Appendix B
BOLTZMANN TRANSPORT
THEORY
Transport of electrons in solids is the basis of many modern technologies. The Boltzmann
transport theory allows us to develop a microscopic model for macroscopic quantities such as
mobility, diffusion coefficient, and conductivity. This theory has been used in Chapter 8 to study
transport of electrons and holes in materials. In this appendix we will present a derivation of this
theory.
B.1 BOLTZMANNTRANSPORTEQUATION ....................
In order to describe the transport properties of an electron gas, we need to know the distribution
function of the electron gas. The distribution would tell us how electrons are distributed in
momentum space ork-space (and energy-space) and from this information all of the transport
properties can be evaluated. We know that at equilibrium the distribution function is simply the
Fermi-Dirac function
f(E)=
1
exp(
E−EF
kBT)
+1
(B.1)
This distribution function describes the equilibrium electron gas and isindependent of any col-
lisions that may be present. While the collisions will continuously remove electrons from one
k-state to another, the net distribution of electrons is always given by the Fermi-Dirac function
as long as there are no external influences to disturb the equilibrium.
To describe the distribution function in the presence of external forces, we develop the Boltz-
mann transport equation. Let us denote byfk(r)the local concentration of the electrons in
statekin the neighborhood ofr. The Boltzmann approach begins with an attempt to determine
howfk(r)changes with time. Three possible reasons account for the change in the electron
distribution ink-space andr-space: