Shallow and deep donors in direct-gap $n$-type ${\mathrm{Al}}_x{\mathrm{Ga}}_{1-x}\mathrm{A}\mathrm{s}:\mathrm{S}\mathrm{i}$ grown by molecular-beam epitaxy

Temperature-dependent Hall-effect measurements are reported and analyzed in detail for $n$-type ${\mathrm{Al}}_x{\mathrm{Ga}}_{1-x}\mathrm{As}$ of composition $0\le x\le 0.40$ grown by molecular-beam epitaxy and highly doped with Si ($N_{\mathrm{Si}}\ge 5\times {10}^{17}$ ${\mathrm{cm}}^{-3\mathrm{}}$). A quantitative analysis using Fermi-Dirac statistics reveals for the composition range $0.20\le x\le 0.40$ the presence of a hydrogenlike shallow Si donor which interacts with the $\Gamma$ valley and a deep Si donor related to the $X$ valley. In contrast to previous results, the thermal activation energy of the deep donor, determined to be $E_{\mathrm{dd}}=140\mathrm{}\pm 10$ meV, does not change significantly with alloy composition. Only the ratio of shallow-to deep-donor concentration depends on composition. For $x\le 0.20$, neither deep-donor nor persistent photoconductivity exists in $n$-type ${\mathrm{Al}}_x{\mathrm{Ga}}_{1-x}\mathrm{A}\mathrm{s}:\mathrm{S}\mathrm{i}$. For $0.20\le x\le 0.40$, however, the deep-donor concentration increases with $x$ while simultaneously the shallow-donor concentration decreases. The proposed interaction of the deep donor with the $X$ valley helps in understanding the persistent photoconductivity found in $n$-type ${\mathrm{Al}}_x{\mathrm{Ga}}_{1-x}\mathrm{As}$ with $0.20\le x\le 0.40$.