Thesis presented June 28, 2024

Abstract:
With the progress of mesoscopic physics we have gained precise control over the environment where electrons propagate. Through careful device design we are able to control the electrostatic environment with a precision within the electrons fermi wavelength and draw actual highways allowing for coherent transport over long distances, going as far as hundreds of µm. This allows for experimental physicists to use nanoelectronic devices to probe complex quantum states, such as fractional quantum hall quasiparticles, or to manipulate single electron states. The theoretical modeling of such devices however is not nearly as advanced, there is no single model capable of quantitatively correlating the volts applied at the electrostatic gates to the quantum transport observed experimentally. Most approaches model the quantum particle behavior independently from their electrostatic environment. They account for the field effect of the gates and charges in the device as an effective potential added to the hamiltonian of the system in the form of a fitting parameter. The actual form of the fitting potential is usually derived from analytical calculations or complex semiconductor material models. A single model capturing both the gate field effect and the quantum particle behavior has to correlate the eV physics at the gates to the meV quantum physics. This is not an easy task and requires solving the Schrödinger equation self-consistently with the Poisson equation (Schrödinger-Poisson problem). In this thesis we propose an algorithm and a software (PESCADO) to solve the Schrödinger-Poisson problem. The Schrödinger-Poisson problem is highly non-linear and most approaches to solve it are unstable at low temperatures and near the device regions where the charge density is depleted. We have managed to stabilized our algorithm by developing a method to first find where the non-linearities lie in the energy space and then isolate them s.t. they can be dealt with appropriately. During this thesis we have also developed a model capable of predicting the experimental gate (pinch-off) voltages required to deplete the two-dimensional electron gas beneath quantum point contact devices. We have applied it to predict the pinch-off voltages of 110 experimental quantum point contacts of 48 different designs and to study the effect of disorder in scanning gate microscopy conductance maps.

Keywords:
Quantum Device, Predictive Modeling, Schrödinger-Poisson