Electrostatic simulations of interactions between a scanning probe and quantum Hall liquid

Abstract

In this thesis we modelled interactions between metal probe and quantum Hall liquid. The setup geometries were similar to those met in scanning capacitance microscopy experiments carried out by Ashoori's group [1]. The main interest was to explore the 2DEG charge densities for a different system geometries, magnetic field applied, and tip voltages. We modelled quantum bubble formation under the metal probe, incompressible strip formation beneath the edge of the metal gate, and 2DEG density profile under the influence of donors in magnetic field. In order to model complex geometry systems, but also optimize running times and memory allocation we developed two electrostatic simulators one for cylindrically symmetric and second for arbitrary 3D geometries. Our electrostatic solver was based on successive over relaxation algorithm, but it was optimized for better stability and faster convergence times.