L1₀ FePt nanostructures have been extensively studied in many magnetic and magnetoelectric devices, such as late-model heat assisted magnetic recording media,^[1–3] permanent magnets,^[3–5] and magnetic heterostructures,^[6–13] due to their large perpendicular magnetic anisotropy (PMA), excellent chemical stability, and good corrosion resistance.^[14–19] The dependence of PMA on the structural domain orientation is vital to the performance of aforementioned applications. Ordered L1₀ FePt thin films include three different orientations of the structural domains, one of which is out-of-plane ordered structural domains (OPOSDs) and the other two are in-plane ordered structural domains (IPOSDs). In general, the OPOSDs are desired in microminiature devices.^[20] However, not all of the structural domains are OPOSDs in the fabricated L1₀ FePt thin films. In order to fabricate high-quality L1₀ FePt thin films, many research groups conducted experimental studies on obtaining the designed OPOSDs by changing temperature, deposition pressure, segregants, deposition technique,^[21] substrates,^[2,22] and so on. Among these, the misfit strain between the thin films and substrate can have a large effect on the formation of the OPOSDs in L1₀ FePt thin films based on the experimental measurements. Wang et al.^[2] and Ho et al.^[20] measured the effect of misfit strains between the L1₀ FePt thin films and substrate on the orientations of structural domains by using a transmission electron microscope (TEM) based automated orientation imaging microscopy technique. The maximal volume of the OPOSDs can only reach 76%,^[2,20] and the rest of the structural domains that result in the in-plane magnetic anisotropy need to be reduced in order to improve the PMA in the L1₀ FePt thin films. One of the important factors to reduce the volume of IPOSDs is to change the strain states between the L1₀ FePt thin films and substrate. Many experimental results^[1,20,22] showed that the large lattice mismatch between the L1₀ FePt thin films and substrates can cause tensile stress to the L1₀ FePt thin films, resulting in a large volume of OPOSDs and a large PMA.

Nevertheless, one experimental work^[23] showed that the value of coercivities increased with the increase of the recoverable compressive misfit strain in the L1₀ FePt (3 nm) thin film deposited on PMN–PT substrate. The reason is that when the PMN–PT substrate is polarized under an applied electric field, a recoverable strain is induced by a converse piezoelectric effect, which can pass on to the L1₀ FePt thin film and thus manipulate the magnetic anisotropy of the film through the converse magnetostriction effect. Their results^[23] also showed that when the electric field increased to ±10 kV/cm, the value of the recoverable compressive strain can reach at least to −0.06%. However, there are few studies on how the misfit strain variations affect the formation and volume of OPOSDs in L1₀ FePt thin films, especially on the mechanism of magnetic domain evolution, which rely on ordered structural domains influenced by strain variations in the ultrathin L1₀ FePt thin films.^[24] Accordingly, it is essential to investigate the mechanisms of the magnetic domain evolutions and magnetic properties in the L1₀ FePt thin films under misfit strains for future applications. To this end, we apply the phase field simulation approach to study the effect of the recoverable strain on the magnetic properties in the L1₀ FePt thin film.

The phase field method is a powerful tool to simulate the magnetic domain structure evolution and the resultant properties in various magnetic materials,^[25–32] which provides useful guidelines for a precise engineering of magnetic domain structures in magnetic thin films.^[33,35] Although the L1₀ FePt thin films are usually polycrystalline or doped for practical applications, a basic study on the single crystal L1₀ FePt thin film deposited on single crystalline PMN–PT substrates can give a better understanding to the physical mechanisms of the strain-induced magnetic domain structure and properties.

In this work, in the light of the experimental work reported in Ref. [23], the effect of the recoverable compressive strains on the PMA, magnetic domains evolution, and out-of-plane magnetic moments components in the single crystal L1₀ FePt (3 nm) thin film were studied in detail by using the three-dimensional phase field model based on Khachaturyan’s microelastic theory.

The vector magnetization M is chosen as the phase field variable to describe the magnetic domain structures in the L1₀ FePt thin film. The spatial and temporal evolutions of M are controlled by the Landau–Lifshitz–Gilbert (LLG) equation^[33–35] 1 where γ₀ is the gyromagnetic ratio and α is the damping constant, with a value of 2.211 × 10⁵ A⁻¹ · s⁻¹ and 0.1 in this simulation, respectively. M = M_s m = M_s (m₁, m₂, m₃), in which M_s is the saturation magnetization, and m_i is the direction cosine. H_eff is the effective magnetic field, and H_eff = −μ₀ M_s δE_total/δM, namely, . Also, E_total is the total magnetic free energy of the L1₀ FePt thin film, which is constructed as 2 where E_anis, E_stat, E_exch, E_elas, and E_exte are the magnetocrystalline anisotropy energy, magnetostatic energy, exchange energy, magnetoelastic energy, and Zeeman energy, respectively. The magnetoelastic energy E_elas is given as^[33–36] 3 where C_ijkl are the elastic coefficients, is the elastic strain, and is the total strain, while is the eigenstrain related to the local magnetization and given by^[31–33] 4 where λ₁₀₀ and λ₁₁₁ are the magnetostrictive coefficients.

The magnetoelastic energy is obtained through combining Khachaturyan’s elastic theory and constraint boundary conditions as reported in Refs. [36] and [37]. The temporal evolution of the local magnetization is obtained by solving the Landau–Lifshitz–Gilbert (LLG) equation. The size of L1₀ FePt thin film is 256 nm × 256 nm × 3 nm. The time step for integration is 2.4 × 10⁻¹³ s. The following parameters were used in the simulation in consistence with the experimental work:^[23] the saturation magnetization M_s of the L1₀ FePt thin film is 1140 emu/cc, the magnetic anisotropic constant K_u is taken as 1.64 × 10⁵ erg/cc, the exchange constant A_ex is 1.0 × 10⁻¹⁰ J/m, the demagnetization factor N_z in the z direction is 1.0, the magnetostriction coefficient λ₁₀₀ is 70 × 10⁻⁶, λ₁₁₁ is 0, and the elastic coefficient C₁₁, C₁₂, and C₄₄ are 304 GPa, 223 GPa, and 107 GPa, respectively.

The shape of magnetic hysteresis loops and magnetic coercivities of the L1₀ FePt thin films play an important role in determining the properties of magnetoelectrical microminiature devices. Figure 1 shows the effect of compressive strain on the magnetic hysteresis loops and magnetic coercivities of the L1₀ FePt (3 nm) thin film. To compare the magnetic coercivities calculated by the phase field method with the magnetic coercivities measured by the anomalous Hall effect experimental method as reported in Ref. [23], area D in the magnetic hysteresis of the L1₀ FePt (3 nm) thin film in Fig. 1(a) is enlarged in Fig. 1(b), which shows that the magnetic coercivity is 2.885 kOe when the strain is 0. The simulated coercivity agrees well with the reported experimental value of 2.88 kOe.^[23] The magnetic coercivity gradually increases with decreasing strain from 0 to −0.06% and reaches to 3.178 kOe when the strain is −0.06%. The increase of magnetic coercivities can be attributed to the contribution of the magnetoelatic energy to the magnetic anisotropy energy and will be explained later in the energy density analysis after afterwards. Table 1 shows the comparison between the experimental^[23] and calculated coercivities. It can be seen that the simulated coercivities are consistent with experimental results. The minor differences might have originated from the inhomogeneous microstructure of the fabricated L1₀ FePt thin film. It is worth mentioning that the magnetic coercivities calculated by the phase field method decrease with the increasing tensile strains. Thus the magnetic properties of the L1₀ FePt thin films can be effectively regulated by using tensile or compressive strains.

Fig. 1.

	Figure Option ViewDownloadNew Window
	Fig. 1. (color online) (a) Simulated magnetic hysteresis loops and (b) the enlargement of area D in the hysteresis loops of the L10 FePt (3 nm) thin film with different compressive strains produced by the PMN–PT substrate under the applied electric field.

Table 1.

Table 1.

Table 1. Comparison of the magnetic coercivities of the L10 FePt (3 nm) thin film between the experimental[23] and the simulation results. . Strain% 0.00 –0.02 –0.03 –0.04 –0.05 –0.06 Experimental coercivities/kOe 2.880 2.983 3.076 3.105 3.127 3.145 Simulated coercivities/kOe 2.885 2.988 3.037 3.085 3.130 3.178

Table 1. Comparison of the magnetic coercivities of the L10 FePt (3 nm) thin film between the experimental[23] and the simulation results. .

Figure 2 shows the magnetic domain structures corresponding to the points A, B, C, D, E, and F in the simulated hysteresis loops of the L1₀ FePt (3 nm) thin film with the strain of −0.06% in Fig. 1(a). In each of the figures of the magnetic domain structures, the upper part is the plane-view image and the bottom part is the cross-section image of the magnetic domain structures in L1₀ FePt thin film and PMN–PT substrate. Figure 2(a) shows that the magnetization is perpendicular to the in-plane direction of the L1₀ FePt thin film. When the switching magnetic field is −1.5 kOe (point B), the magnetization reverses from the [001] direction and forms the [101], , , and [011] magnetic domains, as shown in Fig. 2(b). Figure 2(c) displays that 50.6% of the statistical [001] magnetic domain volume has deviated from [001] direction, while, [101], , , and [011] magnetic domain structures are increasing, and [111], , , and magnetic domain structures are forming with the switch magnetic field increased to −2.0 kOe (point C). It can be seen that the magnetization lies in a random state, and the [001] and magnetic domains are merely around 3% of the magnetic domain volume, as presented in Fig. 2(d), when the switch magnetic field reaches the coercivity −3.178 kOe (point D). In Fig. 2(e), the magnetic domain increases when the switch magnetic field increases to −12.0 kOe, and , magnetic domains have not finished reversing to the direction. Finally, when the switch magnetic field increases to the saturated field −20 kOe, the magnetization completely reverses to the direction, as indicated in Fig. 2(f).

Fig. 2.

	Figure Option ViewDownloadNew Window
	Fig. 2. (color online) Simulated magnetic domain structures, (a)–(f) correspond to the points A–F in the hysteresis loops of the L10 FePt (3 nm) thin film when the compressive strain is −0.06%.

Figure 3 shows the evolution of simulated magnetic domain structures of the L1₀ FePt (3 nm) thin film in different magnetization reversible processes with different compressive strains under the same switch magnetic field H = 2.0 kOe (point C). The magnetic domain structures in Figs. 3(a)–3(f) correspond to the point O, P, Q, R, S, and T in inset of Fig. 1(a), respectively. Figure 3(a) shows that the volume fraction of the [001] magnetic domain is 29.3% without any compressive strain, which means 70.7% of the volume fraction of the [001] magnetic domain changes into other orientation magnetic domains. When there is a trivial compressive strain −0.02% acting on the L1₀ FePt thin film, the volume fraction of the [001] magnetic domain is 35.1% as displayed in Fig. 3(b). The volume fraction of the [001] magnetic domain increases from 38.6% to 49.4% when the compressive strain changes from −0.03% to −0.06%, as shown in Figs. 3(c)–3(f). It demonstrates that when the compressive strain becomes larger, the reversal of the magnetization along the [001] direction becomes slower. In other words, it establishes that when the compressive strain increases, the magnetization perpendicular to the L1₀ FePt thin film plane inclines to align to the [001] direction and reverses tardily compared to the one without compressive strain. This can be explained by considering the magnetic energy contribution to the L1₀ FePt thin film.

Fig. 3.

	Figure Option ViewDownloadNew Window
	Fig. 3. (color online) Simulated magnetic domains structures of the L10 FePt (3 nm) thin film with the same reversible magnetic field H = 2.0 kOe but different compressive strains, (a) 0.00%, (b) −0.02%, (c) −0.03%, (d) −0.04%, (e) −0.05%, and (f) −0.06%, corresponding to the points O, P, Q, R, S, and T in Fig. 1(a).

Figure 4 shows the effect of compressive strain on the magnetic energy densities of the L1₀ FePt (3 nm) thin film at various reverse magnetic fields (20 kOe to −20 kOe). Figure 4(a) shows that the external magnetic field energies increase initially and then decrease with the external magnetic field decreasing from 20 kOe to −20 kOe, and the energy maximum appears at −2.206 kOe. Figures 4(b) and 4(c) show that the magnetic anisotropic energies and magnetic exchange energies are zero in the first place, and then gradually increase from zero to a maximum when the magnetic field decreases from 20 kOe to −3.0 kOe, which finally decrease to zero with the magnetic field decreasing from about −3.0 kOe to −20 kOe. The reason is that the [001] axis is an easy magnetization direction in the L1₀ FePt thin film. When the magnetic fields are large, the magnetic anisotropic energies and magnetic exchange energies are lower for the energy minimization; when the magnetic fields decrease to the coercivities, the magnetic moment deviates from the easy magnetization direction and the magnetic exchange interaction between magnetic moments becomes large. Figure 4(d) shows that the demagnetization energies have the largest values when the magnetization is perpendicular to the plane of the L1₀ FePt thin film along the external magnetic field direction. They start to decrease when the magnetization deviates from the normal direction of the L1₀ FePt thin film and increase again as the magnetization inclines to align perpendicular to the plane of the L1₀ FePt thin film. Figure 4(e) displays that the magnetoelastic energy is zero when there is no compressive strain on the L1₀ FePt thin film under the switching of the magnetic field changing from 20 kOe to −20 kOe. However, when there are different compressive strains acting on the L1₀ FePt thin film, the magnetoelastic energies are constant at high external magnetic fields and the energies peak values appear at coercivities fields, because the magnetoelastic energy can induce a PMA. Thus the magnetoelastic energy is lower when the applied magnetic field is larger, and energies peak values show up as magnetic moments in the L1₀ FePt thin film deviate from the normal direction in the magnetization reversible processes. Moreover, differences of magnetic energies are negligible among different compressive strains, which range from 0 to −0.06%, except magnetoelastic energies. The magnetoelastic energy in the L1₀ FePt thin film increases by leaps with compressive strain strengthening due to the increase of compressive stress in the L1₀ FePt thin film, which is also ascribed to the fact that the magnetoelastic energy affects the magnetic anisotropy so that the distribution of the easy magnetization axis is changed. The total magnetic energies are shown in Fig. 4(f), which determine the reversible magnetization processes. Therefore, the magnetization even inclines to align perpendicular to the plane of the L1₀ FePt thin film and perpendicular anisotropy is improved when the compressive strain becomes large.

Fig. 4.

	Figure Option ViewDownloadNew Window
	Fig. 4. (color online) Calculated magnetic energies densities of the L10 FePt (3 nm) thin film in magnetization reversible processes with different compressive strains. (a) External field energies, (b) anisotropy energies, (c) exchange energies, (d) demagnetization energies, (e) magnetoelastic energies, and (f) total energies.

We apply the three-dimensional phase field model to study the effect of compressive misfit strains on the magnetic properties and magnetization reversible processes of the L1₀ FePt (3 nm) thin film. Results show that the magnetic coercivities increase with the decrease of compressive strains from 0% to −0.06%, which agrees well with the experimental results.^[23] The evolution of the magnetic domain structure of the L1₀ FePt thin film under a compressive strain −0.06 shows that the magnetization changes from [001] single domain to multidomain, and then to single magnetic domain at the reversible magnetization processes. The magnetization perpendicular to the L1₀ FePt thin film plane inclines to along the [001] direction and reverses tardily when the compressive strain becomes larger. The calculated magnetic energy densities show that the main reason for the increase of coercivities as the compressive strain increases is the enhancement of magnetoelastic energy. When different compressive strains act on the L1₀ FePt thin film, the magnetoelastic energies are constant at a high external magnetic field and the energies peak values appear at the coercivities field, which is due to the fact that magnetoelastic energy induces the PMA, thus causing the magnetoelastic energy lower at the larger applied magnetic field. Meanwhile, when the magnetization deviates from the normal direction of the plane of the L1₀ FePt thin film, the energies peaks appear in the magnetization reversible processes.