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. Pyrochlore ruthenates, Pr$_{2}$Ru$_{2}$O$_{7}$ (Ru-$4d$ electron number, $n=4$) and Ca$_{2}$Ru$_{2}$O$_{7}$ ($n=3$), are a Mott insulator and a magnetic bad metal, respectively, due to the strong electron correlation. We investigate magneto-transport properties of (Pr$_{1-x}$Ca$_{x}$)$_{2}$Ru$_{2}$O$_{7}$ in a whole band-filling range, $0<x<1$. With increasing hole-doping $x$, the system undergoes an insulator-metal transition. When Ca$_{2}$Ru$_{2}$O$_{7}$ is doped with electrons ($0.5<x<0.9$), the enhanced coupling among Ru-$4d$ spins produces a ferromagnetic-metal phase with a large anomalous-Hall angle up to 2 %. We discuss the electronic phase transitions in (Pr$_{1-x}$Ca$_{x}$)$_{2}$Ru$_{2}$O$_{7}$ in view of Hund's metal.