To compute the transport properties of a material without full bands it is necessary to know the occupancy of the states within each of the partially filled bands. Under the influence of an applied force the distribution becomes distorted from that obtained at thermal equilibrium (Equation 3-5) and additionally can vary in space (for example in the presence of nonuniform electric fields, temperatures, or material properties) and time. The distribution function (f (m, r, t)) is related to the number of electrons (δn(k, r) in a volume element δ3k of k-space and δ3r of real space at point k, r in the following way:

Here the integral is performed over the first Brillouin zone in k-space. Since f (k, r, t) gives information about the electron occupancy of states, at a given instant in time Equation 3-38 can be rewritten in the following form:

Equation 3-39 is useful when computing the current contribution from a band that is nearly full, as only a small fraction of the band needs to be considered in the integral. The current is exactly the same as that which would be produced if the unoccupied electron states were occupied by positively charged particles (referred to as holes) and the occupied electron states were empty. The hole occupancy factor, f h(k, r, t), is simply given by:

Equation 3-38 shows that the time- and space-dependence of electric currents in a semiconductor is determined by the distribution function f (k, r, t). Within a volume element δ3kδ3r there are f (k, r, t)δ3kδ3r/4π3 electrons. In the presence of an external force the electrons move through real space at a velocity v (k) (given by Equation 3-34) and through k-space at a velocity ∂k/∂t (from Equation 3-35). At a time f (k, r) the electrons that occupy the volume element are those which were at k − δk, r − δr, at time t − δt. Therefore:

Note that the assumption has been made that the size of the volume element (δ3kδ3r) remains unchanged in time — Liouville’s theorem asserts that it is (see Ref. 1). In practice electrons collide with defects in the lattice (impurities) and with lattice distortions caused by mechanical waves in the lattice (phonons). The electrons do not collide with the lattice itself as the effect of the lattice is already incorporated into the Bloch functions. However, any deviation from perfect periodicity produce collisions. The effect of collisions is to add additional time dependence to the equation system, so the full Boltzmann equation is given by:

Using Equation 3-35 and substituting ∂r/∂t = v gives the final form of the Boltzmann equation for electrons:

The hole distribution function can be shown to obey a similar Boltzmann equation: