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Submitted on 1 Jan 1981
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FIRST PRINCIPLES CALCULATION OF THE PHONON SPECTRA OF SOLIDS
P. van Camp, V. van Doren, J. Devreese
To cite this version:
P. van Camp, V. van Doren, J. Devreese. FIRST PRINCIPLES CALCULATION OF THE PHONON SPECTRA OF SOLIDS. Journal de Physique Colloques, 1981, 42 (C6), pp.C6-625-C6- 627. �10.1051/jphyscol:19816181�. �jpa-00221264�
JOURNAL DE PHYSIQUE
CoZZoque C6, suppZ6ment au n012, Tome 42, ddcembre 2981 page c6-625
FIRST PRINCIPLES CALCULATION OF THE PHONON SPECTRA OF SOLIDS"
P.E. Van Camp, V.E. Van Doren and J . T . ~ e v r e e s e * '
University of Antmrpen, Belgium.
A b s t r a c t . - I n t h i s a r t i c l e , a f i r s t p r i n c i p l e s m i c r o s c o p i c c a l c u - l a t i o n o f S i i s p r e s e n t e d a n d c o m p a r e d w i t h r e s u l t s o f o t h e r m e t h o d s .
I n t h e l a s t f e w y e a r s s e v e r a l g r o u p s h a v e a c h i e v e d f i r s t p r i n c i p l e s m i c r o s c o p i c c a l c u l a t i o n s o f p h o n o n f r e q u e n c i e s o f s e m i c o n d u c t o r s . Two d i f f e r e n t a p p r o a c h e s h a v e b e e n u s e d . T h e f i r s t r e s u l t s p u b l i s h e d 1 w e r e a b t a i n e d b y c a l c u l a t i n g t h e t o t a l g r o u n d s t a t e e n e r g y o f t h e c r y s t a l a s a f u n c t i o n o f t h e l a t t i c e p a r a m e t e r . I n t h i s m e t h o d t h e u n i t c e l l o f t h e c r y s t a l h a s t o b e a d a p t e d t o s a t i s f y t h e s y m m e t r y r e q u i r e m e n t s i m p o s e d b y t h e l a t t i c e d i s p l a c e m e n t s o f a p a r t i c u l a r p h o n o n mode. T h i s i m p l i e s t h a t s u c h a m e t h o d i s i n p r a c t i c e r e s t r i c t e d t o p h o n o n f r e - q u e n c i e s o f h i g h s y m m e t r y p o i n t s ( r , X , L ) . R e c e n t l y , r e f i n e m e n t s o f t h i s m e t h o d h a v e l e a d t o c a l c u l a t i o n s o f t h e p h a s e t r a n s f o r m a t i o n s o f c o v a l e n t s e m i c o n d u c t o r s a n d o f a f e w p h o n o n f r e q u e n c i e s o f S i .
I n t h e s e c o n d a p p r o a c h , 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 t h e
d y n a m i c a l m a t r i x i s t r e a t e d b y m e a n s o f t h e d i e l e c t r i c s c r e e n i n g t h e o r y w h i c h d e s c r i b e s t h e l i n e a r r e s p o n s e o f t h e e l e c t r o n i c s y s t e m t o t h e d i s p l a c e m e n t s o f t h e i o n s . T h i s f o r m a l i s m h a s b e e n known f o r a l o n g t i m e 2 . T h e a d v a n t a g e o f t h e d i e l e c t r i c s c r e e n i n g m e t h o d o v e r t h e a b o v e m e n t i o n e d t o t a l e n e r g y d i f f e r e n c e a p p r o a c h i s t h a t i t i s n o t r e s t r i c t e d t o h i g h s y m m e t r y p o i n t s b u t c a n b e u s e d t o c a l c u l a t e t h e c o m p l e t e p h o n o n d i s p e r s i o n c u r v e s .
I n t h i s d i e l e c t r i c s c r e e n i n g m e t h o d , t h e e s s e n t i a l q u a n t i t y i s t h e p o l a r i z a b i l i t y m a t r i x 2. T h e p r e s e n t a u t h o r s h a v e c a l c u l a t e d t h e H a r t r e e e x p r e s s i o n o f t h e d i e l e c t r i c m a t r i x b y e x p a n d i n g 2 i n t e r m s o f m o m e n t s . T h i s p r o c e d u r e i n c o r p o r a t e s t h e e f f e c t o f all c o n d u c t i o n b a n d s , w h i c h i s i m p o s s i b l e t o a c h i e v e i n a d i r e c t s u m m a t i o n o v e r t h e b a n d s . T h e r e s u l t s f o r S i w h e r e t h e z e r o t h moment i s e v a l u a t e d e x p l i - c i t l y a n d t h e r a t i o o f t h e f i r s t t o t h e z e r o t h moment i s a c a l c u l a t e d c o n s t a n t a r e g i v e n i n f i g . 1 . D e t a i l s o f t h i s c a l c u l a t i o n c a n b e f o u n d e l s e w h e r e 3 . I t w a s f o u n d t h a t i n o r d e r t o o b t a i n r e a l i s t i c p h o n o n c u r v e s i t i s c r u c i a l t o a c h i e v e c o n s i s t e n c y b e t w e e n t h e
*Work s u p p o r t e d i n p a r t b y a C D C - g r a n t .
" " ~ l s o a t T . H . E i n d h o v e n , T h e N e t h e r l a n d s .
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19816181
JOURNAL DE PHYSIQUE
0. 2 .4 .6 .8 1.0 - 8 -6 .4 .2 0. .2 .4 .5
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e l e c t r o n - i o n p o t e n t i a l a n d t h e c r y s t a l p o t e n t i a l u s e d t o c a l c u l a t e t h e e l e c t r o n w a v e f u n c t i o n s .
R e c e n t l y , t h e p r e s e n t a u t h o r s h a v e i n v e s t i g a t e d t h e e x c h a n g e - c o r r e l a t i o n c o n t r i b u t i o n t o t h e p h o n o n f r e q u e n c i e s . T h e s a m e e x c h a n g e - c o r r e l a t i o n e f f e c t s (pi" a p p r o x i m a t i o n ) a r e i n c l u d e d i n b o t h t h e
H a m i l t o n i a n a n d i n t h e d i e l e c t r i c m a t r i x ( s e e eq. ( 1 ) i n r e f . 4 ) . T h e s u m m a t i o n s o v e r t h e c o n d u c t i o n b a n d s h a v e b e e n p e r f o r m e d e x p l i c i t l y . F u r t h e r m o r e , t h e l a t t i c e c o n s t a n t i s n o t t a k e n f r o m e x p e r i m e n t b u t i s c a l c u l a t e d Prom m i n i m i z i n g t h e t o t a l e n e r g y o f t h e c r y s t a l - a s i t s h o u l d b e d o n e i n o r d e r t o h a v e a s y s t e m w i t h o u t s t r e s s - t h e f o l l o w - i n g r e s u l t s a r e o b t a i n e d i n 10" Hz
I n c o n c l u s i o n , t h e d i e l e c t r i c s c r e e n i n g a p p r o a c h s h o w s t h a t r e a l i s t i c p h o n o n f r e q u e n c i e s f o r a l l w a v e v e c t o r s a r e o b t a i n e d f r o m a f i r s t p r i n c i q l e s c a l c u l a t i o n .
References
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L.J. Sham, P h y s . R e v . 188, 1 4 3 1 ( 1 9 6 9 ) .
R.M. P i c k , M . H . C o h e n , R.M. M a r t i n , P h y s . R e v . g , 9 1 0 ( 1 9 7 0 ) . P . E . v a n Camp, V . E . Van D o r e n , J . T . D e v r e e s e , ~ h y s . R e v . L e t t . 42,
1 2 2 4 ( 1 9 7 9 ) .
P . E . v a n c a m p , V . E . Van or en, J . T . D e v r e e s e , ~ h y s . R e v . ~ 2 4 , 1 0 9 6 ( 1 9 8 1 ) .