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Introduction to Part II

Carrier transport in modern semiconductor devices is commonly investigated using Boltzmann's transport equation, either directly by means of the Monte Carlo method or by using simplified macroscopic transport models derived from Boltzmann's equation. To obtain a physically accurate picture which includes as many effects with as few approximations as possible, the Monte Carlo method is often the method of choice. However, on an engineering level the Monte Carlo method is inconvenient because it requires large simulation times. In particular, the extraction of some relevant device parameters is very costly, for instant transit frequencies, small-signal parameters, threshold voltages and in general situations where only small current levels are involved. Therefore simpler methods have to be derived which, for instance, focus only on the first few moments of Boltzmann's transport equation.

Since the advent of the first simulator that solved the semiconductor equations on one- or two-dimensional geometries, the drift-diffusion model which comprises the first two moment of Boltzmann's equation has been the workhorse of TCAD device engineers. As classical MOS devices continue to scale, the numerical methods and algorithms need improvement to support the growing complexity of physical phenomena to be addressed by TCAD. In some aspects, such as non-local effects due to impact ionization, band-to-band tunneling, trap-assisted tunneling or channel mobility, the basic physical models have to be refined. For this more accurate treatments of the Boltzmann transport equation than provided by drift-diffusion models are required [RO003].

Higher-order transport models have been proposed to alleviate some of the most stringent assumptions underlying the drift-diffusion equations. However, many doubts regarding the accuracy of fourth-order models like the energy-transport model have been raised. It has often been observed that the additional overhead imposed by these methods is not justifiable. For improved accuracy six moments models [GKGS01] have been proposed which add the kurtosis of the distribution function as additional solution variable.

One point subject to much research are the numerical properties of higher-order transport models. With the first implementation of the six moments model [Gri02] many convergence problems were observed. Trying to achieve a robust implementation we found that the closure relation for the sixth moment is the critical issue with respect to numerical robustness of the solution algorithm for the implemented six moments model. The search for a suitable closure relation is the central topic in this first part of the thesis. This is discussed in Chapter 3. The search for a suitable closure was accompanied by investing much work in the fine-tuning of the nonlinear solver as described in Chapter 4.

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R. Kosik: Numerical Challenges on the Road to NanoTCAD