Fully filling-controlled pyrochlore ruthenates: Emergent ferromagnetic-metal state and geometrical Hall effect

Carrier doping to the Mott insulator is essential to produce highly correlated metals with emergent properties. The pyrochlore ruthenates ${\mathrm{Pr}}_{2}{\mathrm{Ru}}_{2}{\mathrm{O}}_{7}$ (Ru $4d$ electron number $n=4$) and ${\mathrm{Ca}}_{2}{\mathrm{Ru}}_{2}{\mathrm{O}}_{7}$ ($n=3$) are a Mott insulator and a magnetic bad metal, respectively, due to the strong electron correlation. We investigate magnetotransport properties of ${({\mathrm{Pr}}_{1\ensuremath{-}x}{\mathrm{Ca}}_{x})}_{2}{\mathrm{Ru}}_{2}{\mathrm{O}}_{7}$ in a whole band-filling range, $0\ensuremath{\le}x\ensuremath{\le}1$. With increasing hole doping $x$, the system undergoes an insulator-metal transition. When ${\mathrm{Ca}}_{2}{\mathrm{Ru}}_{2}{\mathrm{O}}_{7}$ is doped with electrons ($0.5