The notion of Berry phase has been recognized a powerful and unifying concept in many different fields of physics since its discovery in 19841. In condensed matter physics, application of this concept has brought out geometric and topological viewpoints, generating deeper understandings of various transport phenomena2. Of particular interest is the intrinsic contribution to the anomalous Hall effect (AHE), which is determined by integrating the Berry curvature of each occupied Bloch state over the entire Brillouin zone3. Since such a scattering-independent Hall conductivity only depends on the electronic band structure, the Berry-phase mechanism has established a link between the AHE and the topological nature of materials4. Especially in ferromagnetic conductors with broken time-reversal symmetry and sizable spin-orbit couplings, peculiar features such as avoided crossings or Weyl points near the Fermi energy can act as the source and sink of the Berry curvatures, significantly enhancing the intrinsic contribution and dominating the AHE over other extrinsic mechanisms (i.e. side-jump and skew scattering)3,5,6,7,8.

A prototypical system of this category is the perovskite ruthenate SrRuO3. The dual nature of both itinerancy and localization of Ru 4d electrons makes SrRuO3 a rare example of ferromagnetic metallic oxides9,10. The nontrivial topological properties of SrRuO3 were initially realized as a result of the non-monotonous relationship between its anomalous Hall conductivity σxy and the magnetization M, which could be rationalized well from first-principle calculations by taking into account the Berry phases of electronic band structures11. These were subsequently suggested as the existence of Weyl nodes characteristic of linear band crossings at the Fermi energy12,13,14,15. Lately, a variety of experimental efforts have been put forth to tailor the Berry-phase related features of (001)-oriented SrRuO3 films via epitaxial strain16, helium irradiation17, variation of film thickness18,19,20, ionic liquid gating19, and construction of asymmetric interfaces21. In addition, it was argued in (001) SrRuO3 ultrathin films and heterostructures that topologically nontrivial features can also emerge in real space due to the formation of nonlinear spin textures such as the skyrmions, giving rise to the so-called ‘topological Hall effect’ (THE)22,23,24. Nevertheless, the contribution of THE to the net Hall effect is usually entangled with those from the Berry curvatures in momentum space and practically challenging to be unambiguously identified25,26,27,28.

Recently, inspired by the salient studies on the honeycomb lattice29,30, heterostructures of perovskite-type transition metal oxides along the pseudocubic [111] axis have been extensively investigated by theorists as promising platforms towards exotic correlated and topological phenomena31,32,33,34,35,36,37. Notably, the symmetry of the (111) surface is trigonal and the three octahedral rotation axes lie neither parallel nor perpendicular to the (111) plane, as illustrated in Fig. 1a. As a result, the strategy of strain accommodation in (111) heteroepitaxy can manifest in rather distinct manners, which may essentially affect the topological phase diagrams and/or lead to emergent phenomena that are inaccessible in bulk and (001)-oriented heterostructures34,38,39,40,41. From the experimental perspective, while initiative studies were mainly focused on strongly correlated 3d complex oxides42,43,44,45,46,47,48, it has remained largely unexplored in ruthenates with intermediate strength of correlations and sizable spin-orbit interactions49,50,51.

Fig. 1: Emergence of hidden trigonal SrRuO3 under (111) tensile strain.
figure 1

a Schematic depiction of the octahedral tilt and rotation axes for (001) and (111) orientations, respectively. All indices of the crystallographic planes and directions in this work are defined in the frame of pseudocubic axes unless for special notice. b Energy of structures vs. magnitude of (111) strain by first-principles calculation. Six possible phases are considered on KTaO3 substrate and the lowest five in energy are displayed on the figure. c, d Heteroepitaxial stabilization of the orthorhombic Pbnm and the trigonal \(R\bar{3}c\) SrRuO3 on (111) SrTiO3 and KTaO3 substrates, respectively. The rotation patterns are denoted in the Glazer notations72.

Here, we report the discovery of an emergent trigonal phase of SrRuO3 stabilized epitaxially on (111) KTaO3 substrate. Unlike the original aac+ rotation pattern in bulk, the ~ 1.5% tensile strain triggers the formation of aaa pattern in trigonal SrRuO3. Combined magnetization and transport measurements have revealed its peculiar magnetic and topological properties: (1) three-dimensional (3D) XY ferromagnetism (FM) with sixfold anisotropic magnetoresistance (AMR) below 160 K, (2) coexistence of high-mobility holes ( ~ 104 cm2V−1s−1) and low-mobility electrons as charge carriers, (3) double AHE channels including intrinsic contributions from the Berry-phase mechanism. These results are consistent with our first-principles electronic structure calculations, which suggest the presence of multiple pairs of Weyl nodes within an energy range of 50 meV at the Fermi level. Our findings highlight the opportunity for unveiling hidden correlated topological phenomena by means of (111) strain engineering.

Results and discussion

Heteroepitaxial stabilization of \(R\bar{3}c\) SrRuO3 under (111) tensile strain

Our earlier study about SrRuO3 films on (111) SrTiO3 substrate has uncovered that the compressive strain ( ~ -0.6%) is accommodated by significantly suppressing the degree of the c+ octahedral rotation52. Thus, it is desirable to explore the strategy of strain accommodation and the feasibility of achieving latent phases of SrRuO3 on the tensile strain side. Here, we select KTaO3 as the substrate in that it possesses the same perovskite-type structure as SrRuO3 and can supply a practically large magnitude of tensile strain of ~ 1.5%.

We have synthesized (111)-oriented SrRuO3 thin films ( ~ 15 nm) on KTaO3 substrates using the pulsed laser deposition technique (see ‘Methods’). Scanning transmission electron microscopy (STEM) image confirms the expected epitaxial relationship between film and substrate with abrupt interface (Supplementary Fig. 1). Comprehensive x-ray diffraction characterizations demonstrate that the films have high crystallinity without secondary phases, which are coherently strained on the substrate (Supplementary Fig. 2).

To obtain microscopic information about the octahedral rotation patterns, the schematic crystal structures of SrRuO3 with \(R\bar{3}c\) and Pbnm space groups are projected along the [\(11\bar{2}\)] direction, respectively, highlighting their peculiar distinctions in atomic arrangements on the Sr and O sublattices (Fig. 2a, b). Specifically, in the \(R\bar{3}c\) phase, while the Sr atoms in each column along [111] are aligned straightforwardly (Fig. 2a, black dash lines), the O atoms exhibit pronounced displacements forming opposite zigzag patterns between adjacent columns and the same pattern in every other column (Fig. 2a, red and blue dash lines). In sharp contrast, in the Pbnm phase, displacements of the Sr atoms exhibit the zigzag patterns whereas the O columns are aligned straightforwardly. Hence it can be regarded as the hallmarks for identifying the symmetry of our tensile-strained (111) SrRuO3 thin films.

Fig. 2: Atomic-scale illustration of the trigonal SrRuO3 in (111) epitaxial films on KTaO3 substrate.
figure 2

Characteristic crystal structures of SrRuO3 in (a) \(R\bar{3}c\) and (b) Pbnm space group projected along the [\(11\bar{2}\)] direction. The Sr and the O sublattices are displayed separately. For each symmetry as a result of the corresponding octahedral tilt and rotation, the atoms with relative displacements are indicated by red and blue dash lines, while those of no relative displacements are indicated by black dash lines. c HAADF and d ABF STEM images of the heterostructure. The areas in SrRuO3 enclosed by the yellow boxes on individual figures are enlarged below, where the levels of intensities are converted into different colors using the CalAtom software73. The calculated positions of atoms are marked by black dots. e Layer-resolved evolution of the atomic relative displacements within the Sr columns. Note, the positions of Ta atoms on KTaO3 side are counted as references. The blue lines represent the ideal values in the Pbnm phase. f Layer-resolved evolution of the atomic relative displacements within the O columns across the interface. The blue lines represent the theoretical values in the \(R\bar{3}c\) phase. In (e) and (f), the standard error of each data point is achieved by statistically sampling along the vertical [\(1\bar{1}0\)] direction.

Indeed, these characteristic features are captured by high-precision STEM high-angle annular dark field (HAADF) and annular bright field (ABF) images, as exhibited in Fig. 2c, d. It is evident that the Sr atoms are lined up, meanwhile, the relative displacements of the O atoms in every other column give rise to the same zigzag pattern, all compatible with the \(R\bar{3}c\) characters. This is further corroborated by quantitative analyses on the layer-resolved atomic displacements across the interface. There are overall no experimentally resolvable Sr displacements, far from the ideal magnitude in the Pbnm phase as indicated by the blue lines on Fig. 2e. On the other hand, the single O columns exhibit well-defined zigzag patterns on SrRuO3 side, with the magnitudes of relative displacements in good agreement with the theoretical value (Fig. 2f). These results lead us to conclude that the trigonal SrRuO3 phase that is inaccessible in bulk has been stabilized by means of (111) strain engineering.

Electronic and magnetic properties of trigonal SrRuO3

Next, we unravel the magnetic properties of trigonal SrRuO3. The temperature dependence of resistivity ρ(T) indicates an overall metallic behavior down to 2 K, with a kink present at Tkink ~ 150 K reflecting the para- to ferro-magnetic transition (Fig. 3a, top panel). The onset of ferromagnetism is clearly visible on the temperature-dependent spontaneous magnetization M(T) curve with a Curie temperature TC ~ 160 K (Fig. 3a, bottom panel). These values are in general close to those reported in bulk and (001)-oriented films9. However, unlike the uniaxial magnetic anisotropic behaviors observed in the Pbnm phase53, it is noteworthy that the trigonal SrRuO3 displays planar magnetic anisotropy below TC, and the moments lie in the (111) plane (Supplementary Fig. 3). A good fit to the in-plane spontaneous magnetization, \(M(T)={M}_{0}{(1-T/{T}_{{{{\rm{C}}}}})}^{\beta }\), gives rise to the critical exponent β = 0.36, consistently falling in the three-dimensional (3D) XY universality class54.

Fig. 3: Electronic and magnetic properties of trigonal SrRuO3.
figure 3

a Temperature dependence of resistivity and remnant magnetization. The inset exhibits the derivative curves highlighting individual transition temperature as labeled on the graph (Tkink and TC). The critical scaling fit on M(T) (blue solid line) close to the phase transition determines an exponent β ~ 0.36, falling in the three-dimensional XY universality class54. b Angular dependence of AMR and PHE measured in 8 T and 0.1 T magnetic field, respectively. Data are displayed in open square (circle) for AMR (PHE), and the black curves are fits from the equations given in the main text. The inset shows the geometry of measurements: the current was fixed along the [\(1\bar{1}0\)] direction and the magnetic field was applied within the (111) plane with an angle θ relative to the [\(1\bar{1}0\)] direction. The measurements started at an offset of θ around 67. c The magnitude of each harmonics extracted from the corresponding fits.

More insights into the magnetic state are obtained by measuring the in-plane anisotropic magnetoresistance (AMR) and planar Hall effect (PHE). In general, both AMR and PHE should exhibit dominant twofold oscillations as a function of θ in a full cycle with 45 relative phase shift (i.e. \({R}_{xx} \sim \cos 2\theta\); \({R}_{xy} \sim \sin 2\theta\)), due to rotation of the principle axes of the resistivity tensor55. However, symmetry of the lattice and magnetic structure can induce additional anisotropy, giving rise to higher harmonics of oscillations (fourfold, sixfold, etc.).

As shown in Fig. 3b, the angular dependence of the longitudinal and transverse resistance (Rxx and Rxy) of our sample were recorded simultaneously while the magnetic field was rotated within the (111) plane. At T = 2 K, the sixfold (fourfold) harmonic is clearly observed in AMR (PHE) under 8 T field, as can be more evidently seen from the extracted magnitude of each oscillation (Fig. 3c). While the fourfold Rxx,4 and sixfold Rxx,6 harmonics are well defined in AMR, the sixfold Pxy,6 term is much suppressed in PHE, referring to the existence of C3 rotation axes along the surface normal based on symmetry analyses56.

Note, the higher harmonics of oscillations diminish practically to zero in the absence of magnetic field (Fig. 3b, gray curves) and above TC (see Supplementary Fig. 5 for data at 170 K). Hence, these results confirm the presence of C3 symmetry of the magnetic state, coinciding with the trigonal nature of the underlying lattices47.

Ordinary and anomalous Hall effect

After demonstrating the structural and magnetic features, we turn to explore the Berry-phase manipulations and plausible topological properties of trigonal SrRuO3 from Hall measurements at a set of temperatures across TC (Fig. 4a). At T = 200 K, the Hall resistance is simple and linear with respect to field. Right below TC, nonlinear and hysteretic Rxy(B) curves are captured during 160 - 120 K, as a result of the mixture of AHE with nonlinear ordinary Hall effect (OHE). Strikingly, at lower temperatures from 90 to 2 K, the hysteresis of AHE exhibits a more intricate ‘two-step’ character, indicating the coexistence of two AHE channels with different coercive fields26. To separate each component from the net Hall signal, the data were fitted using a phenomenological model (see ‘Methods’ for details). A representative case at T = 60 K is displayed in Fig. 4b, where the two AHE components and the nonlinear OHE component have been extracted separately.

Fig. 4: Anomalous Hall effect of trigonal SrRuO3.
figure 4

a Evolution of the transverse magnetoresistance Rxy from 200 to 2 K. For each curve, the red branch represents the scan from +6 to -6 T field, and vice versa for the blue one. b A representative fit of data at T = 60 K, using the `two AHEs + nonlinear OHE' model as described in the main text. c The extracted spontaneous anomalous Hall resistivity ρAHE (total signal along with individual components) as a function of temperature. d Comparison of the normalized hysteresis loop between ρAHE of each component and the out-of-plane magnetization MOOP at 10 K. e, Plot of the linear relationship between the spontaneous \({\rho }_{{{{{\rm{AHE}}}}}_{2}}\) and MOOP from 120 to 10 K.

The nonlinear OHE can be rationally described using the two-band model, where both electrons and holes contribute to charge transport. This reflects the coexistence of electron and hole pockets on the Fermi surface, plausibly due to the splitting between majority and minority spin bands below the Curie temperature10. The dominant carriers are electrons with low mobility and high concentration (μe = 0.14 cm2V−1s−1, ne = 7.1 × 1023 cm−3 at 2 K). These values are in common with those reported in bulk or (001) SrRuO3 thin films57,58, indicating the electron pockets from normal quadratic bands. However, it is striking to note the rather high mobility of holes (μh = 4.8 × 103 cm2V−1s−1, nh = 1.1 × 1016 cm−3 at 2 K) in our trigonal SrRuO3. In general, existence of highly mobile carriers is rare in complex oxides due to strong correlations, and the observation of such may suggest its topologically nontrivial origin from the band structures13[,59,60,61. Particularly, electrons of high mobility (~ 104 cm2V−1s−1) have been lately reported in ultra-clean (001) SrRuO3 films, signaling the contributions of Weyl fermions in electrical transport13. We speculate in a similar fashion that the hole pockets in trigonal SrRuO3 originate from linear Weyl bands62.

We now discuss the AHE. Recap that our trigonal SrRuO3 favors the in-plane magnetic anisotropy parallel to the film surface, observation of AHE would be exotic in the geometry of standard Hall measurements. However, two channels of AHE are clearly probed in trigonal SrRuO3. Fig. 4c shows the temperature dependence of the spontaneous Hall resistivity (total ρAHE along with individual components \({\rho }_{{{{{\rm{AHE}}}}}_{1}}\) and \({\rho }_{{{{{\rm{AHE}}}}}_{2}}\)) of trigonal SrRuO3. Essentially, AHE1 is developed right below the Curie temperature, exhibiting non-monotonous evolutions as a function of temperature and significantly broader hysteresis loops than out-of-plane magnetization MOOP (Fig. 4d). These results suggest a momentum-space Berry-phase mechanism for AHE1, where the spontaneous Hall conductivity is intrinsically given by the nonzero Berry curvatures from Weyl pairs near the Fermi energy11. Recall that for the dominant twofold harmonics of AMR (Fig. 3b), the resistance at BI is smaller than BI (i.e., R < R), at adds with that of a trivial ferromagnetic metal where R > R is usually expected as a result of the spin-dependent scattering. This behavior can plausibly be rationalized by the chiral-anomaly induced negative MR effect, in which existence of Weyl nodes of opposite chiralities would lead to additional conducting channels under the geometry of BI13,51.

On the contrary, AHE2 is only visible below ~ 120 K and increases monotonously as temperature decreases. The narrow loops of normalized \({\rho }_{{{{{\rm{AHE}}}}}_{2}}(B)\) and MOOP(B) resemble each other closely with nearly identical coercive fields (Fig. 4d). Especially, the magnitude of spontaneous \({\rho }_{{{{{\rm{AHE}}}}}_{2}}\) exhibits a linear relationship with respect to MOOP at zero field (Fig. 4e). In this manner, the signal of \({\rho }_{{{{{\rm{AHE}}}}}_{2}}\) practically mimics the profile of magnetization M(B, T) that is perpendicular to the Hall plane. Hence we tend to believe that AHE2 is of conventional type as commonly expected in an itinerant ferromagnetic system, resulting from field-induced canting and pinning of magnetization along the [111] direction3.

Electronic band structure calculations

Since intrinsic AHE is a direct manifest of the band topology, the observed distinctive behaviors in trigonal SrRuO3 would refer to dramatically manipulated electronic band structures compared to the orthorhombic or tetragonal SrRuO3 phases that are normally stabilized in (001) thin films16. To highlight these information, we performed density functional theory (DFT) calculations to investigate the possible presence and distribution of Weyl points (see ‘Methods’ for more details).

First, we calculated the energy of SrRuO3 in various space groups. Six most plausible symmetries (cubic \(Pm\bar{3}m\); orthorhombic Pbnm, Imma; trigonal \(R\bar{3}c\); monoclinic C2/c, C2/m) were considered for comparison. To simulate the effect of epitaxial strain, the in-plane lattice parameter of the film was locked to the substrate’s value and the out-of-plane lattice parameter was allowed to relax. As shown in Fig. 1b, it is notable that the trigonal \(R\bar{3}c\) SrRuO3 with aaa rotation pattern exhibits the lowest energy on KTaO3 substrate, rather than the orthorhombic Pbnm with aac+ rotation pattern. By calculating the energies of these two phases as a function of (111) strain, we find that the \(R\bar{3}c\) phase starts to emerge under small tensile strain, whereas Pbnm is still favored under compressive strain. These results agree well with our experimental observations.

Next, we calculated the band structure of trigonal SrRuO3 in the non-magnetic case (Supplementary Fig. 6). According to the calculated compatibility relations of this band structure, there exist several topological Dirac points around the Fermi level along high-symmetry line Γ - T. The pattern of surface states at the Fermi level suggests a C6 rotation symmetry due to combined C3 and C2 symmetries (Supplementary Fig. 7). By taking the spin-orbit coupling (SOC) into account, we calculated the band structure of the ferromagnetic state with an in-plane spontaneous magnetization as suggested from our experiments. Due to breaking of the time-reversal symmetry, the doubly degenerate Dirac points are expected to split into Weyl pairs.

Two key features are captured from the calculations. On one hand, along high-symmetry k-paths, the Fermi energy crosses conventional quadratic bands, contributing the dominant carriers with low-mobility (Fig. 5a). The resultant electron pockets with finite density of states are reflected from the surface state spectrum (Fig. 5b). On the other hand, multiple pairs of Weyl nodes are found near the Fermi energy, distributed along low-symmetry k-paths in the first Brillouin zone (Supplementary Table 1). The linear dispersion of a representative Weyl point (W9) at an energy ~ 0.02 eV above the Fermi energy is shown in Fig. 5c, and all pairs of Weyl nodes within EW − EF < 0.05 eV above and below the Fermi energy are plotted in Fig. 5d. The high-mobility carriers are likely contributed from these Weyl bands. In addition, Weyl points of opposite chiralities near the Fermi energy serve as source and sink of the Berry curvatures, leading to intrinsic anomalous Hall conductivity (AHC) as illustrated in Fig. 5e, f, where the value of AHC depends sensitively on the position of the Fermi energy.

Fig. 5: Calculated electronic and topological properties of trigonal SrRuO3 with in-plane magnetization.
figure 5

a Band structure along the high-symmetry k-paths (Γ: (0 0 0); T: (0.5 0.5 0.5); F: (0.5 0.5 0); L: (0 0.5 0)). b Surface state spectrum (plotted in the logarithmic scale) at the Fermi energy projected on the (111) plane of trigonal SrRuO3. c The enlarged band structure around a linear crossing point denoted as W9, along Γ' (-0.04452,0.10343,0) to W9 (-0.04452,0.10343,-0.43273). d Distribution of the Weyl points over the first Brillouin zone within an energy range EW − EF < 0.05 eV above and below the Fermi energy. The colors represent Weyl points of different energies (W9 is drawn in green). The Cartesian coordinates (kx, ky, kz) are indicated on the figure and Γ, T, F, L, and Z denote the high-symmetry momenta. e Mapping of the Berry curvatures in the (111) plane. f Calculated anomalous Hall conductivity (AHC) of trigonal SrRuO3.

To conclude, our combined experimental and theoretical investigations demonstrate an emergent trigonal phase of SrRuO3 induced by (111) tensile strain, and reveal its peculiar magnetic and topological features which are remarkably different from bulk. These results provide ubiquitous insights into the scenario of film-substrate interplay for perovskite’s (111) heteroepitaxy. Moreover, it also highlights the opportunities for realizing intriguing correlated and topological quantum states of matter in oxides, especially for quantum anomalous Hall effect and topological insulator at the quasi-two-dimensional limit.


Sample fabrication

The (111) oriented SrRuO3 thin films were grown on 5 × 5 mm2 (111) KTaO3 single crystalline substrates by pulsed laser deposition. SrRuO3 ceramic target was ablated using a KrF excimer laser (λ = 248 nm, energy density ~ 2 J/cm2) with a repetition rate of 2 Hz. The deposition was carried out at a substrate temperature of 730 C, under an oxygen atmosphere of 75 mTorr. The films were post-annealed at the growth condition for 15 min, and then cooled down to room temperature. The whole deposition process was in-situ monitored by reflective-high-energy-electron diffraction.

Scanning transmission electron microscopy

The scanning transmission electron microscopy (STEM) measurements were carried out using a double spherical aberration-corrected JEM-ARM200F, operated at 200 kV. The sample was cut and projected onto the \({(11\bar{2})}_{{{{\rm{pc}}}}}\) plane. The high-angle annular dark-field (HAADF) imaging was taken using the collection semi-angle of about 70–250 mrad. Positions of the atoms were extracted using the CalAtom software, which basically normalized the brightness of the image and calculated in the vicinity of each atom the local maximum as the round center. This process gave rise to a simulated matrix where the relative position of each atom is labeled with two coordinates, based on which the bucking of the Sr and O sublattices were analyzed.

Magnetization measurements

The magnetization measurements were carried out using a Magnetic Property Measurement System (MPMS-3, Quantum Design). After zero-field cooling the sample down to 10 K, the temperature dependence of magnetization was measured from 10 K to 300 K in an external field of 1000 Oe applied in the film plane. Field dependence of magnetization was measured at 10 K by applying the magnetic field along [111], [\(1\bar{1}0\)], and [\(11\bar{2}\)] direction, respectively.

Transport measurements

The electrical transport measurements were performed in a Physical Property Measurement System (PPMS, Quantum Design) with the 4-point contact method. During the magneto-transport experiments, the magnetic field was applied along the [111] direction, and the current was driven along the [\(1\bar{1}0\)] direction. At each temperature, the longitudinal and transverse resistance were recorded while sweeping the field in a cycle from +6 T to -6 T to +6 T. Standard symmetrization (anti-symmetrization) treatment was further applied using these two branches of data on the longitudinal (transverse) resistance to achieve pure signals of the magnetoresistance (Hall resistance).

Details of analyses on anomalous Hall effect

At temperatures below the magnetic phase transition, the overall Hall effect of SrRuO3 on KTaO3 is attributed from both the ordinary Hall effect (OHE) and the anomalous Hall effect (AHE), \({R}_{xy}(B)={R}_{xy}^{{{{\rm{OHE}}}}}(B)+{R}_{xy}^{{{{\rm{AHE}}}}}(B)\). In particular, at lower temperatures (e.g. T = 60 K as shown Fig. 3a), the OHE component is nonlinear and the AHE component contains two contributions with different coercive fields. Thus we introduce the phenomenological model ‘two AHEs + nonlinear OHE’ to describe the Hall data. The AHE component \({R}_{xy}^{{{{\rm{AHE}}}}}(B)\) is expressed as:

$${R}_{xy}^{{{{\rm{AHE}}}}}(B)={R}_{xy}^{{{{\rm{AHE}}}}1}(B)+{R}_{xy}^{{{{\rm{AHE}}}}2}(B)={R}_{1}^{{{{\rm{AHE}}}}}\tanh [{\omega }_{1}(B-{B}_{c,1})]+{R}_{2}^{{{{\rm{AHE}}}}}\tanh [{\omega }_{2}(B-{B}_{c,2})]$$

where \({R}_{i}^{{{{\rm{AHE}}}}}\) and Bc,i represent the spontaneous AH resistance and the coercive field of hysteresis loop, respectively; ωi is a parameter describing the slope of loop at Bc,i. The nonlinear OHE component \({R}_{xy}^{{{{\rm{OHE}}}}}(B)\) is expressed based on the conventional two-carrier model:

$${R}_{xy}^{{{{\rm{OHE}}}}}(B)=\frac{B}{d\cdot e}\frac{({n}_{h}{\mu }_{h}^{2}-{n}_{e}{\mu }_{e}^{2})+({n}_{h}-{n}_{e}){\mu }_{h}^{2}{\mu }_{e}^{2}{B}^{2}}{{({n}_{h}{\mu }_{h}+{n}_{e}{\mu }_{e})}^{2}+({n}_{h}-{n}_{e}){\mu }_{h}^{2}{\mu }_{e}^{2}{B}^{2}}$$

where nh (ne), μh (μe), d and e represent the density of holes (electrons), the mobility of holes (electrons), the thickness of films, and the elementary charge, respectively.

To extract the parameters (nh, ne, μh, μe) from the fit, we denote \(\alpha =\frac{{\mu }_{e}}{{\mu }_{h}}\), \(\beta =\frac{{n}_{e}}{{n}_{h}}\) (α, β > 0). Then the \({R}_{xy}^{{{{\rm{OHE}}}}}(B)\) can be rewritten as:

$${R}_{xy}^{{{{\rm{OHE}}}}}(B)=\frac{B}{d\cdot e\cdot {n}_{h}}\frac{1}{1-\beta }\frac{\frac{(1-{\alpha }^{2}\beta )(1-\beta )}{{(1+\alpha \beta )}^{2}}+\frac{{(1-\beta )}^{2}{\alpha }^{2}}{{(1+\alpha \beta )}^{2}}{\mu }_{h}^{2}{B}^{2}}{1+\frac{{(1-\beta )}^{2}{\alpha }^{2}}{{(1+\alpha \beta )}^{2}}{\mu }_{h}^{2}{B}^{2}}$$

In the limits of B →  and B → 0, \({R}_{xy}^{{{{\rm{OHE}}}}}(B\to \infty )=\frac{B}{d\cdot e\cdot {n}_{h}}\frac{1}{1-\beta }\); \({R}_{xy}^{{{{\rm{OHE}}}}}(B\to 0)=\frac{B}{d\cdot e\cdot {n}_{h}}\frac{1-{\alpha }^{2}\beta }{{(1+\alpha \beta )}^{2}}\). We further denote:

$$K=\frac{1}{d\cdot e\cdot {n}_{h}}\frac{1}{1-\beta };$$
$$C=\frac{(1-{\alpha }^{2}\beta )(1-\beta )}{{(1+\alpha \beta )}^{2}};$$
$$D=\frac{{(1-\beta )}^{2}{\alpha }^{2}}{{(1+\alpha \beta )}^{2}}{\mu }_{h}^{2}$$

where K and C can be either positive or negative; D≥0. In this manner, the OHE component is eventually expressed as:

$${R}_{xy}^{{{{\rm{OHE}}}}}(B)=K\cdot B\cdot \frac{C+D\cdot {B}^{2}}{1+D\cdot {B}^{2}}$$

Meanwhile, based on the two-carrier model, the longitudinal resistance is expressed as:

$${R}_{xx}(B)=\frac{1}{d\cdot e}\frac{({n}_{h}{\mu }_{h}+{n}_{e}{\mu }_{e})(1+{\mu }_{h}{\mu }_{e}{B}^{2})}{{({n}_{h}{\mu }_{h}+{n}_{e}{\mu }_{e})}^{2}+{\mu }_{h}^{2}{\mu }_{e}^{2}{({n}_{e}-{n}_{h})}^{2}{B}^{2}}$$

At B = 0, it is rewritten as

$$\lambda ={R}_{xx}(B=0)=\frac{1}{d\cdot e}\frac{1}{{n}_{h}{\mu }_{h}+{n}_{e}{\mu }_{e}}=\frac{1}{d\cdot e}\frac{1}{{n}_{h}{\mu }_{h}}\frac{1}{1+\alpha \beta }$$

Here, λ can be directly obtained from the measured longitudinal resistance at zero field.

In this end, the measured longitudinal and transverse magnetoresistance at individual temperatures were simultaneously fit using equation (3), (9) and (11), from which one can achieve the fitting parameters {\({R}_{i}^{{{{\rm{AHE}}}}}\), ωi, Bc,i} relevant to AHE and those {K, C, D, λ} relevant to OHE. Eventually, the carrier densities and mobilities nh, ne, μh, μe can be achieved by solving the combined four equations of (6), (7), (8), (11) with a simple Mathematica program.

Anisotropic magnetoresistance and planar Hall effect

For the in-plane angle-dependent transport experiments, the current I was applied along the [\(1\bar{1}0\)] direction. The magnetic field B lied within the (111) plane, and rotated with an angle θ relative to the current direction. The measurements were started with an offset of θ ~ 67 , and the longitudinal resistance Rxx and the transverse resistance Rxy were measured simultaneously as a function of the angle. The obtained experimental data on Rxx and Rxy were fit using the following expansions up to the eightfold harmonics:

$${R}_{xx}=\mathop{\sum }\limits_{n=0}^{4}{R}_{xx,2n}={R}_{0}+\mathop{\sum }\limits_{n=1}^{4}{R}_{2n}\cos (2n(\theta +{\theta }_{2n}))$$
$${R}_{xy}=\mathop{\sum }\limits_{n=0}^{4}{P}_{xy,2n}={P}_{0}+\mathop{\sum }\limits_{n=1}^{4}{P}_{2n}\sin (2n(\theta +{\theta }_{2n}))$$

where R2n (P2n) is the magnitude of each harmonics in Rxx (Rxy). Note, due to the slight misalignment of contacts, signals from the transverse resistance could be mixed with small portions of the longitudinal resistance, giving rise to the P0 term in Rxy. The corresponding anisotropic magnetoresistance (AMR) and planar Hall effect (PHE) are defined as AMR = (Rxx − R0)/R0 × 100%; PHE = Rxy − P0, respectively.

Theoretical calculations

We performed first-principles calculations on the electronic properties of magnetic material SrRuO3 utilizing the generalized gradient approximation (GGA) with the revised Perdew-Burke-Ernzerhof for solids (PBEsol)63 and the projector augmented wave method64 as implemented in the VASP65. We constructed equivalent [111] crystalline direction of cubic for orthorhombic (space group:62) and trigonal (space group:167) lattice in order to match the KTO substrate and further tune the strain using fixed relaxed process. Under this substrate effect, the \(R\bar{3}c\) lattice with the point group D3d (-3m) takes up the ground state, which is transformed into primitive cell for following calculations. Note, the eg and t2g orbitals under Oh point group from cubic symmetry viewpoint has been changed into eg and a1g + eg orbitals under D3d, respectively. Meantime, basis function of the new eg orbital is (x2 − y2, xy) or (xz, yz) and the corresponding function of a1g orbital is only z2. The generators of this parent structure include identity, C3, {C2, (0, 0, 1/2)}, inversion, {1, {2/3, 1/3, 1/3}} pure translation group elements. Therefore, without considering the magnetization of Ru atoms, the non-magnetic band structure will keep the Kramer’ double degenerate states due to protection by time-reversal symmetry and space inversion symmetry (Supplementary Fig. 6). The Γ-center k-mesh was set as 7 × 7 × 7 for integration over Brillouin zone and kinetic energy cutoff was equal to 500 eV. The spin-orbital coupling was self-consistently included in the electronic computations. Considering the correlated effect of Ru-d orbital, Hubbard U value66 was set as 2 eV, which gives rise to 2 μB localization magnetic moment. Symmetry analysis based on magnetic topological quantum chemistry theory have been studied together with compatibility relations67,68,69, which suggests enforced semimetal feature. Meantime, tight-binding model was also constructed by projecting Ru-d as well as O-p orbital into localized wannier basis70. Chirality of nodes and anomalous hall effect were calculated using wanniertools software71.