Energy-transport Equations for Semiconductors

Due to the increasing miniaturization of semiconductor devices, temperature and quantum effects become more and more important. Usually, these effects are included in industrial codes for highly integrated semiconductor circuits by compact models or correction terms. However, it is more and more difficult to model the parasitic effects in modern devices by simple corrections, and it seems necessary to include more physical informations into the models including, for instance, the temperature as an independent variable. In this project, we will model, analyze and numerically solve the classical and quantum energy-transport equations. They consist of the conservation laws for the electron mass and thermal energy, together with constitutive relations for the particle and energy fluxes and coupled with the Poisson equation for the electric potential. In this model, the temperature (or, equivalently, the thermal energy) is an independent variable. The classical energy model is mathematically a cross-diffusion system of parabolic, possibly degenerate equations of second order. The quantum energy-transport model contains additional quantum terms derived from the Wigner equation. It is expected to be of parabolic type but with fourth-order differential terms. The advantage of these models is that their parabolic structure allows for an efficient numerical solution, in particular compared with other temperature models, like hydrodynamic or kinetic equations. The main goals of this project are to understand the influence of the electron temperature on the particle current, the heating of the semiconductor crystal, and the interplay of the temperature with quantum effects. More precisely, we plan to analyze the classical and quantum energy-transport models mathematically, perform an asymptotic expansion around the state of constant temperature, and solve the model equations with appropriate boundary and initial conditions in up to three space dimensions using mixed finite-element techniques. Furthermore, ultrasmall field-effect transistors (MOSFET, HEMT) are numerically simulated. We expect that our results will help in the understanding of highly nonlinear cross-diffusion systems and of heat and quantum transport in semiconductor devices. This project is in collaboration with Prof. Pinnau (Kaiserslautern, Germany) who has applied in parallel for a research grant at the German Science Foundation (DFG). Some subprojects will be mainly executed in Vienna, some of them mainly in Kaiserslautern.

The project was concerned with the mathematical understanding of thermal and/or quantum effects in semiconductor devices. These effects can be in principle modelled by the semi- classical Boltzmann transport equation or the quantum-mechanical Wigner equation. However, the numerical solution of these equations is far too difficult, and approximate models need to be devised. In this project, certain approximations have been analyzed and numerically solved. Two main results have been obtained. First, a very efficient implementation of the so-called spherical harmonics expansion of the Boltzmann equation has been suggested by exploiting the special structure of the discretization and by developing a scheme which is adaptive in the approximation order. We have been able to simulate state-of-the-art three-dimensional nano-scale device geometries. Our scheme has the potential to improve significantly existing commercial semiconductor software packages. Second, we made a major progress in the understanding of certain approximate quantum models, in particular quantum fluid models. We developed novel mathematical tools to deal with the highly nonlinear mathematical models, and we revealed the structure of quantum Navier-Stokes systems. Unexpected connections between aspects of classical fluid theory and quantum diffusive equations have been discovered. These connections may help to understand better certain theoretical aspects of quantum mechanics.