Coherence limit due to hyperfine interaction with nuclei in the barrier material of Si spin qubits

On the quest to understand and reduce environmental noise in Si spin qubits, hyperfine interactions between electron and nuclear spins impose a major challenge. Silicon is a promising host material because one can enhance the spin coherence time by removing spinful $^{29}$Si isotopes. As more experiments rely on isotopic purification of Si, the role of other spinful atoms in the device should be clarified. This is not a straightforward task, as the hyperfine interactions with atoms in the barrier layers are poorly understood. We utilize density functional theory to determine the hyperfine tensors of both Si and Ge in a crystalline epitaxial Si/SiGe quantum well as well as Si and O atoms in an amorphous Si/SiO$_2$ (MOS) interface structure. Based on these results, we estimate the dephasing time $T_2^*$ due to magnetic noise from the spin bath and show that the coherence is limited by interactions with non-Si barrier atoms to a few \textmu s in Si/SiGe (for non-purified Ge) and about 100\,\textmu s in Si-MOS. Expressing these numbers alternatively, in Si/SiGe the interactions with Ge dominate below 1000\,ppm of $^{29}$Si content, and, due to low natural concentration of the spinful oxygen isotopes, the interactions with oxygen in Si-MOS become significant only below 1\,ppm of $^{29}$Si content.