Electrically Tuneable Variability in Germanium Hole Spin Qubits

TL;DR

The paper tackles dot-to-dot variability in Ge hole-spin qubits caused by local electrostatics and strain by proposing in-plane squeezing of quantum-dot confinement to imprint a controlled anisotropy and enable on-demand $g$-tensor control. Using a Luttinger–Kohn–Bir–Pikus framework with both homogeneous and inhomogeneous strain and disorder, they compute the qubit $g$-tensor by projecting the magnetic-field Hamiltonian onto the ground-state manifold and analyze disorder ensembles. They find that squeezed QDs align the $g$-tensor along the long axis and significantly suppress both the magnitude and angular variability for selected in-plane magnetic fields, with unstrained Ge offering the strongest robustness; larger, smoother confinements further reduce variability. These results provide practical design guidelines for scalable, reproducible Ge-based spin-qubit processors, highlighting unstrained Ge squeezed dots as a particularly promising route for large-scale quantum computation.

Abstract

Hole spin qubits in planar germanium heterostructures are frontrunners for scalable semiconductor quantum computing. However, their current performance is mostly limited by large dot-to-dot variability that leads to uncontrolled qubit energies and random tilts in the spin quantization axis. Here, we propose a systematic and local method to engineer the spin qubit response by imprinting a controlled anisotropy in the quantum dot confinement, enabling on-demand electric g-tensor control. In particular, we find that both the quantum-dot size and asymmetry allow electrical tuning of the g-tensor and significantly suppress magnitude and angular variability of the spin response for selected magnetic field directions. We confirm this behavior by analyzing single-disorder realizations and statistical ensembles in state-of-the-art strained and unstrained germanium channels, showing that the latter provides an optimal path for $g$-tensor engineering. Our results provide practical design principles for on-demand control of the spin response and mitigating variability, paving the way towards large-scale germanium-based quantum computers.

Figures (10)

• Figure 1: Robust g-tensor engineering by squeezing. (a) Sketch of planar Ge heterostructures with circular and squeezed QDs, including interface disorder (orange dots). Circular QDs yield isotropic spin response, with symmetry randomly broken by electrostatic disorder and fluctuations in the harmonic confinement. In contrast, squeezing QDs pins the $g$-tensor along the long axis, making it more robust against disorder. (b)-(c) Simulated $g$-tensors and standard deviations footnoteFig1, for strained (b) and unstrained (c) Ge, comparing circular and squeezed QDs oriented in different directions. In (b), uncontrolled long-range strain fluctuations pin the circular QD $g$-tensors and are partly compensated by squeezing. In (c), these fields are absent enabling reliable $g$-tensor engineering. We used $\ell=40$ nm, $L_z=20$ nm, and long (short) in-plane lengths of $40$ nm ($20$ nm).
• Figure 2: Variability of circular QDs. (a)-(b)-(c) Standard deviation (a) and probability distribution functions of the relative $g$-factors $\delta g$ of circular QDs in strained Ge with $L_z=10$ nm (b) and $L_z=14$ nm (c). We compare these quantities to unstrained Ge in (d)-(e)-(f). The results in both cases are obtained by averaging over 200 disorder configurations. For circular QDs, the variability is generally lower at smoother confinements both in-plane and out-of-plane, although the trend especially in unstrained Ge is non-monotonous.
• Figure 3: Variability of squeezed QDs. (a) Simulated variability of the relative $g$-factor $\delta g$ for different $\textbf{B}$ field directions $\phi_B$ and (b) probability density function of the tilt angle $\delta\phi$ of $g$-tensors in strained Ge. We use $L_z=20$ nm, $L_x=50$ nm, and squeeze QDs in $y$-direction. Squeezed QDs have generally lower relative variability. (c)-(d) Variability in unstrained Ge with $L_z=14$ nm, where tighter squeezing strongly pins the $g$-tensor and globally reduces variability, especially when $\textbf{B}$ is aligned to the long-QD direction. We exclude circular QDs ($L_x=L_y=50$ nm) in unstrained Ge because of their arbitrary angular fluctuations, caused by the random breaking of the QD symmetry. All results are obtained averaging over 200 disorder realizations.
• Figure S1: Here we show how the strain varies in the $xy$-plane underneath the plunger gate, with radius ${L}_p = 60\ \rm nm$, for the different strain components used in our simulations.
• Figure S2: Inversion point in unstrained germanium. The figure presents the $g$-tensor of a quantum dot with lateral confinement lengths $L_x = 50\ \mathrm{nm}$ and $L_y = 20\ \mathrm{nm}$, evaluated for varying germanium well thicknesses $L_z$. For sufficiently large well thicknesses, the $g$-tensor is pinned along the $x$ direction. As the germanium well thickness is reduced, this pinning is inverted, and the principal axis of the $g$-tensor becomes aligned with the more strongly confined (squeezed) lateral direction of the quantum dot