Analytical base transit time model of a bipolar transistor considreing majority carrier current in the base

Abstract

In deriving an analytical model for the base transit time  B for bipolar junction transistors (BJTs), various non-ideal effects have to be considered. These effects include the bandgap narrowing effects due to heavy doping, the Webster and the Kirk effects due to high injection and the effects due to the position and field dependence of the transport parameters (i.e. carrier mobility and carrier lifetime). The non-uniformity of the doping profile, and the doping levels make the transport parameters to be position and field dependent. The electric field in the base is mainly due to the non-uniformity of the doping profile. However, the field is modulated by the injection levels, the gradient of the transport parameters and the majority carrier current density in the base. For low doping levels, the effects of majority carrier current density are insignificant. When base doping level is heavy ( 1018 3  cm ), the effects of majority carrier current density are no longer negligible. Moreover, at such high base doping, recombination mechanisms and the lateral base injection become significant, which also enhance the effects of majority carrier current density. However, consideration of p J as well as all non-ideal effects results in a nonlinear, nonhomogeneous, variable-coefficient differential equation, the solution of which is intractable. In this work, a modified current equation reflecting the injection-level dependency has been derived for the first time in the literature. The electric field term deduced in this equation is able to identify the effects of the band-gap narrowing, the injection level and the majority carrier current density. Concept of perturbation theory is applied to linearize the governing differential equation. An exponential approximation technique is introduced to address the intractability problem and used to convert this differential equation into a solvable form. The results of the developed model shows that p J has a significant effect on the base transit time. Close match with the numerical simulation results and also with measurement data with two experimental setups justifies the validity of the developed model.