Role of carbon and hydrogen in limitingn-type doping of monoclinic (Al_xGa_1-x)₂O₃

Effective control of the carrier concentration is essential for Ga₂O₃-based high-power electronics. Using density functional theory, we explore the self-compensation of Si donors and the impact of carbon and hydrogen impurities on doping and compensation in (Al_xGa_1-x)₂O₃. We find that Si is an effective donor in (Al_xGa_1-x)₂O₃ up to 70% Al before forming a DX center.  We further demonstrate that Si donors in (Al_xGa_1-x)₂O₃ can be compensated by interstitial H at Al concentrations exceeding 1% and by substitutional C_cation at 5% Al. The diffusivity of H and the likelihood of complex formation are also assessed.  A stable C_cation-H is revealed, which is electrically neutral in n-type Ga₂O₃ but turns into a compensating center at 54% Al in (Al_xGa_1-x)₂O₃. C_O-H incorporates easily in (Al_xGa_1-x)₂O₃, acting as an acceptor. This provides a possible source of C-related compensation in oxygen-poor Ga₂O₃ grown by metal-organic chemical vapor deposition. Our study highlights that, while Si is in principle a suitable shallow donor in (Al_xGa_1-x)₂O₃ alloys up to high Al compositions, control of unintentional impurities is essential to avoid compensation.