Modeling and optimization of shallow and opaque heavily doped emitters for bipolar devices

Different analytical formulations for the minority-carrier current injected into arbitrarily doped semiconductor regions are proposed, covering the case of shallow as well as opaque emitters. Firstly, based on an original transformation of the transport equations, a different derivation and formulation of the Selvakumar and Roulston model is presented. The interesting feature of this formulation is that the expression of the injected current is formally identical to that relative to the constant doping case, thereby facilitating physical insight. All effects related to the doping dependence of transport parameters are embedded into two new physical parameters, namely a characteristic surface recombination velocity and an effective diffusion length. From the study of the dependence of these quantities upon the surface doping, it is possible to obtain a simple expression for the current injected in shallow regions, which is shown to be a useful tool in interpreting the dependence of the minority-carrier current upon all significant technological parameters, and to facilitate the optimization process. The injection in opaque regions is then discussed in detail, and the asymptotic behavior for large region widths is described. A new analytical model is presented, which, unlike all previous approaches, provides an accurate description of both the nonopaque and opaque regimes