Abstract
Voltagemodulated magnetism in magnetic/BiFeO_{3} heterostructures can be driven by a combination of the intrinsic ferroelectricantiferromagnetic coupling in BiFeO_{3} and the antiferromagneticferromagnetic exchange interaction across the heterointerface. However, ferroelectric BiFeO_{3} film is also ferroelastic, thus it is possible to generate voltageinduced strain in BiFeO_{3} that could be applied onto the magnetic layer across the heterointerface and modulate magnetism through magnetoelastic coupling. Here, we investigated, using phasefield simulations, the role of strain in voltagecontrolled magnetism for these BiFeO_{3}based heterostructures. It is predicted, under certain condition, coexistence of strain and exchange interaction will result in a pure voltagedriven 180° magnetization reversal in BiFeO_{3}based heterostructures.
Introduction
It is accepted that voltagemodulated magnetism in magnetic/BiFeO_{3} (BFO) layered heterostructures is based on a combination of intrinsic coupling between the coexisted ferroelectric and antiferromagnetic orders in BFO, and the antiferromagneticmagnetic exchange interaction across the heterointerface^{1,2,3,4}. However, ferroelectric BFO film, if not fully clamped by substrate, would generate voltageinduced strains that could be transferred to the magnetic thin film across the heterointerface and modulate magnetism together with exchange interaction through magnetoelastic coupling. Influence of this strain on voltagemodulated magnetism in BFObased heterostructures has remained largely unexplored since raised by Mathur^{5}, though a giant voltageinduced strain of over 5% has later been observed in BFO thin films with mixed rhombohedral and tetragonal phases^{6,7}.
In this article, we explore how strain affects the voltagemodulated magnetism in BFObased heterostructures by taking the Co_{0.9}Fe_{0.1}(CoFe)/BFO thinfilm heterostructure as an example. (001) BFO thin films were grown on (110) DyScO_{3} substrate, exhibiting twovariant ferroelectric domains with 71° wall due to anisotropic filmsubstrate misfit strains^{8}. In particular, the magnetic domain patterns in the top CoFe film almost copy the inplane projection patterns of the ferroelectric domains at the BFO surface^{2,9}. Such magnetic domain pattern is induced by an effective magnetic field from the canted magnetic moment M_{c} in BFO via DzyaloshinskiiMoriya (DM) exchange interaction^{10,11}, namely, H_{DM}field. The direction of H_{DM}field (also M_{c}) is perpendicular to the plane of the polarization P and the antiferromagnetic axis L^{12,13}, i.e., H_{DM} = P × L, resulting in a nonuniform H_{DM}field distribution based on the twovariant 71° ferroelectric domains. In this case, when electrically rearranging ferroelectric domain configuration to switch the inplane net polarization in BFO by 180°, rotation of H_{DM}field within individual ferroelectric domain could lead to an overall 180° switching of the inplane net magnetization^{2} in CoFe. Detailed experimental analysis^{14} reveals that the H_{DM}field (M_{c}) lies along the and directions corresponding to the and polarizations, while along the and directions corresponding to the and [111] polarizations, respectively (figure 1a).
Furthermore, the first few unit cells at the BFO thin film surface would become stressfree when the thickness of the BFO film was above a certain critical value [t_{c} ~70 nm for the BFO films grown on DyScO_{3}(DSO) substrate^{15,16}] to allow the presence of lattice defects such as dislocation (figure 1b). In this case, sizable and nonuniform strain can be generated associated with local non180° ferroelectric polarization (i.e., ferroelastic) switching under an electricfield. The strain can be further transferred to the top magnetic thin film across the interface and modulate the magnetic domain structure locally via magnetoelastic coupling^{17}. Transfer of such nonuniform strain has recently been demonstrated in BaTiO_{3} single crystalbased heterostructues^{18,19,20,21}, which exhibit a similar onetoone domain pattern match between magnetic thin film and ferroelectric BaTiO_{3} underneath. Particularly, it was proposed that^{22} such nonuniform ferroelastic strain transfer alone could drive a 180° inplane net magnetization reversal similarly to the nonuniform H_{DM}field driven reversal shown in figure 1a. These similarities thus raise one interesting question for BFObased heterostructures: what role does the nonuniform strain in BFO play in the electricfield induced magnetization reversal? A phasefield^{23,24} model is developed herein (see Method section) to address this question, and illustrate new possibility in electricfieldcontrolled magnetism under coexistence of nonuniform H_{DM}field and nonuniform strain.
Results
Inplane net magnetization reversal purely by H_{DM}field
We first examine the influence of the interfacial H_{DM}field on the electricfield driven magnetization reversal in the CoFe/BFO heterostructure. If the electricfield applied along the [100] direction (E_{100}) exceeds the coercive field of BFO (E_{c}), the initial alternating and ferroelectric domains would switch by 71° to the and [111] domains, respectively, during which the polarization vectors would always keep a head to tail configuration (see figure 2a) to reduce the electrostatic energy. This leads to a full reversal of the average polarization along the [100] direction (i.e., P_{100}), as illustrated by the simulated ferroelectric hysteresis loop in figure 2c. Note that individual ferroelectric domain under a certain E_{100} results in one unique local H_{DM}field distribution [Eq. (11)]. Correspondingly, figure 2b presents the inplane projections of the H_{DM}field, e.g., an initial configuration of head to tail inplane and orientations. The induced local magnetization distributions (viz. domain structures) in the CoFe film are almost identical to those of inplane H_{DM}field (not shown here for simplicity), while both distributions are essentially the same as the inplane projections of the ferroelectric domains and thereby accounts for the domain pattern transfer between the CoFe and BFO films^{2}. Furthermore, a full reversal of the net [100] magnetization (M_{100}) in the CoFe film occurs when reversing the average inplane H_{DM} field along the [100] direction () with an electricfield, as shown by their electricfield switching loops in figure 2e and 2d, respectively. The magnitude of the total H_{DM} field [i.e., , see Eqs. (11) and (12) in the Method section] is taken as 100 Oe based on a relevant experimental measumrent^{9}.
It is worth noting that such net magnetization reversal may not be triggered if the was too small, because the striped magnetic domains cannot be stabilized unless exceeds 35 Oe (see figure 3a), where the influence of strains is not incorporated (λ_{s} = 0). As it can be seen, once the striped domains form at 35 Oe, the net inplane magnetization can then be reversed, i.e., the normalized magnetization M_{100}/M_{s} changes from −0.7 to about 0.7, by changing the polarity of the net inplane polarization. Simulated magnetic and ferroelectric domain structures at = 6 Oe and 100 Oe are illustrated in the inset of figure 3a. We argue that the critical value for net magnetization reversal should be strongly dependent on the coercive field of the magnetic thin film, because influence of H_{DM}field is equivalent to an external nonuniform magnetic field [Eq. (9)].
Full magnetization reversal driven by both H_{DM}field and strain
Figure 3b shows the M_{100}/M_{s} variations as a function of before (the upper panel) and after (the lower panel) reversing the net inplane polarization, driven by both nonuniform H_{DM}field and nonuniform ferroelastic strain. Strain distributions before and after polarization reversal are plotted in figure 3c, with an alternating distribution of ±0.6% [also see Eq. (4a)]. Of interest, the magnetic striped domains can be stabilized during the growth of CoFe film by the growthinduced strains [see Eq. 4(b)] imposed by the striped ferroelectric (also ferroelastic) domain pattern at the BFO surface even when = 0 (i.e., no exchange interaction but sole strain effect, see figure 3b). However, these striped domains cannot be switched to achieve an inplane net magnetization reversal when is small (weak exchange interaction). For example, at = 6 Oe, the striped domains only demonstrate a change in pattern periodicity (see corresponding domain structures in the inset of figure 3b) when reversing inplane polarization. Thus, although nonuniform ferroelastic strain contributes to the stabilization of the domain stripes, it alone cannot trigger the net magnetization reversal. On the contrary, this strain could even suppress the reversal compared to the case purely driven by H_{DM}field (figure 3a), as demonstrated by (i) the enhancement of the critical value of from 35 Oe to 45 Oe (see the solid line in figure 3b), and (ii) the different magnetic domain stripes at = 6 Oe (prevailing strain effect) and 1000 Oe (prevailing exchange interaction) with the same FE domain patterns underneath (i.e., both are before electricfield poling, see the upper panel of figure 3b).
These counteractive nonuniform H_{DM}field and nonuniform strain can, however, be exploited to achieve a full rather than net magnetization reversal. As shown in figure 3b, the magnetic striped domains are straightened out to form a uniform single domain when is about 500 Oe. We attribute this to the almost equal contributions from the competing strain effect and exchange interaction at this point. Indeed, taking the coupled magnetic and ferroelectric domain structures at = 0 Oe as the reference, the changes in magnetic free energy induced by strain and H_{DM}field are 8.26 × 10^{4} J/m^{3} and −7.95 × 10^{4} J/m^{3}, respectively, at = 500 Oe. Furthermore, a full 180° magnetization reversal takes place when reversing the inplane polarization (see corresponding domain structures in figure 3b), which should promise higher application potential for lowpower memory applications^{25,26} than the average magnetization reversal in striped multidomain. Further simulations show that the range of required for such voltagedriven 180° magnetization reversal is from 393 Oe to 778 Oe, whereas the rest values that are over 45 Oe only lead to a net magnetization reversal, in which the contribution form H_{DM}field is enough to trigger magnetization reversal but much smaller/larger than contribution from strain. Particularly, the range of for 180° reversal can be tuned by engineering the saturation magnetostriction coefficient λ_{s} (e.g., by using different magnets) and/or the magnitude of ferroelastic strains [e.g., by tuning polarization, see Eq. (4a)]. Detailed analysis can be found in Supplemental materials S1.
Dynamics of magnetic domain switching
So far three different voltageinduced magnetic domain switching paths has been observed under the coexistence of H_{DM}field and strain, including the change in domain pattern periodicity (pattern exchange) when H_{DM}field is weak or absent (< 45 Oe), the full magnetization reversal when contributions from H_{DM}field and strain are comparable (393 Oe < < 778 Oe), and the net magnetization reversal for the remaining values of . To unravel the underlying physics, we track the timedependent changes in the sum of the magnetostatic and exchange energy (i.e., F_{ms} + F_{exch}) for these different paths (figure 4), and their local magnetic vector distributions at various time stages are shown in Supplemental Materials S2. For the net magnetization reversal at = 1000 Oe, an energy barrier is surmounted during its initial stages (<0.04 ns), whereas for the change in domain pattern periodicity at = 6 Oe, a lower energy path is present throughout the evolution. Given the almost identical energy states between these striped domains with opposite net magnetization (i.e., pattern exchange and net reversal in figure 4), and also given their similar energy oscillation trends after surmounting the energy barrier, it can be concluded that the energy barrier can only be overcome by the unidirectional H_{DM}field rather than the uniaxial strain. In particular, for the full magnetization reversal at = 500 Oe, the counteraction between H_{DM}field and strain leads to the lowest energy state before and after switching, however, an energy barrier still needs to be surmounted to complete the reversal.
Discussion
Voltagecontrolled magnetism in magnetic/BiFeO_{3} heterostructures is a complex process with multiple underlying mechanisms, depending on the size of such heterostructures^{27}. In the present article, our focus has been put on the effects of both ferroelastic strains and DM exchange interaction in the CoFe/BiFeO_{3} heterostructures currently investigated. Besides these effects, the modulation of magnetism could also be contributed by the voltageinduced changes in interface charge densities^{28,29}, and/or interface orbital configuration (e.g., via FeO hybridization, refs. 30,31). Such interface effects would become more remarkable as the thickness of the CoFe film (2.5 nm herein) further decreases^{32}. On the other hand, further decrease in the lateral size of the heterostructures may allow us to study the complex interplay among strain, exchange interaction, and possibly the charge/orbital effect in ferroic singledomains. This is not only fundamentally interesting, but also important for the design of highdensity magnetoelectric devices^{25,26}. The model established herein could provide a good starting point for the further study of these issues.
In summary, a phasefield model has been developed to understand how strain affects the voltagemodulated magnetism in BFObased heterostructures. The model is validated by reproducing the experimentally observed voltagedriven inplane net magnetization reversal in a CoFe/BFO heterostructure^{2} driven by nonuniform H_{DM}field. Then we evaluate the stable magnetic and ferroelectric domain structures by adding the influence of the nonuniform ferroelastic strain that arises from the ferroelectric domain patterns at the BFO surface. Under such coexistence of H_{DM}field and strain, three different magnetic domain switching paths are discovered depending on the magnitude of the H_{DM}field, including a full 180° reversal of uniform magnetization when contributions from H_{DM}filed and strain are comparable. Analysis on magnetic domain switching dynamics demonstrates the lowest energy state for such full magnetization reversal, as well as the decisive role of H_{DM}field for both full and net magnetization reversals.
Methods
Phasefield model
In phase field approach, the magnetic and ferroelectric domain structures are described by the spatial distributions of the local magnetization vector M = M_{S}(m_{1},m_{2},m_{3}) and local polarization vector P = (P_{1},P_{2},P_{3}), respectively, where M_{S} and m_{i} (i = 1,2,3) represent the saturation magnetization and the direction cosine^{33}.
The temporal evolution of the ferroelectric domain structure in (001)oriented BFO thin films is governed by the timedependent LandauGinzburg (TDGL) equation^{34}, i.e.,where L is a kinetic coefficient that is related to the domain wall mobility and F_{P} is the total free energy of the FE layer, which can be expressed as,here f_{bulk}, , f_{electric}, and f_{grad} indicate the densities of bulk free energy, elastic energy, electrostatic energy, and gradient energy of the BFO, respectively, with V_{P} representing the volume of the ferroelectric layer in the heterostructure. The mathematical expressions for the f_{bulk}, f_{grad}, and f_{electric} of the BFO (001) thin films are described in literature^{35}. Corresponding to the local inplane electricfields applied across the CoFe/BFO heterostructure via planar poling electrodes^{2}, the electrostatic energy density f_{electric} is obtained under a shortcircuit boundary condition^{36}.
The elastic energy density is calculated as,where is the elastic stiffness tensor of the BFO and e_{ij}(r) the positiondependent elastic strain; ε_{ij}(r) is the total strain and is the spontaneous (stressfree) strain of the BFO arising from the electrostrictive effect, i.e., (i,j,k,l = 1,2,3,4), where Q_{ijkl} is the electrostrictive coefficient tensor. Following Khachaturyan's elastic theory^{37}, the total strain ε_{ij}(r) can be expressed as the sum of homogeneous and heterogeneous strains, i.e. . Among them, the heterogeneous strain δε_{ij}(r) does not cause any macroscopic deformation in a sample, i.e.,, and can be calculated as , where u_{i}(r) is the displacement.
The homogeneous strain represents the macroscopic deformation, whose inplane components equal the film/substrate mismatch strain if the BFO (001) thin films were fully constrained by the DSO (110) substrate, i.e., , , and . The mechanical equilibrium equation, i.e., , is then solved by taking σ_{13} = σ_{23} = σ_{33} = 0^{38}. It is noteworthy that such anisotropic biaxial inplane strains can reduce the ferroelectric domain variants of the rhombohedral BFO from eight to two^{8}, leading to the unique twovariant striped domains with 71° walls as reconstructed in figure 1a. Particularly, in a partly relaxed BFO thin film (figure 1b), a nonuniform ferroelastic strain could be generated across the first few stressfree unit cells of individual ferroelectric domain, which can be expressed as,Here t_{s}( = t_{p}t_{c}) denotes the thickness of the stressfree unit cells at the BFO film surface (see figure 1b), which is approximated to be 40 nm by taking the total thickness t_{p} of BFO as 110 nm^{39}.
Corresponding to the twovariant FE domains at the BFO surface, a twovariant striped strain distribution with alternating ±0.6% can be derived. Such nonuniform strain would further be transferred, at least partially^{18}, to the upper CoFe film that has a much smaller thickness (t_{m} ~2.5 nm, ref. 2) than t_{s}. Note that the twovariant striped domains in the BFO film should still emerge at the thickness t_{p} of 110 nm as observed in experiments^{39}, though the possible interfacial dislocations would somewhat release the anisotropic mismatch strains from the orthorhombic DSO substrate. Nevertheless, even the fully constrained BFO thin films with striped domains can impose structural strains on the upper magnetic thin film during its growth,^{33} similarly to those observed in magnetic/BaTiO_{3} heterostructures.^{18,19} Thus, the average growthinduced strain affects the domain structure of the asgrown magnetic thin film together with the interfacial H_{DM} field, which can be expressed as,where is the spontaneous (or remnant) polarizations under zero external electric fields. Accordingly, when applying an electricfield to the BFO film, the resultant polarization switching would change spatial distributions of both the H_{DM}( = P × L) field and the strain [Eq. (4a)] at the interface.
Magnetic domain structures of the CoFe film will then evolve driven by these nonuniform H_{DM} field and nonuniform (or the for simulating the domain structures of asgrown CoFe film), and can be described by the LandauLifshitzGilbert (LLG) equation, i.e.,Here γ_{0} and α are the gyromagnetic ratio (taken as −2.2 × 10^{5} m·A^{−1}·s^{−1} from ref. 40) and the Gilbert damping constant (~0.01 from ref. 41), respectively, whereby the real time step ΔT (~0.06 ps) for the magnetic domain evolution can be determined by with Δτ = 0.02. H_{eff} is the effective magnetic field, given as , with μ_{0} denoting the vacuum permeability and F_{m} the total free energy of the CoFe layer. The F_{m} is formulated as,where f_{anis}, f_{exch}, f_{ms}, f_{H}, and are the magnetocrystalline anisotropy energy density, exchange energy density, magnetostatic energy density, the H_{DM}field energy density, and elastic energy density, respectively. Among them, the f_{anis} is neglected for simplicity regarding the isotropic nature of the polycrystalline CoFe film. The isotropic f_{exch} is determined by the gradient of local magnetization vectors, i.e.,where J denotes the exchange stiffness constant. The magnetostatic energy density f_{ms} can be written as,Here the stray field consists of a heterogeneous part from the magnetostatic interaction which depends on local magnetization distributions and is obtained by solving magnetostatic equilibrium equation, i.e., , under periodic boundary conditions^{42}. also includes an demagnetization part that relates to the average magnetization by the samplesize dependent demagnetizing factor matrix N^{43}, i.e., .
Furthermore, the H_{DM}field energy density can be expressed, similarly to the Zeeman energy of an external magnetic field, as,The indicates the H_{DM}field imposed on the CoFe film, and is given as,where t_{i} denotes the thickness of the interface creating interfacial magnetic interaction, and the H_{DM}field vector in the BFO layer can be expanded as,The P_{i} and L_{i} (i = 1,2,3) are the components of the polarization vectors and the antiferromagnetic axes along the cubic <100> axes, respectively, and the represents the magnitude of the H_{DM} field that depends on the chemical composition and geometric size of a specific BFObased heterostructure^{9,44}. Note that the antiferromagnetic axis L is expressed as L = (P_{1}, P_{2}, 0) × (0, 0, P_{3}), which indicates an alignment along either the inplane or [110] axis depending on polarization orientations (see figure 1a). Equation (11) clearly indicates that nonuniform nature of the H_{DM}field related to individual ferroelectric domain at the BFO surface, which can propagate across the heterointerface and act on the CoFe film similarly to the case in nonuniform ferroelastic strains [Eqs. (4a) and (4b)]. Combining Eq. (11), Eq. (10) can be rewritten as,Influence of nonuniform on magnetic domain structures is characterized by changes in the elastic energy density of the CoFe, i.e., , which can be calculated similarly to that in ferroelectric BFO [Eq. (3)] but use the elastic stiff constants of CoFe. Note these nonuniform are included in the positiondependent stressfree strain of the magnets,with λ_{s} denoting the saturation magnetostrictive coefficient, taken as −62 ppm herein^{45}. The coupling factor η (0 ≤ η ≤ 1) is introduced to describe the possible loss of transferred strain due to imperfect interface contact, and is assumed to be 1 for a full strain transfer.
Temporal evolutions of the ferroelectric and magnetic domain structures are obtained by numerically solving the TDGL and LLG equations using semiimplicit Fourier spectral method^{46} and GaussSeidel projection method^{47}, respectively. The material parameters used for simulations, including the Landau coefficients, electrostrictive coefficients, elastic constants, gradient energy coefficients of BFO layer, and the saturated magnetization, exchange stiffness constant, elastic constants of CoFe layer, are taken from literatures^{48,49,50}. The discrete grid points of 128Δx × 128Δy × 48Δz with real grid space Δx = Δy = 10 nm, and Δz = 4 nm are employed to describe the BFO film/substrate system, wherein the thickness of BFO t_{p} is taken as 112 nm by setting t_{p} = 28Δz, and the thickness of the interface t_{i} creating the interfacial magnetic interaction is taken as 4 nm by setting t_{i} = 1Δz. While for CoFe thin film, discrete grid points of 128Δx × 128Δy × 20Δz with Δx = Δy = 10 nm, and Δz = 0.5 nm are used, where the thickness of the CoFe t_{m} is set to be 2. 5 nm by taking t_{m} = 5Δz.
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Acknowledgements
This work is supported by the NSF of China (Grant Nos. 51332001, 11234005 and 51221291), Beijing Education Committee (Grant No. 20121000301), and the NSF under grant No DMR1006541.
Author information
Author notes
 J. J. Wang
 & J. M. Hu
These authors contributed equally to this work.
Affiliations
State Key Lab of New Ceramics and Fine Processing, School of Materials Science and Engineering, Tsinghua University, Beijing, 100084, China
 J. J. Wang
 , J. M. Hu
 , M. Feng
 , L. Q. Chen
 & C. W. Nan
Department of Materials Science and Engineering, The Pennsylvania State University, University Park, Pennsylvania, 16802, USA
 J. M. Hu
 , T. N. Yang
 & L. Q. Chen
Department of Physics, Beijing Normal University, Beijing, 100875, China
 J. X. Zhang
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J.J.W., J.M.H. and T.Y. performed the simulations. C.W.N. and L.Q.C. directed the work. J.M.H., J.J.W., L.Q.C. and C.W.N. cowrote the paper. J.J.W., J.M.H., M.F., J.Z. and C.W.N. analyzed the data. All contributed discussion.
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