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arXiv:cond-mat/0210081 · Physical review. B, Condensed matter

Incompressible states in double quantum dots

N. Barberán, Joan Soto

Published Jul 11, 2003• 4 citations
arXivHTMLPDF

Abstract

Incompressible (magic) states that result from many-body effects in vertically coupled quantum dots submitted to strong magnetic fields such that only the lowest Landau level is relevant are studied within an exact diagonalization calculation for $N=3,$ 5, and 6, electrons. We find that the sequences of total angular momentum M for which these incompressible states exist depend on the interplay between the interdot hopping parameter ${\ensuremath{\Delta}}_{t}$ and the interdot distance d. For d of the order of the magnetic length and for all values of ${\ensuremath{\Delta}}_{t},$ we conclude that, in contrast to previous claims, these incompressible states appear at magic values of M which do not differ from those obtained for a single dot: namely, $M=N(N\ensuremath{-}1)/2+jN,$ where j is a positive integer number. For large interdot distance and simultaneously small interdot hopping parameter, new sequences of magic values of M are observed. These new sequences can be easily understood in terms of a transition regime towards a system of two decoupled single dots. However, important differences in the nature of the incompressible ground states are found with respect to those of a single dot.

DOI
10.1103/physrevb.68.045306

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Incompressible (magic) states that result from many-body effects in vertically coupled quantum dots submitted to strong magnetic fields such that only the lowest Landau level is relevant are studied within an exact diagonalization calculation for N=3,N=3,N=3, 5, and 6, electrons. We find that the sequences of total angular momentum M for which these incompressible states exist depend on the interplay between the interdot hopping parameter \ensuremathΔt{\ensuremath{\Delta}}_{t}\ensuremathΔt​ and the interdot distance d. For d of the order of the magnetic length and for all values of \ensuremathΔt,{\ensuremath{\Delta}}_{t},\ensuremathΔt​, we conclude that, in contrast to previous claims, these incompressible states appear at magic values of M which do not differ from those obtained for a single dot: namely, M=N(N\ensuremath−1)/2+jN,M=N(N\ensuremath{-}1)/2+jN,M=N(N\ensuremath−1)/2+jN, where j is a positive integer number. For large interdot distance and simultaneously small interdot hopping parameter, new sequences of magic values of M are observed. These new sequences can be easily understood in terms of a transition regime towards a system of two decoupled single dots. However, important differences in the nature of the incompressible ground states are found with respect to those of a single dot.
Energy spectra, spin configurations, and entanglement characteristics of a system of four electrons in lateral double quantum dots are investigated using exact diagonalization (EXD), as a function of interdot separation, applied magnetic field, and strength of interelectron repulsion. A distinctly different quantum behavior is found compared to that of circular single quantum dots. As a function of the magnetic field, the energy spectra exhibit a low-energy band consisting of a group of six states, with the number six being a consequence of the conservation of the total spin of the four electrons and the ensuing spin degeneracies. These six states appear to cross at a single value of the magnetic field, with the crossing point becoming sharper for larger interdot distances. As the strength of the Coulomb repulsion increases, the six states tend to become degenerate and a well defined energy gap separates them from the higher-in-energy excited states. The appearance of the low-energy band is a consequence of the formation of a Wigner supermolecule, with the four electrons (two in each dot) being localized at the vertices of a rectangular parallelogram. Using the spin-resolved pair-correlation functions, one can map the EXD many-body wave functions onto the spin functions associated with the four localized electrons. Analogies with nanomagnets, such as finite Heisenberg clusters, are discussed. The ability to determine associated spin functions enables investigations concerning entanglement properties of the system of four electrons. In particular, the formation of Wigner supermolecules generates EXD solutions belonging to the class of strongly entangled states referred to as N-qubit Dicke states in the quantum-information literature.
We study the nature of the many-body electron states (and quantum phase transitions between them) in a double-quantum-well structure under a strong external magnetic field as a function of the materials parameters that define the two-layer system, namely, the thickness of individual layers, the separation between them, the individual well potentials confining the electrons, and the potential barrier between them. Motivated by two recent experiments, we consider two different situations, one with (almost) no quantum tunneling between the wells, and the other with substantial interwell quantum tunneling. We use the spherical system finite-size exact diagonalization technique for our calculations. By calculating the overlap of our exact small-system numerical wave functions with various (postulated) analytic wave functions, we comment on the nature of the incompressibility for various Landau-level filling factors (\ensuremath{\nu}). We investigate, in particular details, the possibility of \ensuremath{\nu}=1/2 (i.e., 1/4 average occupancy in each well), and, 1 (i.e., 1/2 occupancy in each well) incompressible states where \ensuremath{\nu} is the total filling factor for the system. We also provide results for the \ensuremath{\nu}=2/3 situation. Our conclusion, based on our use of realistic system parameters in our calculations, is that in both the recent experimental realizations of the \ensuremath{\nu}=1/2 fractional quantum Hall effect, the relevant ground state is the so-called ``331'' state which is stabilized by the competition between intrawell and interwell electron-electron correlations.We calculate for various double-quantum-well structural parameters and magnetic fields the \ensuremath{\nu}=1/2 excitation spectra and the energy gaps, and compare it with the recent experimental data, obtaining excellent quantitative agreement. We provide a detailed quantum phase diagram for the \ensuremath{\nu}=1 state as a function of the interwell Coulomb interaction and the symmetric-antisymmetric single-particle gap, and find that the \ensuremath{\nu}=1 quantum Hall effect observed in the two recent experiments, in contrast to the \ensuremath{\nu}=1/2 effect, belongs in some sense to two different universality classes. We also predict the existence of a reentrant \ensuremath{\nu}=1 quantum Hall effect in double-quantum-well systems.
Exact eigenstates of the parallel coupled double quantum dots attached to the non-interacting leads taken in zero-bandwidth limit are analytically obtained in each particle and spin sector. The ground state of the half-filled system is identified from a four dimensional subspace of the twenty dimensional Hilbert space for different values of tunable parameters of the system viz. the energy levels of the quantum dots, the interdot tunneling matrix-element, the ondot and interdot Coulomb interactions and quantities like spin-spin correlation between the dots, occupancies of the dots are calculated. In the parameter space of the interdot tunneling matrix-element and ondot Coulomb interaction, the dots exhibit both ferromagnetic and antiferromagnetic correlation. There is a critical dependency of the interdot tunneling matrix-element on the ondot Coulomb interaction which leads to transition from the ferromagnetic correlation to the antiferromagnetic correlation as the interdot tunneling matrix-element is increased. The ferromagnetic and antiferromagnetic correlations also exist in the absence of interdot tunneling matrix-element through indirect exchange via the leads. The interdot Coulomb interaction is found to affect this dependency considerably.
We investigate the stability of few-electron quantum phases in vertically coupled quantum dots under a magnetic field of arbitrary strength and direction. The orbital and spin stability diagrams of realistic devices containing up to five electrons, from strong to weak interdot coupling, is determined. Correlation effects and realistic sample geometries are fully taken into account within the full configuration interaction method. In general, the magnetic field drives the system into a strongly correlated regime by modulating the single-particle gaps. In coupled quantum dots different components of the field, either parallel or perpendicular to the tunneling direction, affect single-dot orbitals and tunneling energy, respectively. Therefore the stability of the quantum phases is related to different correlation mechanisms, depending on the field direction. Comparison of exact diagonalization results with simple models allows one to identify the specific role of correlations.
The out-of-equilibrium transport properties of a double quantum dot system in the Kondo regime are studied theoretically by means of a two-impurity Anderson Hamiltonian with interimpurity hopping. The Hamiltonian is solved by means of a nonequilibrium generalization of the slave-boson mean-field theory. It is demonstrated that measurements of the differential conductance dI/dV, for appropriate values of voltages and tunneling couplings, can give a direct observation of the coherent superposition between the many-body Kondo states of each dot. For large voltages and arbitrarily large interdot tunneling, there is a critical voltage above which the physical behavior of the system again resembles that of two decoupled quantum dots.
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.