We present a rigorous analytical treatment of band-band impact ionizat
ion in semiconductor high-field transport. The microscopic electron im
pact ionization scattering time is calculated for the general case of
three different anisotropic parabolic bands (one for the initial valen
ce electron, the other two for the two final conduction electrons) and
an arbitrarily shaped band for the impact-ionizing energetic conducti
on electron. In this derivation the wave vector dependence of the matr
ix element is accounted for in contrast to previous calculations. The
total impact ionization rate in both direct and indirect semiconductor
s and the associated energy relaxation rate in direct semiconductors a
re expressed analytically in a universal scaling form as a function of
the electron temperature and a few band-structure parameters like eff
ective masses, energy gap, and the distance in k-space between the ban
d extrema (in case of an indirect semiconductor). Such expressions can
be used in the hydrodynamic model as approximations for the collision
terms arising in a moment expansion of the Boltzmann equation.