Numerical Green’s Function Modeling of One-Dimensional Quantum Transport

Since the initial development of one-dimensional electron gases (1DEG) two decades ago, there has been intense interest in both the fundamental physics and the potential applications—including quantum computation—of these quantum transport systems. While experimental measurements of 1DEGs reveal the conductance through a system, they do not probe critical other aspects of the underlying physics, including energy eigenstate distribution, magnetic field effects, and band structure. These are better accessed by theoretical modeling, especially modeling of the energy and wavefunction distribution across a system: the local density of states (DOS). In this thesis, a numerical Green’s function model of the local DOS in a 1DEG has been developed and implemented. The model uses an iterative method in a discrete lattice to calculate Green’s functions by vertical slice across a 1DEG. The numerical model is adaptable to arbitrary surface gate geometry and arbitrary finite magnetic field conditions. When compared with exact analytical results for the local DOS, waveband structure, and real band structure, the model returned very accurate results. In zero magnetic field, the local DOS plots from the model behaved as anticipated by theory; under a finite magnetic field, depopulation and waveband separation were present in the model, also, precisely as was expected. The model was also used to investigate imaginary band structure and gave interesting results warranting further investigation. A second numerical model was also developed that measured the transmission and reflection coefficients through the quantum system based on the Landauer-Büttiker formalism. The combination of the local DOS model with the transmission coefficients model was applied to two current research topics: antidot behavior and zero-dimensional to one-dimensional tunneling. These models can be further applied to investigate a wide range of quantum transport phenomena.