Acoustic phonons, spin-phonon coupling, and spin relaxation via the lattice reorientation mechanism in hexagonal germanium nanowires

Spin relaxation via electron-phonon interaction is an important decoherence mechanism for spin qubits. In this work, we study spin relaxation in hexagonal (2H) germanium, a direct-gap semiconductor showing great potential to combine highly coherent spin qubits with optical functionality. Focusing on electrostatically defined quantum dots in hexagonal germanium nanowires, we (i) identify geometries where spin qubit experiments are feasible, (ii) compute the nanowire phonon modes, and (iii) describe spin relaxation of hole spin qubits due to phonon-induced lattice reorientation, a direct spin-phonon coupling mechanism that is absent in cubic semiconductors typically used for spin qubits (GaAs, cubic Si, and cubic Ge). We obtain the spin relaxation time as a function of nanowire cross-section, quantum dot confinement length, and magnetic field. For realistic parameters, we find relaxation times above 10 ms, and reveal that the magnetic field direction maximizing the relaxation time depends on the qubit Larmor frequency. Our results facilitate the design of nanowire quantum dot experiments with long qubit relaxation times.