Magnetization Reversal by Out-of-plane Voltage in BiFeO3-based Multiferroic Heterostructures

Voltage controlled 180° magnetization reversal has been achieved in BiFeO3-based multiferroic heterostructures, which is promising for the future development of low-power spintronic devices. However, all existing reports involve the use of an in-plane voltage that is unfavorable for practical device applications. Here, we investigate, using phase-field simulations, the out-of-plane (i.e., perpendicular to heterostructures) voltage controlled magnetism in heterostructures consisting of CoFe nanodots and (110) BiFeO3 thin film or island. It is predicted that the in-plane component of the canted magnetic moment at the CoFe/BiFeO3 interface can be reversed repeatedly by applying a perpendicular voltage across the bottom (110) BiFeO3 thin film, which further leads to an in-plane magnetization reversal in the overlaying CoFe nanodot. The non-volatility of such perpendicular voltage controlled magnetization reversal can be achieved by etching the continuous BiFeO3 film into isolated nanoislands with the same in-plane sizes as the CoFe nanodot. The findings would provide general guidelines for future experimental and engineering efforts on developing the electric-field controlled spintronic devices with BiFeO3-based multiferroic heterostructures.

using phase field method 16,22,23 , a mesoscale morphological engineering approach to achieving perpendicular voltage controlled magnetization reversal in magnetic/BFO thin-film heterostructures.
Consider the widely investigated Co 90 Fe 10 (abbreviated to CoFe herein)/BFO heterostructure as an example. Compared to the (001)-oriented BFO films in previous CoFe/BFO heterostructures 10,13,16 , a (110)-oriented BFO film is utilized herein, which can be grown on a (110) SrTiO 3 substrate with SrRuO 3 as the bottom electrode 24 . An electric voltage is then applied perpendicularly through the SrRuO 3 and the top CoFe dot that can be cut out from a continuous CoFe film by focused ion beams 25 , 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).

Results
Principles of the Perpendicular Voltage Controlled Magnetization Reversal. By engineering the substrate 26,27 or pre-poling 24 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 μ m × 3 μ m) individual domains can be obtained according to experiments 24,26,27 . 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 + R 3 domain with P along [111] as shown in Fig. 1a. Given that the polarization P, the antiferromagnetic axis L, and the canted magnetic moment M cant induced via the Dzyaloshinskii-Moriya (DM) interaction 28,29 (the contribution of which can be described by an effective magnetic field H DM along the same direction) are orthogonal to each other 30,31 at the CoFe/BFO interface, i.e., M cant (H DM ) = P × L 13,32 , and also given that L is along the [ 24,36 .
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 + R 3 domain, the low-energy polarization switching path is within the ). 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 + R 3 domain (θ = − 58°, φ = 0°) to the metastable (see the saddle point in the energy density profile) − R 1 (θ = 70°, φ = 0°), which would further relax to the − R 3 (θ = 112°, φ = 0°) domain. Although the electric energy increases as the head to tail + R 3 (unpoled region)/ − R 1 (poled region) 109°domain wall changes into the + R 3 / − R 3 domain wall during the latter process, the elastic energy decreases more significantly. Indeed, when  Once removing the voltage, the + R 3 / − R 3 multi-domain will evolve back to the + R 3 single-domain to reduce the electric energy, which is evidenced by the presence of global energy minima at the + R 3 single-domain in the profile shown in Fig. 2b. Such relaxation of high-energy domain structure has been experimentally observed in BFO thin films 36,37 . Nevertheless, the stability of the + R 3 / − R 3 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 + R 3 single domain obtained by applying a positive voltage along [110] 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 structure 38 Fig. 4c,f). As a result, the in-plane magnetization component m x' 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.
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 + R 3 to − R 1 happens (Fig. 6b, (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 BiFeO 3 thin films (i.e., about 4500 ns as shown in Fig. 4).
Another advantage of using BiFeO 3 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 ( − R 3 ) and the unpoled region ( + R 3 ) 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,    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 case 39,40 .

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) BiFiO 3 continuous film or island. In the clamped BiFeO 3 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 H DM field.
Similar non-volatile and repeatable voltage-induced magnetization reversal has been demonstrated when the BiFiO 3 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/BiFeO 3 island heterostructure, bistable 109° ferroelastic switching in single-domain BiFeO 3 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/BiFeO 3 island heterostructure could be overestimated, i.e., the actual switching speed could be faster in the CoFe dot/BiFeO 3 island heterostructure, since we used the same value of the kinetic coefficient L in the TDGL equation for the BiFeO 3 island as for the BiFeO 3 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 BiFeO 3 island than in the clamped continuous thin film. A larger L in the BiFeO 3 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 BiFeO 3 -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 BiFeO 3 has a rhombohedrally distorted perovskite structure with space group R3c, the spontaneous polarization of BiFeO 3 is along the pseudocubic < 111> c in coordinate system xyz with x, y, and z along the [111] 4 c (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], [110], and [110] directions (see Fig. 1a). The polarization vector ′ P in the x'y'z' coordinate system is chosen to be evolved by the time-dependent Landau-Ginzburg (TDGL) equation 41 where L the kinetic coefficient related to the domain wall mobility and F P 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 V P represents the volume of the FE layer in the heterostructure. The expressions for the bulk, elastic, electric, and gradient energy densities were used as before [41][42][43] . The correspondence of the polarization components P j in coordinate system xyz to the counterparts ′ P i in x'y'z' is ′ = P T P i i j j with T ij 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 electrostatic equilibrium equation, the Fast Fourier Transformation method is employed by incorporating the short-circuit surface boundary condition 49  where J denotes the exchange stiffness constant. The magnetostatic energy density f ms can be written as, Here H d denotes 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 cubes 53 . For the H DM -field induced energy density, it is similar to the Zeeman energy of an external magnetic field, and can be expressed as Scientific RepoRts | 5:10459 | DOi: 10.1038/srep10459 in bulk BiFeO 3 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 calculations 33 . Therefore, the preserved antiferromagnetic vector L coinciding with P for thin-film BiFeO 3 in xyz coordinate system can be phenomenologically described by  The P i and L i (i = 1,2,3) are the components of the polarization vectors and the antiferromagnetic vectors in coordinate system xyz, and the h DM 0 represents the magnitude of the H DM field. Note that the expression of H DM field published in our previous paper is just a specific case for studying the in-plane 71° polarization switching in (001) BiFeO 3 thin film 13 . Equation (12) clearly indicates that P-dependent nature of the H DM 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) .
( ) H h t P P P P P P P P P P P P P P P P P dz P P h t P P P P P P P P P P P P P P P P P dz P P The formulation of the elastic energy density f elastic m of CoFe layer is also same as before 16 . 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 literature 16,41,54  setting t i = 4Δ z. While for CoFe dots, discrete grid points of NΔ x × NΔ y × 20Δ z with Δ x = Δ y = 1 nm, and Δ z = 0.5 nm are used, where the thickness of the CoFe t m is set to be 2.5 nm by taking t m = 5Δ z and the in-plane size NΔ x can be given by setting N. When the in-plane size of CoFe layer is larger than 96 × 96 nm 2 , the real size of the heterostructure is achieved by changing Δ x and Δ y but keeping N = 96. For the BFO island, the discrete grid points of 128Δ x × 128Δ y × 48Δ z with real grid space Δ x = Δ y = 2 nm and Δ z = 1 nm are employed with NΔ x × NΔ y × 24Δ z of them are occupied by the BFO island while the rest are the air which allows the lateral relaxation of the island.