A globally convergent algorithm for the solution of the steady-state semiconductor device equations

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Abstract
An iterative method for solving the discretized steady-state semiconductor device equations is presented. This method uses Gummel’s block iteration technique to decouple the nonlinear Poisson and electron-hole current continuity equations. However, the main feature of this method is that it takes advantage of the diagonal nonlinearity of the discretized equations, and solves each equation iteratively by using the nonlinear Jacobi method. Using the fact that the diagonal nonlinearities are monotonically increasing functions, it is shown that this method has two important advantages. First, it has global convergence, i.e., convergence is guaranteed for any initial guess. Second, the solution of simultaneous algebraic equations is avoided by updating the value of the electrostatic and quasi-Fermi potentials at each mesh point by means of explicit formulae. This allows the implementation of this method on computers with small random access memories, such as personal computers, and also makes it very attractive to use on parallel processor machines. Furthermore, for serial computations, this method is generalized to the faster nonlinear successive overrelaxation method which has global convergence as well. The iterative solution of the nonlinear Poisson equation is formulated with energy- and position-dependent interface traps. It is shown that the iterative method is globally convergent for arbitrary distributions of interface traps. This is an important step in analyzing hot-electron effects in metal-oxide-silicon field-effect transistors (MOSFETs). Various numerical results on two- and three-dimensional MOSFET geometries are presented as well.