Short Notes K6 3

phys. stat. sol. (b) 69, K63 (1975)

Subject classification: 15; 22

Institute of Physics, Academy of Sciences of the Azerbaidzhan SSR (a), and S .M. Kirov Azerbaidzhan State University (b), Baku

On the Quantum Theory of the Nernst-Ettingshausen Effect

in Semiconductors with Kane's Dispersion Law

BY

R.G. AGAEVA (a), B.M. ASKEROV (b), and R.F. EMINOV (b)

The present work considers the transverse Nernst-Ettingshausen (N-E) effect

in semiconductors with Kane's dispersion law (1) for the case of electron scatter-

ing by piezo-acoustic phonons, Expressions for the N-E coefficient a r e obtained in

the quantum limit for the case of strongly degenerate semiconductors a t any degree

of nonparabolicity . The calculations a r e done with the nonparabolicity being com-

pletely taken into account both in the density of states and in the matrix elements

of the scattering probability. It i s shown that the nonparabolicity essentially in-

fluences the value of the N-E coefficient, as well a s i ts field dependence. Consi-

deration of nonparabolicity in the density of states gives the essential contribution

to the field dependence.

y $ k Z ' The expression of the electron transition probability from the state a (N, k

1 1 1 Z Y z z

j ) to the state a ' (N' , k , k , j ) was obtained using the operator of electron-

phonon interaction (2 , 3) and the wave functions of the Kane Hamiltonian in quantizing

Fagnetic field (4), thus we get

) 6 ( d Z - k Z - q z ) 6 ( E a 1 - C m - h ~ ) q +

Here A I is a rather complicated j z j z

function depending upon the direction of the to-

ta l momentum. Assuming the scattering to be elastic and taking into account only

the electrons with one direction of spin (since further concern is given only within

the quantum limit, with all electrons placed in one spin subband, i.e. j

we get

= j ' = +1) z z

K64 physica status solidi (b) 69

where

1 2 2 2 2 I

LN - N (u) is the generalized Laguerre polynomial, u = z q1 R , q, = q + q2. The

rest of notations a r e usual (reference (5)). The transition probability (1) has been

used in reference (6) to estimate the transverse magnetoresistivity of degenerate

n-InSb.

N X Y

I

The measured isothermal N-E coefficient is expressed in terms of the compo-

nents of galvanomagnetic d and thermomagnetic P. tensors. ik ik The components d a r e determined using the quantum theory of galvanomag-

netic phenomena (7, 8). The nondiagonal component p tensor in quantizing magnetic field does not depend upon the scattering mechanism.

In the first nonvanishing approximation with respect to degeneracy and in the quan-

tum limit the nondiagonal component

(9).

ik of the thermomagnetic

YX

f3 is given by the expression (14) of reference YX

Short Notes K6 5

The diagonal component f3 of the thermomagnetic tensor depends essentially

upon the scattering mechanism. In the quantum case it i s expressed by the formula

(11) of reference (10). Inserting the expression of the probability (1) into this for-

mula, in the quantum case (N = N = 0, j = j = +1) we obtain after some transfor-

mat i ons

xx

I I z z

Here the following notations a re introduced:

When obtaining equation (4) we took no account of the inelasticity term in the

argument of the 6 -function, further on, in the quantum case we assumed q < q, (11). Z

Inserting (4) and the corresponding expressions for P and dik into equation YX

(2) in reference (10) and using the first nonvanishing approximation with respect to

the degeneracy we obtain

5 physica (b)

K66

Here a! and b

physica status solidi (b) 69

I

F' a re energy derivatives of a. and b at L = 0 1 1

From formula (6) for Q at I L -- we get an expression for the parabolic band g

( l o ) , while at

The latter expression may be applied to semiconductors with the very narrow gap

of the type of the solid solutions Cd Hg

L - 0 , we get an expression for the strong nonparabolicity case. g

Te: x 1-x

2 3 2 Here 7= 2(2T R n) is a dimensionless parameter, s = ( E m*)'I2 is a constant

having the dimension of a velocity. d o

In the quantum limit for semiconductors with degenerate electron gas the ex-

pression for Q is obtained in the case of nonparabolicity being taken account of

only in the density of states. For comparison we present only the result for Q at

E - 0 : g 8 ko (ko Tj2 D

Q = - 12 4 11 5 3

( 2 2 ) fi R n uoc s 0

It should be noted that e and m* do not enter the equations (7) and (8) for g 0

Q at E - 0 individually, but instead, the equations contain their ratio

s - 10

all the semiconductors with the Kane dispersion law.

2 g16 ( c ~ / s ) ~ , the numerical values of the latter being almost the same for

The above-mentioned results a re valid if the conditions of the quantum limit and

strong degeneracy of electron gas a re satisfied. In case of strong nonparabolicity

the conditions take the form

2koT

s5R- l *

2 3 2 2 3 2 1 = ( 2 * R n) (1, (2n R n) >>- 2 (9)

Since a* varies within the interval 4 k T/ shR-' << t < 2 (as follows from 11/2 0

equation (9)), the equations (7) and (8) give an almost similar relationship Q .-.J H

This means that the nonparabolicity effect upon the scattering matrix element is

not important to obtain the Q(H) dependence. Thus to obtain the field dependence of

Q in case of strong nonparabolicity, it i s sufficient to allow for this in the density

. 0

Short Notes K67

of states only. In this case the relationship for Qo to Qm is given by the expression

where Qm is the N-E coefficient for the parabolic band, the vaIue of which is ob-

tained f rom (6) at E. - 0 0 . g

According to equation ( lo ) , the nonparabolicity does not change the temperature

and concentration dependences, but does influence the field dependence considerably.

The latter effect may be accounted for qualitatively by replacing the parabolic band

of effective mass in the expression for Q by the effective mass on the Fe rmi sur -

face calculated for the nonparabolic band:

taken into account that the effective mass en ters Q taken in the third power but at

E - 0 m* - 0,- H ~ / ~ (see equation (11) in reference (9)). g

In conclusion, it should be noted that the relationship Q /Q- N H3l2 holds a s 0

well fo r a l l the other elastic scattering mechanisms. In fact the effective mass m*

enters the expression f o r Q in the same way fo r all the scattering mechanisms (5) .

References

(1) E.O. KANE, J. Phys. Chem. Solids 1, 249 (1957).

(2) H.B. CALLEN, Phys. Rev. 76, 1394 (1949).

(3) H. J . MEIJER and D. POLDER, Physica 2, 255 (1953).

(4) P. BOWERS and Y. YAFET, Phys. Rev. 115, 1165 (1959).

(5) B . M. ASKEROV, Kinetic Effects in Semiconductors, Nauka, Leningrad 1970.

(6) R.G. AGAEVA, B.M. ASKEROV, and F.M. GASHIMZADE, phys. stat. sol.

(b) g, K43 (1973).

(7) S. TITEICA, Ann. Phys. 22, 128 (1935).

(8) E.N. ADAMS and T.D. HOLSTEIN, J. Phys. Chem. Solids lo, 254 (1959).

(9) B.M. ASKEROV and R .F . EMINOV, Fiz. Tekh. Poluprov. 2, 950 (1974).

(10) A.I. ANSELM and B.M. ASKEROV, Fiz. tverd. Tela 2, 31 (1967).

(11) L.E. GUREVICH and G.M. NEDLIN, Fiz. tverd. Tela 3, 2779 (1961).

(Received April 2, 1975)