Stability, electronic quantum states, and magnetic interactions of Er$^{3+}$ ions in Ga$_2$O$_3$

The chemical, structural, mechanical, and dynamical stabilities of the $\alpha$- and $\beta$-Ga$_2$O$_3$ are confirmed from respective negative formation energies, negative cohesive energies, favorable elastic constants, and positive phonon frequencies. The phonon dispersions indicate that the Ga-O bonds are uniform in the $\alpha$-phase, while they vary in the $\beta$-phase due to the anisotropic polyhedral movement. The defect formation energy analysis confirms that both Er-doped phases prefer Er$^{3+}$ state. The underestimated band gaps of the pristine phases from $ab~initio$ calculations are corrected by employing the hybrid functional calculations. The site preference energy analysis indicates partial occupation of Er in the octahedral site of Ga. Anisotropic nature of hyperfine tensor coefficients of Er are similar in both phases. Calculated magnetic exchange interaction between two Er dopants is negative for $\alpha$ and positive for $\beta$, indicating antiferromagnetic ground state in the former and the ferromagnetic ground state in the latter. A large values of Dzyaloshinskii-Moriya interactions (DMIs) are obtained along the $x$ direction in the $\alpha$ and along the $y$ direction in the $\beta$. The analysis of dielectric constants and refractive indices of both pristine and Er doped phases shows a good agreement with available experimental values. The calculated optical anisotropy is slightly higher in $\beta$ than those in $\alpha$, which is due to the involvement of lower symmetry in $\beta$. The crystal field coefficients (CFCs) calculated from DFT are used to analyze 4$f$ multiplets and 4$f$ - 4$f$ transitions. Thus calculated lowest energy level of the first excited state to the lowest energy level of the ground state is about 1.53~$\mu$m, which is in a good agreement with available experiment and it falls within the quantum telecommunication wavelength range.