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arXiv:cond-mat/9910105 · Physical review. B, Condensed matter

Spin interactions and switching in vertically tunnel-coupled quantum dots

Guido Burkard, Georg Seelig, Daniel Loss

Published Jul 15, 2000• 141 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevb.62.2581

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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.
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 report a successful measurement of the magnetic field-induced spin singlet-triplet transition in silicon-based coupled dot systems. Our specific experimental scheme incorporates a lateral gate-controlled Coulomb-blockaded structure in Si to meet the proposed scheme of Loss and DiVincenzo [1], and a non-equilibrium single-electron tunneling technique to probe the fine energy splitting between the spin singlet and triplet, which varies as a function of applying magnetic fields and interdot coupling constant. Our results, exhibiting the singlet-triplet crossing at a magnetic field for various interdot coupling constants, are in agreement with the theoretical predictions, and give the first experimental demonstration of the possible spin swapping occurring in the coupled double dot systems with magnetic field. *Electronic address: jungchoi@chungbuk.ac.kr [1] D. Loss and D. P. DiVincenzo, Phys. Rev. A 57, 120 (1998).
A two-level quantum system is the building block of a quantum computer. This pair of quantum states defines the computational unit of a quantum computer, and it is called a quantum bit or qubit in analogy to the bit that is the binary unit of a classical computer. The spin of a single electron naturally defines such a two-level quantum system. A quantum dot can be tuned to the single-electron regime, and an array of singly occupied quantum dots provides a multi-qubit register. This thesis describes the multielectron encoding of a spin qubit using a pair or a trio of quantum dots, and it analyzes the coherence properties and manipulation protocols of this system. The singlet-triplet qubit, coded using the singlet state and the spinless triplet state of a pair of singly occupied quantum dots, can be controlled all electrically when applying voltages to gates close to the quantum dot structure. Rapid, subnanosecond modifications of the double quantum dot's charge configuration have been realized successfully. A small magnetic field gradient across the double quantum dot enables universal qubit control. Two modifications of the normal singlet-triplet encoding are introduced. (1) The six-electron configuration of a double quantum dot encodes a singlet-triplet qubit in the same way as for the two-electron double quantum dot. Two electrons at each quantum dot are irrelevant for the qubit manipulations because they are paired in a singlet state; the remaining two electrons encode the qubit. The qubit's wave function is immune to charge noise at moderate out-of-plane magnetic fields. (2) A singlet-triplet qubit encoded using two quantum dots of different sizes has an orbital state degeneracy of the singlet state and the spinless triplet state at finite out-of-plane magnetic fields. Spin-orbit interactions lift this state degeneracy, while the magnitude of the state coupling is determined by the size difference of the QDs. This setup enables the manipulations of singlet-triplet qubits without the need for magnetic field gradients. Finally, two-qubit gates between singlet-triplet qubits are proposed that use mediated exchange interactions via one quantum state. These operations are well controlled and highly noise insensitive. Orbital interactions alone can control spin qubits coded in a three electron Hilbert space. For example, the exchange-only qubit is encoded using three singly occupied quantum dots. The exchange interactions of two quantum dot pairs need to be modified to manipulate this qubit. The noise sensitivity of the exchange-only qubit is discussed. Alternatively, a three-electron qubit at a double quantum dot can be operated when single electrons are transferred between the quantum dots. Fast, subnanosecond manipulations of the double quantum dot's charge configuration are required to realize single-qubit gates. A novel two-qubit pulse gate for the three-electron double quantum dot qubit 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.
The exchange interaction in a singlet-triplet qubit defined by two-electron states in the double-quantum-dot system (``two-electron singlet-triplet qubit'') typically varies monotonically with the exchange interaction and thus carries no sweet spot. Here we study a singlet-triplet qubit defined by four-electron states in the double-quantum-dot system (``four-electron singlet-triplet qubit''). We demonstrate, using configuration-interaction calculations, that in the four-electron singlet-triplet qubit the exchange energy as a function of detuning can be nonmonotonic, suggesting existence of sweet spots. We further show that the tuning of the sweet spot and the corresponding exchange energy by perpendicular magnetic field can be related to the variation of orbital splitting. Our results suggest that a singlet-triplet qubit with more than two electrons can have advantages in the realization of quantum computing.