ANL 1

Numerical Modeling of Field-Enhanced Photoemission from Metals and Coated Materials

Numerical Modeling of Field-Enhanced Photoemission from Metals and Coated Materials

ARGONNE NATIONAL LABORATORY DECEMBER 6, 2005 Argonne, IL

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We gratefully acknowledge:FUNDING by Joint Technology Office & Office of Naval Research

INTERACTIONS with (alph). S. Biedron , C. Bohn, C. Cahay, D. Dimitrov, D. Dowell, J. Lewellen, J. Petillo, J. Smedley

UMD:

D. W. FeldmanN. A. Moody

P. G. O’Shea

NRL:

K. L. JensenJ. L. ShawJ. E. Yater

ANL Beams and Applications SeminarHost: John Lewellen ASD

Bldg. 401, Room A1100 Tuesday, 1:30 pm

ANL 2

OUTLINE

Electron Sources Photocathodes and Photocathode Issues

The Dispenser Photocathode Concept

Electron Emission Fundamentals

1st Generation Emission Model & Usage

Next Generation Components & Application Bare Metals

Cesiated Surfaces & Gyftopoulos-Levine Model

Quantum Distribution Function

Quantum Effects on Barrier & Scattering

Conclusion

ANL 3

Thermionic

ELECTRON EMISSION

The Manner in Which Electrons Are Extracted Dictates the Technological Gambits Invoked

Metal

Field

HeatWave Labshttp://www.cathode.com/c_cathode.htm

HeatWave Labshttp://www.cathode.com/c_cathode.htm

ANL-BNL-JLAB Gun: Ilan Ben-Zviwww.agsrhichome.bnl.gov/eCool/MAC_Ilan.pdf

ANL-BNL-JLAB Gun: Ilan Ben-Zviwww.agsrhichome.bnl.gov/eCool/MAC_Ilan.pdf

Photo

Courtesy of C. A. Spindtwww.sri.com/psd/microsys/vacuum/

Courtesy of C. A. Spindtwww.sri.com/psd/microsys/vacuum/

ANL 4

PHOTOINJECTORS & PHOTOCATHODESCritical Components of Free Electron Lasers, Synchrotron Light & X-ray Sources

LinacScale

2" 3"0 1"

Drive laser

Photo-cathode

rf Klystron MasterOscillator

High Power FEL Demands on Photocathode:CHARGE PER BUNCH: 0.1 - 1 nC in 10-50 ps pulseFIELD: 10 - 100 MV/m in pressure of 10-8 Torr (approx)OPERATION: Robust, Prompt, Operate At Longest LIFETIME: Longevity & Reliability Paramount

ANL 5

PHOTOCATHODE RESPONSE TIME

Pulse Shaping Optimal Shape for emittance:

beer-can (disk-like) profile

Laser Fluctuations occur (esp. for higher harmonics of drive laser)

Fast response: laser hash reproduced

Slow response: beer-can profile degraded

Optimal: 1 ps response time

Mathematical Model (n = 2n/T)

I t( )=Ioθ t( )θ T −t( ) cncosnt( )n=0

N∑

Ie t( )=QEτ

I s( )ex −t−sτ

⎛⎝⎜

⎞⎠⎟

⎡

⎣⎢

⎤

⎦⎥−∞

t

∫ ds∝cn

1+ nτ( )2

cosnt( )+nτ sin nt( )( )eT /τ −1⎡⎣ ⎤⎦e−t/τ t <T

eT /τ −1( )e−t/τ t≥T

⎧⎨⎪

⎩⎪n=0

N

∑

Formulation based on model of J. Lewellen

0.0

0.2

0.4

0.6

0.8

1.0

1.2

-5 0 5 10 15 20 25 30 35time [ps]

IdealMetal25.6 ps6.4 ps3.2 ps0.8 ps

ANL 6

CATHODE + LASER NONUNIFORMITY

Laser

Cathode Emittance Is Important (esp. in Gun) Pulse Shape Can Result in Reduction of Emittance

Prediction of Photocathode / Drive Laser Combos & Beam Crucial to Design of Larger Systems

Realistic Initial Distributions Needed: Advanced Codes (Michelle, Argus, Magic, Mafia, Vorpal, Etc.) Need Adequate Emission Models, Photoemission Studies Sparse, Emission Distributions Are Unknown

Laser Nonuniformity

Work FunctionNonuniformity

Laser Non-uniformity + work function non-uniformity(sim)

Combined Nonuniformity

ANL 7

IDENTIFY factors that affect QE (e.g., laser, environment, photocathode material)DEVELOP a custom-engineered controlled porosity photo-dispenser cathode

PROGRAM OBJECTIVE

A Cs-dispenser Photocathode Concept

monolayer

W plug w/ Cs

Al2O3 Potting

Heater

Maryland DispenserCathode

Generic Parameter Range

Assumptions:

• Charge = 1 nC

• Wavelength = 266 nm

• Pulse FWHM = 10 ps

Example Cases of Io(QE)

For Radius of 0.3 cm

• Io(1%) = 0.165 MW/cm2

• Io(0.01%) = 16.5 MW/cm2

For Radius of 0.1 cm

• Io(1%) = 1.48 MW/cm2

• Io(0.01%) = 148 MW/cm2

Io =

hq

ΔQΔtFWHMr2

⎛

⎝⎜⎞

⎠⎟1

QE

Metal

TopView

CsO

SideView

Interpore ≈ 6 µm; Grain Size≈ 4.5 µm; Pore Diam. ≈ 3 µm

Conventional Dispenser

Controlled Porosity

ANL 8

0

2

4

6

8

1 2 3 4 5 6 7 8Grain Index

DISPENSER CATHODE POROSITY ANALYSIS

Central Questions:

Pore-to-Pore

separation &

Grain Size

Desorption &

migration rates of

low work function coating over surface

80

160

240

320

400

480

0 100 200 300 400 500 600 700Pixel X

10 μ = 143 m pixel

SEMS: Nathan Moody, UMD

L(x;μ,s 2 ) ≡1

2sxexp −

12s 2 ln x / μ( )( )

2⎡⎣

⎤⎦

⎧⎨⎩

⎫⎬⎭

Pore to Pore: Log-Normal Distribution

Grain Size: Effective Diameter

dA =14D2∫

0

1

2

3

4

5

6

7

0 16 32 48 64 80 96 112 128 144 160SEPARATION (pixel)

Pore-to-PoreLogNorm(x)

bin size = 8 pixelsLog-Normal Parametersμ = 35.3 ; pixelss = 0.786

- - :Mean pore to pore35.3 (10 x μ / 143 ) = 2.47 m pixel μm

ANL 9

Electron Number Density

Zero Temperature (μ(0 ˚K) = μo = EF)

N does not change with T so μ must:

μ,T( ) =N

V= 2 fFD (Ei )

i∑

⇒2

2π( )3 1+ exp β μ − E(

rk )( )⎡⎣ ⎤⎦( )

−1

d 3k∫

STATISTICAL MECHANICS OF ELECTRON GASElectrons Incident On Barrier Are

Distributed In Energy According To A 1-D

“Thermalized” Fermi Dirac Distribution

Characterized By The Chemical Potential

and Called The “Supply Function”

f(k) obtained by integrating over the

transverse momentum components

(μ o ) =

1

3π 2

2mμ o

h2

⎛⎝⎜

⎞⎠⎟

3/2

=kF

3

3π 2

f (k) =2

2 2

2k⊥dk⊥

1+ exp β(E||+ E⊥ −μ)( )0

∞

∫

=m

βh2 ln 1+ exp β(μ −E||)( )⎡⎣ ⎤⎦μ(T ) = μ o 1−1

12

π

βμ o

⎛

⎝⎜⎞

⎠⎟

2

−1

80

π

βμ o

⎛

⎝⎜⎞

⎠⎟

4

+…⎡

⎣⎢⎢

⎤

⎦⎥⎥

BAND BENDINGMetal vs. Semiconductor

Ec

μoFvac

Metal

Ec

Ev

μo

Fvac

c

μ

Semiconductor

ANL 10

CURRENT - A CLASSICAL APPROACH

f(x,k,t) is the probability a particle is at position x with momentum hk at time t

Conservation of particle number:

N = 2( )

−1f (x,k,t)dxdk∫∫

dn = f (x,k,t)dxdk= f x',k',t'( )dx'dk'=dn'

x'=x+hkm

dt; hk'=hk+ Fdtto order O(dt)

dk

dk’

dx

dx’

dndn’

dx 'dk ' =∂xx' ∂xk'

∂kx' ∂kk'dxdk=dxdk

0 =

f x+ dx,k+ dk,t+ dt( )− f (x,k,t)

dt⇒

∂∂t

+hkm

∂∂x

+Fh

∂∂k

⎧⎨⎩

⎫⎬⎭

f (x,k,t) =0

∂∂t

x,t( ) =∂∂t

12

f (x,k,t)dk−∞

∞

∫⎡

⎣⎢

⎤

⎦⎥=

∂∂x

12

hkm

⎛

⎝⎜⎞

⎠⎟f (x,k,t)dk

−∞

∞

∫⎡

⎣⎢

⎤

⎦⎥=−

∂∂x

J x,t( )

Boltzmann Transport Equation

“Moments” give number density and current density J: Continuity Equation

velocity & acceleration

ANL 11

CURRENT IN SCHRöDINGER REPRESENTATION

Consider a pure state

j(x, t) = x j(x,t) x =h2m

x (t), k{ } x

=h

2miψ † (x,t)∂xψ (x,t)−ψ (x,t)∂xψ

† (x,t){ }

The form most often used in emission theoryBasis for FN & RLD Equations

H ψ k(t) =h2k2

2m+V x( )

⎛

⎝⎜

⎞

⎠⎟ ψ k(t) =E ψ k(t)

(t) = f E(k)( ) ψ k(t)∑ ψ k(t)

∂∂t

t( ) = H , (t)⎡⎣ ⎤⎦=ih2

2m∂∂x

k, (t){ } =−∂∂x

j t( )

Schrödinger’s Equation

Time-dependence of Operators governed by commutators with Hamiltonian

Simple Case: Gaussian

x ψ (0) =Δk

2exp −

12Δk2x2 + ikox

⎡

⎣⎢

⎤

⎦⎥

(x) =Δk2

2exp −Δk2x2⎡⎣ ⎤⎦

J (x) =hko

m(x)

jk (x, t) = t(k) 2 hkm

J (x,t) =12

hkm

T(k) f(k)0

∞

∫

ψ k (x) → t(k)ψ k(x)→

T (k) =jtrans(k)jinc(k)

ANL 12

The most widely used forms of: Field Emission: Fowler Nordheim (FN) Thermal Emission: Richardson-Laue-Dushman (RLD)

THERMIONIC VS FIELD EMISSION

Fowler Nordheim

T (E) ≈ex − bfn / F( )+ cfn μ −E( )( )⎡⎣

⎤⎦

f(E)=mh2 μ −E( )Θ μ −E( )

J FN(F)=AF 2 ex −B / F( )

Richardson

T (E) =Θ E − μ +φ( )⎡⎣ ⎤⎦

f (E) =m

βh2 exp β μ −E( )⎡⎣ ⎤⎦

J RLD(T ) =ARLDT2 exp −φ / kBT[ ]

J(F,T )= 1

2hT E( ) f E( )dE0

∞∫

5

6

7

8

9

0 2 4 6Position [nm]

Field Emission:Work Func = 4.6 eV

Field = 4 GV/m

Thermionic Emission:Work Func = 1.8 eV

Field = 10 MV/m

Fermi Level

High TemperatureLow Field

Low TemperatureHigh Field

Transmission Probability

Electron Supply

Emission Equation

A

rld=120.173

Amp

Kelvin2cm2;Q=0.359991 eV-nm

A =1.38072×10-6

Amp

eV 2exp 9.83624-1/2( )

B=6.399523/2 eVnm

Constants for Work Function in eV, T in Kelvin, F in eV/nm

ANL 13

10-12

10-10

10-8

10-6

10-4

10-2

100

A GENERAL THERMAL-FIELD EQUATION

MaxwellBoltzmann

Regime

0 K-like Regime

Supply Function

General Transmission Coefficient

Define Slope Ratio:

n « 1: Richardson-Laue-Dushman Eqn » 1: Fowler Nordheim Equation

f E( ) =

mβTh2 ln 1+ exp βT μ −E( )⎡⎣ ⎤⎦{ }

T E( ) ≈To 1+ exp βF Eo −E( )⎡⎣ ⎤⎦{ }−1

Eo =μ +bfn

Fcfn

Eo =μ +− 4QF

Field

Thermalf(7,300)

f(0.01,2000)

T(0.01,2000)T(7,300)

X(F[GV/m],T[K])

Fermi

0

0.2

0.4

0.6

0.8

1

1.2

3 4 5 6 7 8 9 10 11Energy [eV]

7 GV/m300 K

2 MV/m1094 K

10 MV/m2000 K

n =βT / βFn =βT / βF

JFN ⇒ ARLD kBβF( )−2

1+ 2

6βF

βT

⎛

⎝⎜⎞

⎠⎟

2

+74

βF

βT

⎛

⎝⎜⎞

⎠⎟

4⎛

⎝⎜

⎞

⎠⎟exp βF (μ −Eo)( )

J RLD ⇒ ARLD kBβT( )−2

1+ 2

6βT

βF

⎛

⎝⎜⎞

⎠⎟

2

+74

βT

βF

⎛

⎝⎜⎞

⎠⎟

4⎛

⎝⎜

⎞

⎠⎟exp βT (μ −Eo)( )

JFN ⇒ ARLD kBβF( )−2

1+ 2

6βF

βT

⎛

⎝⎜⎞

⎠⎟

2

+74

βF

βT

⎛

⎝⎜⎞

⎠⎟

4⎛

⎝⎜

⎞

⎠⎟exp βF (μ −Eo)( )

J RLD ⇒ ARLD kBβT( )−2

1+ 2

6βT

βF

⎛

⎝⎜⎞

⎠⎟

2

+74

βT

βF

⎛

⎝⎜⎞

⎠⎟

4⎛

⎝⎜

⎞

⎠⎟exp βT (μ −Eo)( )

ANL 14

QDF

INTEGRATED SCATTERING & EMISSION MODEL

Quantum Distribution Function (QDF) Simulation Simultaneously Relates Scattering (τ), Barrier Emission (), Thermal () and Density Effects

k

h

z( )

f & U(x)Scattering &

Electron Transport & Emission

Coverage-Dependent Work Function

-24

-20

-16

-12

-8

-4

0

0 5 10 15

z(bohr)

Clean W(001)Potential

-24

-20

-16

-12

-8

-4

0

0 5 10 15

z(bohr)

Ba/O/W(001)

POSITION (x/ao)

EN

ER

GY

W c

ores

W c

ores

Hemstreet, et al. PRB40, 3592 (1989)

W

Ba O

& Relaxation Time &

Thermal Model2

4

6

8

10

12

Literatureln(Tac)ln(Tee)Theory Au

ANL 15

EXP. VALIDATED PHOTOEMISSION MODEL

GOAL Predict Quantum Efficiency From Laser & Material Parameters Analyze Experimental Results from UMD, NRL, Colleagues in FEL Program Emission Model For Beam Code for NRL, SAIC, Tech-X, NIU, Colleagues

GOAL Predict Quantum Efficiency From Laser & Material Parameters Analyze Experimental Results from UMD, NRL, Colleagues in FEL Program Emission Model For Beam Code for NRL, SAIC, Tech-X, NIU, Colleagues

QE =q

hf 1−R()( )PFD h( )

PFD h( ) =U β h −+ 4QF( )⎡

⎣⎤⎦

U βμ[ ]

COMPONENTS:

Work function variation with coating θ)

Gyftopolous-Levine theory

Thermal & Material; laser β, R(), f

Transient heating & heat diffusion

Simple Photoemission Model U-function

Revised Fowler-Dubridge Model

Quantum effects @ hi F & T U, f

barrier due to e- density; scattering

Transmission through barriers U

Relate barrier to emission probability

Photocurrent depends on

• Scattering Factor: f

• Absorbed laser power: (1-R) I

• Escape Probability: U-terms

First Generation Beam Code Model

Next Generation Beam Code Model

ANL 16

DETERMINATION OF R[%] &

Algorithm:

Spline-fit experimental optical data (e.g., CRC, AIP Handbook) for index of refraction (n), damping constant (k)

Designate incident angle = θ Use Equations to determine

Reflectance R[%] and penetration depth of laser for given wavelength

Rs =a2 +b2 −2acosθ + cos2 θa2 +b2 + 2acosθ + cos2 θ

Rp =Rs

a2 +b2 −2asinθ tanθ + sin2 θ tan2 θa2 +b2 + 2asinθ tanθ + sin2 θ tan2 θ

2a2 = n2 −k2 −sin2 θ( )2+ (2nk)2⎡

⎣⎤⎦

1/2

+ n2 −k2 −sin2 θ( )

2b2 = n2 −k2 −sin2 θ( )2+ (2nk)2⎡

⎣⎤⎦

1/2

− n2 −k2 −sin2 θ( )

R =12

Rs + Rp( ); =

4k

0.1

1

10

100

0.1 1 10

n(W)k(W)n(Cu)k(Cu)n(Au)k(Au)

Wavelength [micron]

10

100

0.1 1 10

delta(W)%R(W)delta(Cu)%R(Cu)delta(Au)%R(Au)

[ ]Wavelength micron

…other metals in database

Consider W, Cu, Au…

ANL 17

POST-ABSORPTION SCATTERING FACTOR

Factor (f ) governing proportion of electrons emitted after absorbing a photon: Photon absorbed by an electron at depth x Electron Energy augmented by photon, but

direction of propagation distributed over sphere Probability of escape depends upon electron

path length to surface and probability of collision (assume any collision prevents escape) path to surface &

scattering length

To leading order, k integral can be ignored z θ( ) =

xcos θ( )

; l k( ) =hkτm

f =

f(k)dkko

∞

∫ dθ0

/2

∫ exp −x−

z(θ)l k( )

⎛

⎝⎜⎞

⎠⎟0

∞

∫ dx

f(k)dk dθ0

∫ko

∞

∫ exp −x

⎛⎝⎜

⎞⎠⎟0

∞

∫ dx

k =1h

2m E(k) +h( )

ko: minimum k of e- that can escape after photo-absorption : penetration of laser (wavelength dependent); : relaxation time

θk

h

z(θ)

Average probability of

escape

f ≈12

Gmhkoτ

⎛

⎝⎜⎞

⎠⎟

G cos(y)[ ] =1−2

cot y( ) ln1+ sin(y)

cos(y)⎛⎝⎜

⎞⎠⎟

G sec(y)[ ] =1−2

ysin(y)

⎛⎝⎜

⎞⎠⎟

argument < 1

argument > 1

Ex: Copper: = 266 nm = 12.9 nmτ() = 0.85 fsμ = 7.0 eV• F = 4.3 eV

sec(y) = 7.7f = 0.038

ANL 18

U β(h −φ)( )⇒ U β(h −φ)( )+

2

12βb

⎛⎝⎜

⎞⎠⎟

2

QM contributes for photon E near barrier height, large fields,and cold temperatures

Copper * = 266 nm; F = 5 MV/m; β = 3• R = 33.6%, = 4.3 eV, EF = 7.0 eV• QE [%] (analytic) 1.31E-2• QE [%] (time-sim) 1.36E-2• QE [%] (exp) 1.40E-2

QM-EXTENSION OF FOWLER-DUBRIDGE EQ.

RLD-based Fowler Dubridge Model

• U(x): depends on thermal distribution and barrier μ+ for emission probability

• Reflectivity R and Scattering Factor f depends on material & relaxation time

QE ≈

f 1−R( )μ2 h −φ( )2

QE = f 1−R( )U h −φ( ) / kBT⎡⎣ ⎤⎦

U μ / kBT⎡⎣ ⎤⎦

U x( )= 2

12+ ln 1+ ez( )dz

0

x

∫“Fowler factor”

0

20

40

60

3 4 5 6 7

1064 (x100)532355266

Energy [eV]

0

20

40

60

3 4 5 6 7Energy [eV]

J =

12h

T(E) f(E)dE0

∞

∫ TRLD (E) =Θ E −μ −φ+h( )

TQM (E) = 1+ exp[b Ec(h)−E( ){ }

−1

coating = 2.0 eV

Exp data: T. Srinivasan-Rao, et al., JAP69, 3291 (1991).

ANL 19

SCATTERING & Electrical / Thermal Conductivity

If an electric field F (or temperature gradient T) is removed, then distribution “relaxes” back to equilibrium after a “relaxation time” τ

f x,t( ) ≈ fo x,k( ) + fF x,k( )− fo x,k( ){ }exp −t / τ (k)( )

f

rr ,

rk( )= fo

r ,k( )+

τh

Fg

∇k fo −τ

hkmg∇rT(

r )( )

∂∂T

fo

Distribution for Fermi-Dirac is approximately constant except near Fermi Energy μ

WIEDEMANN-FRANZ LAW

sT

= 2

3kB

q

⎛

⎝⎜⎞

⎠⎟

2

=2.443×10−8 W-ΩKelvin2

⎡

⎣⎢

⎤

⎦⎥

Thermal Conductivity

Electrical Conductivity

Specific Heat

rJ

F(k) =

2q

2( )3

d3khrkm

⎛

⎝⎜⎞

⎠⎟∫ frk( )

=2q

2( )3

d3khkx

m

⎛

⎝⎜⎞

⎠⎟Fhτ

rk( )

⎧⎨⎩

⎫⎬⎭

∫∂∂kx

fo

rk( )

=qτ (μ)kF

3

3 2m

⎛

⎝⎜

⎞

⎠⎟ F ≈s

Fq

⎛

⎝⎜⎞

⎠⎟ Field

rJ

T(k) =

2q

2( )3

∂T∂x

⎛

⎝⎜⎞

⎠⎟d3kτ

k( )

hkx

m

⎛

⎝⎜⎞

⎠⎟

2

∫∂∂T

fo

k( )

=m3τ(μ)

∂T∂x

⎛

⎝⎜⎞

⎠⎟∂∂T

2

2( )3

d3k E(k)−μ⎡⎣ ⎤⎦∫ fo

k( )

⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪

=2μ3m

τ μ( )CV T( )∂T∂x

=(T)∂T∂x Temp

Electric Field Temperature

ANL 20

Diffusion mimics the temporal spread of

Dirac-Delta-like pulses with Do

Do acts as Length2 / time

Length ≈ O(laser penetration depth)

Model captures physics…

as long as there is an estimate for to

LASER HEATING OF ELECTRON GASDifferential Eqs. Relating Electron to Lattice Temp

Electron & LatticeSpecific Heat Laser Energy

AbsorbedPower transfer by electrons to lattice

285.1 GW / K cm3 (W @ RT)

ΔT =1−R( ) IoΔt

DoΔt Ce To( ) +Ci To( )( )

Do ≈2 / to

ΔT =1−R( ) IoΔt

DoΔt Ce To( ) +Ci To( )( )

Do ≈2 / to

∂t y = Do∂x2y ⇒ y(x, t) =

exp -x2 / 4Dot( )⎡⎣ ⎤⎦4π Dot

Ce

∂∂t

Te =∂∂z

(Te,Ti )∂∂z

Te

⎛

⎝⎜⎞

⎠⎟- g Te -Ti( ) +G z,t( )

Ci

∂∂t

Ti =g Te -Ti( )

100 101 102 103

Intensity [MW/cm2]

Copper Δ = 6 t ps t

o = 1.495 ps

10 -1

10 0

10 1

10 2

10 3

10 -3 10 -2 10 -1 10 0

.Time Dep

Model

[ ]Pulse Width ns

Copper

Iμax

= 100 /MW cm2

to = 1.673 ps

ANL 21

PHOTOEMISSION MODULES IN BEAM CODEGoal: modules for 3D RF gun / beam codes for the analysis of beam generation and transport. Present model: high-T scattering operator with T evaluated using Delta-diffusion model as function of laser intensity for

copper; probability of emission factor based on Fowler Dubridge but without QM Next generation to include all-temperature scattering, QM, metal & coating library

SIMULATION CODE: MICHELLE (SAIC)• Photoemission from laser-illuminated Cu hemisphere

Using 1st generation photoemission model• J. Petillo, et al., 8th DEPS, Lihue, HI (2005)

Hemisphere unit cell model

SIMULATION CODE: VORPAL (TECH-X)

• 3D visualization of photo emitted electron particles (white dots) following the beam emission and its evolution at different times from simulations with steady-state photocathode model.

• Cu photocathode at left boundary. Front of laser pulse has reached the photocathode and emission of the electron beam has started.

• D. Dimitrov, et al., 8th DEPS, Lihue, HI (2005)

VORPAL Cu Photocathode Beam Emission and Evolution

ANL 22

-2

0

2

4

6

8

10

12

14

0 1 2 3 4 5 6 7

ln(Tau [fs])ln(Tac)ln(Tee)ln(TOTL)Room TempLiq. Nitrogen

ln(Temperature [Kelvin])

Cu

THEORETICAL EVALUATION OF SCATTERING

Scattering in metals due to collisions with lattice (acoustic phonons & defects) and e-e collisions. If mechanisms independent, then:

τ ac =h3vs

2

mkBTkFΞ2

TD / T( )5

ℜ TD / T( )

ℜ x( )=s5ds

es −1( ) 1−e−s( )0

x

∫

For T > TD (Debye Temp), then τac goes as T, but at low T, goes as T^5. Scattering cross section is also related to

• deformation potential Ξ (related to stress on lattice)

• sound velocity vs

τee =4hKs

2

a 2mc2(kBT)2

1+ΔEkBT

⎛

⎝⎜⎞

⎠⎟

2⎛

⎝⎜⎜

⎞

⎠⎟⎟g

2kF

qo

⎛

⎝⎜⎞

⎠⎟

⎡

⎣

⎢⎢

⎤

⎦

⎥⎥

−1

g(x) =x3

4tan−1 x+

x1+ x2

−tan−1 xy( )

y

⎛

⎝⎜

⎞

⎠⎟

y(x) = 2 + x2

Ag

Pb

Au

W

AgCuAuWPb

Phonons:

Electron-Electron:e-e scattering in simple metals is not simple. Model of Lugovskoy & Bray [1] depends on electron energy above Fermi Level (ΔE) and Thomas-Fermi Screening Wave Number qo (depends on electron density) & dielectric Ks

[1] A. V. Lugovskoy, I. Bray, J Phys D: Appl. Phys. 31, L78 (1998)

Mathiessens Rule τ

−1 =τee−1 +τ ac

−1

ANL 23

10-7

10-6

10-5

10-4

10-3

10-2

200 220 240 260 280 300

Exp (Smedley)TheoryTheory (Extrapolated n,k)

Wavelenth [nm]

Lead

200 220 240 260 280 300

Exp (Dowell)

Theory x 0.7

Wavelength [nm]

Copper

EXPERIMENT VS. THEORY (BULK METALS)

Field Enhancement 3.0 Macroscopic field 1.0 MV/m Work Function 3.97 eV Temperature 300 K Data & Image

Courtesy of J. SMEDLEY (BNL)

BNLBNL SLACSLAC

Field Enhancement 1.0Macro field (MV/m) 0.01Work Function 4.31 eVTemperature 300 KData Courtesy of D. DOWELL (SLAC)

ANL 24

EXPERIMENT VS. THEORY (PART II)

Measured (UMD), calculated (NRL), & literature for various DISPENSER CATHODES(1st Generation model used)B-TYPE:

B. Leblond, NIMA317, 365 (1992)UMD experimental data

M-TYPEUMD experimental data

SCANDATEUMD experimental data

10-5

10-4

10-3

250 300 350 400 450 500 550

WAVELENGTH [nm]

Scandate

B-Type

M-type

ExperimentTheory

0.0

0.5

1.0

1.5

0 20 40 60 80 100

2% Any70˚ p70˚ sTheory

Laser [µJ]

Field = 39 MV/mBeta = 1t = 16 psA = π(0.39 mm)^2 = 266 nm

, ., 341,379 (1994)Rosenzweig et al NIMA

How the comparison is made:

• Time-Dependent Thermal Photoemission Model using τee(E=μ+hn) ran for each incident laser intensity

• Laser pulses were Gaussian in time

• Total energy and charge emitted evaluated via integration over Gaussian pulse (Laser) and Emitted charge profile (electron)

• Use of library values for Copper only (no adjustable constants)

ANL 25

COVERAGE DEPENDENT WORK FUNCTION

Gyftopolous-Levine Theory relates Work Function to coverage factor. Dependent upon Covalent Radii rx, Factors “f” and “w” (Act As “Atoms Per Cell”,

Values of which Depend on Crystal Face). General Surface = “Bumpy [B]”

alkali metal (n = 1)

alkaline-earth metal (n = 1.65)

gm ≡w

2rC( )2; g f ≡

f

2rW( )2

gm[110] = 2gm[100]

gm[B] = 3gm[100]

g f :gm =1: 4 for Cs on W, Mo, Ta

g f :gm =1: 2 for Ba, Sr, Th on W, etc.

θ( )=φ f − φ f −φm( )θ 2 3−2θ( ) 1−G θ( ){ }

G θ( )=

rorC

⎛

⎝⎜⎞

⎠⎟

2

1−2w

rWR

⎛

⎝⎜⎞

⎠⎟

2⎛

⎝⎜⎜

⎞

⎠⎟⎟

1+ nrCR

⎛

⎝⎜⎞

⎠⎟

3⎛

⎝⎜⎜

⎞

⎠⎟⎟

1+9n8

fθ( )3/ 2⎛

⎝⎜⎞

⎠⎟

fθ

Modified Gyftopolous-Levine Theory

β R

Hard Sphere Model of Surface Dipole

W

C

ANL 26

1

2

3

4

5

0 0.2 0.4 0.6 0.8 1

Phi (Wang)(Scale = 0.0889)Phi (Taylor)(Scale = 0.8698)Theory

Coverage θ

Cs on Tungsten1

2

3

4

5

0 0.2 0.4 0.6 0.8 1

Phi (Haas)(Scale = 0.6858)Phi (Longo)(Scale = 1.000)Theory

Coverage θ

Ba Dispenser Cathode

GYFTOPOLOUS-LEVINE MODEL PERFORMANCE

LEAST SQUARES ANALYSIS:Minimize Difference between GL theory & Exp. Data With Regard to scale factor, monolayer work function value, and f coverage factor

C-S Wang, J. Appl. Phys. 44, 1477 (1977)

J. B. Taylor, I. Langmuir, Phys. Rev. 44, 423 (1933).

R. T. Longo, E. A. Adler, L. R. Falce, Tech. Dig. of Int'l. El. Dev. Meeting 1984, 12.2 (1984).

G. A. Haas, A. Shih, C. R. K. Marrian, Applications of Surface Science 16, 139 (1983)

• Tightly constrained parameter variation• Unique determination of theory based on

experimental values• Predictive ability from basic experiments

ANL 27

A n o d e / C a t h o d e

A s s e m b l y

D e p o s i t i o n S e n s o r

C e s i u m S o u r c e

C e s i u m E v a p o r a t i o n

2 0 0 L i t e r / s e c

I o n P u m p

4 0 L i t e r / s e c

I o n P u m p

T o D e p o s i t i o n

M o n i t o r C o n t r o l l e r

F i n e M e t e r i n g

V a l v e

E l e c t r i c a l

F e e d t h r o u g h s

L a s e r W i n d o w

V i e w p o r t W i n d o w

F o u r W a y C r o s s

U H V C h a m b e r

7 0 . 9 9 c m .

EXPERIMENTAL MEASUREMENTS @ UMD

• Test and evaluation chamber @ 2E-10 Torr

• Surface preparation using H-ion beam

• Surface deposition of various coatings

• Deposition monitor (+/- 0.01nm thickness)

• Femto-ampere current measurements (for QE)

• Solid state CW lasers on single-axis robot

• QE as function of time, temp, coverage, wavelength, and laser intensity (AUTOMATED)

ANL 28

0

0.01

0.02

0.03

20 40 60 80 100Theta [%]

QE OF Cs ON W: EXP. VS. THEORYCONDITIONS:

Coverage Is Uniform

Field & Laser intensity low: Schottky barrier lowering & heating negligible

For 407 nm

Cs deposited rapidly: Exp measured mass by a “depth” factor. Therefore: Scale factor = (100%/Atomic diameter)

For 532 nm and 655 nm

Cesium deposited slowly: accumulation rate affected by desorption (scale a)

Peak value affected by residual cesium left on W (center xmax)

Presence of O (hard to remove) affects work function variation

To account for effects:

θ x( ) =a xexp −xmax( ) +θmax

ExperimentTheory

405 nm: x 1 (theory x 1.40)

532 nm: x 4 (theory x 1.40)

655 nm: x 35 (theory x 0.84)

ANL 29

0

0.04

0.08

0.12

0.16

0 20 40 60 80 100COVERAGE [%]

25 K

77 K

300 K

CS ON W @ 405 nm

Io = 10 MW/cm2

F = 10 MV/mβ = 3

600 K

QE OF Cs ON W: EFFECT OF TEMPERATURE

Consider “typical” conditions of:

Laser Intensity10 MW/cm2

Field at cathode 10 MV/m

Field enhancement 2 (generic)

Pulse length10 ps

Relaxation Time Is Temperature-dependent: Impact of Operating Photocathode at Lower Temperatures Is to Raise QE (all other things being equal)

ΔQ 266 nm( ) =QE ×717 nCUnder these conditions

For r = 0.1 cm2:

ANL 30

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0 20 40 60 80 100

TheoryTheory x 0.85Experiment AExperiment B

Coverage [%]

Cs on Ag = 405 nμ = 0.0174 MV/μ

min = 1.6 eV

a = 5.3 Angsτoμs

QE OF Cs ON Ag: EXP. VS. THEORY

Cs ON Ag Exp: INITIAL

COMPARISON METHODS for data taken week of Aug 8 2005

Experimental conditions same as for Cs-W for 532 and 655 nm exp.

Used same scale a as Cs on W

Shift factor aligns peaks for each experiment

Data taken by ANNE BALTER

ANL 31

HYPOTHESES OF EXP-THEORY DIFFERENCESActual surfaces differ from theory models because of Surface geometry (altered by cleaning?)

Reflectivity changes with exposed face (inc. angle)

Crystal Faces can have different work functions (e.g., Cu(111) = 4.86, Cu(110) = 5.61, etc)

Contaminants (e.g., Carbon-based ≈ 5.5 eV) & possibility of only sub-area contributing

Solid Silver 500x

Solid Tungsten 500x

Nathan Moody (UMD)

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0 50 100 150 200

Horizontal (micron)

Surface of Silver

Angle = 88.607 deg

θ

θo

Shadows model: fraction of surface illuminated (f)

f =tanθo

tanθ + tanθo

ANL 32

0

1

2

-2 0 2Distance- +

IMPACT OF COATINGS ON WORK FUNCTION

Conventional View: charged atoms at surface & image charge = dipole

QM View: Electron penetration of barrier determined by height and width

Exchange-Correlation & Poisson’s Eq: Changes in density = changes in V(x)

0

2

4

6

8

10

-12 -9 -6 -3 0 3

Electrons

Lattice

Position [Angstroms]

+

_Cu ParametersF = 1 GV/m

-24

-20

-16

-12

-8

-4

0

0 5 10 15

z(bohr)

Clean W(001)Potential

W

Ba O

L. A. Hemstreet, et al. PRB40, 3592 (1989)

W c

ore

s

W c

ore

s

VBa

(x) =qi

4Q

x−rBa+

exp − x−rBa+

{ } Δφ≈

292

QkF

Friedel Oscillations

o 1+ 3cos ζ( )ζ 2

−3sin ζ( )ζ 3

⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪

ζ =2kF x−xo( )

tanh model

ExCorr+

Poisson-24

-20

-16

-12

-8

-4

0

0 5 10 15

z(bohr)

Ba/O/W(001)

EN

ER

GY

POSITION (x/ao)

ANL 33

0

5

10

15

1 10 100r[]

Al

Fe

Cs

BaAg

Au

Cu

W

Mg

2 18ESi

EXCHANGE CORRELATION ENERGY OF e-

Aspects of system of interacting e- in ground state determined by density

Exch.-Corr. Potential: change in gives rise to change in V = Fermi Level= electron density ao = Bohr radius

H N =hk( )

2

2m+

q2

4e0

e−a rr−r

r '

rr −

rr 'i< j=1

N

∑i=1

N∑ −q2

4e0

drr∫

e−a rr−r

ri

rr −

rrii=1

N

∑ +(rr ) +

q2

8e0

drr∫ d

rr '

e−a rr−r

r '

rr −

rr 'i< j=1

N

∑ +(rr )+(

rr ')

1

V

h2k 2

2mk∑ ⇒2

2( )3

h2k2

2m0

kF∫ 4k2dk=35μo Kinetic

1

4V 2k

1k

2V

ee

rk

1-

rk

2( ) k1k2

(-)

k1k2

'∑ ⇒m

3h2ao

μ2Exchange

Terms with k

1k

2V

ee

rk

1−

rk2( ) k2k1 & higher Correlation

KineticEnergy

e-einteraction

e-latticeinteraction

self-interaction ofbackground

V

xc=−

∂∂

Eexch + Ecorr( )

Vxc

Metal Vacuum

Typical Metals

1

=43 rao( )

3

ANL 34

OTHER CONTRIBUTIONS TO SURF. BARRIER

Simple Theory: Wave function penetration creates dipole & gives Friedel Oscillations

(x) =12

f (k)sin2 ζ / 2⎡⎣ ⎤⎦dk0

∞

∫

limβ→ ∞

x( ) =o 1+ 3cos ζ( )ζ 2

−3sin ζ( )ζ 3

⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪

ζ =2kF x−xo( ); xo2 ∼2mVo / h

2

0

0.02

0.04

0.06

0.08

-12 -9 -6 -3 0 3

Electrons

Background

Position [Angstroms]

Copper ParametersNo BaField = 1 eV/nm

origin of background dictatedby global charge neutrality

+

_

Why bother looking at Friedel Oscillations? For two good reasons:

• Friedel Density profile has analytic V(x) sol’n

• Metal + Ba Density profile can be decomposed into Friedel component + Gaussian add-on: enables an analytic solution to Poisson Eq… or at least a very easily solved solution

How?

• Approximations exist for location of background positive charge

• Poisson’s Eq. easily solved with Friedel Density where = 2kF(xi – xo)

∂xφ( )−∞

∞

∫ dx=− x ∂x2φ( )dx

−∞

∞

∫ =−q2

eo

x−xo( ) e(x) −i (x)( )dx−∞

∞

∫

=q2o

4eokF2

3dds

sin(s)s

⎛

⎝⎜⎞

⎠⎟−∞

0

∫ ds− sΔ

0

∫ ds⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪=

QkF

1632−3 2( )

Dipole due to electron-lattice

difference

ζ ⇒ 2kF x−xo( )

Δφ⇒QkF

1632−3 2( )

Lattice origin in relation to electron

x

i=xo −

38kF

ANL 35

f(x,k,t): quantum phase space distribution acts

like probability distribution function

Contours mimic classical trajectories;

when potentials are smooth and slowly varying, they are

classical trajectories

QUANTUM DISTRIBUTION FUNCTION

Heisenberg Representation

∂∂t

(t) =ih

H , (t)⎡⎣ ⎤⎦

H ψ k(t) =h2k2

2m+V x( )

⎛

⎝⎜

⎞

⎠⎟ ψ k(t) =E ψ k(t)

(t) = f E(k)( ) ψ k(t)∑ ψ k(t)

f x,k,t( ) =2 e2iky x+ y (t) x−y

−∞

∞

∫ dyWigner Distribution function (WDF)

∂∂t

f x,k,t( ) =−hkm

∂∂x

f x,k,t( )

+ V x,k−k'( ) f x,k',t( )dk'−∞

∞

∫V x,k( ) =−

ih

e2iky

−∞

∞

∫ V(x+ y) −V(x−y){ } dy

x,t( ) =12

f (x,k,t)dk−∞

∞

∫

J x,t( ) =12

hkm

⎛

⎝⎜⎞

⎠⎟f (x,k,t)dk

−∞

∞

∫

integrate both sides with respect to momentum (k) get density and current density

vacuum metal

vacuum metal

Copper parametersField = 1 eV/nm

ANL 36

ANALYTICAL WDF MODEL: GAUSSIAN V(x)

How does V(x,k) behave? Consider a solvable case where V(x) is a Gaussian:

V x( ) =Voexp − x / Δx( )2⎡

⎣⎢⎤⎦⎥

V(x+ y) −V(x−y) =−2Vosinh 2xy / Δx2( )exp − x2 + y2( ) / Δx2⎡⎣

⎤⎦

SharpΔx2 = 5.0

BroadΔx2 = 0.1

V x,k( ) =−

2Δx 1/ 2h

Voexp −Δx2k2⎡⎣ ⎤⎦sin kx( ) small x samples f(x,k') far from k

large x samples f(x,k') near k

ANL 37

ANALYTICAL WDF MODEL (II): GAUSSIAN V(x)

The behavior of V(x,k) signals the transition from classical to quantum behavior:

Sharp: classical distribution

Broad: quantum effects

Can V(x,k) give a feel for when thermionic or field emission dominates?

Consider most energetic electron appreciably present(corresponds to E = μ or k = kF)

If sin(kFx) does not “wiggle” muchover range Δx, QM important

Thermionic Emission:Δx is very large - expectclassical description to be good

Field EmissionkFΔx = O(2) implies

V x,k( ) =−

2Δx 1/ 2h

Voexp −Δx2k2⎡⎣ ⎤⎦sin kx( )

4

6

8

10

0 20 40 60 80 100

No Field50 MV/m4 GV/mμ

(sin kF)x

Δisτance [Å]

Image Charge Potential

kF

F=2 ⇒ F ≈3

GVm

ANL 38

ELECTRON DENSITY TO EMISSION BARRIER

1. Numerically solve QDF to Obtain Electron Density

2. Render Density in Terms of Friedel Components

3. Evaluate Potential From Exchange-Correlation Relation & Poisson’s Equation from the Friedel Representation

4. Change in Barrier seen to be due to shift in F and xo terms

-12 -8 -4 0 4

BaO

F =0.629319

xo = +0.143379 Å0

0.02

0.04

0.06

0.08

0.1

-12 -8 -4 0 4

[#/ 3]Rho Ang

[#/ 3]Friedel Ang

Posiτion [Angsτoμs]

No BaO

= 1 /F eV nm

F = 0.598473

xo = 0.055413 Å

-4

-2

0

2

4

6

8

-1 -0.5 0 0.5

NumericalFriedelGaussian

[ ]x nm

Gauss(x)=9.785exp−(x+0.17357)/0.11387⎡⎣⎤⎦2⎧⎨⎪⎩⎪⎫⎬⎪⎭⎪

ANL 39

EMISSION BARRIER TO TRANSMISSION PROB.

Modified Airy Function Approach:

V(x) = Vo + F(x-xn):

V(x) V(xn): N Linear Segments at xn

ψ, ∂xψ Matched at xn, xn+1, etc.

0

4

8

12

-1 -0.5 0 0.5

ExCor

Field

Ion

Total

Position [nm]

No BaO

-1 -0.5 0 0.5

BaO

0

0.2

0.4

0.6

0.8

1

8 9 10 11 12

No BaO

BaO

μ+

[ ]Electron Energy eV

= 2 eV = 4.6 eV

M (x) =

Zi(z) Zi(z)∂xZi(z) ∂xZi(z)

⎡

⎣⎢

⎤

⎦⎥;

rζn =

tn(k)

rn(k)

⎛

⎝⎜

⎞

⎠⎟

M x

n−( ) •

rζn−1 k( ) =M xn

+( ) •rζn k( )

z(x) = f −2/3 v± f (x−xn) −k2 ; v=

2mVo

h2; f =

2mF

h2

Zi = Linear combinations of Airy Functions Ai(z), Bi(z)

ANL 40

CONCLUSION

Components of the Photocathode Program

Analysis of Coated & Bare Metals (extend to semiconductors)

Development Custom Engineered Controlled Porosity Photocathodes

Creation of Photoemission Models Validated By Exp. for Beam Codes

Theory Components included in photoemission code

Work function dependence on coverage & components; local variation

Spatial & Time Dependence of Temperature for laser & material parameters

Fundamental models of scattering, photoemission, QE & Barrier

Validation by bare and coated metal QE (macro) measurements

Status of Modeling Effort

Integrated Simulation Model Framework Without Recourse (Insofar As Possible) to “Fit” Parameters for “Library” Metals Using Quantum Distribution Function, Emission Theory, Coatings Theory

Photoemission Modules Appropriate for Beam Simulation Code (1st Generation Model distributed) From Integrated Simulation Model