Hole Spin in Direct Bandgap Germanium-Tin Quantum Dot

Germanium (Ge) has emerged as a contender for scalable solid-state spin qubits. This interest stems from the numerous attractive properties of hole spin in Ge low-dimensional systems and their compatibility with the standards of silicon processing. Herein, we show that the controlled incorporation of Sn into the Ge lattice enables hole spin quantum dots that retain the same advantages as those made of Ge while also providing bandgap directness. The latter is essential for a more efficient interaction with light, a key feature in the implementation of photon-spin interfaces and quantum memories. We first map the material properties for a range of Ge$_{1-x}$Sn$_x$ planar heterostructures to identify the optimal conditions to simultaneously achieve hole spin confinement and bandgap directness. Although compressive strain is necessary for heavy hole confinement, we estimate that an additional 4.5 at.% of Sn is needed for every 1% increase in the absolute value of compressive strain to preserve the direct bandgap. However, a high compressive strain is found to be detrimental to the Rashba coupling. Moreover, a theoretical framework is derived to evaluate the dipole moment $d$ and the relaxation rate $\Gamma$ of electric dipole spin resonance quantum dot devices. We compare the perturbative and effective values of $d$ with the values obtained from the full 3D Hamiltonian. We find $d$ to be around 1 and 0.01 e pm for the out-of-plane and in-plane configurations, respectively, and $\Gamma\propto B^5$, eventually becoming $\propto B^7$ in the out-of-plane configuration.