Journal of Computational Electronics 1: 359–363, 2002 c  2002 Kluwer Academic Publishers. Manufactured in The Netherlands. Efficient Poisson Solver for Semiconductor Device Modeling Using the Multi-Grid Preconditioned BiCGSTAB Method * G. SPEYER, D. VASILESKA AND S.M. GOODNICK Department of Electrical Engineering, Arizona State University, Tempe, AZ 85287, USA Abstract. This paper presents the performance results of an efficient algorithm for solving the three dimensional Poisson equation. The multi-grid method exploits the efficient oscillatory error reduction of basic iterative methods by smoothing on a set of progressively coarsened grids. When used as a preconditioner for BiCGSTAB method, a computationally demanding solver can be shown to be effective for large scale simulations. Varying the number of grids used and the level of overrelaxation as well as exploring the benefits of semicoarsening in the multi- grid preconditioner reveals the underlying strengths of this combined scheme. The convergence properties of the developed solver are tested on a 3D split-gate silicon on insulator (SOI) device. Keywords: computation, BiCGSTAB, multi-grid, preconditioning, SOI device modeling Introduction Self-consistent semiconductor device modeling re- quires repeated solution of the Poisson equation which describes the potential profile in the device for as given charge distribution. Due to the ever-diminishing scale of state-of-the-art device technology and the increased cost of the fabrication process, the need for accurate de- vice modeling has increased dramatically. Moreover, effects due to discrete impurities (Mizuno, Okamura and Toriumi 1994, Horstmann, Hilleringmann and Goser 1998, Wong and Taur 1993, Gross, Vasileska and Ferry 2000), in addition to the prominence of nar- row channel effects, necessitate accurate and fast three- dimensional solvers. However, with the addition of the third dimension, a tremendous increase in the number of mesh points as well as an increase in the average number of nearest neighbors must be included. As a result, a much larger number of linear equations must be solved, which dominate a much larger share of the overall simulation time. Hence, efficient algorithms are desired. One of the methods includes preconditioned bicon- jugate gradient stabilized (BiCGSTAB) method that ∗ Work supported by NSF under contract Nos. ECS-9802596 and ECS-9875051. has demonstrated efficient, albeit nonmonotonic con- vergence properties (Van der Vorst 1992). Like all Krylov-subspace methods, with each iteration, the BiCGSTAB method minimizes an increasing order polynomial, whose roots, after N (rank of Hessian) steps, correspond to the eigenvalues of the Hessian. Preconditioners enhance convergence by clustering the eigenvalues, thereby condensing the breadth of the eigenspectrum and allowing several roots to be min- imized concurrently (Jennings 1977). In this way, pre- conditioning increases the chance that the minimizing search direction chosen at each iteration will minimize more toward the solution. The second method is the multi-grid method for which it has been shown that for a large number of points in the spatial domain, it offers enhancement via multi-mode reduction of the error (Briggs 1987) and the fact that codes can be easily implemented on par- allel computers (Sandal¸ ci, Koc and Goodnick 1998). However, not all error modes can be reduced with the same efficiency due to the rigid restriction of double coarsening. In addition, aliasing of oscillatory error components onto coarser grids can introduce further error. From the above discussion, it follows that using the multi-grid method as a preconditioner to BiCGSTAB