Numerical Methods for the Wigner-Poisson Equations

Abstract

This thesis applies modern numerical methods to solve the Wigner-Poisson equations for simulating quantum mechanical electron transport in nanoscale semiconductor devices, in particular, a resonant tunneling diode (RTD). The goal of this dissertation is to provide engineers with a simulation tool that will verify earlier numerical results as well as improve upon the computational efficiency and resolution of older simulations. Iterative methods are applied to the linear and nonlinear problems in these simulations to reduce the computational memory and time required to calculate solutions.

Initially the focus of the research involved updating time-integration techniques, but this switched to implementing continuation methods for finding steady-state solutions to the equations as the applied voltage drop across the device varied. This method requires the solution to eigenvalue problems to produce information on the RTD's time-dependent behavior such as the development of current oscillation at a particular applied voltage drop. The continuation algorithms/eigensolving capabilities were provided by Sandia National Laboratories' software library LOCA (Library of Continuation Algorithms). The RTD simulator was parallelized, and a preconditioner was developed to speed-up the iterative linear solver. This allowed us to use finer computational meshes to fully resolve the physics.

We also theoretically analyze the steady-state solutions of the Wigner-Poisson equations by noting that the solutions to the steady-state problems are also solutions to a fixed point problem. By analyzing the fixed point map, we are able to prove some regularity of the steady-state solutions as well provide a theoretical explanation for the mesh-independence of the preconditioned linear solver.