Mathematical advances and horizons for classical and quantum-perturbed drift-diffusion systems: Solid state devices and beyond

The classical drift-diffusion model employed in semi-conductor simulation is now seen as part of a hierarchy of mathematical models designed to capture the intricate patterns of current flow in solid-state devices. These models include those incorporating quantum mechanical effects. Scientific computation has vastly outpaced our mathematical understanding of these models. This article is restricted in its focus, and describes mathematical understanding achieved during the last few decades primarily in terms of Gummel decomposition, as applied to drift-diffusion models and the closely related family of quantum corrected drift-diffusion models. Drift-diffusion models are being employed once again in organic devices, and in bio-chip devices, and a re-examination is now seen as timely, as such studies proceed beyond solid state devices.