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We use hybrid density functional calculations to assess n-type doping inmonoclinic (Al$_x$Ga$_{1-x}$)$_2$O$_3$ alloys. We focus on Si, the mostpromising donor dopant, and study the structural properties, formation energiesand charge-state transition levels of its various configurations. We alsoexplore the impact of C and H, which are common impurities in metal-organicchemical vapor deposition (MOCVD). In Ga$_2$O$_3$, Si$_{Ga}$ is an effectiveshallow donor, but in Al$_2O_3$ Si$_{Al}$ acts as a DX center with a (+/-)transition level in the band gap. Interstitial H acts as a shallow donor inGa$_2$O$_3$, but behaves as a compensating acceptor in n-type Al$_2O_3$.Interpolation indicates that Si is an effective donor in(Al$_x$Ga$_{1-x}$)$_2$O$_3$ up to 70% Al, but it can be compensated by Halready at 1% Al. We also assess the diffusivity of H and study complexformation. Si$_{cation}$-H complexes have relatively low binding energies.Substitutional C on a cation site acts as a shallow donor in Ga$_2$O$_3$, butcan be stable in a negative charge state in (Al$_x$Ga$_{1-x}$)$_2$O$_3$ whenx>5%. Substitutional C on an O site (C$_O$) always acts as an acceptor inn-type (Al$_x$Ga$_{1-x}$)$_2$O$_3$, but will incorporate only under relativelyO-poor conditions. C$_O$-H complexes can actually incorporate more easily,explaining observations of C-related compensation in Ga$_2$O$_3$ grown byMOCVD. We also investigate C$_{cation}$-H complexes, finding they have highbinding energies and act as compensating acceptors when x>56%; otherwise the Hjust passivates the unintentional C donors. C-H complex formation explains whyMOCVD grown Ga$_2$O$_3$ can exhibit record-low free-carrier concentrations, inspite of the unavoidable incorporation of C. Our study highlights that, whileSi is a suitable shallow donor in ALGO alloys, control of unintentionalimpurities is essential to avoid compensation.

Role of carbon and hydrogen in limiting $n$-type doping of monoclinic (Al$_x$Ga$_{1-x}$)$_2$O$_3$