Anomalous Hall effect scaling in ferromagnetic thin films

We propose a scaling law for anomalous Hall effect in ferromagnetic thin films. Our approach distinguishes multiple scattering sources, namely, bulk impurity, phonon for Hall resistivity, and most importantly the rough surface contribution to longitudinal resistivity. In stark contrast to earlier laws that rely on temperature- and thickness-dependent fitting coefficients, this scaling law fits the recent experimental data excellently with constant parameters that are independent of temperature and film thickness, strongly indicating that this law captures the underlying physical processes. Based on a few data points, this scaling law can even fit all experimental data in full temperature and thickness range. We apply this law to interpret the experimental data for Fe, Co, and Ni and conclude that (i) the phonon-induced skew scattering is unimportant as expected; (ii) contribution from the impurity-induced skew scattering is negative; (iii) the intrinsic (extrinsic) mechanism dominates in Fe (Co), and both the extrinsic and intrinsic contributions are important in Ni.

Figure 1

The thickness dependence of the longitudinal resistivity at various temperatures. Curves at different temperatures have different ${\tau }_0$, which is plotted in the inset and can be well described by the BG formula (the red curve). The points are the experimental data for Co from Ref. [11], the curves are plotted from Eq. (14) with $\delta \approx 5.7a_0$.

Figure 2

(a) The thickness and temperature dependence of the longitudinal resistivity. The surface is plotted from Eq. (14). (b) The thickness and temperature dependence of the anomalous Hall resistivity. The surface is plotted from Eq. (18) with ${\alpha }_{\text{I}},{\alpha }_{\text{P}}, \gamma$ as fitting parameters, whose values are listed in Table 1. The points are the experimental data for Co from Ref. [11]. The shaded area at the bottom indicates the absolute difference between the experiment and theory. (c), (d) Thickness dependence of extrinsic and intrinsic contributions at low and high temperatures, respectively. The dotted (dashed) red lines are the side-jump (skew-scattering) contribution due to impurity scattering, and their sum is plotted as the solid red line. The blue solid line is the side-jump contribution due to phonon scattering. The solid green line is the intrinsic contribution.

Figure 3

(Top row) The same as in Fig. 2 for Ni. The points are the experimental data from Ref. [12]. (Bottom row) The same as in Fig. 2 for Fe. The points are the experimental data from Ref. [13].

Figure 4

The ratio of intrinsic contribution to the total AHE: $r_{\text{INT}}={\rho }_{\text{AH}}^{\text{INT}}/{\rho }_{\text{AH}}$ as function of film thickness and temperature for Fe, Co, Ni. For Fe, the AHE is dominated by the intrinsic mechanism with $r_{\text{INT}}>90\%$. For Co, the AHE is dominated by extrinsic mechanisms with $r_{\text{INT}}\sim 20\%$. For Ni, the solid curve, corresponding to $r_{\text{INT}}=50\%$, separates the intrinsic dominating region ($r_{\text{INT}}>50\%$) from the extrinsic dominating region ($r_{\text{INT}}<50\%$).

Figure 5

The fitting of thickness and temperature dependence of the anomalous Hall resistivity for Co, Ni, and Fe in (a), (b), and (c), respectively, by using part of the experimental data. Points used for fitting are indicated by triangles.