Electrical current and carrier density in degenerate materials with nonuniform band structure

This paper reviews and extends the theory of electron and hole motion and density in solids with position-dependent band structure. This includes materials with graded composition, like heterojunctions, with nonuniform temperature or strain, and devices with highly doped regions, like the emitter region of modern bipolar transistors and solar cells. Size effects, such as those in short-channel MOS transistors, are not considered. Changes in the energy band edges due to spatial variations in electron affinity and bandgap and changes in the density of states produce terms in the carrier- and current-density equations in addition to those found in the conventional Shockley model. This paper emphasizes the physical concepts underlying these modifications and provides a theoretical framework from which experimental results can be correctly interpreted. The new effects are derived and discussed. Areas which require further research are pointed out. The paper discusses the validity and meaning of a nonuniform band structure. The general energy-band diagram relating the electrostatic potential, the electron affinity, and the bandgap is given. Expressions for current density are derived from a solution of the Boltzmann equation. Expressions for nonequilibrium carrier densities are presented. The effective mass approximation and the rigid band model are discussed. The concepts of generalized drift and diffusion are established and alternative formulations are given. Various device applications including minority-carrier flow in quasi-neutral regions are also given.