Mathematical Modelling of Semi Conductor Devices and Processes

Abstract

Mathematical models of semiconductor devices are developed using the nearly isotropic approximation to the Boltzmann transport equation. The formalism treats the steady state inhomogeneous cases and an electric field is included in the analysis. As well as developing the equations to describe the charge transport appropriate boundary value problems are discussed. A general equation can be developed which, by the inclusion or exclusion of certain parameters, is able to describe the twelve models that are being considered: the type of scatterers, whether non-polar optical, piezoelectric or acoustic phonons, the presence or absence of an electric field and the order of expansion of the collision integral in terms of the phonon energy. Restricted cases of this equation are considered and general solutions given. One particular model, that of non-polar optical phonon scattering in the presence of an electric field with first order phonon energy expansion is discussed in detail. The electron distribution function and associated current due to an arbitrary injected energy distribution of electrons is determined by novel semi-analytical means. The other possible models, and solutions, are discussed and methods of validating the analysis are mentioned.