Scattering of Holes by Phonons in Germanium

The lattice scattering of holes within and between the two valence bands of germanium, degenerate at k=0, is calculated. Scattering by both acoustical and optical modes is considered. The electron-lattice interaction Hamiltonian is seen to be separable into two parts: the first, associated with acoustical modes, arises from vibrations of the unit cells as a whole and the other, associated with both acoustical and optical modes, arises from the relative motion of the two atoms in the unit cell of the germanium lattice. The matrix elements for scattering are expressible in terms of two constants, $C_1$ and $C_4$, associated respectively with the two parts of the interaction Hamiltonian. The wave functions used to calculate the matrix elements are determined by k·p and spin-orbit perturbations and assume spherical surfaces of constant energy in $k$-space and a parabolic relation between energy and wave number. For the terms in $C_1$ the scattering is treated using both the deformable and rigid ion models. The angular distributions for scattering are such that heavy holes are scattered predominantly in the forward direction and light holes in the backward direction for the deformable ion model, whereas the opposite is true for the rigid ion model. The scattering resulting from transverse and longitudinal phonons is about equally important for deformable ions; for rigid ions scattering by transverse modes is less significant. The matrix elements depending on $C_4$ are obtained from the rigid ion model alone. The transition probabilities for scattering are presented in a form which can be applied readily to the transport properties of germanium.