Mathematical Sciences: Mathematical Analysis of Kinetic Quantum Transport Models for Semiconductors

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DMS-9500852 Arnold The object of the proposed research is the mathematical and numerical analysis of quantum kinetic equations in solid state physics. We will first investigate relaxation-time Wigner-Poisson models (quantum Boltzmann equation). The existence and uniqueness analysis will be based on the reformulation of this nonlinear evolution equation in terms of the density matrix operator. The analysis of the large time behavior and the steady states in thermodynamic equilibrium will be based on compactness methods and a quantum entropy functional. Absorbing boundary conditions for the 1D Wigner equation will be extended to 2D situations, the coupled Wigner-Poisson and relaxation-time models. The well- posedness analysis will be based on considering the Wigner equation as a limit of finite dimensional hyperbolic systems. The models of quantum mechanical transport equations have become the most important basis for accurate simulations of novel, ultra-integrated semiconductor devices (quantum devices, electron wave guides). A sound mathematical understanding of the evolution equation will form the basis for numerical implementations. The numerical simulation of hypersonic gas flows around rigid objects (e.g., space shuttles in the high altitude phase of their re-entry into the atmosphere) is usually accomplished by numerically coupling two different transport models . A refined coupling strategy is proposed here, which will be equally relevant for gas dynamics applications, and for efficient simulations of ultra-integrated semiconductor devices. We will numerically compare it to existing coupling strategies and investigate its stability.