Abstract
Electric polarization is well defined only in insulators not metals, and there is no general scheme to induce and control bulk polarity in metals. Here we circumvent this limitation by utilizing a pseudoelectric field generated by inhomogeneous lattice strain, namely a flexoelectric field, as a means of polarizing and controlling a metal. Using heteroepitaxy and atomicscale imaging, we show that flexoelectric fields polarize the bulk of an otherwise centrosymmetric metal SrRuO_{3}, with offcentre displacements of Ru ions. This further impacts the electronic bands and lattice anisotropy of the flexopolar SrRuO_{3}, potentially leading to an enhancement of electron correlation, ferromagnetism and its anisotropy. Beyond conventional electric fields, flexoelectric fields may be used to create and control electronic states through pure atomic displacements.
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Main
The polarization response of matter to an electric field forms an essential basis for many aspects of basic science and technology, such as ferroelectricity^{1}, piezoelectricity^{2}, magnetoelectricity^{3} and spintronics^{4}. However, according to the modern theory of polarization, macroscopic electric polarization is well defined only in insulating crystals^{5}. Gauss’s law also states that the electrostatic field inside a metal is zero due to the screening by free charge carriers^{6}. 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 topology^{7,8} and realspace spin texture^{9,10}, 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 P_{flexo} in the presence of a strain gradient ∂u/∂z as P_{flexo} = εf_{eff}(∂u/∂z), where ε is the dielectric constant, f_{eff} 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 E_{flexo} = f_{eff}(∂u/∂z). Importantly, E_{flexo} is not a real electric field that obeys Gauss’s law but takes the essence of elastic fields. Thus, E_{flexo} should be regarded as a pseudoelectric 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 metal^{16}. However, despite recent theoretical predictions of a strong flexocoupling effect in metals^{17,18}, the existence of flexoelectricity in metals has yet to be experimentally validated. Here, we show theoretically and experimentally that E_{flexo} not only polarizes the bulk of a ferromagnetic metal SrRuO_{3} but can also control its electronic properties.
Theoretical examination of flexoelectricity in a metal
We begin with a theoretical examination of the existence of flexoelectricity in SrRuO_{3}. We constructed a supercell of cubic SrRuO_{3} in our theoretical simulation, as illustrated in Fig. 1a (see Methods for details). A sinusoidal shear strain u_{31} 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 ∂u_{31}/∂z (yellow squares in Fig. 1b). We found a sizable offcentre displacement of Ru, comparable to 1 pm under a strain gradient of ~9 × 10^{6} m^{−1} (Fig. 1c). We further found that the ionic offcentring persists in SrRuO_{3} 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 SrTiO_{3} yielded similar offcentre displacements of cations (Supplementary Fig. 3). This finding is consistent with a recent theoretical work predicting comparable flexocoupling coefficients of a metal and a dielectric^{17}. Moreover, we have thoroughly examined the polar instability of bulk SrRuO_{3} in a variety of structures by zonecentre phonon calculations (Supplementary Fig. 4). Theoretically, it seems that it is practically impossible to realize a polar structure in bulk ‘homogeneous’ SrRuO_{3}, in sharp contrast to the previously reported polar metals, for example LiOsO_{3} and NdNiO_{3} (refs. ^{19,20}). This result thus emphasizes the critical role of strain gradients in polarizing SrRuO_{3}.
Experimental design of strain gradients in SrRuO_{3}
To achieve large strain gradients in SrRuO_{3}, we consider a heterostructure of SrRuO_{3} epitaxially grown on a SrTiO_{3} (111) substrate (Fig. 2a). In its bulk form, SrRuO_{3} 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 inphase tilting). It forms an orthorhombic structure (Pbnm) below 820 K. Our firstprinciples 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^{−}c^{0} 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) orientation^{19,21}, we expect that the structure of the (111) SrRuO_{3} 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 highquality SrRuO_{3} film on a SrTiO_{3} (111) substrate (Supplementary Fig. 7) exhibits completely suppressed inphase tilting (namely, the c^{+} component) according to the halfinteger Bragg diffraction (Supplementary Fig. 8).
Figure 2b schematically illustrates how to generate a large shear strain gradient, based on the inherent structural distortion in the (111) SrRuO_{3}. 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 SrTiO_{3} to 54.76° in rhombohedral SrRuO_{3} and 56.14° in monoclinic SrRuO_{3}. When the SrRuO_{3} layer is coherently bonded onto a cubic SrTiO_{3} 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 u_{31} can be calculated as: u_{31} = Δx/z = cot α_{1} − cot α_{2}, where Δx is the cation shearing distance along the \([\bar{1}\bar{1}2]\) direction and z is the (111) interplanar constant. α_{1} is the expected tilt angle of SrRuO_{3} that is presumed to be fully coherent with SrTiO_{3}, and α_{2} is the actual tilt angle of lowsymmetry distorted SrRuO_{3}. Therefore, as the interfacial SrRuO_{3} 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 SrRuO_{3} and cubic SrTiO_{3} 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 × 10^{6} m^{−1} can form, possibly polarizing SrRuO_{3} through E_{flexo}.
Atomicscale imaging of flexoelectricity in SrRuO_{3}
To directly visualize the expected, socalled flexopolar SrRuO_{3} 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 SrRuO_{3} away from the SrTiO_{3} [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 × 10^{6} m^{−1} along the \([\bar{1}\bar{1}2]\) direction. Then, such a large strain gradient causes the Ru ions in SrRuO_{3} to be markedly displaced offcentre 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 Bsite cation in a prototypical ferroelectric oxide BaTiO_{3}. The large polarity from Ru offcentring is surprising since Ru ions dominantly contribute electrons at the Fermi level and should endure strong carrier screening of polar displacements. Therefore, the flexopolar metal phase of SrRuO_{3} is sharply distinct from the socalled Anderson–Blount polar metals such as LiOsO_{3} and NdNiO_{3} (ref. ^{19,20}), whose polarity mainly comes from Asite cation displacements in a weak electron–phonon coupling scenario^{22}.
Note that the experimentally observed Ru offcentring 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 SrTiO_{3}/SrRuO_{3}/vacuum slab under a constant strain gradient to mimic the experimental structure. Importantly, Ru offcentre displacements persist and are comparable to those under the sinusoidal strain gradients (Supplementary Discussion and Supplementary Fig. 10). Therefore, the underestimation of Ru offcentre displacements in our calculation should have other origins. For example, theoretical calculations are based on defectfree 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 sample^{23,24}. In addition, huge strain gradients could cause a nonlinear flexoelectric response, leading to an orderofmagnitude enhancement of the flexoelectric effects in the experiments^{25,26}. 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 outofplane direction, yielding additional E_{flexo} along \([\bar{1}\bar{1}\bar{1}]\) (Supplementary Fig. 1). This is a result of the different clattice constants of the monoclinic and rhombohedral structures calculated with the fixed inplane lattice constants of the (111)oriented SrTiO_{3}, 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 × 10^{6} 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 E_{flexo} along \([\bar{1}\bar{1}\bar{1}]\) through longitudinal strain gradients, which should naturally exist, could cooperate with E_{flexo} along \([\bar{1}\bar{1}2]\) through shear strain gradients, thereby facilitating the actual Ru offcentring to align along the direction close to \([\bar{1}\bar{1}0]\) (Supplementary Fig. 2).
We further analysed the OOR of SrRuO_{3} 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 SrRuO_{3} 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 SrRuO_{3} at the interface and support the proposed scenario that the spatial evolution of the SrRuO_{3} structure drives bulk flexopolarity.
Correlation between strain gradient and Ru offcentring
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) SrRuO_{3} film. We first used optical secondharmonic generation (SHG) to quantify the polarity of SrRuO_{3}. Figure 3a shows the threefold symmetric SHG anisotropy pattern of a SrRuO_{3}/SrTiO_{3} (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 I^{PP}(2ω) versus the thickness t in Fig. 3b. The plot exhibits a nonmonotonic 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 flexopolarity. Therefore, these results confirm the emergence of a polar monoclinic phase in the bulk of the SrRuO_{3} film. Moreover, the 5 u.c. SrRuO_{3} 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 flexopolarity.
Then, we performed Williamson–Hall (WH) analysis based on the Xray diffraction (XRD) peak broadening of the SrRuO_{3} film, which allowed us to estimate the inhomogeneous strain across the film^{27}. 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 outofplane direction from the WH 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) SrRuO_{3} film evolves nonmonotonically with the film thickness (the blue guideline in Fig. 3c). By simply assuming a linear strain profile, a similarly nonmonotonic evolution of the strain gradient with the film thickness can be revealed (Supplementary Fig. 15). Comparing this result with the thicknessdependent SHG (Fig. 3b), we thus reveal a strong correlation between the strain gradient and the polarization.
Flexopolar phase transition in SrRuO_{3}
As the monoclinic structural distortion plays a critical role in the flexopolarity in SrRuO_{3}, conceivably, a polartononpolar phase transition would be coupled to a structural transition from the monoclinic to a highersymmetry structure at high temperatures. The highersymmetry structure is probably centrosymmetric rhombohedral according to the energy gain calculated for different structures. To investigate this, we measured the temperaturedependent SHG intensity of the (111) SrRuO_{3} film (Fig. 4a). With increasing temperature, the polartononpolar transition seems to occur at ~630 K, followed by a temperatureindependent 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 T_{f} 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 ferroelectrics^{28}.
Furthermore, temperaturedependent XRD measurements confirm that the polartononpolar phase transition is coupled to a structural transition, as evidenced by a kink at ~630 K in the temperaturedependent (111) lattice constant of SrRuO_{3} (Fig. 4b). The structural anomaly resembles the temperaturedriven, symmetryelevating structure transition observed in (001)oriented SrRuO_{3} films^{29}. The nonlinear evolution of the lattice constant below 630 K is a consequence of the electrostrictivelike expansion caused by Ru offcentring. 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 SrRuO_{3} film and the nonpolar bulk SrRuO_{3}, 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 SrRuO_{3} 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 temperaturedependent WH analysis (Supplementary Fig. 17), which further corroborates the inferred flexopolar phase transition.
Flexoelectric control of electronic properties in SrRuO_{3}
SrRuO_{3} is a wellrecognized correlated electron system that exhibits strong coupling among lattice, charge, spin and orbital degrees of freedom. Therefore, the markedly large Ru offcentre displacement is expected to profoundly impact the electronic properties of SrRuO_{3}, due to the change in the Ru–O bonding environment and the dominant contribution of Ru to electrons at the Fermi level. Indeed, our firstprinciples calculations show that Ru offcentring induces considerable reduction of the t_{2g} 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 SrRuO_{3} 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 SrRuO_{3} may be impacted by an inverse magnetostriction effect and the enhanced electron correlation originating from Ru offcentring (Supplementary Discussion and Supplementary Fig. 20). Although the detailed coupling mechanism between the electron/spin properties and Ru offcentring requires further extensive work, our results indicate the tremendous potential of flexoelectricity as a crucial tuning knob in metallic and magnetic materials.
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 flexopolarity in a metallic ferromagnet SrRuO_{3} could even enhance ferromagnetism while preserving the metallicity with little loss. Given the universality of flexoelectricity^{11}, 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 thicknessdependent 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 established^{32}, with several outstanding characteristics, including universality, nondestructiveness 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
Firstprinciples 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 augmentedwave method was used with the Ceperley–Alder exchange–correlation functional^{34}. The onsite Coulomb interaction considered by rotationally invariant LDA + U was parameterized by U = 3 eV and J = 0.75 eV for Ru d orbitals^{35}. The choice of the parameters was consistent with that in other studies^{36,37}. We used the chargeonly LDA exchange–correlation functional with the plus U extension, giving increasing exchange splitting proportional to J for SrRuO_{3} (ref. ^{38}). With our choice of the exchange–correlation functional and parameters for onsite Coulomb interaction, the calculated lattice constants and magnetic moments of the bulk SrRuO_{3} 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 inplane lattice constants of SrTiO_{3} was 2μ_{B}/Ru with a moderate overestimation relative to the experimental value of 1.58μ_{B}/Ru (Supplementary Fig. 19). The increase in the magnetic moment may be due to the static meanfield approximation of DFT, as the DFT plus dynamical meanfield theory study of SrRuO_{3} (ref. ^{39}) shows a decreased magnetic moment of 1.6μ_{B}/Ru. We used an energy cutoff of 500 eV. The kpoint samplings of 6 × 6 × 4 for the 20atom \(\sqrt{2}\times \sqrt{2}\times 2\) unit cell, 6 × 6 × 3 for the 30atom \(\sqrt{2}\times \sqrt{2}\times 2\sqrt{3}\) unit cell and 3 × 3 × 3 for the 120atom \(2\sqrt{2}\times 2\sqrt{2}\times 2\sqrt{3}\) unit cells were used for bulk and strainedbulk calculations. To evaluate the flexoelectric distortions, we used 6 × 6 × 1 kpoint samplings for the 150atom \(\sqrt{2}\times \sqrt{2}\times 10\sqrt{3}\) (111)oriented supercells and also for the 120atom (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 fiveatom cubic SrRuO_{3} (space group \({Pm}\bar{3}m\)) and 20atom orthorhombic SrRuO_{3} (space group Imma) structures with a^{0}a^{0}a^{0} and a^{−}a^{−}c^{0} 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 outofplane bulk lattice constant, z_{i} is the position along the direction of the strain gradient and h is a parameter controlling the strain gradient^{40}. The parameter h was set to produce a maximum shear strain gradient of 9 × 10^{6} m^{−1}, comparable to the experimental value (3.5 × 10^{6} m^{−1}). A maximum longitudinal strain gradient of 4.5 × 10^{6} m^{−1} was applied, based on the change in the outofplane lattice constant of 0.75% between rhombohedral and monoclinic structures in strainedbulk 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 offcentre displacements of Ru atoms from the initial positions were measured. The flexoelectric distortion of SrTiO_{3} 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 offcentring, 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^{−}c^{0}) 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 SrRuO_{3}.
Sample fabrication and characterization
SrRuO_{3} films were grown on (001) or (111) SrTiO_{3} substrates using a pulsedlaser deposition system with a KrF excimer laser (248 nm). A highpressure reflection highenergy electron diffraction system was used to monitor the growth. Before deposition, SrTiO_{3} 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 unitcell 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 SrRuO_{3} 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 (MFP3D, Asylum Research) with IrPtcoated tips (PPPEFM, Nanosensors). A roomtemperature θ–2θ scan and reciprocal space mapping by XRD were performed with a Bruker D8 Discover. Halfinteger Bragg diffraction and temperaturedependent ω–2θ scans were carried out using synchrotron XRD at the 3A beamline of Pohang Accelerator Laboratory. The (111) lattice constants of SrRuO_{3} and SrTiO_{3} were extracted from a Gaussian fitting of the (222) peaks with uncertainty. For the temperaturedependent WH analysis, we used a highresolution XRD (Panalytical X’pert Pro MRD) equipped with a fourbounce Ge 220 cut monochromator giving pure Cu Kα_{1} 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:
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. WH 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 Xray wavelength and K is a geometrical constant close to 1 (usually 0.9). Therefore, we performed the WH analysis using the quadratic relation. From the slope of linear fits of (βcosθ)^{2} versus (4sinθ)^{2}, we estimated the inhomogeneous strain u_{i}. 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 secondharmonic wave was controlled by a halfwave plate and a Glan–Taylor polarizer, respectively. We isolated the SHG wave from the fundamental wave by adopting lowpass and bandpass 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. Temperaturedependent measurements were carried out in an incidentplanerotating setup with a hightemperature 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 thirdorder susceptibility tensor components, respectively. For the roomtemperature polar phase of SrRuO_{3}, 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 hightemperature 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 secondorder susceptibility tensors for the monoclinic point group m are
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 threedimensional 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:
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
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 thirdorder susceptibility tensors are
The analytical expressions for SHG used for fitting are
STEM measurement and analyses
The crosssectional STEM specimen was prepared by first thinning the sample using a focused ion beam milling workstation (Helios NanoLab 650, FEI Co) with lowenergy ion beams (<2 kV), followed by focused lowenergy (<500 eV) Arion milling (NanoMill 1040, E.A. Fischione Instruments). Zone axis [\(1\bar{1}0\)] was predetermined by highresolution XRD.
Atomicresolution ABFSTEM experiments were performed at room temperature using spherical aberration probecorrected STEM with an acceleration voltage of 200 kV (JEMARM 200F, JEOL Ltd). The instrument was equipped with a coldfield emission gun installed at the National Center for InterUniversity Research Facilities, Seoul National University, South Korea. The TEM specimen was cleaned using an ion cleaner (JEC4000DS, 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 signaltonoise 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 results^{42}. The atomic positions were extracted using a twodimensional 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.
Lowtemperature magnetism and transport measurements
Conventional photolithography and ion milling were used to pattern the SrRuO_{3} films into the Hall bar geometry. The channel size was minimized to 50 × 50 μm^{2}. 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 angledependent Hall effect measurements were carried out with a homemade rotational stage.
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Data availability
Source data for the main figures are provided with this paper. These data are available at the figshare repository (https://doi.org/10.6084/m9.figshare.22586689). All other data related to this study are available from the corresponding authors on request.
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Acknowledgements
This work was supported by the Research Center Program of the Institute for Basic Science in Korea (IBSR009D1) and by the National Research Foundation of Korea (NRF) as funded by the Korean Ministry of Science and ICT (MSIT; Grant Nos. 2021R1A5A1032996, 2023R1A2C1007073, 2022R1A3B1077234 and 2022M3H4A1A04074153). The STEM measurements were supported by the National Center for InterUniversity Research Facilities at Seoul National University. This work was supported by Samsung Electronics Co, Ltd (No. IO201211–0806101). This paper is supported by the Basic Science Research Institute Fund (NRF Grant No. 2021R1A6A1A10042944). S.Y.P. was supported by the NRF as funded by MSIT (Grant No. 2021R1C1C1009494) and by the Basic Science Research Program through the NRF as funded by the Ministry of Education (Grant Nos. 2021R1A6A1A03043957 and 2021R1A6A1A10044154). H.J. and J.S.L. were supported by MSIT (Grant No. 2022R1A2C2007847). Y.J.J. and T.H.K. acknowledge support from the Priority Research Centers Program through the NRF as funded by the Ministry of Education (Grant No. 2019R1A6A1A11053838). Experiments at PLSII were supported in part by MSIT and the Pohang University of Science and Technology. L.W. is supported by the CAS Project for Young Scientists in Basic Research No. YSBR084, the National Basic Research Program of China (Grant Nos. 2023YFA1406404 and 2020YFA0309100), the National Natural Science Foundation of China (Grant Nos. 12374094 and 12074365), and the USTC Center for Micro and Nanoscale Research and Fabrication for sample fabrications. The nonambient XRD measurements were made using equipment housed within the Xray Diffraction Research Technology Platform at the University of Warwick. D.W. acknowledges funding from the Engineering and Physical Sciences Research Council (Grant No. EP/V007688/1).
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W.P., T.W.N. and D.L. conceived the idea and designed the experiments. W.P. grew the materials, fabricated the devices and performed the laboratory XRD, atomic force microscopy and magnetotransport measurements with help from E.K.K. D.W. assisted with the laboratory nonambient XRD. S.Y.P. performed the firstprinciples calculations. C.J.R., H.J. and J.S.L. performed the SHG measurements. J.M., M.K., A.M.S. and S. Hindmarsh performed the STEM measurements. W.P., J.K., E.K.K., Y.J.J. and T.H.K. performed the synchrotron XRD. S. Hahn, L.S. and C.K. performed the band structure analysis. W.P., Z.L., J.Z. and L.W. performed the magnetization measurements. Y.J. contributed to the data analysis. W.P. and D.L. analysed the data and wrote the manuscript, with input from all authors. D.L. directed the overall research.
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Peng, W., Park, S.Y., Roh, C.J. et al. Flexoelectric polarizing and control of a ferromagnetic metal. Nat. Phys. 20, 450–455 (2024). https://doi.org/10.1038/s41567023023338
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DOI: https://doi.org/10.1038/s41567023023338
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