Low-Energy Electron Diffraction (LEED)

LEED is (still) themost frequentlyused surface-structural method

Why??

surface sensitivity and wavelength at the same energyideal

Me

an

fre

epa

th(Å

)

1

5

10

50

electron energy (eV)10 100 1000

LEED

E = 150 eV => = 1�

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

Penetrationabout 10Å=

>ideal surfacesensitivity

ideal because of the order of atomicspacings => large diffraction angles

1. Introduction

fromP.R. Watson,M.A. Van HoveK. Hermann,NIST Surface StructureDatabase, Version 5.0

, Low Energy Electron Diffraction, Academic Press 1974J.B. Pendry

M.A. Van Hove, W.-H. Weinberg, C.-M. Chan, Low-Energy Electron Diffraction,Springer 1986

K. Heinz, LEED and DLEED as modern tools for quantitative surface structuredetermination, Rep. Progr. Phys. 58 (1995) 637

01 02

1897 “Discovery” of electron beams (J.J. Thomson)

Historical in short

1927 Prove of electron diffraction at anatomic lattice (Ni(111)-surface)

(C.J. Davisson /L.H. Germer)

1924 Wave mechanics postulated (L. de Broglie) = h/pë

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

Faraday cup

(1937 Nobel Prize for Davisson and Thomson)

I( ) at E = 54 eV� �������

(3-fold rotational symmetry)

from:C.J. Davisson,

, Dec 13, 1937Nobel Lecture

(experiments carried out in 1927)

Ni sample(crystallizedafter “accident”)

until 1960 no further significant development!

Why??

LEED is dominated by multiple (=dynamic) scattering

XRD is dominated by single (= kinematic) scattering

OR: Full SGL must be solved

OR: 1st Born approximation of SGL is sufficient

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

03 042. Experimental (again in very short)

Needed: electron gun + (clean) sample + energy dispersive detector

request for UHV

pri

mary

peakdiscrete losses

(phonons, plasmons, Auger effect)

seco

nd

ary

ele

ctr

on

s

energy

only about 1% of electronsis elastically scattered

"(any TV sethas one)

easy"

luminescent

screen

screenhigh voltage

suppressorvoltage

sample

primarybeam

diffractedbeamsLEED optics

Since about 1960:electron gun and detector are accom-modated in a so called LEED opticswith electrostatic grids to repell inelasticelectrons and a screen to make elasticelectrons visible ( ):display analysator

Lst. Festkörperphysik / Erl.-Nbg.

LEED opticsin UHV vessel

commercial optics

TV

LEED optics (UHV) video data acquisition

3.Basics of diffraction-pattern formationRemember:

e-gun

sam

ple

win

dow

spo

tin

ten

sity

electron enenergy

LEED spectrum

unitcell

lattice (2-atomic basis)incident wave

A eikr

0

�

�

x

y

�

k

�

�k

= scattering factor ofatom at

�

ri

t r k k ti i

( , , )�

� �

� �

from to�

k

�

�k

=> amplitude before scattering at :�

ri

A eikri

0

�

�

=> amplitude at detector:

At A e e

R ri

i

ikr ik R r

i

i i

��

� �

0

�

�

� �

�

�

�

( )

�

r

=> amplitude after scattering at :�

ri t A e

i

ikri

0

�

�

�

R

�

�k

�

�

R ri

�

detector

eik R� �

�= only constant phase factor;

�

�

R r Ri

� � ;� � �

k k k'� � �

= momentum transfer

A

A

e

Rt e

i

ik R

i

i kri

0

��

�

� �

�

�

�=>Note:

� �

k k� �

(diffraction is equivalent to coherentscattering)elastic

05 06

A

A

e

Re t e

ik R

i kR

R

i

i ki

i

i

i0

��

� �� �

� �

�� �

�

� � �

�

I

I

A

A Re t e

i kR

R

i

i ki

i

i

i0 0

2

2

2 2

1� � � �� �� �

�� �

�

�

�} }

lattice factor G structure factor (form factor)

G e e ei ka n

n

N

i ka n

n

N

i ka n

n

N

� �

�

�

�

�

�

� � �( ) ( ) ( )� � ��

�

�

�

�

�

1

1

2

2

3

3

1

2

1

2

1

2

=>�

� � �

R n a n a n ai � � �1 1 2 2 3 3

�

� �

ga a

VEZ

12 32�

� ;

�

� �

ga a

VEZ

23 12�

� ;

�

� �

ga a

VEZ

31 22�

� ;

��

� � � �

k hg kg g� � � �1 2 3lg “Laue condition”

V a a aEZ � �� � �

1 2 3( )

�

�1�

�2

�

Ri

|Fourier transformation|2

summation over all atoms at�

��

r Ri i i

� � �i unit cellth

Intensity:

When Ni 1 only ifG � 0then ei ka

i�

�

�

� 1 � ���

�

ka ni 2

=>

reciprocal unit-cell vectorsvolume of realspace unit cell

Intensities can appear only in directions given by �� � �

�

k k k g� � �'

�

g1

�

g2

�

g3

(0,0,0)

�

k

�

k '�

g

reciprocal lattice

Ewaldsphere

� �

g3 0

�

a2�

a1

ideal two-dimensionality:only a single atomic layer

�

a3 �

� � � �V a a aEZ

� � �

1 2 3( )

reciprocal latticeconsists of rods

�

g1

�

g2

=>

always

reciprocal latticecuts the

surface of theEwald sphere

For rectangularreal space lattice:

� � � �

ga

e ga

e11

1 22

2

2 2� �

;

surface reality:The LEED beam doesnot only “see” a singlelayer but several l(according to the electronpenetration depth)

ayers

<=

Consequences: - LEED pattern = through the reciprocal latticecut

=> reciprocal lattice“seen” by LEED is amixture betweencontinuous rodsand discrete points

- LEED pattern reflects the and of thereal space unit cell

size shape

- LEED pattern says about the positions ofatoms in the real space unit cell, i.e. the structure

nothing

I

I

A

A Re t e

i kR

R

i

i ki

i

i

i0 0

2

2

2 2

1� � � �� �� �

�� �

�

�

� }

structure factor

The atomic positionsare “buried” in thediffraction intensitiesdetermined by thestructure factor

This is why surface structure determination by LEEDrequires the measurement and analysis of intensities.

07 08

09 105. Atomic scattering

t E l e P tk l

i

ll ll

l( , ) ( ) sin (cos )� � �� �� � � �� �2 2 1

is also denoted by (diagonal)scattering matrixt E tll lll l

( , ) ' ', '� �� �

atom

�representation by phase shifts

The electron energy is high above the Fermi-energy:

�

��

l l

ikr

l

ikr

re

r

e

r( ) � �

�

2 2incoming outgoing

� �

l l

ie l� 2 �

lphase shift�

(simplifiedpicture)

8

4

010-10

8

4

0

40302010

4

05040302010

�

35 eV

100 eV

4

0

5040302010

300 eV

600 eV

Pt1.0

0.01.00.0

3

2

1

0

86420-2-4

6

5

4

3

2

1

0

151050-5

5

4

3

2

1

0

151050-5-10

10

8

6

4

2

0

403020100

1H

6C

16S

26Fe

77Ir

scattering factor

�

k

� t E( , )�

�

k '

=> Scattering is mainly by inner shells and nucleus

=> Scattering potential is spherically symmetric

=> Represent in- and outgoing wave by a set ofsespherical waves. U angular momentum conservation.

=> Solve Schrödinger equ. with ansatz

100 eV

-4

-3

-2

-1

0

1

2

phase

shift

600500400300200100

energy (eV)

l =0 1 2

34 5

6 78 9

( )r �

4. Structure determination procedure

atomicscattering layer diffraction

a) imagine a structural model (one usually knows the type of theclean surface (e.g. Cu(111)) and the adsorbate (e.g. C H )6 6

b) calculate the intensity spectra of all spots for the model

c) compare the spectra computed with those measured

d) when- the comparison is not satisfying => modify model

- the comparison is satisfying => correct model is found

Howe to calculate intensities?

Solve Schrödinger equation! ???

This would be equivalent to a band structure calculationat high energies!! Forget it !!

Instead: Make use of scattering hierarchy!

full surface diffraction

a) calculate the scattering of the single atom

b) assemble atoms to layers and calculate their diffraction

c) stack layers to build the surface and compute its diffraction

11 12

�

k

full intralayer scattering

no intralayer scattering

500400300200100

energy (eV)

�

k

�

k

6. Layer Diffraction

È

Pt(100): a=0.276 nm

300 eV:400 eV:500 eV:

100 eV:145 eV:200 eV:

È=153.5°

È=158.3°

È=161.6°

È=16 .0°5

È=167.1°

È=16 .5°8

scattering angle varies with energy

Pt(100):layer diffractioninto (10) spot

0.8

0.4

0.010

-sp

ot-

Pt(1

00

)

500400300200100

energy (eV)

100 eV10

86420

100 50

10

8

6

4

2

0403020100

140ev200 eV

8642

3020100

Yet, there is multiple sacttering

wave from outside:total impinging wave:total wave scattered:

A0

A

A X� }A A A X A X A� � � � � � �0 01( )

� � � � �A A X0

11( )

self-consistency <=> matrix inversion

� � � � �A A X0

11( )

rang X l( ) ( )max� � �1 1002

X Xlmk l m k

� , ' ' '

kinematic diffraction (each layer diffracts only once)

+

+ ....+ ....+

+

full dynamical interlayer diffraction

7. Full surface diffraction

Problem: Diffraction amplitude of layers depends onlayer depth due to exponential attenuation

Formal solution: k k ik e e er i

ikr k r ik ri r� � � � �

equivalently: E E iVi

� � 0 <= optical potential

typically: V eVi0 5�

BUT: there is also multiple interlayer diffraction

+ + ...+ ...

+ + ...+ ...

+

+

.010

.005

0500400300200100 eV

(10) spot of Pt(100)I/I0

500400300200100 eV

I/I0 (10) spot of Pt(100)

note: again this multiple diffraction problemis solved by a matrix inversion (withthe layer diffraction matrix involved)

13 148. Multiple scattering and its benefits

The multiple scattering comes by the fact that the atom's cross section forelectron scattering is of the order of its geometrical cross section:

�e atom

cm A� ��10 16 2

=> whenever an electron "meets" an atom it scattered.is

[ : ]XRD cm Aatom

�� � ���10 24 2

Dense atomic packing => strong multiple scattering

=> a dynamical spectrum has muchmore peaks than a kinematicspectrum

=> more structural information

d =20 Å=> 5d /100=0.10 Å

d12

d0

d0

d0

25

20

15

10

5

0

x10

-3

500400300200100

Pt(100)00-spot

energy (eV)

I/Io

(d -d )/d =12 0 0

0%

-5%

�

500400300200100 eV

I/I0 (10) spot of Pt(100)

R-factor: compares two spectra, e.g. by the shifts { } of peaks�

var( )/

R R

Nm

�1

2var( )

/R R

Nm

�1

2

N = total number of peaksR

Rm

var(R)

d12d12error for

the more peaks N, the smaller var(R)=> the smaller the error

errors as low as 0.01Å can be reached

Sensitivity:

Precision:

Erlangen calc.

exp.(10)

100 200 300 400 eV

Precision (by R-factor variance) :

a = 2.545±0.014 Åp

d= 1.770±0.015 Å12 2.50 2.52 2.54 2.56

0.07

0.08

0.09

0.10

0.11

RP

var(R)

statisticalerror width

d12

a (Å)p

simple example: Cu(100)

9. Structural Search

Note: If we have M to be determined with N values for each to try, thenmust be calculated + compared to exp.Z=N sets of intensity spectra

M

even for a simple structure (e.g. M=3, N=10) a huge value results: Z = 103

=> Don't scan parameter space, but apply some search strategy

Concept of simulated annealing:

- 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

-0,4-0,2

0,00,2

0,4-0,4

-0,2

0,00,2

0,40,2

0,4

0,6

0,8

1,0P

endry

R-f

act

or

(Å)

2dÄd

1Ä

(Å)

0.400.50

0.60

0.700.70

0.80

0.800.80

0.90

0.90

0.90

0.90

-0,4 -0,2 0,0 0,2 0,4

-0,4

-0,2

0,0

0,2

0,4

d 1Ä

(Å)

(Å)2dÄ

full symbols: successful jumpsopen symbols: trial jumps, discarded

(Å)2dÄ

d1Ä (Å)

start fromdifferentpoints

15 1610. Tensor LEED

0 100 200 300 400

d12 (Å)Ni(100) (1 1)

1.56

1.66

1.76

1.86

1.96

(eV)

Experience: when structural parameters are changed gradually,the spectra change gradually, too

d12

Ni(100)

Can we get the changing

spectra by the perturbation

of a reference spectrum?

Scattering of a displaced atom

��

rj Scattering factor of the undisplaced atom: t

j

Scattering factor ofthe atom:displaced

In short notation:

t t t rj j j j' ( )� � � �

�

� � � �t r P r t P r tj j j j j j( ) ( ) ( )

� � �

� � �

propagate to thenew position!

scatter!propagate back tothe old position!

change of the total amplitide (1. order perturbation): � � � �A tj f j i

�� �| |

For many dis-placed atoms:

In angular momentum representation this writes as � �A T tj jLL

LL

jLL� � '

'

'

with the depending only on the unperturbed structuretensor T

� �A T tjLL

j LL

jLL� � '

; '

' I A A� �0

2�

Once the tensor is computed, the new intensity results just

by matrix multiplications for many test structures

Even with a structural search applied many structures need to becalculated from scratch - though they might be rather close

new intensity:

0 100 200 300 400

d12 (Å)Ni(100) (1 1)

1.56

1.66

(ref.)

1.76

1.86

1.96

(eV)

full dyn. TensorLEED

In many cases atoms can be displaced byas much as 0.5 off the position in thereference structure

Å

(Note: By such displacementsbonds are usually broken)

even though the spectra can changedramatically, 1. order perturbationis sufficient

Chemical Tensor LEED

5

4

3

2

1

0151050-5-10

OS

100 eV

Can a scatterer be replaced by another one?

x 4

(1/2 1/2)-spot

p (2 x2 )O -1 .0 0

p (2 x2 )S -1 .2 5

p (2 x 2 )S -1 .0 0

0 1 0 0 2 0 0 3 0 0 4 0 0 (e V )

full dynamicalTLEED

Å

Å

Å

O/NiO/Ni

S/NiS/NiS/Ni

TLEEDfull dynamical

reference

A A A AB

�t tB

�t t tB A

� � �t tB

�t t tB A

� �

17 18

Br/Pt(110)-(3x1)-2Br

dBr

d12

d23

d34

d45

bBr

b1b2b3b4

50 100 150 200 250 300 350

Inte

nsi

ty(a

.u.) (1,0) calc.

exp.

energy (eV)

expcalc

inte

nsi

ty(a

.u.)

energy (eV)100 200 300 400

( 0) spot35

20 Å

FeFe

Ir

IrFe Fe Fe FeIr

FeIrFe-nanostructureon Ir(100)

10 parameters determined

16 parameters determined Europhys. Lett. 65 (2004) 830

Phys. Rev. B 69 (2004) 195405

13

b1

13

d12

d23

d34

b1

2 3' '

b3

2 3'

b4

13'

b1

34

b2

13

b3

23

b4

23

p1

3p1

2

p2

3p2

2

42'

1

31

12 3 2'

2 3 3'2' 1

p1

4p1

2'

p1

3' p1

1

1'

p1

1

p2

2'p2

3'p2

1 p2

1

d45

b1

23

b2

23

b1

43'

b1

13'

b2

2 3' 'b2

13'

b3

13b3

1 3'

1b4

13

b4

2 3' '

3'

3'

2'21

2

0 100 200 300 400energy (eV)

expcalc

inte

nsi

ty(a

.u.)

3H / Ir(100)-(5x1)-hex

33 parameters determined Phys. Rev. B (submitted)

11. Examples

3x3 unit mesh

27

8 410

56 9 129

612

1

3

0.13AO

0.04AO

0.27AO

1.45AO

1.04AO

2.34AO

0.63AO

48

5

9

1

10

2 12

37 6

�

12

200100200100energy [eV]

Inte

nsi

ty[a

rb.

un

its]

expcalc

(01) ( )2 32 3/ /

6H-SiC(0001)-(3x3)

� 100 parameters determined

Phys. Rev. B 62 (2000) 10335Phys. Rev. Lett. 79 (1997) 4818 + 80 (1998) 758

demanding data anlaysis(strong multiple scattering)

Rather demanding experimentalset-up (synchrotron)

LEED XRD

12. LEED vs XRD

Relatively simple and cheapexperimental set-up ( 100 k€)�

Advantages

Easy, i.e. kinematic data analysis(no multiple scattering)

Drawbacks

High surface sensitivity

Easy information on symmetryand shape of surface unit-cell

Atomic structure can be retrievedwith high accuracy

UHV essential

no insulators accessible

electron stimulated processesmay take place

No problems with surface charging(access to insulators)

High structural accuracy

Access to buried interfaces

surface sensitivity by slantingincidence - high accuracy needed