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Journal of Applied Physics

On the nuclear magnetic resonance frequency of phosphorus donor atom in a silicon-based quantum computer

Hamidreza Mirzaei, Hon Tat Hui

Published Nov 1, 2010• 3 citations
Publisher

Abstract

The nuclear magnetic resonance (NMR) frequency of a single qubit structure of Kane’s solid-state quantum computer is investigated by using the perturbation theory. With higher-order excited states (up to 3d modes) included in our calculation, the perturbation frequencies and energies are obtained numerically. To compute for arbitrary A gate geometries, the perturbation potential inside the qubit structure is determined through an electromagnetic simulation method. Calculations show that the potential distributions for realistic A gate geometries are far from linear ones. Our results show that the A gate voltage has a much more effective control over the NMR frequency of the phosphorus nucleus than that previously shown. Using our method, arbitrary A gate structures of any shapes or geometries can be engineered for the realization of a solid-state scalable quantum computer. We also investigate an alternative A gate structure using SiGe as the insulation barrier. Our study shows that this A gate structure offers a much more efficient utilization of the control voltage than the original A gate structure using SiO2 as the insulation barrier.

DOI
10.1063/1.3504653

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The nuclear magnetic resonance (NMR) frequency of a single qubit structure of Kane’s solid-state quantum computer is investigated by using the perturbation theory. With higher-order excited states (up to 3d modes) included in our calculation, the perturbation frequencies and energies are obtained numerically. To compute for arbitrary A gate geometries, the perturbation potential inside the qubit structure is determined through an electromagnetic simulation method. Calculations show that the potential distributions for realistic A gate geometries are far from linear ones. Our results show that the A gate voltage has a much more effective control over the NMR frequency of the phosphorus nucleus than that previously shown. Using our method, arbitrary A gate structures of any shapes or geometries can be engineered for the realization of a solid-state scalable quantum computer. We also investigate an alternative A gate structure using SiGe as the insulation barrier. Our study shows that this A gate structure offers a much more efficient utilization of the control voltage than the original A gate structure using SiO2 as the insulation barrier.
A numerical method is introduced to determine the nuclear magnetic resonance frequency of a donor (31P)(^{31}\mathrm{P})(31P) doped inside a silicon substrate under the influence of an applied electric field. This phosphorus donor has been suggested for operation as a qubit for the realization of a solid-state scalable quantum computer. The operation of the qubit is achieved by a combination of the rotation of the phosphorus nuclear spin through a globally applied magnetic field and the selection of the phosphorus nucleus through a locally applied electric field. To realize the selection function, it is required to know the relationship between the applied electric field and the change of the nuclear magnetic resonance frequency of phosphorus. In this study, based on the wave functions obtained by the effective-mass theory, we introduce an empirical correction factor to the wave functions at the donor nucleus. Using the corrected wave functions, we formulate a first-order perturbation theory for the perturbed system under the influence of an electric field. In order to calculate the potential distributions inside the silicon and the silicon dioxide layers due to the applied electric field, we use the multilayered Green's functions and solve an integral equation by the moment method. This enables us to consider more realistic, arbitrary shape, and three-dimensional qubit structures. With the calculation of the potential distributions, we have investigated the effects of the thicknesses of silicon and silicon dioxide layers, the relative position of the donor, and the applied electric field on the nuclear magnetic resonance frequency of the donor.
The nuclear spin quantum computer proposed by Kane1 exploits as a qubit array 31P dopants embedded within a silicon matrix. Single-qubit operations are controlled by the application of electrostatic potentials via a set of metallic A gates, situated above the donors, on the silicon surface, that tune the resonance frequency of individual nuclear spins, and a globally applied RF magnetic field that flips spins at resonance. Coupling between qubits is controlled by the application of potentials via a set of J gates, between the donors, that induce an electron-mediated coupling between nuclear spins. We report the results of a study of the electric field and potential profiles arising within the Kane device from typical gate operations. The extent to which a single nuclear spin can be tuned independently of its neighbours, by operation of an associated A-gate, is examined and key design parameters in the Kane architecture are addressed. Implications for current fabrication strategies involving the implantation of 31P atoms are discussed. Solution of the Poisson equation has been carried out by simulation using a TCAD modeling package (Integrated Systems Engineering AG).
In the Kane silicon-based electron-mediated nuclear spin quantum computer architecture, phosphorous is doped at precise positions in a silicon lattice, and the P donor nuclear spins act as qubits. Logical operations on the nuclear spins are performed using externally applied magnetic and electric fields. There are two important interactions: the hyperfine and exchange interactions, crucial for logical qubit operations. Single qubit operations are performed by applying radio frequency magnetic fields resonant with targeted nuclear spin transition frequencies, tuned by the gate-controlled hyperfine interaction. Two qubit operations are mediated through the exchange interaction between adjacent donor electrons. It is important to examine how these two interactions vary as functions of experimental parameters. Here we provide such an investigation. First, we examine the effects of varying several experimental parameters: gate voltage, inter donor separation, donor depth below the silicon oxide interface and back gate depth, to explore how these variables affect the donor electro density. Second, we calculate the hyperfine interaction and the exchange coupling as a function of these parameters. These calculations were performed using an anisotropic effective mass Hamiltonian. The electric field potential was obtained using Technology Computer Aided Design software, and the interfaces were modelled as a barrier using a step function. We aim to provide relevant information for the experimental design of these devices and highlight the significance of environmental factors other than gate potential that affect the donor electron.
As an ensemble scheme of solid-state NMR quantum computers the extension of Kane's many-qubits silicon scheme based on the array of 31 P donor atoms are spaced lengthwise of the strip gates is considered. The possible planar topology of such ensemble quantum computer is suggested. The estimation of the output NMR signal was performed and it was shown that for the number N>=10^5 of ensemble elements involving L~10^3 qubits each, the standard NMR methods are usable. As main mechanisms of decoherence for low temperature (<0.1K), the adiabatic processes of random modulation of qubit resonance frequency determined by secular part of nuclear spin hyperfine interaction with electron magnetic moment of basic atom and dipole-dipole interaction with nuclear moments of neighboring impurity atoms was considered, It was made estimations of allowed concentrations of magnetic impurities and of spin temperature whereby the required decoherence suppression is obtained. Semiclassical decoherence model of two qubit entangled states is also presented. As another variant of the solid-state ensemble quantum computer, the gateless architecture of cellular-automaton with antiferromagnetically ordered electron spins is also discussed here.
A new NMR quantum computer scheme based on semiconductor nanostructures is discussed theoretically. We also propose using an infrared laser as a new independent instrument for manipulating qubits. Methods of single- and many-qubit gate realization are discussed. Resonant transfer of the electron to several quantum dots helps in implementing complex many-qubit gates. The energy spectrum of the spin-qubit system is obtained. This variant of a quantum computer is simpler to control and more coherent. The distance between qubits and hence the structural requirements are more achievable using modern nanotechnology.
Spin is a quantum mechanical property that describes the magnetic orientation of elementary and composite particles such as electrons and nuclei. For spin-half particles placed in a magnetic field, two states with differing energies are defined. Proposals exist for developing a revolutionary kind of computer that encodes bits of information on the states of a simple quantum two-level system - like those of a spin in a magnetic field. These quantum computers exploit the fundamental principles of quantum mechanics in order to perform certain calculations exponentially faster than their classical counterparts. Key to the development of such a device is the identification of potential systems that are suitable for employment as a quantum bit (qubit). This thesis explores quantum bits constructed from the spin states of single atoms hosted in silicon semiconductor devices. Silicon is an exceptional platform for spin-based quantum computing - in addition to being the dominant material in the microelectronic industry, it is also predominantly composed of nuclear spin-zero isotopes, providing very little background magnetic disturbance. The donor-atom phosphorus (31P) is investigated here. It possesses a nucleus that binds an additional electron to its positive charge, providing two distinct spin qubits. Here, several world-first experimental milestones in spin-based quantum information processing are reported. First, the readout of the electron spin is demonstrated in single-shot, using a silicon device which combines implanted phosphorus donors with a sensitive nanoscale electrometer. Then, the electron spin is coherently operated as a qubit by delivering on-chip microwave excitations. Combining the electron spin detection and control capabilities enables a quantum non-demolition measurement of the single 31P nuclear spin. Experiments yield exceptional nuclear readout fidelities - the highest for any solid-state qubit. Through an application of nuclear magnetic resonance, high-fidelity quantum control of the nuclear spin qubit is demonstrated. Finally, a single spin-bearing silicon nucleus (29Si) coupled to the electron spin is detected in single-shot and operated as a qubit. These results, along with the clear pathways identified for improving device performance, illustrate that single-atom electron and nuclear spin qubits in silicon constitute an excellent basis on which to construct a quantum computer.