NON-LOCAL THEORY FOR THE FAR-INFRARED ABSORPTION IN SMALL METAL PARTICLES

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NON-LOCAL THEORY FOR THE FAR-INFRARED ABSORPTION IN SMALL METAL PARTICLES

A. Mal’Shukov

To cite this version:

A. Mal’Shukov. NON-LOCAL THEORY FOR THE FAR-INFRARED ABSORPTION IN SMALL METAL PARTICLES. Journal de Physique Colloques, 1983, 44 (C10), pp.C10-371-C10-374.

�10.1051/jphyscol:19831075�. �jpa-00223533�

JOURNAL DE PHYSIQUE

Colloque C10, supplkment au n012, T o m e 44, d k e m b r e 1983 page CI 0-371

NON-LOCAL THEORY FOR THE FAR-INFRARED ABSORPTION I N SMALL METAL PARTICLES

A . G . Mal' shukov

I n s t i t u t e o f Spectroscopy, USSR Academy o f Sciences, T r o i t s k , Moscow r-n, 142092, U.S.S.R.

Resum6

-

O n propose une th6orie non-locale d e l'absorption dans l'infra-rouge lointain dans l e s p e t i t e s particules metalliques.

O n m o n t r e que l'absorption magn6tique dipolaire depend forte- ment de l a nature d e la d i f f u s i o n electronique par l a limite d e la particule. O n predit u n e importante a u g m e n t a t i o n d e c e t t e absorption p o u r l a reflexion speculaire des electrons par u n e surface spherique o u par une surface consistant e n microfacettes orientees au hasard.

A b s t r a c t

-

^A n o n l o c a l theory of f a r - i n f r a s e d a b s o r p t i o n i n s m a l l m e t a l p a r t i c l e s has been proposed. The magnetic-dipole a b s o r p t i o n h a s been shown t o depend s t r o n g l y on t h e n a t u r e of e l e c t r o n i c s c a t t e r i n g from t h e p a r t i c l e boundary. A l a r g e i n c r e a s e o f t h e magnetic-dipole a b s o r p t i o n h a s been p r e d i c t e d f o r t h e s p e c u l a r r e f l e c t i o n of e l e c t r o n s from t h e s p h e r i c a l s u r f a c e o r t h e s u r f a c e c o n s i s t i n g o f randomly o r i e n t e d m i c r o f a c e t s .

The low-frequency r e s p o n s e of s m a l l - p a r t i c l e composites has some s p e c i f i c p e c u l i a r i t i e s , which a r e caused by t h e s m a l l s i z e o f t h e p a r t i c l e s . Two parameters determine t h e r e s p o n s e o f a n e l e c t r o n i c system: t h e e l e c t r o n i c mean f r e e - p a t h 1 and t h e e l e c t r o n i c p a t h d u r i n g one o s c i l l a t i o n of the e x t e r n a l f i e l d VF/W * I f b o t h of them exceed t h e p a r t i c l e s i z e a , t h e s c a t t e r i n g o f - e l e c t r o n s by t h e p a r t i c l e b o w d a r y becomes important and t h e r e l a t i o n between c u r r e n t s and f i e l d s a c q u i r e s non-local form. It becomes e v i d e n t from t h e s e arguments why t h e c l a s s i c a l l o c a l t h e o r y successful i n d e s c r i p - t i o n of t h e o p t i c a l p r o p e r t i e s of t h e s m a l l - p a r t i c l e composites i n v i s i b l e , where V,#W

4

a , f a i l s t o p r e d i c t e x p e r i m e n t a l l y obser- v a b l e magnitude o f t h e f a r - i n f r a r e d a b s o r p t i o n /I-3/.

E a r l i e r t h e theory h a s been proposed which t a k e s i n t o account c o l l i s i o n s o f e l e c t r o n s w i t h t h e p a r t i c l e boundary and t h e n o n l o c a l r e l a t i o n between c u r r e n t s and e l e c t r i c f i e l d s

/4/.

The g e n e r a l form o f t h i s n o n l o c a l r e l a t i o n

ia:

J ~ C

0 , ?)

= e i j (

0 , F, T ' )

E ~ ( u , ?')d3r'. (1) The n o n l o c a l c o n d u c t i v i t y ' G i j was c a l c u l a t e d by use of t h e k i n e t i c e q u a t i o n f o r e l e c t r o n s i n t h e p a r t i c l e . This q u a s i c l a s s i c a l approach c a n be ' s a f e l y a p p l i e d t o a s m a l l p a r t i c l e i f one i g n o r e s quantum o s c i l l a t i o n e f f e c t s which occure i n very s m a l l p a r t i c l e and a t very low f r e q u e n c i e s .

The e q . ( l ) can be s u b s t i t u t e d t o t h e Ataxwell e q u a t i o n s t o y i e l d a n i n t e g r a l e q u a t i o n f o r d i p o l e components of f i e l d s . The s o l u t i o n o f t h e s e e q u a t i o n s i n t h e f a r - i n f r a r e d regime

a03

/vp 4 1 gave f i n a l l y f o r a b s o r p t i o n :

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19831075

JOURNAL DE PHYSIQUE

where f i s t h e p a r t i c l e velum; f r a c t i o n i n t h e composite system and

u p

^{i s} t h e plasma frequency. T h i s e x p r e s s i o n i s very s i m i l a r t o t h a t g i v e n by t h e c l a s s i c a l t h e o r y based on t h e l o c a l Drude formula f o r t h e d i e l e c t r i c f u n c t i o n . However, a n e s s e n t i a l d i f f e r e n c e i s t h a t g e n e r a l l y t h e e f f e c t i v e s c a t t e r i n g time

ze

^is n o t e q u a l t o

Tm.

These times have been shown t o be of one o r d e r of magnitude only f o r d i f f u s e s c a t t e r i n g of e l e c t r o n s from t h e p a r t i c l e boundary. I t was

found t h a t :

-

I n t h i s c a s e t h e n o n l o c a l theory gave q u a l i t a t i v e l y t h e same r e s u l t a s t h e c l a s s i c a l one.

S i t u a t i o n d r a s t i c a l l y changes i n t h e o t h e r l i m i t i n g c a s e of s p e c u l a r r e f l e c t i o n a t t h e s p h e r i c a l boundary. I t f o l l o w s from t h e n o n l o c a l t h e o r y t h a t

zm

i n c r e a s e s e s s e n t i a l l y i n t h i s c a s e . I n f a c t ,

/Cm

c a n be as l o n g a s s c a t t e r i n g i n "bulk" m a t e r i a l a l l o w s , i . e .

x m -

l / v F , where 1 i s t h e mean f r e e p a t h i n t h e bulk metal.

However,

xe

remains of t h e o r d e r a/vp. The s t r o n g i n c r e a s e of

Tm

a t t r a n s i t i o n from d i f f u s e s c a t t e r i n g t o s p e c u l a r one i n d i c a t e s t h a t i t could e x p l a i n t h e p e a t observable absorption. A t l e a s t i t c l e a r l y demonstrates t h a t s c a t t e r i n g c o n d i t i o n s a t t h e boundsry a r e i m p o r t a n t f o r t h e f a - i n f r a r e d a b s o r p t i o n i n small p a r t i c l e s yhen t h e second (ma n e t i c - d i p o l e ) term i n ( 2 ) dominates ( a

&

30 A f o r t y p i c a l metalsy.

A cause o f such a b e h a v i o w o f

/G

^{i s} r a t h e r simple. The second term i n ( 2 ) d e s c r i b e s t h e magneticgdipole a b s o r p t i o n , i. e. energy l o s s e s b y eddy c u r r e n t s induced i n t h e p a r t i c l e . Therefore t h e

magnetic-dipole a b s o r p t i o n is c h a r a c t e r i z e d mostly by a n g u l a r e l e c t - r o n i c motion. Indeed i t was shown /4,5/ t h a t

T m

i s p r o p o r t i o n a l t o c o r r e l a t i o n f uslction of e l e c t r o n i c a n g u l a r momenta taken a t d i f f e r e n t times. This can e a s i l y be s e e n from t h e g e n e r a l c l a s s i c a l e x p r e s s i o n f o r t h e n o n l o c a l c o n d u c t i v i t y

+ +* 2 +, 4.5

Ckj

( r , r , t - t e ) = 3 ~ { n . n : ~ ( r , n ,

+.*.

^{1 J} t ; r , n , t *)& d i i x q r l 2 ( 3 ) where ~ ( g , n , tLrAn, t

'1

i s thz,p+~obabj,lity of a n e l e c t r o n t r a n s i t i o n from a s t a t e r , n v , t t o r , n v , t i n t h e phase space. The magnetic-dipole p o ? a r i z a b i l i t y

i E

g i v e n by /4/

where

em(,,,; ⁾ =

J ^G

^qg'dt Y ? ~ ~ ( Z )

6 . .

( r ~ , r p ~ , t ) x

1 J

3

x Y;,(@ e x p ( i W t )

w i t h YI0 = (2)-'I2 [ Z - ~ Y , , ( ' ; ) ] and ;= ?/r

.

Using (3) and ( 5 ) t h e e x p r e s s i o n (4) c a n be reduced t o

where 111 i s t h e a n g u l m momentum of an e l e c t r o n . The b r a c k e t s mean a v e r a g i n g o v e r t h e e l e c t r o n i c t r a j e c t o r i e s

.

The e x p l i c i t form f o r

X

i s /5/:

The e x p r e s s i o n ( 6 ) c a n e a s i l y be analysed. Indeed, a f t e r t h e f i r s t c o l l i s i o n w i t h t h e boundary t h e d i f f u s i v e l y s c a t t e r e d e l e c t r o n

" f o r g e t s w which momentum i t had b e f o r e c o l l i s i o n . It means t h a t

t ^.v a/vF i n (6) and

1;

cv a/vp i n ( 7 ) . Thus we g o t t h e c l a s s i c a l r e s u l t f o r t h e magnetic-dipole p a r t of a b s o r p t i o n i n t h e c a s e of

d i f f u s e s c a t t e r i n g . When e l e c t r o n s a r e r e f l e c t e d s p e c u l a r i l y

,

^{t h e}

angulslr momentum i s conserved a t each c o l l i s i o n w i t h t h e s u r f a c e . T h e r e f o r e ,

X,*

^{02 a t W-*} 0. I f one t a k e s into_jtcco t t h e s c a t t e r i n g from d e f e c t s i n s i d e t h e p a r t i c l e , t h e n (Dl(t) 8 0 ) ) N exp(-t/ b ) , where

2

i s t h e time of t h e mean F r e e p a t h i n bulk m e t a l , which exceeds c o n s i d e r a b l y t h e e f f e c t i v e r e l a x a t i o n time

( cu a/vv) e n t e r i n g i n t h e c l a s s i c a l e x p r e s s i o n f o r t h e a b s o r p t i o n . Thus, f o r a p e c u l a r r e f l e c t i o n one has /4,5/;

Z m =

z b / P +

xbw)

T h i s r e s u l t f o l l o w s a l s o from c a l c u l a t i o n s o f TBODAHL / 6 / , who considered i n a d d i t i o n t h e e l e c t r o m a g n e t i c s c r e e n i n g of c u r r e n t s i n t h e p a r t i c l e (skin-depth) which can be i m p o r t a n t f o r

a+

200-300 A . I t has been demonstrated above t h a t t h e magne t i c - d i p o l e a b s o r p t i o n i s v e r y s e n s i t i v e t o s c a t t e r i n g o f e l e c t r o n s fyom t h e p m t i c l e boundary. Tvio extreme c a s e s were c o n s i d e r e d : t o t a l l y d i f f u s e s c a t - t e r i n g and speculscr r e f l e c t i o n . The a b s o r p t i o n was shown t o i n c r e a s e by s e v e r a l o r d e r s o f magnitude upon t r a n s i t i o n from d i f f u s e s c a t t e r i n g regime t o s p e c u l a r one. However, t h e q u e s t i o n i s what t h e r e a l

s c a t t e r i n g c o n d i t i o n s ape i n e x p e r i m e n t a l l y s t u d i e d systems. It s e e m r e a s o n a b l e t o assume t h e model i n which t h e p a r t i c l e boundary c o n s i s t s o f randomly o r i e n t e d m i c r o f a c e t s . Each of t h e s e f a c e t s was,supposed t o r e f l e c t e l e c t r o n s s p e c u l z r i l y ; o r i e n t a t i o n of a f a c e t N (normal t o t h e f a c e t s u r f a c e ) was t a k e n t o be d i s t r i b u t e d a s

-C

dp =

-P

S-' 71-j

exp

[-(i; - Z0)'/ s

^{2 ] dii}

where T? i s t h e l o c a l normal-to he s p h e r i c a l s m f a c e . When

gg

I

,

t h e pso%lem o f c a l c u l a t i o n

<u(

t ) ~

3

⁽⁰⁾⁾ can be reduced +to a d i f f e r e n - t i a l e q u a t i o n which d e s c r i b e s d i f f u s i o n o f t h e a n g u l a r momentum i n t h e phase space. S o l v i n g this e q u a t i o n we have o b t a i n e d /5/ t h a t

< Z ( t ) Z ( O ) > ~ exp(- t / S f )

,

where = (2/3)a/vF

S

2. It i s e v i d e n t t h a t even f o r r e l a t i v e l y wide d i s t r i b u t i o n o f f a c e t o r i e n t a t i o n s ( f o r example

8

⁼ 0.3) t h e a b s o r p t i o n e x h i b i t s approximately a t e n f o l d i n c r e a s e w i t h r e s p e c t t o t h e c a s e o f t o t a l l y d i f f u s e s c a t t e r i n g .

C10-374 JOURNAL DE PHYSIQUE

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(1981) 632.

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A.G.~al~shukov, Sol.State Cornmun. (1982) 1257.

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