Detailed introductions to the physics of semiconductors are available in Refs 1 to 5. Ref. 6 is an introduction to semiconductor device modeling using the drift-diffusion method used by COMSOL Multiphysics. Ref. 45 provides a comprehensive review of the density-gradient (DG) theory which adds the effect of quantum confinement to the drift-diffusion method in a computationally efficient fashion. The DG-Confinement theory is implemented by COMSOL and the corresponding equations are pointed out when they differ from the conventional drift-diffusion method.

A brief summary of the physics of semiconductors is included here, to a level sufficient to derive the drift-diffusion equations and to develop a basic understanding of semiconductor band structures. The derivation of the density-gradient theory is not included. See Ref. 45 if interested.

To completely describe the dynamics of electrons within a solid, the many body Schrödinger equation must be solved in the periodic structure defined by the crystal structure of the solid. In practice this is not possible and approximations must be made. Solid state physicists have devised a number of methods to solve simplified forms of this equation and these methods have been validated through experiment (see, for example, chapters 9 to 11 of Ref. 1 and chapters 4 and 5 of Ref. 2). A starting point for many of these methods is to consider only the motion of the electrons through an essentially stationary lattice of nuclei (known as the adiabatic approximation). Then the many electron wave function is simplified into a form in which it reduces to a set of one electron wave functions. The effect of the nuclei and of electron-electron interactions can be incorporated into these one electron models by modifications to the effective potential. Models within the so-called one-electron approximation have been very successful at predicting the properties of semiconductors and semiconductor transport. Although the one electron model explained in this section seems very simplistic (particularly when considering the complexity of electron-electron and electron-ion interactions), in practice this model can be used to develop a very detailed understanding of transport in semiconductors.