  We present an overview of electronic device modeling using non-equilibrium
Green function techniques. The basic approach developed in the early 1970s has
become increasingly popular during the last 10 years. The rise in popularity
was driven first by the experimental investigations of mesoscopic physics made
possible by high quality semiconductor heterostructures grown by molecular beam
epitaxy. The theory has continuously been adapted to address current systems of
interest moving from the mesoscopic physics of the late 1980s to single
electronics to molecular electronics to nanoscaled FETs. We give an overview of
the varied applications. We provide a tutorial level derivation of the polar
optical phonon self-energy [1]. Then, focusing on issues of a non-orthogonal
basis used in molecular electronics calculations, we derive and the basic Green
function expressions starting from their definitions in second quantized form
in a non-orthogonal basis. We derive the equations of motion for the retarded
Green function G^R and the correlation function G^<, and we derive the standard
expressions for the electron density and the current that are in widespread
use. We point out common approximations and open questions of which one finds
little discussion in the literature.
