HAL Id: jpa-00223534
https://hal.archives-ouvertes.fr/jpa-00223534
Submitted on 1 Jan 1983
HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
QUANTUM-SIZE EFFECTS ON THE OPTICAL PROPERTIES OF SMALL PARTICLES
P. Ahlqvist, P. de Andrés, R. Monreal, F. Flores
To cite this version:
P. Ahlqvist, P. de Andrés, R. Monreal, F. Flores. QUANTUM-SIZE EFFECTS ON THE OPTICAL
PROPERTIES OF SMALL PARTICLES. Journal de Physique Colloques, 1983, 44 (C10), pp.C10-
375-C10-378. �10.1051/jphyscol:19831076�. �jpa-00223534�
QUANTUM-SIZE EFFECTS ON THE OPTICAL PROPERTIES OF SMALL PARTICLES
P . Ahlqvist, P . d e ~ n d r d s * , R . onr real* and F . F l o r e s r Ericson ~ a d i o System AB, Box 1002, S-431 26 Md'ZndaZ, Sweden
* ~ e ~ a r t m e n t o de Fisica de Z Estado Sb Zido, Universidad Authnoma, CantobZanco, Madrid 34, Spain
R6sum6.- On p r 6 s e n t e une m6thode p e r m e t t a n t d ' o b t e n i r p o u r l e s c o u c h e s m i n c e s une f o n c t i o n d i e l e c t r i q u e s i m p l i f i E e
i n c l u a n t d e s e f f e t s d e t a i l l e q u a n t i q u e s p o u r l e s e x c i t a t i o n s d e p a i r e s 6 l e c t r o n - t r o u . P a r une g 6 n 6 r a l i s a t i o n c o n v e n a b l e a u c a s d e s s p h g r e s , on a n a l y s e l a p h o t o a b s o r p t i o n de c e s s y t s m e s . Les r 6 s u l t a t s m o n t r e n t d ' i n t e r e s s a n t s e f f e t s d e t a i l l e q u a n t i q u e s a s s o c i 6 s aux r 6 s o n a n c e s d e s p a i r e s B l e c t r o n - t r o u .
A b s t r a c t . - A method i s p r e s e n t e d t o o b t a i n a s i m p l i f i e d d i - e l e c t r i c f u n c t i o n f o r t h i n f i l m s , i n c o r p o r a t i n g q u a n t u m - s i z e e f f e c t s f o r e l e c t r o n - h o l e p a i r s e x c i t a t i o n s . By a p r o p e r g e n e r a l i s a t i o n t o s p h e r e s , we a n a l y s e t h e p h o t o a b s o r p t i o n o f t h e s e s y s t e m s . Our r e s u l t s show i n t e r e s t i n g q u a n t u m - s i z e e f f e c t s a s s o c i a t e d w i t h r e s o n a n c e s o f e l e c t r o n - h o l e p a i r s . T h e r e h a s b e e n r e c e n t l y a l o t o f work a d d r e s s e d t o u n d e r s t a n - d i n g t h e r o l e o f s u r f a c e r o u g h n e s s
[I]
and o f s m a l l p a r t i c l e s [2,3]on d i f f e r e n t e l e c t r o m a g n e t i c e x p e r i m e n t s . The l o s s o f c e r t a i n s p a t i a l s y m m e t r i e s a l l o w s d i f f e r e n t modes ( s u r f a c e plasmons a n d e l e c t r o n - h o l e p a i r s ) t o b e e x c i t e d b y t h e i n c i d e n t f i e l d , g i v i n g i m p o r t a n t e f f e c t s on p r o p e r t i e s s u c h as p h o t o a b s o r p t i o n h o t o y i e l d , Raman s c a t t e r i n g , e t c . I n p a r t i c u l a r , Penn a n d R e n d e l l [2J h a v e s t r e s s e d t h e i m p o r t a n c e o f e l e c t r o n - h o l e p a i r s i n t h e p h o t o a b s o r p t i o n and p h o t o y i e l d o f s m a l l s p h e r e s . T h e i r a n a l y s i s u s e d a c o n t i n u u m e x c i t a t i o n s p e c t r u m f o r p a i r s n e g l e c t i n g q u a n t u m - s i z e e f f e c t s w h i c h , up t o o u r knowledge, h a v e b e e n o n l y i n c l u d e d i n t h e a n a l y s i s o f t h i n f i l m s [4]
.
The a i m o f t h e p r e s e n t work i s t o g i v e a scheme by means o f which o b t a i n i n g q u a n t u m - s i z e e f f e c t s on t h e o p t i c a l p r o p e r t i e s o f s m a l l s p h e r e s .
F i r s t o f a l l , it i s c o n v e n i e n t t o show how t h e method g i v e n by A p e l l a n d A h l q v i s t [4] (AA) t o a n a l y s e t h i n f i l m c$n b e r e c o v e r e d by i n t r o d u c i n g a n a p p r o x i m a t e d i e l e c t r i c f u n c t i o n ~ ~ ( k , w ) . To t h i s e n d , n o t e t h a t t h e AA-method g i v e s a d i e l e c t r i c f u n c t i o n f o r t h e f i l m h a v i n g a q u a s i - d i s c r e t e e n e r g y s p e c t r u m . T h i s s p e c t r u m i s a s s o c i a t e d w i t h t h e q u a n t i s a t i o n o f t h e e l e c t r o n i c l e v e l s i n t h e d i r e c t i o n
p e r p e n d i c u l a r t o t h e s u r f a c e , i n s u c h a way t h a t t h e number o f s i n g l e - p a r t i c l e e x c i t a t i o n s o f t h e f i l m i n c r e a s e s w i t h i t s w i d t h . Our method t o r e c o v e r t h $ r e s u l t s g i v e n b y AA c o n s i s t s i n w r i t i n g t h e d i e l e c t r i c f u n c t i o n , c L ( k , w ) , o f t h e film i n t h e f o l l o w i n g way:
where t h e number o f p o l e s a p p e a r i n g i n (1) i s d i r e c t l y c o n n e c t e d w i t h t h e number o f s i n g l e p a i r s e x c i t a t i o n s o f t h e f i l m . ( I n e q . (1)
,
e n e r g i e s a n d momentum a r e n o r m a l i s e d t o EF a n d k F , t h e Fermi e n e r g y and t h e Fermi momentum f o r t h e f i l m
[LC]
) .W e h a v e c h e c k e d o u r a p p r o x i m a t i o n , by c o n s i d e r i n g t h e c a s e o f a K - f i l m on a c o p p e r s u b s t r a t e , as w a s done by AA. I n F i g u r e 1 we
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19831076
JOURNAL DE PHYSIQUE
FIGURE 1.- P h o t o a b s o r p t a n c e v e r s u s f r e q u e n c y f o r a 15
a
th;ckp o t a s i u m f i l m o n a c o p p e r s u b s t r a t e . Angle o f i n c i d e n c e f o r l i g h t i s 75'.
g i v e t h e a b s o r p t a n c e f o r a 1 5
2
t h i c k f i l m a s a f u n c t i o n o f w , as o b t a i n e d w i t h e q . ( l ) f o r 62 = 5 . 2 2 , 6 ' 2 = 7 . 0 ,2 2 2
y: = 5 . 5 , y 2 = 2 . 2 , y 3 = - 1 . 4 9 , y4 = - 4 . 3 6 , 2
Y 5 = - 5 . 3 2 ,
Note t h a t t h e s p e c t r u m g i v e n i n t h i s f i g u r e i s q u i t e s i m i l a r t o t h e one o b t a i n e d by AA, e x c e p t f o r two p e a k s a p p e a r i n g a t v e r y low f r e q u e n c i e s .
Our r e s u l t s show t h a t t h e d i e l e c t r i c f u n c t i o n g i v e n above i s a r e a s o n a b l e a p p r o x i m a t i o n t o t h e d i e l e c t r i c f u n c t i o n o f t h e f i l m i n o r d e r t o a n a l y s e EM p r o p e r t i e s . Now, t h i s a p p r o x i m a t i o n g i v e s a p r o c e d u r e t o i n c l u d e quantum e f f e c t s f o r a m e t a l l i c s h e r e . I n t h i s c a s e , we f o l l o w t h e scheme p r o p o s e d by o t h e r a u t h o r s f 2 , 3 ] a n d assume t h e s p h e r e t o b e embedded i n a n i n f i n i t e medium. We m o d i f y , h o w e v e r , o t h e r a p p r o a c h e s [2,3] by u i n g a d i e l e c t t t i c d u n c t i o n
0 6
t h e t y p e g i v e n a b o v e , i n o r d e r t o i n c l u d e quantum e f f e c t s . I n o u r a p p r o x i m a t i o n we t a k e t h e s p h e r e d i a m e t e r t o b e e q u a l t o t h e w i d t h o f t h e f i l m u s e d t o o b t a i n e q . ( l ) .The i n t e r a c t i o n b e t w e e n a n EM-plane wave a n d t h e s p h e r e i s o b t a i n e d by u s i n g t h e e x t e n d e d media a p p r o a c h [ 5 ] . The i n f i n i t e medium h a s a d i e l e c t r i c r e s p o n s e f u n c t i o n g i v e n by e q . ( 1 ) ; i n o r d e r t o c o m p l e t e l y s p e c i f y t h e model, i t i s n e c e s s a r y f o d e f i n e t h e EM f i e l d o u t s i d e t h e s p h e r e . Penn a n d R e n d e l l [2] u s e d a p a r t i c u l a r a n s a t z ; i n s t e a d o f t h i s , we h a v e u s e d d i f f e r e n t c o n d i t i o n s :
( i ) F i r s t , we h a v e c o n s i d e r e d a c o m p l e t e l y d i f f u s e s u r f a c e . T h i s i s e q u i v a l e n t t o t a k i n g
?
= 0 f o r r > R 161.( i i ) Our s e c o n d c o n d i t i o n t r i e s t o s i m u l a t e a p e r f e c t l y s p e c u l a r s u r f a c e , p =
1.
F o r t h i s c a s e , we h a v e u s e d t h e f o l l o w i n g c o n d i t i o n f o r t h e longitudinal p a r t o f2:
m a t c h i n g t h a t s o l u t i o n t o t h e vacuum f i e l d a t r = R. D e t a i l s a b o u t t h i s p r o c e d u r e w i l l b e ~ u b l i s h e d e l s e w h e r e , b u t t h e a n a l y s i s f o l l o w s r o u g h l y Ref. [6]
:
We g i v e I n F i g u r e 2 t h e a b s o r p t a n c e f o r a s p h e r e o f D = 15
a
a s c a l c u l a t e d w i t h p = 0 , and by u s i n g t h e f o l l o w i n g d i e l e c t r i c f u n c t i o n s : ( i ) hydrodynamic a p p r o x i m a t i o n ; ( i i ) a s i n g l e p a i r s t r u c t u r e a p p r o x i m a t i o n ; ( i i i ) e q . ( l ) .
FIGURE 2 . - P h o t o a b s o r p t a n c e v e r s u s f r e q u e n c y f o r a s p h e r e o f D = 1 5
a
and p = 0 ( s e e t e x t ) . D i f f e r e n t c u r v e s c o r r e s p o n d t o : ( i ) d o t t e d l i n e : hydrodynamic a p p r o x i m a t i o n ; ( i i ) d a s h e d l i n e : a s i n g l e p a i r s t r u c t u r e ; ( i i i ) f u l l l i n e : e q . ( 1 ) .FIGURE 3 . - P h o t o a b s o r p - t a n c e v e r s u s f r e q u e n c y f o r a s p h e r e o f D = 30.8 and p = 1 ( s e e t e x t ) . A s i n g l e p a i r s t r u c t u r e f o r t h e d i e l e c t r i c f u n c t i o n h a s b e e n u s e d i n t h i s f i g u r e . n = o / w
P'
JOURNAL DE PHYSIQUE
Our r e s u l t s show t h a t t h e a b s o r p t a n c e , f o r w < w i n c r e a s e s w i t h t h e number o f p a i r s i n c l u d e d i n E L - i n s i m i l a r i t y P ' w i t h Penn and R e n d e l l ' s r e s u l t s - b u t show n o s t r u c t u r e a s s o c i a t e d w i t h t h e p a i r s . S u r f a c e r o u g h n e s s washes o u t a l l t h e p a i r s t r u c t u r e . We h a v e
f o u n d t h e same r e s u l t s f o r a f i l m .
F i n a l l y , i n F i g u r e 3 we show t h e r e s u l t s a s o b t a i n e d by u s i n g c o n d i t i o n p
=
1, and a s i m p l i f i e d d i e l e c t r i c f u n c t i o n w i t h o n l y a s i n g l e p a i r e x c i t a t i o n (D = 30a ) .
T h i s c a l c u l a t i o n t r i e s t o i l l u s t r a t e t h e e f f e c t o f t h e s i n g l e e x c i t a t i o n s i n t h e o p t i c a l a d s o r p t i o n f o r s u r f a c e s p e c u l a r c o n d i t i o n s ; a s was o b t a i n e d f o r t h e c a s e o f a f i l m f o r p = 1, 'p a i r s t r u c t u r e a p p e a r s a s t h e s u p e r p o s i t i o n o f d i f f e r e n t p a i r s excitation. Our r e s u l t s show t h e a p p e a r a n c e o f a r e s o n a n t peak a s s o c i a t e d t o t h e p a i r mode.RE FE REN CE S
[l] BURSTEIN E , LUNDQVIST S and MILLS D L , i n S u t d a c e enhanced Raman n c a t t e h i n g
,
R.K.Chang a n d T . E . F u r t a k Eds. (Plenum P r e s s , 1982) PENN D R a n d RENDELL R W , ' ~ h y s . R e v . B ,3
( 1 9 8 2 ) 3047RUPPIN R , Phys.Rev.B,
11
( 1 9 7 5 ) 2871P , P h y s i c a S c r i p t a , 22 ( 1 9 8 1 ) 659
FLORES F , l n t h o d u c t i ~ t o t h e t h e u t y
0 6
h o e i d n uhdacen (Cambridge Uni.v. P r e s s , 1 9 79 )[6] MONREAL R , GARCIA-MOLINER F and FLORES F , J . P h y s i q u e , ( 1 9 8 2 ) 9 0 1