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Submitted on 1 Jan 1981
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THE DIELECTRIC FUNCTION MATRIX AND THE
LATTICE DYNAMICS OF KC1
M. Ball, W. Leung
To cite this version:
M. Ball, W. Leung.
THE DIELECTRIC FUNCTION MATRIX AND THE LATTICE
JOURNAL
DE
PHYSIQUECo Lloque
C6,suppliment au
n o 12,Tome
4 2 ,ddcembre
1981page
C6-911THE DIELECTRIC FUNCTION MATRIX A N D THE LATTICE DYNAMICS OF KC1
M.A. B a l l andW.N.
LeungD.A.M.
T.P.,University o f Liverpool, Liverpool
L69 3BX, U .K.
A b s t r a c t .
-
The RPA d i e l e c t r i c f u n c t i o n m a t r i x of KCL is c a l c u l a t e d w i t h a model i n which t h e v a l e n c e band c o n s i s t s of p - s t a t e s on t h e Ci atoms and t h e c o n d u c t i o n bands a r e e i t h e r ( i ) a s i n g l e OPW band o r ( i i ) two OPW bands. Local f i e l d e f f e c t s a r e a p p r o x i m a t e l y i n c l u d e d . R e s u l t s f o r t h e d i a g o n a l , some o f f - d i a g o n a l e l e m e n t s and t h e e f f e c t i v e c h a r g e s a r e p r e s e n t e d . The long-range p a r t s of t h e dynamical m a t r i x a r e g o t from t h e c a l c u l a t i o n s ; t h e s h o r t - r a n g e p a r t s a r e o b t a i n e d by f i t t i n g t o t h e t r a n s v e r s e f r e q u e n c i e s .We r e p o r t some model c a l c u l a t i o n s of t h e d i e l e c t r i c f u n c t i o n m a t r i x
e ( q
-
+
g,- - -
q+
g') o f t h e i o n i c c r y s t a l KC&, i n c l u d i n g r e s u l t s f o r t h e d i a g o n a l and some o f f - d i a g o n a l e l e m e n t s . We a l s o show how t h e dynamical m a t r i x c a n b e e x p r e s s e d i n t e r m s of t h e s e r e s u l t s .The d i e l e c t r i c f u n c t i o n E i s g i v e n i n t h e R.P.A. by
where t h e sum i s o v e r t h e f i r s t B r i l l o u i n zone and t h e o c c u p i e d and unoccupied bands n2 and n l r e s p e c t i v e l y . h e most i m p o r t a n t o f t h e s e bands a r e t h e uppermost v a l e n c e
(p-) band and t h e l o w e s t c o n d u c t i o n band. Fry /1/ and L i p a r i / 2 / u s e f o r t h e v a l e n c e band t i g h t - b i n d i n g wave-functions w i t h p - l i k e s t a t e s c e n t r e d on t h e CP.
atoms.
The exponent 6 i n U ( r ) i s a p a r a m e t e r . PO
-
For t h e c o n d u c t i o n band e n e r g i e s and wave-functions two s e p a r a t e models w e r e used:
-
( i ) Only o n e band was c o n s i d e r e d , t h i s b e i n g a p a r a b o l i c ( s - ) band d e r i v e d from a s i n g l e o r t h o g o n a l i z e d p l a n e wave (OPW)
E (k) = E:
+
a l k I 2C ( 4 )
C6-9 12 JOURNAL DE PHYSIQUE
where N(k) is a n o r m a l i z a t i o n f a c t o r and pkm i s t h e o r t h o g o n a l i z a t i o n c o e f f i c i e n t . ( i i ) I n t h i s model two bands (s- and d-) were i n c l u d e d ;
-
'sk = ak,g'ck(f) + 6k,g'c(k
+
g)(')The r e c i p r o c a l - l a t t i c e v e c t o r g i n ( 6 ) and ( 7 ) depends on k.
T i g h t - b i n d i n g e x p r e s s i o n s were u s e d f o r t h e e n e r g i e s a n d t h e c o e f f i c i e n t s a and 6 were d e t e r m i n e d by f i t t i n g t o t h e s e e n e r g i e s . The a d v a n t a g e of t h i s model o v e r ( i ) is t h a t i t t a k e s B r i l l o u i n Zone boundary e f f e c t s i n t o a c c o u n t .
The e n e r g y p a r a m e t e r s i n ( i ) and ( i i ) were d e t e r m i n e d by f i t t i n g t o t h e energy- band c a l c u l a t i o n s of Kunz / 3 / . The d i e l e c t r i c f u n c t i o n c is t h e n c a l c u l a t e d b y n u m e r i c a l i n t e g r a t i o n o v e r t h e f i r s t B r i l l o u i n Zone. T h i s i s more r e a l i s t i c t h a n t h e p r o c e d u r e s of Fry /1/ and L i p a r i 1 2 1 .
To i n v e r t t h e d i e l e c t r i c m a t r i x , t h e f o l l o w i n g a p p r o x i m a t e f o r m u l a e a r e used:-
The p a r a m e t e r 6 i n U ( r ) is d e t e r m i n e d by s e t t i n g 1 / ~ - ~ ( 0 , 0 ) e q u a l t o t h e e x p e r i - PO
-
m e n t a l v a l u e . A s t h e second term i n ( 7 ) i n c o r p o r a t e s some of t h e l o c a l - f i e l d c o r r e c t i o n s , o u r c a l c u l a t i o n d o e s i n c l u d e t h e s e a p p r o x i m a t e l y . Using 52 r e c i p r o c a l l a t t i c e v e c t o r s , t h e s e c o r r e c t i o n s c h a n g e ' t h e r e s u l t s by a p p r o x i m a t e l y 6% and 7% w i t h models (i) and ( i i ) r e s p e c t i v e l y . Some of o u r c a l c u l a t i o n s a r e p r e s e n t e d i n F i g s . 1 & 2. I n t h e s e exchange and c o r r e l a t i o n have a s y e t been n e g l e c t e d .
A sensible test of our calculations of E is provided by the lattice dynamics. The effective charge tensor
zeff(,)
-
of an ion of type K is got from the q+O limit of Z(~,K), the effective charge vector, where. -
and W ( q , ~ ) is the unscreened pseudopotential. As a first approximation this has been taken as -Z(K)V(~) with Z(CI?) =
5
and Z(K) = 1. Our calculations give Zeff(c%) = -.96(i), -1.24(ii) andzeff(~)
= 1.21(i), 1.20(ii). These results compare favourably, considering the approximations made, with the experimental values of '1.123.The dynamical matrix is
in the notation of Sham 141. In (10) the first and second terms represent the long- and short-range effects respectively. The latter can be described by the Lorentz field effect and some short-range force constants. These determine completely the transverse phonons, which can almost be fitted with two (the
nearest-neighbour) parameters and can be fitted to within
5%
by using5
parameters. IAt present we are trying to calculate the longitudinal frequencies using the above short-range forces and our calculated values of Z(q,r) and ~-l(~,~). We have
- -
-. . . -added a small constant to Z(q,CI1) to ensure that the acoustic sum rule is satisfied.
- -
At small q our results are reasonable but at larger values there is a large varia- tion in Z(q,CI?), giving rise to imaginary frequencies. We are investigating the- -
causes of this./I/ Fry, J.L., Phys. Rev.
