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arXiv:0908.2961 · Physical Review Letters

Theory of Anisotropic Exchange in Laterally Coupled Quantum Dots

Fabio Baruffa, Peter Stano, Jaroslav Fabian

Published Mar 23, 2010• 56 citations
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Abstract

The effects of spin-orbit coupling on the two-electron spectra in lateral coupled quantum dots are investigated analytically and numerically. It is demonstrated that in the absence of magnetic field, the exchange interaction is practically unaffected by spin-orbit coupling, for any interdot coupling, boosting prospects for spin-based quantum computing. The anisotropic exchange appears at finite magnetic fields. A numerically accurate effective spin Hamiltonian for modeling spin-orbit-induced two-electron spin dynamics in the presence of magnetic field is proposed.

DOI
10.1103/physrevlett.104.126401

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The effects of spin-orbit coupling on the two-electron spectra in lateral coupled quantum dots are investigated analytically and numerically. It is demonstrated that in the absence of magnetic field, the exchange interaction is practically unaffected by spin-orbit coupling, for any interdot coupling, boosting prospects for spin-based quantum computing. The anisotropic exchange appears at finite magnetic fields. A numerically accurate effective spin Hamiltonian for modeling spin-orbit-induced two-electron spin dynamics in the presence of magnetic field is proposed.
In the last years, the electron spin in quantum dots has become a most promising candidate for the realization of qubits, the basic element for quantum computation. In quantum dot spin qubits the exchange interaction implements a fundamental two qubit gate, which originates in the Coulomb interaction and the Pauli principle. Compared to single qubit gates, the exchange based gates are much faster and easier to control locally, motivating the solely exchange energy based quantum computation. The control is based on the exponential sensitivity of the exchange energy on the interparticle distance. Manipulation then can proceed, for example, shifting the single particle states electrically or compressing them magnetically. The practical manipulation schemes require quantitative knowledge of the exchange energy. The spin-orbit interaction plays an important role in this nano-system. Indeed the usual exchange interaction changes its magnitude, while a new term, called anisotropic exchange, appears. In this thesis we propose an accurate effective spin Hamiltonian for modeling spin-orbit-induced spin dynamics in the presence of the magnetic field. We have also performed numerically exact calculations of the isotropic and anisotropic exchange coupling in realistic two-electron GaAs coupled quantum dots in the presence of both Dresselhaus and Bychkov-Rashba spin-orbit interactions. The numerics allow us to establish the goodness of our model and to see the limits of previous statements. Furthermore we find that in zero magnetic field the second-order spin-orbit effects are absent at all interdot couplings.
We perform time-dependent simulations of spin exchange for an electron pair in laterally coupled quantum dots. The calculation is based on configuration interaction scheme accounting for spin-orbit (SO) coupling and electron-electron interaction in a numerically exact way. Noninteracting electrons exchange orientations of their spins in a manner that can be understood by interdot tunneling associated with spin precession in an effective SO magnetic field that results in anisotropy of the spin swap. The Coulomb interaction blocks the electron transfer between the dots but the spin transfer and spin precession due to SO coupling is still observed. The electron-electron interaction additionally induces an appearance of spin components in the direction of the effective SO magnetic field which are opposite in both dots. Simulations indicate that the isotropy of the spin swap is restored for equal Dresselhaus and Rashba constants and properly oriented dots.
We study the spin-spin coupling between two single-electron quantum dots due to the Coulomb and spin-orbit interactions, in the absence of tunneling between the dots. We find an anisotropic XYXYXY spin-spin interaction that is proportional to the Zeeman splitting produced by the external magnetic field. This interaction is studied both in the limit of weak and strong Coulomb repulsion with respect to the level spacing of the dot. The interaction is found to be a nonmonotonic function of interdot distance a0{a}_{0}a0​ and external magnetic field and, moreover, vanishes for some special values of a0{a}_{0}a0​ and/or magnetic field orientation. This mechanism thus provides a new way to generate and tune spin interaction between quantum dots. We propose a scheme to measure this spin-spin interaction based on the spin-relaxation-measurement technique.
We consider a quantum-gate mechanism based on electron spins in coupled semiconductor quantum dots. Such gates provide a general source of spin entanglement and can be used for quantum computers. We determine the exchange coupling JJJ in the effective Heisenberg model as a function of magnetic (B)(B)(B) and electric fields, and of the interdot distance aaa within the Heitler-London approximation of molecular physics. This result is refined by using sp\mathrm{sp}sp hybridization, and by the Hund-Mulliken molecular-orbit approach, which leads to an extended Hubbard description for the two-dot system that shows a remarkable dependence on BBB and aaa due to the long-range Coulomb interaction. We find that the exchange JJJ changes sign at a finite field (leading to a pronounced jump in the magnetization) and then decays exponentially. The magnetization and the spin susceptibilities of the coupled dots are calculated. We show that the dephasing due to nuclear spins in GaAs can be strongly suppressed by dynamical nuclear-spin polarization and/or by magnetic fields.
We theoretically explore the possibility of creating spin quantum entanglement in a system of two electrons confined respectively in two vertically coupled quantum dots in the presence of Rashba-type spin-orbit coupling. We find that the system can be described by a generalized Jaynes-Cummings model of two-mode bosons interacting with two spins. The lower excited states of this model are calculated to reveal the underlying physics of the far-infrared absorption spectra. The analytic perturbation approach shows that an effective transverse coupling of spins can be obtained by eliminating the orbital degrees of freedom in the large-detuning limit. Here, the orbital degrees of freedom of the two electrons, which are described by two-mode bosons, serve as a quantized data bus to exchange the quantum information between the two electron spins. Then a nontrivial two-qubit logic gate is realized and spin entanglement between the two electrons is created by virtue of spin-orbit interactions.
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.