Abstract
Electricfield control of magnetism requires deterministic control of the magnetic order and understanding of the magnetoelectric coupling in multiferroics like BiFeO_{3} and EuTiO_{3}. Despite this critical need, there are few studies on the strain evolution of magnetic order in BiFeO_{3} films. Here, in (110)oriented BiFeO_{3} films, we reveal that while the polarization structure remains relatively unaffected, strain can continuously tune the orientation of the antiferromagneticspin 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 singleion anisotropy wherein the former dominates at small strains and the two are comparable at large strains. Finally, strong coupling between the BiFeO_{3} and the ferromagnet Co_{0.9}Fe_{0.1} exists such that the magnetic anisotropy of the ferromagnet can be effectively controlled by engineering the orientation of the antiferromagneticspin axis.
Introduction
Antiferromagnetic materials play a critical role in the growing field of antiferromagnetic spintronics and more broadly, electricfield control of magnetism^{1}. Regardless of the applications, knowledge of the antiferromagneticspin 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 spintronics^{2,3}. 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, BiFeO_{3} is particularly interesting because it exhibits robust roomtemperature multiferroism (ferroelectricity and antiferromagnetism)^{4,5} and magnetoelectric coupling that allows one to use electric fields to manipulate magnetic order^{6}. In turn, researchers have demonstrated the potential for electricfield control of ferromagnetism in exchangecoupled ferromagnet/BiFeO_{3} heterostructures, making BiFeO_{3} a prime candidate for lowpower spintronics and nanoelectronics^{7}. Despite this potential, BiFeO_{3} is a complex material with many variables that must be controlled. For example, bulk BiFeO_{3} 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 pathways^{8,9}. The magnetic order of bulk BiFeO_{3} crystals is also complex, as it exhibits Gtype antiferromagnetism with a superimposed longwavelength 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 vector^{12,13}. Antiphase oxygen octahedra rotations permit canting of the antiferromagnetic lattice through the Dzyaloshinskii–Moriya interaction (DMI) resulting in a local, weak canted moment M^{14,15}, while the spincycloid structure results in the cancellation of net macroscopic magnetization and linear magnetoelectric coupling in bulk BiFeO_{3}^{16,17,18,19}. This said, it is reported that epitaxial constraints in thin films can suppress the spin cycloid^{20,21,22}, and drive a transition toward a homogenous, weaklyferromagnetic 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 term^{23}. The DMI term has the form E_{DM} = −D·(L × M), where E_{DM} 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 BiFeO_{3} films, reported that compressive strain drives L to exhibit the largest outofplane 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 antiferromagneticspin structure, including reports that L tends to orient along inplane \(\left[ {1\overline 1 0} \right]\) at large compressive strains and along the outofplane [001] under tensile strains^{25}. 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 observations^{24,25}. Finally, despite these observations, the mechanism responsible for the straininduced spin structure change has not been well developed; precluding further understanding and control of magnetism in BiFeO_{3}based heterostructures.
Here, we employ angle and polarizationdependent soft Xray absorption spectroscopy (XAS) and Xray linear dichroism (XLD) together with computational approaches to investigate the influence of epitaxial strain on the antiferromagneticspin structure in (110)oriented BiFeO_{3} 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 inplane to outofplane 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 singleion 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 Co_{0.9}Fe_{0.1} and BiFeO_{3}, we demonstrate that the anisotropy of the ferromagnet can be tuned by 90° by controlling the strain state of the BiFeO_{3}. Ultimately, this work provides new insights on the origin of magnetic anisotropy in BiFeO_{3} 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 BiFeO_{3}based heterostructures.
Results
Growth and characterization of (110) BiFeO_{3} films with a single structural domain
To understand the effect of epitaxial strain on the antiferromagneticspin structure of BiFeO_{3}, 12–70 nm thick BiFeO_{3} films were grown via pulsedlaser deposition on SrTiO_{3} (110) and GdScO_{3} (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 BiFeO_{3} such that only one P and L projection on the (110) is possible^{8}. For brevity, we focus on four heterostructure variants encompassing two representative BiFeO_{3} thicknesses: 12nmthick films (which are coherently strained to both the SrTiO_{3} and GdScO_{3} substrates) and 70nmthick 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 SrTiO_{3} (GdScO_{3}) (Supplementary Fig. 2). Offaxis 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: magneticlinear dichroism and crystalfieldinduced linear dichroism^{26}. In BiFeO_{3}, temperaturedependent XLD studies have found that the intensity of the XLD signal near T_{N} is much smaller than at 300 K, especially for XLD at the FeL_{2} edge which essentially vanishes (Supplementary Figs. 4 and 5); indicating that the XLD in BiFeO_{3} is largely dominated by a magnetic origin^{27}. Representative pairs of XAS (taken in normal incidence with the polarization vector E of the incoming Xrays parallel to the [001] (blue curves) and \(\left[ {1\overline 1 0} \right]\) (red curves); Fig. 1c) and XLD spectra (Fig. 1d) taken on 70nmthick BiFeO_{3} heterostructures reveal an opposite polarization dependence and a reversal of linear dichroism between films grown on GdScO_{3} and SrTiO_{3} substrates. The Fe L_{2,3} 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 L^{28}. 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 polarizationdependent XAS studies with various incident Xray directions for the BiFeO_{3} films on both SrTiO_{3} and GdScO_{3} substrates. The relative angle θ between E and a specified crystallographic axis is varied by rotating the samples about the Xray Poynting vector with different incident angles of the Xrays (Fig. 2a). The L_{2A}/L_{2B} peak intensity ratios, calculated from the XAS as a function of θ, were used to extract the magnetic nature of the films. The L_{2A}/L_{2B} ratio exhibits a strong angle and polarization dependence (Fig. 2b–e), suggesting a uniaxial anisotropy (i.e., an easy axis) in the BiFeO_{3} films, since an easyplane magnetic structure would give rise to a smaller dichroism and weaker angle and polarization dependence^{24}. In addition, the L_{2A}/L_{2B} 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 12nm BiFeO_{3}/GdScO_{3} 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 inplane \(\left[ {1\overline 1 0} \right]\) (lines, Fig. 2b). For the 70nm BiFeO_{3}/GdScO_{3} 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 outofplane [110] (lines, Fig. 2c). For the 12nm BiFeO_{3}/SrTiO_{3} 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 inplane \(\left[ {00\overline 1 } \right]\) (lines, Fig. 2d). Finally, for the 70nm BiFeO_{3}/SrTiO_{3} 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 inplane \(\left[ {00\overline 1 } \right]\) by ~35° in the outofplane direction (solid lines, Fig. 2e). What these analyses suggest is that, in BiFeO_{3}, L is highly sensitive to the strain state of the material and gradually reorients from inplane to outofplane directions over a wide angular space with increasing tensile strain.
Firstprinciples calculations
To understand the fundamental origin of this straindriven 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 strainfree 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 studies^{23}. Under compressive strain, we find that the easyplane degeneracy is gradually lifted resulting in L continuously rotating to point along the inplane \(\left[ {1\overline 1 0} \right]\) at large compressive strains. Under tensile strain, the easyplane 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 outofplane [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 calculations^{29,30}. 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 Berryphase method under different strain conditions (Fig. 3c) and find that the inplane \({\mathbf{P}}_{[1\overline 1 0]}\) component is maintained essentially constant at ~0 C/m^{2} while the perpendicular, inplane P_{[001]} and outofplane 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 straindriven 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 straindriven 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 BiFeO_{3}/GdScO_{3} heterostructure, Fig. 3e). Magnified brightfield 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 polarizationelectric field hysteresis loop measurements, which reveal that the outofplane P of the (110)oriented films is ~90 μC/cm^{2} (Supplementary Fig. 10), as expected^{31}. Ultimately, this confirms that P and L can deviate markedly from the classically expected perpendicular configuration.
Previous studies on the magnetic structure of BiFeO_{3} 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 FeO_{6} octahedra (i.e., Bi–Fe distance^{30}), 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 strainedinduced 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) effects^{32}. 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 10times smaller than D^{30}; 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 Gtype antiferromagnet becomes robust against an incommensurate magnetic structure with an easymagnetic plane^{30,33}. This is consistent with our calculations wherein the disappearance of the easymagnetic plane in the strained films can be explained by the enhancement of the K value via straininduced 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 righthanded coordinate relationship, L will prefer to remain in the (001). Furthermore, even though D_{[110]} is small, the fact that it is nonzero breaks any degeneracy and leads to L preferring to point along the inplane \(\left[ {1\overline 1 0} \right]\). This is augmented by the SIA since, in the case of compressive strain, the outofplane Fe–O bonding strength is weakened while the inplane term is enhanced, which results in the SIA further driving L towards the inplane \(\left[ {1\overline 1 0} \right]\)^{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 outofplane Fe–O bonding strength is enhanced while the inplane Fe–O bonding is weakened (Supplementary Fig. 12), meaning that K will rotate outofplane, resulting in L pointing along the outofplane [110] rather than inplane \(\left[ {1\overline 1 0} \right]\)^{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 BiFeO_{3} films can be sufficient to suppress the spin cycloid^{20,21,22}.
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 straininduced antiferromagnetic spin reorientation on the exchange coupling with a ferromagnet. 2.5 nm Pt/2.5 nm Co_{0.9}Fe_{0.1} heterostructures were grown on the 70nmthick BiFeO_{3} films under a field (H_{g} = 200 Oe) applied either along the inplane \(\left[ {00\overline 1 } \right]\) or \(\left[ {1\overline 1 0} \right]\) (Methods). Representative magnetooptical Kerr effect (MOKE) hysteresis loops taken from the Co_{0.9}Fe_{0.1}/BiFeO_{3} heterostructures where H_{g} 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 H_{g}, the ferromagnetic easy axis is always along \(\left[ {00\overline 1 } \right]\) and \(\left[ {1\overline 1 0} \right]\) for the heterostructures grown on GdScO_{3} (Fig. 5a) and SrTiO_{3} (Fig. 5b), respectively. Without the BiFeO_{3} layer, the easy axis of the Co_{0.9}Fe_{0.1} film is always along the \(\left[ {00\overline 1 } \right]\) on the two substrates; again irrespective of the direction of H_{g} (blue and green curves, Fig. 5a, b, and Supplementary Fig. 16). All Co_{0.9}Fe_{0.1}/BiFeO_{3} heterostructures show an enhancement of the coercive field, compared to Co_{0.9}Fe_{0.1} grown on bare substrates, indicating a robust exchange coupling. The small exchange bias observed in our Co_{0.9}Fe_{0.1}/BiFeO_{3} heterostructures is consistent with previous studies, which suggest that exchange bias is related to pinned uncompensated spins at BiFeO_{3} domain walls^{22,35,36}. The angular evolution of the remanent magnetization of the Co_{0.9}Fe_{0.1} layers deposited on the BiFeO_{3} obtained from the hysteresis loops are plotted in a polar curve (Fig. 5c). Similar to that observed in permalloy/BiFeO_{3} single crystal structures wherein the spincycloid structure is present^{37}, the Co_{0.9}Fe_{0.1} layers deposited on the BiFeO_{3} 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 SrTiO_{3}, the exchange coupling between L and the ferromagnetic spin axis in the Co_{0.9}Fe_{0.1} is strong enough to overcome the substrateasymmetry and growthfield induced anisotropy and set the ferromagnetic spin along the \(\left[ {1\overline 1 0} \right]\). Since the inplane projection of L on SrTiO_{3} is along the \(\left[ {00\overline 1 } \right]\), one can conclude that the BiFeO_{3} and Co_{0.9}Fe_{0.1} spins are perpendicularly coupled. The perpendicular coupling at the antiferromagnetic/ferromagnetic interface is due to the spinflop coupling mechanism. According to the spinflop 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 ferromagnet^{38}. The XLD measurements have also shown that the L of the 70 nm BiFeO_{3}/GdScO_{3} heterostructures rotates towards the outofplane direction. Thus the inplane anisotropy is reduced, consistent with magnetic torque results which reveal that the inplane uniaxial anisotropy constant in heterostructures grown on SrTiO_{3} is much larger than those on GdScO_{3} (Fig. 5d). All told, our studies demonstrate that a strong exchange coupling between the ferromagnetic Co_{0.9}Fe_{0.1} and antiferromagnetic BiFeO_{3} exists such that the anisotropy direction of the Co_{0.9}Fe_{0.1} is set by the orientation of L of the underlying BiFeO_{3} layer.
Conclusions
In summary, we demonstrate the ability to tune the antiferromagneticaxis orientation from inplane to outofplane across as wide angular space on (110)oriented BiFeO_{3} 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 antiferromagneticspin orientation, in turn, we can effectively tune the magnetic anisotropy of exchangecoupled Co_{0.9}Fe_{0.1} layers. Our results enable a deeper understanding of the magnetic nature of BiFeO_{3} and exchange interaction at the BiFeO_{3}/ferromagnet interface, and will help to design nextgeneration spintronic devices, such that electricfield control of magnetic spin orientation may be more readily achieved.
Methods
Sample preparation
Epitaxial BiFeO_{3} thin films were grown on SrTiO_{3} (110) and GdScO_{3} (010)_{O} singlecrystal substrates via pulsedlaser deposition at 680 °C in a dynamic oxygen pressure of 100 mTorr^{39}. Following growth, the BiFeO_{3} films were cooled in ~700 Torr of oxygen to room temperature at a rate of 5 °C/min. Detailed structural information was obtained using highresolution Xray diffraction (X’Pert MRD Pro, Panalytical) including θ–2θ scans and reciprocal space maps (RSMs).
Soft Xray absorption spectroscopy
Xray 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 totalelectronyield (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 invacuum 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 Xray 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 ellipticallypolarizing 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 polarizationdependent XAS measurements were independently performed at room temperature for the BiFeO_{3} 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 Xray Poynting vector. Such experimental geometry allows for the Xray 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 θ^{40}.
Configuration interaction cluster calculations
To extract the orientation of the antiferromagnetic axes of the various (110)oriented BiFeO_{3} films, we have simulated the experimental spectra using configuration interaction calculations with the octahedral FeO_{6} cluster, based on atomic multiplet theory and the local effects of the solid^{41,42}. It takes into account the intraatomic 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 crystalfield of Fe 3d orbital interaction^{43}. The simulations were carried out using the program XTLS 8.3, and the parameters used in the FeO_{6} cluster calculations for BiFeO_{3} film are: Δ = 2.0 eV; U_{dd} = 5.0 eV; U_{dp} = 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 values^{29}.
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 Arion 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. Highangle annular dark field (HAADF)STEM and brightfield (BF)STEM imaging was performed on the Cscorrected TEAM1 FEI Titan microscope at 300 kV. To enable determination of the atomic positions and Fe^{3+} 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 script^{44}. The Fe^{3+} ion displacement vector was calculated as a vector between each Fe^{3+} and the center of mass of its two nearest neighbors Bi^{3+}.
Co_{0.9}Fe_{0.1} deposition and magnetic properties measurements
For exchangecoupling studies, after the growth of the BiFeO_{3} 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 Co_{0.9}Fe_{0.1} bilayer was deposited on the BiFeO_{3} 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 inplane [001] and \(\left[ {1\overline 1 0} \right]\). Magnetic hysteresis loops and magnetic anisotropy measurements were carried out using the longitudinal MOKE and rotation MOKE^{45,46}.
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 (5d^{10} 6s^{2} 6p^{3}), 14 for iron (3p^{6} 3d^{6} 4s^{2}), and 6 for oxygen (2s^{2} 2p^{4}). Local spindensity approximation with an additional Hubbard parameter (LSDA+U) was used for the exchangecorrelation 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. Spinorbit coupling (SOC) was included to calculate the noncollinear magnetic energy landscape. For the summation of charge densities over the Brillouin zone, a 3 × 3 × 3 kpoint 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 inplane \(\left[ {1\overline 1 0} \right]\) and [001] axes of supercell are fixed while the outofplane [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:
where S_{i} is the ith spin vector; J_{ij} is exchange parameter; D_{ij} is the DMI vector; K is the SIA constant, and n_{i} = (sin θ_{i} cos φ_{I}, sin θ_{i} cos φ_{i}, cos θ_{i}) is the SIA unit vector in spherical coordinates. H_{0} includes all other interactions, such as the lattice elastic energy. Here, we consider only the spin interaction between nearest neighbors. Three different DMI vectors D_{1}, D_{2}, and D_{3} 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 Fe_{1} and Fe_{2} are oriented along the y and z directions, respectively: (1) S_{1} = (0, S, 0), S_{2} = (0, 0, S); (2) S_{1} = (0, S, 0), S_{2} = (0, 0, −S); (3) S_{1} = (0, −S, 0), S_{2} = (0, 0, S); (4) S_{1} = (0, −S, 0), S_{2} = (0, 0, −S). The spin of the other six atoms are the same and are along the x direction: S_{others} = (S, 0, 0). By computing the energy of the four spin configurations, we have:
The other two components of D_{1} and the vectors D_{2} and D_{3} can be computed similarly. The evolution of the three vectors under strain is plotted (Supplementary Fig. 14). Their average vector D = D_{i} = (D_{1} + D_{2} + D_{3})/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. DEAC0205CH11231 within the Materials Project program KC23MP for the synthesis of complexoxide functional materials, the NonEquilibrium Magnetic Materials program (MSMAG) for the firstprinciples 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. DEAC0205CH11231. 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. DEAC0205CH11231. 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 W911NF1410104. L.R.D. acknowledges support U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences under Award Number DESC0012375. 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.
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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 Xray structural characterizations and PFM studies. The MOKEbased 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 Xray 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.
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Chen, Z., Chen, Z., Kuo, CY. et al. Complex strain evolution of polar and magnetic order in multiferroic BiFeO_{3} thin films. Nat Commun 9, 3764 (2018). https://doi.org/10.1038/s41467018061905
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DOI: https://doi.org/10.1038/s41467018061905
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