A simple formulation of the saturation current density in heavily doped emitters

In a non-uniformly and heavily doped emitter region of a bipolar transistor, the continuity equation and the minority-current equation cannot be solved exactly in closed form. This paper shows that the calculation of minority-carrier current density can be calculated by a simple approach. This approach is based on the average value of the equilibrium hole density p0, diffusion constant Dp, and lifetime τp of minority carriers and leads to two coupled differential equations of the first order. These equations can be solved easily and can give a simple expression for the current density. Three definitions of the average values of p0, Dp, and τp are used and lead to three expressions for the emitter current density. The latter is identical to the one established by Rinaldi using another mathematical analysis and gives very accurate results for a shallow emitter (W < 1 µm), irrespective of the range peak doping level N(W) and surface-recombination velocity S. On the other hand, the other two expressions lead also to accurate results for the current density depending on the value of the surface-recombination velocity, but cannot be used when N(W) is greater than 1020 cm–3 and W is superior to 0.1 µm. PACS Nos.: 72.10.–d, 72.20.–i