A two-dimensional numerical analysis of a silicon npn-transistor

For today's transistors made by sophisticated technology the first order transistor theory leads to results which do not compare well with experimental results. In an effort to overcome this situation Gummel for the first time used a digital computer to give a unified exact treatment of one dimensional device performance. This paper treats the two dimensional case, which is to be considered if lateral current effects are to be explained. A set of three nonlinear partial differential equations describing the flow of carriers within the transistor under steady state conditions is formulated and solved in an iterative scheme. Realistic geometries, doping profiles, recombination parameters and terminal voltages are used in the device description. The idealized subdivision into emitter, base and collector regions separated by space charge layers is avoided. The potential distribution and the electron and hole distribution within the transistor are calculated. Two dimensional plots of these quantities are shown. A number of questions, such as lateral base voltage drops and inhomogenous collector current density, which arise from spreading effects and call for a solution in at least two spatial dimensions are treated.