New integral equation approach for the numerical analysis of MOS devices in VLSI

Date of Award


Degree Type


Degree Name

Doctor of Philosophy (PhD)


Electrical Engineering and Computer Science


Tapan K. Sarkar


method of moments, Poisson equation

Subject Categories

Electrical and Computer Engineering


Metal Oxide Semiconductor (MOS) transistors continue to be the predominant building block in Very Large Scale Integrated (VLSI) circuits. Today VLSI circuits with over 3 million transistors in a chip with effective channel widths of 0.5 $\mu m$ are in production. With such small size, however, device characteristics change and no longer can be predicted by conventional analytic theories. As a consequence full numerical simulation of such a small device with complex structure becomes indispensable for accurate modeling of its electrical characteristics. Most conventional numerical simulations are based on finite difference or finite element methods.

In this thesis a simplified approach based on the Method of Moments (MoM) is developed as a suitable new numerical tool for the characterization of a small MOS device with arbitrary geometrical shape. The new method combines the advantage of finite element methods, i.e. flexibility in handling nonrectangular structures, with the ease and simplicity of the finite difference formulation, i.e. using rectangular grids. The new method is novel in the sense that it can be applied to highly irregular domains, yet only rectangular grids are used to discretize the nonlinear Poisson equation. The method is easy to implement and relieves the user from the effort involved in generating grid points which need to be talented according the varying structure and the doping profile of the MOS devices.

The one-dimensional threshold voltage problem for a nonuniformly doped channel is discussed and attacked by transforming Poisson's equation into an integral equation. This equation is then solved using composite Gaussian quadratures. It was found that the accuracy of our solution is independent of the steepness of the substrate implantation profile.

By coupling the double Fourier series with MoM, a general method is obtained for solving the two-dimensional linear Poisson equation in a highly arbitrary domain without needing any domain discretization.

The general nonlinear Poisson equation is converted to a sequence of Helmholtz equations. Then a new technique to solve the general Helmholtz equation is presented. The technique is based on the MoM Laplacian solution without resorting to a different formulation using Hankel functions. The method is used to model the potential function in a MOS transistor. To illustrate the validity of the present method, various other practical engineering problems are solved and the solutions are compared with results reported in the literature or obtained by other means using ELLPACK.


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