Strong spin-orbit interaction and $g$-factor renormalization of hole spins in Ge/Si nanowire quantum dots

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Strong spin-orbit interaction and $g$ -factor renormalization of hole spins in Ge/Si nanowire quantum dots

F. N. M. Froning, M. J. Rančić, B. Hetényi, S. Bosco, M. K. Rehmann, A. Li, E. P. A. M. Bakkers, F. A. Zwanenburg, D. Loss, D. M. Zumbühl, and F. R. Braakman

Phys. Rev. Research 3, 013081 – Published 26 January 2021

Abstract

The spin-orbit interaction lies at the heart of quantum computation with spin qubits, research on topologically nontrivial states, and various applications in spintronics. Hole spins in Ge/Si core/shell nanowires experience a spin-orbit interaction that has been predicted to be both strong and electrically tunable, making them a particularly promising platform for research in these fields. We experimentally determine the strength of spin-orbit interaction of hole spins confined to a double quantum dot in a Ge/Si nanowire by measuring spin-mixing transitions inside a regime of spin-blockaded transport. We find a remarkably short spin-orbit length of $\sim 65 nm$ , comparable to the quantum dot length and the interdot distance. We additionally observe a large orbital effect of the applied magnetic field on the hole states, resulting in a large magnetic field dependence of the spin-mixing transition energies. Strikingly, together with these orbital effects, the strong spin-orbit interaction causes a significant enhancement of the $g$ factor with magnetic field. The large spin-orbit interaction strength demonstrated is consistent with the predicted direct Rashba spin-orbit interaction in this material system and is expected to enable ultrafast Rabi oscillations of spin qubits and efficient qubit-qubit interactions, as well as provide a platform suitable for studying Majorana zero modes.

Received 17 July 2020
Revised 9 November 2020
Accepted 24 December 2020
Corrected 19 April 2021

DOI:https://doi.org/10.1103/PhysRevResearch.3.013081

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)

Quantum computation Spin blockade Spin-orbit coupling

Quantum Information, Science & TechnologyCondensed Matter, Materials & Applied Physics

Corrections

19 April 2021

Correction: Reference [56] has been corrected to reflect that the Supplemental Material now includes raw data files and a plotting script used for the work.

Authors & Affiliations

F. N. M. Froning^1,*, M. J. Rančić^1,2,*, B. Hetényi ¹, S. Bosco ¹, M. K. Rehmann¹, A. Li³, E. P. A. M. Bakkers³, F. A. Zwanenburg⁴, D. Loss¹, D. M. Zumbühl^1,†, and F. R. Braakman ^1,‡

¹Department of Physics, University of Basel, Klingelbergstrasse 82, 4056 Basel, Switzerland
²Total S.A., Nano-INNOV, Bât.861 8, Boulevard Thomas Gobert, 91120 Palaiseau, France
³Department of Applied Physics, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands
⁴NanoElectronics Group, MESA + Institute for Nanotechnology, University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands

^*These authors contributed equally to this work.
^†Corresponding author: dominik.zumbuhl@unibas.chch
^‡Corresponding author: floris.braakman@unibas.ch

Article Text

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Supplemental Material

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References

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Vol. 3, Iss. 1 — January - March 2021

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Images

Figure 1
Device and Pauli spin blockade. (a) False-colour scanning elecron micrograph of the device, used for all the measurements of this work. The finger gates $g_{1-5}$ (red: barrier gates, green: plunger gates) are biased with positive voltages $V_{g1-5}$ in order to create a double quantum dot in the Ge/Si core/shell nanowire (yellow). The source (S) and drain (D) contacts are defined on either side of the nanowire. Dashed ellipses indicate the approximate locations of the two quantum dots. (b) Schematic illustration of Pauli spin blockade, with zero magnetic field. When the double dot is occupied by holes in a triplet (1,1) state, the current is blocked until mixing with a singlet state takes place. The double dot detuning is indicated by $ɛ$ . (c) Bias triangles taken at $V_{SD} = 2 mV$ showing signatures of Pauli spin blockade, through a suppression of current, in the area delineated by the dashed white lines. The blue arrow indicates the direction of the detuning axis. (d) Current as a function of detuning, swept along the arrow in (c), without (red) and with (green) applied magnetic field.
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Figure 2
Measured leakage current as a function of magnetic field for detunings covering the entire bias triangle, as shown by the arrow in Fig. 1. The dashed white lines delineate the spin-blockaded region also shown in Fig. 1. Here, $V_{g3} = 3820 mV$ . Dotted green curves are guides to the eye, indicating $ɛ_{-} (B)$ and $ɛ_{+} (B)$ .
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Figure 3
Level diagram and magnetic-field dependencies. (a) Double dot energy level diagrams for different values of the magnetic field. For $B = 0 T$ , the spin-conserving tunnel coupling $t_{c}$ is maximum and there is no singlet-triplet mixing due to spin-orbit interaction. For large enough magnetic fields ( $B > \tilde{B}$ ), avoided crossings (highlighted by dashed circles) appear when the triplet (1,1) states cross a singlet state with (0,2) component, corresponding to spin-flip tunneling due to spin-orbit interaction. The size of all avoided crossings becomes smaller with increasing magnetic field, as can be understood from (c) and Eq. (4). Moreover, due to the magnetic field dependence of the addition energy $U$ (see (b)), as well as the Zeeman energy, all avoided crossings move to higher detuning with magnetic field. Parameters used to plot the diagrams were extracted from the data set shown in Fig. 2, using the model described in the text. (b) Calculated magnetic field dependence of the addition energy $U$ [See Eq. (B10) of Appendix pp2]. (Inset) Schematic illustration of the effect of increasing magnetic field $B$ on dot size and separation leading to the observed changes in $U$ , $t_{c}$ and $g$ . Quantities change qualitatively with $B$ as indicated by the arrows. (c) Calculated magnetic field dependence of the spin-conserving tunnel coupling $t_{c}$ [see Eq. (B5a) of Appendix pp2]. (d) Calculated magnetic field dependence of the $g$ factor [see Eq. (2)]. For the plots in (b)–(d), the relevant parameters correspond to those of the measurement of Fig. 2.
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Figure 4
Spectroscopy measurements and modeling. [(a)–(c)] Measured leakage current as a function of magnetic field and detuning $ɛ < ɛ_{Δ}$ , for $V_{g3} = 3820$ , 3830, and $3840 mV$ . The green curves are fits of each data set to Eq. (1), with (solid) and without (dashed) taking into account $g$ -factor renormalization with magnetic field. [(d)–(f)] Simulated leakage current as a function of magnetic field and detuning. Here, we used the model discussed in Sections 6, 7, 8 of the main text, with relevant parameters determined from fits of the data shown in (a)–(c). The green curves are identical to the curves in (a)–(c).
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Figure 5
Measured leakage current as a function of detuning, for $V_{g3} = 3820 mV$ and $B = - 3.45 T$ . The black curve is a fit of Eq. (5) to the data.
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