The point of semiconductor device analysis is to find useful predictors of device performance. With these we can design to specifications, uncover promising directions for improving device operation, and understand device limitations. An actual device has no trouble finding its operating point, but we must make a succession of simplifying approximations to achieve even a partial understanding. Useful concepts have been developed for describing semiconducting materials: band gap, densities of states, hole and electron effective masses, dielectric tensor, and so on. With some additional approximations, such as parabolic energy bands, energy independent relaxation times, Boltzmann statistics, etc. then Poisson's equation and the current continuity equations form a set of coupled nonlinear PDEs that can be solved numerically: Only for the very simplest problems can these equations be solved analytically. The response of semiconductor dcvices to magnetic fields is inherently 2-dimensional. Carriers suffer deflections from their usual courses, producing effects in directions perpendicular to their normal flows. Exact analytical techniques in 2 dimensions are reasonably accurate for Hall devices [I], because it is usually appropriate to assume equilibrium carrier concentrations, reducing Poisson's equation to Laplace's equation. When more complex devices such as bipolar transistors are biased into operation, carrier concentrations are perturbed significantly from their equilibrium conditions. Analytical methods have rather limited validity even in the absence of magnetic fields. One of the first examples of 2-d bipolar device modeling  was used to disprove earlier results calculated from an inadequate analytical model. Given the fact that no good 2-d analytical models exist for bipolar device operation, and that modelling in 2 dimensions is necessary to understand the workings of the VCDM, we are naturally led to undertake numerical simulation. In this chapter we will review the effects of magnetic fields on semiconductor devices, and after reviewing 2-dimensional semiconductor simulation, discuss the changes that would be necessary to incorporate a magnetic field into a simulation program.
May 31, 1989
Burns, B. S. (1989). Analysis and Characterization of the Vertical Carrier Domain Magnetometer: Research Project. United States: University of California, Berkeley.