Electrical control of the $g$ tensor of the first hole in a silicon MOS quantum dot

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Electrical control of the $g$ tensor of the first hole in a silicon MOS quantum dot

S. D. Liles, F. Martins, D. S. Miserev, A. A. Kiselev, I. D. Thorvaldson, M. J. Rendell, I. K. Jin, F. E. Hudson, M. Veldhorst, K. M. Itoh, O. P. Sushkov, T. D. Ladd, A. S. Dzurak, and A. R. Hamilton

Phys. Rev. B 104, 235303 – Published 17 December 2021

Abstract

Single holes confined in semiconductor quantum dots are a promising platform for spin-qubit technology, due to the electrical tunability of the $g$ factor of holes. However, the underlying mechanisms that enable electric spin control remain unclear due to the complexity of hole-spin states. Here, we study the underlying hole-spin physics of the first hole in a silicon planar metal-oxide-semiconductor (MOS) quantum dot. We show that nonuniform electrode-induced strain produces nanometer-scale variations in the heavy-hole–light-hole (HH-LH) splitting. Importantly, we find that this nonuniform strain causes the HH-LH splitting to vary by up to 50% across the active region of the quantum dot. We show that local electric fields can be used to displace the hole relative to the nonuniform strain profile, allowing a mechanism for electric modulation of the hole $g$ tensor. Using this mechanism, we demonstrate tuning of the hole $g$ factor by up to 500%. In addition, we observe a potential sweet spot where $d g_{(1 \bar{1} 0)} / d V = 0$ , offering a configuration to suppress spin decoherence caused by electrical noise. These results open a path towards a technology involving engineering of nonuniform strains to optimize spin-based devices.

Received 30 March 2021
Revised 11 November 2021
Accepted 18 November 2021

DOI:https://doi.org/10.1103/PhysRevB.104.235303

Physics Subject Headings (PhySH)

Quantum information with solid state qubits Spin-orbit coupling Spintronics Strain

Quantum dots Semiconductors

Spin

Luttinger–Kohn model

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

S. D. Liles ^1,*, F. Martins ^1,2, D. S. Miserev^1,3, A. A. Kiselev⁴, I. D. Thorvaldson¹, M. J. Rendell¹, I. K. Jin¹, F. E. Hudson ⁵, M. Veldhorst^5,6, K. M. Itoh⁷, O. P. Sushkov ¹, T. D. Ladd ^1,4, A. S. Dzurak ⁵, and A. R. Hamilton ¹

¹School of Physics, University of New South Wales, Sydney, New South Wales 2052, Australia
²Hitachi Cambridge Laboratory, J. J. Thomson Avenue, Cambridge CB3 0HE, United Kingdom
³Department of Physics, University of Basel, Klingelbergstrasse 82, CH-4056 Basel, Switzerland
⁴HRL Laboratories, LLC, 3011 Malibu Canyon Road, Malibu, California 90265, USA
⁵School of Electrical Engineering and Telecommunications, University of New South Wales, Sydney, New South Wales 2052, Australia
⁶QuTech and Kavli Institute of Nanoscience, TU Delft, 2600 GA Delft, The Netherlands
⁷School of Fundamental Science and Technology, Keio University, Yokohama 223-8522, Japan

^*Corresponding author: s.liles@unsw.edu.au

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Vol. 104, Iss. 23 — 15 December 2021

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Images

Figure 1
A single hole in a silicon quantum dot. (a) False-color SEM image of a device gate stack. A single quantum dot is formed in the region indicated by the white dashed circle by using gate G2 as the quantum dot plunger gate, while gates G1 and G3 and the C gate provide the electrostatic confinement. SLB and SRB are the left and right sensor barrier gates, respectively. The in-plane crystal orientations are indicated, where the sample $x$ axis is the crystal axis [110] and the sample $y$ axis is the crystal axis $[1 \bar{1} 0]$ . The out-of-plane direction is the sample $z$ axis, corresponding to crystal axis [001]. The horizontal white scale bar is 250 nm. (b) Charge stability diagram showing operation down to the last hole, where the grayscale is $d I_{sens} / d V_{G 2}$ . A series of charge transitions can be observed as negative spikes in $d I_{sens} / d V_{G 2}$ . (c) The charging energy $E_{C}$ measured at the $V_{G 4}$ indicated by the colored horizontal arrows in (b). Schematics on the right indicate a line cut of the electrostatic potential energy along the sample $x$ axis. A hole quantum dot is formed at the potential maxima. The black horizontal dashed line indicates the Fermi energy, and the ellipse above each schematic represents the single-hole probability density, which is displaced and elongated as $V_{G 4}$ finely tunes the local electrostatic environment. (d) The measured effective $g$ factor for a magnetic field applied along the sample $y$ axis $[1 \bar{1} 0]$ . The dashed red line is a guide to the eye.
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Figure 2
Electrical control of the $g$ -tensor orientation. Measurements of the hole $g$ tensor for (a) $V_{G 4} = - 0.9$ V and (b) $V_{G 4} = - 0.7$ V. (i) The measured effective $g$ factor for magnetic field rotations around the sample $z$ (black), $y$ (blue), and $x$ (red) axes. Solid lines are a best fit of all data to Eq. (1). Typical uncertainty in the $g$ factor is 0.4; see Sec. S5.2 of the Supplemental Material [50]. Angles $θ$ and $φ$ define the orientation of the magnetic field $\vec{B}$ as indicated on the adjacent sphere. (ii) The shaded blue surface is the 3D $g$ -tensor surface defined by the appropriate parameters in Table 1. For reference the experimental data (circles or triangles) and best fit (solid lines) from (i) are included. (iii) Reproduction of the $g$ -factor measurements for the $y$ -axis rotation (blue) ( $θ = 0$ ) as a polar plot. The radial axis is | $g$ |, while the angle corresponds to $φ$ . The axes of symmetry of the $g$ tensor in the $x z$ plane are indicated by the dashed black lines. The magnitude of the tilting into the $x$ plane is indicated by the blue arrow, where the tilt in the $x z$ plane corresponds to $ϕ_{0}$ of Eq. (1). (iv) The sample schematic with the $x z g$ -tensor surface, highlighting the tilted $g$ -tensor orientation with respect to the $Si / {SiO}_{2}$ interface. Circles (triangles) are used for raw data in (a) [in (b)].
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Figure 3
Numerical simulations of the electrostatic confinement, strain, and single-hole eigenstates. (a) Electrostatic confinement profile (contour lines) and hole ground-state probability density (color map) in the $x y$ plane. The contours are spaced by 20 meV. The in-plane confinement is strongest along the sample $y$ axis $[1 \bar{1} 0]$ , consistent with the strong influence of the large C gate. Confinement along the sample $x$ axis (crystal [110] axis) is weaker and is provided by gates G1 and G3. This simulation is for the experimentally applied voltages, as described in Appendix pp3. (b) Hole probability density in the sample $x z$ plane. The footprint of each gate is shown (the vertical height of the Al gates is $30 +$ nm). (c) Spatial profile of the strain-induced HH-LH splitting in energy, with the hole probability density overlaid as a red color map and HH-LH splitting shown as contours spaced by 2 meV.
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Figure 4
Tilted $g$ tensor and electrode-induced strain. (a) Simulated $g$ -tensor surface for the wave-function position and strain profile indicated in (b). The hole is localized directly below the G2 gate, where the local strain gradients are low and shear strains are minimal. The orientation of the $g$ tensor is primarily defined by the orientation of the confinement, with the largest $g$ factor occurring for approximately out-of-plane magnetic field; more details are given in (e). The intersection of the $g$ -tensor surface with the sample $x y$ (black), $x z$ (blue), and $y z$ (red) planes is indicated by the respective solid lines. (c) Same as (a), except forcing the hole to the position indicated in (d). The contour lines in (b) and (d) are spaced by 2 meV. (e) The $g$ -tensor tilting $ϕ_{0}$ as a function of the artificial shift of the electrostatic confinement along the sample $x$ axis. Red circles indicate the $x$ -axis position and $ϕ_{0}$ value for the cases in (a) and (c). Above (e) we schematically indicate the location of gates G1, G2, and G3. We find that the regions near gate edges correspond to the regions of largest tilting $ϕ_{0}$ . (f) The simulated strain-induced $Δ_{HH-LH}$ for a line cut along the $x$ axis. The $Δ_{HH-LH}$ is not symmetric along the sample $x$ axis, since the lithography and confinement potential are not symmetric along the sample $x$ axis (see Fig. 3).
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