SEMICONDUCTOR DEVICE PHYSICS AND DESIGN

528 APPENDIX B. BOLTZMANN TRANSPORT THEORY

semiconductors. For most scattering processes, one finds that it is possible to express the energy
dependence of the relaxation time in the form

τ(E)=τ 0 (E/kBT)s (B.46)

whereτ 0 is a constant andsis an exponent which is characteristic of the scattering process.
We will be calculating this energy dependence for various scattering processes in the next two
chapters. When this form is used in the averaging of equation B.45, we get, using a Boltzmann
distribution forf^0 (k)

〈〈τ〉〉=τ 0

∫∞

0 [p

(^2) /(2m∗kBT)]sexp[−p (^2) /(2m∗kBT)]p (^4) dp
∫∞
0 exp[−p^2 /(2m∗kBT)]p^4 dp

(B.47)

wherep=kis the momentum of the electron.
Substitutingy=p^2 /(2m∗kBT),weget

〈〈τ〉〉=τ 0

∫∞

0 y

s+(3/2)e−ydy

∫∞

0 y

3 / (^2) e−ydy (B.48)
To evaluate this integral, we useΓ-functions which have the properties
Γ(n)=(n−1)!
Γ(1/2) =

√

π

Γ(n+1) = nΓ(n) (B.49)

and have the integral value

Γ(a)=

∫∞

0

ya−^1 e−ydy (B.50)

In terms of theΓ-functions we can then write

〈〈τ〉〉=τ 0

Γ(s+5/2)

Γ(5/2)

(B.51)

If a number of different scattering processes are participating in transport, the following ap-
proximate rule (Mathiesen’s rule) may be used to calculate mobility:

1

τtot

=

∑

i

1

τi

(B.52)

1

μtot

=

∑

i

1

μi

(B.53)

where the sum is over all different scattering processes.