Voltage-modulated magnetism in magnetic/BiFeO3 heterostructures can be driven by a combination of the intrinsic ferroelectric-antiferromagnetic coupling in BiFeO3 and the antiferromagnetic-ferromagnetic exchange interaction across the heterointerface. However, ferroelectric BiFeO3 film is also ferroelastic, thus it is possible to generate voltage-induced strain in BiFeO3 that could be applied onto the magnetic layer across the heterointerface and modulate magnetism through magnetoelastic coupling. Here, we investigated, using phase-field simulations, the role of strain in voltage-controlled magnetism for these BiFeO3-based heterostructures. It is predicted, under certain condition, coexistence of strain and exchange interaction will result in a pure voltage-driven 180° magnetization reversal in BiFeO3-based heterostructures.
It is accepted that voltage-modulated magnetism in magnetic/BiFeO3 (BFO) layered heterostructures is based on a combination of intrinsic coupling between the coexisted ferroelectric and antiferromagnetic orders in BFO, and the antiferromagnetic-magnetic exchange interaction across the heterointerface1,2,3,4. However, ferroelectric BFO film, if not fully clamped by substrate, would generate voltage-induced 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 voltage-modulated magnetism in BFO-based heterostructures has remained largely unexplored since raised by Mathur5, though a giant voltage-induced strain of over 5% has later been observed in BFO thin films with mixed rhombohedral and tetragonal phases6,7.
In this article, we explore how strain affects the voltage-modulated magnetism in BFO-based heterostructures by taking the Co0.9Fe0.1(CoFe)/BFO thin-film heterostructure as an example. (001) BFO thin films were grown on (110) DyScO3 substrate, exhibiting two-variant ferroelectric domains with 71° wall due to anisotropic film-substrate misfit strains8. In particular, the magnetic domain patterns in the top CoFe film almost copy the in-plane projection patterns of the ferroelectric domains at the BFO surface2,9. Such magnetic domain pattern is induced by an effective magnetic field from the canted magnetic moment Mc in BFO via Dzyaloshinskii-Moriya (DM) exchange interaction10,11, namely, HDM-field. The direction of HDM-field (also Mc) is perpendicular to the plane of the polarization P and the antiferromagnetic axis L12,13, i.e., HDM = P × L, resulting in a non-uniform HDM-field distribution based on the two-variant 71° ferroelectric domains. In this case, when electrically rearranging ferroelectric domain configuration to switch the in-plane net polarization in BFO by 180°, rotation of HDM-field within individual ferroelectric domain could lead to an overall 180° switching of the in-plane net magnetization2 in CoFe. Detailed experimental analysis14 reveals that the HDM-field (Mc) lies along the and directions corresponding to the and polarizations, while along the and directions corresponding to the and  polarizations, respectively (figure 1a).
Furthermore, the first few unit cells at the BFO thin film surface would become stress-free when the thickness of the BFO film was above a certain critical value [tc ~70 nm for the BFO films grown on DyScO3(DSO) substrate15,16] to allow the presence of lattice defects such as dislocation (figure 1b). In this case, sizable and non-uniform strain can be generated associated with local non-180° ferroelectric polarization (i.e., ferroelastic) switching under an electric-field. The strain can be further transferred to the top magnetic thin film across the interface and modulate the magnetic domain structure locally via magnetoelastic coupling17. Transfer of such non-uniform strain has recently been demonstrated in BaTiO3 single crystal-based heterostructues18,19,20,21, which exhibit a similar one-to-one domain pattern match between magnetic thin film and ferroelectric BaTiO3 underneath. Particularly, it was proposed that22 such non-uniform ferroelastic strain transfer alone could drive a 180° in-plane net magnetization reversal similarly to the non-uniform HDM-field driven reversal shown in figure 1a. These similarities thus raise one interesting question for BFO-based heterostructures: what role does the non-uniform strain in BFO play in the electric-field induced magnetization reversal? A phase-field23,24 model is developed herein (see Method section) to address this question, and illustrate new possibility in electric-field-controlled magnetism under coexistence of non-uniform HDM-field and non-uniform strain.
In-plane net magnetization reversal purely by HDM-field
We first examine the influence of the interfacial HDM-field on the electric-field driven magnetization reversal in the CoFe/BFO heterostructure. If the electric-field applied along the  direction (E100) exceeds the coercive field of BFO (Ec), the initial alternating and ferroelectric domains would switch by 71° to the and  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  direction (i.e., P100), as illustrated by the simulated ferroelectric hysteresis loop in figure 2c. Note that individual ferroelectric domain under a certain E100 results in one unique local HDM-field distribution [Eq. (11)]. Correspondingly, figure 2b presents the in-plane projections of the HDM-field, e.g., an initial configuration of head to tail in-plane and orientations. The induced local magnetization distributions (viz. domain structures) in the CoFe film are almost identical to those of in-plane HDM-field (not shown here for simplicity), while both distributions are essentially the same as the in-plane projections of the ferroelectric domains and thereby accounts for the domain pattern transfer between the CoFe and BFO films2. Furthermore, a full reversal of the net  magnetization (M100) in the CoFe film occurs when reversing the average in-plane HDM field along the  direction () with an electric-field, as shown by their electric-field switching loops in figure 2e and 2d, respectively. The magnitude of the total HDM field [i.e., , see Eqs. (11) and (12) in the Method section] is taken as 100 Oe based on a relevant experimental measumrent9.
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 (λs = 0). As it can be seen, once the striped domains form at 35 Oe, the net in-plane magnetization can then be reversed, i.e., the normalized magnetization M100/Ms changes from −0.7 to about 0.7, by changing the polarity of the net in-plane 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 HDM-field is equivalent to an external non-uniform magnetic field [Eq. (9)].
Full magnetization reversal driven by both HDM-field and strain
Figure 3b shows the M100/Ms variations as a function of before (the upper panel) and after (the lower panel) reversing the net in-plane polarization, driven by both non-uniform HDM-field and non-uniform 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 growth-induced 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 in-plane 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 in-plane polarization. Thus, although non-uniform 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 HDM-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 electric-field poling, see the upper panel of figure 3b).
These counteractive non-uniform HDM-field and non-uniform 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 HDM-field are 8.26 × 104 J/m3 and −7.95 × 104 J/m3, respectively, at = 500 Oe. Furthermore, a full 180° magnetization reversal takes place when reversing the in-plane polarization (see corresponding domain structures in figure 3b), which should promise higher application potential for low-power memory applications25,26 than the average magnetization reversal in striped multi-domain. Further simulations show that the range of required for such voltage-driven 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 HDM-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 λs (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 voltage-induced magnetic domain switching paths has been observed under the coexistence of HDM-field and strain, including the change in domain pattern periodicity (pattern exchange) when HDM-field is weak or absent (< 45 Oe), the full magnetization reversal when contributions from HDM-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 time-dependent changes in the sum of the magnetostatic and exchange energy (i.e., Fms + Fexch) 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 HDM-field rather than the uniaxial strain. In particular, for the full magnetization reversal at = 500 Oe, the counteraction between HDM-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.
Voltage-controlled magnetism in magnetic/BiFeO3 heterostructures is a complex process with multiple underlying mechanisms, depending on the size of such heterostructures27. In the present article, our focus has been put on the effects of both ferroelastic strains and DM exchange interaction in the CoFe/BiFeO3 heterostructures currently investigated. Besides these effects, the modulation of magnetism could also be contributed by the voltage-induced changes in interface charge densities28,29, and/or interface orbital configuration (e.g., via Fe-O hybridization, refs. 30,31). Such interface effects would become more remarkable as the thickness of the CoFe film (2.5 nm herein) further decreases32. 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 single-domains. This is not only fundamentally interesting, but also important for the design of high-density magnetoelectric devices25,26. The model established herein could provide a good starting point for the further study of these issues.
In summary, a phase-field model has been developed to understand how strain affects the voltage-modulated magnetism in BFO-based heterostructures. The model is validated by reproducing the experimentally observed voltage-driven in-plane net magnetization reversal in a CoFe/BFO heterostructure2 driven by non-uniform HDM-field. Then we evaluate the stable magnetic and ferroelectric domain structures by adding the influence of the non-uniform ferroelastic strain that arises from the ferroelectric domain patterns at the BFO surface. Under such coexistence of HDM-field and strain, three different magnetic domain switching paths are discovered depending on the magnitude of the HDM-field, including a full 180° reversal of uniform magnetization when contributions from HDM-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 HDM-field for both full and net magnetization reversals.
In phase field approach, the magnetic and ferroelectric domain structures are described by the spatial distributions of the local magnetization vector M = MS(m1,m2,m3) and local polarization vector P = (P1,P2,P3), respectively, where MS and mi (i = 1,2,3) represent the saturation magnetization and the direction cosine33.
The temporal evolution of the ferroelectric domain structure in (001)-oriented BFO thin films is governed by the time-dependent Landau-Ginzburg (TDGL) equation34, i.e.,where L is a kinetic coefficient that is related to the domain wall mobility and FP is the total free energy of the FE layer, which can be expressed as,here fbulk, , felectric, and fgrad indicate the densities of bulk free energy, elastic energy, electrostatic energy, and gradient energy of the BFO, respectively, with VP representing the volume of the ferroelectric layer in the heterostructure. The mathematical expressions for the fbulk, fgrad, and felectric of the BFO (001) thin films are described in literature35. Corresponding to the local in-plane electric-fields applied across the CoFe/BFO heterostructure via planar poling electrodes2, the electrostatic energy density felectric is obtained under a short-circuit boundary condition36.
The elastic energy density is calculated as,where is the elastic stiffness tensor of the BFO and eij(r) the position-dependent elastic strain; εij(r) is the total strain and is the spontaneous (stress-free) strain of the BFO arising from the electrostrictive effect, i.e., (i,j,k,l = 1,2,3,4), where Qijkl is the electrostrictive coefficient tensor. Following Khachaturyan's elastic theory37, the total strain εij(r) can be expressed as the sum of homogeneous and heterogeneous strains, i.e. . Among them, the heterogeneous strain δεij(r) does not cause any macroscopic deformation in a sample, i.e.,, and can be calculated as , where ui(r) is the displacement.
The homogeneous strain represents the macroscopic deformation, whose in-plane 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 σ13 = σ23 = σ33 = 038. It is noteworthy that such anisotropic biaxial in-plane strains can reduce the ferroelectric domain variants of the rhombohedral BFO from eight to two8, leading to the unique two-variant striped domains with 71° walls as reconstructed in figure 1a. Particularly, in a partly relaxed BFO thin film (figure 1b), a non-uniform ferroelastic strain could be generated across the first few stress-free unit cells of individual ferroelectric domain, which can be expressed as,Here ts( = tp-tc) denotes the thickness of the stress-free unit cells at the BFO film surface (see figure 1b), which is approximated to be 40 nm by taking the total thickness tp of BFO as 110 nm39.
Corresponding to the two-variant FE domains at the BFO surface, a two-variant striped strain distribution with alternating ±0.6% can be derived. Such non-uniform strain would further be transferred, at least partially18, to the upper CoFe film that has a much smaller thickness (tm ~2.5 nm, ref. 2) than ts. Note that the two-variant striped domains in the BFO film should still emerge at the thickness tp of 110 nm as observed in experiments39, 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/BaTiO3 heterostructures.18,19 Thus, the average growth-induced strain affects the domain structure of the as-grown magnetic thin film together with the interfacial HDM field, which can be expressed as,where is the spontaneous (or remnant) polarizations under zero external electric fields. Accordingly, when applying an electric-field to the BFO film, the resultant polarization switching would change spatial distributions of both the HDM( = P × L) field and the strain [Eq. (4a)] at the interface.
Magnetic domain structures of the CoFe film will then evolve driven by these non-uniform HDM field and non-uniform (or the for simulating the domain structures of as-grown CoFe film), and can be described by the Landau-Lifshitz-Gilbert (LLG) equation, i.e.,Here γ0 and α are the gyromagnetic ratio (taken as −2.2 × 105 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. Heff is the effective magnetic field, given as , with μ0 denoting the vacuum permeability and Fm the total free energy of the CoFe layer. The Fm is formulated as,where fanis, fexch, fms, fH, and are the magnetocrystalline anisotropy energy density, exchange energy density, magnetostatic energy density, the HDM-field energy density, and elastic energy density, respectively. Among them, the fanis is neglected for simplicity regarding the isotropic nature of the polycrystalline CoFe film. The isotropic fexch is determined by the gradient of local magnetization vectors, i.e.,where J denotes the exchange stiffness constant. The magnetostatic energy density fms 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 conditions42. also includes an demagnetization part that relates to the average magnetization by the sample-size dependent demagnetizing factor matrix N43, i.e., .
Furthermore, the HDM-field energy density can be expressed, similarly to the Zeeman energy of an external magnetic field, as,The indicates the HDM-field imposed on the CoFe film, and is given as,where ti denotes the thickness of the interface creating interfacial magnetic interaction, and the HDM-field vector in the BFO layer can be expanded as,The Pi and Li (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 HDM field that depends on the chemical composition and geometric size of a specific BFO-based heterostructure9,44. Note that the antiferromagnetic axis L is expressed as L = (P1, P2, 0) × (0, 0, P3), which indicates an alignment along either the in-plane or  axis depending on polarization orientations (see figure 1a). Equation (11) clearly indicates that non-uniform nature of the HDM-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 non-uniform ferroelastic strains [Eqs. (4a) and (4b)]. Combining Eq. (11), Eq. (10) can be rewritten as,Influence of non-uniform 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 non-uniform are included in the position-dependent stress-free strain of the magnets,with λs denoting the saturation magnetostrictive coefficient, taken as −62 ppm herein45. 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 semi-implicit Fourier spectral method46 and Gauss-Seidel projection method47, 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 literatures48,49,50. 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 tp is taken as 112 nm by setting tp = 28Δz, and the thickness of the interface ti creating the interfacial magnetic interaction is taken as 4 nm by setting ti = 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 tm is set to be 2. 5 nm by taking tm = 5Δz.
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 DMR-1006541.
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Understanding, Predicting, and Designing Ferroelectric Domain Structures and Switching Guided by the Phase-Field Method
Annual Review of Materials Research (2019)