Short Notes K45

phys. stat, sol. (b) 2, K45 (1975)

Subject classification: 13.1; 22.2.1; 22.2.3

Institute of Radio physics and Electronics, University College of Science and Technology, Calcutta

On a Simplified Form of Kane’s Dispersion Relation for Semiconductors

By

B.R. NAG and A.N. CHAKRAVARTI

In recent years, Kane’s dispersion relation has been widely used for taking

into account the non-parabolic nature of the energy bands in calculations on many

of the semiconductors including some important III-V compounds. However, since

the dispersion relation of Kane in its original form is somewhat complicated and is

not very much convenient to be handled in most of the cases, an approximate simpli-

fied form of the relation has been mostly used in the literature in which the coeffi-

cient of non-parabolicity has been taken to be equal to the reciprocal of the band

gap. It may be noted that the e r ror introduced by this approximate form is within reasonable limits as long as the valence band spin-orbit splitting is either much

greater than o r is much less than the band gap. The e r ror increases very much

when the two quantities a re comparable to each other. Thus, with the growihg inter-

est in ternary semiconductars in which the valence band spin-orbit splitting and the

band gap are both functions of the alloy composition leading often to comparable

values, the necessity has been created in obtaining a different simplified form which

gives better accuracy in all the cases. In the present communication such a simpli-

f ied form is given.

For non-parabolic bands, the E(k) relation may be expressed (reference (1))

from Kane’s theory (2) in a simplified manner as mostly used in the literature by

1 5fik 2m = - E ( l + & E ) , a , = ~ , n is

2 2

where m is the effective mass at the band edge, E is the band gap, and the energy n g

E is measured from the band edge. This simplified form of Kane’s relation is widely

used in the literature and i s valid within tolerable limits when the valence band spin-

orbit splitting A is either much greater than or is much less than the band gap E g ‘

K46 phylica status solidi (b) 71

However, the er ror increases very much when A is comparable to E as is often

the case. In what follows, a different simplified form is first derived and is then

demonstrated to have better accuracy in at least three semiconductors than the

form given above.

g

According to Kane's theory (2), the E (k) relation is given by

2 2 g g g 3

( E - E o ) ( E - E o + E ) ( E - E o + E + A ) - P k ( E - E o + E + - A ) = O , (2)

2 2 where E =h k /2m0, m being the free electron mass, and P being the momentum

matrix element. Near the band edge (i.e. when k -. 0), E =fi k /2m and we can 2 2 0 0

n write from equation (2)

E l - z ) [ . ( 1 - $ ) + E g ] [ E ( 1 - 2 ) + E g + h ]

(3) 2 P k =

Now, a s k- 0 , E - 0 and from equation (3) we get

m E (E + A ) P k 2 =-(1- h2k2

2 m e) ' '2 . E + - A g 3

n (4)

Further, since the non-parabolicities in the energy bands a re significant when

m /m << 1, we can neglect (Eo/E)2 and higher order terms and write equation (2)

as n o

2 2 g ' g g 3

E ( E + E ) ( E + E + A ) - P k ( E + E + - A ) - E o

2 + E(E + E + A ) + (E + E )(E + E + A ) - P k ] = o . g g g

2 Replacing P k in the third term of equation(5) by its first order approximate value

given by equation (3) with mn/mo << 1, we get

g 2 2

E ( E + E ) ( E + E + A ) - P k ( E + E + - A ) - Eo ) + E ( E + E +A)+ g g g 3

E(E + E )(E + E + A )

E + E + - A + ( E + E ) ( E + E ) + A ) - 2 I=..

g g g 3

Short Notes K47 2

Substituting the value of P from equation (4) and neglecting (E/E ) compared to g

unity, we can further write from the above equation

Expanding binomially and keeping terms upto (E/E ) , we finally get from equation

(7) g

2 2

n = E(1+ dE) , (8)

fik 2 m

where

Further, if instead of taking a w 1/E , we write from equation (1) g

I% =(e E - 1)/E (9)

and determine this quantity from the original relation of Kane, then we can com-

pare the accuracy obtained from the two simplified forms with that obtained from

the original relation. For this, we equate the values of P k as obtained from

equations (2) and (4) and by putting (E - 5. k /2 m ) = El, we get

2

2 2 0

2 p f i k =

n

E1(E1 + E )(El + E + A ) (E + z A ) 2

m E (E + A )

m 2 n g g

2 m (El i- E + - A ) ( l - -

0 g 3

Thus, from equation (10) we can determine the diffemnt values of u, a8 given by

equation (9) for different values of E , since E = (El+ ( 5 k /2m n n o )(m /m ))-These 2 2

K48 physica status solidi (b) 71

Fig. 1. Plots of the energy dependence of a according to KBne’s dispersion relation

for three different semiconductors together with the values of a,’ and u, = 1/E

g

values of u as obtained from the original

relation of Kane tagether with the value of a

given by 1/E are compared in Fig. 1 for

three different semiconductors with the

value of a’ given under equation (8). The

parameters used for these semiconductors

a re a s follows (reference (3)): (i) InSb:

g

I I I I

0.1 0.2 I o(1 E =0.265eV, m n =0.013m 0’ and A = g =0 .9 eV, (ii) InAs: E =0.46 eV, m = E-

g n =0.022mo, and A =0.43eV,and(iii)GaAs:E =1.58eV, m =0.0655m and g n 0’

A = 0.35 eV . It is apparent from the figure that the simplified form derived here

is a better approximation of Kane’s dispersion relation when A is comparable to

E as in InAs and even when A is much greater than or is much less than E a s g g

in InSb and GaAs , respectively.

Acknowledgement

The authors a re indebted to Prof. J.N. Bhar for his keen interest in the work.

References

(1) E.M. CONWELL and M.O. VASSELL, Phys. Rev, 166, 797 (1968).

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

(3) M. NEUBERGER, III-V Semiconducting Compounds, Plenum Publ. Co., New

York 1971 (p. 77, 93, and 45).

(Received April 10, 1975)