Classical and quantum energy-transport models for semiconductors: modeling, analysis and numerical approximation
Final Report Abstract
Due to the increasing miniaturization of semiconductor devices the existing, classical semiconductor models have to be refined, since temperature and quantum effects become more and more important. This project was concerned with the mathematical understanding of thermal and/or quantum effects in semiconductor devices. These effects can be in principle modeled by the semiclassical 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. The main goals of this project were 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. In particular, we analyzed classical and quantum energy-transport models mathematically and numerically. We provided the existence and uniqueness of solutions for several approximate models, including stationary and transient ones. Here, we made major progress and were able to provide physical bounds for the temperature and electron density. Further, we studied the asymptotic behavior of the models and their stability properties. Moreover, numerical discretizations and iterative schemes for the solution of the nonlinear systems were derived and analyzed. Finally, ultra-small field-effect transistors (MOSFET, HEMT) were numerically simulated and the influence of temperature and quantum effects systematically studied.
Publications
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Global weak solutions to compressible Navier-Stokes equations for quantum fluids. SIAM J. Math. Anal. 42:1025-1045 (2010)
A. Jüngel
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Matrix compression for spherical harmonics expansions of the Boltzmann transport equation for semiconductors. J. Comput Phys. 229: 8750- 8765 (2010)
K. Rupp, A. Jüngel, and T. Grasser
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Analysis of a bipolar energy-transport model for a metaloxide-semiconductor diode. JMAA 378(2):764-774 (2011)
A. Jüngel, R. Pinnau, E. Röhrig
