Abstract – The solution of large sparse unsymmetric linear systems is a critical and challenging component of semiconductor device and circuit simulations. The time for a simulation is often dominated by this part. The sparse solver is expected to balance different, and often conflicting requirements. Reliability, a low memory-footprint, and a short solution time are a few of these demands. Currently, no black-box solver exists that can satisfy all criteria. The linear systems from both simulations can be highly ill-conditioned and therefore quite challenging for direct and iterative methods. In this paper, it is shown that algorithms to place large entries on the diagonal using unsymmetric permutations and scalings greatly enhance the reliability of both direct and preconditioned iterative solvers for unsymmetric linear systems arising in semiconductor device and circuit simulations. The numerical experiments indicate that the overall solution strategy is both reliable and cost effective.
Index Terms – Semiconductor device simulation, circuit simulation, sparse linear solvers, numerical linear algebra, sparse unsymmetric matrices, preconditioning
I. INTRODUCTION
I
N 1950, Van Roosbroeck [1] introduced the drift-diffusion equations, which are the commonly used model in semiconductor device simulation. The drift-diffusion equations are a system of three coupled, nonlinear partial differential equations (PDEs), which describe the relation between the electrostatic potential and the densities of the charge carriers in a semiconductor device. The coupling between the different equations is highly nonlinear and implies numerical difficulties. More sophisticated models have evolved since the beginnings of semiconductor device simulation to describe the increasingly complex devices. The equations of the drift-diffusion model are:
-V – (eVV>) , = q (p – n + C)
Dn.
Q-fo – V – J” = –
qR,
Dp.
Q-fo + V-Jp = – qR,
Manuscript submitted to IEEE TCAD. This work was supported by the Swiss Commission of Technology and Innovation under contract number 5648.1, the IBM T. J. Watson Research Center, and the Strategic Excellence Positions on Computational Science of the Swiss Federal Institute of Technology, Zurich
O. Schenk is with the Computer Science Department of the University of Basel, Basel, Switzerland. S. Rollin is with Integrated Systems Laboratory of the Swiss Federal Institute of Technology and Innovation (ETH), Zurich, Switzerland. Anshul Gupta is affiliated with the Mathematical Sciences Department of the IBM T. J. Watson Research Center, Yorktown Heights, USA.
Where ф is the electrostatic potential, n and p the electron and hole densities. These are the unknown variables. The other quantities are given: e is the dielectrical permittivity, C the doping concentration, R the total recombination rate and q the elementary charge. The carrier current densities jc in equations (I) – (I) are substituted with the equations
Jn = q (DnVn – ^”nVф), jp = – q (DpVp + MpPVф) .
Here, ij>c are the carrier mobilities and Dc are the carrier diffusivities. Typical challenges in solving the drift-diffusion equations include a large range ofn and p, the steep gradients of the solution variables, and the necessity to ensure that n, p > 0 in the numerical computations.
Different discretizations are used to solve the drift-diffusion equations numerically. The Scharfetter-Gummel box method and finite element discretizations are both used in modern semiconductor device simulators [2], [3].
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