Few-electron semiconductor quantum dots in magnetic field: Theory and methods

Semiconductor quantum dots represent nanoscale systems with few electrons confined in a semiconductor host crystal. The importance of semiconductor quantum dots lies primarily in their tunability and sensitivity to external parameters as electrons are confined in all dimensions. The bulk of semiconductor quantum dots are fabricated by applying a lateral confinement potential to a two-dimensional electron gas. Quantum confinement profoundly affects the way electrons interact with each other and with external parameters, such as a magnetic field. Quantum confinement of electrons is just one of several ways quantum mechanics reveals itself. Another pure quantum phenomena associated with electrons is their spin. An external magnetic field affects both orbital and spin motion of electrons. External control of the full quantum wave function in a semiconductor quantum dot may lead to novel technological application involving both charge and spin. From a theoretical point of view, semiconductor quantum dots represent a unique opportunity to study fundamental quantum theories in a tunable atomic like set-up. In this work, we review some of the theoretical approaches used to study two-dimensional few-electron semiconductor quantum dots. The main emphasis is to clarify the relations between different theories and methods for few-electron semiconductor quantum dots in an external parameter, a perpendicular magnetic field. Properties of few-electron semiconductor quantum dots in the weak magnetic regime are explained well through single-electron theory concepts. However, challenges do exist when considering stronger external magnetic fields. A strong magnetic field, when applied perpendicular to the quantum dot, changes the quantum nature of the electronic correlations and spin-polarizes the electrons. As the strength of the external magnetic field increases, the confined electrons start to manifest collective quantum behavior as seen in the integer and fractional quantum Hall effect regime. Theoretical and computational challenges to studies of semiconductor quantum dots as the magnetic field changes from weak to strong are reviewed. Specific examples are introduced to illustrate the transformation of the quantum wave function into a Laughlin-like one as the magnetic field increases. © 2008 by Nova Science Publishers, Inc. All rights reserved