Probing details of spin-orbit coupling through Pauli spin blockade

Jørgen Holme Qvist and Jeroen Danon

Phys. Rev. B 106, 235312 – Published 19 December 2022

Abstract

Spin-orbit interaction (SOI) plays a fundamental role in many low-dimensional semiconductor and hybrid quantum devices. In the rapidly evolving field of semiconductor spin qubits, SOI is an essential ingredient that can allow for ultrafast qubit control. The exact manifestation of SOI in a given device is, however, often both hard to predict theoretically and probe experimentally. Here, we develop a detailed theoretical connection between the leakage current through a double quantum dot in Pauli spin blockade and the underlying SOI in the system. We present a general analytic expression for the leakage current, which allows to connect experimentally observable features to both the magnitude and orientation of an effective spin-orbit field acting on the moving carriers. Motivated by the large recent interest in hole-based quantum devices, we further zoom in on the case of Pauli blockade of hole spins, assuming a strong transverse confinement potential. In this limit we also find an analytic expression for the current at low external magnetic field, that includes the effect of hyperfine coupling of the hole spins to randomly fluctuating nuclear spin baths. This result can be used to extract information about both hyperfine and spin-orbit coupling parameters for hole spins in devices with a significant fraction of nonzero nuclear spins.

Received 2 May 2022
Revised 28 November 2022
Accepted 13 December 2022

DOI:https://doi.org/10.1103/PhysRevB.106.235312

Physics Subject Headings (PhySH)

Magnetic interactions Quantum computation Quantum information with solid state qubits Spin dynamics Spin-orbit coupling

Quantum dots

Master equation

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Jørgen Holme Qvist and Jeroen Danon

Center for Quantum Spintronics, Department of Physics, Norwegian University of Science and Technology, NO-7491 Trondheim, Norway

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Issue

Vol. 106, Iss. 23 — 15 December 2022

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Images

Figure 1
Illustration of the two tunnel-coupled quantum dots connected to two leads, showing the orientation of the different fields: the spin-orbit vector is assumed to be pointing along $\hat{z}$ , whereas the Zeeman fields $E_{Z, i}$ and Overhauser fields $K_{i}$ are arbitrary.
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Figure 2
(a) For two Zeeman fields with the same orientation $b = B / B$ but different magnitudes the current only vanishes when the two fields are oriented along the spin-orbit vector marked in red. (b) Tuning the Zeeman fields away from the spin-orbit vector, the current vanishes when the two fields have the same latitude but a difference in longitude of $δ ϕ = 2 η$ , which happens along the red dashed lines.
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Figure 3
(a) Calculated current as a function of a uniformly applied external field $B_{ext}$ , assuming two different $g$ tensors on the two dots. We used $B_{L} = {0.76, 0.32, 0.34} B_{ext}$ and $B_{R} = {1, 0, 0} B_{ext}$ with $Γ_{s} = 0.1 μ eV$ . The blue line shows the case with no nuclear spins present, and the red line shows how adding two small random nuclear fields $K_{L, R}$ , drawn from a normal distribution with an r.m.s. value of $0.1 μ eV$ , drastically changes the behavior of the current at small fields. (b) Current as a function of $ϕ_{R}$ with $B_{L} = 0.9, B_{R} = 1, θ_{L} = θ_{R} = 3 π / 8$ , and $ϕ_{L} = 0$ . In the absence of nuclear fields (blue line) the current vanishes when the relative azimuthal angle $δ ϕ$ of two fields of different magnitude is equal to $2 η$ . After averaging the current over random nuclear fields (red line) with the same distribution as used in (a) the current still has its minimum at $δ ϕ = 2 η$ .
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Figure 4
Illustration of the orientation of the fields assumed for the analytical derivation in Sec. 4b: The spin-orbit vector $t_{SO}$ and both nuclear fields $K_{L, R}$ are assumed to be pointing along $\hat{z}$ . The external Zeeman field $B_{ext}$ is equal on the two dots, but can point in any direction.
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Figure 5
The current as given by Eq. (10), as a function of the magnitude of the applied field $B_{ext}$ . The four plots have an increasing magnitude of spin-orbit interaction: (a) $η = 0$ , (b) $η = 0.02$ , (c) $η = 0.1$ , and (d) $η = 0.5$ . In each plot the four traces correspond to four different orientations of the applied field, the corresponding polar angles of $B_{ext}$ are indicated in (a) (same colors represent same orientations in all plots).
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Figure 6
The current found from numerical integration of Eq. (9) where the spin-orbit vector $t_{SO}$ is now in plane, i.e., perpendicular to the nuclear fields. We again plot the current as a function of the magnitude of the applied field $B_{ext}$ and for different strengths of spin-orbit interaction: (a) $η = 0$ , (b) $η = 0.02$ , (c) $η = 0.1$ , and (d) $η = 0.5$ . The color coding is the same as in Fig. 5, where $θ$ again gives the angle between $B_{ext}$ and $K_{L, R}$ , i.e., the out-of-plane direction. The angle $θ = π / 2$ corresponds in this case to $B_{ext} ∥ t_{SO}$ . We further used $Γ_{s} = 15 μ eV$ and $K = 0.1 μ eV$ .
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