Computer-aided two-dimensional analysis of bipolar transistors

A method for solving numerically the two-dimensional (2D) semiconductor steady-state transport equations is described. The principles of this method have been published earlier [1]. This paper discusses in detail the method and a number of considerable improvements. Poisson's equation and the two continuity equations are discretized on two networks of different rectangular meshes. The 2D continuity equations are approximated by a set of difference equations assuming that the hole and electron current density components along the meshlines are constant between two neighboring meshpoints in a way similar to that used by Gummel and Scharfetter [2] for the one-dimensional (1D) continuity equations. The resulting difference approximations have generally a much larger validity range than the conventional difference formulations where it is assumed that the change in electrostatic potential between two neighboring points is small compared with k T/q . Therefore, a much smaller number of meshpoints is necessary than for the conventional difference approximations. This reduces considerably the computation time and the required memory space. It will be shown that the matrix of the coefficients of this set of difference equations is always positive definite. This is an important property and guarantees convergence and stability of the numerical solution of the continuity equations. The way in which the difference approximations for the continuity equations are derived gives directly consistent expressions for the current densities that can be used for calculating the currents. In order to demonstrate the kind of solutions obtainable, steady-state results for a bipolar n-p-n silicon transistor are presented and discussed.

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