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

TL;DR

This work analyzes spin relaxation of hole-spin qubits in electrostatically defined quantum dots within hexagonal Ge nanowires, focusing on a lattice reorientation spin-phonon coupling unique to hexagonal symmetry. The authors develop a continuum-elasticity model to compute acoustic phonon modes (analytic long-wavelength and numerical validation) and derive analytical expressions for spin-relaxation rates driven by g-tensor rotations, revealing mode-specific and field-direction dependent behavior. They identify geometric constraints that enable conventional quantum-dot spin qubits, compute relaxation times, and demonstrate sweet-spot directions that can suppress decoherence, with relaxation times exceeding $10$ ms and potential qubit lifetimes above $1$ s under optimal conditions. These results provide design principles for hex-Ge nanowire quantum devices and motivate further microscopic studies of spin-phonon coupling in anisotropic hexagonal materials.

Abstract

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 novel 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, 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.

Figures (5)

• Figure 1: The geometry of the rectangular cross-section hexagonal germanium nanowire compared to the crystal orientation. (a) The unit cell of hexagonal germanium. The $c$ axis is the sixfold screw axis of the space group of the crystal. (b) The nanowire geometry. The cross-section is rectangular and the nanowire is infinite in the $z$ direction, in which direction the confinement is assumed to be parabolic with $\ell_\textrm{d}$ confinement length. The growth direction of the nanowire is perpendicular to the $c$ axis of the crystal.
• Figure 2: Orbital level spacing frequencies for a quantum dot hosted in a rectangular cross-section hexagonal germanium nanowire. The $x$ and $z$ axes show the width of the nanowire, the y axis shows the width of the harmonic confinement in the longitudinal direction. Green surface corresponds to $E_\mathrm{orb}/h = 100\, \mathrm{GHz}$ orbital splitting, the red to the 20 GHz ($\approx1$ K thermal broadening) orbital splitting. The first excited orbital state needs to be well separated from the ground state compared to the thermal broadening ($k_\textrm{B} T$) and the qubit energy splitting ($\hbar\omega_\textrm{q}$). Inside the green surface ($E_\textrm{orb}/h > 100$ GHz), we expect the usual spin-qubit functionality of the system.
• Figure 3: Shape of the phonon modes in a hexagonal germanium nanowire. Four gapless phonon modes of the hexagonal nanowire: (a) longitudinal, (b) torsional, (c) $x$ directional flexural and (d) $y$ directional flexural.
• Figure 4: Dispersion relation of phonons in a hexagonal germanium nanowire with square cross section. Width of the nanowire is set to be $a = b = 30$ nm. Thin solid lines: analytical, approximate dispersion relations obtained in the long-wavelength limit for longitudinal (purple), torsional (blue) and flexural (red) modes. Circles (forming thick solid lines): numerical results. The raw data and the source code corresponding to this figure can be found in a dedicated GitHub repository gitrepo.
• Figure 5: Relaxation rate of a hole spin qubit in a hexagonal germanium nanowire due to the lattice reorientation mechanism. (a) Angle parametrization of the magnetic field with respect to the nanowire. The orientation of the $c$ axis of the crystal points in the $y$ direction as it is in Fig. \ref{['fig:nw_geom']}. (b) Analytical results (dashed) for the spin qubit relaxation rate, based on Eq. \ref{['eq:gammaT']} and Eq. \ref{['eq:gammaF']}, for three special magnetic field directions. Numerical results (solid) obtained by evaluating Eq. \ref{['eq:gamma']} using the numerical results for phonons from Sec. \ref{['subsec:numerical_phonons']}. Numerical results exhibit van Hove singularities for $f_\textrm{L} \gtrsim 45$ GHz. The power-law dependence shows that for $B\,||\,z$ the torsional and for $B\,||\,x$ the flexural modes are decoupled from the qubit. Vertical dashed lines for the frequencies plotted in the d-e) panels are marked with triangle, star and cross, respectively. (c) Relaxation sweet spot as the function of the Larmor frequency. The sweet spot is in the $zy$ plane, along the magnetic field angle $\theta$ shown in the graph. Above ca. 1 GHz, relaxation sweet spot is at the $y$ direction, due to the large $g$ factor in that direction. Below ca. 1 GHz, the flexural mode contribution becomes negligible besides the torsional mode, therefore the sweet spot approaches the $z$ axis, where the torsional mode decouples from the qubit. [(d) and (e)] Relaxation rate dependence on the magnetic field orientation for different qubit frequencies. Markers indicate sweet spot. In (f), the sweet spot is in the $y$ direction, therefore there is no extra purple marker. As it is seen in panel (d), the sweet spot points close to the $z$ axis for $f_\textrm{L} = 0.1$ GHz. By carefully choosing the orientation of the magnetic field, the relaxation time can be kept above 1 second. The source code used to generate the (b)-(f) panels can be found in a dedicated GitHub repository gitrepo.