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RESONANT BRILLOUIN SCATTERING IN SPATIALLY DISPERSIVE MEDIUM
E. Albuquerque, C.E.T. da Silva
To cite this version:
E. Albuquerque, C.E.T. da Silva. RESONANT BRILLOUIN SCATTERING IN SPATIALLY DISPERSIVE MEDIUM. Journal de Physique Colloques, 1984, 45 (C5), pp.C5-61-C5-64.
�10.1051/jphyscol:1984507�. �jpa-00224125�
JOURNAL DE PHYSIQUE
Colloque C 5 , supplement au n ° 4 , Tome 4 5 , avril 1984 page C5-61
R E S O N A N T B R I L L O U I N S C A T T E R I N G IN S P A T I A L L Y D I S P E R S I V E M E D I U M *
E.L. Albuquerque and C E . T . Gonçalves da Silva*
Departamento de Fisiaa, Universidade Fédéral do Rio Grande do Norte, 59 000 - Natal, EN, Bvaz.il
*Instituto de Fisiea, VNICAMP, 13 100 - Campinas, SP, Brazil
Résumé - Nous présentons quelques aspects qualitatifs, qui découlent de la symétrie, de la diffusion Brillouin à l'interface vide-milieu cubique disper- sif. Nous démontrons l'absence d'effet du polariton d'exciton sur le pic dû à 1'onde de Rayleigh.
Abstract - We discuss qualitative, symmetry dependent, features of Brillouin scattering by an interface vacuum - cubic dispersive medium. The peak due to Rayleigh wave is shown to be unaffected by exciton polariton formation.
Spatial dispersion (SD) effects due to wavevector dependence of the dielectric function have been extensively studied since the phenomenological model of Pekar /I/
and Hopfield and Thomas / 2 / . Recently, the great development oftunable lasers stimulated a very thorough investigation of these effects and led to the elucidation of several important features of SD materials. In particular, resonant Brillouin scattering (RBS) is a very useful tool to clarify their fundamental properties, mainly in reasonably pure, direct-gap semiconductors at sufficiently low temperatures /3, 4, 5, 6/.
The purpose of this communication is to present some theoretical considerations about RBS via exciton polaritons in a SD medium. Due to space limitations, we discuss here only qualitative features of the spectra. Complete details, including numerically evaluated Brillouin spectra, are given in a forthcoming paper by the authors.
Let an electromagnetic wave, of frequency to-, and wavevector k, be incident, at an angle eT, from the vacuum into a semi-infinite medium which shows SD effect. The SD medium is assumed to occupy the half-space z < 0 with a flat surface in the z = 0 plane. The SD medium can be described by a dieletric function e((o, k) given by
where E . is the background dielectric constant, and
is the exciton susceptibility. In (2), s is the oscillator streght, u is the frequency of the uncoupled mode, r is the damping coefficient and D = n.io /m*, where m* is the exciton effective mass.
In the SD medium, the exciton polariton interacts with a crystal excitation of wavevector Q and frequency u to produce a scattered wave of frequency to = M-, - to.
This interaction is mediated by a deformation potential, where an i-polarizea (i can be x, y, or z) acoustic mode
changes the band gap of the SD medium by an amount
Work partially supported by the Brazilian Agencies CNPq and FINEP
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1984507
JOURNAL DE PHYSIQUE
. .
where u T J means
a u ~ / a
j.
T h i s e f f e c t can be c o n s i d e r e d i n t h e e q u a t i o n o f m o t i o n o f t h e e x c i t o n f i e l d p bJ augmenting i t , as a p e r t u r b a t i o n , w i t h t h e t e r mHere, oijkl i s a f o u r t h - r a n k t e n s o r which measures t h e c o u p l i n g between and
z;
i,j a r e l a b e l s r e l a t e d w i t h t h e p o l a r i z a t i o n o f t h e i n c i d e n t and s c a t t e r e d l i g h t r e s p e c t i v e l y and k, 1 a r e l a b e l s r e l a t e d w i t h t h e p o l a r i z a t i o n o f t h e a c o u s t i c phonons. I n ( 5 ) , t h e c o n v e n t i o n t h a t r e p e a t e d s u p e r s c r i p t a r e summed o v e r i s adopted.
I f we c o n s i d e ~ ~ 9 ~ S D myqium i t h c u b i c ~ ~ ~ $ t ~ y , ~ ~ $ h e o n l y n o n - v a n i s h i n g components o f n a r e / 7 / n = r l , n X ~ Y Y = " 1 2 , 11 n and t h e i r symmetric c o u n t e r p a r t s deduced b y p e r m u t a t i o n o f t h e i n d i c e s x, y and z.
F o r an i n c i d e n t l i g h t wave, s - p o l a r i z e d , we have two t r a n s v e r s e e x c i t o n p o l a r i t o n branches e x c i t e d , w h i l e f o r an i n c i d e n t l i g h t wave, p - p o l a r i z e d , t h e l o n g i t u d i n a l mode i s a l s o e x c i t e d . Therefore, f o r s c a t t e r i n g by b u l k a c o u s t i c phonons, t h i s means t h a t we have, f o r a p p r o p r i a t e i n c i d e n c e f r e q u e n c y w I , ( a ) f o u r stokes ( o r a n t i - s t o k e s ) r e s o n a n t l i n e s i n s + s s c a t t e r i n g , ( b ) s i x r e s o n a n t l i n e s i n e i t h e r s + p o r p + s s c a t t e r i n g , and ( c ) n i n e r e s o n a n t l i n e s i n p -t p s c a t t e r i n g .
Table 1 shows a l l t h e s e r e s u l t s , where we have c o n s i d e r e d t h e e l e c t r i c f i e l d o f t h e l i g h t = (0, E ~ , 0 ) f o r s - p o l a r i z a t i o n and
2
= (E', 0, E') f o r p - p o l a r i z a t i o n .TABLE 1 Non-vanishing
t e n s o r YYXX
"YYZZ
n x ~ x ~
"
ZY ZY"xxxx
"xxzz
"zzzz
" Z Z X X
"zxzx
"xzxz
rl YXYX
n YZYZ
rl
The l a s t column o f Table 1 g i v e s i n f o r m a t i o n about t h e p o s s i b i l i t y o r n o t t o have t h e p r o p a g a t i o n o f s u r f a c e waves o f t h e R a y l e i g h type. One i m p o r t a n t q u e s t i o n about t h i s mode i s whether i t g i v e s i n f o r m a t i o n abobt t h e e x c i t o n p o l a r i t o n i n t h e B r i l l o u i n spectrum. To answer t h i s q u e s t i o n l e t us c o n s i d e r t h e p r o j e c t i o n on t h e x-y p l a n e o f t h e s c a t t e r e d wave v e c t o r
ks
( s e e F i g u r e 1 ) .The R a y l e i g h wave i s c h a r a c t e r i z e d b y a 2D wavevector
5,
w i t h f r e q u e n c y W=VI;[.
The k i n e m a t i c e q u a t i o n f o r s t o k e s s c a t t e r i n g i s R Type of
s c a t t e r 1 ng s + s s + s S + P S + P P ' P P ' P P ' P P ' P P ' P P + P P + S P + S
T h e r e f o r e , f r o m ( 6 ) we have t h a t
"'I 2 W W
q2=
(3f
s i n 2 es+
(-) s i n 2 e1-
2-
I s i n es s i n eI cos 6C C c2
Phonon p o l a r i z a t i o n x z
-
p l a n e x z-
p l a n e y-
d i r e c t i o n y-
d i r e c t i o n x z-
p l a n e x z-
p l a n e x z-
p l a n e xz-
p l a n e x z-
p l a n e x z-
p l a n e y-
d i r e c t i o n y-
d i r e c t i o nAs q = W / V and w = wI
-
w we have, t o g e t h e r w i t h ( 7 ) , t h r e e simultaneous equations f o r q, W, 8nd4.
The s o l u ? i o n o f t h e s e e q u a t i o n s determines t h e f r e q u e n c y a tRay l e l g h wave
No Yes No No No Yes Yes No Yes Yes No No
which t h e s h a r p s c a t t e r i n g peak due to+Rayieigh+wave i s s e e n , and+hence is determined s o l e l y by x-y plane values k and k
.
However, a s k a n d s i a r e t h e same f o r external 1 ig h t and a1 l exc! 4 6 n - p o l a ~ f i o n branches, t h & l d i s nosl#l i tting of Rayleigh peak due t o e x c i t o n - p o l a r i t o n formation! This i s a very importantinformation s i n c e the Ray1 eigh peak could mask some resonant peaks.
Figure 1: To i l l u s t r a t e t h e p r o j e c t i o n s of t h e i n c i d e n t + ( a I ) and s c a t t e r e d ( z S ) wavevetor and t h e Rayleigh wavevector q.
So f a r we have discussed RBS considering t h a t t h e i n t e r f a c e between t h e SD medium and vacuum i s f l a t . However, as i t was pointed out by Loudon /8/ t h e r e i s another
mechanism t h a t can a l s o s c a t t e r 1 ig h t : t h e s o - c a l l ed s u r f a c e - r i p p l e mechanism. The g e n e r a l f o r n of t h e c r o s s s e c t i o n i n t h i s case i s / 8 , 9/
where t h e constant o f p r o p o r t i o n a l i t y includes f a c t o r s depending on
e
e @ and t h e o p t i c a l p r o p e r t i e s o f t h e two media. The power spectrum <..
.> c o n t i i n ? 'a& - f u n c t i o n peak from t h e Rayleigh wave and broadened s t r u c t u r e from t h e bulk acoustic modes. However, n e i t h e r t h e Rayleigh peak nor t h e broad s t r u c t u r e can be s p l i t by t h e e x c i t o n - p o l a r i t o n s branches, although t h e c o n s t a n t of p r o p o r t i o n a l i t y must depend on p r o p e r t i e s of e x c i t o n i c medium.
In summary, we have drscussed t h e q u a l i t a t i v e , symmetry dependent, p r o p e r t i e s of B r i l l o u i n s c a t t e r i n g ;jf l i g h t by a cubic s p a t i a l l y d i s p e r s i v e medium with a s i n g l e exci ton branch coupled t o t h e e l e t r o m a g n e t i c modes. Q u a n t i t a t i v e r e s u l t s , a s we1 1 a s t h e n a t u r a l extensions of t h i s work t o include r e a l i s t i c exc i tor] d i s p e r s i o n / l o / w i l l be presented elsewhere. Of g r e a t i n t e r e s t a l s o i s t h e study of layered semiconductor s t r u c t u r e s , e . g . , s u p e r - l a t t i c e s / 1 1 / , which we a r e undertaking.
The a u t h o r s would l i k e t o acknowledge s t i m u l a t i n g d i s c u s s i o n s with Dr. D . R . T i l l e y .
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(1963) 5 6 3 ) . / 3 / T I L L E Y , D . R . , J . Phys. C - 13 (1980) 781./4/ SO, V . C . Y . , SIPE, J . E . , FUKUI, M . , and STEGEPAN, G.I., 3 . Phys. C 14 (1981)
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/5/ FUKUI, M . and TADA, O . , J . Phys. Soc. Japan
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(1982) 172.C5-64 JOURNAL DE PHYSIQUE
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/8/ LGUDON, R., Phys. Rev. L e t t . - 40 (1978), 581.
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/ l o /
SERMAGE, B. and FISHMAN, G., Phys. Rev. L e t t .43
(1979) 1043./11/ See, f o r i n s t a n c e , Proccedings o f t h e 1 6 t h I n t e r n a t i o n a l Conference on t h e Physics o f Semiconductors, Averous, M., ed., Physica 1178 and (1983).
