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 $α$- and $β$-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