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Wave function methods for quantum dots in magnetic field

Sami Siljamaki

Published Sep 26, 2003• 2 citations

Abstract

This thesis investigates the use of wave-function methods for the study of quantum-dot systems. It investigates single dots, using quantum Monte Carlo techniques in a wide range of magnetic field values, and a double-dot system, using the exact-diagonalization method.\n\nThe thesis proposes simple yet accurate many-particle wave functions for various angular-momentum and spin states, for both weak and strong magnetic fields. Using these trial wave functions, it evaluates various properties of dots and studies Wigner crystallization and spin polarization for the weak-field limit. For strong magnetic fields, the thesis investigates ground states of different spin polarizations as a function of the magnetic field and tests the commonly used lowest-Landau-level approximation. The results are compared to calculations from the density-functional theory.\n\nFinally, the thesis presents a method that combines the accuracy of the exact-diagonalization method and the scalability of Monte Carlo methods. The Monte Carlo-based diagonalization is a promising tool for use in situations that cannot be handled with a simple trial wave function, and have too many particles for an exact-diagonalization treatment. For quantum dots, methods for efficiently evaluating the high-magnetic-field basis functions and their gradients are demonstrated.

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This thesis investigates the use of wave-function methods for the study of quantum-dot systems. It investigates single dots, using quantum Monte Carlo techniques in a wide range of magnetic field values, and a double-dot system, using the exact-diagonalization method.\n\nThe thesis proposes simple yet accurate many-particle wave functions for various angular-momentum and spin states, for both weak and strong magnetic fields. Using these trial wave functions, it evaluates various properties of dots and studies Wigner crystallization and spin polarization for the weak-field limit. For strong magnetic fields, the thesis investigates ground states of different spin polarizations as a function of the magnetic field and tests the commonly used lowest-Landau-level approximation. The results are compared to calculations from the density-functional theory.\n\nFinally, the thesis presents a method that combines the accuracy of the exact-diagonalization method and the scalability of Monte Carlo methods. The Monte Carlo-based diagonalization is a promising tool for use in situations that cannot be handled with a simple trial wave function, and have too many particles for an exact-diagonalization treatment. For quantum dots, methods for efficiently evaluating the high-magnetic-field basis functions and their gradients are demonstrated.

An efficient variational many-body wave function for electrons in a quantum dot is introduced. We show that the proposed bosonic two-body correlation factor is able to capture nearly exactly the missing correlations of the lowest-Landau-level many-body wave function. We compare the results with exact diagonalizations. The comparison shows that quantum Monte Carlo techniques combined with the proposed wave function are a powerful tool for studying quantum dots. As further evidence for this, we apply the scheme to a maximum density droplet state for up to 40 electrons.

The ground states of parabolically confined electrons in a quantum dot are studied by both direct numerical diagonalization and quantum Monte Carlo (QMC) methods. We present a simple but accurate variational many-body wave function for the dot in the limit of a weak magnetic field. The wave function has the center-of-mass motion restricted to the lowest-energy state and the electron-electron interaction is taken into account by a Jastrow two-body correlation factor. The optimized wave function has an accuracy very close to the state-of-the-art numerical diagonalization calculations. The results and the computational efficiency indicate that the presented wave function combined with the QMC method suits ideally for studies of large quantum dots.

We use a variational Monte Carlo algorithm to solve the electronic structure of two-dimensional semiconductor quantum dots in external magnetic field. We present accurate many-body wave functions for the system in various magnetic field regimes. We show the importance of symmetry, and demonstrate how it can be used to simplify the variational wave functions. We present in detail the algorithm for efficient wave function optimization. We also present a Monte Carlo-based diagonalization technique to solve the quantum dot problem in the strong magnetic field limit where the system is of a multiconfiguration nature. PACS numbers: 71.10.-w, 73.21.La, 02.70.Ss 1.

A system of N=2--4 interacting electrons in a quantum dot, with applied magnetic field, is treated using a quantum Monte Carlo method. The standard fixed-node Monte Carlo approach is modified to facilitate the inclusion of a magnetic field giving rise to a fixed-phase approximation which is introduced here. By making use of this approximation, results whose accuracy is comparable to that of exact diagonalization can be obtained. A simple set of coordinates for the four-particle problem is also used which facilitates finding a symmetrized basis. \textcopyright{} 1996 The American Physical Society.

The exact diagonalization technique is used to study many-particle properties of interacting electrons with spin, confined in a two-dimensional harmonic potential. The single-particle basis is limited to the lowest Landau level. The results are analysed as a function of the total angular momentum of the system. Only at angular momenta corresponding to the filling factors 1, 1/3, 1/5, etc is the system fully polarized. The lowest energy states exhibit spin waves, domains, and localization, depending on the angular momentum. Vortices exist only at excited polarized states. The high angular momentum limit shows localization of electrons and separation of the charge and spin excitations.

Using a variational quantum Monte Carlo method, the effect of Landau-level mixing on the lowest-energy--state diagram of small quantum dots is studied in the magnetic-field range where the density of magnetic flux quanta just exceeds the density of electrons. An accurate analytical many-body wave function is constructed for various angular momentum and spin states in the lowest Landau level, and Landau-level mixing is then introduced using a Jastrow factor. The effect of higher Landau levels is shown to be significant; the transition lines are shifted considerably towards higher values of magnetic field and certain lowest-energy states vanish altogether.