The authors consider the static, one-dimensional modelling of a semiconductor device (namely the pn-junction) when a bias is applied. The governing equations are the well-known equations describing carrier transport in a semiconductor which consist of a system of ordinary differential equations subject to boundary conditions imposed at the contacts. Because of the different orders of magnitude of the solution components at the boundaries, the authors scale the components individually and obtain a singular perturbation problem. They analyze the equilibrium case (zero bias applied) and set up approximate models, posed as singularly perturbed second order equations, by neglecting the hole and electron current densities. For the full problem they prove an a priori estimate on the number of electron-hole carrier pairs and derive asymptotic expansions (as the perturbation parameter tends to zero) by setting up the reduced system and the boundary layer system. They prove existence theorems for both systems and use the asymptotic expansion to solve the model equations numerically and analyze the dependence of the solutions on the applied bias.
ASJC Scopus subject areas
- Applied Mathematics