Non-quasi-static models including all injection levels and DC, AC, and transient emitter crowding in bipolar transistors

Two-dimensional equivalent-circuit models for bipolar junction transistors are systematically derived by solving the continuity equations for DC, AC, and transient excitations. These models take into account carrier propagation delay, all injection levels, as well as exponential doping profiles. They include analytically DC, AC, and transient emitter crowding in a more detailed and accurate manner than previously available. Extensions of the models to accommodate arbitrarily doped and heavily doped quasi-neutral layers and to include energy-gap narrowing due to the electron-hole plasma present at high current density are described. The analysis leads to compact large- and small-signal equivalent-circuit lumped models, suitable for use in circuit simulators such as SPICE. The analytical solutions obtained reveal the two-dimensional distribution of the current and carrier densities in the intrinsic base layer and the onset of emitter crowding. They also provide information for the extraction of the intrinsic base resistance. Several assumptions made in the derivations are assessed by the computer program PISCES. The methods presented apply to both homojunction and heterojunction bipolar transistors