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arXiv:0712.1675 · Journal of Physics Condensed Matter

Effect of confinement potential shape on exchange interaction in coupled quantum dots

A Kwaśniowski, Janusz Adamowski

Published Apr 18, 2008• 26 citations
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Abstract

Exchange interaction has been studied for electrons in coupled quantum dots (QDs) by a configuration interaction method using confinement potentials with different profiles. The confinement potential has been parametrized by a two-centre power-exponential function, which allows us to investigate various types of QDs described by either soft or hard potentials of different ranges. For the soft (Gaussian) confinement potential the exchange energy decreases with increasing interdot distance due to the decreasing interdot tunnelling. For the hard (rectangular-like) confinement potential we have found a non-monotonic behaviour of the exchange interaction as a function of distance between the confinement potential centres. In this case, the exchange interaction energy exhibits a pronounced maximum for the confinement potential profile which corresponds to the nanostructure composed of the small inner QD with a deep potential well embedded in the large outer QD with a shallow potential well. This effect results from the strong localization of electrons in the inner QD, which leads to the large singlet–triplet splitting. Implications of this finding for quantum logic operations have been discussed.

DOI
10.1088/0953-8984/20/21/215208

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The effect of an external electric field on the exchange interaction has been studied by an exact diagonalization method for two electrons in laterally coupled quantum dots (QDs). We have performed a systematic study of several nanodevices that contain two gate-defined QDs with different shapes and sizes located between source and drain contacts. The confinement potential is modeled by two potential wells with a variable range and softness. In all the considered nanodevices, the overall dependence of exchange energy J on electric field F is similar, i.e. for low fields J increases with increasing F, while for intermediate fields J reaches a maximum and then abruptly falls to zero if F exceeds a certain critical value. However, the J(F) dependence also shows certain characteristic properties that depend on the nanodevice geometry. We have found that the low- and intermediate-field behavior can be accurately parameterized by a linear function J(F) = αF+β, where α is independent of the nanodevice geometry and softness of the confinement potential. We have shown that the linear J(F) relation appears only if the tunnel coupling between the QDs is weak, i.e. the interdot separation is sufficiently large. This relation becomes nonlinear for the strong interdot coupling. For specific nanodevices we have found that the J(F) dependence exhibits a plateau in a broad electric-field regime. The properties of the exchange energy found in the present paper can be applied to all electrical manipulation of electron spin qubits.

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Single-Particle State Mixing in Two-Electron Coupled Quantum Dots

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The properties of an electron pair confined in a coupled quantum dot (QD) system are currently being intensively investigated both experimentally and theoretically. Indeed the manipulation of the two-electron spin states is paramount for the realization of controllable entanglement so as to produce the fundamental quantum gate operations in a solid state quantum computer. The works usually concentrate on the properties of a two-electron system without going into detailed analysis of the contributions from various single-particle states or the effects produced by the constituent single QDs. Zeeman splitting is neglected as it affects only the magnitude of the exchange energy but not the single-particle state mixing. In summary, after numerically diagonalizing the two-electron Schrodinger equation in the model confinement potential of both single and laterally coupled double quantum dots, we computed the contributions from the various single-particle states to the final two-particle wave function.

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Exchange interaction has been studied for electrons in coupled quantum dots (QDs) by a configuration interaction method using confinement potentials with different profiles. The confinement potential has been parametrized by a two-centre power-exponential function, which allows us to investigate various types of QDs described by either soft or hard potentials of different ranges. For the soft (Gaussian) confinement potential the exchange energy decreases with increasing interdot distance due to the decreasing interdot tunnelling. For the hard (rectangular-like) confinement potential we have found a non-monotonic behaviour of the exchange interaction as a function of distance between the confinement potential centres. In this case, the exchange interaction energy exhibits a pronounced maximum for the confinement potential profile which corresponds to the nanostructure composed of the small inner QD with a deep potential well embedded in the large outer QD with a shallow potential well. This effect results from the strong localization of electrons in the inner QD, which leads to the large singlet–triplet splitting. Implications of this finding for quantum logic operations have been discussed.
The effect of an external electric field on the exchange interaction has been studied by an exact diagonalization method for two electrons in laterally coupled quantum dots (QDs). We have performed a systematic study of several nanodevices that contain two gate-defined QDs with different shapes and sizes located between source and drain contacts. The confinement potential is modeled by two potential wells with a variable range and softness. In all the considered nanodevices, the overall dependence of exchange energy J on electric field F is similar, i.e. for low fields J increases with increasing F, while for intermediate fields J reaches a maximum and then abruptly falls to zero if F exceeds a certain critical value. However, the J(F) dependence also shows certain characteristic properties that depend on the nanodevice geometry. We have found that the low- and intermediate-field behavior can be accurately parameterized by a linear function J(F) = αF+β, where α is independent of the nanodevice geometry and softness of the confinement potential. We have shown that the linear J(F) relation appears only if the tunnel coupling between the QDs is weak, i.e. the interdot separation is sufficiently large. This relation becomes nonlinear for the strong interdot coupling. For specific nanodevices we have found that the J(F) dependence exhibits a plateau in a broad electric-field regime. The properties of the exchange energy found in the present paper can be applied to all electrical manipulation of electron spin qubits.
The exchange coupling of the spins of two electrons in double well potentials in a semiconductor background is calculated within the Heitler-London (HL) approximation. Atomic and quantum dot types of confining potentials are considered, and a systematic analysis for the source of inaccuracies in the HL approach is presented. For the strongly confining coulombic atomic potentials in the H2 molecule, the most dramatic failure occurs at very large interatomic distances, where HL predicts a triplet ground state, both in 3D and in 2D, coming from the absence of electron-electron correlation effects in this approach. For a 2D double well potential, failures are identified at relatively smaller interdot distances, and may be attributed to the less confining nature of the potential, leading to larger overlap and consequently an inadequate representation of the two-particle states written, within HL, in terms of the ground state orbital at each isolated well. We find that in the double dot case, the range of validity of HL is improved (restricted) in a related 3D (1D) model, and that results always tend to become more reliable as the interdot distance increases. Our calculated exchange coupling is of relevance to the exchange gate quantum computer architectures in semiconductors.
The exchange coupling of the spins of two electrons in double-well potentials in a semiconductor background is calculated within the Heitler-London (HL) approximation. Atomic and quantum dot types of confining potentials are considered, and a systematic analysis for the source of inaccuracies in the HL approach is presented. For the strongly confining Coulombic atomic potentials in the H2{\mathrm{H}}_{2}H2​ molecule, the most dramatic failure occurs at very large interatomic distances, where the HL approximation predicts a triplet ground state, in both three and two dimensions, coming from the absence of electron-electron correlation effects in this approach. For a two-dimensional (2D) double-well potential, failures are identified at relatively smaller interdot distances, and may be attributed to the less confining nature of the potential, leading to larger overlap. We find that in the double-dot case, the range of validity of the HL approximation is improved (restricted) in a related 3D (1D) model, and that results always tend to become more reliable as the interdot distance increases. Our analysis of the exchange coupling is of relevance to the exchange-gate quantum computer architectures in semiconductors.
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.
We theoretically investigate the properties of a two-electron system confined in the three-dimensional potential of coupled quantum dots formed in a quantum wire. For this purpose, we implement a variational Heitler-London method that minimizes the system energies with respect to variational parameters in electron trial wave functions. We find that tunnel and exchange couplings rapidly decay with increasing interdot distance and interdot barrier height. In the quasi-one-dimensional limit achieved by reducing the wire diameter, we find that the overlap between the dots decreases, which results in a drop in the exchange coupling. We also discuss the validity of our variational Heitler-London method with respect to the model potential parameters; we find good agreement between our results and available experimental data.
The properties of an electron pair confined in a coupled quantum dot (QD) system are currently being intensively investigated both experimentally and theoretically. Indeed the manipulation of the two-electron spin states is paramount for the realization of controllable entanglement so as to produce the fundamental quantum gate operations in a solid state quantum computer. The works usually concentrate on the properties of a two-electron system without going into detailed analysis of the contributions from various single-particle states or the effects produced by the constituent single QDs. Zeeman splitting is neglected as it affects only the magnitude of the exchange energy but not the single-particle state mixing. In summary, after numerically diagonalizing the two-electron Schrodinger equation in the model confinement potential of both single and laterally coupled double quantum dots, we computed the contributions from the various single-particle states to the final two-particle wave function.