Bethe theory of low energy electron diffraction applied to computation of diffracted beam intensity as function of wavelengt

Smith, H. P., Jr.

English

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Space Sciences Laboratory 
University of California 
Berkel ey , Cal i f  orni a 94720 
DYNAMICAL THEORY OF LOW ENERGY 
ELECTRON DIFFRACTION 
Final Report on 
NASA Grant 
NGR 05-003-275 
Principal Investigator: Professor Harold P.  Smi th ,  J r .  
Space Sciences Laboratory Series 10, Issue 30 
2 
Abstrac t  
An extended version of the  Bethe theory of low energy 
electron d i f f r ac t ion  (LEED) is presented and successfully 
applied t o  a computation of d i f f r ac t ed  beam in t ens i ty  as a 
function of wavelength of the incident e lectron.  The 
r e s u l t s  show consis tent  behavior with respect  t o  parameter 
var ia t ion  and are i n  reasonable agreement with measured 
beam i n t e n s i t i e s .  
I. INTRODUCTION 
The theory of Low Energy Electron Diffract ion (LEED) has been the  
object  of a number of recent  investigations.  Most of these s t r e s s  
the importance of dynamical e f f e c t s  and point  out the inadequacy cf 
simple kinematic theory t o  describe accurately the dependence of the  
d i f f r ac t ed  beam i n t e n s i t i e s  on electron energy. 
of LEED t h a t  have been proposed so f a r  may be divided in to  two broad 
categories:  ' (1) self-consis tent  f i e l d  methods, and (2) "matching" 
formalisms . 2-5 
belongs to  the  lat ter group. I t  assumes t h a t  the  incident and the  
d i f f r ac t ed  electron beams are plane waves and t h a t  the c r y s t a l  can be 
represented by a three-dimensional, semi-infinite,  periodic poten t ia l ,  
cut  off  abruptly a t  a plane surface. The present paper cons is t s  of three 
par t s :  f i r s t ,  a b r i e f  review of the theory; then, a discussion of 
r e s u l t s  of beam in t ens i ty  calculat ions fo r  a fcc  l ak t i ce ;  and f ina l ly ,  
a comparison with recent  experimental r e su l t s .  
The dynamical theories  
6 O u r  method, which is an extension of the Bethe theory, 
7 
3 
11. THEORY 
Consider a per fec t  c r y s t a l ,  represented by a complex, per iodic  
-f 
potent ia l  , W (r) , 
-t 
W ( r )  = 0 
-f 
where b is  the reciprocal  l a t t i c e  vector. The wave function, 
$(r)  , inside the c r y s t a l  (i.e. , for  z < 0) may be wri t ten i n  the  form 
+ R,m,n 
and 'R,m,n where the in t eg ra l  extends over the f i r s t  Bri l louin zone, 
is  a function of the vector p. 
e lectron beam, represented by a plane wave, t h e  Bloch theorem may be 
invoked fo r  the two surface dimensions (x and y ) .  Thus, equation (2) 
-t 
In  the case of a s ingle  incident  
becomes 
-t 
where the project ion of p onto the surface normal is  a function of the 
sca la r  quant i ty ,  q. However, the components of p p a r a l l e l  t o  the 
surface a re  independent of q. The in t eg ra l  i n  equation ( 3 )  may be 
-f 
4 
approximated by a sum, 
-f -t -f 
and, f i n a l l y ,  a vector k = p + 2Tb may be defined 
such t h a t  
R,m,n,q q R,m,n 
Equation ( 5 )  is  a so lu t ion  of the Schrodinger equation, 
-9- -f v2$(z) + (K2 + W ( r ) ) l l f ( r )  = 0,  i f  the  coe f f i c i en t s  llf s a t i s f y  the R r m  In ,q 
compatibi l i ty  r e l a t i o n s ,  
R’,m’,n’ 
o r  i n  matrix form, 
K2$ - + Allf - = 0 
5 
where $ is  
and A is a 
d i f f r ac t ed  
vacuum may 
,., a vector whose 
square matrix. 
e lec t ron  beams 
be wr i t ten  as 
T- 
components are the coe f f i c i en t s  $ 
,m,n ,q' 
Assuming t h a t  both t h e  incident  and the 
are plane waves, the wave function i n  the  
where 
z > o  - 
The cont inui ty  of the wave function and i t s  der iva t ive  with 
respec t  t o  the surface normal (2-direct ion) ,  a t  the surface,  z = 0 ,  
may be s t a t e d  as 
6 
3 
where 1 is a u n i t  vector i n  the  z-direction. Equations (10) and (11) 
must be s a t i s f i e d  f o r  a l l  values of x and y, and, therefore ,  
z 
-7 7 
k R,m,n,q 'x .& R,m,n -x I 
A t  this point ,  it is convenient t o  consider a spec i f ic  case,  
e.g., a cubic l a t t i c e ,  i n  order t o  simplify the  equations. This can be 
done without l o s s  of genera l i ty ,  s ince the  extension of the treatment 
t o  other  la t t ices  is straightforward. For a cubic l a t t i c e ,  with 
constant a ,  w e  have, 
Combining equations (12) and ( 1 3 )  yie lds  
3 -+ 2mR 
hfi,m,n 'x = px + 7 
-+ -+ 2mm + -  h R,m,n * l y  e py a 
3 
where p and p are the  x and y components o f  p (which.are independent 
of q) , and equation (9) then reads X Y q 
7 
The index, n,  i n  the  above equation may be dropped, and the  wave 
function i n  the  vacuum, equation (8) , is wr i t ten  as 
- + ?  
where h is a vector whose components are iden t i ca l  t o  those of 
h [equations (14) and (15)], with the exception t h a t  the  super- 
s c r i p t  i n  h indicates  the  sign of i t s  z-component. I f  there  i s  
only one incident  e lectron beam, whose amplitude, 0- 
unity,  equation (16) reduces t o  
+- R r m  
R,m,n -+ If: 
r m  
, is equal t o  
010 
where the  f i r s t  and second terms on the  r i g h t  hand s ide  represent  t he  
incident  and the  d i f f r ac t ed  electron beams, respect ively.  
It  follows from equations (14) and (17) t h a t  the real p a r t s  of 
px and py are determined by the  d i rec t ion  of incidence of the primary 
electron beam, whereas t h e i r  imaginary p a r t s  must be zero, due t o  the  
boundary conditions a t  i n f in i ty .  The z-components of the vectors  
-% 
on the  other  hand, do not depend on the  boundary conditions.  Their 
pqr 
real p a r t s  may, i n  pr inc ip le ,  be chosen a r b i t r a r i l y ,  within the  in t e rva l  
1 0 < Re(PqJ < 2 and are usual ly  assumed t o  be uniformly d is t r ibu ted  
throughout t h a t  i n t e rva l .  I f  however, the  number of values of q, 
(i.e.,  the  number of p 'SI is s m a l l ,  a uniform d i s t r ibu t ion  may not 
92 
8 
be the optimum choice. 
for the real parts of the p 
will be discussed below. 
Methods for finding the best possible values 
' s ,  as well as their imaginary parts, 
92 
Introducing equation (17) into equations (10) and (ll), and 
decomposing into a set of equations relating the coefficients of the 
various harmonics in x and y ,  leads to 
R # O  or m # O  
9 
The quantity, 0' may be eliminated from equations (18) and (19) , R ,m' 
and we obtain 
R # O  or m # O  
R = m = O  
or in matrix form, 
The compatibility equations (7) and the boundary condition 
equations (21) may be combined into one single equation, 
G$ = H - -  (22) 
-+ 
If the series representing the potential W(r) and the electron wave 
function in the crystal and in vacuum are approximated by truncation, 
the matrix G has more rows than columns and, therefore, equation (22) 
has no exact solution. The best solution is found, according to the 
principle of least squares, by minimizing 
10 
S = (Gt - H )  D(G$ - H ) *  - - -  (23) 
where D is a positive, real, diagonal matrix, containing suitable 
weighting coefficients, and * indicates complex conjuqation. Since S 
is positive and quadratic in any of the unknowns, Re($$ 
~ ~ ( J I R ,  m , n, g 
) and 
,m,n,q 
) ,  the minimum is characterized by the conditions 
or, equivalently, in matrix form, 
P$ = Q 
? ' y  
where P is now a square matrix. Equation (25) is solved, not by 
straightforward inversion of P, but by an iterative technique, which is 
not affected by round-off errors during the computation. 
It should be noted that the elements of both P and Q contain the - + 
components of the vectors p . As was pointed out above, the x and y 
components of these vectors are independent of q and determined by the 
angles of incidence of the primary electron beam. If the z-components 
of the p ' s  are treated as unknowns, they are determined by solving the 
q 
equations 
q 
-+ 
simultaneously with equation ( 2 5 ) .  I f  we choose t o  represent  the wave 
function i n  the  c r y s t a l  by a s ingle  Bloch wave, i .e . ,  i f  w e  consider 
only one q, we f ind  t h a t  R e  p i s  very near ly  equal t o  
( 9 1  
I ]  1'2 O,O?O 
, whereas I m  p approaches - [K2 + Re(w O r 0 , O  1 J ( q,] 
f o r  la rge  values of Im(w ) . 
Obviously, once the  vector  IJJ is known, the  i n t e n s i t i e s  of the  d i f f r a c t -  
ed beams, i .e.,  lQl,ml 2, are computed r ead i ly  from equations (18) and 
(19) - 
- 
111. BEAM INTENSITY CALCULATIONS 
Using t h e  theory out l ined  above, ca lcu la t ions  were performed under 
the  following conditions:  
(1) A primary e lec t ron  beam of u n i t  i n t e n s i t y  is  inc ident  
normally on the  (100) surface of a fcc  l a t t i c e ,  with 
la t t ice  constant ,  a = 4.04 A (aluminum) . 0 
(2) The c r y s t a l  lat t ice is represented by a complex po ten t i a l .  
The shape of i t s  real part, i.e.,  t he  r e l a t i v e  magnitude 
[equation (111, i s  of the  expansion coe f f i c i en t s ,  
obtained by Fourier-analyzing the  "se l f  consistent ' '  
WR,m,n 
12 
aluminum po ten t i a l  given i n  reference 8. 
includes a l l  t e r m s  fo r  which R2 + m2 + n2 < 4 ,  and the  
imaginary p a r t  of the  po ten t i a l  is assumed independent of 
posi t ion and electron energy. 
The expansion 
- 
( 3 )  The c r y s t a l  surface,  where the  po ten t i a l  abruptly drops t o  
zero, l i es  half-way between two layers  of atoms. 
(4 )  The wave function i n  the c r y s t a l  is  approximated by a 
-f 
s ing le  Bloch wave [only one p i n  equation ( 5 ) ]  whose expansion 
i s  truncated such t h a t  R2 + m 
I 
7 + n2 < 34. 
Figure  1.  i l l u s t r a t e s  the  dependence of the beam in t ens i ty  vs  
energy curves on the  assumed inner po ten t ia l  [Re(wo ,O,o~]  . Decreasing 
t h e  inner  po ten t i a l  by 2.5 v o l t s  t r ans l a t e s  the curves t o  a commensu- 
rate,  higher energy. 
Figure  2.  shows t h e  dependence of the beam i n t e n s i t i e s  on the  
magnitude of the osc i l l a to ry  p a r t  of the poten t ia l ,  i .e . ,  on the  
e l a s t i c  s ca t t e r ing  c ross  section. The la t te r  quant i ty  is var ied by 
introducing a scal ing fac tor  t h a t  mul t ip l ies  a l l  coe f f i c i en t s ,  
W except w For s m a l l  c ross  sect ions,  only the ordinary 
Bragg peaks are present.  A s  the  cross  sect ion is made l a rge r ,  secondary 
Bragg peaks and resonance phenomena, due t o  multiple s ca t t e r ing  e f f e c t s ,  
appear. 
R,m,n '  0,010 
Figure 3 .  demonstrates the dependence of the  in t ens i ty  curves on 
the  magnitude of t he  imaginary poten t ia l ,  i .e . ,  on the i n e l a s t i c  
sca t te r ing  cross  section. A s  the  imaginary po ten t i a l  decreases,  the  
amount of "s t ructure"  (peak-to-valley r a t i o s )  increases considerably. 
This behavior is t o  be expected. A decrease i n  the  i n e l a s t i c  
s ca t t e r ing  enhances the  penetration of the primary beam and, therefore ,  
enhances a l l  three-dimensional e f fec ts .  Clearly,  the very existence 
of Bragg peaks o r  s t ruc tu re  is a three-dimensional e f f e c t  par excellence. 
The disappearance of t h e  secondary Bragg peaks fo r  la rge  values of t h e  
imaginary po ten t i a l ,  as seen i n  Figure 3 ,  indicates  t h a t  these peaks 
may be a t t r i bu ted  t o  multiple sca t te r ing  events involving atoms i n  
d i f f e r e n t  layers ,  ra ther  than atoms i n  a s ingle  layer .  
13 
IV. COMPARISON WITH EXPERIMENT 
Beam in t ens i ty  measurements on the  (100) surface of aluminum s ingle  
c r y s t a l s  are being performed by D r .  S. M. Bedair a t  our laboratory.  
Figure 4 compares h i s  measurements7 with a typical set of our theo re t i ca l  
curves. All three experimental curves are normalized t o  the  same 
primary electron beam current .  
are, therefore ,  represented cor rec t ly .  
In tens i ty  r a t i o s  between d i f f e r e n t  beams 
When considering t h i s  comparison between theory and experiment 
(Figure 41, the  following poin ts  should be borne i n  mind: 
N o  attempt has been made t o  optimize the  parameters of t h e  
po ten t i a l ,  i .e.,  t he  expansion coef f ic ien ts  w R,m,n' 
The imaginary p a r t  of t he  poten t ia l  has been assumed constant,  
whereas it should vary with electron energy in  order t o  
represent  the  known var ia t ion  of i n e l a s t i c  s ca t t e r ing  cross  
sect ions with energy. 
The theo re t i ca l  r e s u l t s  reported here are based on a s ingle  
Bloch function representation. I t  remains t o  be invest igated 
whether t h i s  is a good approximation or there  are cases 
where a more rigorous multi-Bloch function treatment must be 
used. 
A thorough parameter optimization, along the  l i n e s  indicated above, 
might improve the  agreement between theory and experiment considerably. 
This would, however, represent a very la rge  investment i n  computer time. 
V. CONCLUSIONS 
W e  have shown t h a t  an extended Bethe theory (Section I1 of t h i s  
paper) may be applied successfully t o  the  computation of LEED intensi-  
t ies.  The r e s u l t s  are i n  reasonable agreement with measured7 beam 
i n t e n s i t i e s  (Figure 4), although we have used a s ingle  Bloch  function 
approximation i n  order t o  avoid an excessively lengthy ca lcu la t ion  and 
1 4  
there  are a t  least two l imi t a t ions  inherent  i n  our model t h a t  have t o  
be ser ious ly  questi0ned.l These are the  uses of an incident  plane 
wave and the  assumed pe r fec t  per iodic i ty  with abrupt cutoff a t  t he  
c r y s t a l  surface.  
References 
'E. G. McRae, J. Chem. Phys. 45, 3258 (1966). 
2K. Hirabayashi and Y. Takeishi, Surface S c i .  4 ,  150 (1966). 
3D. S. Boudreaux and V. Heine, Surface Sc i .  8, 426 (1967). 
4F. Hofmann and Harold P. Smith, Jr., Phys. Rev. Letters 19, 1472 
(1967). 
5P. M. Marcus and D. W. Jepsen, Bull. Am. Phys. SOC., Series 11, 13, 
367 (1968). 
6H. Bethe, Ann. Physik 87, 55 (1928). 
'S. M. Bedair, p r iva t e  communication. 
*E. C. Snow, Phys. Rev. 158, 683 (1967). 
PARTICIPATING PERSONNEL 
Professor Harold P. Smith, Jr. -P r inc ipa l  Invest igator  
Ferdinand Hofmann 
Yeong-du Song - Research Assis tant  
I 
- Assis tant  Research Engineer 
15  
I. I I I I I I 1 
(ole) BEAM - 
N =5  
- 
/ 
---- - 
id4 I I I I I I 
IO' I 1 I I I I 
icT2 - - 
lo3 - - 
( I  I )  BEAM 
- - 
--- 
I& I I 1 I 1 I 
IO2 I I I I I I 
io3- ~ - 
lo4 - - 
io5 - - 
IO6 I I I I I I 
- Re(wolo,ol = IO VOLTS 
_ _ _ _  Re(VVOpIO) = 7.5 VOLTS 
10 20 30 4 0 .  50 60 70 80 
cn 
2 
Q 
W 
rn 
c3 
W 
I- o 
LL 
2 
4 
n 
LL 
0 
> 
cn z 
W 
I- 
W > 
!= 
z 
i= 
4 
W 
FIGURF. 1. Calculated beam i n t e n s i t i e s  vs  energy, showing the  de- 
pendence on the  inner  po ten t i a l .  Conditions: imaginary p a r t  of 
p o t e n t i a l  Im(wo,o,o) = 2.5 v o l t s  and e l a s t i c  s c a t t e r i n g  c ross  
sec t ion  sca l ing  f ac to r  = 1.0. 
of order  N where the  inner  p o t e n t i a l  cor rec t ion  has been included. 
Beams with mixed ind ices  are absent due t o  symmetry of t he  fcc 
l a t t i c e .  
The arrows ind ica t e  Bragg r e l ec t ions  
16 
I 
Id' 
IOi 
16: 
1 I I 1 I I 
(0,O) BEAM 
N=4 N=5 
I I I I I 5 la 
2 102- - 
& rb3- - 
9 lo4- I 
16' I I I I I I 
( I ,  I 1 BEAM c3 W 
I- 
fE 
4 
c3 
). 
cn t 
z 
w 
-
IO 20 . 30 40 50 60 70 80 
PRIMARY BEAM ENERGY, eV 
FIGURE 2. Calculated beam i n t e n s i t i e s  vs energy, showing the  
dependence on the  e l a s t i c  s c a t t e r i n g  cross  sec t ion  ( sca l ing  f a c t o r ) .  
Conditions: inner  p o t e n t i a l  Re(Wo,o,o) = 10 v o l t s ,  imaginary p a r t  
of p o t e n t i a l  Im(wo,o,o)  = 2.5 vo l t s .  
f l ec t ions  i s  ind ica ted  by N. The letters S and R designate 
secondary Bragg and resonance peaks. 
The order  of the  Bragg re-  
17 
1 
(0,O) BEAM N=4 
N=5 
U I I 1 I I I I 
4 a 
LL 
0 
>- 
v, z 
I- 
P z 
w > 
i= 
5 
w 
rK 
PRIMARY BEAM ENERGY , e V  
FIGURE 3. Calculated beam intensities vs energy, showing the 
dependence on the imaginary potential. Conditions: inner 
potential R ~ ( W ~ , ~ , ~ )  = 10 volts, scaling factor = 0.707. The 
letter N indicates the order of the Bragg reflections, and S 
denotes a secondary Bragg peak. 
18 
.- I 
0 I- 
( I ,  I )  BEAM 
lo3 
104 - 
r 1 I 1 I I I 
(0,2) BEAM 
lo5 
- CALCULATED 
\ 
1 I .  I I I I f 
IO 20 30 40 50 60 70 I 
PRIMARY BEAM ENERGY, eV 
FIGURE 4.  
i n t e n s i t i e s  vs energy. 
Figure 1 ana experimental conditions are aluminum (100) surface,  
normal incidence. 
Comparison between measured7 and calculated beam 
Theoretical  conditions are as noted i n  


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