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