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1.4.1 Higher Order Moment Methods

MINIMOS-NT is a device simulator developed at the Institute for Microelectronics in TU-Vienna in a tradition going back more than 20 years. It is a complex software package consisting of over $ 500.000$ lines of code being the result of over 100 man-years. MINIMOS-NT is made available free of charge to the public. To keep pace with the needs of the latest technology nodes, the simulator was enhanced by the addition of higher order transport models.

The six moments model [GKGS01] is a consistent extension of standard energy-transport models and allows for a detailed study of hot-carrier effects. MINIMOS-NT aims to be applicable to real-world examples from semiconductor industry. However, the implementation of the six moments method [Gri02] did not work reliably as measured by industrial TCAD software quality standards.

It was not clear at the beginning of this work what caused this failure. The job assigned to this author can be summarized in one slogan: ``Robustness''. This is an important issue which marks the difference between a software package for industrial application and a computer program mainly used as an aide in producing figures for academical research papers.

In the course of numerical troubleshooting we expected to make lots of changes to the code. To avoid the overhead of implementing this in MINIMOS-NT it was decided to reimplement the method for the one dimensional case stand-alone in Matlab, which is a very good development platform for prototyping. By comparing the two implementations many problems could be eliminated, resulting in a stable implementation.

Our initial working hypothesis was that some improvements to the nonlinear solver were needed. But during the course of the project it was found that there did not exist a single showstopper but a variety of numerical issues which needed to be addressed.

Chapter 2 starts with introducing the Boltzmann-Poisson system and the one-dimensional model which we used for developing the stand-alone simulator. The solution of the Boltzmann equation by the method of moments is discussed. The hierarchy of moments equations is derived and the ``closure problem'' is defined. We study the closure in the scattering operator using mobilities and relaxation times.

The closure problem for the highest order moment is the topic of Chapter 3. We review three methods from the literature: First, the use of cumulants instead of moments for the description of the distribution function [WSYM98] leads to a generalized Gaussian closure. Second, the maximum entropy principle in the diffusion approximation was applied to solve the closure problem. Finally, the closure relations proposed in [GKGS01] are considered, where the sixth moment is modeled as a function of the variance and the kurtosis of the distribution function using a real number $ c$ as parameter. In the last case the closure problem is reduced to the choice of $ c$. This indeterminacy can be eliminated by requiring consistency with bulk Monte Carlo data.

Chapter 4 describes the tuning of the nonlinear solver. The success and practical applicability of numerical simulation depends critically on the convergence rate as well as the control of any numerical instability arising from discretization error. As a first step in the investigation a variety of generalized Scharfetter Gummel schemes were implemented. We discuss the discretization schemes we tried for the six moments model.

Compared with the simple drift-diffusion model higher order methods give stronger coupling between the device equations as the physical models for mobility are functions of the local carrier temperature. There are several critical points which influence the convergence behavior: A basic one is the choice of variables. This is already an issue for the drift-diffusion systems where different sets of variables can be used. To find the optimal set of variables convergence behavior is compared experimentally.

Correctness of the implementation has been checked by comparing simulation results with exact analytical results known for the bulk case. Furthermore simulation results are compared with Monte Carlo data, which finishes this part of the thesis.

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R. Kosik: Numerical Challenges on the Road to NanoTCAD