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

We measure the anomalous Hall effect (AHE) resistivity ${\rho }_{xy}$ in thin films of the itinerant ferromagnet SrRuO${}_3$. At low temperatures, the AHE coefficient $R_s$ varies with ${\rho }_{xx}^2$, and at higher temperatures, $R_s$ reaches a peak and then changes sign just below $T_c$. We find that for all films studied, $R_s$ scales with resistivity in the entire ferromagnetic phase. We attribute the observed behavior to the contribution of the extrinsic side-jumps mechanism and the intrinsic Karplus-Luttinger (Berry phase) mechanism, including the effect of finite scattering rates.

Figure 1

(a) Remanent AHE resistivity (${\rho }_{xy}^{\mathrm{AHE}}$) of eight films vs temperature. (b) The AHE coefficient ($R_s$) vs temperature derived from ${\rho }_{xy}^{\mathrm{AHE}}$ and $M_{\perp }$ using Eq. (1). Inset: Scaling of the perpendicular magnetization ($M_{\perp }$) normalized by its low temperature value as a function of temperature for five films with thickness between 6 and 90 nm.

Figure 2

(a) Longitudinal resistivity (${\rho }_{xx}$) of the eight films presented in Fig. 1 vs temperature. (b) $R_s$ from Fig. 1b vs resistivity (${\rho }_{xx}$).

Figure 3

$R_s$ normalized by its absolute maximum value (${R_s}^{*}$) vs ${\rho }_{xx}$ normalized by its value when $R_s$ changes its sign (${\rho }^{*}$). Inset: ${R_s}^{*}$ vs ${\left({\rho }^{*}\right)}^2$.

Figure 4

Normalized resistivity (${\rho }^{*}$) vs temperature. Inset: Film thickness ($d$) divided by its resistivity when $R_s$ changes its sign (${\rho }_0$) vs film thickness ($d$).

Figure 5

AHE coefficient R${}_s$ as a function of resistivity ${\rho }_{xx}$ for a thick film (500 Å). The dashed line is a fit to Eq. (8).