THE EFFECT OF NONMETAL-METAL TRANSITION DUE TO DOPING ON LATTICE THERMAL CONDUCTIVITY : APPLICATION TO Ge-DOPED InSb

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THE EFFECT OF NONMETAL-METAL

TRANSITION DUE TO DOPING ON LATTICE THERMAL CONDUCTIVITY : APPLICATION TO

Ge-DOPED InSb

P. Sharma, K. Roy, V. Radhakrishnan

To cite this version:

P. Sharma, K. Roy, V. Radhakrishnan. THE EFFECT OF NONMETAL-METAL TRANSITION DUE TO DOPING ON LATTICE THERMAL CONDUCTIVITY : APPLICATION TO Ge-DOPED InSb. Journal de Physique Colloques, 1981, 42 (C6), pp.C6-268-C6-270. �10.1051/jphyscol:1981678�.

�jpa-00221614�

JOURNAL DE PHYSIQUE

CoZZoque C6, suppZ6ment au n022, Tome 42, d6cembre 1981 page C6-268

THE EFFECT OF NONMETAL-METAL TRANSITION DUE TO DOPING ON LATTICE THERMAL CONDUCTIVITY : APPLICATION TO Ge-DOPED InSb

P.C. S h a m , K.P. Roy and V. Radhakrishnan

Physics Department, Banaras Hindu University, Varanasi-221005, India

Abstract.- I n t h i s work we e x p l a i n t h e phonon thermal conducti- v i t y r e s u l t s of Ge-do ed InSb from 2 t o 50 K f o r impurity

c o n c e n t r a t i o n s 7.2x10i7 and 5.0x1017 cm-3 . Since t h e s e l i e i n t h e i n t e r m e d i a t e c o n c e n t r a t i o n r e g i o n of t h e metal-nonmetal t r a n s i t i o n t h e theory of s c a t t e r i n g o f phonon by h o l e s i n t h e mixed s t a t e , i.e. both i n t h e l o c a l i s e d non-metallic and i n t h e m e t a l l i c s t a t e i s used. The h o l e c o n c e n t r a t i o n s i n t h e non- m e t a l l i c and m e t a l l i c r e g i o n s a r e c a l c u l a t e d f o r each sample by using Mikoshibal s inhomogeneity model, t h e theory of both bound hole-phonon and f r e e h o l e phonon s c a t t e r i n g a r e a p p l i e d t o

e x p l a i n t h e r e s u l t s . The v a l u e of density-of- s t a t e s e f f e c t i v e mass a r e kept c o n s t a n t (=0.23) with t h e v a r i a t i o n i n temperature.

The v a l u e of t h e d i l a t a t i o n and shear deformation p o t e n t i a l c o n s t a n t s obtained from o u r c a l c u l a t i o n s a r e i n good agreement with previously determined values. The e f f e c t of impurity

s c a t t e r i n g due t o doped i m p u r i t i e s alongwith i s o t o p e s c a t t e r i n g i s a l s o taken i n t o account.

It i s r e p o r t e d t h a t i n t h e intermediate c o n c e n t r a t i o n range, t h e mixed hole-phonon s c a t t e r i n g must be included t o e x p l a i n t h e temperature dependeqce o f thermal c o n d u c t i v i t y of Ge-doped InSb.

I n t r o d u c t i o n . - The phonon thermal c o n d u c t i v i t y o f p-InSb doped w i t h Ge, h a s been s t u d i e d experimentally Ll,a f o r acceptor h o l e concentra- t i o n s from 2 . 7 ~ 1 0 ' ~ t o 6x10'~ ~ m - ~ . I n t h e low c o n c e n t r a t i o n r e g i o n w e l l below t h e c r i t i c a l c o n c e n t r a t i o n f o r nonmetal-metal t r a n s i t i o n

(Nc) t h e h o l e s a r e bound to t h e impurity atoms and i n t h e high

impurity c o n c e n t r a t i o n region, they a r e f r e e i n valence band. I n t h e i n t e r m e d i a t e doping c o n c e n t r a t i o n n e a r Nc, t h e h o l e s e x i s t i n a mixed s t a t e . Using Mikoshibals ninhomogeneity model^^, t h e a c c e p t o r h o l e c o n c e n t r a t i o n i n both non-metallic and m e t a l l i c r e g i o n can be

c a l c u l a t e d . I n t h e p r e s e n t work we have chosen two samples of p-InSb with ~ = 7 . 2 x 1 0 ' ~ and 5x10'~ ~ m - ~ . Using M o t t l s expression t h e

c r i t i c a l c o n c e n t r a t i o n f o r metal-nonmetal t r a n s i t i o n (Nc) can be g i v e n a s ~ ~ = ( . ~ i j / a * ) ~ where a* i s t h e e f f e c t i v e Bohr r a d i u s o f t h e

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1981678

i m p u r i t y and t h e c o n c e n t r a t i o n a t which Fermi l e v e l merges w i t h v a l e n c e band i s Nvb = Using a* = 38

A , t h e v a l u e s Nc and Nvb a r e c a l c u l a t e d to be 2 . 8 4 ~ 1 0 ' ~ and 1 . 4 5 ~ 1 0 ~ ~ f o r p- InSb.

using Miko shiba

s inhomogeneity model t h e h o l e c o n c e n t r a t i o n s i n n o n m e t a l l i c and m e t a l l i c r e g i o n s can be c a l c u l a t e d a s Nn = N exp (- 2 r r ~ r ~ 3 ^{) and Nm} = (N-Nn) where r, = a*. Fbr t h e

sample o f p-InSb w i t h N = 7 . 2 ~ 1 0 ~ ~ t h e v a l u e s o f ~ ~ = 0 . 6 4 x 1 0 ~ ~ and Nm= 6 . 5 5 ~ 1 0 ' ~ c i 3 and f o r t h e sample with N = 5x1017 cm-3,

Nn = 0.936x1017 and Nm = 4 . ~ 6 ~ 1 0 ' ~ ~ m ' ~ . Using t h e r e l a x a t i o n r a t e of bound hole-phonon s c a t t e r i n g , f o r t h e h o l e s i n nonmetallic r e g i o n and t h e r e l a x a t i o n r a t e f o r f r e e h o l e phonon s c a t t e r i n g f o r t h e h o l e s i n m e t a l l i c region, t h e phonon c o n d u c t i v i t y v a l u e s o f t h e s e two

samples i n t h e i n t e r m e d i a t e doping c o n c e n t r a t i o n a r e e x p l a i n e d w e l l between 2 and 50 K.

TheorzFlg" The r e l a x a t i o n r a t e f o r t h e bound hole-phonon s c a t t e r i n g i s g i v e n a s

4 4

-1 - - ^{',u2(r/3 D: , ) f (q) 20D4 + l20$ + 6 0}

(%p ) bound

407r e%2 ^{V I 3}

where f ( q ) = (1+1/4 a r 2 q21m2 and D = D:/D:,

D : and D: , a r e t h e s h e a r deformation p o t e n t i a l s f o r t h e s t r a i n s along p 0 1 J a n d S l q d i r e c t i o n s , q i s t h e phonon wave v e c t o r , v i s t h e average phonon v e l o c i t y and i s t h e d e n s i t y o f t h e c r y s t a l .

The a c c e p t o r h o l e s i n t h e m e t a l l i c r e g i o n can be c o n s i d e r e d t o be f r e e i n t h e v a l e n c e band and t h e r e l a x a t i o n r a t e f o r t h e s c a t t e r i n g o f phonons by f r e e h o l e s f o r q ,< 2kF i s g i v e n a s

-1 - m* c2 kB%ft ln l+exp( ?*- N / T - P T X ~ + x/2) ( f p )q62kF - 2 ~ ~ t i 4 e ^v l + e x p ( T*- N/T-mx2 - ^{4 2 )}

712 . h 2 ( 3 ~ ~ / 7 r 2/3 where N = m* v2/2kg ; P = k$8m*v2 and ?* =

2m* kgT

m * i s t h e d e n s i t y o f - s t a t e s e f f e c t i v e mass and C i s t h e d i l a t a t i o n deformation p o t e n t i a l . For phonon wave v e c t o r q ) 2kF, t h e

r e l a x a t i o n r a t e f o r t h e s c a t t e r i n g o f phonons by f r e e h o l e s i s g i v e n a s

The phonon thermal c o n d u c t i v i t y can be c a l c u l a t e d u s i n g t h e s e

e x p r e s s i o n s by t h e method d e s c r i b e d elsewhere 133 .

C6-270 JOURNAL

DE PHYSIQUE

R e s u l t s and Discussion.- By c o n s i d e r i n g t h e r e a l i s t i c hole-phonon s c a t t e r i n g r e l a x a t i o n r a t e s f o r t h e doped samples of p-InSb i n t h e i n t e r m e d i a t e c o n c e n t r a t i o n r e g i o n , t h e phonon c o n d u c t i v i t y v a l u e s a r e e x p l a i n e d s a t i s f a c t o r i l y between 2 and 50 K a s shown i n t h e f i g u r e . The t h e o r e t i c a l a n a l y s i s o f h o l e phonon s c a t t e r i n g e s t a b l i s h e s t h e e x i s t e n c e o f h o l e s i n a mixed s t a t e i.e. both i n bound and f r e e impurity s t a t e s i n t h e r e g i o n o f p r e s e n t i n t e r e s t . The v a l u e s o f s h e a r deformation p o t e n t i a l s determined by o u r

t h e o r e t i c a l a n a l y s i s a r e D: = 2.8 eV and D:, = 4.2 eV; which a r e i n agreement w i t h experimental values. The d i l a t a t i o n deformation p o t e n t i a l c o n s t a n t s a r e 1.4 eV and 1.3 eV, f o r t h e two samples w i t h N = 7 . ~ 1 0 ~ ~ and 5 x 1 0 ~ ~ The s c a t t e r i n g due to doped i m p u r i t i e s i s a l s o c o n s i d e r e d i n a d d i t i o n t o i s o t o p e s c a t t e r i n g i n t h e c a l c u l a - t i o n f o r p o i n t d e f e c t s c a t t e r i n g parameter. The v a l u e s of t h e parameters a r e g i v e n i n t h e Table.

Table.- Parameters used i n t h e c a l c u l a t i o n

-

Figure.- Thermal conductiwity o f p-InSb samples a s a f u n c t i o n o f Temperature.

References

1. V.V.Kosarev e t 61. Physica S t a t - S o l i d ( ~ ) 4 4 , 5 2 5 ( 1 9 7 1 ) .

2. S.S.Skalyt, P.V.Tamarin and V.S.Ivloa Phys.Lett. =,29(1970).

3. V-Radhakrishnan & P.C.Sharma Phys.Rev. =,3004(1981) .