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LOCAL-FIELD EFFECTS AND ZONE-CENTER
PHONONS IN POLAR AND COVALENT CUBIC
SEMICONDUCTORS
R. Resta, A. Baldereschi
To cite this version:
JOURNAL DE PHYSIQUE
CoZZoque C6, suppldment au n o 2 2 , Tome 42, de'cembre 1982 page C6-661
LOCAL-FIELD EFFECTS AND ZONE-CENTER PHONONS I N POLAR AND COVALENT CUBIC SEMICONDUCTORS
R . Resta and A . Baldereschi
I n s t i t u t de Physique AppZiqu&e, EPFL, 1025 Lausanne, SwitzerZand
Abstract.- Microscopic lattice-dynamics c a l c u l a t i o n s based on f i r s t - p r i n c i p l e d i e l e c t r i c matrices a r e presented f o r S i , Ge, GaAs, and ZnSe. Phonon f r e - quencies a r e c a l c u l a t e d i n t h e l i m i t o f l o n g wavelength and compare w e l l w i t h experimental data. L o c a l - f i e l d e f f e c t s a r e e s s e n t i a l t o e x p l a i n t h e trends w i t h i n c r e a s i n g i o n i c i t y i n t h e Ge i s o e l e c t r o n i c s e r i e s . The e l e c - t r o n i c p o l a r i z a t i o n d e n s i t y associated t o zone-center phonons i s analyzed i n r e a l space.
1. I n t r o d u c t i o n . - Recent f i r s t - p r i n c i p l e i n v e s t i g a t i o n s o f t h e l a t t i c e dynamics o f s o l i d s a r e based e i t h e r on the s e l f - c o n s i s t e n t scheme1 o r on t h e linear-response approach2. Advantages and shortcomings o f both techniques a r e p r e s e n t l y being studied.
The microscopic linear-response scheme i s based on t h e i n v e r s e d i e l e c t r i c ma- t r i x (IDM) o f a s o l i d and amounts t o evaluate t h e e l e c t r o n i c response t o t h e l a t t i c e d i s t o r t i o n o f a f r o z e n - i n phonon. The few f i r s t - p r i n c i p l e c a l c u l a t i o n s a v a i l a b l e t o date3 r e f e r t o S i o n l y and g i v e c o n f l i c t i n g r e s u l t s on the accuracy a t t a i n a b l e .
Here we present a f i r s t - p r i n c i p l e linear-response study o f zone-center phonons i n b o t h c o v a l e n t ( S i and Ge) and p o l a r (GaAs and ZnSe) cubic semiconductors. Be- s i d e s phonon frequencies, we a l s o compute i n r space t h e e l e c t r o n i c p o l a r i z a t i o n associated t o phonon-modes. This q u a n t i t y , up t o now a v a i l a b l e f o r S i only4, i s e s s e n t i a l f o r a b e t t e r understanding o f t h e e l e c t r o n i c c o n t r i b u t i o n t o i n t e r a t o m i c f o r c e constants and f o r developing more r e a l i s t i c models i n l a t t i c e dynamics. 2. Phonon frequencies a t B r i l l o u i n - z o n e center.- We have c a l c u l a t e d phonon frequen-
2
c i e s a t q P O u s i n g the linear-response scheme based on the Appelbaum-Hamann i o n i c
5 6
pseudopotentials and r e c e n t l y c a l c u l a t e a IDM's o f o r d e r 113
.
The l a t t e r have beet1 extended by adding t h e L i n d h a r t f r e e - e l e c t r o n c o n t r i b u t i o n o f 116 a d d i t i o n a l r e c i p r o c a l l a t t i c e vectors. The r e s u l t i n g phonon frequencies a r e r e p o r t e d i n Table I t o g e t h e r w i t h experimental data. The o v e r a l l r e l a t i v e e r r o r i s about 10 % (20 % i n ZnSe).C6-662 JOURNAL DE PHYSIQUE
The TA frequency i s vanishing and only the TO one i s of i n t e r e s t . We analyze 2
vO
in Table I1 and find t h a t ( i ) local f i e l d s ( i .e. off-diagonal IDM elements) are more important i n more ionic materials, indeed they a r e responsible f o r the lowering of wTo with i o n i c i t y in the Ge isoelectronic s e r i e s , and ( i i ) local-field e f f e c t s a r e more relevant i n Ge than in S i .Opposite t o the transverse case, the dynamical matrix f o r longitudinal phonons a t q = 0 depends a l s o on the nonanalytic part of the IDM through the Born e f f e c t i v e
2
charges Z i and ZE ( t h e A,C labels indicate anion or cation) which should s a t i s f y the relationship Z i
+
Z;
= 0 as required by the acousticsum
r u l e (ASR). In our f i r s t - p r i n c i p l e calculation without adjustable parameters the ASR i s not exactly f u l f i l l e d ( c f . the calculated values of Z* given in the lower portion of Table I ) and t h i s i s why we obtain wLA#
0. The l a r g e s t ASR violation i s 16 % i n ~ n ~ e ; simi- l a r deviations were previously obtained f o r For the LO frequency we obtain wLO = wTO in covalent c r y s t a l s (a r e s u l t of symmetry alone) and we predict reasonably well t h e LO
-
TO s p l i t t i n g in polar materials. Due t o the ASR violation, the calculated longitudinal eigenvectors a r enot
exactly the acoustic and optic nodes. The mixing however i s small.S i Ge GaAs ZnSe : Polarization electron density
3. E l e c t r o n i c p o l a r i z a t i o n associated t o zone-center phonons
.-
We do n o t consider here q = 0 a c o u s t i c phonons since they correspond t o r i g i d t r a n s l a t i o n s o f t h e l a t t i c e and induce t h e r e f o r e the same r i g i d t r a n s l a t i o n o f the e l e c t r o n d e n s i t y . The q 2 0 LO and TO modes d i f f e r on a macroscopic scale b u t t h e i r i o n i c displacements-? -1 -? -? +
i n t h e u n i t c e l l a r e t h e same f o r q -? 0 and are given by uA =
-
B u and uc = B uwhere 8 = MAIMC i s t h e nuclear mass r a t i o , and
;
i s an e f f e c t i v e amp1 i tude of +t h e f r o z e n - i n phonon which i n the computations has been chosen as b = 0.005 ( 1 ,I ,1) i n a.u.. E l e c t r o n i c p o l a r i z a t i o n d e n s i t i e s induced by these displacements have been c a l c u l a t e d by i n s e r t i n g the 1 inear-response e l e c t r o n i c screening p o t e n t i a l i n t o Poisson equation.
I n c o v a l e n t m a t e r i a l s t h e p o l a r i z a t i o n d e n s i t i e s associated t o LO and TO modes 4 a r e t h e same f o r q + 0 and they a r e s i m i l a r t o t h e r e s u l t already known f o r S i
.
E l e c t r o n i c screening mostly corresponds t o a t r a n s f e r from s t r e t c h e d t o compressed bonds.
I n p o l a r m a t e r i a l s t h e LO and TO modes f o r q = 0 induce t h e same macroscopic p o l a r i z a t i o n b u t a d i f f e r e n t microscopic response (Fig. I ) , t h e d i f f e r e n c e being t h e same e l e c t r o n i c p o l a r i z a t i o n d e n s i t y as t h e one induced by an e f f e c t i v e e x t e r - n a l u n i f o r m f i e l d
8
=-
4~ ( Z iZA
+
Zt
Zc)
/ R where R 'is t h e u n i t c e l l volume. The e l e c t r o n i c response i s s t r o n g e r f o r t h e TO than f o r t h e LO mode i n agreement w i t h energy considerations. E l e c t r o n i c response i n ZnSe i s mostly given by d i p o l e s a t the atomic s i t e s as one expects i n an i o n i c m a t e r i a l and c o n t a i n s a very l a r g e l o c a l - f i e l d c o n t r i b u t i o n . The r e s u l t s f o r GaAs are i n t e r m e d i a t e between those o f Ge and ZnSe.4. Acknowledgement.- Work supported i n p a r t by t h e Swiss National Foundation, and t h e GNSM
-
CNR, Pisa, I t a l y .References
1. H. Wendel and R.
M.
Martin, Festki3rperprobleme2, 21 (1979); M.T. Y i n and M.L. Cohen, Phys. Rev. L e t t .-
45, 1004 (1980).2. R.M. Pick, M.H. -Cohen and R.M. M a r t i n , Phys. Rev.
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42, 1224 (1979).4. A. Baldereschi and K. Maschke, Proc. I n t . Conf. L a t t i c e Dynamics (Flammarion, P a r i s , 1978) p. 36.
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