, 255

Fermi momentum, 10

hkF, 8, 182

Fermi-Dirac (FD), 7

distribution function, 27

integral, 9–10

integral circles, 9

integrals related to, 304–305

Fermi’s golden rule, 174–177

Feynman diagrams, 29, 180

Fick’s law, 225

Field emission, 47–146, 273

current density, 47–64

in Bohm approach, 62–64

in classical distribution function

approach, 47–49

in Gaussian potential

barrier, 61–62

in Schrodinger/Heisenberg

representations, 47, 49–53

in Wigner distribution function

approach, 52–62

exactly solvable models of, 65–85

numerical methods of, 94–101

numerical treatment of image

charge potential, 95–99

numerical treatment of quadratic

potential, 95

resonant tunneling, 99–102

recent revisions of standard

thermal/field models

in, 131–139

emission in thermal-field

transition region


Forbes approach to evaluation of

elliptical integrals, 131–136

revised FN-RLD equation/

inference of work function

from experimental data

in, 118–130

mixed thermal-field

conditions, 123–126

slope-intercept methods applied

to field emission, 127–130

thermionic emission, 121–123

thermal/equation, 102–118

emission equation integrals/their

approximation, 106–110

Field emitter, 265

Field enhancement, 264–269

INDEX 329

Field operator notation, 23

Field-dependent area factor, 120

Field/thermionic emission

fundamentals, 4–46

free electron gas, 5–11

image charge approximation, 40–46

nearly free electron gas, 11–22

surface barriers, 22–40

unit note, 4–5

Finite diVerence approximation, 241

FN. See Fowler-Nordheim

Forbes approach, for elliptical

integrals, 131–136

Forbes approximation, 137

Forbes expression, 140

Force-free evolution, 56

Formula

analytical, 279

asymptotic limit, 109

Fourier components, 175

Fourier transform, 157

Fowler-Dubridge approximation, 257

Fowler-Dubridge formulation

(FD), 257

Fowler-Dubridge function, 107, 152,

220, 265

comparison of, 153, 154

Fowler-Dubridge model, 150, 219,

222, 269

revisions to modified, 253–255

Fowler-Dubridge probability

ratio, 259

Fowler-Nordheim (FN), 45

Fowler-Nordheim

equations, 104–106, 143

Fowler-Nordheim factors, 131

Fowler-Richardson-Laue-Dushmann

equation, 117

Fractional monolayer

coverage, 285

Free electron gas, 5–11

chemical potential of, 9–11

density of states with respect to

nearly, 264

quantum statistical mechanics

of, 5–8

Free electron laser (FEL), 147

Free electron model, 8

Frequencies, 191

Debye, 191, 206

optical, 162

resonance, 162–164

Friedel approximation, 44

Friedel oscillations, 36, 37, 42

Full-width-at-half-max (FWHM), 115

Functions

airy, 71–80, 74, 75, 76, 101

Bessel, 33

Bloch-Grneisen, 208, 210

classical distribution,

approach, 47–49

crystal faces, 261–263

Delta-function-like pulse, 227

Dirac delta, 14, 101, 169,

176, 205

distribution, 48, 257

Drude model, 156–162

error, 228

Fermi-Dirac distribution, 27

Fowler-Dubridge, 107, 152, 153,

154, 220, 265

general distribution, 48

ground-state Wigner, 59–60

Heaviside step, 27, 33, 151

Kronecker delta, 18

Riemann zeta, 9, 108, 152,

305–306

Wigner, 158

Wigner distribution, 52–62, 61

Fundamental constants, 4

FWHM. See Full-width-at-half-max

G

Gas

electron, 197, 238

free electron, 5–11

nearly free electron, 11–22

zero-temperature electron, 151

Gaussian distribution, 48

Gaussian distribution density, 56

Gaussian energy analyzer, 126

330 INDEX

Gaussian laser pulse, 253

Gaussian potential barrier, 61–62

General distribution, 48

General emission equation, 105

General thermal-field equation,

131, 139

Gold parameters, 238

Ground-state Wigner function, 59–60

Gyftopoulos-Levine model, 286–292

photoemission compared

to, 296–304

thermionic comparison to, 292–296

Gyftopoulos-Levine theory, 13, 40,

283

H

Harmonic number, 223

Harmonic oscillator, 57–62, 187

Harmonic oscillator

approximation, 191

Heat of solids, 171–174

Heat/corresponding

temperature, 225–227

Heaviside step function, 27, 33, 151

Heisenberg picture, 50

Heisenberg representations, 49–52

Hermite polynomials, 60

High electric field gradients, 297

High-energy photons, 215

Higher-order derivatives, 241

High-power free-electron lasers, 222

High-temperature representation, 207

Hyperbolic tangent

approximation, 41, 44, 94

Hyperellipsoid, 143

I

IEDM. See International Electron

Devices Meeting

Image charge approximation, 40–46

analytical image charge potential

of, 43–46

classical treatment of, 40–42

expansion near E ¼ of, 88–90

expansion near E ¼ : quadraticbarrier of, 90–91

quantum mechanical treatment

of, 42–43

reflection above barrier maximum

in, 91–94

Image charge barrier, 87–94

Image charge potential, 95–99

Image pulse, 227

Implicit schemes, 247

Incidence angle, 268

Increased absorption, 267

index of refraction/

reflectivity, 154–156

Inference of work function from

experimental data, 118–130

Integrand components, 253

Integrations

angular, 183, 205, 221

time, 231

International Electron Devices

Meeting (IEDM), 282

Intrinsic emittance, 145

Inversion invariant, 203

Ion core eVects, 31–33

Ion-electric cloud, 156

Ionized scattering site, 177

Isotropic crystal, 172

Isotropic system, 188

J

JeVreys-Wenzel-Kramers-Brillouin

(JWKB), 62

Jellium, 13

JWKB. See JeVreys-Wenzel-Kramers-

Brillouin

K

Kinetic energy, 27, 188

component, 219

operator, 23

Kronecker delta function, 18

Kronig-Penney model, 13–20, 31, 47

INDEX 331

L

Large argument case, 78–79

Laser(s)

drive, 222–223, 250

free electron, 147

heating, 223

high-power free-electron, 222

intensities, 304

intensity, 232

pulse, 252, 300

temperature of, illuminated

surface, 222–239

Laser pulse

Gaussian, 253

maximum, 249

simple model of temperature

increase due to, 223–225

Lattice temperature tracks, 250

LDA. See Local-density

approximation

Least-squares analysis, 295

LEUTL. See Low-energy undulator

test line

LINAC. See Linear accelerator

Linear accelerator (LINAC), 144

Linear beam, 281

Linear segment potential, 100

Linearized Boltzmann

equation, 165

Liouville’s theorem, 144

Liquid nitrogen temperature, 218

Local-density approximation

(LDA), 31

Longo approximation, 285

Lorentzian components, 163

Low field thermionic emission

studies, 103

Low-energy undulator test line

(LEUTL), 270

Lower-work function

cathode, 282

Low-temperature behavior, 213

Low-temperature leading-order

limit, 221

Low-work function coatings, 279–304

less simple model of, 282–286

simple model of, 281–282

M

Macroscopic fields, 163, 226

Macroscopic surface, 268

Macroscopic viewpoint, 226

Magnetron-sputtered lead, 267

Matrix notation, 16

Matthiessen’s rule, 212–215

Maximum temperature, 229

Maxwell-Boltzmann statistics, 6–7

Metallic-like parameters, 255

Metals

alkaline-earth, 290, 291

bare, 148–150, 261–274

photocathodes, 150, 260

Meter-kilogram-second-ampere

(MKSA), 4

Microcrystalline surfaces, 270

Microfabricated emitter arrays, 270

Micron-scale resolution, 265

Microscopic scale, 265

Millikan’s erroneous

conjecture, 106

MKSA. See Meter-kilogram-

second-ampere

Molecular dipole moments, 289

Moments-based approach, 215,

255–261

Momentum, 177

delta function, 200

eigenstate, 51

like variables, 33

relaxation time, 206, 211

Monatomic case, 186

Monatomic linear chain of

atoms, 186–194

Monatomic system, 188

Monolayer coverage, 283, 287

Monte Carlo simulations, 216

Multielectron atoms, 13

Multiple pulses, 227–233

332 INDEX

Multiple reflections, 264–269

Multiple square barriers, 69–71

N

NANO units, 5

Nanotubes, 120

National Accelerator Facility, 296

Naval Research Laboratory

(NRL), 279, 298

NEA. See Negative Electron AYnity

Nearly free electron gas, 11–22

band structure/Kronig-Penney

model and, 13–22

hydrogen atom and, 11–12

Negative Electron AYnity (NEA), 148

Nondegenerate, 20

Nontrivial matrix, 245

Normal emission distribution, 126

Normal incidence, 277

Normalization, 59

Normalized brightness, 145

Nottingham heating, 113

NRL. See Naval Research

Laboratory

Numerical treatment of quadratic

potential, 95

Numerically evaluated transmission

probability, 84

O

Ohm’s law, 160

Operators

annihilation, 25, 200

collision, 180

creation/annihilation, 27, 58

D’Alambertian, 198

field, notation, 23

formalism, 202

kinetic energy, 24

Optical frequencies, 162

Optical phonons, 193

Orbitals, 12

Order unity, 185

Order zero, 33

Ordinary diVerential


Oscillators

harmonic, 57–62, 191

strength term, 162

P

Parabolic equations, 240

Parameters

copper, 238

copper-like, 93, 131, 233

coverage factor, 239

dimensionless, 28

gold, 238

Thomas-Fermi, 195

tungsten, 223, 293

Particular finite duration, 227

Pauling radius, 39

Pauling units, 287

Penetration depth, 154–164

dielectric constant/index of

refraction/reflectivity

and, 154–156

Drude model and, 156–162

quantum extension/resonance

frequencies, 162–164

Periodic permanent magnet

focusing, 281

Phase space description, 11

Phonons

acoustic, 193, 208, 212

interaction terms, 201

optical, 193

relaxation time, 212

Photocathodes, 147, 222–223, 225

drive laser combinations, 250

emittance/brightness of,

274–279

metal, 150, 260

simulation algorithm, 232

surface, 278

Photoemission, 147–279

background of, 147–148

conductivity, 165–174

electrical conductivity, 165–167

heat of solids, 171–174

INDEX 333

Photoemission (Cont.)

thermal conductivity, 167–170

Wiedemann-Franz Law, 170–171

emittance/brightness of

photocathodes and, 274–279

modified Gyftopoulos-Levine

model compared to, 296–304

numerical solution of coupled

thermal equations of, 239–253

probability of, 151–154

quantum eYciency of bare metals

and, 148–150, 260–273

quantum eYciency revisited/

moments-based approach

and, 255–259

reflection/penetration depth

of, 154–164

revision to modified Fowler-

Dubridge model and,

253–255

scattering factor of, 215–222

scattering rates of, 174–215

temperature of laser-illuminated

surface of, 222–239

wavelengths of, 212

Photoemitted electrons, 221

Photoexcitation, 214

Photoexcited electrons, 219, 236

energy, 278

Photons

energy, 303

high-energy, 215

Poisson’s equation, 14, 45, 195

Polarization diagrams, 29

Polarized atom, 281

Polycrystalline form, 293

Polynomial airy functions, 75, 76

Polynomial approximation, 137

Pore-to-pore separation, 283

Post-conditioning current-voltage

plots, 129

Potassium, 283

Power tubes, 279

Predictor-corrector methods, 246

Predictor-corrector schemes, 247

Probability

absorption, 267

density, 13

Fowler-Dubridge, 259

numerically evaluated

transmission, 84

of photoemission, 151–154

single barrier transmission, 70

transmission, 52, 68, 69, 101

Proper method, 99

Pseudopotential approximation, 32

Pulses

adjacent, 231

delta-function-like, 227

Gaussian laser, 253

image, 227

laser, 223–225, 252, 300

laser, maximum, 249

multiple, 227–233

Pyramidal depressions, 40

Q

QE. See Quantum eYciency

Quadratic barrier, 85–86

Quadratic method, 99

Quadratic potential, 93, 95

Quantum eYciency (QE), 148–150,

222, 263

of bare metals, 260–273

contamination/eVective emission

area, 269–273

density of states with respect to

nearly free electron gas

model, 264

surface structure/multiple

reflections/field

enhancement, 264–269

variation of work function with

crystal face, 261–263

plots, 272

revisited/moments-based

approach, 255–259

Quantum extension, 162–164

Quantum mechanical

treatment, 42–43

Quantum potential, 63

Quantum statistical

mechanics, 5–8

334 INDEX

Quantum trajectory, 57

Quartz crystal balance, 292

R

Radial hydrogen atom, 13

Random-phase approximation

(RPA), 42, 179

Reflections, 154–164

above barrier maximum, 91–94

multiple, 264–269

Relaxation time, 165, 176

charged impurity, 177–179

phonons, 212

Requisite coeYcients, 254

Residual resistivity, 212

Resistivity values, 211

Resonance frequencies, 162–164

Resonant tunneling, 99–102

Resonant tunneling diode (RTD), 63

Revised FN, 110–118

Revised FN-RLD equation, 118–130

RHS. See right-hand side

Richardson constant, 103

Richardson equation, 166

Richardson-Laue-Dushman

(RLD), 45

Riemann zeta function, 9, 108, 152,

305–306

Right-hand side (RHS), 11

RLD. See Richardson-Laue-

Dushman

RLD equations, 104–106, 110–118

RPA. See Random-phase

approximation

RTD. See Resonant tunneling diode

Runge-Kutta method, 246

Rydberg energy, 29, 185

S

Scattering

acoustic phonon, 208

electron-electron, 180–185

factor, 215–222

rates, 174–215, 255

charged impurity relaxation

time, 177–179

electron-electron


electron-phonon


Fermi’s golden rule, 174–177

Matthiessen’s rule/specification

of scattering terms, 212–215

monatomic linear chain of

atoms, 186–194

number of sites, 179

sinusoidal potential, 185–186

terms, specification of, 212–215

Schematic representation, 52

Schottky emission cathodes, 105

Schottky-barrier-lowering,

222, 304

Schrodinger’s equation, 12,

15–16, 162

Schrodinger’s representations, 47,

49–52

Screened Coulomb potential, 15

SDDS. See Second-order downwind

diVerence scheme

Second order upwind diVerence

scheme (SUDS), 242

Second-order downwind diVerence

scheme (SDDS), 243

Semiconductors, 20, 179

Single barrier transmission

probability, 70

Sintered tungsten dispenser

cathode, 262

Sinusoidal potential, 185–186

SLAC. See Stanford Linear

Accelerator

Slater determinant, 25, 181

Slope factor ratio, 114

Small argument case, 79–80

Solid lead, 266

Sought-for linear dependence, 238

Sound velocity, 192, 196

Spatial Fourier transform, 226

Spatial inversion, 201

INDEX 335

Specific heat capacity, 167

Specification of scattering

terms, 212–215

Spindt quadratic approximation, 116

Spindt-type emitter, 128

Spin-orbit coupling, 12

Square barrier, 67–69

Stanford Linear Accelerator

(SLAC), 147

Step function potential, 81

Stirling’s approximation, 132

Stupidity energy, 29

Submonolayer coverage, 285

SUDS. See Second order upwind

diVerence scheme

SuYciently high temperatures, 194

Surface barriers, 33–40

Surface eVects, 22–32

Surface of surface heating, 233

Surface self-diVusion, 129

Surface structure, 264–269

T

Taylor expansion, 159, 169

Temperature

calculated peak, 225

Debye, 208

excursions, 226

heat/corresponding, 225–227

liquid nitrogen, 218

maximum, 229

rise, 227–233

for tungsten surfaces, 249

suYciently high, 194

Temperature of laser-illuminated

surface

coupled heat equations,

234–235

diVusion of heat/corresponding

temperature, 225–227

drive lasers, 222–223

electron-phonon coupling

factor, 236–239

multiple pulses/temperature

rise, 227–233

photocathodes, 222–223

simple model of temperature

increase due to laser

pulse, 223–225

Temperature-dependent

coeYcients, 247–253

TFA. See Thomas-Fermi

approximation

Theoretical energy

distribution, 127

Theoretical intrinsic emittance, 278

Theoretical quantum eYciency

model, 303

Thermal conductivity, 167–170, 214

Thermal emission, 47–146

current density, 47–64

in Bohm approach, 62–64

in classical distribution function

approach, 47–49

in Gaussian potential

barrier, 61–62

in Schrodinger/Heisenberg

representations, 49–52

in Wigner distribution function

approach, 52–62

exactly solvable models of,

65–85

numerical methods of, 94–101

numerical treatment of image

charge potential, 95–99

numerical treatment of quadratic

potential, 95

resonant tunneling, 99–102

recent revisions of standard

thermal/field models in

emission in thermal-field

transition region


Forbes approach to evaluation of

elliptical integrals, 131–136

revised FN-RLD equation/

inference of work function

from experimental data

in, 118–130

mixed thermal-field

conditions, 123–126

336 INDEX


to field emission, 127–130

thermionic emission, 121–123


to, 127–130

thermal/equation, 102–118

emission equation integrals/their

approximation, 106–110

Fowler-Nordheim/Richardson-

Laue-Dushman


triangular barrier of, 80–85

WKB area under curve

models, 85–94

image charge barrier, 87–94

quadratic barrier, 85–86

Thermal emittance, 48, 143–146

Thermal equilibrium, 234, 235

Thermal-field equation, 139–146

completion of, 142

thermal emittance, 143–146

Thermal-field transition

region, 136–139

Thermionic data, 292–296

Thermionic emission, 279

Thermionic emitter, 270

Thomas-Fermi approximation

(TFA), 42

Thomas-Fermi parameter, 195

Three-step model, 219

Time

acoustic phonon, 212

charged impurity

relaxation, 177–179

dependent electric field, 157

electron-phonon relaxation, 238

independent shield Coulomb

potential, 177

integration, 231

phonon relaxation, 212

Top-down perspective, 268

Total energy per unit volume, 29

Total relaxation time, 214

Trace space, 143

Transmission probability, 52, 68, 69,

101

Triangular barrier, 80–85

Triangular barrier emission

probability, 85

Trivial multidimensional

generalization, 198

Tungsten, 293, 296

argon-cleaned polycrystalline, 300

clean, emitter, 119

parameters, 223, 293

sintered, dispenser cathode, 262

temperature rise for, 249

U

Ultraviolet (UV), 150

Uniform electron density, 31

UV. See Ultraviolet

UV illumination, 153

W

Wave function, 16, 51, 202

methodology

airy function approach, 71–80

for constant potential

segment, 65–67

large argument case, 78–79

multiple square barriers, 69–71

small argument case, 79–80

square barrier, 67–69

Wronskians of airy functions, 80

Wave packet spreading, 54–57

WDF. See Wigner distribution

function approach

Weighted areas, 284

Wentzel-Kramers-Brillouin

(WKB), 62

Width, 35

Wiedemann-Franz Law, 170–171

Wigner distribution function

approach (WDF), 52–62, 61

INDEX 337

Wigner function, 158

Wigner trajectory case, 63

WKB. See Wentzel-Kramers-

Brillouin

Work function, 46, 284

with crystal face, 261–263

from experimental data, 118–130

reduction, Gyftopoulos-Levine

model of, 286–292

Wronskians of airy functions, 80

Y

YLF. See Yttrium-lithium-fluoride

Yttrium-lithium-fluoride

(YLF), 147

Z

Zero-temperature electron gas, 151

338 INDEX

Advances in Imaging and Electron Physics, Volume 149: Electron Emission Physics

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Transcript of Advances in Imaging and Electron Physics, Volume 149: Electron Emission Physics