Measurements and atomistic theory of electron $g$ factor anisotropy for phosphorus donors in strained silicon

This work reports the measurement of electron $g$ factor anisotropy ($| \Delta g |$=$| g_{001} - g_{1 \bar 1 0} |$) for phosphorous donor qubits in strained silicon (sSi = Si/Si${}_{1-x}$Ge${}_x$) environments. Multi-million-atom tight-binding simulations are performed to understand the measured decrease in $\mathrm{\mid }\mathrm{\Delta }g\mathrm{\mid }$ as a function of $x$, which is attributed to a reduction in the interface-related anisotropy. For $x<$7%, the variation in $\mathrm{\mid }\mathrm{\Delta }g\mathrm{\mid }$ is linear and can be described by ${\eta }_{x}x$, where ${\eta }_x\approx$1.62$\times$ 10${}^{-3}$. At $x$=20%, the measured $\mathrm{\mid }\mathrm{\Delta }g\mathrm{\mid }$ is 1.2 $\pm$ 0.04 $\times$ 10${}^{-3}$, which is in good agreement with the computed value of 1$\times 10^{-3}$. When strain and electric fields are applied simultaneously, the strain effect is predicted to play a dominant role on $\mathrm{\mid }\mathrm{\Delta }g\mathrm{\mid }$. Our results provide useful insights on spin properties of sSi:P for spin qubits, and more generally for devices in spintronics and valleytronics areas of research.