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Spatial uniformity of g-tensor and spin-orbit interaction in germanium hole spin qubits

Inga Seidler, Bence Hetényi, Lisa Sommer, Leonardo Massai, Konstantinos Tsoukalas, Eoin G. Kelly, Alexei Orekhov, Michele Aldeghi, Stephen W. Bedell, Stephan Paredes, Felix J. Schupp, Matthias Mergenthaler, Gian Salis, Andreas Fuhrer, Patrick Harvey-Collard

TL;DR

This work maps the full 3D $g$-tensor landscape of Ge/SiGe hole spin qubits in two multi-qubit devices with a $Y$ geometry, revealing average out-of-plane tilts of ~$1.1^{\circ}$ and strong, neighbor-correlated in-plane $g$-tensor anisotropy. By combining three-dimensional $g$-tensor tomography with spin-flip tunneling measurements, the authors extract a Dresselhaus-like spin-orbit field, challenging simple Rashba pictures and suggesting a global mechanism beyond local confinement or strain. The observed long-range correlations in both $g$-tensors and spin-orbit interactions have major implications for scaling Ge hole-qubit architectures, particularly for all-electrical control and uniform qubit addressing across large arrays. The results emphasize the need to account for global or growth-related effects when engineering and operating large Ge-based quantum dot arrays. Mathematical expressions are denoted in $...$ in the text, and all key quantities are described within a 3D tensor framework for the $g$-tensor and a vector model for spin-orbit coupling.

Abstract

Holes in Ge/SiGe heterostructures are now a leading platform for semiconductor spin qubits, thanks to the high confinement quality, two-dimensional arrays, high tunability, and larger gate structure dimensions. One limiting factor for the operation of large arrays of qubits is the considerable variation in qubit frequencies or properties resulting from the strongly anisotropic $g$-tensor. We study the $g$-tensors of six and seven qubits in an array with a Y geometry across two devices. We report a mean distribution of the tilts of the $g$-tensor's out-of-plane principal axis of around $1.1 °$, where nearby quantum dots are more likely to have a similar tilt. Independently of this tilt, and unlike simple theoretical predictions, we find a strong in-plane $g$-tensor anisotropy with strong correlations between neighboring quantum dots. Additionally, in one device where the principal axes of all g-tensors are aligned along the [100] crystal direction, we extract the spin-flip tunneling vector from adjacent dot pairs and find a pattern that is consistent with a uniform Dresselhaus-like spin-orbit field. The Y arrangement of the gate layout and quantum dots allows us to rule out local factors like electrostatic confinement shape or local strain as the origin of the preferential direction. Our results reveal long-range correlations in the spin-orbit interaction and $g$-tensors that were not previously predicted or observed, and could prove critical to reliably understand $g$-tensors in germanium quantum dots.

Spatial uniformity of g-tensor and spin-orbit interaction in germanium hole spin qubits

TL;DR

This work maps the full 3D -tensor landscape of Ge/SiGe hole spin qubits in two multi-qubit devices with a geometry, revealing average out-of-plane tilts of ~ and strong, neighbor-correlated in-plane -tensor anisotropy. By combining three-dimensional -tensor tomography with spin-flip tunneling measurements, the authors extract a Dresselhaus-like spin-orbit field, challenging simple Rashba pictures and suggesting a global mechanism beyond local confinement or strain. The observed long-range correlations in both -tensors and spin-orbit interactions have major implications for scaling Ge hole-qubit architectures, particularly for all-electrical control and uniform qubit addressing across large arrays. The results emphasize the need to account for global or growth-related effects when engineering and operating large Ge-based quantum dot arrays. Mathematical expressions are denoted in in the text, and all key quantities are described within a 3D tensor framework for the -tensor and a vector model for spin-orbit coupling.

Abstract

Holes in Ge/SiGe heterostructures are now a leading platform for semiconductor spin qubits, thanks to the high confinement quality, two-dimensional arrays, high tunability, and larger gate structure dimensions. One limiting factor for the operation of large arrays of qubits is the considerable variation in qubit frequencies or properties resulting from the strongly anisotropic -tensor. We study the -tensors of six and seven qubits in an array with a Y geometry across two devices. We report a mean distribution of the tilts of the -tensor's out-of-plane principal axis of around , where nearby quantum dots are more likely to have a similar tilt. Independently of this tilt, and unlike simple theoretical predictions, we find a strong in-plane -tensor anisotropy with strong correlations between neighboring quantum dots. Additionally, in one device where the principal axes of all g-tensors are aligned along the [100] crystal direction, we extract the spin-flip tunneling vector from adjacent dot pairs and find a pattern that is consistent with a uniform Dresselhaus-like spin-orbit field. The Y arrangement of the gate layout and quantum dots allows us to rule out local factors like electrostatic confinement shape or local strain as the origin of the preferential direction. Our results reveal long-range correlations in the spin-orbit interaction and -tensors that were not previously predicted or observed, and could prove critical to reliably understand -tensors in germanium quantum dots.

Paper Structure

This paper contains 11 sections, 18 equations, 18 figures.

Table of Contents

  1. Introduction
  2. Results
  3. Discussion
  4. Conclusion
  5. Device tuning
  6. Remanent magnetic field calibration
  7. g-tensor fitting
  8. Sample tilt evaluation and impact
  9. g-tensor tunability
  10. Spin-orbit vector fitting
  11. Probability normalization

Figures (18)

  • Figure 1: Device geometry and $g$-factor measurements. (a) Colored scanning electron micrograph of a Y junction device. The QDs used for qubit formation are labeled QD$i$, and those used as charge sensors are labeled S$i$. (b-c) Qubit frequencies of QD1 and QD2 for varying in-plane magnetic field angle $\phi_B$ (b) and out-of-plane angle $\theta_B$ (c). The sweeps are performed at a constant magnetic field strength of $100 \ \text{mT}$.
  • Figure 2: $g$-tensor orientations. (a) Schematic of a $g$-tensor, with the lab frame in black, the principal axis frame in green, and a unit vector $\hat{\boldsymbol{z}}'$ along the $z'$ axis representing the tilt (red). The $zyz$ Euler rotation angles $\zeta$, $\theta$ and $\phi$ describe the transformation between the two frames. (b) In-plane $g$-tensor cross-sections of $g^*$ in the lab frame (black) and in the individual tilt frame of each $g$-tensor (color) for device A. The individual tilt frames are tilted such that the new $z$ axis aligns with every individual $g$-tensor's $z'$ axis without including additional rotations (SI Sec. \ref{['sec:tilt']}). The cross-section plots are arranged according to the device layout. See SI Fig. \ref{['supfig:gs_A']} and Fig. \ref{['supfig:gs_B']} for numerical values. (c) Tilt visualization for device A by projecting of the $\hat{\boldsymbol{z}}'$ unit vectors on the $x{-}y$ plane with matching colors. The solid inverted triangle represents the projection of the average tilt. (c-d) In-plane $g^*$ (d) and tilts (e) for device B.
  • Figure 3: Correlations between $g$-tensors. (a) Correlation between $g_\mathrm{in1}$ and $g_\mathrm{in2}$, revealing that they are likely to increase together. (b) Correlation between the tilt angle $\theta$ and $g_\mathrm{in2}$, revealing that larger tilt angles are associated with larger $g_\mathrm{in2}$. Here, $\theta$ is calculated in the average tilt frame (i.e., not the lab frame). (c-d) Relative change in $g_\mathrm{in2}$ (c) and absolute change in orientation angle $\zeta + \phi$ ((d); Eq. \ref{['eq:oriangle']}) as a function of distance in units of QD pitch. The mean ($\blacktriangle$) and root-mean-square ($\blacktriangledown$) values are plotted with a slight offset alongside, revealing a trend that closer QDs are more likely to have similar $g_\mathrm{in2}$ and in-plane orientation $\zeta + \phi$.
  • Figure 4: Spin-orbit fields in device A. (a) Energy diagram of two spins in a DQD. The ST$_{-}$ anti-crossing with size $2\Delta_{ST}$ is probed with detuning ramps indicated with the grey arrows. (b) Schematic of the applied pulse sequence for fixed $t_\mathrm{ramp\, in}$ and variable $t_\mathrm{ramp\, out}$. (c) Measured and simulated magnetic field angle dependence of the return probability of a blocked spin state for a tunneling between QD1 and QD2 with $t_\mathrm{ramp\, in}=1\ \upmu\text{s}$ (below: QD5 and QD4, QD7 and QD6, with $t_\mathrm{ramp\, in}=30\ \upmu\text{s}$) and variable $t_\mathrm{ramp\, out}$ with an external magnetic field of $7\ \text{mT}$ applied in-plane. (d-f) Fitted spin-orbit vectors in momentum space assuming a momentum aligned with the design DQD axis (color according to the referenced DQD). For comparison a linear Rashba-like (d), cubic Rashba-like (e) and Dresselhaus-like (f) spin-orbit field is plotted (black).
  • Figure S1: Charge stability diagrams for the outer (a-c) and inner (d-f) DQDs with the charge occupation marked. The occupation label $(n_i,n_j)$ corresponds to axis labels $\varepsilon_{ij}$ and $u_{ij}$ for DQD $i,j$. The inner loop of the measurement is the detuning voltage sweep ($\varepsilon_{ij}$), which is swept from negative to positive voltages for measurements a-c and reversed for measurements d-f.
  • ...and 13 more figures