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Nanotechnology Perceptions

Few-electron systems in a quantum dot in a magnetic field: Wigner phase and brokensymmetry spin-singlet state

Artak A. Avetisyan

Published Jul 30, 2008• 0 citations
Publisher

Abstract

The nature of the ground state of a few-electron system in a parabolic twodimensional (2D) quantum dot (QD) is theoretically investigated on the basis of a variational many-body approach within a second quantized formulation. The resulting analytical expressions lead to various phase transitions: maximum density droplet, Wigner phase, etc., in a broad range of magnetic fields. It is observed that in the case of relatively strong QD confinement and weak magnetic fields, a rotationally symmetric spin-singlet state is the ground state of the system. However, in a strong magnetic field and for the same QD confinement, a broken-symmetry spinsinglet state appears to be energetically favoured over the symmetric spin-singlet state. The method developed is generalized for the investigation of possible novel phases for electron-hole interacting systems.

DOI
10.4024/n31av07.ntp.04.02

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The nature of the ground state of a few-electron system in a parabolic twodimensional (2D) quantum dot (QD) is theoretically investigated on the basis of a variational many-body approach within a second quantized formulation. The resulting analytical expressions lead to various phase transitions: maximum density droplet, Wigner phase, etc., in a broad range of magnetic fields. It is observed that in the case of relatively strong QD confinement and weak magnetic fields, a rotationally symmetric spin-singlet state is the ground state of the system. However, in a strong magnetic field and for the same QD confinement, a broken-symmetry spinsinglet state appears to be energetically favoured over the symmetric spin-singlet state. The method developed is generalized for the investigation of possible novel phases for electron-hole interacting systems.
Exchange energies and charge densities for few electrons in electrically tunable quantum dot (QD) systems are investigated by using the variational Monte Carlo (VMC) method in the presence of magnetic fields. For N = 2 electrons in double QD, the singlet Coulomb energy decreases in magnetic fields as a consequence of magnetic localization in both strongly and weakly coupled QDs. However, the triplet Coulomb energy reaches a maximum value at intermediate magnetic fields before decreasing. We observe good agreement between the results for VMC with s-p orbital trial wavefunction and exact diagonalization (ED) method in the exchange energies. For N = 2 electrons in triple quantum dot (TQD), we observe a discontinuity in the J-derivative with detuning voltage ( dJdVT ) in triangular TQDs at B = 0 T as crossing of the eigenenergy levels leads to abrupt spatial symmetry change of the triplet densities (density rotation) and relocalization. For B ≠ 0 T, the angular momentum provided to the electrons quenches this effect. The density rotation is absent in collinear TQD for all magnetic fields as the lowest excited state remains the px state and as such, no change in symmetry is possible. By varying the triangular TQD configuration, we show the discontinuity in ( dJdVT ) persists for top angle between 20° and 70°. For N = 3 electrons in symmetric triangular TQD, the monotonicity of the exchange curve from B = 0 to 4 T remains intact for decoupled QDs but not for coupled QDs. Finally, additional energy for the triangular TQD system is computed for up to N = 3 electrons.
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.
Three low lying energy levels of a 3D two-electron quantum dot (QD) with parabolic confinement are obtained by the variational method. The proposed interpolation formulas for the variation parameters allow one to recover the energy levels in a wide range of the Coulomb interaction constant. The quantum states of the QD are divided into the para- and ortho-states like in the theory of helium atom. The quantum transitions from the ortho-state to the para-state are possible only with account of the spin-orbit interaction. At low temperatures, an ensemble of two-electron QDs contains dots in the ground para-state and in the first excitedortho-state, which is metastable. These QDs have the entangled spin wave functions that are related to the Einstein–Podolsky–Rosen (EPR) states desirable for the quantum information protocol.
Moiré superlattices can bind several electrons to each potential minimum, forming few-electron "moiré atoms." Using neural quantum states that obey magnetic periodic boundary conditions, include full Landau-level mixing, and treat both spin sectors, we map the ground state of two-electron moiré atoms in a superlattice as a function of moiré depth and magnetic field strength. At strong field each atom is a spin-triplet maximum-density droplet with canonical angular momentum ⟨Lz⟩=−1\langle L_z\rangle=-1⟨Lz​⟩=−1 and a persistent circulating current: the rotating counterpart of the "Wigner molecular crystal" recently imaged in moiré superlattices. Deepening the wells drives a singlet-triplet transition to non-rotating spin-singlet atoms whose only current is a weak diamagnetic counter-circulation. The transition moves to deeper wells as the field grows, so depth and field together open a window for the formation of rotors. This is the collective analog of the singlet-triplet transition of two-electron quantum dots. At the transition, the orbital magnetic moments of the atoms reverse sign as their spin polarization switches, a joint signature accessible to magnetometry. If the atoms are instead polarized by other means, such as magnetic proximity coupling, the rotors lock at weak field into a spontaneously ordered crystal of dimers in a pinwheel pattern.
We consider an exactly solvable two-dimensional model of a two-electron parabolic quantum dot (QD) under the perpendicular magnetic field. We perform a detailed study of the degree of the orbital entanglement of the ground and first excited singlet state depending on the shape of QD, the strength of the effective interaction and the magnetic field. It was found that a degree of entanglement increases with the growth of interaction between electrons. The magnetic field destroys the entanglement of electrons. However, the entanglement in deformed QDs is more stable to the effect of the magnetic field.