1.2 Extensions of Drift Diffusion Models

In industrial applications of TCAD there is a strong need for fast and efficient models, preferably compact models. In conventional device simulation efficient models are derived by considering the first few moments of the Boltzmann equation. These models eliminate many of the assumptions required for the applicability of compact models.

However, in modern devices large electric fields which rapidly change over small length scales give rise to non-local and hot-carrier effects which begin to dominate device performance. Accurate description of non-local effects is of utmost importance for these modern semiconductor devices. In particular, the distribution function has to be modeled properly as it is inherently linked with hot-carrier effects. In the framework of hydrodynamic and energy-transport models only the average energy is known which does not provide enough information about the shape of the distribution function.

At the high end of device simulation, there are the Monte Carlo codes which solve the full Boltzmann equation. They play a dominant role in computational device physics. Within TCAD they are important as a tool for calibration and development of simplified models. These codes achieve the highest accuracy and implement physically advanced models at the price of very long simulation times.

Higher order moment method are designed to keep the balance between efficiency and required accuracy of the simulation. To overcome the limitations of the available energy-transport models a consistent transport model based on six moments of Boltzmann's equation has been developed in [GKGS01]. In addition to the concentration and the carrier temperature, as provided by the energy-transport models, we obtain the average of the square energy which we map to a new solution variable, representing the kurtosis of the distribution function.

The aim of our research was to increase the numerical robustness of the solution algorithm for the six moments model. As higher order moment systems are complex nonlinear systems, robust solution requires a lot of fine tuning and experimenting with both the discretization and the nonlinear solver.