HARMONIC AND ANHARMONIC PROPERTIES OF SILICON

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HARMONIC AND ANHARMONIC PROPERTIES OF SILICON

K. Wanser, R. Wallis

To cite this version:

K. Wanser, R. Wallis. HARMONIC AND ANHARMONIC PROPERTIES OF SILICON. Journal de

Physique Colloques, 1981, 42 (C6), pp.C6-128-C6-130. �10.1051/jphyscol:1981638�. �jpa-00221574�

JOURNAL DE PHYSIQUE

CoZZoque C6, suppZe'ment au n O 1 2, Tome 42, de'cembre 1981 page C6-128

HARMONIC AND ANHARMONIC PROPERTIES OF SILICON

K.H. Wanser and R.F. Wallis

Department o f P h y s i c s , U n i v e r s i t y of C a Z i f o m i a , I r u i n e , C a Z i f o r n i a 92717, U.S.A.

Abstract

.-

properties such as the low-lying transverse acoustic modes at the X and L points of the Brillouin zone, negative Gruneisen para- meters, negative thermal expansion and anomalous acoustic attenuation. In an attempt to understand these properties, we have developed a lattice dynamical model for silicon employing

long-range, non-local, dipole-dipole interactions. Several interesting features of this interaction are found and dis- cussed. We present analytic expressions for the Gruneisen parameters that explain how the negative Gruneisen parameters arise. Application of this model to the calculation of the thermal expansion of silicon is made.

It is well known that comparatively long range interactions are necessary to explain the elastic constants as well as the low-lying TA[100] and [ill] modes in silicon. Several models have been developed to explain this behavior. Attempts 9'' to compute

anharmonic properties have utilized first neighbor anharmonicity or first and second neighbor anharmonicity, one of the authors2

concluding that this was sufficient to explain the mode Gruneisen parameters and thermal expansion coefficient. That this is not the case can be seen from the fact that the Gruneisen parameters for these models are in major disagreement with experiment for several modes.

The model we use employs first thru fourth neighbor central potentials, nearest neighbor angle bending and displacement induced non-local dipoles. An interesting feature of the non-local dipole interaction is that it does not affect the elastic constants or the Raman frequency. This is in contrast to the local quadrupoles used by

~ a x . 3

A ten independent parameter preliminary fit of our model to the experimental data is shown in Fig. 1. The major feature of this dipole model is the dramatic lowering of the TA modes at the X point.

Proceeding to the anharmonic properties, we note that silicon h a s negative thermal expansion and thus negative Gruneisen parameters for some modes. The volume thermal expansion coefficient is given to

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1981638

lowest order in the anharmonicity by

Here B is the bulk modulus, Vo the volume of the bare crystal, o(cj) the bare harmonic frequency, C (w) is the Einstein specific heat

+ ^E

function and y(kj) the mode Gruneisen parameter.

Fig. 1. Preliminary fit of dipole model to

experimental data for silicon.

To illustrate the origin of the negative mode gammas, consider the analytic expressions we have found at the X point.

In Eqs. (2) a is the conventional cube edge, E the static dielectric constant, pl and p2 are harmonic dipole parameters, p(')1 and p(l) 2 are anharmonic dipole parameters, 0 " ) is an anharmonic angle-bending

/I/

parameter and (ro), $;(ro), ((r ) are the derivatives of the

C6- 1 30 JOURNAL DE PHYSIQUE

f i r s t n e i g h b o r p o t e n t i a l f u n c t i o n e t c . T h e f o u r t h n e i g h b o r i n t e r a c t i o n d o e s n o t c o n t r i b u t e a t t h e X p o i n t . T h e i n t e r e s t i n g f e a t u r e o f E q s . ( 2 ) i s t h a t t h e n e a r e s t n e i g h b o r t h i r d d e r i v a t i v e i s n o t p r e s e n t i n E q . ( 2 a ) b u t i s i n E q . ( 2 b ) . T h i s i s t h e e x p l a n a t i o n f o r why t h e TAX m o d e h a s a n e g a t i v e mode gamma a n d t h e TOX mode d o e s n o t . I t i s d u e t o t h e f a c t t h a t t h e d i a m o n d s t r u c t u r e h a s a n e a r e s t n e i g h b o r c e n t r a l f o r c e i n s t a b i l i t y . I n f a c t , i t i s j u s t t h o s e m o d e s t h a t h a v e t h e i n s t a b i l i t y t h a t e x h i b i t t h e n e g a t i v e mode gamma. T h i s p o i n t h a s n o t b e e n r e c o g n i z e d i n t h e l i t e r a t u r e p r e v i o u s l y . We e m p h a s i z e a g a i n t h a t s i n c e 4 m ( r ) i s o f o p p o s i t e s i g n t o @ # ( r o ) , t h e c a n c e l l a t i o n o f

,,, ^{1 0 1}

( r ) i n c e r t a i n m o d e s m a k e s t h e mode gamma e x t r e m e l y l i k e l y t o b e n e g a t i v e .

A p r e l i m i n a r y c a l c u l a t i o n o f t h e t h e r m a l e x p a n s i o n i s s h o w n i n F i g . 2 . I n t h i s c a l c u l a t i o n t h e f o u r t h i r d o r d e r p o t e n t i a l

F i g . 2 . V o l u m e t h e r m a l e x p a n s i o n c o e f f i c i e n t o f s i l i c o n . S o l i d l i n e i s p r e s e n t c a l c u l a t i o n , c i r c l e s a r e e x p e r i m e n t a l

T(K) d a t a .

d e r i v a t i v e s w e r e f i t t o t h e e x p e r i m e n t a l mode gammas a t X , L a n d t h e e l a s t i c r e g i o n . T h e d i p o l e a n d a n g l e - b e n d i n g a n h a r m o n i c p a r a m e t e r s w e r e f i x e d b y t h e o r e t i c a l

consideration^.^

A c k n o w l e d g e m e n t - Work s u p p o r t e d i n p a r t b y NASA C o n t r a c t No. NSG-7472 a n d NSF G r a n t No. DMR-7809430.

R e f e r e n c e s

1 . G . D o l l i n g a n d R . A. C o w l e y , P r o c . P h y s . S o c .

88,

45,

3 . M . L a x , " L a t t i c e D y n a m i c s , " E d . b y R. F . W a l l i s , ( P e r g a m o n , O x f o r d , 1 9 6 5 1 , p . 1 7 9 .

4 . K . H . W a n s e r , Ph.D. T h e s i s , UC I r v i n e , 1 9 8 1 ( u n p u b l i s h e d ) .