Quantum Transport Equations: Kinetic, Relativistic, and Diffusive Phenomena

This project is concerned with the mathematical and numerical analysis of quantum transport models (partial differential equations and integro-differential equations) that appear in solid state physics and semiconductor device simulations. It focuses on three closely related topics: Quantum kinetic models (analysis of stationary Wigner equations and development of efficient discretizations); Quantum relativistic models (derivation, analysis, and numerical implementation of absorbing boundary conditions for the Dirac equation); Quantum fluid models (analysis of quantum Euler and quantum drift-diffusion models). The first part of this proposal is concerned with the stationary inflow boundary value problem for the Wigner equation, a model used for simulating the ballistic quantum transport through resonant tunneling structures. We will analyze the well-posedness of this boundary value problem and develop an efficient numerical algorithm for its solution. Furthermore, we will analyze steady states of the Wigner-Fokker-Planck equation with non-quadratic confinement potentials. Using the entropy method of parabolic equations we will investigate the large-time behavior of transient solutions, and expect to prove exponential convergence to the steady state under appropriate conditions on the prescribed potential. In the second part we propose a mathematical formulation and subsequent numerical implementation of open (or absorbing) boundary conditions for the time-dependent Dirac equation in one and two spatial dimensions. The goal is to perform numerical simulations of the interaction of relativistic particles with matter. Such boundary conditions will be derived both for a Crank-Nicholson finite difference scheme and a time-splitting spectral method. Numerical tests will include scattering of wave packets at potential steps and potential discs. The third part is concerned with the analysis of quantum fluid models, which can be derived from quantum kinetic equations and which seem to provide a reasonable compromise between physical accuracy and numerical feasibility. Motivated by recent results on macroscopic approximations of Wigner-Fokker-Planck and Wigner-BGK equations, new isothermal Euler equations including quantum and viscous corrections will be examined. In particular, novel entropy-type estimates for general viscous third-order Euler equations will be derived and we aim at new insights into the entropic structure of quantum drift-diffusion equations. Furthermore, we will analyze internal and boundary layers in these models. The project has a duration of 3 years will be carried out in close cooperation between the Austrian and Chinese teams. The Austrian Team requests funding for 3 Ph.D. students (at 75%), travel costs to China, and partial funding for a Project-Workshop to be held at TU Vienna in the middle of the project duration.

This project was concerned with mathematical and physical aspects of quantum transport models that are crucial for the numerical simulation in solid state physics and in semiconductor devices simulations. Such models are important in laser physics, for Bose-Einstein condensates (i.e. a coherent state of matter of a dilute bason gas at ultra-low temperatures), and for the development of novel, nano-scale semiconductor devices (like diodes and transistors). For many such models we proved their mathematical correctness, their stability and large time behavior (like return to equilibrium for large times). Concerning numerical aspects we developed novel discretization schemes (e.g. for the quantum-relativistic Dirac equation) that preserves qualitative features of the original, continuous model (like correct propagation direction of wave packets). Moreover we developed new transparent boundary conditions for the Dirac and Schrödinger equations. These are mathematical tricks that allow to reduce the computational domains (and hence the simulation effort) - but without changing the model.