B.1. BOLTZMANN TRANSPORT EQUATION 521
Substituting these terms and retaining terms only to second-order in electric field (i.e., ignoring
terms involving productsgk·E), we get, from equation B.11,
−∂f0
∂Ek·vk·[
−(EkT−μ)∇T+eE−∇μ]
=−∂f∂t)
scattering+vk·∇rgk+e(vk×B)·∇kgk.(B.14)
The equation derived above is the Boltzmann transport equation.
We will now apply the Boltzmann equation to derive some simple expressions for conductivity,
mobility, etc., in semiconductors. We will attempt to relate the microscopic scattering events to
the measurable macroscopic transport properties. Let us consider the case where we have a
uniform electric fieldEin an infinite system maintained at a uniform temperature.
The Boltzmann equation becomes
−
∂f^0
∂Ekvk·eE=−∂fk
∂t)
scattering(B.15)
Note that only the deviationgkfrom the equilibrium distribution function above contributes to
the scattering integral.
As mentioned earlier, this equation, although it looks simple, is a very complex equation which
can only be solved analytically under fairly simplifying assumptions. We make an assumption
that the scattering induced change in the distribution function is given by
−
∂fk
∂t)
scattering=
gk
τ(B.16)
We have introduced a time constantτwhose physical interpretation can be understood when
we consider what happens when the external forces have been removed. In this case the pertur-
bation in the distribution function will decay according to the equation
−∂gk
∂t=
gk
τor
gk(t)=gk(0)e−t/τ (B.17)
The timeτthus represents the time constant for relaxation of the perturbation as shown
schematically in figure B.2 The approximation which allows us to write such a simple relation is
called the relaxation time approximation (RTA).
According to this approximation
gk = −∂fk
∂t)
scattering·τ=
−∂f^0
∂Ekτvk·eE (B.18)