This paper examines the solution map of the stationary version of the Van Roosbroeck model for the flow of electrons and holes in a crystalline semiconductor. Thermodynamic equilibrium and the Einstein relations linking mobility and diffusion are assumed to hold for this model. Incorporated into the model is the quantum-statistical assumption that the carrier densities satisfy Fermi-Dirac statistical laws with localization in the 'Boltzmann tail'. This latter approximation, which is seen to be inessential, characterizes the conduction and valence bands as nondegenerate. The boundary conditions are of two types, and correspond to applied potential differences and doping on the contact portions of the device, and insulation on the remainder. The complete system, then, involves a coupled set of three nonlinear partial differential equations, with their boundary conditions, for the electrostatic potential, and for the quasi-Fermi levels corresponding to the electron and hole concentrations.
ASJC Scopus subject areas
- Applied Mathematics