Integral-equation solution of minority-carrier transport problems in heavily doped semiconductors

The problem of minority-carrier transport in quasi-neutral regions of heavily doped semiconductors, presented in an integral-equation form, is discussed with reference to bipolar diffused-junction transistors. The procedure avoids any regional simplification of the coefficients, and builds all of them, along with the boundary conditions, into only two terms of the integral equation. This makes it easy to point out the reciprocal trade-off effects of the coefficients, particularly those deriving from the heavy doping and finite surface-recombination velocity at ohmic contacts. In addition, the integral equation can be solved by using a well-known iterative technique, the convergence of which can be determined a priori by examining the kernel. The results show that in many cases a single iteration is sufficient, yielding a closed-form expression for the minority-carrier distribution, the minority current injected into the emitter, and the emitter transparency. In those cases where the convergence is slow or fails, an alternative solution technique is suggested, based upon the expansion of the unknown into a set of orthogonal functions.