  We develop a theory of energy relaxation in semiconductors and insulators
highly excited by the long-acting external irradiation. We derive the equation
for the non-equilibrium distribution function of excited electrons. The
solution for this function breaks up into the sum of two contributions. The
low-energy contribution is concentrated in a narrow range near the bottom of
the conduction band. It has the typical form of a Fermi distribution with an
effective temperature and chemical potential. The effective temperature and
chemical potential in this low-energy term are determined by the intensity of
carriers' generation, the speed of electron-phonon relaxation, rates of
inter-band recombination and electron capture on the defects. In addition,
there is a substantial high-energy correction. This high-energy 'tail' covers
largely the conduction band. The shape of the high-energy 'tail' strongly
depends on the rate of electron-phonon relaxation but does not depend on the
rates of recombination and trapping. We apply the theory to the calculation of
a non-equilibrium distribution of electrons in irradiated GaN. Probabilities of
optical excitations from the valence to conduction band and electron-phonon
coupling probabilities in GaN were calculated by the density functional
perturbation theory. Our calculation of both parts of distribution function in
gallium nitride shows that when the speed of electron-phonon scattering is
comparable with the rate of recombination and trapping then the contribution of
the non-Fermi 'tail' is comparable with that of the low-energy Fermi-like
component. So the high-energy contribution can affect essentially the charge
transport in the irradiated and highly doped semiconductors.
