ELECTRON ENERGY LOSS SPECTRA BY RICOCHET SCATTERING

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ELECTRON ENERGY LOSS SPECTRA BY RICOCHET SCATTERING

S. Bose, P. Longe

To cite this version:

S. Bose, P. Longe. ELECTRON ENERGY LOSS SPECTRA BY RICOCHET SCATTERING. Jour-

nal de Physique Colloques, 1987, 48 (C9), pp.C9-901-C9-905. �10.1051/jphyscol:19879159�. �jpa-

00227272�

JOURNAL DE PHYSIQUE

Colloque C9, supplbment au n 0 1 2 , Tome 48, dbcembre 1987

ELECTRON ENERGY LOSS SPECTRA BY RICOCHET SCATTERING

S.M. BOSE _and P. LONGE*

Department of P h y s i c s and Atmospheric S c i e n c e , D r e x e l U n i v e r s i t y , P h i l a d e l p h i a , PA 19104, U.S.A.

" ~ n s t i t u t d e P h y s i q u e , Bdt. 5, U n i v e r s i t e d e L i e g e , S a r t Tilman, B-4000 L i e g e , Belgium

RESUME.

-

L o r s q u ' u n f a i s c e a u d 1 6 1 e c t r o n s a t t e i n t l a s u r f a c e d ' u n m 6 t a l avec une i n c i d e n c e q u a s i - p a r a l l G l e , l e s 6 l e c t r o n s o n t une g r a n d e p r o b a b i l i t 6 d l E t r e d i f f u s 6 s F a r r i c o c h e t e n e x c i t a n t un p l a s - mon d e s u r f a c e . La s e c t i o n e f f i c a c e d e d i f f u s i o n p r 6 s e n t e un p i c pour d e s d i r e c t i o n s c r i t i q u e s d86mergence d6pendant d e l V 6 n e r g i e E d e s 6 l e c t r o n s d i f f u s 6 s . La d s t e r m i n a t i o n e x p e r i m e n t a l e d e c e s d i r e c t i o n s e n f o n c t i o n d e E p e u t f o u r n i r une i n f o r m a t i o n d i r e c t e s u r l e s r e l a - t i o n s d e d i s p e r s i o n , c e c i pour d i v e r s plasmons b i - ou monodimension- n e l s .

ABSTRACT.

-

When a beam of e l e c t r o n s i s i n c i d e n t q u a s i - p a r a l l e l t o t h e s u r f a c e of a m e t a l t h e e l e c t r o n s have a h i g h p r o b a b i l i t y t o under- go a r i c o c h e t s c a t t e r i n g w i t h t h e e x c i t a t i o n of a s u r f a c e plasmon. The s c a t t e r i n g c r o s s s e c t i o n peaks a l o n g c r i t i c a l d i r e c t i o n s of emergence depending on t h e e n e r g y E of t h e s c a t t e r e d e l e c t r o n s . Experimental d e t e r m i n a t i o n of t h e s e d i r e c t i o n s a s a f u n c t i o n o f E c a n y i e l d d i r e c t i n f o r m a t i o n on t h e d i s p e r s i o n r e l a t i o n s f o r a v a r i e t y of two- and one- d i m e n s i o n a l plasmons.

I n a r e c e n t t h e o r e t i c a l p a p e r [I] we have proposed a photoemis- s i o n experiment which could y i e l d t h e v a l u e s of t h e p a r a m e t e r s

( w g ,

p and n) of t h e s u r f a c e plasmon d i s p e r s i o n r e l a t i o n ,

ws(q) = wg+pqh

,

of a metal: We have shown t h a t t h e p h o t o e l e c t r o n s e m i t t e d alonq t h e m e t a l s u r f a c e from atoms l o c a t e d a t a s m a l l d i s t a n c e o u t s i d e t h e s u r f a c e have a h i q h p r o b a b i l i t y t o r i c o c h e t w i t h t h e e x c i t a t i o n of a s u r f a c e plasmon i f t h e a n g l e cf emergence a i s l e s s t h a n a c r i t i c a l a n g l e a M ( E ) , where ^E = k2/2m i s t h e e n e r g y of t h e o u t g o i n g e l e c t r o n . Moreover, t h e l i n e s h a p e I ( E , ~ ) of t h e s u r f a c e plasmon s a t e l l i t e h a s a peak a t a = a M ( € ) . By l o c a t i n g t h i s peak one d e t e r m i n e s t h e f u n c t i o n a M ( ~ ) which y i e l Q d i r e c t i n f o r m a t i o n on t h e form and p a r a m e t e r s of t h e above d i s p e r s i o n r u l e . The proposed expe- r i m e n t , however, p r e s e n t s two d i f f i c u l t i e s . F i r s t , t h e p r o d u c t i o n r a t e of t h e p h o t o e l e c t r o n s i s weak i n t h e d i r e c t i o n of t h e m e t a l s u r f a c e when e m i t t e d s o c l o s e t o t h e s u r f a c e , and s e c o n d , t h e spec- t r a l r e s o l u t i o n of t h e p h o t o e l e c t r o n s i s d e t e r m i n e d by t h e l i n e w i d t h of t h e i o n i z e d c o r e l e v e l .

I n t h i s p a p e r , we a r e p r o p o s i n q y e t a n o t h e r experiment i n which t h e s e d i f f i c u l t i e s can b e avoided. I n t h i s e x p e r i m e n t , t h e photo- e l e c t r o n s a r e r e p l a c e d by an e x t e r n a l beam of monoenergetic e l e c t r o n s which a r e allowed t o be s c a t t e r e d by a sample p l a c e d w i t h i t s s u r f a c e q u a s i - p a r a l l e l t o t h e beam. We have c a l c u l a t e d t h e s c a t t e r i n g c r o s s s e c t i o n f o r r i c o c h e t energy l o s s e s by s u r f a c e plasmon p r o d u c t i o n and have shown t h a t such a r e l a t i v e l y more f l e x i b l e e x p e r i m e n t c o u l d y i e l d t h e same i n f o r m a t i o n on t h e d i s p e r s i o n r e l a t i o n s of a wider v a r i e t y of two- and one-dimensional plasmons.

I n t h i s e x p e r i m e n t , t h e o n l y i n p u t d a t a a r e t h e e n e r g y p2/2m of t h e incoming e l e c t r o n beam and t h e s m a l l a n g l e '3 it makes w i t h t h e s u r f a c e of t h e sample. I n o t h e r words, t h e two components

ifil

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19879159

JOURNAL DE PHYSIQUE

f i g . 1

-

T h i s f i g u r e shows t h e geometry t h a t t h e v e c t o r s $, j; and q must s a t i s f y . Vector % l i e s a l o n g t h e i n t e r s e c t i o n of a s p h e r e of r a d i u s k and a c i r c u l a r c y l i n d e r of r a d i u s q w i t h i t s a x i s through t h e p o i n t

6 ,

A The d i r e c t i o n s of c o p o l a r a n g l e a and a z i m u t h a l a n g l e f? of k a r e p o i n t e d o u t ( t h i s f i g u r e i s n o t t o s c a l e ) .

F i g . 2

-

T h i s f i g u r e shows t h r e e p o s s i b l e s h a p e s of t h e i n t e r s e c - t i o n , mentioned i n F i g . 1, f o r t h e v a r i o u s p o s s i b l e c h o i c e s of R and P. The i n t e r s e c t i o n i s r e p r e s e n t e d when looked toward t h e d i r e c t i o n of t h e o u t g o i n g beam ( t h i s f i g u r e i s n o t t o s c a l e ) .

and

Sz

^of ( p a r a l l e l and p e r p e n d i c u l a r t o t h e sample s u r f a c e z=o, r e s p e c t i v e l y ) a r e g i v e n and chosen such t h a t 0 = p,/p << 1. One

measures t h e a n g u l a r d i s t r i b u t i o n ( d i f f e r e n t i a l c r o s s s e c t i o n ) of t h e s c a t t e r e d e l e c t r o n s f o r a g i v e n o u t g o i n g e n e r g y E = k2/2m. T h i s s c a t t e - r i n g must s a t i s f y t h e momentum c o n s e r v a t i o n i n a p l a n e p a r a l l e l t o t h e s u r f a c e , which i s

-+ +

P,, = j:/i + q

(momentum q of t h e plasmon l i e s a l o n g t h e s u r f a c e ) . We must a l s o have -t

t h e e n e r g y c o n s e r v a t i o n

p2 = k +2mws ( q ) 2 ( 2 )

These two c o n d i t i o n s r e q u i r e t h a t a l l t h e r e l e v a n t p a r a m e t e r s a r e d e f i n e d i n r a t h e r narrow r a n g e s a s can be s e e n by c o n s i d e r i n g t h e geometry. Once p,,, p z and a l s o k a r e f i x e d , t h e magnitude of q i s d e f i n e d by E q . ( 2 ) . A s shown i n F i g . 1, t h e v e c t o r

I?

h a s t o l i e a t t h e i n t e r s e c t i o n of a s p h e r e of r a d i u s k and a t h i n c i r c u l a r c y l i n d e r of r a d i u s q , w i t h i t s a x i s p e r p e n d i c u l a r t o t h e sample s u r f a c e a t a p o i n t d e f i n e d by ,!!j

.

The c o n d i t i o n under which t h e c o n s e r v a t i o n laws

(1) and ( 2 ) a r e s a t i s f i e d s i m u l t a n e o u s l y i s t h a t t h i s i n t e r s e c t i o n e x i s t s . T h i s r e q u i r e s t h a t

5

^and ?? a r e l a r g e compared t o ?j and q u a s i - p a r a l l e l t o t h e s u r f a c e of t h e sample ( r i c o c h e t c o n d i t i o n s ) . F i g u r e 2 shows t h e t h r e e p o s s i b l e s h a p e s of t h i s i n t e r s e c t i o n , when looked toward t h e d i r e c t i o n of t h e o u t g o i n g beam. S i n c e t h e c o p o l a r a n g l e a and t h e a z i m u t h a l a n g l e B d e f i n i n g t h e d i r e c t i o n of v e c t o r

2

a r e b o t h s m a l l , it can be shown t h a t ( 1 ) and ( 2 ) can be combined t o g i v e

w i t h

and q d e f i n e d by ( 2 ) . Note t h a t r e l a t i o n ( 3 ) , e q u i v a l e n t t o ( 2 1 , a p p e a r s i n t h e e x p r e s s i o n f o r t h e s c a t t e r i n q c r o s s s e c t i o n a ( a , B , ~ ) i n t h e f a c t o r 6 ( p a 2 + B 2 - ~ ) r e l a t e d t o t h e e n e r q y c o n s e r v a t i o n .

R e l a t i o n (3), i n f a c t , d e f i n e s t h e r e l e v a n t p a r t s of t h e above men- t i o n e d i n t e r s e c t i o n ( t h i c k l i n e s i n F i q . 2 ) , i . e . it r e p r e s e n t s an e l l i p s e ( F i g . 2a) o r two t y p e s of h y p e r b o l a s ( F i q . 2 b , c ) a c c o r d i n g t o t h e s i g n s of R and P ( t h e r e i s no i n t e r s e c t i o n i f R<O and P>O).

S i n c e t h e c y l i n d e r h a s a r a d i u s much s m a l l e r t h a n t h a t of t h e s p h e r e , t h e i n t e r s e c t i o n s a r e much more narrow t h a n r e p r e s e n t e d i n F i g . 2.

From o b s e r v a t i o n a l v i e w p o i n t it i s i n t e r e s t i n g t o c a l c u l a t e a a s a f u n c t i o n of 6 f o r s m a l l v a l u e s of a i n t h e s i t u a t i o n s d e p i c t e d by F i g . 2a and 2b, and a s a f u n c t i o n a i n t h e s i t u a t i o n of F i g . 2c

(where a n g l e B i s s m a l l ) .

L e t u s f i r s t c o n s i d e r t h e s i t u a t i o n of F i g . 2a,b. I n t r o d u c i n g a s m a l l a n q l e Aa, we have found

where S i s a f a c t o r which depends on p and k , A i s t h e a r e a of t h e sample and O t h e s t e p f u n c t i o n . T h i s c r o s s s e c t i o n ~ ( B , E ) p r e s e n t s two i n t e r e s t i n g f e a t u r e s . F i r s t , it h a s a s h a r p peak a t B =

JR

JOURNAL DE PHYSIQUE

I I I I

- -

-

_I

: ^{4 - -}

.-,

1

5 - -

V

> 4

_W

^{2 - -}

C

m

V

-

b

I I I I

0 1" 2" 3" 4" 5"

F i g . 3

-

S c a t t e r i n g c r o s s s e c t i o n

B

u ( E , B ) p l o t t e d a s a f u n c t i o n o f B f o r f o u r v a l u e s o f o u t g o i n q e n e r g y E . H e r e s u r f a c e p l a s m o n s a r e e x c i t e d o n a n aluminum s a m p l e , t h e d i s p e r s i o n r e l a t i o n b e i n g w s ( q ) = ( 1 1 . 1 6

+

9 . 7 5 a p q ) e V . T h e i n c i d e n t e l e c t r o n e n e r g y i s 1 0 0 eV w i t h 8 = 2', a n d t h e o u t g o i n q e n e r g y E i s t h e t h r e s h o l d e n e r g y ( 1 0 0

-

1 1 . 1 6 ) e V m i n u s a d e c r e m e n t 1 . 7 5 ( 1

-

0 . 2 r ) e V w i t h r = 1 , 2 , 3 a n d 4. T h e v e r t i c a l l i n e s c o r r e s p o n d t o t h e c r i t i c a l a n g l e s B M ( € ) = (2.40') Jr.

h . 4

1

$ 9

.+

a

I

%

, 6

0 0

.

0

V

4

3

\

W

-

8

V

0 2" 3" 4" 5" 6" 7"

a

F i g . 4

-

S c a t t e r i n g c r o s s s e c t i o n o ( a , E ) p l o t t e d a s a f u n c t i o n o f a f o r f o u r v a l u e s o f o u t q o i n q e n e r q y E . H e r e t w o - d i m e n s i o n a l p l a s m o n s a r e e x c i t e d o n a 2DEG ( s e e R e f . 4 1 , t h e d i s p e r s i o n r e l a t i o n b e i n g us (q) = 1 . 6 1 6 f a B ¶ ) " eV. T h e i n c i d e n t e l e c t r o n e n e r g y i s 400 eV w l t h 8 = 4 " , a n d t h e o u t g o i n g e n e r p y E i s t h a t i n c i d e n t e n e r g y m i n u s 1 . 7 7 ( 1 + 0 . 2 r ) e V . T h e c u r v e s a r e p l o t t e d f o r r = 1 , 2 , 3 a n d 4 . T h e v e r t i c a l l i n e s c o r r e s p o n d t o t h e c r i t i c a l a n g l e s C ( M ( E ) = ( 2 . 4 1 ' ) Jr.

above (or below) which it cancels. Second the cross section is large when 6 is small because of the factor 1/(e5+p2)

.

Thus these results are qualitatively similar to those of Ref. 1, in the sense that here also there is a critical angle given by show the zngular dependence of cross sectlon o(p,~) B

v

(E) =

JR.

for four outgoing In Fig. 3, we energies for the case where the electron beam is scattered by the surface plasmon of a metal sample with the dispersion relation

@ ( q ) = UsfUq. Here it can be shown that the conditions of Fig. 2a are satisfled and the cross section is nonzero on an elli~se.Obvious- ly, in this case ~ ( B , E ) will be nonzero for p<$ (E) as shown in Fig.3.

In Fig. 4 another situation is considered.M~ere the electron beam is scattered by a plasmon of a two-dimensional electron gas

(2DEG) with the dispersion relation wS(q) = uJq. As is well known the 2DEG can be realized in an MOS invers~on layer [3], in the accumu- lation layer of a semiconductor such as ZnO [4] and on the surface of

a

liquid helium sample [S]. For this dispersion relation, the condition of Fig. 2c is satisfied and the cross section is nonzero on a hyperbola, i.e. for a > a M ( ~ ) = (R/P) ifi. Indeed one has

In Fig. 4 we plot the angular dependence of o ( a , ~ ) for four values of E and we propose experimental parameters adapted to the data of Ref. 4.

From the experimental determination of the curves of Figs. 3 or 4 we can obtain pM(&) or aM(€) as functions of the energy of the outgoing electrons E. As in Ref. 1 frorr. t h e w Functions we can find the nature and form of the plasmon dispersion relations in these two cases. The theory presented here can be applied to other

situations where the plasmon dispersion has two- or one-dimensional nature. For example, the theory could easily be applied to the case of a quasi-one dimensional metal studied by several authors [6].

The details of these calculations will be presented in a more complete forthcoming paper.

REFERENCES

[I] LONGE, P. and BOSE, S.M., P h y s . Rev. L e t t . 57 (1986) 2307.

[2] LONGE, P. and BOSE, S.M., S o l i d S t a t e cornmu=

54

(1985) 879.

131 VINTER, B., P h y s . R e v . L e t t .

35

(1975) 1044.

[4] GOLDSTEIN, Y., MANY, A., WAGNER, I. and GERSTEN, J., S u r f . S c i . 98 (1980) 599.

[51 GRIMES,C.C., S u r f . S c i . 73 (1978) 379.

[6] CAMPOS, V.B., HIPOLITO, 07and LOBO, R., P h y s . S t a t . S o l i d i ( b ) 81 (1977) 657; FRIESEN, W.I. and BERGERSEN, B., J. P h y s . C

-

13 (1980) 6627.

-