Scaling of the paramagnetic anomalous Hall effect in SrRuO${}_3$

We extract the paramagnetic anomalous Hall effect (AHE) resistivity ${\rho }_{xy}^{\mathrm{AHE}}$ in thin films of the itinerant ferromagnet SrRuO${}_3$ and show that the AHE coefficient $R_s$ scales with the longitudinal resistivity ${\rho }_{xx}$. We fit the resistivity dependence of ${\rho }_{xy}^{\mathrm{AHE}}$ assuming two mechanisms: side jumps and Karplus-Luttinger (Berry phase) mechanism. For the latter, we consider the effect of a finite scattering time with a fractional power-law relation to ${\rho }_{xx}$.

Figure 1

The Hall effect ${\rho }_{xy}^{\mathrm{HE}}$ (a) and the AHE resistivity ${\rho }_{xy}^{\mathrm{AHE}}$ (b) as a function of temperature for different fields.

Figure 2

(a) Resistivity ${\rho }_{xx}$ vs temperature for different magnetic fields applied along the easy axis. (b) Magnetic resistivity $\Delta \rho$, extracted from (a), vs temperature for different fields. (c) $M_{\mathrm{calc}}/M_s$ defined in the text vs $\sqrt{\Delta \rho }$ for different $T_c^{\mathrm{eff}}$ at $T=165$ K. (d) $M^{MR}/M_s$ vs $g{\mu }_B{H}_{\mathrm{eff}}/k_{B}T$ defined in text for $T\ge 180$ K (circles) and $T<180$ K (squares). The solid line is the Brillouin function. Inset: extracted optimal $T_c^{\mathrm{eff}}$ as a function of $T$.

Figure 3

(a) Extracted AHE coefficient $R_s$ as a function of temperature for different fields. (b) AHE coefficient $R_s$ extracted at different temperatures and fields as a function of resistivity.

Figure 4

(a) The AHE coefficient $R_s$ normalized by the absolute value of its negative peak value $R_s^{*}$ as a function of resistivity normalized by the resistivity at which $R_s$ vanishes, ${\rho }^{*}$. The solid lines are for the extreme $R_s$ values. (b) AHE coefficient $R_s$ vs resistivity ${\rho }_{xx}$ for a 50-nm sample; the solid line is a fit to Eq. (8). Inset: fit error $E$ as a function of the assumed exponent $\alpha$.