Introduction

Multiferroic magnetoelectric (ME) heterostructures are comprised of at least two different ferroic (ferromagnetic, ferroelectric, antiferromagnetic, etc.) materials that are artificially combined with well-defined interface1,2,3,4,5. In multiferroic ME heterostructures, the ability to switch magnetization directly using an electric voltage rather than electric current (i.e., magnetoelectric coupling) can potentially be implemented to solve the overheating problem in spintronic devices6,7,8. Among the various multiferroic ME heteostructures, the magnet/BiFeO3 (BFO) heterostructure is attracting increasing research efforts9,10,11,12,13,14,15,16,17,18, in which the ferroelectric (FE) polarization P is coupled to the magnetization M in the overlaying magnet, based on the intrinsic room-temperature ME coupling between P and the antiferromagnetic axis L in the BFO and the exchange coupling between M and L. In particular, in Co90Fe10 (abbreviated to CoFe)/BFO thin-film heterostructures, voltage controlled net magnetization reversal has been demonstrated driven by this interfacial exchange interaction10,13. A voltage controlled uniform (i.e., magnetic single-domain) magnetization reversal is further predicted under the action of both exchange interaction and ferroelastic strain16. However, in these CoFe/BFO heterostructures, the voltage was applied through in-plane electrodes grown on top of the BFO film10,13. These in-plane electrode configuration introduces unfavorable factors including too large coercive voltage, too large size limitation and inconvenient integration into conventional microelectronics circuits. Therefore, an out-of-plane (i.e., perpendicular) voltage configuration across the BFO film is highly desired, for the design of high-performance spintronic devices19,20. Soon after we submitted the present article, a wonderful experimental work on the magnetization reversal by out-of-plane voltage in the CoFe/(110) BFO heterostructure was reported21, further indicating this future research tendency in the BFO-based multiferroic heterostructures. In this article, we demonstrate, using phase field method16,22,23, a mesoscale morphological engineering approach to achieving perpendicular voltage controlled magnetization reversal in magnetic/BFO thin-film heterostructures.

Consider the widely investigated Co90Fe10 (abbreviated to CoFe herein)/BFO heterostructure as an example. Compared to the (001)-oriented BFO films in previous CoFe/BFO heterostructures10,13,16, a (110)-oriented BFO film is utilized herein, which can be grown on a (110) SrTiO3 substrate with SrRuO3 as the bottom electrode24. An electric voltage is then applied perpendicularly through the SrRuO3 and the top CoFe dot that can be cut out from a continuous CoFe film by focused ion beams25, as shown in Fig. 1a. In order to demonstrate the perpendicular voltage-induced magnetization reversal, a phase field model is developed to understand and predict the switching behaviors of the polarization and magnetization in the (110) BFO thin film/island and the CoFe dot (see Methods).

Figure 1
figure 1

Voltage-induced ferroelectric and antiferromagnetic switching in CoFe/(110) BiFeO3 heterostructure.

(a) Schematic illustration of the proposed CoFe/(110) BiFeO3 multiferroic heterostructure under perpendicular voltage modulation. (b) The lattice structures of (110) BiFeO3 film in phase and under perpendicular electric field along [110] direction, the polarization may switch by (c) 180° to phase, or (d) 109° to phase.

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Results

Principles of the Perpendicular Voltage Controlled Magnetization Reversal

By engineering the substrate26,27 or pre-poling24 the (110) BFO film using piezoelectric force microscopy (PFM) before sputtering the CoFe layer, remnant polarization distributions with single-domain state or large surface-area (usually larger than 3 μm × 3 μm) individual domains can be obtained according to experiments24,26,27. Therefore, it is very likely that the patterned nanoscale CoFe dot lies on an individual domain of BFO with much larger size over microns, for instance, the domain with P along as shown in Fig. 1a. Given that the polarization P, the antiferromagnetic axis L and the canted magnetic moment Mcant induced via the Dzyaloshinskii-Moriya (DM) interaction28,29 (the contribution of which can be described by an effective magnetic field HDM along the same direction) are orthogonal to each other30,31 at the CoFe/BFO interface, i.e., Mcant (HDM) = P × L13,32 and also given that L is along the direction for the (i.e., P→) domain according to experimental observations33, the HDM field should be along the direction. Under a negative perpendicular voltage applied along the [110] direction between the SrRuO3 and CoFe electrodes, the polarization may either switch by 180° from to (i.e., P→) (Fig. 1c) or by 109° from to ( i.e., P→[111]c) (Fig. 1d). Experiments also show that the antiferromagnetic axis L along does not rotate30 for both switching paths, though the antiferromagnetic plane rotates during the 109° switching33. As a result, the HDM field rotates from to or after the 109° or 180° switching, respectively. Such reversal of the in-plane component of HDM field may further induce a reversal of in-plane net magnetization as demonstrated later. If the CoFe nanodot does not lie on an individual domain but on the domain wall of BFO, the electric field control of magnetization would be weakened because the interfacial exchange coupling can be neutralized due to the inhomogeneity of the polarization as well as the DM field at the BFO domain wall.

Features of polarization/magnetization reversal in CoFe dot /(110) BFO film and island

As shown in Fig. 2a, by applying a negative voltage (i.e., with upward electric fields) of −2Vc (Vc being the coercive voltage) across the (110) BFO film through the CoFe dot with an in-plane size of 80 nm × 80 nm and SrRuO3 bottom electrode, the volume fraction of domain in the region underneath the CoFe drops dramatically after about 5  ns. Meanwhile, the volume fraction of domain increases greatly and reaches a maximum value of 81% (with respect to the poled region) after about 200 ns and then decreases rapidly. This is accompanied by the dramatic increase in the volume fraction of domain. After 1500 ns, the polarizations underneath the CoFe dot are stabilized almost as a single-domain with a volume fraction of 96.3%. This sub-μs ferroelectric switching time is consistent with experimental observations in BiFeO3 and Pb(Zr,Ti)O3 films34,35. Such polarization reversal via successive 109° switching from to and 71° switching from to under perpendicular voltage has been experimentally observed in (110) and (001) BFO films24,36.

Figure 2
figure 2

Mechanism of voltage-induced 180° ferroelectric switching in CoFe /(110) BiFeO3 heterostructure.

For (110) BiFeO3 thin film with 80 × 80 nm2 CoFe dots grown above as the top electrode and ferromagnetic layer, the [110] electric field induced (a) phase evolutions and (b) the switching path from first to and then to , obtained from the total energy profile as a function of polarization orientation defined by θ and ϕ in inset schematic coordinate system. The total energy in (b) is illustrated by both the 3-Dimensional grid curves and 2-Dimensional color contour map. The inset lattice structures illustrate the polarization orientation of , and phase (i.e., P→for , P→ for and P→ for , see Fig. 1). (c) Phase evolutions of after removing the voltage for CoFe/(110)-film BiFeO3 multiferroic heterostructures with different CoFe size.

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To understand the underlying physics for this ferroelectric switching path, we calculate the total free energy density profile by setting the polarization in the region underneath the CoFe dot of 192 nm × 192 nm pointing along every directions (see the orientation angles θ and ϕ) in the x’y’z’ coordinate system. As shown in Fig. 2b, for an initial domain, the low-energy polarization switching path is within the plane (ϕ = 0°). An electric field along [110] direction is required to overcome the energy barrier, which mainly results from the elastic and Landau-type bulk free energy, to switch the domain (θ = −58°, ϕ = 0°) to the metastable (see the saddle point in the energy density profile) (θ = 70°, ϕ = 0°), which would further relax to the (θ = 112°, ϕ = 0°) domain. Although the electric energy increases as the head to tail (unpoled region)/(poled region) 109°domain wall changes into the / domain wall during the latter process, the elastic energy decreases more significantly. Indeed, when the polarization underneath the CoFe dot is , and , the corresponding elastic and electric energy densities (felastic, felectric) are (6.91 MJ/m3, −4.01 MJ/m3), (14.1 MJ/m3, −4.18 MJ/m3) and (5.97 MJ/m3, 2.64 MJ/m3), respectively.

Figure 2c further shows the time-dependent evolution of the domain volume fraction in the BFO region underneath the CoFe electrode after removing the negative voltage of −2Vc, with the in-plane size of the CoFe dot varying from 32 nm × 32 nm to 192 nm × 192 nm. Once removing the voltage, the /multi-domain will evolve back to the single-domain to reduce the electric energy, which is evidenced by the presence of global energy minima at the single-domain in the profile shown in Fig. 2b. Such relaxation of high-energy domain structure has been experimentally observed in BFO thin films36,37. Nevertheless, the stability of the / multi-domain can be improved by increasing the in-plane size of the CoFe electrode (e.g., to 192 nm × 192 nm), as shown in Fig. 2c.

Now turn to discuss how the perpendicular electric field modulates the magnetization distribution in CoFe dots through the interfacial exchange interaction mechanism. As discussed above, the (110) BFO thin film can be pre-poled to become a single domain over a micron scale range, for instance, the single domain obtained by applying a positive voltage along direction (see Fig. 3a). In this case, the magnetization distribution in the CoFe dot of 192 nm × 192 nm × 2.5 nm exhibits a typical ‘leaf’-like ground state structure38 with average magnetization component mx’ = 0.90, my’ = 0.31 under the interface HDM field along the direction (see the top part of Fig. 3b). As the electric field reverses to be along [110] direction under negative voltages (on the right of Fig. 3a), the local domain in the region underneath the CoFe dot transforms into a local domain by successive 109° and 71° switching, leading to a reoriented interface HDM field along the direction (see the bottom part of Fig. 3b). Accordingly, the average magnetization components gradually changes to mx’ = −0.90, my’ = 0.31, i.e., it occurs a 150° reversal of the net magnetization along the in-plane x’ axis. Note that such reversal of in-plane net magnetization requires a sufficiently large magnitude of the interface HDM field (namely, ). The critical value of is dependent on the in-plane size of the CoFe dot. As shown in Fig. 3c, the critical value increases with increasing in-plane lateral size L (the length and width being the same) as a power-function. Specifically, for the CoFe dots with in-plane size below 192 nm × 192 nm, the critical value is about 100 Oe, quite close to the experimentally measured value (about 100 Oe) in a similar layered heterostructure of polycrystalline CoFe thin film and (001) BFO thin films15.

Figure 3
figure 3

Voltage-controlled magnetism in CoFe/(110) BiFeO3 heterostructure.

(a) Perpendicular electric field induced ferroelectric and ferromagnetic domains switching in dot CoFe/(110)-thin-film BFO multiferroic heterostructure mediated by (b) the interfacial DM fields. (c) Dependence of the critical DM field strength which can make the net magnetization switch 150° under perpendicular voltage on the in-plane lateral size (length and width) of CoFe dots.

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Figure 4 further shows the evolution of the distributions of polarization, interface HDM field and magnetization under successive square-wave bipolar voltage pulses with a constant duration of 4500 ns (Fig. 4a) in the CoFe dot (192 nm × 192 nm × 2.5 nm)/(110) BFO heterostrucuture. As it can be seen, non-volatile behaviors are exhibited for the 180° ferroelectric reversal (Fig. 4b,e) and subsequent reversal of the interface HDM field from to (see Fig. 4c,f). As a result, the in-plane magnetization component mx’ can be reversed back and forth between the bistable states of −0.9 and 0.9 (i.e., a 150° reversal, see Fig. 4d,g) under perpendicular voltage. Moreover, as the magnetization switching (~75 ns) in the CoFe dot is much faster than the ferroelectric switching (~4500 ns) in the BFO thin film, the ferroelectric switching is the time-determining step during such perpendicular voltage-induced magnetization reversal.

Figure 4
figure 4

Polarization and magnetization dynamics in CoFe dot/(110)-BiFeO3 film under voltage.

(a) Perpendicular voltage pulse applied on the poly-dot CoFe/(110)-thin-film BFO multiferroic heterostructure. (b) Voltage pulses induced ferroelectric phase evolutions of , , and in BFO thin-film layer. (c) Voltage pulses induced interfacial HDM field and hence (d) the magnetization evolution in CoFe dots. Voltage-induced corresponding (e) polarization domain-vector morphology in x’-z’ middle cross-sectional plane of BFO-substrate layers, (f) the and (g) magnetization domain-vector morphologies in x’-y’ plane middle cross-sectional plane of CoFe layer, respectively. The color wheel indicates the P , , or m orientation in the their correspondingly exhibited planes.

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The entire time of reversal, however, can be reduced by cutting out the continuous (110) BFO thin film into isolated islands so that the overlaying CoFe dot covers the whole surface of BFO (see Fig. 5). In this case, the polarization time can be greatly reduced because of the significantly released elastic energy. As demonstrated in Fig. 6, upon a negative voltage pulse of 495 ns (see the second stage of Fig. 6a), a single-domain 109° polarization switching from to happens (Fig. 6b,e), leading to a reversal of the in-plane component of interface HDM field from to (Fig. 6c,f). A similar 150° reversal of in-plane net magnetization occurs accordingly (Fig. 6d,g). Note that the time of 109° polarization switching is about 45 ns, which is the same as the magnetization switching time of 45 ns. As a result, the overall magnetization switching time driven by the perpendicular voltage is also 45 ns, which is about 100 times faster than the case in the heterostructures involving BiFeO3 thin films (i.e., about 4500 ns as shown in Fig. 4).

Figure 5
figure 5

Patterned CoFe/BiFeO3 island heterostructure.

Dot CoFe/(110)-island BFO multiferroic heterostructures grown on (110) SrTiO3 substrate with CoFe and SrRuO3 as the top and bottom electrodes, respectively.

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Figure 6
figure 6

Polarization and magnetization dynamics in CoFe dot/(110)-BiFeO3 island under voltage.

(a) Perpendicular voltage pulse applied on the poly-dot CoFe/(110)-island BFO multiferroic heterostructure. (b) Voltage pulses induced ferroelectric phase evolutions of and in BFO island layer. (c) Voltage pulses induced interfacial HDM field and hence (d) the magnetization evolution in CoFe dots. Voltage-induced corresponding (e) polarization domain-vector morphology in x’-z’ middle cross-sectional plane of BFO-substrate layers, (f) the and (g) magnetization domain-vector morphologies in x’-y’ plane middle cross-sectional plane of CoFe layer, respectively. The color wheel indicates the P , , or m orientation in the their correspondingly exhibited planes.

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Another advantage of using BiFeO3 island based heterostructure is the significantly improved non-volatility (and thermal stability). Unlike the case of a continuous BFO thin film where the 180° domain wall between the poled region () and the unpoled region () leads to high electric energy, the 109° switching in the present single-domain BFO island (Fig. 6b) is thermodynamically stable. Figure 7a shows the total free energy density profile of a single-domain BFO island by assuming the polarization pointing along every direction in the x’y’z’ coordinate system. As it can be seen, , , and are thermodynamically degenerate with equal depth at equilibrium states. The barrier (Δf) between and or and determines the stability of polarization under thermal fluctuation. For the CoFe dot only with in-plane magnetization, the energy barrier Δf between the global energy minima at f2 and the metastable state f1 (induced by the interface HDM field along the direction) determines the thermal stability of magnetization, as shown in Fig. 7b. Accordingly, the thermal stability factors of BFO island and CoFe dot can be calculated as , where kB and T are the Boltzmann constant and temperature in Kelvin, respectively and Vi (i = BFO, CoFe) is the volume.

Figure 7
figure 7

Thermal stability analysis in CoFe dot/(110)-BiFeO3 island.

(a) Energy profile as function of polarization orientation for single-domain BFO islands. (b) Energy polar plot for an in-plane magnetized CoFe dot with a size of 192 nm × 192nm × 192 nm when the polarization in the bottom single-domain BFO island is along the direction. (c) Thermal stability factors as function of the respective lateral sizes (length and width) of BFO and CoFe. The thermal stability can be obtained from the energy profile by extracting the barrier (Δf) between the adjacent energy minimal.

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Figure 7c further shows the thermal stability factor of both BFO islands and CoFe dots as a function of their in-plane sizes, with their thicknesses fixed at 24 nm and 2.5 nm, respectively. As seen, the stability factors of the BFO islands are at least two orders of magnitude higher than those of the CoFe dots, indicating that the thermal stability of the heterostructure is determined by the latter. Such high thermal stability of polarization in single-domain BFO islands results from the high potential barrier between the degenerate polarization states from the Landau-type bulk free energy. Nevertheless, the thermal stability factor of the CoFe dot can still be larger than 60 as its in-plane size exceeds 32 nm × 32 nm, suggesting a long timescale retention of magnetization states up to 10 years in the ideal case39,40.

Discussion

In summary, perpendicular voltage-driven reversal of in-plane magnetization reversal has been demonstrated by phase-field simulations in multiferroic magnetoelectric heterostructures composed of polycrystalline CoFe dots and (110) BiFiO3 continuous film or island. In the clamped BiFeO3 thin film, the 180° ferroelectric reversal occurs by successive 109°and 71° ferroelastic switching. The non-volatility of such ferroelectric reversal can be enhanced by increasing the in-plane size of the overlaying CoFe dot, to alleviate the energy competition between the poled region underneath the CoFe dot and the rest region. Associated with repeatable polarization reversal, a repeatable 150° reversal of in-plane net magnetization reversal in the CoFe dot has been further demonstrated due to the reversal of the in-plane component of the interface HDM field.

Similar non-volatile and repeatable voltage-induced magnetization reversal has been demonstrated when the BiFiO3 thin film is etched into islands to release the substrate clamping and to eliminate the competition between the poled and unpoled regions. In such CoFe dot/BiFeO3 island heterostructure, bistable 109° ferroelastic switching in single-domain BiFeO3 has been demonstrated, which leads to 100 times faster ferroelectric switching (and hence faster overall response). As the switching time is estimated according to the Kolmogorov–Avrami–Ishibashi model (see Methods) which assumes that the polarization reversal occurs by domain wall nucleation and propagation, the calculated time (Fig. 6) for uniform switching in the CoFe dot/BiFeO3 island heterostructure could be overestimated, i.e., the actual switching speed could be faster in the CoFe dot/BiFeO3 island heterostructure, since we used the same value of the kinetic coefficient L in the TDGL equation for the BiFeO3 island as for the BiFeO3 thin film due to unknown for the island. Actually, the L values are different for continuous thin film and isolated island due to their different strain conditions and should be larger in the BiFeO3 island than in the clamped continuous thin film. A larger L in the BiFeO3 island would yield a higher switching speed, though it is hard to estimate the actual switching speed in the island at present. Furthermore, the island heterostructure also shows good thermal stability even when the in-plane size of the heterostructure decreases down to 32 nm × 32 nm. These predictions would provide further directions for experimental studies of the BiFeO3-based multiferroic heterostructures for potential spintronic device applications.

Methods

Phase-field model

In phase field modeling of the magnetic/BFO multiferroic heterostructures, the spatial distributions of local polarization and magnetization vectors are used to describe the ferroelectric and ferromagnetic domain structures, respectively. As the ferroelectric phase of BiFeO3 has a rhombohedrally distorted perovskite structure with space group R3c, the spontaneous polarization of BiFeO3 is along the pseudocubic <111>c in coordinate system xyz with x, y and z along the [100]c, [010]c and [001]c directions, respectively, giving rise to the formation of eight possible polarization variants, i.e., , , , , , , (see Fig. 1). For studying the ferroelectric domain structure in (110) BFO thin films, we introduce another coordinate system x’y’z’ with x’, y’, z’ along the [001], and [110] directions (see Fig. 1a). The polarization vector in the x’y’z’ coordinate system is chosen to be evolved by the time-dependent Landau-Ginzburg (TDGL) equation41, i.e.,

where L the kinetic coefficient related to the domain wall mobility and FP the total free energy of the FE layer, respectively. The total free energy of the FE layer includes the bulk, elastic, electric and the gradient energies, i.e.,

where VP represents the volume of the FE layer in the heterostructure.

The expressions for the bulk, elastic, electric and gradient energy densities were used as before41,42,43. The correspondence of the polarization components Pj in coordinate system xyz to the counterparts in x’y’z’ is with Tij the transformation matrix given as follows:

Note that when calculating the elastic, electric and gradient energy densities, the related tensors including the electrostrictive coefficient tensor, background dielectric constant tensor, gradient energy coefficient tensor have to be performed transformation from coordinate system xyz to the x’y’z’ (see Ref. 41). For solving the mechanical equilibrium equation, Khachaturyan’s microelastic theory44 is employed by incorporating the thin-film45 or the isolated-island43,46 boundary conditions for the cases of BFO continuous film and island, respectively. For (110) BFO film grown on (110) SrTiO3 substrate under full constraint, the film/substrate mismatch strain are and 47,48.

For solving the electrostatic equilibrium equation, the Fast Fourier Transformation method is employed by incorporating the short-circuit surface boundary condition49 through which the externally applied voltage on BFO by CoFe and SrRuO3 bottom electrode is introduced into the phase field model.

For determining the real time scale corresponding to each iteration step of the TDGL equation, the switching dynamics as a function of iteration step is obtained and compared to the Kolmogorov–Avrami–Ishibashi (KAI) model50,51 as following

where P(t) the switched polarization, t the real time, PS the saturated polarization, t0 the characteristic switching time and n the effective dimension of domain growth. For (110) BFO thin film, n has a value of 2 and t0 is about 1.5 × 10−7s with the applied electric field of 250 kV/cm34. Our simulation shows that a real time interval of Δt = 8.25 × 10−10s for the polarization domain evolution is determined corresponding to the each iteration step of TDGL equation.

The evolution of the magnetic domain structures of the FM layer can be described by the Landau-Lifshitz-Gilbert (LLG) equation, i.e.,

where γ0 the gyromagnetic ratio and α the Gilbert damping constant, respectively. The real time interval for the magnetic domain evolution determined by Δτ(1+α2)/(γ0Ms) corresponding to each iteration step of Eq. (5) is about 0.06 ps with (from Ref. 52), α = 0.01 (from Ref. 32) and 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 the magnetocrystalline anisotropy, exchange, magnetostatic, the HDM-field and elastic energy densities, respectively. Regarding that the CoFe layer is grown in isotropically polycrystalline or amorphous states, all the above energy density expressions are independent of the coordinate system. Among them, the fanis is neglected for simplicity due to the isotropic nature of the polycrystalline 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 Hddenotes the stray field and it can be numerically calculated by employing a finite-size magnetostatic boundary condition previously developed for a 3D array of ferromagnetic cubes53.

For the HDM-field induced energy density, it is similar to the Zeeman energy of an external magnetic field and can be expressed as

Here the indicates the HDM-field that penetrates onto the CoFe film through the interface and is given by,

where ti denotes the thickness of the interface creating interfacial magnetic interaction and the represents the HDM-field vector in the BFO layer in coordinate system x’y’z’. can be obtained from HDM using coordinate transformation , wherein the HDM can be obtained through HDM = P × L. With regard to the antiferromagnetic vector L, it was predicted with a sixfold degeneracy in bulk BiFeO3 system. However, in thin-film system, this sixfold degeneracy is broken and only one easy axis remains, according to the experimental observations and first-principle calculations33. Therefore, the preserved antiferromagnetic vector L coinciding with P for thin-film BiFeO3 in xyz coordinate system can be phenomenologically described by

or the correspondingly equivalent directions. From Eq. (11) one can see that the antiferromagnetic vector L remains for the perpendicular 71° and 180° polarization switching under perpendicular voltage, while it rotates 90° for the in-plane 71° polarization switching under in-plane voltage in (001) BiFeO3 thin films, in consistent with the experimental results13,33. Thus, the DM effect induced HDM field can be written as

The Pi and Li (i = 1,2,3) are the components of the polarization vectors and the antiferromagnetic vectors in coordinate system xyz and the represents the magnitude of the HDM field. Note that the expression of HDM field published in our previous paper is just a specific case for studying the in-plane 71° polarization switching in (001) BiFeO3 thin film13. Equation (12) clearly indicates that P-dependent nature of the HDM field related to individual ferroelectric domain at the BFO surface, which can propagate across the hetero-interface and act on the CoFe dots. Combining Eqs. (3) and (12), Eq. (10) can be rewritten as,

The formulation of the elastic energy density of CoFe layer is also same as before16. Note that for a (110) BFO film under perpendicular voltage approaching to saturation the in-plane ferroelastic strain remains unchanged whenever the polarization vector is up or down, thus only the structural strain introduced during growth of CoFe is affecting the initial magnetization.

Temporal evolutions of the ferroelectric and magnetic domain structures are obtained by numerically solving the TDGL and LLG equations using semi-implicit Fourier spectral method and Gauss-Seidel projection method, respectively. The material parameters used for simulations, including the Landau coefficients, electrostrictive coefficients, elastic constants of BFO layer and the saturated magnetization, exchange stiffness constant, elastic constants of CoFe layer can be found in the literature16,41,54 and are listed as following: (where the temperature T is in K), , , , , , , , ; For isotropically amorphous CoFe film, , , , , . The discrete grid points of 128Δx × 128Δy × 48Δz with real grid space Δx = Δy = Δz = 1 nm are employed to describe the BFO film/substrate system, wherein the thickness of BFO tp is taken as 24 nm by setting tp = 24Δz and the thickness of the interface ti creating the interfacial magnetic interaction is assigned a value of 4 nm by setting ti = 4Δz. While for CoFe dots, discrete grid points of NΔx × NΔy × 20Δz with Δx = Δy = 1 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 and the in-plane size NΔx can be given by setting N. When the in-plane size of CoFe layer is larger than 96 × 96 nm2, the real size of the heterostructure is achieved by changing Δx and Δy but keeping N = 96. For the BFO island, the discrete grid points of 128Δx × 128Δy × 48Δz with real grid space Δx = Δy = 2 nm and Δz = 1 nm are employed with NΔx × NΔy × 24Δz of them are occupied by the BFO island while the rest are the air which allows the lateral relaxation of the island.

Additional Information

How to cite this article: Wang, J. J. et al. Magnetization Reversal by Out-of-plane Voltage in BiFeO3-based Multiferroic Heterostructures. Sci. Rep. 5, 10459; doi: 10.1038/srep10459 (2015).