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VIBRATIONAL FREQUENCIES VIA FROZEN
PHONONS
B. Harmon, W. Weber, D. Hamann
To cite this version:
JOURNAL DE PHYSIQUE
CoZZoque C6, suppte'ment au n o 12, Tome 42, de'cembre 1981 page C6-628
I
VIBRATIONAL FREQUENCIES VIA FROZEN PHONONS
B.N. Harmon *(')
,
W. Weber*and D .R. ~amann*" iKernforschungszentrwn KarZsruhe, I n s t i t u t
fiir
Angewandte Kernphysik I , 0-7500 KarZsruhe, F.R.G.**
Be22 LaSoratories, Murray H i l l ,
NJ
07974, U.S.A.Abstract.- We have used a f i r s t p r i n c i p l e s l i n e a r combination of atomic o r b i t a l s (LCAO) method t o c a l c u l a t e t h e t o t a l ground s t a t e energy f o r c r y s t a l s of S i , Nb and Mo i n v o l v i n g l a t t i c e d i s t o r t i o n s . From t h e s e cal- c u l a t i o n s t h e e q u i l i b r i u m l a t t i c e c o n s t a n t , cohesive energy, and bulk modulus a s well a s t h e v i b r a t i o n a l f r e q u e n c i e s f o r s e l e c t e d phonons were
determined.
1. I n t r o d u c t i o n . - Band t h e o r e t i c a l methods a r e f i n d i n g i n c r e a s e d use i n t h e s t u d y of e l e c t r o n i c response t o l a t t i c e d i s t o r t i o n (caused by compression, s t r e s s , phonons, e t c . ) . These techniques provide a t o o l f o r a c c u r a t e l y calcu- l a t i n g from f i r s t p r i n c i p l e s t h e frequency and charge d e n s i t y response f o r phonons a t s e l e c t e d wavevectors. These t e c h n i q u e s have a l r e a d y been a p p l i e d t o S i w i t h c o n s i d e r a b l e s u c c e s s u s i n g t h e p s e u d o p o t e n t i a l and LCAO methods. Here we b r i e f l y p r e s e n t o u r LCAO method and our r e s u l t s f o r S i and then d i s c u s s t h e a p p l i c a t i o n of t h e method t o m e t a l s g i v i n g p r e l i m i n a r y r e s u l t s f o r Nb and Mo
.
2. Method.- We have used t h e l o c a l d e n s i t y approximation f o r exchange and cor- r e l a t i o n 4 combined w i t h a f i r s t p r i n c i p l e s t i g h t binding method, t h e d e t a i l s of which have been described e l s e ~ h e r e . ~ , 5 ' ~ The method employs an atomic
b a s i s composed o f Gaussian f u n c t i o n s which a l l o w easy a n a l y t i c e v a l u a t i o n of a l l t h r e e c e n t e r i n t e g r a l s . The p o t e n t i a l i s expanded i n a second Gaussian b a s i s s e t and i s g e n e r a l ( i . e . , no muffin-tin approximation is made). The c a l c u l a - t i o n s a r e i t e r a t e d u n t i l t h e t o t a l energy is s t a b l e t o seven s i g n i f i c a n t d i g i t s . Absolute e r r o r s , f o r example t h o s e a s s o c i a t e d with t h e approximate treatment of exchange and c o r r e l a t i o n , a r e of course l a r g e r , but t h e y a r e expected t o cancel s i n c e we c o n s i d e r only energy d i f f e r e n c e s .
3. A p p l i c a t i o n t o Si.- S i was used as a t e s t case t o f i r s t avoid complications caused by a Fermi s u r f a c e . Using t h e f r o z e n c o r e approximation t h e t o t a l energy was e v a l u a t e d a t e i g h t v a l u e s of t h e l a t t i c e c o n s t a n t and l e a s t s q u a r e s f i t w i t h XOperated f o r t h e U.S. Department' of Energy by Iowa S t a t e U n i v e r s i t y under c o n t r a c t
no. W-7405-Eng-82. This work was supported by t h e O f f i c e of B a s i c Energy S c i e n c e s . (')Also Ames Laboratory and Department of Physics,Iowa S t a t e U n i v e r s i t y , Ames, Iowa
50011, U.S.A
a f o u r t h o r d e r polynomial. The c o r r e s p o n d i n g e q u i l i b r i u m l a t t i c e c o n s t a n t , b u l k modulus and c o h e s i v e energy a r e l i s t e d i n Table 1. The t o t a l energy was a l s o
c a l c u l a t e d a s a f u n c t i o n of displacement f o r l a t t i c e d i s t o r t i o n s corresponding t o p a r t i c u l a r normal v i b r a t i o n a l modes. These y i e l d e s s e n t i a l l y c l a s s i c a l p o t e n t i a l w e l l s whose c u r v a t u r e g i v e s t h e phonon f o r c e c o n s t a n t o r frequency. The f r e q u e n c i e s f o r t h e t r a n s v e r s e o p t i c phonon a t
r
[TO(F)] and t h e t r a n s v e r s e a c c o u s t i c phonon a t X [TA(X)] a r e a l s o l i s t e d i n T a b l e 1. There i s good agree- ment w i t h experiment, and t h e d e t a i l e d a n a l y s i s of t h e c o n t r i b u t i o n s t o t h e t o t a l energy a g r e e w i t h p r e v i o u s s t u d i e s . l y 2 The c h a r g e d e n s i t y f o r t h e TO(T) phonon i s shown i n F i g u r e 1.Fig. 1 : The charge d e n s i t y i n t h e ( 1 1 0 ) p l a n e f o r t h e l a t t i c e d i s t o r - t i o n corresponding t o t h e TO(r) phonon w i t h a displacement of -0.1371 a l o n g t h e 111) d i r e c t i o n .
a
(atomic u n i t s x 1 0 ). 4 . A p p l i c a t i o n To Metals.- For m e t a l s s m a l l d i s t o r t i o n s of t h e l a t t i c e c a u s e changes i n t h e band o c c u p a t i o n n e a r t h e Fermi l e v e l which must be accounted f o r i n c a l c u l a t i n g changes i n t o t a l energy. Indeed such e f f e c t s f r e q u e n t l y g i v e r i s e t o phonon anomalies i n t r a n s i t i o n m e t a l s . 7 One approach f o r m e t a l s i s t o-f
simply keep i n c r e a s i n g t h e number of k p o i n t s sampled u n t i l convergence is reached, however, t h i s is c o s t l y and n o t n e c e s s a r y . Our approach has been t o d i v i d e t h e i r r e d u c i b l e B r i l l o u i n zone i n t o a number of l a r g e t e t r a h e d r o n s (32 f o r t h e H p o i n t phonon i n Nb and Mo) and t a k e t h e c e n t e r of mass
d
v e c t o r s a s a sample g r i d . At each i t e r a t i o n a t i g h t b i n d i n g (TB) f i t ( s e e Ref. 7 ) i s made t o t h e e i g e n v a l u e s on t h i s g r i d and is used t o determine an a c c u r a t e Fermi energy and s u r f a c e . The occupied volume a s determined by t h e TB f i t u s i n g 64 s m a l l e r t e t r a h e d r o n s i n s i d e eacn l a r g e one is used t o weight t h e%
p o i n t s . C a l c u l a t i o n s u s i n g a mixed b a s i s p s e u d o p o t e n t i a l t e c h n i q u e f o r phonons i n Nb and Mo a l s o-f
confirm t h e importance of c a r e f u l l y w e i g h t i n g t h e k p ~ i n t s . ~
C6-630 JOURNAL DE PHYSIQUE
a p a r a b o l a t o t h e energy c a l c u l a t e d f o r displacements of 0.0646, 0.078%, and 0.0926, which a r e comparable t o t h e displacement caused by t h e r e a l phonon. Very mnall d i s p l a c e m e n t s l e d t o numerical problems because of t h e l i n e a r i n t e r - p o l a t i o n used i n s i d e t h e s m a l l t e t r a h e d r o n s , and much l a r g e r displacements caused p r e v i o u s l y unoccupied p o r t i o n s of bands t o d i p below t h e Fermi l e v e l , g i v i n g r i s e t o anharmonic o r non-parabolic behavior i n t h e t o t a l energy v s d i s - placement curve.
5. Conclusion.- Our encouraging r e s u l t s s u g g e s t t h a t modem band t h e o r y tech- n i q u e s a r e c a p a b l e of becoming a u s e f u l t o o l i n s t u d y i n g t h e d e t a i l s of e l e c - t r o n i c r e s p o n s e t o c e r t a i n h i g h symmetry l a t t i c e d i s t o r t i o n s .
6. Acknowledgment.- One of t h e a u t h o r s (B.N.H.) would l i k e t o thank t h e s t a f f of t h e I n s t i t u t f c r Angewandte Kernphysik I , Kernforschungszentrum, K a r l s r u h e , f o r t h e i r kind h o s p i t a l i t y d u r i n g h i s s t a y .
T a b l e 1. C a l c u l a t e d and e x p e r i m e n t a l p r o p e r t i e s
a B
L a t t i c e Bulk Cohesive Phonon
Constant Modulus Energy Frequency
( a . u.) (Mbar) (evlatom) (THz)
Si c a l c . 10.40 exp. 10.26 Nb c a l c . 6.32 1.62 6.63 6.6* H p o i n t exp. 6.23 1.74 7.57 6.4 Mo c a l c . 5.99 2.57 6.28 5.7 H p o i n t exp. 5.95 2.63 6.82 5.5
*This is a p r e l i m i n a r y v a l u e based on a fewer number of small t e t r a h e d r o n s and b o t h l a r g e and z e r o displacements s o t h a t it i s l e s s p r e c i s e than t h e value f o r Mo ( s e e t e x t ) .
7. References
.-
1. H. Wendel and R. M. Martin Phys. Rev.
g ,
5251 (1979). 2. M. T. Yin and M. L. Cohen Phys. Rev. L e t t .65,
1004 (1980). 3. B. N. Harmon, W. Weber, and D. R. Hamann Phys. Rev. B ( i n p r e s s ) . 4. The approximate c o r r e l a t i o n f u n c t i o n a l was o b t a i n e d from E. Wigner,Phys. Rev.
46,
1002 (1934).5. H. Sambe and R. H. F e l t o n ,
J.
Chem. Phys.62,
1122 (1975). 6. J. A. Appelbaum and D. R. Hamann, p. 111, T r a n s i t i o n Metals,Eds. M. J. G. Lee, 3 . M. Perz and E. Fawcett ( I n s t . of P h y s i c s , B r i s t o l , 1978), and P. J. Feibelmann, J. A. Appelbaum and
D. R. Hamann, Phys. Rev.
