Anisotropic spin-blockade leakage current in a Ge hole double quantum dot

TL;DR

This work analyzes leakage in Pauli spin blockade for a Ge hole double quantum dot under an out-of-plane magnetic field, combining a $H=H_0+H_{\text{SO}}+H_{\text{Z}}+V_{\text{C}}$ Hamiltonian with parabolic confinement and $k^3$-Rashba SOC. Using Hund-Mulliken molecular orbitals and a Lindblad quantum kinetic equation, the authors map how the PSB leakage current depends on detuning $\delta$, interdot distance, dot ellipticity, and $B_z$, revealing a sharp maximum leakage at a calculable $\delta_{\text{max}}$ and a strong anisotropy as the elliptical dot is rotated. The key contributions are (i) an explicit link between leakage current and geometric tuneables (ellipse aspect ratio and orientation), (ii) a perturbative-analytic view of when leakage vanishes, and (iii) practical guidelines to optimize SOC probing and PSB readout fidelity in hole qubits, with potential extension to silicon-based platforms. Overall, the work provides a concrete framework tying spin–orbit coupling, dot geometry, and magnetic-field orientation to PSB leakage, enabling geometry-assisted control and SOC characterization in group-IV hole qubits.

Abstract

Group IV quantum dot hole spin systems, exhibiting strong spin-orbit coupling, provide platforms for various qubit architectures. The rapid advancement of solid-state technologies has significantly improved qubit quality, including the time scales characterizing electrical operation, relaxation, and dephasing. At this stage of development, understanding the relations between the underlying spin-orbit coupling and experimental parameters, such as quantum dot geometry and external electric and magnetic fields, has become a priority. Here we focus on a Ge hole double quantum dot in the Pauli spin blockade regime and present a complete analysis of the leakage current under an out-of-plane magnetic field. By considering a model of anisotropic in-plane confinement and $k^3$-Rashba spin-orbit coupling, we determine the behaviour of the leakage current as a function of detuning, magnetic field magnitude, interdot distance, and individual dot ellipticities. We identify regions in which the leakage current can be suppressed by quantum dot geometry designs. Most importantly, by rotating one of the quantum dots, we observe that the quantum dot shape induces a strongly anisotropic leakage current. These findings provide guidelines for probing the spin-orbit coupling, enhancing the signal-to-noise ratio, and improving the precision of Pauli spin blockade readout in hole qubit architectures.

Figures (7)

• Figure 1: A schematic view of a Ge double quantum dot system. a) The full double quantum device. The substrate includes a fully strain-relaxed SiGe layer at the bottom. The middle of the heterostructure consists of an epitaxially grown layer of strained germanium, hosting the hole qubit, and another layer of relaxed SiGe atop the Ge layer. Gates B1, P1, B2, P2, and B3 are used to confine two quantum dots, while gate B2 and T1 can control the inter-dot tunneling and detuning parameter $\delta$. b) A planar view of the double quantum dot system in the $xy$-plane. The two quantum dots are depicted as shaded solid circles. The left quantum dot, located at $x=-d_0$, is a circular dot with radius $R_{a,0}$, while the right quantum dot, located at $x=d_0$, is elliptical, with semi-minor axis of $R_{b,x}$ and semi-major axis of $R_{b,y}$. The two quantum dots are separated by a distance of $2d_0$, referred to as the interdot distance in the following text. c) A three-dimensional view of the double quantum dot potential function described in Eq. \ref{['Eq: DOD Potential']}. The ellipticity of the confinement potential, corresponding to b), is reflected in the contour projection on the $V(x,y)=0$ surface.
• Figure 2: The energy levels of $H$ as a function of the out-of-plane magnetic field at $\delta=0$. At $B_z$=0, the energy levels for the three (1,1)-triplet states are degenerated having the same energy. The $\ket{S_{(1,1)}}$ state is below the $\ket{T_0}$ due to the finite exchange interaction. As $B_z$ increase, both the orbit magnetic field term and the Zeeman term are activated, however, the orbital magnetic field terms only have magrinal effect and are further dimmed in the range of the magnetic field we considered. The Zeeman terms will create a splitting between $\ket{T_+}$ and $\ket{T_-}$ of magnetude $\varepsilon_{\text{Z}}$. The parameters used to generate this plots are as the following: $R_{a,0}$=15 nm, $e=0.8$, $\delta=0$ meV.
• Figure 3: The energy levels of $H$ as a function of the detuning $\delta$. These energy levels are obtained by assuming the left quantum dot is circular, with a radius of $R_{a,0}$=15 nm, and the right quantum dot is elliptical, with a semi-minor axis (along the x-axis) of $R_{b,x}$=15 nm and an aspect ratio of $e$=0.8. The two quantum dots are separated by 105 nm. The magnetic field is fixed at $B_z$=10 mT, with $g_L=6.5$ and $g_R=4.5$. The parameters used to generate plots in the following text are selected from Ref. Jirovec2022.
• Figure 4: The leakage currents as the function of the detuning $\delta$ and the magnitude of the out-of-plane magnetic field $B_z$. a) The leakage current as a function of the detuning $\delta$ in different magnetic fields. The solid blue curve is for $B_z$=15 mT case, where the dotted red curve is for $B_z$=10 mT case. b) The leakage current as a function of the magnetic field in different detunings. In both plots, the left quantum dot is circular with a radius of $R_{a,0}$=15 nm, and the right quantum dot is elliptical with a semi-minor axis (along the x-axis) of $R_{b,x}$=15 nm and an aspect ratio of $e$=0.8, the interdot distance is $2d_0$=105 nm.
• Figure 5: The leakage currents as the function of $d_0$. The interdot distance is $2d_0$. a) The leakage current as a function of $d_0$ in different detuning. b) The leakage current as a function of $d_0$ in different magnetic fields. In both plots, the left quantum dot is circular with a radius of $R_{a,0}$=15 nm, and the right quantum dot is elliptical with a semi-minor axis (along the x-axis) of $R_{b,x}$=15 nm and an aspect ratio of $e$=0.8.