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Semiconductor Science and Technology

Spin-split excitation gap and spin entanglement of a pair of interacting electrons in a quantum dot

Godfrey A. Gumbs, Danhong Huang, M. D. Blumenthal, Samuel J Wright, M. Pepper, Yonatan Abranyos

Published Oct 6, 2009• 7 citations
Publisher

Abstract

We calculate the energy eigenvalues, the spin-split excitation gap (energy separation between the spin-triplet excited state and the spin-singlet ground state) and the concurrence for two interacting electrons captured in a quantum dot (QD) formed by a gigahertz electron pump which is modeled by harmonic confining potentials. We find from our calculations a peak in the QD size dependence of the energy level for the spin-singlet ground state, indicating the effect due to Coulomb blockade. In addition, we observe a local minimum in the QD size dependence of the spin-split excitation gap for a relatively narrow quasi-one-dimensional (1D) channel formed from an etched wire, but a strong positive peak for the spin-split excitation gap in its QD size dependence with a relatively wide 1D channel. From the existence of a robust spin-split excitation gap against both thermal fluctuation due to finite (low) temperatures and the nonadiabatic effect due to fast barrier variations, we predict a spin-entangled electron pair inside the QD with a weak coupling to external leads. An interference-type experiment which employs a gate-controlled electron pump and a beam splitter is proposed to verify this prediction. For the electron pump, a sinusoidal radio-frequency signal is applied to the entrance gate of a two-gated system over a narrow channel etched in a GaAs/AlGaAs heterostructure, where the measured current within the channel shows plateaus at Nef with N = 1, 2,... being the number of captured electrons in a QD and f the frequency of the sinusoidal signal.

DOI
10.1088/0268-1242/24/11/115001

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We calculate the energy eigenvalues, the spin-split excitation gap (energy separation between the spin-triplet excited state and the spin-singlet ground state) and the concurrence for two interacting electrons captured in a quantum dot (QD) formed by a gigahertz electron pump which is modeled by harmonic confining potentials. We find from our calculations a peak in the QD size dependence of the energy level for the spin-singlet ground state, indicating the effect due to Coulomb blockade. In addition, we observe a local minimum in the QD size dependence of the spin-split excitation gap for a relatively narrow quasi-one-dimensional (1D) channel formed from an etched wire, but a strong positive peak for the spin-split excitation gap in its QD size dependence with a relatively wide 1D channel. From the existence of a robust spin-split excitation gap against both thermal fluctuation due to finite (low) temperatures and the nonadiabatic effect due to fast barrier variations, we predict a spin-entangled electron pair inside the QD with a weak coupling to external leads. An interference-type experiment which employs a gate-controlled electron pump and a beam splitter is proposed to verify this prediction. For the electron pump, a sinusoidal radio-frequency signal is applied to the entrance gate of a two-gated system over a narrow channel etched in a GaAs/AlGaAs heterostructure, where the measured current within the channel shows plateaus at Nef with N = 1, 2,... being the number of captured electrons in a QD and f the frequency of the sinusoidal signal.
The recent developments in the experimental realization of quasi-one-dimensional (1D) systems exhibit many interesting features. These include current quantization which has the potential application for a current standard as well as quantum information and security schemes. In this paper, we investigate the effect of spin-orbit interaction (SOI) on the energy levels of electrons confined to quantum dots on the surface of nanotubes. The radius of a nanotube is a few nanometers and is quasi-1D. The energy levels play a crucial role in determining the electron transport properties.The SOI may arise from the electrostatic confining potential due to gates applied perpendicular to the axis of the nanotube. The quantum computation scheme which we are suggesting consists of a nano-circuit of nanotubes on which electrons are confined within dots. The qubit operation is based on the exchange interaction between a pair of spins occupying states within the quantum dots. We employ a simple model for the electron confinement to obtain the energy eigenstates. Our simplified calculation was able to show that the SOI splits the energy levels which are then used to obtain the exchange energy of a pair of spins with the s-wave Heitler-London approach. We calculate the exchange energy of the entangled electrons on a pair of coaxial and parallel nanotubes as a function of separation between the nanotubes and show that the SOI enhances the entanglement.
It has been suggested (Giavaras et al 2007 Phys. Rev. B 75 085302) that a quantum dot structure, which consists of a small open dot formed within a larger dot, can generate spin entanglement between two electrons and filter singlet–triplet states as a consequence of Coulomb interaction and the Pauli exclusion principle. In this work, we show via numerical modelling how to form the dot structure in an AlGaAs/GaAs/AlGaAs quasi-one-dimensional channel using top and back gates. We calculate the confining potential of the dot and the corresponding low-lying two-electron states and demonstrate a way of controlling the degree of entanglement and filtering by tuning the dot potential with the voltage applied to a back gate.
Currently there is strong interest in realizing implementations of quantum computation and quantum cryptography in a solid state environment. One of the systems that are actively studied are semiconductor quantum dots (QDs). Due to their discrete energy level structure, they are often called artificial atoms, and they attract immediately interest of quantum information science since they allow to mimic the design developed for atomic physics systems such as ions in traps or atoms in cavities. However, despite of the similarities, one has to keep in main that any elementary excitation in a QD has a generic many-body character. An essential building block of a quantum processor is a quantum gate which entangles the states of two quantum bits. Recently it has been proposed that a pair of vertically aligned QDs could be used as an optically driven quantum gate: The quantum bits are individual carriers either on dot zero or dot one. The different dot indices play the same role as a "spin," therefore we term them "isospin." Quantum mechanical tunneling between the dots rotates the "isospin" and leads to superposition of two quantum dot states. The quantum gate is built when two different particles, an electron and a hole, are created optically. The two particles form entangled isospin states. The entanglement can be controlled by application of an electric field along the heterostructure growth direction. Here we present spectroscopic studies on single quantum dot molecules (QDMs) with different vertical separation between the dots that support the feasibility of this proposal. The comparison of the evolution of the excitonic recombination spectrum with the results of calculations allows us to demonstrate coherent tunneling of electrons and holes across the separating barrier and the formation of entangled exciton states. For a given barrier width, we find only small variations of the tunneling induced splitting between the entangled states demonstrating a good homogeneity within the obtained QDM ensembles.
Entanglement is a valuable resource for quantum computing and information technologies. A promising way to obtain spin-entangled electrons is the splitting of Cooper pairs residing in superconductors. This is realized by a nanodevice called Cooper pair splitter. A Cooper pair splitter is realized by tunnel-coupling a superconductor to two quantum dots, which are each further tunnelcoupled to separate terminal leads. This setup enables the extraction and splitting of Cooper pairs from the superconductor into the terminals via the quantum dots. The quantum dots are commonly fabricated from semiconducting materials like InAs and InSb, that can manifest spin-orbit interaction. This thesis studies the effect of spin-orbit interaction on the spin state of the electron pair in the Cooper pair splitter. Activating the spin-orbit effects in a Cooper pair splitter requires a static magnetic field to be applied to the quantum dots. Together with the gate electrodes controlling the electrostatic energy of the quantum dots, the external magnetic field provides a handle for addressing different two-particle spin states. The spin-orbit interaction can be activated by various combinations of the gate voltage and Zeeman magnetic field, that create resonances between different states of the system. The most interesting coherent evolution, that can be invoked in this way, involves the simultaneous activation of the two spin-polarized triplet states. This gives a rise to an entangled Bell state, (|↑↑>+|↓↓>) / √2, that moreover exhibits spin blockade and thus stabilizes entanglement in the Cooper pair splitter. Secondly, the spin-orbit interaction can be used to bypass suppressed Cooper pair tunneling at a high superconductor-quantum dots detuning. Introducing a large Zeeman magnetic field allows Cooper pairs to tunnel in to the dots given that their spin state flips to the triplet |↓↓>. Although this regime lacks spin-entanglement, it is of interest for creating triplet Cooper pairs and spin-polarized superconducting currents.
Since the development of microsized devices is moving forward at enormous speed, there is huge amount of new industrial opportunities. However such devices also require high precision and understanding of the operating of their constituent parts up to the quantum level. The device of the purely quantum nature being developed so far is quantum computer. However the physical realization of it is still not performed, as the requirements for it are very rigorous. \n \nIn the pioneering work by Loss and DiVincenzo it was suggested to use a spin of electron placed in a quantum dot as an information qubit. Following this work the study of electron or hole spin qubits developed. Both experimental and theoretical tools for studying them greatly advanced. \n \nIn the first part of the thesis we study the phonon-induced decoherence and relaxation of singlet-triplet qubits in the double quantum dots. First of all we consider AlGaAs/GaAs double quantum dots. The important result we present here is the strong dephasing that occurs at large detuning. This dephasing is due to two-phonon process that affects mainly singlet state of the qubit and consequently changes the splitting between singlet and triplet leading to dephasing. Remarkably at small detuning this dephasing process is suppressed and the decoherence time is by orders of magnitude longer than in case of large detuning and is mainly defined by one-phonon process. We also present the dependence of relaxation time and decoherence time on the strength of spin-orbit interaction and different directions of the system. Our results provide a deeper insight into the recently obtained experimental data. \n \nWe also studied Si/SiGe quantum dots as a potential candidate for a qubit. Apart from the absence of hyperfine interaction and bulk spin-orbit interaction in the isotopically purified 28^{28}28Si, its electron-phonon interaction is different from GaAs that also leads to longer qubit lifetimes. We study SSS-T−T_-T−​ qubit near the anticrossing of the basis states. This particular region is interesting due to possibilities in operating the qubit. We show that the type of singlet plays a crucial role, i.e. whether it is a singlet with each dot singly occupied or a singlet with only one dot doubly occupied. Depending on the type of singlet the qubit lifetimes change by several orders of magnitude. We also study the influence of a micromagnet, usually used in experiments to operate the qubit, on the relaxation time and decoherence time and present the regime where its effect is negligible. We suggest how to test experimentally our theory of one-phonon and two-phonon processes separately. We also show how the relaxation and decoherence time depend on different system parameters for SSS-T0T_0T0​ qubits. \n \nThe second part of the thesis is devoted to the other important constituent part of microsized devices, namely nanowires. We are interested in the dynamic of polarization of localized spins in the nanowires, as it can affect such important device characteristics as e.g. conductance. \n \nWe studied Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction-induced polarization in the nanowire in case when the voltage is applied to it. It was already proposed theoretically that in the ground state the localized spins in 1D systems align in a helix due to RKKY interaction. This polarization is still present until some critical temperature. The presence of such polarization acts as a spin filter for electrons, that most likely affects the conductance of the nanowire. Therefore we studied how this helical polarization changes when the voltage is applied. The key result is the appearance of the uniform polarization perpendicular to the helix plane, that occurs due to backscattering of electrons that is accompanied by flip-flop process with localized spins. When this uniform polarization is formed, the helix starts to rotate as a whole around the direction of the uniform polarization. We present the dependence of polarization of the localized spins on temperature and voltage. Remarkably the uniform polarization grows both with voltage and temperature in the given range of parameters. \n \nWe also considered the electron-induced relaxation of the nuclear spins. As the electron spins and nuclear spins interact via hyperfine interaction, the nuclear relaxation time reflects some properties of electron bath. Namely, we see a strong dependence of nuclear relaxation time on spin-orbit interaction strength. We present here the dependence of the nuclear relaxation time on the external magnetic field and chemical potential of the wire, that can be experimentally varied via gate. The dependences for the strong spin-orbit interaction and for the weak one are substantially different. Moreover, they have distinct peaks, that allow to get the value of spin-orbit interaction amplitude with the high precision.
We theoretically study a double-quantum-dot hydrogen molecule in the GaAs conduction band as the basic elementary gate for a quantum computer, with the electron spins in the dots serving as qubits. Such a two-dot system provides the necessary two-qubit entanglement required for quantum computation. We determine the excitation spectrum of two horizontally coupled quantum dots with two confined electrons, and study its dependence on an external magnetic field. In particular, we focus on the splitting of the lowest singlet and triplet states, the double occupation probability of the lowest states, and the relative energy scales of these states. We point out that at zero magnetic field it is difficult to have both a vanishing double occupation probability for a small error rate and a sizable exchange coupling for fast gating. On the other hand, finite magnetic fields may provide finite exchange coupling for quantum computer operations with small errors. We critically discuss the applicability of the envelope-function approach in the current scheme, and also the merits of various quantum-chemical approaches in dealing with few-electron problems in quantum dots, such as the Hartree-Fock self-consistent-field method, the molecular-orbital method, the Heisenberg model, and the Hubbard model. We also discuss a number of relevant issues in quantum dot quantum computing in the context of our calculations, such as the required design tolerance, spin decoherence, adiabatic transitions, magnetic-field control, and error correction.