Investigation of O interstitial diffusion in $\beta$-Ga$_2$O$_3$: direct approach via master diffusion equations

Monoclinic $\beta$-Ga$_2$O$_3$, a promising wide band gap semiconducting material, exhibits complex, anisotropic diffusional characteristics and mass transport behavior as a results of its low symmetry crystal structure. From first-principles calculations combined with master diffusion equations, we determine three-dimensional diffusion tensors for neutral ($\text{O}_{\text{i}}^{0}$) and 2- charged oxygen interstitials ($\text{O}_{\text{i}}^{2-}$). Systematic exploration of the configurational space identifies stable configurations in these two dominant charge states and their corresponding formation energies. By connecting every pair of low-energy configurations considering both interstitial or interstitialcy hops, we construct three-dimensional diffusion networks and evaluate hopping barriers of all transition pathways in networks. Combining the collection of (i) defect configurations and their formation energies and (ii) the hopping barriers that link them, we construct and solve the master diffusion equations for $\text{O}_{\text{i}}^{0}$ and $\text{O}_{\text{i}}^{2-}$ separately through the Onsager approach, resulting in respective three-dimensional diffusion tensors D$_{\text{O}_{\text{i}}}^{0}$ and D$_{\text{O}_{\text{i}}}^{2-}$. Both $\text{O}_{\text{i}}^{0}$ and $\text{O}_{\text{i}}^{2-}$ present the fastest diffusion along the $b$-axis, demonstrating significant anisotropy. The predicted self-diffusivities along [100] and [$\overline{2}01$] align well with previously reported values from isotopically labeled oxygen tracer experiments, highlighting the reliability of the approach in capturing complex diffusion mechanisms.