Symmetry breaking structural relaxation and optical transitions of native defects and carbon impurities in LiGa$_5$O$_8$

LiGa$_5$O$_8$ in a spinel type structure has recently been claimed to be an unintentional p-type ultra-wide-band-gap oxide semiconductor. While previous computational work did not yet identify the origin of p-type doping and in fact predicted insulating behavior by compensation of deep acceptors by shallow donors, defect characterization in terms of its optical signatures remains important. Rather than focusing on thermodynamics transition levels, as in earlier work, this present paper focuses on the vertical transitions in a defect configuration diagram of defects in different charge states, representing absorption and emission processes involving carrier capture/emission from/to band edges. In addition, the structural relaxation of several native defects is revisited by allowing for more complex symmetry-breaking distortions in an effort to reconcile conflicting results in the previous literature. Special attention is given to the Li vacancy because it is the shallowest native acceptor. For this defect, the previously reported transition levels are revised on the basis of symmetry-breaking relaxations. The structural relaxations, band structures, and densities of states are compared between the symmetry-broken polaronic and symmetry-conserving non-polaronic states. Finally, we also study carbon impurities, which are likely to originate from growth methods involving organic precursors.