Valley Filtering in Spatial Maps of Coupling between Silicon Donors and Quantum Dots

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Valley Filtering in Spatial Maps of Coupling between Silicon Donors and Quantum Dots

J. Salfi, B. Voisin, A. Tankasala, J. Bocquel, M. Usman, M. Y. Simmons, L. C. L. Hollenberg, R. Rahman, and S. Rogge

Phys. Rev. X 8, 031049 – Published 27 August 2018

Abstract

Exchange coupling is a key ingredient for spin-based quantum technologies since it can be used to entangle spin qubits and create logical spin qubits. However, the influence of the electronic valley degree of freedom in silicon on exchange interactions is presently the subject of important open questions. Here we investigate the influence of valleys on exchange in a coupled donor–quantum-dot system, a basic building block of recently proposed schemes for robust quantum information processing. Using a scanning tunneling microscope tip to position the quantum dot with sub-nm precision, we find a near monotonic exchange characteristic where lattice-aperiodic modulations associated with valley degrees of freedom comprise less than 2% of exchange. From this we conclude that intravalley tunneling processes that preserve the donor’s $\pm x$ and $\pm y$ valley index are filtered out of the interaction with the $\pm z$ valley quantum dot, and that the $\pm x$ and $\pm y$ intervalley processes where the electron valley index changes are weak. Complemented by tight-binding calculations of exchange versus donor depth, the demonstrated electrostatic tunability of donor–quantum-dot exchange can be used to compensate the remaining intravalley $\pm z$ oscillations to realize uniform interactions in an array of highly coherent donor spins.

Received 5 July 2017
Revised 14 June 2018

DOI:https://doi.org/10.1103/PhysRevX.8.031049

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Coulomb blockade Quantum interference effects Spintronics Valleytronics

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

J. Salfi¹, B. Voisin¹, A. Tankasala², J. Bocquel¹, M. Usman³, M. Y. Simmons¹, L. C. L. Hollenberg³, R. Rahman², and S. Rogge¹

¹Centre for Quantum Computation and Communication Technology, School of Physics, The University of New South Wales, Sydney, New South Wales 2052, Australia
²Purdue University, West Lafayette, Indiana 47906, USA
³Centre for Quantum Computation and Communication Technology, School of Physics, University of Melbourne, Parkville, Victoria 3010, Australia

Popular Summary

Hybrid systems of donor atoms and quantum dots in silicon have recently been proposed as building blocks for robust quantum computers. Interconnecting donor qubits in these arrays requires interactions influenced by the local energy minima (or “valleys”) occupied by each electron. This degree of freedom is predicted to cause an oscillatory dependence of exchange interactions on the electron position, a dependence that is intimately tied to the uniformity of qubit couplings that can be obtained in a quantum computer. However, establishing the strength of this effect has proven elusive. Here, we experimentally investigate exchange interactions between a single donor and quantum dot.

In our setup, the quantum dot is formed using the electric field applied by an atomically sharp tip and can be moved relative to the donor with subnanometer accuracy. We find that the rapid oscillatory behavior is absent for displacements perpendicular to the direction of strongest spatial confinement of the quantum dot. This occurs because the quantum numbers associated with the valley degree of freedom differ for the donor atom and the quantum dot and because exchange processes where electrons change their valley quantum number are very weak. Adjusting the tunnel barrier between the quantum dot and donor can compensate for the remaining oscillation in exchange interactions.

The use of quantum dots in donor-based schemes for quantum computing opens a new pathway to controllably couple the highly coherent spins of donor-based qubits via their mutual interactions with quantum dots.

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Vol. 8, Iss. 3 — July - September 2018

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Figure 1
Schematic of single-electron tunneling from buried reservoir, to coupled donor-QD state, to the tip. The donor wave function parameters are taken from Ref. [58], and the $\pm x$ and $\pm z$ valleys participating in the six-valley superposition are labeled. The $\pm y$ valleys of the donor are omitted for clarity. $λ_{μ} = 2 π / k_{μ}$ , where $k_{μ} = 0.82 (2 π / a_{0})$ is the conduction band minimum wave vector, and $a_{0}$ is silicon’s lattice constant. (b) Schematic diagram of isolated QD resonance created by tip-induced band bending. (c) Measured $d I / d V$ spectrum labeling the 0,1,2 (donor-QD hybrid) and 3 electron regions along the 110 crystal direction. Inset: Measured topography where dashed line shows locations for $d I / d V$ data. (d) Energy diagram for transitions between 0, 1, 2, and 3 electron states labeled with green, blue, and red lines, respectively. Tunnel coupling $t$ between the donor and QD hybridizes the two-electron states $S (2, 0)$ and $S (1, 1)$ , where $S (i, j)$ is the singlet with $i$ electrons on the donor and $j$ on the QD. Unhybridized two-electron states are shown as black lines and hybridized states as blue lines. The star (box) denotes the QD position for strong (weak) hybridization of two-electron states. Temperature is 4.2 K.
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Figure 2
(a) Bias dependence of current $I$ (green) and conductance $d I / d V$ (blue) over the donor, taken at $- 1.42 nm$ in Fig. 2, and fit to rate-equation model (black lines). (b) Measured current map of the neutral donor. Right: Tunnel junction energy diagram. (c) Measured current map of coupled donor-QD singlet. Right: Tunnel junction energy diagram. Data for the donor and donor-QD resonances were acquired in the spatial region of the inset of Fig. 1(c). The donor ion position is marked with a red cross.
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Figure 3
(a) Donor-QD energy $E (x^{'})$ as a function of tip position $x$ . Inset: Measured topography. (b) Spatial Fourier decomposition of $E (x^{'})$ where lattice-aperiodic components are less than 1% of value at $q_{110} = 0$ . Inset: Intravalley (green) and intervalley (white) tunneling for donor and QD. (c) One-dimensional spatial Fourier decomposition of $2 e$ and $1 e$ tunneling current along a $[110]$ direction, with lattice-aperiodic oscillations due to valley superpositions in donor bound states at $q_{110} = 0.85 (2 π / a_{110})$ and harmonics, where $a_{110} = \sqrt{2} a_{0}$ . Left-hand inset: Two-dimensional Fourier decomposition of one-electron tunneling probability at electric field $8 \pm 2 MV / m$ and bias $V = - 1.010 V$ . Right-hand inset: Line cut through two-dimensional decomposition evidencing negligible valley repopulation of donor just below the $2 e$ resonance.
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Figure 4
(a) Calculated electron density for donor and quantum dot, where only the donor contains lattice incommensurate components. Inset: Predicted decay of exchange as a function of donor-QD distance. (b) Calculated dependence of donor-QD exchange on donor depth, for different displacements along 110 between 18.43 and 23.04 nm.
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