4.1 Semiconductor Equations

The Boltzmann transport equation (BTE) provides the fundament for the semiclassical description of carrier transport in semiconductor devices. As a numerical solution is prohibitive due to the extraordinary high computational expenses, several approximations to derive simpler solutions exist. Those differ widely in their computational demand and physical accuracy. This work focuses on the drift-diffusion (DD) transport model and the hydrodynamic (HD) transport model. The former is a simple model, well established today in TCAD tools, while the latter is a higher-order model capable of describing non-local effects. Several other approaches also exist, like the Monte Carlo approach [261] or the method of spherical harmonics [262] for instance, however those are beyond the scope of this engineering-oriented work.

The basic semiconductor equations include Poisson's equation, the two current continuity equations and the two current relation equations. The first three are shared by both the DD and HD transport models, however the current relations have different formulations for each model.