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Submitted on 1 Jan 1981
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LATTICE DYNAMICS OF GRAPHITE
INTERCALATION COMPOUNDS - MODELLED BY THE PHONON DISPERSION OF LINEAR CHAINS
A. Magerl, H. Zabel
To cite this version:
A. Magerl, H. Zabel. LATTICE DYNAMICS OF GRAPHITE INTERCALATION COMPOUNDS - MODELLED BY THE PHONON DISPERSION OF LINEAR CHAINS. Journal de Physique Collo- ques, 1981, 42 (C6), pp.C6-329-C6-331. �10.1051/jphyscol:1981696�. �jpa-00221633�
JOURNAL DE PHYSIQUE
CoZloque C6, suppZ4ment au n o 12, Tome 42, d4cembre 1981 page C6-329
L A T T I C E D Y N A M I C S O F G R A P H I T E I N T E R C A L A T I O N COMPOUNDS - MODELLED B Y T H E PHONON D I S P E R S I O N O F L I N E A R C H A I N S
A. Magerl and H. ~ a b e l *
I n s t i t u t Laue-Langevin, 156X, 38042 GrenobZe Cedex, Dance.
*Loomis Laboratory of Physics and MateriaZs Research Laboratory, University of IZlinois a t Urbana-Champaign, U r b m , IL 61801, U.S.A.
A1;stract.-The c h a r a c t e r i s t i c f e a t u r e s of t h e phonon d i s p e r s i o n of a monatomic l i n e a r chain i n a Born - von JGrmSn and a s h e l l model formalism a r e o u t l i n e d . Comparing model c a l c u l a t i o n s w i t h measurements of t h e [001]L phonon d i s p e r s i o n of i n t e r c a l a t e d g r a p h i t e , we f i n d t h a t a s h e l l model h a s t o be used f o r a n a p p r o p r i a t e d e s c r i p t i o n .
The displacement p a t t e r n of [001]L phonon modes i n g r a p h i t e and g r a p h i t e i n t e r - c a l a t e s i s determined by i n t e r p l a n a r f o r c e s only. Because t h e s e modes show a one- dimensional c h a r a c t e r , a l i n e a r c h a i n w i t h two d i f f e r e n t masses r e p r e s e n t i n g i n t e r - c a l a t e d and g r a p h i t e l a y e r s p r o v i d e s a n a p p r o p r i a t e model. The e f f e c t i v e r e p e a t d i s t a n c e f o r a s t a g e n compound extends over n g r a p h i t e and 1 i n t e r c a l a t e d l a y e r . Correspondingly, t h e [001] d i s p e r s i o n c o n s i s t s of 1 a c o u s t i c a l and n o p t i c a l b r a n c h e s
Fig.1: Comparison of t h e phonon d i s p e r s i o n of a monatomic c h a i n c a l c u l a t e d by a Born - von KQrmgn model and by a s h e l l model.
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1981696
C6-330 JOURNAL DE PHYSIQUE
Complete [001]L p h o n o n d i s p e r s i o n s a r e now a v a i l a b l e f o r s t a g e 1 and 2 K, Rb and C s and f o r s t a g e 3 K compounds.' I n a r e c e n t p u b l i c a t i o n t h e t h r e e lower branches of C36K could be w e l l d e s c r i b e d by a Born - von K5rm6n model (BvK), w i t h only n e a r e s t neighbor i n t e r a c t i o n s . * However, t h i s model f a i l s , when i n c l u d i n g
t h e t h i r d o p t i c a l branch (compare F i g . 2 a ) .
To improve t h e f i t , we have extended t h e BvK model t o i n c l u d e l o n g e r ranged i n t e r a c t i o n s : i . ) an i n t e r a c t i o n between two a l k a l i l a y e r s , i i . ) an i n t e r a c t i o n between t h e bonding g r a p h i t e p l a n e s on b o t h s i d e s of a n a l k a l i l a y e r , i i i . ) a n i n t e r a c t i o n between a n a l k a l i l a y e r and t h e second n e a r e s t neighbor g r a p h i t e p l a n e s . However, w i t h t h e exception of s t a g e 1 Rb, t h e measurements could not b e d e s c r i b e d reasonably w e l l by any one of t h e s e f o u r BvK models.
S h e l l models (SM), which t r e a t an i n t e r a c t i o n between two mass p o i n t s v i a an e l e c t r o n i c s h e l l w i t h mass z e r o ( a d i a b a t i c a p p r o x i m a t i o n ) , r e v e a l a d i s t i n c t i v e l y d i f f e r e n t behavior. The phonon d i s p e r s i o n of a monatomic c h a i n , c a l c u l a t e d by
Fig.2: [OQlIL phonon d i s p e r s i o n of s t a g e 3 C3#. The s o l i d l i n e s r e p r e s e n t f i t s by ( a ) a Born - von Grm6n model and (€11 by a s h e l l model. F i t parameters a r e i n d i c a t e d i n t h e i n s e r t i o n .
a two-parameter BvK model and a two-parameter SN a r e compared i n Fig. 1. Decreasing t h e second neighbor f o r c e c o n s t a n t $2 t o z e r o f o r t h e BvK model o r i n c r e a s i n g t h e c o r e s h e l l i n t e r a c t i o n +K t o i n f i n i t y f o r t h e SK only l e a v e s one f o r c e c o n s t a n t f o r b o t h models. The phonon d i s p e r s i o n s a r e t h e n i d e n t i c a l , r e v e a l i n g s i n u s o d i a l f u n c t i o n s f o r w(q) . I n c r e a s i n g $ 2 i n t h e BvK model i n f l u e n c e s t h e d i s p e r s i o n over a l a r g e regime of t h e B r i l l o u i n zone (BZ). P a r t i c u l a r l y , t h e i n i t i a l s l o p e of t h e d i s p e r s i o n i s changed, whereas t h e v a l u e a t t h e edge of t h e BZ remains c o n s t a n t . However, a f i n i t e v a l u e of OK i n t h e SM mainly r e d u c e s t h e phonon modes a t t h e
edge of t h e BZ, w h i l e l e a v i n g t h e i n i t i a l s l o p e unchanged.
A s an example, i n Fig. 2 a f i t of t h e [001]L d i s p e r s i o n w i t h t h e BvK model and SM f o r s t a g e 3 C K i s shown. The SM p r o v i d e s a l s o a v e r y good d e s c r i p t i o n
3 6
f o r s t a g e 1 and 2 compounds which i s p r e s e n t e d i n r e f . 4 . We t h e r e f o r e b e l i w e t h a t t h e i n t e r a c t i o n of a l k a l i and g r a p h i t e p l a n e s v i a an " e l e c t r o n i c l a y e r v i s a n e s s e n t i a l f e a t u r e of t h e [001]L phonon d i s p e r s i o n of a l k a l i i n t e r c a l a t e s .
One of u s (H.Z.) acknowledges s u p p o r t by t h e U. S. Department of Energy under Contract DE-AC02-76ER01198.
References
1. A. Magerl and H. Zabel, in Proceedings of t h e " ~ n t e r n a t i o n a l Conference on P h y s i c s of I n t e r c a l a t i o n Compounds," J u l y 1981, T r i e s t e , I t a l y .
2. A. Magerl and H. Zabel, Phys. Rev. L e t t . 46, 444 (1981).
3 . B. G. Dick Jr. and A. W. Overhauser, Phys. Rev. 112, 90 (1958).
4. H. Zabel and A. Magerl, s e e c o n t r i b u t i o n t o t h i s volume.
