[Paper Review] Effective out-of-plane g-factor in strained-Ge/SiGe quantum dots

Andrew J. Miller, Mitchell Brickson+7 more|arXiv (Cornell University)|Feb 2, 2021

Semiconductor Quantum Structures and Devices4 citations

TL;DR

This study measures the out-of-plane g-factor in single-hole Ge/SiGe quantum dots for the first time, revealing a value of 15.7(22) with extreme anisotropy (>50) relative to the in-plane g-factor (~0.3). Using magnetospectroscopy and Luttinger Hamiltonian simulations, it demonstrates strong spin-orbit coupling and tunable g-factors via charge state and magnetic field, enabling low-field spin readout in quantum computing platforms.

ABSTRACT

Recently, lithographic quantum dots in strained-Ge/SiGe have become a promising candidate for quantum computation, with a remarkably quick progression from demonstration of a quantum dot to qubit logic demonstrations. Here we present a measurement of the out-of-plane $g$-factor for single-hole quantum dots in this material. As this is a single-hole measurement, this is the first experimental result that avoids the strong orbital effects present in the out-of-plane configuration. In addition to verifying the expected $g$-factor anisotropy between in-plane and out-of-plane magnetic ($B$)-fields, variations in the $g$-factor dependent on the occupation of the quantum dot are observed. These results are in good agreement with calculations of the $g$-factor using the heavy- and light-hole spaces of the Luttinger Hamiltonian, especially the first two holes, showing a strong spin-orbit coupling and suggesting dramatic $g$-factor tunability through both the $B$-field and the charge state.

Motivation & Objective

• To measure the out-of-plane g-factor in single-hole Ge/SiGe quantum dots, avoiding orbital effects present in multi-hole systems.
• To verify g-factor anisotropy between in-plane and out-of-plane magnetic fields in a single-hole regime.
• To investigate how the g-factor varies with hole occupancy and magnetic field orientation.
• To compare experimental results with Luttinger Hamiltonian simulations to understand spin-orbit coupling and orbital effects.

Proposed method

• Conducted magnetospectroscopy with out-of-plane magnetic fields up to 3 T in a dilution refrigerator at 30 mK.
• Used a lithographic quantum dot device with dual dots, where the lower dot served as a charge sensor and the upper dot was tuned to the single-hole regime.
• Measured gate coupling (α = 75.4(35) mV/μm) and extracted g-factors from energy level slopes in the 0–0.2 T range.
• Performed theoretical simulations using the Luttinger Hamiltonian with strain from the Bir-Pikus model, including realistic quantum dot potential.
• Calculated theoretical g-factors from Zeeman splitting (ΔE/μB B) at 0.2 T and compared with experiment.
• Accounted for orbital effects and spin-orbit coupling by analyzing the sign and magnitude of energy level slopes with increasing B-field.

Experimental results

Research questions

• RQ1 What is the value of the out-of-plane g-factor in a single-hole Ge/SiGe quantum dot, and how does it compare to the in-plane g-factor?
• RQ2 How does the g-factor vary with hole occupancy in the quantum dot under an out-of-plane magnetic field?
• RQ3 To what extent do orbital effects obscure the intrinsic g-factor in the out-of-plane configuration for p-like orbitals?
• RQ4 Can Luttinger Hamiltonian simulations accurately reproduce the observed g-factor and its dependence on occupation and B-field?
• RQ5 What is the role of strain and material parameters (e.g., Luttinger parameters) in determining the g-factor anisotropy?

Key findings

• The measured out-of-plane g-factor for a single hole is 15.7(22), significantly higher than the in-plane g-factor of ~0.3, resulting in a g-factor anisotropy exceeding 50.
• For hole occupancies above two, the apparent g-factor shows large variations due to dominant orbital effects, particularly in p-like orbitals.
• Theoretical simulations using the Luttinger Hamiltonian reproduce the experimental g-factor for the first two hole states with good agreement, yielding a predicted g-factor of 21.25 at 0.2 T.
• The simulations correctly predict the sign and relative magnitude of energy level slopes but overestimate the magnitude for p-like states, suggesting many-body effects may play a role.
• The orbital angular momentum alignment with the B-field causes energy level curvature and sign reversals in slopes, explaining deviations from pure Zeeman splitting.
• The results confirm that strong spin-orbit coupling enables dramatic g-factor tunability via both charge state and magnetic field direction, supporting low-field spin readout in quantum computing.