Metamagnetism and anomalous magnetotransport properties in rare-earth-based polar semimetals $R\mathrm{AuGe}$ $(R=\mathrm{Dy}, \mathrm{Ho}, \text{and} \mathrm{Gd})$

We report the magnetic, magnetoelastic, and magnetotransport properties of single crystals of polar magnets $R\mathrm{AuGe}$ $(R=\mathrm{Dy}, \mathrm{Ho}, \text{and} \mathrm{Gd})$, grown by Au-Ge self-flux. Magnetization and magnetostriction measurements reveal multistep metamagnetic transitions for the $c$-axis magnetic field $(H\parallel c)$ for DyAuGe and HoAuGe, suggesting magnetic frustration in the triangular lattice of $R$ ions. The magnetic phase diagrams have clarified a close connection between the magnetoelastic property and the emergence of the intermediate metamagentic phase. The magnetic-field dependence of the resistivity and Hall resistivity reveal the semimetallic transport dominated by hole-type carriers, consistent with the behavior in a nonmagnetic analog YAuGe. We also identify a signature of an anomalous Hall effect (AHE) proportional to the field-induced magnetization in $R=\mathrm{Dy}$, Ho, and Gd. GdAuGe shows magnetic and transport behavior as reported in a previous study using Bi-flux grown single crystals, while the self-flux grown crystal shows larger magnetoresistance $(\sim 345\%, \text{at} 1.8 K \text{and} 9 T)$ due to higher hole-type carrier mobility $[\sim 6400{\mathrm{cm}}^2/\left(Vs\right)]$. Using the two-band model analysis considering the mobility change during the magnetization process, we extract the anomalous Hall conductivity: $\sim 1200$ and $\sim 530$ S/cm for $R=\mathrm{Dy} \text{and} \mathrm{Ho}$, respectively, at 1.8 K with 9 T for $H\parallel c$. The magnitude of conductivity suggests a contribution of intrinsic origin, possibly related to the Berry curvature in the electron bands induced by the time-reversal symmetry breaking and the polar lattice.