The $\nu =5\mathrm{/}2$ Fractional Quantum Hall State in the Presence of Alloy Disorder

We report quantitative measurements of the impact of alloy disorder on the $\nu =5\mathrm{/}2$ fractional quantum Hall state. Alloy disorder is controlled by the aluminum content $x$ in the Al${}_x$Ga${}_{1-x}$As channel of a quantum well. We find that the $\nu =5\mathrm{/}2$ state is suppressed with alloy scattering. To our surprise, in samples with alloy disorder the $\nu =5\mathrm{/}2$ state appears at significantly reduced mobilities when compared to samples in which alloy disorder is not the dominant scattering mechanism. Our results highlight the distinct roles of the different types of disorder present in these samples, such as the short-range alloy and the long-range Coulomb disorder.