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# Complex strain evolution of polar and magnetic order in multiferroic BiFeO3 thin films

## Abstract

Electric-field control of magnetism requires deterministic control of the magnetic order and understanding of the magnetoelectric coupling in multiferroics like BiFeO3 and EuTiO3. Despite this critical need, there are few studies on the strain evolution of magnetic order in BiFeO3 films. Here, in (110)-oriented BiFeO3 films, we reveal that while the polarization structure remains relatively unaffected, strain can continuously tune the orientation of the antiferromagnetic-spin axis across a wide angular space, resulting in an unexpected deviation of the classical perpendicular relationship between the antiferromagnetic axis and the polarization. Calculations suggest that this evolution arises from a competition between the Dzyaloshinskii–Moriya interaction and single-ion anisotropy wherein the former dominates at small strains and the two are comparable at large strains. Finally, strong coupling between the BiFeO3 and the ferromagnet Co0.9Fe0.1 exists such that the magnetic anisotropy of the ferromagnet can be effectively controlled by engineering the orientation of the antiferromagnetic-spin axis.

## Introduction

Antiferromagnetic materials play a critical role in the growing field of antiferromagnetic spintronics and more broadly, electric-field control of magnetism1. Regardless of the applications, knowledge of the antiferromagnetic-spin structure and manipulation of the spin axis are essential for both fundamental understanding of exchange interactions between antiferromagnetic and ferromagnetic layers and for enabling  deterministic performance in spintronics2,3. But despite this fact, compared with ferromagnets, there are relatively few studies on controlling the spin structure of antiferromagnets mainly due to the limited probes of spin structure in these materials that lack macroscopic net magnetization.

Among antiferromagnets, BiFeO3 is particularly interesting because it exhibits robust room-temperature multiferroism (ferroelectricity and antiferromagnetism)4,5 and magnetoelectric coupling that allows one to use electric fields to manipulate magnetic order6. In turn, researchers have demonstrated the potential for electric-field control of ferromagnetism in exchange-coupled ferromagnet/BiFeO3 heterostructures, making BiFeO3 a prime candidate for low-power spintronics and nanoelectronics7. Despite this potential, BiFeO3 is a complex material with many variables that must be controlled. For example, bulk BiFeO3 has a rhombohedral R3c structure with spontaneous polarization (P) and antiphase octahedral rotations along the 〈111〉 (pseudocubic indices are used throughout unless otherwise specified), which gives rise to complex domain patterns and switching pathways8,9. The magnetic order of bulk BiFeO3 crystals is also complex, as it exhibits G-type antiferromagnetism with a superimposed long-wavelength cycloidal modulation along the $$\left\langle {1\overline 1 0} \right\rangle$$10,11; that is, the spins rotate within the $$\left\{ {11\overline 2 } \right\}$$ containing the direction of the spontaneous polarization P and the cycloid modulation vector12,13. Antiphase oxygen octahedra rotations permit canting of the antiferromagnetic lattice through the Dzyaloshinskii–Moriya interaction (DMI) resulting in a local, weak canted moment M14,15, while the spin-cycloid structure results in the cancellation of net macroscopic magnetization and linear magnetoelectric coupling in bulk BiFeO316,17,18,19. This said, it is reported that epitaxial constraints in thin films can suppress the spin cycloid20,21,22, and drive a transition toward a homogenous, weakly-ferromagnetic order with a preferred antiferromagnetic spin axis (L) in {111}, which is perpendicular to the oxygen octahedral rotation axis and the direction of P (Fig. 1a)6,7,23,24. This can be understood using a phenomenological Hamiltonian consisting of a DMI term and a spin–spin exchange interaction term23. The DMI term has the form EDM = −D·(L × M), where EDM is the DMI energy and D is the DM vector. Due to symmetry arguments, D is determined by the sense of rotation of the oxygen octahedra and is thus oriented along 〈111〉23; i.e., parallel to P. A perfect antiferromagnetic order as preferred by the exchange interaction term will have zero DMI energy, while canting of the magnetic moment can make the DMI energy negative, and the most efficient way to have such canting and to reach the lowest energy is when L is perpendicular to D (and, in turn, P)23. Previous studies, mainly focused on (001)-oriented BiFeO3 films, reported that compressive strain drives L to exhibit the largest out-of-plane component while remaining in the plane perpendicular to P (i.e., $$\left[ {11\overline 2 } \right]$$); while tensile strain favors L lying in the film plane (i.e., $$\left[ {1\overline 1 0} \right]$$)24. Other studies found a more complex strain evolution of the antiferromagnetic-spin structure, including reports that L tends to orient along in-plane $$\left[ {1\overline 1 0} \right]$$ at large compressive strains and along the out-of-plane [001] under tensile strains25. Regardless of the approach, such analyses are complicated by the existence of multiple ferroelastic domains in (001)-oriented films, which could be partially responsible for the diversity of observations24,25. Finally, despite these observations, the mechanism responsible for the strain-induced spin structure change has not been well developed; precluding further understanding and control of magnetism in BiFeO3-based heterostructures.

Here, we employ angle- and polarization-dependent soft X-ray absorption spectroscopy (XAS) and X-ray linear dichroism (XLD) together with computational approaches to investigate the influence of epitaxial strain on the antiferromagnetic-spin structure in (110)-oriented BiFeO3 thin films. We demonstrate from both experiment and theory that, while epitaxial strain has relatively little impact on the orientation of P, it can drive a continuous reorientation of L from in-plane to out-of-plane directions over a wide angular space, such that P and L are no longer perpendicular when the films are under tensile strain. Our calculations suggest that spin–spin exchange coupling and the DMI dominate at low strain, giving rise to a perpendicular relationship between P and L. At large strain values, however, the single-ion anisotropy (SIA) increases in magnitude until it is comparable with the DMI, leading to a gradual deviation of the perpendicular relationship between P and L. Leveraging the strong exchange coupling between the ferromagnet Co0.9Fe0.1 and BiFeO3, we demonstrate that the anisotropy of the ferromagnet can be tuned by 90° by controlling the strain state of the BiFeO3. Ultimately, this work provides new insights on the origin of magnetic anisotropy in BiFeO3 films, demonstrates a pathway to break the expected perpendicular relationship between P and L, and provides fundamental understanding to enable controllable tuning of spin orientation in BiFeO3-based heterostructures.

## Results

### Growth and characterization of (110) BiFeO3 films with a single structural domain

To understand the effect of epitaxial strain on the antiferromagnetic-spin structure of BiFeO3, 12–70 nm thick BiFeO3 films were grown via pulsed-laser deposition on SrTiO3 (110) and GdScO3 (010)O (where the subscript O denotes orthorhombic indices) substrates such that (110)-oriented films are produced (Methods and Supplementary Fig. 1). (110)-oriented films were chosen to reduce the domain variants in BiFeO3 such that only one P and L projection on the (110) is possible8. For brevity, we focus on four heterostructure variants encompassing two representative BiFeO3 thicknesses: 12-nm-thick films (which are coherently strained to both the SrTiO3 and GdScO3 substrates) and 70-nm-thick films. These show anisotropic strain relaxation such that the films are coherently strained only along the [001] ([001]O) and relaxed along the $$\left[ {1\overline 1 0} \right]$$ ([100]O) for growth on SrTiO3 (GdScO3) (Supplementary Fig. 2). Off-axis reciprocal space mapping (Fig. 1b and Supplementary Fig. 2) and piezoresponse force microscopy (Supplementary Fig. 3) studies show that the films are monodomain.

### Study of antiferromagnetic spin axis via XLD

The magnetic structure was probed with XLD which arises from two different origins: magnetic-linear dichroism and crystal-field-induced linear dichroism26. In BiFeO3, temperature-dependent XLD studies have found that the intensity of the XLD signal near TN is much smaller than at 300 K, especially for XLD at the Fe-L2 edge which essentially vanishes (Supplementary Figs. 4 and 5); indicating that the XLD in BiFeO3 is largely dominated by a magnetic origin27. Representative pairs of XAS (taken in normal incidence with the polarization vector E of the incoming X-rays parallel to the [001] (blue curves) and $$\left[ {1\overline 1 0} \right]$$ (red curves); Fig. 1c) and XLD spectra (Fig. 1d) taken on 70-nm-thick BiFeO3 heterostructures reveal an opposite polarization dependence and a reversal of linear dichroism between films grown on GdScO3 and SrTiO3 substrates. The Fe L2,3 XAS consists of two absorption peaks because of multiplet effects (denoted as A and B; Fig. 1c), and the spectral shape depends on the relative orientation of E, the crystallographic axes, and L28. The reversed dichroism (Fig. 1d) suggests that the orientation of L for these two heterostructures is different.

To better understand the reason for and nature of these changes in the XLD for the different heterostructures, we completed angle- and polarization-dependent XAS studies with various incident X-ray directions for the BiFeO3 films on both SrTiO3 and GdScO3 substrates. The relative angle θ between E and a specified crystallographic axis is varied by rotating the samples about the X-ray Poynting vector with different incident angles of the X-rays (Fig. 2a). The L2A/L2B peak intensity ratios, calculated from the XAS as a function of θ, were used to extract the magnetic nature of the films. The L2A/L2B ratio exhibits a strong angle and polarization dependence (Fig. 2b–e), suggesting a uniaxial anisotropy (i.e., an easy axis) in the BiFeO3 films, since an easy-plane magnetic structure would give rise to a smaller dichroism and weaker angle and polarization dependence24. In addition, the L2A/L2B ratios show a markedly different trend with the polarization angle for the four different film variants (Fig. 2b–e); indicating that strain effectively modifies the orientation of L.

To extract the orientation of L for the different heterostructure variants, we have simulated the experimental XAS spectra using configuration interaction cluster calculations (Supplementary Figs. 6 and 7)29. For the 12-nm BiFeO3/GdScO3 heterostructures (coherently strained to the substrate with a compressive strain of −2% along the $$\left[ {1\overline 1 0} \right]$$ and a tensile strain of 0.1% along $$\left[ {00\overline 1 } \right]$$), the experimental results (points, Fig. 2b) can be well reproduced by simulations with L along the in-plane $$\left[ {1\overline 1 0} \right]$$ (lines, Fig. 2b). For the 70-nm BiFeO3/GdScO3 heterostructures (relaxed and strained along the $$\left[ {1\overline 1 0} \right]$$ and $$\left[ {00\overline 1 } \right]$$, respectively), the experimental results (points, Fig. 2c) can be well reproduced by simulations with L along the out-of-plane [110] (lines, Fig. 2c). For the 12-nm BiFeO3/SrTiO3 heterostructures (coherently strained to the substrate with a compressive strain of −1.4% along both the $$\left[ {1\overline 1 0} \right]$$ and $$\left[ {00\overline 1 } \right]$$), the experimental results (points, Fig. 2d) can be reproduced by simulations with L along the in-plane $$\left[ {00\overline 1 } \right]$$ (lines, Fig. 2d). Finally, for the 70-nm BiFeO3/SrTiO3 heterostructures (relaxed and strained along the $$\left[ {1\overline 1 0} \right]$$ and $$\left[ {00\overline 1 } \right]$$, respectively), the experimental results (points, Fig. 2e) can be reproduced by simulations with L rotated from the in-plane $$\left[ {00\overline 1 } \right]$$ by ~35° in the out-of-plane direction (solid lines, Fig. 2e). What these analyses suggest is that, in BiFeO3, L is highly sensitive to the strain state of the material and gradually reorients from in-plane to out-of-plane directions over a wide angular space with increasing tensile strain.

### First-principles calculations

To understand the fundamental origin of this strain-driven reorientation of L, we performed density functional theory (DFT) calculations (Methods) to explore the evolution of both L and P under different strains. For each strain state, we calculate the energy landscape when L points in different directions (Fig. 3a). In a strain-free film, we find that L is energetically degenerate within a plane perpendicular to P (here we set P to be along the [111]); consistent with previous studies23. Under compressive strain, we find that the easy-plane degeneracy is gradually lifted resulting in L continuously rotating to point along the in-plane $$\left[ {1\overline 1 0} \right]$$ at large compressive strains. Under tensile strain, the easy-plane degeneracy also disappears and L is gradually changed as the axis first converges to point approximately along the $$\left[ {11\overline 2 } \right]$$. Upon further increasing the magnitude of the tensile strain, L rotates toward the out-of-plane [110]. The change of L with strain is summarized (red lines, Fig. 3b). Note that the DFT results are calculated at 0 K and the magnetic anisotropy energies (sub meV) are at the resolution of the DFT calculations29,30. Thus, as is common practice, a relatively large strain is used to demonstrate the trend, which agrees well with the experiments.

Having observed that L rotates in a continuous manner as the strain is varied, this begs the question: Is P also rotating considerably during application of strain to maintain the classical perpendicular coupling to L? To explore this, we have evaluated P using the Berry-phase method under different strain conditions (Fig. 3c) and find that the in-plane $${\mathbf{P}}_{[1\overline 1 0]}$$ component is maintained essentially constant at ~0 C/m2 while the perpendicular, in-plane P[001] and out-of-plane P[110] components increase and decrease in magnitude, respectively, as the strain varies from compressive to tensile in nature. This suggests that P rotates only by an amount of ±15° from the [111] within the $$\left( {1\overline 1 0} \right)$$ (blue lines, Fig. 3b). Importantly, this gradual strain-driven rotation of P is not synchronized with the rapid rotation of L (Fig. 3a, b). As a result, P and L will no longer be perpendicular under some stain states and the angle between P and L will vary nonlinearly with strain starting from 90° and reaching a minimum of 46° at large tensile strains (Fig. 3d). To confirm that the strain-driven polarization rotation is not synchronized with the rotation of L, we completed polarization mapping with scanning transmission electron microscopy (STEM and Methods) (data shown here for the 70 nm BiFeO3/GdScO3 heterostructure, Fig. 3e). Magnified bright-field images (Fig. 3f) allow for direct measurement of the cation and anion displacements. Polarization maps were produced by extracting these displacements and reveal a uniform polarization direction close to the expected [111] (Fig. 3g and Supplementary Figs. 8 and 9). This observation is further supported by polarization-electric field hysteresis loop measurements, which reveal that the out-of-plane P of the (110)-oriented films is ~90 μC/cm2 (Supplementary Fig. 10), as expected31. Ultimately, this confirms that P and L can deviate markedly from the classically expected perpendicular configuration.

Previous studies on the magnetic structure of BiFeO3 have generally considered only the DMI contribution which is related to the Fe–O–Fe bond angle (inset, Fig. 4a), while the SIA, which is related to the distortion of the FeO6 octahedra (i.e., Bi–Fe distance30), has been less studied. Using an ab initio derived spin Hamiltonian (Methods and Supplementary Fig. 11), we find that the SIA is very sensitive to the strained-induced lattice distortion and that the change of L with strain results from a change of the balance between the DMI (as represented by D, Fig. 4a) and the SIA (as represented by the SIA constant K, Fig. 4b) effects32. When the DMI energy dominates over the SIA energy, the L and P will be approximately perpendicular, otherwise, this relationship will be broken. Closer inspection of the trends reveals a number of important observations. At zero strain, K is more than 10-times smaller than D30; therefore, the magnetic anisotropy is dictated by the DMI and thus L is predicted to be constrained within an easy plane perpendicular to D and, ultimately, P. It has been reported that when K becomes larger than a critical value, a simple G-type antiferromagnet becomes robust against an incommensurate magnetic structure with an easy-magnetic plane30,33. This is consistent with our calculations wherein the disappearance of the easy-magnetic plane in the strained films can be explained by the enhancement of the K value via strain-induced structural distortion. K is dramatically enhanced in films under large compressive or tensile strain and becomes comparable in magnitude to D. At large compressive strains, $${\mathbf{D}}_{\left[ {1\overline 1 0} \right]}$$ is essentially zero, and D[110] is considerably smaller than $${\mathbf{D}}_{\left[ {00\overline 1 } \right]}$$; suggesting that D is aligned along the $$\left[ {00\overline 1 } \right]$$ (Fig. 4a). Because D, L, and M form a right-handed coordinate relationship, L will prefer to remain in the (001). Furthermore, even though D[110] is small, the fact that it is non-zero breaks any degeneracy and leads to L preferring to point along the in-plane $$\left[ {1\overline 1 0} \right]$$. This is augmented by the SIA since, in the case of compressive strain, the out-of-plane Fe–O bonding strength is weakened while the in-plane term is enhanced, which results in the SIA further driving L towards the in-plane $$\left[ {1\overline 1 0} \right]$$34. At large tensile strains, however, things are more complicated. $${\mathbf{D}}_{\left[ {1\overline 1 0} \right]}$$ is again essentially zero, and $${\mathbf{D}}_{\left[ {00\overline 1 } \right]}$$ is now marginally larger than D[110], indicating that D should be within the $$\left( {1\overline 1 0} \right)$$, and is driven closer to the $$\left[ {00\overline 1 } \right]$$, thus L tends to be in the $$\left( {00\overline 1 } \right)$$. Note that K (~40 μeV) is also comparable to D (<90 μeV). Thus under tensile strain, the out-of-plane Fe–O bonding strength is enhanced while the in-plane Fe–O bonding is weakened (Supplementary Fig. 12), meaning that K will rotate out-of-plane, resulting in L pointing along the out-of-plane [110] rather than in-plane $$\left[ {1\overline 1 0} \right]$$34. P and D point to the same [111] under zero strain, but they deviate under strain. This is because D is related to the Fe–O–Fe bonding angle while P is related to the relative displacement of the positive (Bi, and to a lesser extent, Fe) and negative (O) charge centers which experience only minor changes (Supplementary Fig. 13) under strain; thus leading to the gradual rotation of P and its asynchrony with the rotation of L. The fine sensitivity of SIA to structural distortion may also explain how even small misfit strains in BiFeO3 films can be sufficient to suppress the spin cycloid20,21,22.

### Controlling magnetic anisotropy of a coupled ferromagnetic layer

Having developed a picture of the fundamental nature of magnetic structure evolution with strain and the underlying mechanism for those changes, we continue to probe the effect of strain-induced antiferromagnetic spin reorientation on the exchange coupling with a ferromagnet. 2.5 nm Pt/2.5 nm Co0.9Fe0.1 heterostructures were grown on the 70-nm-thick BiFeO3 films under a field (Hg = 200 Oe) applied either along the in-plane $$\left[ {00\overline 1 } \right]$$ or $$\left[ {1\overline 1 0} \right]$$ (Methods). Representative magneto-optical Kerr effect (MOKE) hysteresis loops taken from the Co0.9Fe0.1/BiFeO3 heterostructures where Hg was applied along the $$\left[ {00\overline 1 } \right]$$ (Fig. 5a, b) and the $$\left[ {1\overline 1 0} \right]$$ (Supplementary Fig. 15) illustrate that, irrespective of the orientation of Hg, the ferromagnetic easy axis is always along $$\left[ {00\overline 1 } \right]$$ and $$\left[ {1\overline 1 0} \right]$$ for the heterostructures grown on GdScO3 (Fig. 5a) and SrTiO3 (Fig. 5b), respectively. Without the BiFeO3 layer, the easy axis of the Co0.9Fe0.1 film is always along the $$\left[ {00\overline 1 } \right]$$ on the two substrates; again irrespective of the direction of Hg (blue and green curves, Fig. 5a, b, and Supplementary Fig. 16). All Co0.9Fe0.1/BiFeO3 heterostructures show an enhancement of the coercive field, compared to Co0.9Fe0.1 grown on bare substrates, indicating a robust exchange coupling. The small exchange bias observed in our Co0.9Fe0.1/BiFeO3 heterostructures is consistent with previous studies, which suggest that exchange bias is related to pinned uncompensated spins at BiFeO3 domain walls22,35,36. The angular evolution of the remanent magnetization of the Co0.9Fe0.1 layers deposited on the BiFeO3 obtained from the hysteresis loops are plotted in a polar curve (Fig. 5c). Similar to that observed in permalloy/BiFeO3 single crystal structures wherein the spin-cycloid structure is present37, the Co0.9Fe0.1 layers deposited on the BiFeO3 thin films all present a uniaxial anisotropy; however, the coupling mechanism could be different since the spin cycloid has been suppressed in our films. Interestingly, for heterostructures grown on SrTiO3, the exchange coupling between L and the ferromagnetic spin axis in the Co0.9Fe0.1 is strong enough to overcome the substrate-asymmetry and growth-field induced anisotropy and set the ferromagnetic spin along the $$\left[ {1\overline 1 0} \right]$$. Since the in-plane projection of L on SrTiO3 is along the $$\left[ {00\overline 1 } \right]$$, one can conclude that the BiFeO3 and Co0.9Fe0.1 spins are perpendicularly coupled. The perpendicular coupling at the antiferromagnetic/ferromagnetic interface is due to the spin-flop coupling mechanism. According to the spin-flop mechanism, it is energetically preferred for a small canted moment in the antiferromagnet to couple parallel to the ferromagnetic magnetization, giving rise to a uniaxial magnetic anisotropy in the ferromagnet38. The XLD measurements have also shown that the L of the 70 nm BiFeO3/GdScO3 heterostructures rotates towards the out-of-plane direction. Thus the in-plane anisotropy is reduced, consistent with magnetic torque results which reveal that the in-plane uniaxial anisotropy constant in heterostructures grown on SrTiO3 is much larger than those on GdScO3 (Fig. 5d). All told, our studies demonstrate that a strong exchange coupling between the ferromagnetic Co0.9Fe0.1 and antiferromagnetic BiFeO3 exists such that the anisotropy direction of the Co0.9Fe0.1 is set by the orientation of L of the underlying BiFeO3 layer.

### Conclusions

In summary, we demonstrate the ability to tune the antiferromagnetic-axis orientation from in-plane to out-of-plane across as wide angular space on (110)-oriented BiFeO3 thin films via epitaxial strain. A deviation of the classical perpendicular relationship between the antiferromagnetic axis and the polarization vector was found in both experiments and ab initio calculations. This phenomenon arises due to the interplay of the DMI and SIA effects. By engineering the antiferromagnetic-spin orientation, in turn, we can effectively tune the magnetic anisotropy of exchange-coupled Co0.9Fe0.1 layers. Our results enable a deeper understanding of the magnetic nature of BiFeO3 and exchange interaction at the BiFeO3/ferromagnet interface, and will help to design next-generation spintronic devices, such that electric-field control of magnetic spin orientation may be more readily achieved.

## Methods

### Sample preparation

Epitaxial BiFeO3 thin films were grown on SrTiO3 (110) and GdScO3 (010)O single-crystal substrates via pulsed-laser deposition at 680 °C in a dynamic oxygen pressure of 100 mTorr39. Following growth, the BiFeO3 films were cooled in ~700 Torr of oxygen to room temperature at a rate of 5 °C/min. Detailed structural information was obtained using high-resolution X-ray diffraction (X’Pert MRD Pro, Panalytical) including θ–2θ scans and reciprocal space maps (RSMs).

### Soft X-ray absorption spectroscopy

X-ray spectroscopy measurements were carried out at beamline 4.0.2 of the Advanced Light Source (ALS) at Lawrence Berkeley National Laboratory and beamline 08B of the National Synchrotron Radiation Research Center (NSRRC) in Taiwan. The measurements were performed in total-electron-yield (TEY) geometry. The XLD measurements were obtained from the difference of horizontal and vertical polarized light absorption spectra. The sample temperature was controlled using an in-vacuum resistive heater. After heating to 600 K, we checked the XAS at 300 K again and found no surface degradation caused by the heating (Supplementary Fig. S5). The X-ray beam was incident on the sample at an angle of 20° and 90° from the sample surface for grazing incidence and normal incidence, respectively; the light polarization was selected using an elliptically-polarizing undulator. Spectra were captured with the order of polarization rotation reversed (e.g., horizontal, vertical, and then vertical, horizontal) so as to eliminate experimental artifacts. The angle- and polarization-dependent XAS measurements were independently performed at room temperature for the BiFeO3 films using TEY geometry at NSRRC. The relative angle θ between light polarization vector E and the crystallographic axes are varied by rotating the samples about the X-ray Poynting vector. Such experimental geometry allows for the X-ray penetration path length of the incoming beam to be independent of the polarization angle θ, guaranteeing a reliable comparison of the spectral line shapes as a function of θ40.

### Configuration interaction cluster calculations

To extract the orientation of the antiferromagnetic axes of the various (110)-oriented BiFeO3 films, we have simulated the experimental spectra using configuration interaction calculations with the octahedral FeO6 cluster, based on atomic multiplet theory and the local effects of the solid41,42. It takes into account the intra-atomic 3d–3d and 2p–3d Coulomb and exchange interactions, the atomic 2p and 3d spin–orbit coupling of the Fe ion, the oxygen 2p–Fe 3d hybridization, and the octahedral crystal-field of Fe 3d orbital interaction43. The simulations were carried out using the program XTLS 8.3, and the parameters used in the FeO6 cluster calculations for BiFeO3 film are: Δ = 2.0 eV; Udd = 5.0 eV; Udp = 6.0 eV; 10Dq = 0.8 eV; pdσ = −1.47 eV; pdπ = 0.68 eV; and the Slater integrals reduced to 70% of the Hartree–Fock values29.

### Scanning transmission electron microscopy

STEM work was performed in National Center for Electron Microscopy (NCEM), Molecular Foundry, Lawrence Berkeley National Laboratory. The samples for the STEM experiments were prepared by slicing, gluing, grinding, dimpling, and, finally, ion milling. Samples were subsequently Ar-ion milled using a Gatan Precision Ion Milling System II (PIPS II) with starting energies of 4 keV stepped down to an energy of 1 keV for the final milling. Before ion milling, the samples were dimpled down to less than 20 μm. High-angle annular dark field (HAADF)-STEM and bright-field (BF)-STEM imaging was performed on the Cs-corrected TEAM1 FEI Titan microscope at 300 kV. To enable determination of the atomic positions and Fe3+ ion displacement vectors, noise in the HAADF images was filtered by Wiener filtering. The atom positions were determined accurately by fitting them as 2D Gaussian peaks by using a Matlab script44. The Fe3+ ion displacement vector was calculated as a vector between each Fe3+ and the center of mass of its two nearest neighbors Bi3+.

### Co0.9Fe0.1 deposition and magnetic properties measurements

For exchange-coupling studies, after the growth of the BiFeO3 films, they were broken into two pieces and immediately inserted into a vacuum sputtering chamber with a base pressure of 8 × 10−8 Torr. A 2.5 nm Pt/2.5 nm Co0.9Fe0.1 bilayer was deposited on the BiFeO3 films and bare substrates by DC sputtering in 8 × 10−4 Torr of Ar at room temperature with a 20 mT growth field along the in-plane [001] and $$\left[ {1\overline 1 0} \right]$$. Magnetic hysteresis loops and magnetic anisotropy measurements were carried out using the longitudinal MOKE and rotation MOKE45,46.

### Ab initio calculations

Ab initio calculations were performed using the projector augmented wave (PAW)47,48 formalism and a plane wave basis set, as implemented in the Vienna ab initio simulation package (VASP)49,50. The exchange and correlation potential were treated in the framework of generalized gradient approximation (GGA) of Perdew–Burke–Ernzerhof (PBE)51. The PAW potentials used explicitly treat 15 valence electrons for bismuth (5d10 6s2 6p3), 14 for iron (3p6 3d6 4s2), and 6 for oxygen (2s2 2p4). Local spin-density approximation with an additional Hubbard parameter (LSDA+U) was used for the exchange-correlation functional. The Hubbard parameter U and the exchange interaction J that treat the Fe d electrons were set to U = 2 eV and J= 0 eV. Spin-orbit coupling (SOC) was included to calculate the non-collinear magnetic energy landscape. For the summation of charge densities over the Brillouin zone, a 3 × 3 × 3 k-point mesh is adopted in the calculation of the total energy and force. The wave functions are expanded in plane waves up to a cutoff of 550 eV and the convergence precision of the total energy is set to be lower than 1 × 10−6 eV. Symmetry was switched off to remove any artificial constraints on the possible spin ordering.

The supercell was made of 2 × 2 × 2 cubic perovskite units, containing 40 atoms in total. Its three axes are $$\left[ {1\overline 1 0} \right]$$, [001], and [110], respectively, so as to simulate films of a (110)-oriented material (Supplementary Fig. S9). Under each strain (depicted by the misfit of crystal parameters), the in-plane $$\left[ {1\overline 1 0} \right]$$ and [001] axes of supercell are fixed while the out-of-plane [110] axis and the atomic positions are fully relaxed; a conjugate gradient algorithm is used and force precision is lower than 0.005 eV/Å. The polarization vector is evaluated by the Born effective charges using the Berry phase method.

The Hamiltonian containing exchange coupling, DMI, and SIA is written as:

$$H = - \mathop {\sum }\limits_{ij} J_{ij}{\mathbf{S}}_i \cdot {\mathbf{S}}_j - \mathop {\sum }\limits_{ij} {\mathbf{D}}_{ij} \cdot {\mathbf{S}}_i \times {\mathbf{S}}_j - K\mathop {\sum }\limits_i \left| {{\mathbf{S}}_i \cdot {\mathbf{n}}_i} \right|^2 + H_0$$
(1)

where Si is the ith spin vector; Jij is exchange parameter; Dij is the DMI vector; K is the SIA constant, and ni = (sin θi cos φI, sin θi cos φi, cos θi) is the SIA unit vector in spherical coordinates. H0 includes all other interactions, such as the lattice elastic energy. Here, we consider only the spin interaction between nearest neighbors. Three different DMI vectors D1, D2, and D3 are used for neighborhoods along x, y, and z direction, respectively (Supplementary Fig. 12).

We used the method proposed by Xiang et al.52 to calculate the DMI parameters. Specifically, to calculate $${\mathbf{D}}_1^x$$, we considered four spin configurations, in which the spin of Fe1 and Fe2 are oriented along the y and z directions, respectively: (1) S1 = (0, S, 0), S2 = (0, 0, S); (2) S1 = (0, S, 0), S2 = (0, 0, −S); (3) S1 = (0, −S, 0), S2 = (0, 0, S); (4) S1 = (0, −S, 0), S2 = (0, 0, −S). The spin of the other six atoms are the same and are along the x direction: Sothers = (S, 0, 0). By computing the energy of the four spin configurations, we have:

$${\mathbf{D}}_1^x = \frac{{{\mathbf{E}}_1 + {\mathbf{E}}_4 - {\mathbf{E}}_2 - {\mathbf{E}}_3}}{{4{\mathbf{S}}^2}}$$
(2)

The other two components of D1 and the vectors D2 and D3 can be computed similarly. The evolution of the three vectors under strain is plotted (Supplementary Fig. 14). Their average vector D = |Di| = (D1 + D2 + D3)/3 is also plotted (Fig. 3b).

The SIA term is calculated by considering only the Fe ion with spin while all the other ions without spin. We performed constrained calculations with various directions of this isolated spin, so as to resolve the energy surface. The SIA constant K is then fitted to these data points.

## Data availability

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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## Acknowledgements

The authors thank Prof. P. Yu for fruitful discussions. This work was funded by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, Materials Sciences and Engineering Division under Contract No. DE-AC02-05-CH11231 within the Materials Project program KC23MP for the synthesis of complex-oxide functional materials, the Non-Equilibrium Magnetic Materials program (MSMAG) for the first-principles calculations, and the van der Waals heterostructures program (KCWF16) for the magnetic studies of materials. This research used resources of the Advanced Light Source, which is a DOE Office of Science User Facility under contract no. DE-AC02-05CH11231. STEM work at the Molecular Foundry was supported by the Office of Science, Office of Basic Energy Sciences, of the U.S. Department of Energy under Contract No. DE-AC02-05CH11231. This research used the resources of the National Energy Research Scientific Computing Center (NERSC) and Oak Ridge Leadership Computing Facility (OLCF) that are supported by the Office of Science of the U.S. Department of Energy, with the computational time allocated by the Innovative and Novel Computational Impact on Theory and Experiment (INCITEE) project. The study of multiferroic thin film devices was supported by the Army Research Office under grant W911NF-14-1-0104. L.R.D. acknowledges support U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences under Award Number DE-SC-0012375. L.W.M. & R.R. acknowledge support from the Gordon and Betty Moore Foundation’s EPiQS Initiative, Grant GBMF5307 and Intel, Corp. Z.H.C. acknowledges a startup grant from Harbin Institute of Technology, Shenzhen, China, under project number DD45001017.

## Author information

Authors

### Contributions

Z.H.C. conceived of and designed the research, and analyzed the results with R.R. and L.W.M. Z.H.C. and L.R.D. synthesized the samples. Z.H.C. performed the X-ray structural characterizations and PFM studies. The MOKE-based magnetic properties measurements were carried out by Z.H.C. with the assistance of Q.L., M.M.Y. and Z.Q.Q. Z.H.C., C.Y.K., A.F., L.Z. and C.K. performed the soft X-ray spectroscopy measurements with the assistance of P.S., E.A. and A.S., and interpreted the data with Z.W.H. and L.-H.T. Y.L.T. performed the STEM experiments and analyzed the data. Z.C. and L.-W.W. contributed to the theoretical studies. Z.H.C. and L.W.M. prepared the manuscript with the assistance of Z.C. and L.-W.W. All authors read and contributed to the manuscript and the interpretation of the data.

### Corresponding authors

Correspondence to Zuhuang Chen or Lane W. Martin.

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### Competing interests

The authors declare no competing interests.

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Chen, Z., Chen, Z., Kuo, CY. et al. Complex strain evolution of polar and magnetic order in multiferroic BiFeO3 thin films. Nat Commun 9, 3764 (2018). https://doi.org/10.1038/s41467-018-06190-5

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• DOI: https://doi.org/10.1038/s41467-018-06190-5

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