Overview

As carriers traverse a device, their motion is frequently interrupted by collisions with impurity atoms, phonons, crystal defects, or with other carriers. In this chapter we examine carrier scattering and evaluate the transition rate, S (p, p ?), which is the probability per unit time that a carrier with crystal momentum p scatters to a state with crystal momentum p ?. Our approach is based on Fermi s Golden Rule as described in Section 1.7. The first step is to identify the scattering potential then to evaluate the matrix element,

(2.1)

In this chapter, we keep the mathematics to a minimum by evaluating matrix elements using plane wave electron wave functions rather than the actual Bloch functions. The overlap integral due to the cell periodic part of the Bloch function [recall eq. (1.116)] is unity when the constant energy surfaces are spherical and the energy varies parabolically with momentum (which we will simply call spherical, parabolic bands). In practice, however, the energy bands are rarely parabolic, and overlap integrals have to be evaluated [1.5].

Having evaluated the matrix element, we find the transition rate as

(2.2)

where ?E is the change in energy (if any) caused by the scattering event. (Note that the order of p and p ? in H _{p ?p} is, by definition, opposite to their order in S (p,p').)

The effects of scattering are conveniently summarized by evaluating characteristic times from the transition rate,