Two-channel anomalous Hall effect in ${\mathrm{SrRuO}}_3$

The Hall effect in ${\mathrm{SrRuO}}_3$ thin films near the thickness limit for ferromagnetism shows an extra peak in addition to the ordinary and anomalous Hall effects. This extra peak has been attributed to a topological Hall effect due to two-dimensional skyrmions in the film around the coercive field; however, the sign of the anomalous Hall effect in ${\mathrm{SrRuO}}_3$ can change as a function of saturation magnetization. Here we report Hall peaks in ${\mathrm{SrRuO}}_3$ in which volumetric magnetometry measurements and magnetic force microscopy indicate that the peaks result from the superposition of two anomalous Hall channels with opposite sign. These channels likely form due to thickness variations in ${\mathrm{SrRuO}}_3$, creating two spatially separated magnetic regions with different saturation magnetizations and coercive fields. The results are central to the development of strongly correlated materials for spintronics.

Figure 1

(a) Illustrations of the peak structure in the Hall effect describing how the peaks may arise due to (b) the superposition of the AHE and THE, or (c) two AHE channels with different signs and coercive fields.

Figure 2

(a) Plane-leveled AFM image of a 4.5-UC-thick ${\mathrm{SrRuO}}_3$ film, showing a step-and-terrace structure typical of step-flow growth. (b) The corresponding height profile along the line drawn in (a); the horizontal dotted lines are separations of 1 UC ($c=0.395$ nm, as measured in XRD on thicker films). (c) Illustration of thickness variations in the film.

Figure 3

Transport and magnetometry of a 10-nm-thick film. (a) $R_{xy}\left(H\right)$ at different $T$. The OHE is removed by a linear fitting at high field (5–6 T). (b) The spontaneous anomalous Hall effect extracted from (a) as a function of temperature. This is nonmonotonic with a sign change at $T^{\prime }\approx 121$ K. (c) Magnetization as a function of temperature. Field-cooling and field-warming curves have substrate curves subtracted and are shifted vertically so $M\left(170\right)=0$; the dotted curve shows the derivative of the 5-mT field-warming curve with arbitrary units to highlight the magnetic transition. Points from $M\left(H\right)$ curves are spontaneous magnetization taken as average moment under 1 T for saturated quadrants. There is no magnetic transition evident at the AHE sign change. (d) Spontaneous anomalous Hall resistivity of the ${\mathrm{SrRuO}}_3$ as a function of spontaneous magnetization. The anomalous Hall effect is positive for small $M_s$ and negative for large $M_s$, switching at $M_s^{\prime }\approx 0.7{\mu }_B/\mathrm{Ru}$. (e) Spontaneous anomalous Hall conductivity as a function of spontaneous magnetization, compared to previous literature values [31].

Figure 4

Transport and magnetometry of a 4.5-UC-thick ${\mathrm{SrRuO}}_3$ film. (a) $R_{xy}(H$) at different $T$. The data (blue points) are fitted to a function of two sigmoids as an approximation of two AHEs (black line). (b) $R_{xy}\left(H\right)$ of the film at 10 K; the black line is the two-AHE fit. (c) The two fit components from (b). (d) Moment ($m$) versus $H$ of the film at 10 K, showing two switches at the same coercive fields as in the Hall resistance. For the fit, the widths and coercive fields from (b) and (c) are fixed, and the scalings are fit parameters. (e) The two fit components from (d). (f) $T$ dependence of the spontaneous anomalous Hall resistance of the two channels, defined as the mean and standard deviation of anomalous Hall resistance in the 0–100 mT range of the saturated quadrant of an individual fit component. (g) $T$ dependence of the magnetization in field cooling (FC) in 100 mT, and field warming (FW) in 5 mT after saturating at 7 T. Data have substrate curves subtracted and are shifted vertically so $M\left(170\right)=0$. The dotted curve shows the derivative of the 5-mT field-warming curve with arbitrary units to highlight the two magnetic transitions.

Figure 5

Transport measurements of a nominally 4.0-UC-thick ${\mathrm{SrRuO}}_3$ film. (a) $T$ dependence of $R_{xy}\left(H\right)$, showing a transition from the usual “peak” shape below $T_C$ to a “step” shape at 50 K and below. (b) $R_{xy}\left(H\right)$ at 60 K showing the usual “peak” shape, with the fitting shown in black. (c) The two components of the fit at 60 K: one positive and one negative AHE. (d) $R_{xy}\left(H\right)$ at 40 K showing the “step” shape, with the fitting shown in black. (e) The two components of the fit at 40 K: two negative AHEs. (f) $T$ dependence of the two AHE components. One component experiences a sign change near 53 K, where the anomalous Hall signal changes from steplike at low $T$ to peaklike at high $T$. (g) $T$ dependence of the OHE, found from a linear fit at 2–3 T; the error is the difference between the OHE estimated by high-field and low-field fits. The error becomes very large above 90 K, where it becomes difficult to separate the broad paramagnetic AHE from the OHE. The OHE does not change sign near 53 K.

Figure 6

MFM images at 10 K, 0.1 T after applying different fields. Scale bars are $1\mu \mathrm{m}$. $\mathrm{\Delta }f$, the change in resonant frequency, is proportional to out-of-plane stray field gradient. The sign of $\mathrm{\Delta }f$ is opposite to the sign of magnetization. Panels (a)–(e) show the progression of the magnetic structure of the film through the two transitions. Red/yellow is negatively magnetized, blue/cyan is positively magnetized. Panel (f) shows the anomalous Hall resistance (blue line) of the film as a function of field, with points labeled where the presented MFM images were measured. The RMS deviation of the MFM signal (green squares) shows a peak in inhomogeneity corresponding to the peak in the Hall effect. (g) The change in MFM signal through the first transition, showing only one set of stripes switching. Black lines are added as a guide to the eye. (h) The change in MFM signal through the second transition, showing the other set of stripes switching. The same black lines are a guide to the eye. (i) The images from (g) and (h) combined additively. The MFM signal either changes in the first transition or the second rather than both, meaning there are two spatially magnetic regions with two different coercive fields.