Modeling of minority-carrier transport in semiconductor regions with position-dependent material parameters at arbitrary injection levels

An analytical treatment of minority-carrier transport in bipolar transistors under arbitrary injection levels is presented. The analysis is not restricted to particular doping profiles and applies also to SiGe devices. As a first result, it is demonstrated that the minority-carrier transport equation is exactly soluble at high-injection (HI) levels, yielding closed-form expressions for the injected current, transit time, and sheet resistance. Contrary to the presently available formula which is recovered here as a particular case, our result reveals that the HI transit time is strongly affected by band-gap narrowing effects. It is also found that the transit time increase due to velocity saturation is more pronounced at HI levels than at low-injection (LI) levels. An analytical formulation for the collector current density, base transit time, and base sheet resistance, valid at any injection level, is then proposed. The analysis is based on the construction of approximate solutions of the transport equation. Finally, a simple expression for the effective electric field is derived, which allows one to more clearly study its variation with the injection level, and to easily take into account the electric field-dependence of the mobility