Robust skyrmion-bubble textures in $\mathrm{Sr}\mathrm{Ru}{\mathrm{O}}_3$ thin films stabilized by magnetic anisotropy

Topological spin textures in the itinerant ferromagnet $\mathrm{Sr}\mathrm{Ru}{\mathrm{O}}_3$ are studied combining Hall transport measurements and numerical simulations. We observe characteristic signatures of the topological Hall effect associated with skyrmions. A relatively large thickness of our films and the absence of heavy-metal layers make the interfacial Dzyaloshinskii-Moriya interaction an unlikely source of these topological spin textures. In addition, the transport anomalies exhibit an unprecedented robustness to magnetic field tilting and temperature variations. Our numerical simulations suggest that this unconventional behavior results from compact magnetic bubbles with skyrmion topology stabilized by magnetodipolar interactions and higher-order magnetocrystalline ansiotropies in an unexpected region of parameter space. This work presents a comprehensive understanding of the origin of the topological Hall effect in $\mathrm{Sr}\mathrm{Ru}{\mathrm{O}}_3$, incorporating the complex angular dependence of the magnetic anisotropy, intrinsic to this material.

Figure 1

(a) $M\text{−}H$ curves measured at 10 K for film B in out-of-plane (black) and in-plane (red) directions, respectively. (b) Temperature dependence of anomalous Hall resistivity (${\rho }_{\text{AHE}}$) for films A and B. Film A shows a sign change in ${\rho }_{\text{AHE}}$ unlike film B. (c) $M\text{−}H$ for out of plane (black) and in plane (red) at different temperatures for film A. Between each measurement, the film was warmed to 300 K and thereafter cooled to the measurement temperature. (d) Magnetic-field-dependent Hall resistivity measured at different temperatures for film A. The inset shows the Hall resistivity at 10 K which changes sign beyond 70 K.

Figure 2

Field- and angle-dependent Hall resistivity of SRO thin films. (a) Hall bar schematic showing longitudinal ($V_{xx}$) and transverse contact voltage contacts ($V_{yx}$). The tilt angle $\theta$ with the magnetic field $B$ in the out-of-plane direction and the current ($I$) along the $x$ direction are also shown. (b) Field-dependent Hall resistivity at different temperatures for film B. The linear contribution from OHE has been subtracted (except for 120 K). (c), (d) Angular-dependent Hall resistivity at 75 K for films B and A, respectively.

Figure 3

Arrays of bubbles (a) with topological charge $-1$ and (b) carrying zero topological charge and an in-plane magnetic moment. In-plane components of $\mathit{m}$ are shown with arrows, and $m_z$ is color coded. Distances are given in units of the film thickness $h$.

Figure 4

Magnetic field $H$ vs quality factor $Q=\frac{K_1}{4\pi }$, phase diagrams, which include the skyrmion crystal (SkX), stripe domain (SD), an array of nontopological bubbles (NTBs, yellow), and uniform (U) states. Red color intensity indicates $m_z$ in the uniform state with a tilted magnetization. The state with the magnetization normal to the film is shown with white color. These diagrams are calculated for (a) $R=\frac{K_2}{4\pi }=0$ and (b) $R=0.4$. Phase diagrams in a tilted magnetic field, $\theta$ being the tilt angle, for (c) $Q=0.87$ and $R=0$ and (d) $Q=0.65$ and $R=0.60$. The insets show the dependence of the dimensionless anisotropy energy density $e_{\mathrm{a}}$ in the uniform state on the magnetization tilt angle ${\theta }_{\mathrm{m}}$ for the corresponding parameter sets.

Figure 5

Schematic picture of ferromagnetic (FM) and antiferromagnetic (AFM) phases and transitions in an external magnetic field as calculated by DFT $+$ $U$ (further details in Ref. [39]). The $\uparrow$ and $\downarrow$ spins are on the Ru sites, and the (blue) circles are nonmagnetic Ti sites (shown are only the Ru-Ti intermixed and the neighboring Ru layer). $E$ is the zero-field energy of the phase and the $M$ magnetic moment per Ru atom. The gray box indicates the possible coexistence of the AFM-5 and FM-low-spin (LS) phase.