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arXiv:cond-mat/0606582 · Physical Review B

Quantum phases of correlated electrons in artificial molecules under magnetic fields

Devis Bellucci, Massimo Rontani, Guido Goldoni, Elisa Molinari

Published Jul 24, 2006• 4 citations
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Abstract

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.

DOI
10.1103/physrevb.74.035331

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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.
We determine the spin-exchange coupling J between two electrons located in two vertically tunnel-coupled quantum dots, and its variation when magnetic (B) and electric (E) fields (both in-plane and perpendicular) are applied. We predict a strong decrease of J as the in-plane B field is increased, mainly due to orbital compression. Combined with the Zeeman splitting, this leads to a singlet-triplet crossing, which can be observed as a pronounced jump in the magnetization at in-plane fields of a few T, and perpendicular fields of the order of 10 T for typical self-assembled dots. We use harmonic potentials to model the confining of electrons, and calculate the exchange J using the Heitler-London and Hund-Mulliken techniques, including the long-range Coulomb interaction. With our results we provide experimental criteria for the distinction of singlet and triplet states, and therefore for microscopic spin measurements. In the case where dots of different sizes are coupled, we present a simple method to switch the spin coupling on and off with exponential sensitivity using an in-plane electric field. Switching the spin coupling is essential for quantum computation using electronic spins as qubits.
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 ${\ensuremath{\Delta}}_{t}$ 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.
We predict large regions of the charge stability diagram using a multiband and multielectron configuration interaction model of a double quantum dot system. We account for many-body interactions within each quantum dot using full configuration interaction and solve for single-particle density operators. This allows charge states to be predicted more accurately than the extensively used classical capacitance model or the single-band Hubbard model. The resulting single-particle mixed states then serve as inputs into an atomic orbital picture that allows for the explicit calculation of the underlying Hubbard model parameters by performing the appropriate integrals. This numerical approach allows for arbitrary choices of electrostatic potential and gate geometry. A common assumption when calculating charge stability diagrams from the Hubbard model is that the charge stability diagrams are periodic, but we find that the tunnel couplings for valence electrons in dots with N=3N=3N=3 electrons are significantly enhanced when compared to single-electron dots. This difference is apparent in the charge stability diagram for higher occupancy Coulomb diamonds. We also quantitatively explore how the barrier gate strength and dot pitch impact this behavior. Our work should help improve the future realistic modeling of semiconductor-dot-based quantum circuits.
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.