On calculating the current gain of junction transistors with arbitrary doping distributions

In transistors made by such techniques as solid-state diffusion, the doping density is not constant in the emitter and base regions. Tanenbaum and Thomas have given an expression for the emitter efficiency of such a transistor. The present work extends that treatment, and gives an expression for the base transport factor. Thus the current gain can be calculated for any arbitrary distribution of doping density. It is found that, for constant doping densities, the results from this treatment reduce to those usually given, (at least to the usual approximation), as they should. Two cases of some practical interest are that in which the doping density varies monotonically from one side of the base to the other, as in the "drift transistor," and that in which the conductivity in the base region has a maximum somewhere within that region. Results for simplified forms of these two cases are presented in the form of graphs for convenience. Also, the result given by the present approximation for an exponential variation of doping density is compared with the exact calculation which can be made for this case.