Effects of the spin-orbit coupling and confinement geometry on germanium hole spin qubits

Hole spin qubits are among the most promising platforms for scalable quantum computing in semiconductor systems, offering ultrafast and fully electrical gate operations enabled by strong spin-orbit coupling. Recent progress in quantum confinement and strain engineering of germanium heterostructures has accelerated the development of high-performance hole spin qubit devices. However, the complex valence band structure that underlies the spin-orbit interaction in these systems presents significant theoretical challenges. Most theoretical studies of germanium holes rely on numerical solutions of the Luttinger-Kohn Hamiltonian. While this approach yields highly accurate results, it often lacks the intuitive, analytical insight necessary to guide the design and optimization of qubit devices. In this work, we develop an analytical model that captures key physical properties of germanium hole systems, particularly those arising from the interplay between spin-orbit coupling and quantum dot confinement geometry. We consider a hole spin qubit confined in an anisotropic harmonic potential subjected to in-plane electric and magnetic fields. By incorporating an effective Rashba spin-orbit interaction—originating from broken structural inversion symmetry at the quantum well interface—we derive an effective Hamiltonian for the qubit subspace. This leads to compact analytical expressions for important qubit parameters, such as the effective g-factor and Rabi frequency. Our results provide clear explanations for the observed dependence of the effective g-factor on the direction of the applied magnetic field and the symmetry of the quantum dot potential, as seen in both experiments and numerical simulations. This analytical framework offers a more intuitive understanding of the underlying physics and serves as a practical tool to optimize qubit performance through control of gate-induced confinement and magnetic field orientation.