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

TL;DR

This paper presents a numerical Green's function model for analyzing the local density of states and transport properties in one-dimensional quantum systems, providing accurate insights into quantum behavior under various conditions.

Contribution

It introduces an adaptable iterative numerical Green's function approach for 1DEG systems, capable of modeling complex geometries and magnetic fields, with validation against analytical results.

Findings

Model accurately reproduces analytical local DOS results.

Successfully simulates transmission and reflection coefficients.

Applicable to studying antidot behavior and tunneling phenomena.

Abstract

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 Greens 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 Greens 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. A second numerical model was also developed that measured the transmission and reflection coefficients through the quantum system based on the Landauer-Buttiker 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.