  Boltzmann relations are widely used in semiconductor physics to express the
charge-carrier densities as a function of the Fermi level and temperature.
However, these simple exponential relations only apply to sharp band edges of
the conduction and valence bands. In this article, we present a generalization
of the Boltzmann relations accounting for exponential band tails. To this end,
the required Fermi-Dirac integral is first recast as a Gauss hypergeometric
function, followed by a suitable transformation of that special function, and a
zeroth-order series expansion using the hypergeometric series. This results in
simple relations for the electron and hole densities that each involve two
exponentials. One exponential depends on the temperature and the other one on
the band-tail parameter. The proposed relations tend to the Boltzmann relations
if the band-tail parameters tend to zero. This work comes timely for the
modeling of classical semiconductor devices at cryogenic temperatures for
large-scale quantum computing.
