The Boltzmann equation

The carriers in a metal or semiconductor can be affected by external fields, and by temperature gradients. They also suffer scattering from impurities, lattice waves, etc. These effects have to be balanced against each other—we have to consider situations in which the electron is accelerated by a field, but loses its extra energy and momentum by scattering. In this chapter we shall consider the ‘Ordinary’ transport properties, such as are observed when constant fields are applied.

Much the simplest approach to this problem, in general, is to set up the transport equation or Boltzmann equation. We study a quantity fk(r), the local concentration of carrier in the state k in the neighbourhood of the point r in space. Strictly speaking, this quantity can only be defined in terms of fine-grained distributions, ensemble averages, density matrices, etc. There is a considerable literature upon this point—but it belongs more to the formal theory of quantum statistical mechanics than to the theory of solids.