Computational identification of Ga-vacancy related electron paramagnetic resonance centers in $\beta$-Ga$_2$O$_3$

A combined experimental/theoretical study of the EPR in irradiated $\beta$-Ga$_2$O$_3$ is presented. Four EPR spectra, two $S=1/2$ and two $S=1$, are observed after high-energy proton or electron irradiation. One of the S=1/2 spectra (EPR1) can be observed at room temperature and below and is characterized by the spin Hamiltonian parameters $g_b=2.0313$, $g_c=2.0079$, $g_{a*}=2.0025$ and a quasi isotropic hyperfine interaction with two equivalent Ga neighbors of $~\sim$14 G on $^{69}$Ga. The second (EPR2) is observed after photoexcitation (with threshold 2.8 eV) at low temperature and is characterized by $g_b=2.0064$, $g_c=2.0464$, $g_{a*}=2.0024$ and a quasi isotropic hyperfine interaction with two equivalent Ga neighbors of 10 G. A spin $S=1$ spectrum with a similar g-tensor and a 50\% reduced hyperfine splitting accompanies each of these, which is indicative of a defect of two weakly coupled $S=1/2$ centers. DFT calculations of the magnetic resonance fingerprint of a wide variety of native defect models are carried out to identify these EPR centers in terms of specific defect configurations. The EPR1 center is proposed to correspond to a complex of two tetrahedral $V_\mathrm{Ga1}$ with an interstitial Ga in between them. This model was previously shown to have lower energy than the simple tetrahedral Ga vacancy and has a $2-/3-$ transition level higher than other $V_\mathrm{Ga}$ related models, which would explain why the other ones are already in their diamagnetic $3-$ state and are thus not observed if the Fermi level is pinned approximately at this level. The EPR2 spectra are proposed to correspond to the octahedral $V_\mathrm{Ga2}$. Models based on self-trapped holes and oxygen interstitials are ruled out because they would have hyperfine interaction with more than two Ga nuclei and because they can not support a corresponding $S=1$ center.