SEMICONDUCTOR DEVICE PHYSICS AND DESIGN

B.1. BOLTZMANN TRANSPORT EQUATION 523

eτE

h

fk

0

0 )

eτE

h

f^0 k)–

Figure B.3: The displaced distribution function shows the effect of an applied electric field.

and negative velocities. When the field is applied, there is a net shift in the electron momenta
and velocities given by

δp = δk=−eτE

δv = −

eτE

m∗

(B.22)

This gives, for the mobility,

μ=

eτ

m∗

(B.23)

If the electron concentration isn, the current density is

J = neδv

=

ne^2 τE

m∗

or the conductivity of the system is

σ=

ne^2 τ

m∗

(B.24)

This equation relates a microscopic quantityτto a macroscopic quantityσ.
So far we have introduced the relaxation timeτ, but not described how it is to be calculated.
We will now relate it to the scattering rateW(k,k

′
), which can be calculated by using the Fermi
golden rule. We have, for the scattering integral,

∂f

∂t

)

scattering

=

∫ [

f(k

′

)(1−f(k))W(k

′

,k)−f(k)(1−f(k

′

))W(k,k

′

)

]d (^3) k′
(2π)^3
Let us examine some simple cases where the integral on the right-hand side becomes simplified.