Numerical Methods for the Wigner - Poisson Equations

LASATER, MATTHEW SCOTT. Numerical Methods for the Wigner-Poisson Equa-tions. (Under the direction of C.T. Kelley).This thesis applies modern numerical methods to solve the Wigner-Poisson equa-tions for simulating quantum mechanical electron transport in nanoscale semicon-ductor devices, in particular, a resonant tunneling diode (RTD). The goal of thisdissertation is to provide engineers with a simulation tool that will verify earlier nu-merical results as well as improve upon the computational efficiency and resolutionof older simulations. Iterative methods are applied to the linear and nonlinear prob-lems in these simulations to reduce the computational memory and time require tocalculate solutions.Initially the focus of the research involved updating time-integration techniques,but this switched to implementing continuation methods for finding steady-state so-lutions to the equations as the applied voltage drop across the device varied. Thismethod requires the solution to eigenvalue problems to produce information on theRTD’s time-dependent behavior such as the development of current oscillation at a particular applied voltage drop. The continuation algorithms/eigensolving capabili-ties were provided by Sandia National Laboratories’ software library LOCA (Libraryof Continuation Algorithms). The RTD simulator was parallelized, and a precondi-tioner was developed to speed-up the iterative linear solver. This allowed us to usefiner computational meshes to fully resolve the physics.We also theoretically analyze the steady-state solutions of the Wigner-Poissonequations by noting that the solutions to the steady-state problem are also solutionsto a fixed point problem. By analyzing the fixed point map, we are able to provesome regularity of the steady-state solutions as well provide a theoretical explanationfor the mesh-independence of the preconditioned linear solver.