[Paper Review] Anomalous zero-field splitting for hole spin qubits in Si and Ge quantum dots

Bence Hetényi, Stefano Bosco+1 more|arXiv (Cornell University)|May 5, 2022

Quantum and electron transport phenomena85 references12 citations

TL;DR

This paper identifies cubic spin-orbit interaction (SOI) as the origin of large zero-field splitting (ZFS) in hole spin qubits within Si and Ge quantum dots. Using analytical and numerical models, it shows that this ZFS—reaching µeV-scale energies—arises from momentum-cubic SOI, is tunable via electric fields, and dominates over other mechanisms like short-range Coulomb corrections. The findings explain recent experimental observations and have critical implications for quantum computing architectures based on tunnel-coupled quantum dots.

ABSTRACT

An anomalous energy splitting of spin triplet states at zero magnetic field has recently been measured in germanium quantum dots. This zero-field splitting could crucially alter the coupling between tunnel-coupled quantum dots, the basic building blocks of state-of-the-art spin-based quantum processors, with profound implications for semiconducting quantum computers. We develop an analytical model linking the zero-field splitting to spin-orbit interactions that are cubic in momentum. Such interactions naturally emerge in hole nanostructures, where they can also be tuned by external electric fields, and we find them to be particularly large in silicon and germanium, resulting in a significant zero-field splitting in the $\mu$eV range. We confirm our analytical theory by numerical simulations of different quantum dots, also including other possible sources of zero-field splitting. Our findings are applicable to a broad range of current architectures encoding spin qubits and provide a deeper understanding of these materials, paving the way towards the next generation of semiconducting quantum processors.

Motivation & Objective

• To identify the microscopic origin of anomalous zero-field splitting (ZFS) in hole spin qubits of Si and Ge quantum dots, as recently observed experimentally.
• To explain why ZFS is significantly larger in hole systems than in electron systems, particularly in the absence of external magnetic fields.
• To develop a general analytical and numerical framework that accurately predicts ZFS across diverse quantum dot architectures.
• To demonstrate that cubic spin-orbit interaction is the dominant source of ZFS, outperforming other mechanisms like short-range Coulomb corrections.
• To clarify the implications of ZFS for quantum computing, including its impact on two-qubit gates, qubit initialization, and the potential for encoding qubits at zero magnetic field.

Proposed method

• Developed an effective 1D Hamiltonian for hole quantum dots in nanowires, including linear and cubic spin-orbit coupling terms, using Schrieffer-Wolff perturbation theory.
• Derived a transformed Hamiltonian where linear SOI is eliminated via a momentum shift, isolating the cubic SOI as the primary source of ZFS through mixed center-of-mass and relative coordinate terms.
• Used a harmonic confinement potential with effective mass and Bohr radius to model quantum dot geometry, with Coulomb interaction approximated as a 1D potential with a short-range cutoff.
• Performed numerical simulations on both long and short quantum dots using basis states for center-of-mass and relative motion, validating results across different confinement regimes.
• Applied group theory to analyze triplet state degeneracy breaking under various point group symmetries (D4h, D2h, C2v), linking symmetry breaking to ZFS magnitude and direction.
• Compared contributions from cubic SOI, short-range Coulomb corrections, and other effects, showing that cubic SOI dominates by orders of magnitude.

Experimental results

Research questions

• RQ1 What is the microscopic origin of the large zero-field splitting observed in Ge quantum dots with hole occupation?
• RQ2 Why is the zero-field splitting significantly larger in hole-based systems compared to electron-based systems, especially in the absence of magnetic fields?
• RQ3 How does cubic spin-orbit interaction contribute to the lifting of triplet state degeneracy in hole quantum dots?
• RQ4 To what extent can the zero-field splitting be tuned or engineered using external electric fields or quantum dot geometry?
• RQ5 What is the relative contribution of cubic spin-orbit interaction versus other mechanisms (e.g., short-range Coulomb corrections) to the observed ZFS?

Key findings

• Cubic spin-orbit interaction, which is naturally large in hole systems due to strong heavy- and light-hole mixing, is the dominant source of zero-field splitting in Si and Ge quantum dots.
• The zero-field splitting reaches µeV-scale energies—orders of magnitude larger than alternative mechanisms such as short-range Coulomb corrections (estimated at a few neV).
• The splitting is tunable via external electric fields and can be engineered through quantum dot design, particularly in long, quasi-1D nanostructures.
• The direction of the exchange anisotropy is directly linked to the direction of the spin-orbit interaction, consistent with experimental observations under small magnetic fields.
• Numerical simulations confirm the analytical predictions across different quantum dot geometries and confinement regimes, showing good agreement between long- and short-dot models.
• Symmetry breaking (e.g., by electric fields or strain) lifts the triplet degeneracy in a way that matches the predicted behavior from group theory, with D4h, D2h, and C2v point groups showing distinct splitting patterns.