A Finite Element Approximation Theory for the Drift Diffusion Semiconductor Model

Abstract

Two-sided estimates are derived for the approximation of solutions to the driftdiffusion steady-state semiconductor device system which are identified with fixed points of Gummel's solution map. The approximations are defined in terms of fixed points of numerical finite element discretization maps. By use of a calculus developed by Krasnosel'skii and his coworkers, it is possible both to locate approximations near fixed points in an "a priori" manner, as well as fixed points near approximations in an '% posteriori" manner. These results thus establish a nonlinear approximation theory, in the energy norm, with rate keyed to what is possible in a standard linear theory. This analysis provides a convergence theory for typical computational approaches in current use for semiconductor simulation.1. Introduction. The drift-diffusion model of a steady-state semiconductor device is formed by a system of three coupled partial differential equations (PDEs).This system of PDEs is solved by a solution vector of three function components. A fixed point mapping T :x Tx can be defined by solving each of these PDEs for its corresponding component and substituting these components in successive PDEs in a Gauss-Seidel fashion. Fixed points of such a mapping T then coincide with solutions to the drift diffusion model. Iteration with the mapping T defines an algorithm for the solution of the drift-diffusion model in which the PDEs are decoupled. The mapping T, termed Gummel's map [5] in the literature, is defined through solution for the potential u, for given electron and hole quasi-Fermi levels v and w, as a fractional step, and subsequently through solution for the electron and hole quasi-Fermi levels. This definition specifies the range of the mapping T. Principal properties, including fixed points and maximum principles, have been studied in increasing generality in [21],[23], and [8]. For this mapping, and for the slightly different mapping that operates in the space of the Slotboom variables V e -v and W e , the Lipschitz constant L T has been examined in detail in [15],[9],[17],[16]. Employing either quasi-Fermi levels or Slotboom variables, the appropriate formulation for device applications is the mixed Dirichlet/Neumann boundary value problem, taken over the physical device ( assumed to be a polyhedral domain, with possible solution gradient singularities at boundary transition points [15],[9].A companion approximation map is induced by piecewise linear finite elements, if the convex minimization, inherent in defining the successive gradient equations, is taken over finite dimensional affine subspaces. The fixed points of the companion map are clearly candidates for approximation of the fixed points of the solution map for the original system of PDEs. In this paper, we deduce an approximation theory, described by two-sided estimates, for this discretization procedure. Our theory is based upon an operator calculus developed by Krasnosel'skii and his collaborators (cf. [20]) in which

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