Variable and Orbital-Dependent Spin-Orbit Field Orientations in an InSb Double Quantum Dot Characterized via Dispersive Gate Sensing

Lin Han, Michael Chan, Damaz de Jong, Christian Prosko, Ghada Badawy, Sasa Gazibegovic, Erik P.A.M. Bakkers, Leo P. Kouwenhoven, Filip K. Malinowski, and Wolfgang Pfaff

Phys. Rev. Applied 19, 014063 – Published 24 January 2023

Abstract

Utilizing dispersive gate sensing (DGS), we investigate the spin-orbit field ( $B_{SO}$ ) orientation in a many-electron double quantum dot (DQD) defined in an $In Sb$ nanowire. While characterizing the interdot tunnel couplings, we find the measured dispersive signal depends on the electron-charge occupancy, as well as on the amplitude and orientation of the external magnetic field. The dispersive signal is mostly insensitive to the external field orientation when a DQD is occupied by a total odd number of electrons. For a DQD occupied by a total even number of electrons, the dispersive signal is reduced when the finite external magnetic field aligns with the effective $B_{SO}$ orientation. This fact enables the identification of $B_{SO}$ orientations for different DQD electron occupancies. The $B_{SO}$ orientation varies drastically between charge transitions, and is generally neither perpendicular to the nanowire nor in the chip plane. Moreover, $B_{SO}$ is similar for pairs of transitions involving the same valence orbital, and varies between such pairs. Our work demonstrates the practicality of DGS in characterizing spin-orbit interactions in quantum dot systems, without requiring any current flow through the device.

Received 1 April 2022
Revised 26 October 2022
Accepted 13 December 2022

DOI:https://doi.org/10.1103/PhysRevApplied.19.014063

Physics Subject Headings (PhySH)

Metrology Quantum information with solid state qubits Quantum transport Rashba coupling

Double quantum dots III-V semiconductors Nanowires

Capacitance spectroscopy

Condensed Matter, Materials & Applied PhysicsQuantum Information, Science & Technology

Authors & Affiliations

Lin Han^1,*, Michael Chan ¹, Damaz de Jong¹, Christian Prosko ¹, Ghada Badawy², Sasa Gazibegovic², Erik P.A.M. Bakkers², Leo P. Kouwenhoven¹, Filip K. Malinowski^1,†, and Wolfgang Pfaff³

¹QuTech and Kavli Institute of Nanoscience, Delft University of Technology, Delft 2600 GA, Netherlands
²Department of Applied Physics, Eindhoven University of Technology, MB Eindhoven 5600, Netherlands
³Department of Physics and Frederick Seitz Materials Research Laboratory, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA

^*[email protected]
^†[email protected]

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Vol. 19, Iss. 1 — January 2023

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Images

Figure 1
(a) False-colored SEM image of the device and the circuit schematics for DGS. The barrier and plunger gates are indicated in pink and blue, respectively, while the metallic leads are indicated in green. The gray gate is electrostatically floating (unused in this work). (b) Charge-stability diagram of the DQD at zero magnetic field. The inset shows the maximum phase response in the $V_{LP}$ range marked by the dashed black rectangle. (c) The position of four neighboring ICTs along detuning axis, as a function of external field pointing in an arbitrary direction. Markers in (b) indicate the relative charge occupancy of the ICTs, although they do not correspond to the same ICTs as in (c).
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Figure 2
(a) The magnitude of the reflection signal as a function of probing frequency, as an example of the reference data. The black dots represents the raw data, while the blue curve shows the fitting result. (b) Parametric plot of the resonator reflection measurement in $I$ - $Q$ plane, and the corresponding fit result. (c) The reflected phase signal of the entire CSD. (d) The corresponding color maps of $C_{q}$ values.
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Figure 3
(a) Illustration of spherical coordinates with respect to the nanowire and the substrate. The top and bottom panel correspond to (d),(e), respectively. (b),(c) Schematic energy diagrams of a DQD for (b) $B ⊥ B_{SO}$ and (c) $B ∥ B_{SO}$ . The (avoided) crossing between the two lowest states are highlighted. $C_{q, \max} = 0$ when they cross, due to the flat shape of $| T_{+} (1, 1) ⟩$ . (d),(e) The extracted $C_{q, \max}$ values plotted on an external magnetic field angle map. Polar projection of the map is shown in (d) when $0^{\circ} \leq θ \leq 90^{\circ}$ , and (e) when $90^{\circ} \leq θ \leq 180^{\circ}$ . The blue crosses mark the characterized $\pm B_{SO}$ orientations, which is the center of the region where $C_{q, \max}$ is suppressed.
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Figure 4
Evolution of the ICT in external magnetic field $B$ for (a)–(c) $B ⊥ B_{SO}$ , and (d)–(f) $B ∥ B_{SO}$ . (a),(d) $C_{q}$ as a function of $B$ and $ε$ . The inset presents the numerical simulation. Black curves in both the main figure and the inset indicate the degeneracy point between $| S (2, 0) ⟩$ and $| T_{+} (1, 1) ⟩$ . (b),(e) Line cuts of (a),(d) for several magnitudes of $B$ . For clarity, they are separated in the $C_{q}$ axis by 200 aF. Black curves come from the simulation. (c),(f) $C_{q, \max}$ as a function of $B$ . Gray dots are extracted from a phenomenological Gaussian fit of the data. Black curves indicate the $C_{q, \max}$ taken from the insets of (a),(d).
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Figure 5
(a)–(p) The external magnetic field angle map of $C_{q, \max}$ ( $0^{\circ} \leq θ \leq 90^{\circ}$ , with $| B | = 50 mT$ ) for 16 neighboring ICTs, with their extracted $B_{SO}$ directions marked by colored crosses. These ICTs are labeled with the corresponding charge occupations ( $n_{L}^{'}, n_{R}^{'}$ ) with respect to even charge numbers ( $N_{L}^{'}, N_{R}^{'}$ ). (q) Summary of the extracted $B_{SO}$ directions for the even-occupied ICTs under study. Crosses in the same color correspond to the ICT pairs with the same valence orbital. Squares mark the ICTs when the other ICT from a pair is not measured.
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