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Submitted on 1 Jan 1981
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PHONONS AT METAL SURFACES
C. Schwartz, W. Schaich
To cite this version:
JOURNAL DE PHYSIQUE
CoZloque
C6,supplkment au n o
1 2 ,Tome
42,de'cernbre
1981page
C6-828PHONONS A T METAL SURFACES
C. Schwartz and W.L. S c h a i c h
Physics Department, Indiana University, Bloomington,
IN 47405, U . S . A.A b s t r a c t . - W e examine phonons a t m e t a l s u r f a c e s u s i n g hydrodynamic e q u a t i o n s . The p r i n c i p l e f o c u s i s on t h e s t r e n g t h o f t h e e l e c t r i c f i e l d produced o u t s i d e t h e m e t a l by s u c h o s c i l l a t i o n s . We f i n d t h a t t h e m a g n i t u d e s o f t h e s e c o u p l i n g s t r e n g t h s f o r b o t h s u r f a c e and b u l k modes depend c r u c i a l l y on t h e boundary c o n d i t i o n s imposed a t t h e s u r f a c e .
1 . Hydrodynamic
model.
- Our l i n e a r i z e d continuum model of t h e c o u p l e d motion o f e l e c t r o n s and i o n s i s d e f i n e d by t h e e q u a t i o n s :*
where e a n d i s u b s c r i p t s r e f e r , r e s p e c t i v e l y , t o e l e c t r o n s and i o n s ; E is t h e ( l o n g i t u d i n a l ) e l e c t r i c f i e l d ; D , t h e ( c o n s t a n t ) e q u i l i b r i u m i o n c h a r g e d e n s i t y , fi and w a r e t h e i o n and e l e c t r o n plasma f r e q u e n c i e s ; t h e c ' s and Be p a r a m e t r i z e
P P s h o r t r a n g e r e s t o r i n g f o r c e s ; and t h e i ' s a r e d i s p l a c ~ e n t f i e l d s r e l ~ t e d by a
*
-b + t i m e d e r i v a t i v e t o t h e c u r r e n t d e n s i t i e s j : j i = P O -i and j = - p ,%
.
Notea t
a t
t h a t we h a v e n e g l e c t e d r e t a r d a t i o n and d i s c r e t e l a t t i c e e f f e c t s and t h a t o n l y t h e i o n s s e n s e t r a n s v e r s e f o r c e s , v i a cT. I n b u l k m a t e r i a l a t a g e n e r a l f r e q u e n c y w,
one h a s e l e c t r o n i c l o n g i t u d i n a l waves and i o n i c l o n g i t u d i n a l and t r a n s v e r s e waves. When o n e c o n s i d e r s e x c i t a t i o n s n e a r a s u r f a c e , where P , d r o p s d i s c o n t i n u o u s l y t o z e r o , l i n e a r c o m b i n a t i o n s o f t h e s e b u l k modes p l u s e x c i t a t i o n s v a r y i n g a seig''e-Qlxl must b e u s e d . Here
X
(and t h e w a v e v e c t o r Q ) l i e s i n t h e s u r f a c e p l a n e w h i l e x i s normal t o i t . The c o u p l e d modes a r e l a b e l e d by w,9,
and p o l a r i z a t i o ni n d i c e s . To d e t e r m i n e t h e eigenmodes of ( 1 ) r e q u i r e s i n g e n e r a l t h e i m p o s i t i o n of f i v e i n d e p e n d e n t s u r f a c e boundary c o n d i t i o n s . For t h e f i r s t two, we u s e c o n t i n u i t y
-+
of t h e p o t e n t i a l 0
(E=-6~)
and c o n t i n u i t y of t h e normal component of t h e d i s p l a c e --+
..
rnent f i e l d . T h i s l a s t i s e q u i v a l e n t t o A (E.x)=4nos
,
where os is t h e induced s u r f a c e c h a r g e d e n s i t y andA (...) means t h e d i s c o n t i n u i t y i n (...). For t h e o t h e r boundary c o n d i t i o n s we have examined s e v e r a l p o s s i b i l i t i e s . C o n s i d e r a t i o n o f m e t a l o r plasma p h y s i c s s u g g e s t s we s e t cT=O, which e l i m i n a t e s t h e t r a n s v e r s e modes andone boundary c o n d i t i o n , and f o r t h e r e m a i n i n g two c o n d i t i o n s r e q u i r e a t t h e s u r f a c e ,
which we c a l l c a s e C . Note t h a t (2) i m p l i e s us=O. On t h e o t h e r hand, e l a s t i c i t y t h e o r y s u g g e s t s we keep cT#
0
and i n s t e a d r e q u i r e ( c a s e S)C-,
where a i s t h e s t r e s s t e n s o r t h a t l e a d s t o (1).
2. R e s u l t s .
-
The d i f f e r e n t boundary c o n d i t i o n s (2) and (3) l e a d t o d r a m a t i c a l l y d i f f e r e n t p r e d i c t i o n s . C o n s i d e r f i r s t p o s s i b l e s u r f a c e modes. With (2) t h e r e i sa s u r f a c e mode j u s t below ( a t each Q) t h e b u l k l o n g i t u d i n a l modes. T h i s mode h a s been found a l s o i n l e s s g e n e r a l (cL=O) With (3) a s u r f a c e mode o n l y a p p e a r s below t h e b u l k t r a n s v e r s e modes. T h i s s u r f a c e mode i s i n e s s e n c e a
( s t i f f e n e d ) R a y l e i g h wave. It r e q u i r e s a f i n i t e c T and t h e e f f e c t i v e l o n g i t u d i n a l sound speed i s v i = ci+c$ where fi =c k w i t h k
.
To d i s c u s s t h e p o s s i b l eP S S Be
e x t e r n a l c o u p l i n g t o t h e s e modes we have q u a n t i z e d them and w r i t e t h e e x t e r n a l p o t e n t i a l as
8(2)=1
A eiQeX e-QX(a +a +) where a t ( a ) is a c r e a t i o n ( a n n i h i l a t i o n )9
9
-9
o p e r a t o r f o r t h e & f a c e mode
9 .
A s i m i l a r e x p r e s s i o n h o l d s f o r t h e c o u p l i n g t o b u l k modes. I n f i g u r e 1 we p l o t A v s . Q f o r t h e c a s e s C and S and a l s o a n a n a l o g o u s q u a n t i t y from t h e t h e o r y of Rahman and ~ i l l s ~ , c a s e R.F i g . 1: E x t e r n a l c o u p l i n g s t r e n g t h p a r a m e t e r A f o r s u r f a c e modes r e s u l t i n g from d i f f e r e n t boundary c o n d i t i o n s v e r s u s s u r f a c e wavevector Q. A l l A's a r e i n t h e same a r b i t r a r y u n i t s . O t h e r symbols a r e d e f i n e d i n t h e t e x t . I n f i g u r e 2 ve p l o t a q u a n t i t y B , p r o p o r t i o n a l t o t h e sum of t h e s q u a r e s of a l l t h e f i n i t e A ' s f o r t h e b u l k modes a t e a c h w and Q. Simple i n t e g r a l s of B
JOURNAL DE PHYSIQUE F i g . 2: E x t e r n a l c o u p l i n g s t r e n g t h p a r a m e t e r B ( d i m e n s i o n l e s s ) f o r b u l k modes s u b j e c t t o d i f f e r e n t boundary c o n d i t i o n s v e r s u s s u r f a c e wavevector Q a t f i x e d f r e q u e n c y w. See t e x t f o r d e f i n i t i o n of t h e o t h e r symbols.
The most s t r i k i n g f e a t u r e of b o t h f i g u r e s (1) and (2) i s t h e s m a l l r e l a t i v e s i z e of t h e A ' s f o r c a s e S. They a r e reduced by r o u g h l y Q2/u2 compared t o t h o s e
