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Submitted on 1 Jan 1981
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TIME REVERSAL SYMMETRY AND THE RAMAN
SCATTERING BY CRYSTALS
A. Ramdas, S. Rodriguez
To cite this version:
JOURNAL DE PHYSIQUE
CoZZoque C6, suppZEment a u n012, Tome 42, ddcernbre 1981 page C6-887
TIME REVERSAL SYMMETRY AND THE RAMAN SCATTERING BY CRYSTALS A . K . Ramdas and S. Rodriguez
Department o f P h y s i c s , Purdue U n i v e r s i t y , W e s t L a f a y e t t e , I n d i a n a 47907,
1'. S. A.
Abstract.- A survey of the e f f e c t o f time r e v e r s a l symmetry on t h e phonon spectra o f c r y s t a l s i s presented. The l i n e a r wave v e c t o r dependence o f t h e d i s p e r s i o n curves o f t h e zone c e n t e r p o l a r modes as w e l l as s p l i t t i n g o f shear acoustic modes a r e accessible t o o b s e r v a t i o n i n t h e Raman and B r i l l o u i n spectra o f selected l i n e s i n s u i t a b l e c r y s t a l s .
I n p e r f e c t c r y s t a l s , the symmetry of the l a t t i c e imposes r e s t r i c t i o n s on t h e n a t u r e of t h e normal modes of v i b r a t i o n . For example, phonons o f i n f i n i t e wave l e n g t h can be e i t h e r non-degenerate o r a t most doubly degenerate except f o r c r y s t a l s belonging t o one o f the f i v e cubic classes i n which case, i n a d d i t i o n t o t h e above, there can e x i s t t r i p l y degenerate phonons. These conclusions f o r these modes as w e l l as t h e i r eigenvectors can be deduced from t h e knowledge o f t h e symmetry o f t h e c r y s t a l and t h e content o f i t s p r i m i t i v e c e l l .
I n a d d i t i o n t o t h e geometrical symmetry o p e r a t i o n s of t h e group of t h e c r y s t a l under i n v e s t i g a t i o n , t h e equations o f motion o f t h e atoms ( c l a s s i c a l o r q u a n t a l ) are i n v a r i a n t under t h e o p e r a t i o n o f time r e v e r s a l . This a d d i t i o n a l r e - s t r i c t i o n has i n t e r e s t i n g consequences f o r the d i s p e r s i o n (dependence o f t h e f r e - quency o f the phonons on t h e i r wave vectors 6 ) o f the normal modes.
L e t each p r i m i t i v e c e l l o f the c r y s t a l be denoted by a t r a n s l a t i o n v e c t o r and l e t a = 1,2,
...
3 f l a b e l the degrees o f freedom o f t h e f atoms i n each c e l l . I n the harmonic approximation, t h e equations o f motion o f t h e displacements una from t h e e q u i l i b r i u m c o n f i i u r a t i o n areMains
=-
1
Caa,(i;
-
E l ) Un 1 a l I n ' a 'where Caa,(G
-
6 ' )
are the f o r c e constants and Ma t h e mass o f the atom associated w i t h the a degree o f freedom; t r a n s l a t i o n a l symmetry r e q u i r e s t h a t Caal(; - + n u ) depend o n l y on6
-
if'.
Furthermore Caa, (if-
E l ) i s symmetric i n a,al and;,So.
The propagating s o l u t i o n s o f these equations w i t h wave v e c t o r4
and angular f r e - quency~(4)
obey t h e eigenvalue equation+ 2 +
1
CCaal(q)-
w ( 9 ) A a a 1 I Uao = 0,
a 'JOURNAL DE PHYSIQUE
where
+
c ~ ~ , ( ) = (M,M,,)-~
caa.
( f-
n l ) e x pC-ib
.
( 5-
f l ) l.
( 3 )
n '
C(q), t h e dynamical m a t r i x , i s Hermitian and p o s i t i v e d e f i n i t e . Time-reversal symnetry imposes, i n a d d i t i o n ,
E*(-b) = C(3)
.
( 4 )Pine and ~ r e s s e l h a u s " ~ have shown t h a t t h i s r e q u i r e s t h e frequency o f a l l o p t i c a l phonons t o vary q u a d r a t i c a l l y w i t h wave v e c t o r i n the v i c i n i t y o f
5
= 0 w i t h t h e exception o f t h e doubly degenerate p o l a r phonons o f c r y s t a l classes C3,C4,C6,D3,D4. and D6 and the t r i p l y degenerate p o l a r phonons of c r y s t a l s w i t h p o i n t group s y m t r i e s T and 0. The frequencies o f these phonons vary l i n e a r l y w i t h6
near +q = 0. The symmetry groups enumerated above possess o n l y proper r o t a t i o n s . The presence o f an improper o p e r a t i o n i s s u f f i c i e n t t o i n v a l i d a t e the p o s s i b i l i t y o f t h i s behavior.
For a c o u s t i c a l phonons, s i n c e w(G) tends t o zero as
3
+ 0 and t h e eigenvalue equation (2) gives w2, s o l u t i o n s f o r a l l c r y s t a l classes a r e l i n e a r i n wave v e c t o r near{
= 0. However, shear a c o u s t i c waves propagating along t h e o p t i c axes o f c r y s t a l s o f symmetry C3,C4,C6,D3,D4, o r D6 and along t h e (100) o r (111) d i r e c t i o n s o f c r y s t a l s o f classes T and 0, corresponding t o r i g h t - and l e f t - c i r c u l a r p o l a r - i z a t i o n s have d i f f e r e n t phase v e l o c i t i e s . This gives r i s e t o the phenomenon o f a c o u s t i c a l a c t i v i t y . 3Raman and B r i l l o u i n s c a t t e r i n g are i d e a l techniques t o i n v e s t i g a t e t h e be- h a v i o r o f phonons described above. The wave v e c t o r conservation r u l e e s t a b l i s h e s t h a t t h e phonon i n v o l v e d i n a given i n e l a s t i c s c a t t e r i n g has a wave v e c t o r equal t o
fi
kL f o r r i g h t - a n g l e s c a t t e r i n g and 2 kL f o r b a c k - s c a t t e r i n g wherexL
i s t h e wave v e c t o r o f t h e i n c i d e n t r a d i a t i o n . The energy d i f f e r e n c e s between t h e d i f f e r e n t phonon branches i n t o which t h e zone center p o l a r modes s p l i t s are small f o r 0 < q < 2 kL. This n e c e s s i t a t e s the study o f a p p r o p r i a t e narrow l i n e s i n t h e Raman s p e c t r a r e q u i r i n g temperatures i n t h e l i q u i d helium range and h i g h r e s o l u t i o n spectroscopy w i t h a Fabry-Perot i n t e r f e r o m e t e r .Studies on s u i t a b l e Raman l i n e s i n a-quartz and B i 12Ge020 ,5 have c l e a r l y demonstrated the above phenomena. The l a t t e r m a t e r i a l i s c u b i c w i t h p o i n t group symmetry T. The t h r e e - f o l d degenerate p o l a r o p t i c a l phonons a t
4
= 0 a r e s p l i t i s o t r o p i c a l l y according t o an e f f e c t i v e Hamiltonian i n t e r a c t i o n o f t h e form-+
H ' = A $ . I
,
-+
( 5 )We n o t e t h a t i n t h e presence o f a magnetic f i e l d ,
2,
time r e v e r s a l symnetry breaks down unless t h e time r e v e r s a l operator i s a p p l i e d a l s o t o the sources o f the magnetic f i e l d . Then, i n s t e a d o f Eq. ( 4 ) we havet*(-$,
-8)
=t($,E)
.
( 6 ) A t h e o r e t i c a l study o f t h e e f f e c t s o f a magnetic f i e l d on the phonon modes o f c r y s t a l s a t6
= 0 has been c a r r i e d o u t by Anastassakis9
f12E
The consequences regarding l i n e a r - $ dependence o f phonon frequencies a r e s i m i l a r t o those f o r t h e linear-; dependence discussed above, b u t occurs i n a l a r g e r number o f c r y s t a l classes. I n general t h e changes i n phonon energies due t o t h e presence o f a m a g n e t i c f i e l d are small. However, i f an i n t e r a c t i o n o f t h e resonant type occurs, i - e . , i f t h e r e are magnetic and phonon e x c i t a t i o n s o f t h e c r y s t a l having approximately equal7 energies, the e f f e c t s can be l a r g e . This i s i l l u s t r a t e d i n the work o f Schaack on r a r e e a r t h compounds.
F i n a l l y , an i n t e r e s t i n g case o f 1 inear-; dependence o f phonon frequencies i n h e l i c a l chains has been i n v e s t i g a t e d by ~ m a i n o . ~ I n a model study he e x h i b i t e d t h e l i n e a r dependence o f doubly degenerate o p t i c a l phonons i n l i n e a r h e l i c a l chains con- t a i n i n g t h r e e atoms p e r t u r n w i t h equal f o r c e constants between t h e f i r s t t h r e e nearest neighbors.
References
A. S. Pine and G. Dresselhaus, Phys. Rev.
188,
1489 (1969). A. S. Pine and G. Dresselhaus, Phys. Rev. B4, 356 (1971).A. S. Pine and G. Dresselhaus, Proceedinqs o f t h e I n t e r n a t i o n a l School of Physics "Enrico Fermi", Course 52, E. B u r s t e i n , ed., (Academic Press, New York, 1972), pp. 325-344.
M. H. Grimsditch, A. K. Ramdas, S. Rodriguez, and V. J. Tekippe, Phys. Rev. B E , 5869 (1977).
W. Imaino, A. K. Ramdas, and S. Rodriguez, Phys. Rev. B z , 5679 (1980). E. Anastassakis, E. B u r s t e i n , A. A. Maradudin, and R. Minnick, J. Phys. Chem. Sol i d s ,
33,
519, 1091 (1972).G. Schaack, Proceedinqs, V I I t h I n t e r n a t i o n a l Conference on Raman Spectroscopy, Ottawa, Canada (1980); W. F. Murphy, ed., (North Holland, Amsterdam, 1980), pp. 18-21.
