Main

The polarization response of matter to an electric field forms an essential basis for many aspects of basic science and technology, such as ferroelectricity1, piezoelectricity2, magnetoelectricity3 and spintronics4. However, according to the modern theory of polarization, macroscopic electric polarization is well defined only in insulating crystals5. Gauss’s law also states that the electrostatic field inside a metal is zero due to the screening by free charge carriers6. These have been fundamental challenges in understanding and manipulating bulk polarity in metals. Despite these limitations, the artificial control of bulk polarity in metals through an appropriate external field holds a great potential for scientific and technological endeavours, based on recently emerging quantum materials. For example, by modifying electronic band topology7,8 and real-space spin texture9,10, the bulk polarity can be coupled with conduction electrons, leading to novel electronic and spin–orbitronic functionalities. Exploring a general scheme for controlling the bulk polarity in metals is, thus, of great interest and demand.

To this end, we focus on flexoelectricity that describes the generation of electric polarization Pflexo in the presence of a strain gradient ∂u/∂z as Pflexo = εfeff(∂u/∂z), where ε is the dielectric constant, feff is the flexocoupling coefficient and u is the strain (Fig. 1a)11,12,13,14,15. This effect is not limited to a certain crystal symmetry, so flexoelectricity is a universal property of all materials. Conceptually, flexoelectricity can be perceived as the polarization response of a medium to an applied flexoelectric field Eflexo = feff(∂u/∂z). Importantly, Eflexo is not a real electric field that obeys Gauss’s law but takes the essence of elastic fields. Thus, Eflexo should be regarded as a pseudo-electric field. It may be free of electrostatic screening and could serve as a unique means for controlling the physical properties, including bulk polarity, of a metal16. However, despite recent theoretical predictions of a strong flexocoupling effect in metals17,18, the existence of flexoelectricity in metals has yet to be experimentally validated. Here, we show theoretically and experimentally that Eflexo not only polarizes the bulk of a ferromagnetic metal SrRuO3 but can also control its electronic properties.

Fig. 1: First-principles calculation of flexoelectricity of SrRuO3.
figure 1

a, Schematics showing the flexoelectric field (Eflexo) generated from a gradient of shear strain (u31) in perovskite oxides ABO3. b, Plot of u31 (green triangles) and its gradient (du31/dz; yellow squares) in the theoretical SrRuO3 supercell with u31 varying spatially by artificially imposing Sr shearing in a sinusoidal form. c, Calculated off-centre displacements of Ru atoms, extracted from the SrRuO3 supercell under the shear strain gradient in b.

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Theoretical examination of flexoelectricity in a metal

We begin with a theoretical examination of the existence of flexoelectricity in SrRuO3. We constructed a supercell of cubic SrRuO3 in our theoretical simulation, as illustrated in Fig. 1a (see Methods for details). A sinusoidal shear strain u31 was imposed by artificially shearing Sr along the \([\bar{1}\bar{1}2]\) axis while relaxing the remaining ions (green triangles in Fig. 1b). Accordingly, a strain gradient was derived from ∂u31/∂z (yellow squares in Fig. 1b). We found a sizable off-centre displacement of Ru, comparable to 1 pm under a strain gradient of ~9 × 106 m−1 (Fig. 1c). We further found that the ionic off-centring persists in SrRuO3 regardless of the oxygen octahedral rotation (OOR) and the types of strain gradients, suggesting the robustness of flexoelectricity (Supplementary Figs. 1 and 2). Interestingly, a simulation of dielectric SrTiO3 yielded similar off-centre displacements of cations (Supplementary Fig. 3). This finding is consistent with a recent theoretical work predicting comparable flexocoupling coefficients of a metal and a dielectric17. Moreover, we have thoroughly examined the polar instability of bulk SrRuO3 in a variety of structures by zone-centre phonon calculations (Supplementary Fig. 4). Theoretically, it seems that it is practically impossible to realize a polar structure in bulk ‘homogeneous’ SrRuO3, in sharp contrast to the previously reported polar metals, for example LiOsO3 and NdNiO3 (refs. 19,20). This result thus emphasizes the critical role of strain gradients in polarizing SrRuO3.

Experimental design of strain gradients in SrRuO3

To achieve large strain gradients in SrRuO3, we consider a heterostructure of SrRuO3 epitaxially grown on a SrTiO3 (111) substrate (Fig. 2a). In its bulk form, SrRuO3 has a distorted, centrosymmetric perovskite structure with an OOR pattern described by a−a−c+ according to Glazer notation (where a minus sign represents antiphase tilting and a plus sign represents in-phase tilting). It forms an orthorhombic structure (Pbnm) below 820 K. Our first-principles calculations predicted that, under moderate compressive (111)pc strain (the subscript pc refers to pseudocubic; henceforth, the subscript will be omitted below), a monoclinic structure (C2/c) with an OOR pattern of a−a−c0 and a rhombohedral structure (\(R\bar{3}c\)) with an OOR pattern of a−a−a− can be stabilized with small energy cost, compared to the orthorhombic structure (Supplementary Fig. 5). Further considering the strong interfacial structure coupling along the (111) orientation19,21, we expect that the structure of the (111) SrRuO3 evolves from rhombohedral at the interface to monoclinic within a certain thickness range (Supplementary Fig. 6). In line with the expected monoclinic and rhombohedral structures, the high-quality SrRuO3 film on a SrTiO3 (111) substrate (Supplementary Fig. 7) exhibits completely suppressed in-phase tilting (namely, the c+ component) according to the half-integer Bragg diffraction (Supplementary Fig. 8).

Fig. 2: Atomic-scale imaging of the shear strain gradient and flexo-polar phase in SrRuO3.
figure 2

a, Illustration of the SrRuO3/SrTiO3 (111) heterostructure with the cations aligned coherently. b, Schematic showing the formation of a shear strain and its gradient in the SrRuO3/SrTiO3 (111) heterostructure. c, Cross-sectional ABF-STEM image of the SrRuO3/SrTiO3 (111) heterostructure. The red dashed line is drawn along the SrTiO3 [110] direction for reference when visualizing the Sr shear in SrRuO3. d, Zoomed-in images of representative SrTiO3 and SrRuO3 regions marked by the green and blue boxes in c, respectively. The bottom schematic shows the B-site cation displacement δB and the projected octahedral tilting angle θ. e,f, Mapping of δB (e) and θ (f) in c.

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Figure 2b schematically illustrates how to generate a large shear strain gradient, based on the inherent structural distortion in the (111) SrRuO3. The main crystalline axis [110] tilts away from the (111) plane, which is characterized by α. The theoretical value of α varies from 54.74° in cubic SrTiO3 to 54.76° in rhombohedral SrRuO3 and 56.14° in monoclinic SrRuO3. When the SrRuO3 layer is coherently bonded onto a cubic SrTiO3 substrate, the competition between inherent structural distortion and the epitaxial coherence enforces a shear strain along the \([\bar{1}\bar{1}2]\) direction of the (111) plane. Accordingly, the shear strain u31 can be calculated as: u31 = Δx/z = cot α1 − cot α2, where Δx is the cation shearing distance along the \([\bar{1}\bar{1}2]\) direction and z is the (111) interplanar constant. α1 is the expected tilt angle of SrRuO3 that is presumed to be fully coherent with SrTiO3, and α2 is the actual tilt angle of low-symmetry distorted SrRuO3. Therefore, as the interfacial SrRuO3 is expected to adopt the rhombohedral structure, it should be subject to negligible shear strain as well as the associated gradient. Within a sufficient distance from the interface, the large α difference between monoclinic SrRuO3 and cubic SrTiO3 gives rise to a substantial shear strain of around 3.5%. Assuming structural relaxation within a thickness of 10 nm, a giant strain gradient of 3.5 × 106 m−1 can form, possibly polarizing SrRuO3 through Eflexo.

Atomic-scale imaging of flexoelectricity in SrRuO3

To directly visualize the expected, so-called flexo-polar SrRuO3 at the atomic scale, we used scanning transmission electron microscopy (STEM) with the annular bright field (ABF) technique. Figure 2c clearly shows the shearing of Sr cations in SrRuO3 away from the SrTiO3 [110] axis (red dotted line). By extracting the Sr positions along the [110] atomic row (Supplementary Fig. 9), we evaluated the shear angle α and the shear strain. Although the shear strain is negligibly small in the interfacial region (~5 unit cells (u.c.)), it gradually develops to a maximum of ~1.5% at the top, yielding an averaged shear strain gradient across the film as large as 2.5 × 106 m−1 along the \([\bar{1}\bar{1}2]\) direction. Then, such a large strain gradient causes the Ru ions in SrRuO3 to be markedly displaced off-centre along \([\bar{1}\bar{1}0]\) (Fig. 2d,e), as they are subject to a vector component of the strain gradient. This is consistent with our calculation, which shows the lowest energy cost for the ionic displacement along [110] or \([\bar{1}\bar{1}0]\) (see Methods for details). The remarkable Ru displacement, greater than 25 pm locally (15 pm on average), is on par with that of the B-site cation in a prototypical ferroelectric oxide BaTiO3. The large polarity from Ru off-centring is surprising since Ru ions dominantly contribute electrons at the Fermi level and should endure strong carrier screening of polar displacements. Therefore, the flexo-polar metal phase of SrRuO3 is sharply distinct from the so-called Anderson–Blount polar metals such as LiOsO3 and NdNiO3 (ref. 19,20), whose polarity mainly comes from A-site cation displacements in a weak electron–phonon coupling scenario22.

Note that the experimentally observed Ru off-centring is a few tens of times larger than the theoretically predicted value under a comparable strain gradient in Fig. 1. To check if the discrepancy is from the choice of cell geometry for calculation and the form of strain gradients, we performed further calculations with a SrTiO3/SrRuO3/vacuum slab under a constant strain gradient to mimic the experimental structure. Importantly, Ru off-centre displacements persist and are comparable to those under the sinusoidal strain gradients (Supplementary Discussion and Supplementary Fig. 10). Therefore, the underestimation of Ru off-centre displacements in our calculation should have other origins. For example, theoretical calculations are based on defect-free crystals without temperature effects, whereas the experimental measurements were conducted at room temperature and could have been strongly influenced by the amount of point defects and the electronic conductivity of the sample23,24. In addition, huge strain gradients could cause a nonlinear flexoelectric response, leading to an order-of-magnitude enhancement of the flexoelectric effects in the experiments25,26. Also note that, although our analysis of the STEM image focused on the shear strain gradient, a longitudinal strain gradient should naturally accompany the structural evolution along the out-of-plane direction, yielding additional Eflexo along \([\bar{1}\bar{1}\bar{1}]\) (Supplementary Fig. 1). This is a result of the different c-lattice constants of the monoclinic and rhombohedral structures calculated with the fixed in-plane lattice constants of the (111)-oriented SrTiO3, which are 13.492 and 13.392 Å, respectively. This gives rise to a longitudinal strain of approximately 0.75%, corresponding to a strain gradient of 1.3 × 106 m−1, which is around half of the experimental value of the shear strain gradient. The longitudinal strain is not apparent in the STEM image, since the expected lattice variation is at most around ~1 pm locally, which is below the detection limit of STEM. However, we suggest that Eflexo along \([\bar{1}\bar{1}\bar{1}]\) through longitudinal strain gradients, which should naturally exist, could cooperate with Eflexo along \([\bar{1}\bar{1}2]\) through shear strain gradients, thereby facilitating the actual Ru off-centring to align along the direction close to \([\bar{1}\bar{1}0]\) (Supplementary Fig. 2).

We further analysed the OOR of SrRuO3 to examine its local structure change. As shown in Fig. 2d, the OOR manifests as a zigzag pattern of oxygen atoms and is characterized by the zigzag angle θ. In Fig. 2f, we map the θ of the same region by unit cell. Noticeably, both the OOR and the polarity are simultaneously suppressed in the interfacial SrRuO3 region of ~5 u.c. In addition, we found that the interfacial region shows a negligible strain gradient (Supplementary Fig. 11). These results are in line with the theoretically predicted rhombohedral SrRuO3 at the interface and support the proposed scenario that the spatial evolution of the SrRuO3 structure drives bulk flexo-polarity.

Correlation between strain gradient and Ru off-centring

To shed further light on the flexoelectric origin of the observed polar structure, we investigated the correlation between the polarity and strain gradient of the (111) SrRuO3 film. We first used optical second-harmonic generation (SHG) to quantify the polarity of SrRuO3. Figure 3a shows the threefold symmetric SHG anisotropy pattern of a SrRuO3/SrTiO3 (111) heterostructure, which is well fitted with the contribution of three electric dipoles in point group m (see Methods and Supplementary Fig. 6 for details). We plotted the SHG intensity IPP(2ω) versus the thickness t in Fig. 3b. The plot exhibits a non-monotonic evolution, first increasing with thickness up to ~13 u.c. and then gradually decreasing. This trend suggests that the measured SHG is dominantly contributed by the film bulk, consistent with the scenario of flexo-polarity. Therefore, these results confirm the emergence of a polar monoclinic phase in the bulk of the SrRuO3 film. Moreover, the 5 u.c. SrRuO3 yielded rather distorted SHG patterns with considerably suppressed intensity (Supplementary Fig. 13), which is consistent with our STEM observation (Supplementary Fig. 12). This result again supports the proposed rhombohedral structure at the interface, which should lack a strain gradient and the associated flexo-polarity.

Fig. 3: Correlation between strain gradient and polarity.
figure 3

a, SHG anisotropic patterns of a 12 u.c. SrRuO3/SrTiO3 (111) heterostructure acquired with the Pin–Pout (upper) and Pin–Sout (lower) configurations. Here, P and S stand for P- and S-polarised waves, respectively. The subscripts refer to incident (in) and outgoing (out) waves. b, SHG intensity IPP(2ω) of (111) SrRuO3/SrTiO3 versus the SrRuO3 film thickness, where IPP(2ω) is the average intensity of the peaks in the SHG pattern acquired in the Pin–Pout configuration from two different samples for each thickness. The error bars represent the standard deviation. c, Inhomogeneous strain versus thickness from the W-H analysis (red squares) and the theoretically predicted values for the ultrathin film (≤5 u.c.; black square). The blue line is guidance for the eye.

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Then, we performed Williamson–Hall (W-H) analysis based on the X-ray diffraction (XRD) peak broadening of the SrRuO3 film, which allowed us to estimate the inhomogeneous strain across the film27. The analysis was done on films with a thickness of over 10 u.c., which yielded a sufficient XRD signal for a peak analysis (Supplementary Figs. 14 and 15). The extracted inhomogeneous strain versus the film thickness is plotted as red squares in Fig. 3c. To validate the results, we also extracted the coherence length along the out-of-plane direction from the W-H analysis, which exhibits a good correlation with the film thickness (Supplementary Fig. 15). Besides, the ultrathin films (≤5 u.c.) are predicted to possess negligible strain gradients (black square in Fig. 3c), which is also supported by our STEM measurement (Supplementary Fig. 12). Based on these results, the inhomogeneous strain in the (111) SrRuO3 film evolves non-monotonically with the film thickness (the blue guideline in Fig. 3c). By simply assuming a linear strain profile, a similarly non-monotonic evolution of the strain gradient with the film thickness can be revealed (Supplementary Fig. 15). Comparing this result with the thickness-dependent SHG (Fig. 3b), we thus reveal a strong correlation between the strain gradient and the polarization.

Flexo-polar phase transition in SrRuO3

As the monoclinic structural distortion plays a critical role in the flexo-polarity in SrRuO3, conceivably, a polar-to-nonpolar phase transition would be coupled to a structural transition from the monoclinic to a higher-symmetry structure at high temperatures. The higher-symmetry structure is probably centrosymmetric rhombohedral according to the energy gain calculated for different structures. To investigate this, we measured the temperature-dependent SHG intensity of the (111) SrRuO3 film (Fig. 4a). With increasing temperature, the polar-to-nonpolar transition seems to occur at ~630 K, followed by a temperature-independent plateau of weaker intensity. The finite value of the latter part can be explained by the electric quadrupole contribution of a centrosymmetric rhombohedral phase (Supplementary Fig. 16a). The transition can be described by a critical behaviour: \({I}^\mathrm{PP}(2\omega )\propto {({T}_\mathrm{f}-T)}^{\mu }\), where the transition temperature Tf and the critical exponent μ are fitted as ~630 K and 1.4, respectively (Supplementary Fig. 16b). The rather large μ is like those observed in improper ferroelectrics28.

Fig. 4: Flexo-polar phase transition in SrRuO3.
figure 4

a, Temperature-dependent SHG intensity IPP(2ω), obtained for the Pin–Pout configuration. b, Temperature-dependent evolution of the (111) d spacing of SrRuO3 (blue circles) and SrTiO3 (grey squares). The uncertainty of the d spacing in our measurement (Methods) is approximately 3 × 10−5 and 1 × 10−6 Å for SrRuO3 and SrTiO3, respectively, but not shown here. The red solid line shows the predicted lattice evolution of the polar SrRuO3. The red and black dashed lines show the lattice evolution, depending linearly on temperature, of nonpolar SrRuO3 and SrTiO3, respectively.

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Furthermore, temperature-dependent XRD measurements confirm that the polar-to-nonpolar phase transition is coupled to a structural transition, as evidenced by a kink at ~630 K in the temperature-dependent (111) lattice constant of SrRuO3 (Fig. 4b). The structural anomaly resembles the temperature-driven, symmetry-elevating structure transition observed in (001)-oriented SrRuO3 films29. The nonlinear evolution of the lattice constant below 630 K is a consequence of the electrostrictive-like expansion caused by Ru off-centring. The gradual evolution reflects the dependence of phase competition on temperature, for which the monoclinic and the rhombohedral phases are preferred below and above the structural transition temperature, respectively. Therefore, the energy landscape between the two phases is expected to be gradually flattened in the approach to the structural transition with increasing temperature, resulting in metastable structures of intermediate distortion with less shearing and consequently less strain gradients. The evolution can be described by a thermodynamically derived equation:30 \({d}_{(111)}(T\,)={d}_{(111)}^{{\rm{bulk}}}(T\,)\times [1+aP{(T\,)}^{2}]\), as shown by the red solid line. In the equation, \({d}_{(111)}(T\,)\) and \({d}_{(111)}^{{\rm{bulk}}}(T\,)\) are the lattice constants of the SrRuO3 film and the nonpolar bulk SrRuO3, respectively. \({d}_{(111)}^{{\rm{bulk}}}(T\,)\) linearly depends on temperature due to thermal expansion (red dashed line). P(T) is the polarity evolution of the SrRuO3 film, which can be extracted from Fig. 4a. a is a constant related to elastic constants. We also verified the disappearance of the strain gradient above 630 K by temperature-dependent W-H analysis (Supplementary Fig. 17), which further corroborates the inferred flexo-polar phase transition.

Flexoelectric control of electronic properties in SrRuO3

SrRuO3 is a well-recognized correlated electron system that exhibits strong coupling among lattice, charge, spin and orbital degrees of freedom. Therefore, the markedly large Ru off-centre displacement is expected to profoundly impact the electronic properties of SrRuO3, due to the change in the Ru–O bonding environment and the dominant contribution of Ru to electrons at the Fermi level. Indeed, our first-principles calculations show that Ru off-centring induces considerable reduction of the t2g bandwidth (Fig. 5 and Supplementary Fig. 18), thereby enhancing electron correlation. This characteristic can be manifested in the reduced electron mobility, as suggested by our transport measurements (Supplementary Fig. 18 and Discussion). The enhanced electron correlation is also supposed to strengthen the magnetic ordering of SrRuO3 according to Stoner theory (Supplementary Discussion)31, which can be supported by our magnetization measurements (Supplementary Fig. 19). Furthermore, we suggest that the uniaxial magnetic anisotropy of SrRuO3 may be impacted by an inverse magnetostriction effect and the enhanced electron correlation originating from Ru off-centring (Supplementary Discussion and Supplementary Fig. 20). Although the detailed coupling mechanism between the electron/spin properties and Ru off-centring requires further extensive work, our results indicate the tremendous potential of flexoelectricity as a crucial tuning knob in metallic and magnetic materials.

Fig. 5: Flexoelectric control of the electronic band structure.
figure 5

Total density of states of the Ru d orbitals calculated for the nonpolar bulk SrRuO3 (blue line) and the polar SrRuO3 (red line) with a Ru off-centre displacement of 20 pm. a.u., arbitrary units.

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Outlook

Bulk polarity tends to be incompatible with many technologically important electronic properties, such as metallicity, superconductivity, ferromagnetism and strong spin–orbit coupling. Challenging this, our study reveals that bulk flexo-polarity in a metallic ferromagnet SrRuO3 could even enhance ferromagnetism while preserving the metallicity with little loss. Given the universality of flexoelectricity11, our approach is generally applicable to a wide range of materials. Note that although our analysis has focused on the gradients of shear and longitudinal strain originating from the thickness-dependent structural transition, other sources of strain gradients may also contribute and require further microscopic structure characterization for clarification, for example, the ferroelastic domain walls (Supplementary Fig. 6). Moreover, an important question left for future theoretical and experimental research is the actual flexoelectric coefficients of metals. Importantly, experimental methods for applying flexoelectric fields have recently been well established32, with several outstanding characteristics, including universality, non-destructiveness and high speed. Thus, these recent advances in experimental techniques further improve the prospects for exploring exotic physical properties using flexoelectric fields.

Methods

Theoretical calculation

First-principles density functional theory (DFT) calculations were performed within the local density approximation plus U (LDA + U) scheme using the Vienna ab initio simulation package (VASP, ref. 33). The projector augmented-wave method was used with the Ceperley–Alder exchange–correlation functional34. The on-site Coulomb interaction considered by rotationally invariant LDA + U was parameterized by U = 3 eV and J = 0.75 eV for Ru d orbitals35. The choice of the parameters was consistent with that in other studies36,37. We used the charge-only LDA exchange–correlation functional with the plus U extension, giving increasing exchange splitting proportional to J for SrRuO3 (ref. 38). With our choice of the exchange–correlation functional and parameters for on-site Coulomb interaction, the calculated lattice constants and magnetic moments of the bulk SrRuO3 in Pbnm space group are in good agreement with experimental data (Supplementary Table 3 and Supplementary Fig. 21). Moreover, the calculated magnetic moment of a (111)-oriented Imma structure relaxed with fixed hexagonal in-plane lattice constants of SrTiO3 was 2μB/Ru with a moderate overestimation relative to the experimental value of 1.58μB/Ru (Supplementary Fig. 19). The increase in the magnetic moment may be due to the static mean-field approximation of DFT, as the DFT plus dynamical mean-field theory study of SrRuO3 (ref. 39) shows a decreased magnetic moment of 1.6μB/Ru. We used an energy cutoff of 500 eV. The k-point samplings of 6 × 6 × 4 for the 20-atom \(\sqrt{2}\times \sqrt{2}\times 2\) unit cell, 6 × 6 × 3 for the 30-atom \(\sqrt{2}\times \sqrt{2}\times 2\sqrt{3}\) unit cell and 3 × 3 × 3 for the 120-atom \(2\sqrt{2}\times 2\sqrt{2}\times 2\sqrt{3}\) unit cells were used for bulk and strained-bulk calculations. To evaluate the flexoelectric distortions, we used 6 × 6 × 1 k-point samplings for the 150-atom \(\sqrt{2}\times \sqrt{2}\times 10\sqrt{3}\) (111)-oriented supercells and also for the 120-atom (111)-oriented vacuum/slab geometries. For the slab calculations, the dipole correction implemented in VASP was used. The atomic positions were relaxed with a force threshold of 0.02 eV/Å.

The flexoelectric distortions were calculated for two different supercell orientations. For the strain gradient in the [111] direction, two \(\sqrt{2}\times \sqrt{2}\times 10\sqrt{3}\) hexagonal supercells were constructed based on the fully relaxed five-atom cubic SrRuO3 (space group \({Pm}\bar{3}m\)) and 20-atom orthorhombic SrRuO3 (space group Imma) structures with a0a0a0 and a−a−c0 Glazer symbols, respectively. We found no ferroelectric instability in the cubic and orthorhombic structures. The initial atomic structure of each supercell was prepared with the shear and/or longitudinal displacement of the ith atom with an amount of \(h\times {Nc}\sin \left(\frac{2\pi * {z}_{i}}{{Nc}}\right)\), where N is the number of unit cells in the z direction, c is the hexagonal out-of-plane bulk lattice constant, zi is the position along the direction of the strain gradient and h is a parameter controlling the strain gradient40. The parameter h was set to produce a maximum shear strain gradient of 9 × 106 m−1, comparable to the experimental value (3.5 × 106 m−1). A maximum longitudinal strain gradient of 4.5 × 106 m−1 was applied, based on the change in the out-of-plane lattice constant of 0.75% between rhombohedral and monoclinic structures in strained-bulk calculations (Supplementary Table 2), corresponding to approximately half of the shear strain gradient. When fixing the Sr atoms at the initial positions, the atomic positions of the Ru and O atoms were relaxed. The off-centre displacements of Ru atoms from the initial positions were measured. The flexoelectric distortion of SrTiO3 was calculated in the same way for a (111)-oriented supercell, constructed based on the relaxed cubic structure.

To investigate the preferred direction of Ru off-centring, we calculated the energy cost of shifting the Ru atoms by 0.05 Å along different pseudocubic directions: [100], [010], [001], [110], [101], [011], [111] and [\(1\bar{1}\bar{1}\)]. The calculated energy cost per formula unit for these directions relative to the [110] direction is 0.61, 0.61, 2.46, 0, 1.37, 1.37, 0.71 and 1.29 meV, respectively.

The change in the partial density of states as a function of Ru displacement was evaluated by gradually shifting the Ru positions along the pseudocubic \([\bar{1}\bar{1}0]\) direction from those in the relaxed Imma (a−a−c0) structure. We calculated the nonmagnetic partial density of states of Ru d orbitals, which dominantly contribute the electronic states at the Fermi level and are mainly responsible for the electronic properties of SrRuO3.

Sample fabrication and characterization

SrRuO3 films were grown on (001) or (111) SrTiO3 substrates using a pulsed-laser deposition system with a KrF excimer laser (248 nm). A high-pressure reflection high-energy electron diffraction system was used to monitor the growth. Before deposition, SrTiO3 substrates (miscut <0.1°) were etched with a buffered hydrofluoric acid solution and then annealed in air at 1,050 °C for 1 h to produce an atomically flat surface with a unit-cell step terrace structure. The substrates obtained were further leached in deionized water for 1 h to remove excess Sr on the surface. During deposition, the temperature of the substrate was maintained at 675 °C. Ultrathin SrRuO3 films were grown under an oxygen pressure of 100 mTorr with a laser fluence of 2 J cm−2.

Atomic force microscopy was performed using a Cypher scanning probe microscope (MFP-3D, Asylum Research) with Ir-Pt-coated tips (PPP-EFM, Nanosensors). A room-temperature θ–2θ scan and reciprocal space mapping by XRD were performed with a Bruker D8 Discover. Half-integer Bragg diffraction and temperature-dependent ω–2θ scans were carried out using synchrotron XRD at the 3A beamline of Pohang Accelerator Laboratory. The (111) lattice constants of SrRuO3 and SrTiO3 were extracted from a Gaussian fitting of the (222) peaks with uncertainty. For the temperature-dependent W-H analysis, we used a high-resolution XRD (Panalytical X’pert Pro MRD) equipped with a four-bounce Ge 220 cut monochromator giving pure Cu Kα1 radiation and an Anton Paar DHS 1100 domed hot stage. The sample was aligned at each temperature to account for the thermal expansion of the stage. We used a Pearson VII function to fit the XRD peaks:

$$I(2\theta )={I}_{\max }\frac{{w}^{2m-1}}{{[{w}^{2}+4({2}^{1/m}-1){(2\theta -2{\theta }_{0})}^{2}]}^{m}}.$$

This function becomes a Lorentzian as m → 1 and a Gaussian as m → ∞. As we mostly fitted the XRD data with m > 50, our fits reasonably correspond to Gaussian fits. W-H analysis uses a quadratic relation, that is, \({(\beta \cos \theta )}^{2}={(K\frac{\lambda }{D})}^{2}+{(4{u}_{i}\sin \theta )}^{2}\), for Gaussian peaks and a linear relation, that is, \(\beta \,\cos \theta =K\frac{\lambda }{D}+4{u}_{i}\,\sin \theta\), for the Lorentzian, where D is the coherence length, λ is the X-ray wavelength and K is a geometrical constant close to 1 (usually 0.9). Therefore, we performed the W-H analysis using the quadratic relation. From the slope of linear fits of (βcosθ)2 versus (4sinθ)2, we estimated the inhomogeneous strain ui. Also, from the intercept at the origin of linear fits of (βcosθ)2 versus (4sinθ)2, we estimated the coherence length D.

SHG measurement

SHG measurements were performed with two femtosecond wave sources possessing central wavelengths of 800 and 840 nm and repetition rates of 80 MHz and 250 kHz, respectively. The polarization of the fundamental wave and the second-harmonic wave was controlled by a half-wave plate and a Glan–Taylor polarizer, respectively. We isolated the SHG wave from the fundamental wave by adopting low-pass and band-pass filters, and the isolated SHG wave was detected by a photomultiplier tube. The laser fluence for the SHG measurement was 0.127 mJ cm−2, which did not induce any visible damage during the experiment. We found that surface burning occurred once the laser fluence was increased to several mJ cm−2 with continuous illumination for 30 min. Temperature-dependent measurements were carried out in an incident-plane-rotating set-up with a high-temperature stage (HCP621G; Instec; ref. 41).

The SHG results were analysed with both electric quadrupole (EQ) and electric dipole (ED) contributions, which are given by \({I}_{\rm{EQ}}^{2\omega }\propto{\left|{E}_{\rm{EQ}}^{2\omega }\right|}^{2}={\left|{\chi }_{{ijkl}}{E}_{j}^{\omega }{\partial }_{k}{E}_{l}^{\omega }\right|}^{2}\) and \({I}_{\rm{ED}}^{2\omega}\propto{\left|{E}_{\rm{ED}}^{2\omega }\right|}^{2}={\left|{\chi }_{{ijk}}{E}_{j}^{\omega }{E}_{k}^{\omega }\right|}^{2}\), respectively. Here, \({E}_{i}^{\omega }\) denotes the electric field component of the fundamental wave with optical polarization along the i axis. \({\chi }_{{ijkl}}\) and \({\chi }_{{ijk}}\) represent the second- and third-order susceptibility tensor components, respectively. For the room-temperature polar phase of SrRuO3, we used the electric dipole contribution of three equivalent polar monoclinic domains with point group m for fitting in consideration of the observed polar axis. For the high-temperature nonpolar phase, we considered the electric quadrupole contribution of a nonpolar rhombohedral structure with point group \(\bar{3}m\) for fitting. In the following, we provide the nonlinear susceptibility tensor components allowed for the considered crystal symmetry and the details of the analysis.

For the electric dipole contribution of three equivalent polar monoclinic domains (point group m; Supplementary Fig. 6), the second-order susceptibility tensors for the monoclinic point group m are

$${\chi }_{{xxx}},\,{\chi }_{{xyy}},\,{\chi }_{{xzz}},\,{\chi }_{{xzx}},\,{\chi }_{{xxz}},\,{\chi }_{{yyz}},\,{\chi }_{{yzy}},$$
$${\chi }_{{yxy}},\,{\chi }_{{yyx}},\,{\chi }_{{zxx}},\,{\chi }_{{zyy}},\,{\chi }_{{zzz}},\,{\chi }_{{zzx}},\,{\chi }_{{zxz}}.$$

To consider the SHG wave in the sample coordinates (x', y', z'), we decomposed the tangential components of the fundamental wave in the laboratory coordinates (x, y, z). The two coordinate systems can be transformed into each other according to the conventional rule of three-dimensional coordinate rotation that is described by two angles α and β, where (x, y, z) is first rotated around the y axis by α and then around the z axis by β to obtain (x', y', z'). Therefore, the electric field components of the fundamental wave inside the thin film (\({E}_{{x}^{{\prime} }}^{\omega },{E}_{{y}^{{\prime} }}^{\omega },{E}_{{z}^{{\prime} }}^{\omega }\)) are:

$${E}_{{x}^{{\prime} }}^{\omega }=\left({E}_{x}^{\omega }\cos \alpha +{E}_{z}^{\omega }\sin \alpha \right)\cos \beta -{E}_{y}^{\omega }\sin \beta$$
$${E}_{{y}^{{\prime} }}^{\omega }=\left({E}_{x}^{\omega }\cos \alpha +{E}_{z}^{\omega }\sin \alpha \right)\sin \beta +{E}_{y}^{\omega }\sin \beta$$
$${E}_{{z}^{{\prime} }}^{\omega }=-{E}_{x}^{\omega }\sin \alpha +{E}_{z}^{\omega }\cos \alpha$$

Then, the electric field components of the SHG wave contributed by the electric dipole (\({E}_{{x}^{{\prime} }}^{2\omega },{E}_{{y}^{{\prime} }}^{2\omega },{E}_{{z}^{{\prime} }}^{2\omega }\)) are

$${E}_{x}^{2\omega }=\left({E}_{{x}^{{\prime} }}^{2\omega }\cos \beta +{E}_{{y}^{{\prime} }}^{2\omega }\sin \beta \right)\cos \alpha -{E}_{{z}^{{\prime} }}^{2\omega }\sin \alpha$$
$${E}_{y}^{2\omega }={E}_{{y}^{{\prime} }}^{2\omega }\cos \beta -{E}_{{x}^{{\prime} }}^{2\omega }\sin \beta$$
$${E}_{z}^{\omega }=\left({E}_{{x}^{{\prime} }}^{2\omega }\cos \beta +{E}_{{y}^{{\prime} }}^{2\omega }\sin \beta \right)\sin \alpha +{E}_{{z}^{{\prime} }}^{2\omega }\cos \alpha$$

The SHG of three equivalent monoclinic domains, the orientations of which in the laboratory coordinates are rotated by 120° around the z axis from each other, can then be obtained by coherently adding up the contribution of each domain: \({I}_{\rm{total}}^{2\omega }={\left|{E}_{{\rm{\theta }}}^{2\omega }+{E}_{{\rm{\theta }}+120^\circ }^{2\omega }+{E}_{{\rm{\theta }}+240^\circ }^{2\omega }\right|}^{2}\).

For the electric quadrupole contribution of a centrosymmetric rhombohedral structure (point group \(\bar{3}m\)), the third-order susceptibility tensors are

$${\chi }_{{zzzz}},\,{\chi }_{{xxxx}}={\chi }_{{yyyy}}={\chi }_{{xxyy}}+{\chi }_{{xyyx}}+{\chi }_{{xyxy}},$$
$${\chi }_{{xxyy}}={\chi }_{{yyxx}},{\chi }_{{xyyx}}={\chi }_{{yxxy}},{\chi }_{{xyxy}}={\chi }_{{yxyx}},$$
$${\chi }_{{yyzz}}={\chi }_{{xxzz}},{\chi }_{{zzyy}}={\chi }_{{zzxx}},{\chi }_{{zyyz}}={\chi }_{{zxxz}},{\chi }_{{yzzy}}={\chi }_{{xzzx}},$$
$${\chi }_{{xxxz}}=-{\chi }_{{xyyz}}=-{\chi }_{{yxyz}}=-{\chi }_{{yyxz}},$$
$${\chi }_{{xxzx}}=-{\chi }_{{xyzy}}=-{\chi }_{{yxzy}}=-{\chi }_{{yyzx}},$$
$${\chi }_{{xzxx}}=-{\chi }_{{xzyy}}=-{\chi }_{{yzxy}}=-{\chi }_{{yzyx}},$$
$${\chi }_{{zxxx}}=-{\chi }_{{zxyy}}=-{\chi }_{{zyxy}}=-{\chi }_{{zyyx}},$$
$${\chi }_{{zyzy}}={\chi }_{{xzxz}},{\chi }_{{zyzy}}={\chi }_{{zxzx}}$$

The analytical expressions for SHG used for fitting are

$$\begin{array}{l}{I}^{\rm{PP}}\left(2\omega \right)={A}_{1}\left[{\left({\chi }_{{zyzy}}-{2\chi }_{{zzyy}}+{\chi }_{{zzzz}}+{\chi }_{{zyyy}}\sin \left(3\phi \right)\right)}^{2}\right.\\\left.+{\left({2\chi }_{{xx}yy}+{\chi }_{xy{xy}}-{2\chi }_{{xxxy}}+{\chi }_{{yzyz}}+\left({\chi }_{{yyzy}}-{\chi }_{{yy}{yz}}\right)\sin \left(3\phi \right)\right)}^{2}\right]\end{array}$$
$${I}^{\rm{PS}}\left(2\omega \right)={A}_{2}{\left({\chi }_{{xyxy}}-{\chi }_{yyzy}\sin \left(3\phi \right)\right)}^{2}+{\left({\chi }_{zyzy}+{\chi }_{{zyyy}}\sin \left(3\phi \right)\right)}^{2}$$
$${I}^{\,\rm{SP}}\left(2\omega \right)={A}_{3}{\left(\left({\chi }_{yyzy}-{2\chi }_{{yyyz}}\right)\cos \left(3\phi \right)\right)}^{2}$$
$${I}^{\,\rm{SS}}\left(2\omega \right)={A}_{4}{\left({\chi }_{{yyzy}}\cos \left(3\phi \right)\right)}^{2}$$

STEM measurement and analyses

The cross-sectional STEM specimen was prepared by first thinning the sample using a focused ion beam milling workstation (Helios NanoLab 650, FEI Co) with low-energy ion beams (<2 kV), followed by focused low-energy (<500 eV) Ar-ion milling (NanoMill 1040, E.A. Fischione Instruments). Zone axis [\(1\bar{1}0\)] was predetermined by high-resolution XRD.

Atomic-resolution ABF-STEM experiments were performed at room temperature using spherical aberration probe-corrected STEM with an acceleration voltage of 200 kV (JEM-ARM 200F, JEOL Ltd). The instrument was equipped with a cold-field emission gun installed at the National Center for Inter-University Research Facilities, Seoul National University, South Korea. The TEM specimen was cleaned using an ion cleaner (JEC-4000DS, JEOL Ltd) before the STEM experiments. For STEM analyses, the semiconvergence angle was set to 24 mrad and the semicollection angle range was set to 12–24 mrad.

To minimize scan distortion and enhance the signal-to-noise ratio, 20 frames of serial STEM images were acquired with a short dwell time of 2 μs px−1. Each image was 1,024 × 1,024 in size. The image series was registered using both rigid and nonrigid methods, which yielded similar results42. The atomic positions were extracted using a two-dimensional Gaussian fitting method with seven parameters based on a customized MATLAB script. To eliminate the artefact induced by the tail of heavy elements, atomic positions were calculated in atomic mass order after removing the larger atomic peaks.

Low-temperature magnetism and transport measurements

Conventional photolithography and ion milling were used to pattern the SrRuO3 films into the Hall bar geometry. The channel size was minimized to 50 × 50 μm2. Pt (50 nm) electrodes were sputtered onto the Hall bar contacts to reduce contact resistance. Magnetization measurements were performed using a superconducting quantum interference device magnetometer (MPMS; Quantum Design). The longitudinal and transverse resistance were measured using a physical properties measurement system (Quantum Design) on standard Hall bars. Magnetic field angle-dependent Hall effect measurements were carried out with a homemade rotational stage.

Reporting summary

Further information on research design is available in the Nature Portfolio Reporting Summary linked to this article.