1.1 Motivation: Why are surface structures important?1. Background

Surface Structure Determination by Low-Energy Electron Diffraction (LEED)

J.B. Pendry, Low Energy Electron Diffraction, Academic Press 1974M.A. Van Hove, W.-H. Weinberg, C.-M. Chan, Low-Energy Electron Diffraction, Springer 1986K. Heinz, LEED and DLEED, Rep. Progr. Phys. 58 (1995) 637

Similar to the volume: Structure very influential w.r.t. physical properties![e.g.: β-Sn (tetragonal, metallic) ↔ α-Sn (cubic, semiconducting)Fe (bcc, ferromagn.) ↔ Fe (fcc, paramagnet-ferromagn.)C (hexagonal, graphite)↔ C (cubic, diamond) ]

Surface = Interface to the environment, important for (e.g.) - contacts

- corrosion properties - catalysis- crystal growth- epitaxy (new materials!)

Semiconductor miniaturisation: Surface area / volume gets ever larger

Nanophysics

basic science: What happens on creation of a surface?

surface

reconstruction

Ir(100)-hex

1.2 Methods for structure determination - advantages/disadvantagesa) MicroscopiesElectron Microscopy (imaging)for atomic resolution: lens aberrations must be reduced=> high electron energies (>50 keV)

=> high sample penetration => only projection visible => low surface sensitivity. => Radiation damage!

„Bright field imaging“ of (220) net planes of Au

Special:Low Energy Electron Microscope (LEEM)Electrons are retarded just before reaching the surface, so that there is only the surface penetration wanted. Resolution reached: ≈5 nm.

Scanning Electron Microscopyresolution depends on (effective) beam diameter: 1-10 nm.

Special: Scanning Tunnelling Microscope (STM)Atomic resolution well achievableYet: - only top layer imaged

- electronic surface corrugation is imaged (rather than atoms).

Pt(110)missing row

Field Ion Microscopy (FIM)

atomic resolution well achievable.Yet: only atoms positions of high electric field or low work function are imaged. Also: application limited by the high fields applied; no subsurface atomic layers imaged.

b) Diffraction methodsonly intensities can be measured (“phase problem”) .=> at best, only the structural autocorrelation results

(only when the Fourier-transformation applies. Yet, even then there are no real direct methods) .

=> “trial-and-error" methods to find the structure.

Atom diffraction (thermal He, Ne, Ar)Sensitive to surface corrugation of electron density. When correlated to atomic positions, accuracy < 0.1 Å achievable.Yet: only top-layer diffraction (no sample penetration).

X-Ray Diffraction (XRD)Atomic positions with accuracy < 0.1 Å well achievable.Yet: Very oblique incidence of the primary beam necessary for surface

sensitivity. Very accurate sample alignment necessary.

Electron Diffraction (and related methods)Low Energy Electron Diffraction (LEED = E≲ 600 eV) ideal surface penetration (5-10 Å), experimentally “easy”.

Yet: Determination of atomic positions complicated by multiple electron scattering. Accuracy < 0.1 Å.

c) Scattering methods (ions)............... (structural triangulation)

Reflection High-Energy Electron Diffraction (RHEED) (E≤10 keV). Multiple scattering less pronounced => simpler data evaluation compared to LEED (many beams!).

Yet: to get surface sensitivity, (very) oblique incidence necessary; many beams; intensity measurement difficult.

Photo-Electron Diffraction (PED ) Angular Resolved Photo-Electron Spectroscopy (ARPES) Similar to LEED, except that a surface atom acts as electron source. Accuracy similar to LEED.

Surface Extended X-ray Absorption Fine Structure (SEXAFS)Only sensitive to bond lengths

1.3 Comparison of the different methods as applied

LEED is still the dominating tool.

SSD: structures per author < 2001

2527272729293030303033333335353637

4243454749525658

83100102

107114118

0 20 40 60 80 100 120

H.L. Davis

R.D. Diehl

A. Paton

D.A. King

D.W. Jepsen

Y. Gauthier

M. Lindroos

K.-M. Schindler

K. Mueller

K.A.R. Mitchell

L. Hammer

S.Y. Tong

A.M. Bradshaw

M.A. Van Hove

F. Jona

K. Heinz

from P.R. Watson, M.A. Van Hove K. Hermann, NIST Surface Structure Database, Version 5.0

1.4 Historical in short1897 “Discovery” of electron beams (J.J. Thomson)

1924 Wave mechanics (L. de Broglie) λ = h/p

1927 Prove of electron diffraction at a (Ni(111)-surface (C.J. Davisson / L.H. Germer, Nobel Prize 1937)

Davisson (l.) / Germer (r.) 1927

I( ) at E = 54 eVϕ θ = 30°

(3-fold rotational symmetry)

in contrast to X-ray diffraction: 1912 first prove for X-ray diffraction1913 first quantitative bulk-structure analysis

until about 1960 no further significant development!

Why?? LEED is dominated by multiple (=dynamic) scatteringOR: Full SGL for a complex potential must be solved

XRD is dominated by single (= kinematic) scattering

2. Experimental (in short)

luminescent

screen

screenvoltage

suppressor voltage

sample

primarybeam

diffractedbeams

prim

ary

peak

discrete losses (phonons, plasmons, Auger effect)

seco

ndar

y el

ectro

ns

energy

LEED optics(commerciallyavailable)

2.1 Experimental realization

Why is LEED used so often??

surface sensitivity and wavelength are almost ideal at the same energy

Mea

n fre

e pa

th (

Å)

1

5

10

50

electron energy (eV)10 100 1000

LEED

surface sensitivity dueto inelastic processes(plasmon generation, electron-hole excitation)

Penetrationabout 10Å=>

ideal surfacesensitivity

E = 150 eV => = 1Å 8ideal because of the order of atomic spacings => large diffraction angles

2.2 Structural information by LEED

LEED pattern reciprocalunit-cell

real spaceunit-cell

spot intensity

135 eV3C-SiC(111)-(3x3)

=>

energy (eV)

spot profile

statisticaldistributionof defects

atomic positionsIn the unit-cell= STRUCTURE

It is not alwayseasy to find theunit cellNiAl(110)+10 000L O2

tv camera

digitization / storage

UHVwindowLEED optics

500400300200 energy (eV)

inte

nsity

unitcontrol

2.3 Modern intensity measurement sinα= λ / g

spot moves when energy is swept

2.4 Evidence for multiple scattering and diffraction

inte

nsity

(a.u

.)

300200100energy (eV)

I kin

I dyn

+ ++

+ + +

+

Multiple scattering modifiesspectra drastically

3. The Schrödinger equation for LEED3.1 The Schrödinger equation and translational symmetry

sample

rk0 eik r

r r0 SE V E: − ∇ + =1

22ψ ψ ψ atomic units

h = = = ⇒ = =m e E k k1 12

2 12 0

2

constructive interference only for angles

sin /α λ= n a

rk0

α

a

r′k

k k k k na

nga a= → ′ − = = =2 20π λ α

π/ ( ) sinr r

r

equivalently: ( ) sinr r

r′ − = = =k k k mb

mgb b02

βπ } ( )||

r r

r r

r r

′ − =

+

= +

k kng e mg e

ng mga x b y

a b

0⇒the parallel momentumtransfer of the electrons is given by vectors reciprocal to real space vectors: recipr. lattice vectors

=>The wave function can be developed in terms of plane waves using rec. lattice vectors:

ψ = + ∑e e a z eik r ik rg

igr

g

r r r r rr0 0|| || ( )

ψ in

} }

ψ ψsc scattered=for z<0 (outside the crystal) with beingthe amplitude of the beam scattered into the direction given by

a z a constg g( ) = = ag

g hg kghk a b h k= + →r r (sin ,sin )α β (h,k) labels the beam

translationalsymmetry:

v r rt na mb= +

ab

xy

V r t V r( ) ( )v v v+ =

( )sin( / )

r rr′ − =

=k k

kk n a

a0

αλ

}

sample

ab

xy

kx

ky

2πb

2πa

a'b'

xy

2x1 superstructure

kx

ky2 2π πb b'

=

2π πa a'

=

sample

rk0||

r rk k0

202− ||

rk0

IN

Note: The number of beams is limited by energy conservation

sample

r rk g0|| +r r rk k g0

20

2− +( )||

rKg

−OUT

beams with ( )||

r r rk g k0

202+ >

are strongly attenuated.(evanescent beams)

Note: The calculation of reciprocal lattice vectors is more complicated for non-rectangular lattices:

ra a ax y= ( , )rb b bx y= ( , )

real space

ra

rb

reciprocal space

rA A A

b ba b b a

x y

y x

x y x y

=

= −−

( , )( , )2π

rB B B

a aa b b a

x y

y x

x y x y

=

= −−

( , )( , )2π

r r r r

r r r ra A b B

a B b A

⋅ = ⋅ =

⋅ = ⋅ =

2

0

π

3.2 Born series and kinematic approximation

For the calculation of the wave function we can use that at each point r' of the potential V(r') a spherical wave is emitted:

er r

ik r r| '|

| ' |

r r

r r−

−2πThe amplitude of this wave is proportional to the potential strength V(r') and to the amplitde of the wave function at r' which is to be scattered: V r r( ' ) ( ' )r rψ

ψ ψ ψπ( ) ( )| '|

( ' ) ( ' ) '| '|

r rr r

r r rr r

r r er r

V r r drin

ik r r

= −−

−

∫12=>for r k r r kr k r→ ∞ − ≈ − ⋅: | ' | 'r r r r

ψ ψ

ψπ

( ) ( ) ( )

( ) ( ') ( ') ''

r r r

r r r rrr

r r er

f k

with f k e V r r dr

in

ikr

ikr

= −

= − −∫12

f k( )r

=scattering factor

up to now no approximations!!

Ψin

Ψin

Ψsc

1. Born approximation(single scatt. approach,kinematic approximation,valid for small enough V)

Ψin

Ψin

Ψsc

2. term of 2. Born approximation(double scattering approach)

f k e V r dr V r V r dr dri k k r( ) ( ' ) ' ...... ( ' ) ( ' ' )... ' ' ' ......( ) 'r r r r r r rr r

= − + +− −∫ ∫∫12

14

02π π

*

Use equ. repeatedly:*

Kinematic (= 1. Born) approximation f k f k k V r e dri k k r1 1 0

12

0( ) ( ) ( ) ( )r r r r rr r r

= − = − −∫π

V r t r Vj i iji

= − +∑∑ δ ( ( ))r r

rt j

rr1

rr2

⇒ = − − −∑ ∑f k e V ei k k t

ji

i

i k k rj i1

12

0 0( ) ( ) ( )r r r r r r r

π}

}} }

G=lattice factor F=form factor

a) lattice factor: r r r rt t ma nbj mn= = + ⇒ − = =

=

−

=

−

∑ ∑G k k G q e eiqam

m

Miqbn

n

N

( ) ( )r r r rr rr

00

1

0

1

use x xx

nN

n

N

=−−=

−

∑ 110

1

⇒ =−−

⋅−−

G q ee

ee

iqaM

iqa

iqbN

iqb( )rrr

rr

r r

r r11

11

⇒ = ⋅G q qaMqa

qbNqb

( ) sin ( / )sin ( / )

sin ( / )sin ( / )

rr r

r r

r r

r r2

2

2

2

2

22

22

strong maxima forr r r rqa m qb n= =2 2π π;„Laue condition“

=> The lattice periodicity determines where spots can appeararrangement of spots <=> size and shape of the unit cell

(no information about atomic positions)

1 222

2

2MMqaqa

sin ( / )sin ( / )

number of unit cells

M = 5

M = 10

M = 20

M = 50

M = 100qaπ

LEED: M ~ 50eff

b) Form factor (structure factor): F k V eii

i k k ri( ) ( )r r r r

= ∑ −0

ρ1 ρ2r r rr r nci in i= = + ρ F e V ei k k c

ni

i

i k k i= −

=

∞−∑ ∑( ) ( )

r r r r r r0 0

0

ρ=> }}

smmation overall layers

layer diffractionM k k( , )

r r0

electron attenuation: p e pp

i k k c n

n= < ⇒ =

−−

=

∞

∑| |( )r r r

0 1 110

described by optical potential: 12

20 0 0

12

2k E V E V iV k ikr i r i= − = − − = +( ) ( )

E'

}k E V E V E V ki i i i r= + − ≈ ≈4 2 22

02

0 0' ' / ' /

k E V Er i= + ≈4 2202' '

} e e eikz ik z k zr i= −=>penetration depth:λp i ik E V E= = ∝1 2 0

1 2/ ' / /

slope = 1/2

frequently: energy dependent is usedV i0

examples:

V eV A at E eVi p0 5 55 50= − ⇒ ≈ =λ . '

⇒ ≈ =λp A at E eV11 200'

for normal incidence and g=0, i.e. ( )r r rk k c kc0 2− = , on gets:

| | | ( , ) | | |( ) cos( )

F M k ke

Mi k k c

2 0 22

21 1 20=

−=

+ −−

r r

r r rβ β α

with α β= = −2 2 2 2E c e V c Eoi' , / '

Bragg maxima forα π

ππ

=⇒ =

= =

2

12

2 2 2

nk n cE k n

n

n n c

/( )

<= Different optical potentials (different attenu-tion) cause different peak heights anddifferent peak widths

peak half width = 2Voi(independent ofg energy)

BUT:Experimental spectrumtotal different fromkinematic approximation

⇒ this is no„approximation“

=> We need to consider multiple scattering / diffraction

4. Multiple scattering

muffin-tin potential of the surface

Instead: on uses hierarchy ofscattering/diffraction:

- atomic scattering- layer diffraction- crystal diffraction

The solution of the SE wouldcorrespond to a band structurecalculation at high energies!Very demanding!

4.1 Atomic scattering

appro-xima-tions:

(a) The LEED electron leaves inner shells undistorted

(b) the (weak) scattering of outer shells is considered by the inner potential Vo

(c) if wave functions overlap at the muffin-tin radius they are folded back to the inner of the atomic sphere

=> The atomic potential is spherically symmetric: V r V r( ) ( )r =

ψ α β απ

βπl l l l l l

ikr

l

ikr

r h kr h kr er

er

( ) ( ) ( )= + → +−

1 2

2 2incoming outgoing

incoming spherical partis unaffected: => α γl l=

flux conservation: outgoing and incoming wave must have the same modulus:

β α δl l

ie l= 2

δl phase shift=" "

=> The wave function can be developed in spherical waves: ψ ψ( ) ( ) ( ),

r rr a r Y rlm l lm

l m= ∑

The incoming wave writes as: e Y r with h hik r

l lm

l l l l

r r r0 0 0 1 2= = +∑ψ ψ γ( ) ( )and i Y kl

llm

m l

l

γ π== −∑2 0

*( )r

e L h R h R h R L h RiR l l l R l

l2 2 2 1 1δ = − −[ ( ) ' ( )] / [ ' ( ) ( )]

with L R RRi i= Ψ Ψ' ( ) / ( )

The calculation of requires the knowledge of the wave function and its derivative at r=R:

δl

simple picture for the role of the phase shifts:

The scattered part of is:ΨΨ Ψ Ψsc l l l l

ile h krl

. ( ) ( )= − = −0 2 11α δ

⇒ = −∑ψ π δ2 120

1i e Y k Y r h krl

lm

ilm

lm

ll( ) ( ) ( ) ( )* r r = ++∑ i e l P h krl

ll

il l

l1 2 12 1sin ( ) (cos ) ( )δ θδ

→ → ∞−

t E e for rikr

( , ) ( )θπ2

t E l e P tk li

ll

ll

l( , ) ( ) sin (cos )θ δ θπ δ= − + =∑ ∑2 2 1

- the weight of a phase shift grows with energy- the number of phase shifts to be considered

grows with energy- estimation:

l k R E Rmax ≈ ⋅ = ⋅2example:R A aPt B= ≈138 2 6. .E eV H= ≈300 11 }

}

lmax ≈ 12

Polar angle representation more intuitive:

rather largescatteringanisotropy

The higher the energy the morepronounced is forward scattering(with backscatteringusually reduced)

chemical sensitivity

4.2 Layer diffraction

r r rK k g Kg z± = + ±( , )||0

back scattering

forwardscattering

K E k gz = − +2 02( )||

r r

each atom scatters into the direction given by thereciprocal lattice vector g:

total amplitude:ψ δ

sclm j

il

ik rlm

g goe h kr e Y K Ml j∝ − =∑ ± +( ) ( ) ( ),

2 11 0r r

reality: intralayer multiple scattering

-the scattering of a certain atom depends on the total wave field impinging on that atom

-this wave field consists of the wave coming from out-side plus that from the neighbours

-even more: there is a con-tribution which depends on the scattering of the very atom itself

=> impinging wave field

A A A X jlm lm l m lml m

l m j

= + ∑0' '

' '

' '

( )

describes all non-outside waves

matrix/vector notation:

[ ] [ ] [ ]{ }A A A X= +0

⇒ − =[ ]{ { }} [ ]A X A1 0

⇒ = − −[ ] [ ]{ { }}A A X0 11

=> the consideration of full intralayermultiple scattering requires the inversion of a huge matrix (for each energy separately)

rang X

N l

{ }

( )max

=

+1 2

(for N atoms per cell)

e g l. . max = 12N = 5

=>rang X{ } = 845

4.3 Surface diffraction

There is multiple interlayer diffractionfor 2 layers

The procedure can be iterated: „Layer doubling“

5. Structural search5.1 Location of intensity maxima <=> structure

Simple example: Top spacing differs from bulk value, all layers scatter equally

M M Mg gi

g g' '+− +−= = ⇒ = +

−A M M e

e

i kd

i kd

∆

∆

r r

r r

1

01r r rg k k ik ei z= ⇒ = +0 2 2∆ ( )

r r r rd d e d d ez z1 1 0 0= =, } ∆r rkd kd ik di1 1 12 2= +

∆r rkd kd ik di0 0 02 2= +

⇒ =+ −

−AM

be aeae

ikd ikd

ikd1

1

2 2

2

1 0

0

a e b ek d k di i= =− −2 20 1; ; k V ki i= 0 0/

for d d a b1 0= =( ):

I AM a a kd

= =+ −

2

20

11 2 2cos

Bragg-Maxima at

k n dn = π / 0

E n dn = 12

20

2( / )π

for d d d a b kk

dd

a EE

kk

n

n

n

n

n

n1 0

0

21 2= + ≈ ≈ − − ⇔ =δ δ δ δ δ( ): ( )

Example: d A a u0 196 3 7= ≈. . . .V eV Hi0 5 018= − = − .δd d/ 0 5%= −

Pt(100)

E eV H= ≈350 13 ⇒ =δE eV2 0,E eV H= ≈90 3 3. ⇒ =δE eV1 5,

Note: − δE E dependsonV via ai/ 0− ≠ =δE E const because a a E/ ( )

- for a reduced one needs a shorterd d( )δ < 0wave length to reach constructive interfrence

δd < 0 δE > 0

Pt(100) - kinematic (V = -5eV)0iSimilar shiftsfor full dyna-mical spectra

when peaks overlap groups of peaks can change their spectral appearance

NOTE: The situation becomes even more complex when -the top layer scatters differently from subsurface layers

(vibrations, different chemical species)- there are multilayer relaxations (as usually is the case)

5.2 R-factors and error estimation

To find the correct structure, the exp. spectra must be reproduced by model calculations

⇒one needs a quantitative measure to compare spectra

⇒R-factors (”reliability factors”,

as widely used in XRD)

At the time being, mainly two different R-factors are used:

a) mean square deviation

Similar mean square deviation,though spectral features are betterreproduced on the left.

b) Pendry R-factorConceptional idea: The location of both maxima and minima reflect the structure because they mirror de- and constructive interference. The height of maxima is less important, all maxima (small or high) are equally important!!

=> use transformed spectra

I EI

dIdE

L E( ) ( )→ =1or better, to avoid singularities

Y E L EV Li

( ) ( )( )

=+1 0

2

RY Y

Y YPg E g E

th

g E g Eth

g E

=−+∑

( )( ) ( )

,exp

,

,exp

,,

2

2 2

Perfect agreement:

Y Y RthP

exp = ⇒ = 0

No exp/th-correlation:

Y Y RthP

exp∑ = ⇒ =0 1

exp/th-anticorrelation:

Y Y RthP

exp = − ⇒ = 2

number of peaks:N E Voi= ∆ / 4N E Voi= ∆ / 4

multiple diffraction => more peaks => higher precision

R-factor R = R( ) ))= deviation in positions when p is a model

parameter: ) = (p))

Even for correct structure,R>0 results (errors in exp.and theory!)

When errors meet a Gaussianprobability distribution, theStandard deviation is

var( )/

R RN

RVEm moi= =

12

8∆

var( )/

R RN

RVEm moi= =

12

8∆

pstatistical error of

)p )

var( )/

R RN

RVEm moi= =

12

8∆

var( )/

R RN

RVEm moi= =

12

8∆

=> the larger the data base, the smaller the error

When N parameters must be determined, one needs at least N independent data points,that is, N peaks. Each peak has a full width 4Voi, so the minimum data base is 4NVoi.

Yet, one needs much more data because of errors of measurement and parameterCorrelations. Experience tells, that one shoud have five as many data points(5 = „redundance factor“)

So, for e.g. N=20 and Voi=5eV: ∆E = 5x400 eV = 2000 eV

Note: the data base scales with the number of beams, i.e. inversely with the unit mesh size!

BUT: if M values have to be tried for each parameter, the number of trial structures is

M = 10 trial structures !!N 20

=> structural search strategies needed!

5.3 Automated structural search

Fact: for many parameters the R-factor is a hyper-surface with many minima and there is no criterion whether or not a minimum is local or absolute!

p pN1....R p pN( .... )1

Example: Rh(100)-c(4x2)Cs

„multiple coincidences“

Example: Ir(110)-(1x2)-missing row

Structural search by simulated annealing:Concept:- jump,statistically around in parameter space- determine R-factor at each landing- if R has decreased accept jump, if not try another one- start from different starting points in parameter space- reduce jump distance with decreasing R-factor

(but allow for occasional long-distance jumps)

Pt(110)-(2x1)-missing row

full symbols: successful jumpsopen symbols: trial jumps, discarded search from different

starting points

5.4 Accuracy and Precision

Precision: What are the error limits of the method?

=> Precision (by variance of R-factor): a = 2.545±0.014 Å d= 1.770±0.015 Å

Accuracy: Are the parameters determined correct?

Cu(100)

(Could only be answered if one would knew the correct value!!)

A) compare with results of other groups

B) compare with inde-pendent methods and/ortheory

Strong versus weak scatterers

error for b : ± 0.08 ÅC

± 0.08 Åerror for b : ± 0.04 Å

The challenge: (1x1)H/W(110)

Note: Detection of H via LEED is only possible, when- a low R-factor is achieved- no serious substrate reconstruction is induced

6. Tensor LEEDExperience: if structural parametersare changed gradually, the spectra change gradually, too

Can we get the changing spectra by the perturbation of a reference spectrum?

6.1 Scattering of a displaced atom

Scattering factor of the undisplaced atom: t j

Scattering factor of the displaced atom: t t t rj j j j' ( )= + δ δr

How to calculate ?δ δt rj j( )r

In short notation: δ δ δ δt r P r t P r tj j j j j j( ) ( ) ( )r r r= −

propagate to the new position!

scatter! propagate back to the old position!

(All this is done in angular momentum space, so the quantities P and t are matrices)

6.2 Scattering of the surface

The atomic displacement cause a change in the scattering amplitude of the total surface. In 1. order perturbation this can be written as δ ψ δ ψA tj f j i=< >| |

In angular momentum representation this writes as δ δA T tj jLLLL

jLL= ∑ ''

'

with the tensor T depending only on the unperturbed structure (=reference structure)

For many displaced atoms: δ δA T tjLLj LL

jLL= ∑ '; '

'(This is 1. Born (i.e. kinematic) approximation with respect to the scattering atδt j

The intensity of the new structure is simply:

I A A= +02δ

Once the Tensor has been computed, the new intensity results just by matrix multiplications

In many cases atoms can be displaced byas much as 0.5 Å offthe position in thereference structure

6.3 Examples for geometrical Tensor LEED

Ir(100)-(5x1)-hex

6.4 Chemical Tensor LEED Or: Can a scatterer be replaced by another one?

=>

′ =t tB′ =t tB

δt t tB A= −δt t tB A= −

R=1.14

Substitutionally disordered alloys, here: Mo Re (100)X 1-X

6.5 Thermal Tensor LEED

When the atoms vibrate statistically independently from each other this can be described by a “Debye-Waller-Factor”:

′ = −t t e M T( ; )Θ′ = −t t e M T( ; )Θ

the larger the vibrational amplitude (i.e. the higher the temperature T or the smaller the Debye-Temperature ) the larger is MΘ

The scattering change with temperature or Debye-temperature can be described by Tensor LEED:

=> δt t t= ′ −δt t t′ −

One can determine the layer-dependent Debye-temperature:

best fit: 1 = 250K1 = 350K

12

( = 440K)1b

7. Direct methods

Our lack of imagination for the correct MODEL CLASS is THE obstacle for the solution of complex structures.

As long as we have no modelwe cannot vary parameters!

=>All thesemodelsagree withtheLEEDpattern

We have a lotof spots andtheir Intensitiesbut cannotuse themwhen there isno model!

The holographic approach

BUT: Multiple scattering processes must be filtered out

Idea : (Tong & Li PRL 67 (’91) 3102, Barton ibid. 3106)

Use a stationary phase for the ideal process and integrate over many energies

Phase of ideal process:

k r k ri i−r r

⇒ = − ⊥∫A r I k i kr kr dk dk( ) ( ) exp{ ( )} ||r r rr r

Holographic reconstructionYet, only atoms near the beam-splitter atomcan bee reconstructed (electr. attenuation!)

Conventionally completed structure:

PRL 80 (1998) 758PRL 79 (1997) 4818

Holographic approach using the bulk contribution as reference

Milwaukee-Erlangen PRL 88 (’02) 115507

fit to exp. intensities via maximum-entropy method

Literature

Pendry, Low Energy Electron Diffraction

Academic Press, London 1974

Van Hove & Tong ,Surface Crystallography by LEED

Springer, 1979

Van Hove & Weinberg &Chen , Low Energy Electron Diffraction

Springer, 1979

Woodruff & Delchar,Modern Techniques of Surface Science

Cambridge University Press, 1986

Heinz: LEED and DLEED as modern tools of quantitative surface structure determination

Rep. Progr. Phys. 58 (1995) 637-704