Modeling and numerics for two partial differential equation systems arising from nanoscale physics

Abstract

This thesis focuses on the mathematical analysis of two partial differential equation systems. Consistent improvement of mathematical computation allows more and more questions to be addressed in the form of numerical simulations. At the same time, novel materials arising from advances in physics and material sciences are creating new problems which must be addressed. This thesis is divided into two parts based on analysis of two such materials: organic semiconductors and graphene. In part one we derive a generalized reaction-drift-diffusion model for organic photovoltaic devices -- solar cells based on organic semiconductors. After formulating an appropriate self-consistent model (based largely on generalizing partly contradictory previous models), we study the operation of the device in several specific asymptotic regimes. Furthermore, we simulate such devices using a customized 2D hybrid discontinuous Galerkin finite element scheme and compare the numerical results to our asymptotics. Next, we use specialized asymptotic regimes applicable to a broad range of device parameters to justify several assumptions used in the formulation of simplified models which have already been discussed in the literature. We then discuss the potential applicability of the simulations to real devices by discussing which parameters will be the most important for a functioning device. We then give further generic 2D numerical results and discuss the limitations of the model in this regime. Finally, we give several perspectives on proving existence and uniqueness of the model.

In part two we derive a second-order finite difference numerical scheme for simulation of the 2D Dirac equation and prove that the method converges in the electromagnetically static case. Of particular interest is the application to electrons in graphene. We demonstrate this convergence numerically with several examples for which explicit solutions are known and discuss the manner in which errors appear and propagate. We furthermore extend the Dirac system with Poisson's equation to investigate interesting electronic effects. In particular, we show that our numerical scheme can successfully simulate a beam-splitter and Veselago lens, both of which have been predicted analytically to appear in graphene.