Mathematical Modelling of Nano-Electronic Systems

Abstract

Double-Qdots (DQDs) are attractive in light of their potential application to quantum computing and other electronic applications, e.g. as specialized sensors. We consider the electronic properties of a system consisting of two quantum dots in physical proximity, which we will refer to as the DQD. Our main goal is to derive the essential properties of the DQD from a model that is rigorous yet numerically tractable, and largely circumvents the complexities of an ab initio simulation. To this end we propose a class of novel Hamiltonians that captures the dynamics of a bi-partite quantum system, wherein the interaction is described via a convolution or a Wiener-Hopf type operator. We subsequently describe the density of states function and derive the electronic properties of the underlying system. Our analysis shows that the model captures a plethora of electronic profiles which serves as evidence for the versatility of the proposed framework for DQD channel modelling. A massive body of mathematical physics results, dating mostly to the last half a century, give evidence to the claim that the statistical characteristic of fluctuations in the level structure of a quantum system provides essential information about its dynamic properties, e.g. in some instances these statistical parameters show whether or not the underlying classical dynamics is integrable or chaotic. Following this tradition we have conducted statistical analysis of the data generated numerically from the model at hand. In this way we have characterized the fine-scale fluctuations of the spectra for several choices of the constituents. In conclusion, we have found that the model is versatile enough to produce several statistically distinct types of level structure. In particular, the model is capable of reproducing very complex level structures, such as those of the resonant microwave cavities that have been obtained experimentally in the 1990s