Density-functional approaches to interacting electrons in quantum dots

Abstract

Quantum dots are man-made nanoscale structures. As they show typical atomic properties they are often referred to as artificial atoms. The wave functions, shell structure, and energy levels are usually reminiscent of real atomic systems. A wide variety of geometries is possible by choosing appropriate materials and external confinement: one-dimensional rods, two-dimensional pancakes, or three-dimensional spheres. Since quantum dots are nanoscale systems, quantum mechanics is required for their accurate description. However, the electronic structure of these systems is very hard or even impossible to solve exactly even in the case of a few electrons, and approximations must be used.

This thesis concentrates on electronic structure calculations of two-dimensional quantum dot systems, using the density-functional approach. The spin-density-functional theory (SDFT) and the current-spin-density-functional theory (CSDFT) are applied to study the ground-state properties of quantum dot systems in zero and finite magnetic fields. Especially the effects of complex electron-electron interactions are studied.

Emphasis has also been set on developing and testing various methods and approximations. This is done by comparing the ground-state energy and other observables to those obtained using the variational quantum Monte Carlo method. The Kohn-Sham equations of the density-functional theories are solved in real-space by using the Rayleigh quotient multigrid method. This approach is compared to the traditional plane-wave solving methods.

The systems under consideration in this work include single quantum dots with different confining potentials, double quantum dot 'hydrogen' molecule, and a superconductor-normal quantum dot-superconductor (SNS) structure. Symmetry-breaking solutions emerge in these calculations. These include spin-density-wave-like solutions, charge-density-wave-like solutions, Wigner molecule formation, and solutions with vortex structures. The structure and properties of these solutions have been calculated and the interpretation of the broken symmetry is discussed.