High-Field Energy Distribution, Mobility, and Diffusion of Heavy Holes inp-Germanium

A novel approximation scheme for hot-carrier distribution functions is introduced and employed in a calculation of the high-field mobility and diffusion coefficients of heavy holes in $p$-Ge at 77°K. It is assumed that the heavy-hole band has an isotropic and momentum-independent effective mass, and that the holes are scattered elastically by acoustic phonons and inelastically by optical phonons. Interaction with the light-hole and split-off bands is neglected. The principal results of the calculation are as follows: Over the decade $1 \mathrm{kV}/\mathrm{cm}<E<\mathrm{}10\mathrm{} \mathrm{kV}/\mathrm{cm}$ the mobility obeys the power-law relation $\mu \propto e^{-0.8\mathrm{}}$. The diffusion tensor is moderately anisotropic with $D_{\mathrm{II}}>D_{\perp }$, but neither coefficient departs greatly from the zero-field diffusion constant, 250 ${\mathrm{cm}}^2$/sec, in the field range up to 35 kV/cm. The calculation method makes use of a parametrized model of the distribution function which characterizes the energy dependence of its angular average by two distinct Maxwellians intersecting at the optical-phonon energy, but which makes no a priori assumptions about the angular dependence of the distribution function. Solution for the Maxwellian temperatures is effected by means of a special set of "anisotropy balance equations." These equations involve only the isotropic part of the distribution function and may be used to obtain the parameter values of any parametrized energy distribution, of which the present model is but a special case. Following a procedure originally outlined by Wannier, a derivation of these equations is given first for isotropic scattering and a spherical, constant mass, as required for the $p$-Ge calculation. An alternative method is used to derive a more general set of balance equations valid for scattering probabilities of the form $P\left(\right|\mathrm{}\mathrm{k}-{\mathrm{k}}^{\prime }\mathrm{}\left|\right)\mathrm{}$ and spherical bands of arbitrary dispersion law. An error-estimate criterion is formulated. This criterion permits evaluation of the influence on calculated transport quantities of distribution-function parametrizations with one additional parameter.