In Chapter 4 we have derived a number of important mathematical relations necessary to calculate transport properties. A key ingredient of the theory is the scattering rate W(k, k′) which tells us how an electron in a state k scatters into the state k′. We will now evaluate the scattering rates for a number of important scattering mechanisms. As noted in Chapter 4, the approach used by us is semiclassical—the electron is treated as a Bloch wave while calculating the scattering rate, but is otherwise treated as a particle. The Fermi golden rule is used to calculate the scattering rate.
In Fig. 5.1 we show an overview of how one goes about a transport calculation. Once the various imperfections in a material are identified the first and most important ingredient is an understanding of the scattering potential. This may seem like a simple problem, but is, in fact, one of the most difficult parts of the problem. Once the potential is known, one evaluates the scattering matrix element between the initial and final state of the electron. This effectively amounts to taking a Fourier transform of the potential since the initial and final states are essentially plane wave states. With the matrix element known one carries out an integral over all final states into which the electron can scatter and which are consistent with energy conservation. This kind of integral provides the various scattering times.