2.6 Procedure for Evaluating Phonon Scattering Rates

In the next two sections, we evaluate the scattering and momentum relaxation rates for phonon scattering. Because the same general procedure is used for all types of phonon scattering, we present the general procedure first, then examine the various types of phonon scattering separately. Recall first from Section 2.2, that we can express the perturbing potential for phonon scattering as

(2.58)

where u _? is a Fourier component of the lattice vibration as given by eq. (2.18). From the results of Section 2.2.2, we have

(2.59a)

(2.59b)

(2.59c)

(2.59d)

Because the perturbing potential is a traveling wave, we can evaluate the matrix element by analogy with eq. (1.127) in Section 1.7 to find

(2.60)

To evaluate the transition rate, we insert eq. (2.60) in eq. (2.2) to find

(2.61)

The two ?-functions in this equation are simply expressions of conservation of momentum and energy. To deal with the product of two ?-functions, we re-express them as a single ?-function expressing conservation of both momentum and energy. Beginning with

(2.62)

and taking its dot product with itself, we find

(2.63)

which, assuming parabolic energy bands ( E=p ²/2 m*), we use to write

(2.64)

or

(2.65)

By inserting this expression inside the ?-function in eq. (2.61), we obtain

(2.66)

(here, we made use of the fact that ?(ax) = ?(x)/a). Equation (2.66) replaces the product of two ?-functions, which express momentum and energy...