The drift-diffusion current relations can, amongst others, be derived from the
Boltzmann transport equation by the method of moments or
from basic principles of irreversible thermodynamics under the assumption
T_{} = T_{L}. The electron and hole current densities are given by

Here,
and
denote the carrier mobilities, and T_{L} is the
lattice temperature.
These current relations account for position-dependent band edge energies,
E_{C} and E_{V}, and position-dependent effective masses, which are included in
the effective density of states, N_{C, 0} and N_{V, 0}. The index 0 indicates
that N_{C, 0} and N_{V, 0} are evaluated at some (arbitrary) reference temperature,
T_{0}, which is constant in real space regardless of what the local
values of the lattice and carrier temperatures are.