Non-Equilibrium Green Functions in Electronic Device Modeling
  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. 
Roger K. Lake and Rajeev R. Pandey