Abstract
In this paper we describe, analyse and implement a parallel iterative method for the solution of the steady-state drift diffusion equations governing the behaviour of a semiconductor device in two space dimensions. The unknowns in our model are the electrostatic potential and the electron and hole quasi-Fermi potentials. Our discretisation consists of a finite element method with mass lumping for the electrostatic potential equation and a hybrid finite element with local current conservation properties for the continuity equations. A version of Gummel's decoupling algorithm which only requires the solution of positive definite symmetric linear systems is used to solve the resulting nonlinear equations. We show that this method has an overall rate of convergence which only degrades logarithmically as the mesh is refined. Indeed the (inner) nonlinear solves of the electrostatic potential equation converge quadratically, with a mesh independent asymptotic constant. We also describe an implementation on a MasPar MP-1 data parallel machine, where the required linear systems are solved by the preconditioned conjugate gradient method. Domain decomposition methods are used to parallelise the required matrix-vector multiplications and to build preconditioners for these very poorly-conditioned systems. Our preconditioned linear solves also have a rate of convergence which degrades logarithmically as the grid is refined relative to subdomain size, and their performance is resilient to the severe layers which arise in the coefficients of the underlying elliptic operators. Parallel experiments are given.
Zusammenfassung
Diese Arbeit enthält die Beschreibung, Analyse und Implementierung eines parallelen iterativen Verfahrens zur Lösung der stationären Drift-Diffusions-Gleichung für ein zweidimensionales Halbleitermodell. Die Unbekannten unseres Modells sind das elektrostatische Potential und die Quasi-Fermi-Potentiale der Elektronen und Löcher. Unsere Diskretisierung verwendet die Finite-Element-Methode mit ‘mass limping’ für das elektrostatische Potential und hybride Elemente mit lokaler Stromerhaltung für die Kontinuitätsgleichung. Zur Lösung der entsprechenden nichtlinearen Gleichungen wird eine Version des Gummel-Algorithmus verwendet, der lediglich die Lösung positiv definiter, symmetrischer linearer Gleichungssysteme erfordert. Wir zeigen, daß diese Methode eine Konvergenzrate besitzt, die nur logarithmisch von der Schrittweite abhängt. Die (inneren) nichtlinearen Löser der elektrostatischen Potentialgleichung konvergieren gitterunabhängig quadratisch. Wir beschreiben auch eine Implementierung auf einem MasPar MP-1-Parallelrechner, wobei die auftretenden linearen Systeme mit einem vorkonditionierten cg-Verfahren approximiert werden. Gebietszerlegungsmethoden werden eingesetzt, um die notwendige Matrix-Vektor-Multiplikation zu parallelisieren und Vorkonditionierer dieser schlechtkonditionierten Systeme zu erstellen. Die vorkonditionierten linearen Löser haben ebenfalls eine Konvergenzrate, die logarithmisch vom Verhältnis Teilgebietsgröße zu Schrittweite abhängt und welche robust ist bezüglich stark variierenden Koeffizientenfunktionen des zugrundeliegenden elliptischen Operators. Experimente auf einem Parallelrechner werden diskutiert.
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Coomer, R.K., Graham, I.G. Massively parallel methods for semiconductor device modelling. Computing 56, 1–27 (1996). https://doi.org/10.1007/BF02238289
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DOI: https://doi.org/10.1007/BF02238289