Many-Body Approaches to Quantum Dots

Abstract

In this thesis, we studied numerically systems consisting of several interacting electrons in two-dimensions, confi ned to small regions between layers of semiconductors. These arti cially fabricated electron systems are dubbed quantum dots in the literature. Quantum dots provide a new challenge to theoretical calculations of their properties using many-body methods. The size of these arti cial atoms is several orders of magnitude larger than that of atoms, leading to a much greater sensitivity to magnetic fields. The full many-body problem of quantum dots is truly complex and simulating a quantum dot constrained by a magnetic field may be even more complicated. Of particular interest is the reliability of the Hartree-Fock (HF) method for studies of quantum dots in two-dimensions as a function of the external magnetic field. In order to achieve this goal, we developed a Hartree-Fock code for electrons trapped in a single harmonic oscillator potential in two-dimensions. We also developed a code implementing many-body perturbation theory (MBPT) up to third order either directly applied to the harmonic oscillator basis or as a correction to the Hartree-Fock energy. A discussion of the results compared with large-scale diagonalisation methods indicated a quadractic error growth of HF and MBPT as the interaction strength increases. We tested also the reliability of a single Slater determinant approximation for the ground state of closed shell systems as a function of varying interaction strength. We found that the Hartree-Fock method, compared with large-scale diagonalization methods, has a limited range of applicability as function of the interaction strength and increasing number of eletrons in the dot, indicating a break of the computational technique before entering the limit of validity of the closed-shell model. Our study also showed that the HF approximation might become less accurate compared to MBPT as the number of electrons in the dot increases.