  The singular density of states and the two Fermi wavevectors resulting from a
ring-shaped or "Mexican hat" valence band give rise to unique trends in the
charged impurity scattering rates and charged impurity limited mobilities. Ring
shaped valence bands are common features of many monolayer and few-layer
two-dimensional materials including the III-VI materials GaS, GaSe, InS, and
InSe. The wavevector dependence of the screening, calculated within the random
phase approximation, is so strong that it is the dominant factor determining
the overall trends of the scattering rates and mobilities with respect to
temperature and hole density. Charged impurities placed on the substrate and in
the 2D channel are considered. The different wavevector dependencies of the
bare Coulomb potentials alter the temperature dependence of the mobilities.
Moving the charged impurities 5 $\AA$ from the center of the channel to the
substrate increases the mobility by an order of magnitude.
