  Many challenges faced in today's semiconductor devices are related to
self-heating phenomena. The optimization of device designs can be assisted by
numerical simulations using the non-isothermal drift-diffusion system, where
the magnitude of the thermoelectric cross effects is controlled by the Seebeck
coefficient. We show that the model equations take a remarkably simple form
when assuming the so-called Kelvin formula for the Seebeck coefficient. The
corresponding heat generation rate involves exactly the three classically known
self-heating effects, namely Joule, recombination and Thomson-Peltier heating,
without any further (transient) contributions. Moreover, the thermal driving
force in the electrical current density expressions can be entirely absorbed in
the diffusion coefficient via a generalized Einstein relation. The efficient
numerical simulation relies on an accurate and robust discretization technique
for the fluxes (finite volume Scharfetter-Gummel method), which allows to cope
with the typically stiff solutions of the semiconductor device equations. We
derive two non-isothermal generalizations of the Scharfetter-Gummel scheme for
degenerate semiconductors (Fermi-Dirac statistics) obeying the Kelvin formula.
The approaches differ in the treatment of degeneration effects: The first is
based on an approximation of the discrete generalized Einstein relation
implying a specifically modified thermal voltage, whereas the second scheme
follows the conventionally used approach employing a modified electric field.
We present a detailed analysis and comparison of both schemes, indicating a
superior performance of the modified thermal voltage scheme.
