Self-compensation by silicon $DX$ centers in ultrawide-bandgap nitrides

\textit{DX} behavior limits $n$-type carrier concentrations in ultrawide-bandgap nitrides such as aluminum nitride (AlN) and cubic boron nitride ($c$-BN). Instead of acting as effective-mass donors, \textit{DX} centers capture two electrons, stabilizing a negative charge state that leads to self compensation. Silicon is the most effective $n$-type dopant in this class of materials; in AlN, its \textit{DX} level [(i.e., the (+/$-$) transition level] is $\sim$270 meV from the conduction-band minimum. This implies that many silicon impurities incorporated into AlN will be negatively charged and compensate the intended $n$-type doping. By combining density functional theory calculations of temperature-dependent band gaps and Si dopant transition levels, we show here that significant compensation occurs in silicon-doped AlN, even in the absence of any other defects. This compensation strongly limits free electron concentrations which become independent of doping concentration, and donor activation is only significant for light doping scenarios. Higher free carrier concentrations can be achieved in AlGaN alloys or in $c$-BN, where the \textit{DX} level sits closer to the conduction-band minimum.