Thesis presented June 26, 2019

Abstract:
Electrostatic energy is very often the largest energy scale in quantum nanoelectronic systems. Yet, in theoretical work or numerical simulations, the electrostatic landscape is equally often taken for granted as an external potential, which may result in a wrong physical picture. Developing numerical tools that can properly handle the electrostatics and its interplay with quantum mechanics is of utter importance for the understanding of quantum devices in e.g. semi-conducting or graphene like materials.
This thesis is devoted to the self-consistent quantum-electrostatic problem. This problem (also known as Poisson-Schrödinger) is notoriously difficult in situations where the density of states varies rapidly with energy. At low temperatures, these fluctuations make the problem highly non-linear which renders iterative schemes deeply unstable. In this thesis, we present a stable algorithm that provides a solution to this problem with controlled accuracy. The technique is intrinsically convergent including in highly non-linear regimes. Thus, it provides a viable route for the predictive modeling of the transport properties of quantum nanoelectronics devices.
We illustrate our approach with a calculation of the differential conductance of a quantum point contact geometry.
We also revisit the problem of the compressible and incompressible stripes in the integer quantum Hall regime. Our calculations reveal the existence of a new ”hybrid” phase at intermediate magnetic field that separate the low field phase from the high field stripes.
In a second part we construct a theory that describes the propagation of the collective excitations (plasmons) that can be excited in two-dimensional electron gases. Our theory, which reduces to Luttinger liquid in one dimension can be directly connected to the microscopic quantum-electrostatic problem enabling us to make predictions free of any free parameters. We discuss recent experiments made in Grenoble that aim at demonstrating electronic flying quantum bits. We find that our theory agrees quantitatively with the experimental data.

Keywords:
Transport, Electrostatics, Kwant

On-line thesis.