Electrical operation of planar Ge hole spin qubits in an in-plane magnetic field

Hole spin qubits in group-IV semiconductors, especially Ge and Si, are actively investigated as platforms for ultrafast electrical spin manipulation thanks to their strong spin-orbit coupling. Nevertheless, the theoretical understanding of spin dynamics in these systems is in the early stages of development, particularly for in-plane magnetic fields as used in the vast majority of experiments. In this work, we present a comprehensive theory of spin physics in planar Ge hole quantum dots in an in-plane magnetic field, where the orbital terms play a dominant role in qubit physics, and provide a brief comparison with experimental measurements of the angular dependence of electrically driven spin resonance. We focus the theoretical analysis on electrical spin operation, phonon-induced relaxation, and the existence of coherence sweet spots. We find that the choice of magnetic field orientation makes a substantial difference for the properties of hole spin qubits. Specifically, we find that (i) EDSR for in-plane magnetic fields varies nonlinearly with the field strength and weaker than for perpendicular magnetic fields. (ii) The EDSR Rabi frequency is maximized when the a.c. electric field is aligned parallel to the magnetic field, and vanishes when the two are perpendicular. (iii) The orbital magnetic field terms make the in-plane g-factor strongly anisotropic in a squeezed dot, in excellent agreement with experimental measurements. (iv) Focusing on random telegraph noise, we show that the effect of noise in an in-plane magnetic field cannot be fully mitigated, as the orbital magnetic field terms expose the qubit to all components of the defect electric field. These findings will provide a guideline for experiments to design ultrafast, highly coherent hole spin qubits in Ge.