Phase field simulations are performed to investigate the domain structures of soft ferromagnetic nanodots. It is found that the stability of the domain state is sensitive to its lateral dimensions. As the lateral dimensions increase, the stable domain state gradually changes from polar to vortex, with a transitional region where both the two ordered states are stable. Interestingly, the phase diagram is also a strong function of mechanical loads. By appropriately choosing the lateral dimensions, transformations between polar and vortex states can be induced or controlled by mechanical loads. The study provides instructive information for the applications of ferromagnetic nanostructures.

Ferromagnetic nanostructures have attracted increasing attentions in recent years for their potential in applications of ultrahigh density magnetic memories, spintronics and sensors, etc. The micromagnetic domain states in ferromagnetic nanostructures have great effect on their properties.^{1} Polar domain structures and vortex domain structures are two possible domain structures at the nanometer scale. For a ferromagnetic nanosystem, its domain structure has been shown to strongly depend on its shape,^{2} size^{3} and material parameters.^{4} When the size of the magnetic element is small enough, its domain structure is expected to be the polar state because the exchange energy is larger than the magnetostatic energy. With increasing the element size (still at the nanoscale), the vortex state is energetically more favorable due to the prevailed magnetostatic energy. These phenomena are very interesting for the people attempting to control polar-vortex transformation of ferromagnetic materials. Polar state and vortex state in the ferromagnetic nanosystem can also be used in novel memories, sensors and actuators, etc., so in-depth and systematic studies on the transformations between polar and vortex states are significantly important.^{5,6}

A number of experimental and theoretical works studied the domain state stability and possible polar-vortex transformations in ferromagnetic nanosystems through size/geometry design,^{7–9} current,^{10,11} magnetic field^{12,13} and thermal excitation.^{14,15} These works provide us some useful information on the control of polar-vortex transformations in ferromagnetic nanostructures, mainly via electrical, magnetical and thermal means, etc. It is noteworthy that mechanical stress or strain also have a significant influence on the domain structures of ferromagnetics,^{16} ferroelectrics^{17–20} and multiferroics.^{21} The mechanical stress and strain commonly reside in the system due to the lattice mismatch with substrate, grain boundaries and defects, and can also be introduced via external stimulus such as magnetoelastic and piezoelectric actuations.^{22} The effect of external strain on the evolution of magnetic vortex domain structure in nanoscale ferromagnetic platelets has been discussed through a phase field simulation.^{23} However, the influence of mechanical loads on the control of polar-vortex state stability and transformations (see Fig. 1(a) and Fig. 1(b)) in ferromagnetic nanostructures has remained elusive.

In this letter, to explore such an issue, we employ a micromagnetic phase field approach to simulate the domain structure evolution of soft ferromagnetic nanodots (here “soft” means the system is associated with a small magnetocrystalline anisotropy). Similar approaches have been used to predict the domain structure evolution of bulk ferromagnets,^{24} ferroelectric films,^{25–27} ferromagnetic shape memory alloys^{28} and ferromagnetic platelets.^{23} Fe_{81.3}Ga_{18.7} with strong magnetoelastic coupling is chosen as the representative soft magnetic material. We calculate the phase diagram of the nanodot as a function of the lateral dimensions under various mechanical loads. By choosing appropriate size of the nanodot, our simulations show that polar-to-vortex and vortex-to-polar transformations can be induced regularly by mechanical loads. Our result also indicates the possibility of other domain state transformations such as polar-polar transformations.

In the phase field model, the domain structure is described by the spatial distribution of a magnetization field **M** = *M*_{s}(*m*_{1}, *m*_{2}, *m*_{3}), where *M*_{s} and *m*_{i} (*i* = 1, 2, 3) represent the saturation magnetization and the direction cosines. The total free energy of magnetostricitive materials is given by *F*_{tot} = *F*_{anis} + *F*_{exch} + *F*_{mag} + *F*_{elas}. Here $ F anis =\u222b [ K 1 ( m 1 2 m 2 2 + m 1 2 m 3 2 + m 2 2 m 3 2 ) + K 2 m 1 2 m 2 2 m 3 2 ] dV$ is the magnetocrystalline anisotropy energy, *K*_{1} and *K*_{2} are the magnetocrystalline anisotropy constants, *V* is the volume of the system. The magnetic exchange energy is described as $ F exch =A\u222b ( m 1 , 1 2 + m 1 , 2 2 + m 1 , 3 2 + m 2 , 1 2 + m 2 , 3 2 + m 3 , 1 2 + m 3 , 2 2 + m 3 , 3 2 ) dV$, where *A* is the exchange stiffness constant, the commas denote the spatial derivatives. The magnetostatic energy is computed by $ F mag =\u222b ( \u2212 1 2 \mu 0 ( H 1 2 + H 2 2 + H 3 2 ) \u2212 \mu 0 ( H 1 M 1 + H 2 M 2 + H 3 M 3 ) dV$, *μ*_{0} is the permeability of vacuum, *H*_{i} (i=1,2,3) are the total magnetic field. $ F elas =\u222b ( \u2212 1 2 s 11 ( \sigma 11 + \sigma 22 + \sigma 33 ) \u2212 1 2 s 12 ( \sigma 11 \sigma 22 + \sigma 22 \sigma 33 + \sigma 11 \sigma 33 ) \u2212 1 2 s 44 ( \sigma 12 2 + \sigma 13 2 + \sigma 23 2 ) \u2212 3 2 \lambda 100 ( m 1 2 \u2212 1 3 ) \sigma 11 \u2212 3 2 \lambda 100 ( m 2 2 \u2212 1 3 ) \sigma 22 \u2212 3 2 \lambda 100 ( m 3 2 \u2212 1 3 ) \sigma 33 \u2212 3 2 \lambda 111 m 1 m 2 \sigma 12 \u2009 \u2212 \u2009 3 2 \lambda 111 m 2 m 3 \sigma 23 \u2212 3 2 \lambda 111 m 1 m 3 \sigma 13 ) dV$ is the elastic energy, where *s*_{11}, *s*_{12}, *s*_{44} are the elastic compliance tensor, *σ*_{ij} is the stress tensor, *λ*_{111} and *λ*_{100} are the magnetostrictive constants.

The temporal evolution of the magnetization configuration, thus the domain structure, is described by the Landau-Lifshitz-Gilbert (LLG) equation

where *γ*_{0} is the gyromagnetic ratio, *α* is the damping constant, and **H**_{eff} is the effective magnetic field, which can be represented as a variational derivative of the total free energy with respect to magnetization **H**_{eff} = − *δF*_{tot}/(*μ*_{0}*M*_{s}*δ***m**).

The temporal evolution of the local magnetization and thus the domain structures in the Fe_{81.3}Ga_{18.7} nanodot are obtained by numerically solving the LLG equation. The simulated nanodots are divided by a three-dimensional meshing of *n*_{x}Δ*x* × *n*_{y}Δ*y* × *n*_{z}Δ*z* at a scale of Δ*x* = Δ*y* = Δ*z* = 1 nm, and *n*_{x}, *n*_{y} and *n*_{z} are the number of meshing elements along the *x*, *y* and *z* axis, respectively. To fully characterize various domain patterns that might be in polar or vortex states, we give plots of the domain patterns and calculate their average magnetization vector $ m \xaf $ and the toroidal moment, i.e., $g= 1 V \u222br\xd7 ( m \u2212 m \xaf ) dV$, where **r** is the position vector and *V* is the volume of the system.^{29} We introduce the following notation for the different equilibrium phases that may exist in the nanodot: (i) the *a*_{1}-phase, where $ m 1 2 =1$ and $ m 2 2 = m 3 2 =0$; (ii) the *a*_{2}-phase, where $ m 2 2 =1$ and $ m 1 2 = m 3 2 =0$; (iii) the *c*-phase, where $ m 3 2 =1$ and $ m 1 2 = m 2 2 =0$; (iv) the *r*-phase, where $ m 1 2 \u22600$, $ m 2 2 \u22600$ and $ m 3 2 \u22600$. For illustration, the calculations were performed for the Fe_{81.3}Ga_{18.7} nanodots and the corresponding materials parameters are listed in Refs. 23 and 24.

To investigate the effect of lateral dimensions on the polar and vortex states, the evolution of magnetic domain states in nanodots with dimensions of *n _{x}* ×

*n*× 6nm

_{y}^{3}is firstly simulated at the free standing state. Simulations of magnetization evolution in the nanodot from initial random perturbations are conducted to obtain the phase diagram of stable magnetic domain states. Note that, since we want to find the possible domain states that can be formed in the nanodot, we generally try more than ten different random perturbations for a given set of simulation conditions (i.e., lateral dimensions

*n*and

_{x}*n*and stress). The phase diagram of the free standing nanodots as a function of

_{y}*n*and

_{x}*n*dimensions is depicted in Fig. 2(a). Actually we calculate the different vertical dimensions of the nanostructures. We find that those phase diagrams have a similar shape but the phase boundary has a little shift when the vertical dimensions are less than 16 nm. And the effects that we study in the below also occur at the different vertical dimensions (less than 16 nm). But when the vertical dimension exceeds 16 nm, the polar/vortex region disappears and the phase diagram is different from the phase diagram in our article. This phase diagram provides us an insight into the size and geometry effect on the domain state stability of the free standing nanodots. When the lateral size of the nanodot is small, the polar state is stable (see the left-bottom region of the phase diagram). Meanwhile, vortex state is stable at the large lateral size (see the right-top region of the phase diagram). Interestingly, besides these two regions, there is a large transitional region where the domain state is either the polar state or the vortex state. In the simulations, the competition between the exchange energy and the magnetostatic energy governs the stability between polar and vortex states. As a result, the free energy of the nanodot only has a single minimum where polar (vortex) state is stable as the nanodot lateral size is small (large) enough. At moderate lateral size, the free energy of the nanodot has two minimums corresponding to the polar state and the vortex state. The results agree well with the calculations in polycrystalline Ni films, wherein the film tends to form single domain at small size and tends to form multi-domain at large size.

_{y}^{30}

Fig. 2(b) shows the detailed domain morphologies of three nanodots in different regions of the phase diagram. The three nanodots are in size of 12 × 12 × 6nm^{3}, 15 × 15 × 6nm^{3} and 20 × 20 × 6nm^{3}, respectively. We can see that the nanodot in size of 12 × 12 × 6nm^{3} can only form a polar state, meanwhile the nanodot in size of 20 × 20 × 6nm^{3} can form a vortex state. For the nanodot in size of 15 × 15 × 6nm^{3}, which lies in the transitional region of the phase diagram, the stable domain state can be either the polar state or the vortex state, due to the comparable effects of the exchange energy and magnetostatic energy on the domain stability. According to results of Fig. 2(a) and 2(b), we clearly see how the polar and vortex domain states of the nanodot are controlled by its size and geometry.

In order to study the effect of external mechanical stress on the domain state stability of the nanodots, we further simulate the stable domain structures of nanodots under isotropic biaxial in-plane stresses. Both tensile and compressive stresses are considered. Similar to the previous simulations conducted on the free standing nanodots, the possible domain states of the nanodots under the isotropic biaxial in-plane tensile and compressive stresses are also explored by trying more than ten initial random perturbations of magnetization field. The calculated phase diagrams of the nanodots under isotropic in-plane biaxial tensile stress of 0.5Gpa and 1.0Gpa are depicted in Fig. 2(c). One can see that the phase diagrams under biaxial tensile stress are similar to that at free standing condition, with only small shifting appearing at the phase boundaries. Specifically, under biaxial stress of 0.5GPa, there is a slight shrinkage of the transitional region where both polar and vortex states can be stable. Meanwhile, the transitional region slightly expands under biaxial stress of 1.0Gpa. This result indicates that the effect of biaxial tensile stress on the phase diagram is not obvious. In contrast, the phase diagram is changed significantly by isotropic biaxial in-plane compressive stress. As shown in Fig. 2(d), the transitional region disappears after applying the compressive stress of -0.5Gpa and -1.0Gpa. Moreover, as the compressive stress increases from -0.5Gpa to -1.0Gpa, the phase boundary between the polar state and vortex state has an upward shifting, and the region of the polar state becomes larger. Therefore, it shows that the stability of polar and vortex states is more sensitive to the compressive stress than the tensile stress. This result indicates that we may induce novel transformations between the polar and vortex states via compressive in-plane stress loads. This large effect of in-plane compressive stress on the phase diagram is believed due to the following fact. On the one hand, polar state along the *z*-axis (in-plane axis) is favored at the compressive (tensile) in-plane stress condition. On the other hand, as the nanodots are generally in prolate shape, the vortex state of the nanodots has the vortex axis along the *z*-axis and a net vortex core magnetization along the *z*-axis as well. This *z*-directed vortex core magnetization links a transformation path (acting as a nucleation region) between the vortex state and the *z*-directed polar state. In other words, due to this feature of vortex state, the energy barrier between the vortex state and the *z*-directed polar state is much smaller than that between the vortex state and the in-plane polar states.

While the above results indicate a strong effect of isotropic biaxial in-plane compressive stress on the stability of polar state and vortex state of the nanodots, it is interesting to see whether mechanical-load-induced transformations between the polar and vortex states can be achieved. To do so, we focus on a nanodot with its dimensions being 15 × 15 × 6nm^{3}, i.e., lying in the transitional region. Initially, the nanodot is free standing state and has an existent polar state or vortex state as already depicted in Fig. 2(b). Then we apply the isotropic biaxial in-plane compressive stress to the nanodot, and simulate its domain evolution to explore the possible transformations between polar and vortex states. We first simulate the domain evolution of the nanodot with an existent polar state and under -0.5GPa in-plane stress. As shown in Fig. 3(a), 3(b) and 3(c), the nanodot indeed takes a transformation from the polar state into vortex state under the in-plane compressive stress. Fig. 3(a) and 3(b) respectively depict the evolution curves of the average magnetization and the toroidal moment during this polar-to-vortex transformation. At the initial state, the nanodot adopts a *y*-directed polar state (*a*_{2}-phase) with the *y*-component of the average magnetization $ m 2 \xaf $ being equal -1 and the other two components being zero. After applying the stress, the domain state of the nanodot gradually changes from the *y*-directed polar state (*a*_{2}-phase) to another polar state with its magnetization along neither *x*, *y* or *z* axis (*r*-phase). During this stage, the magnitude of $ m 2 \xaf $ decreases and those of $ m 1 \xaf $ and $ m 3 \xaf $ increase. At the second stage, $ m 3 \xaf $ keeps increasing in magnitude until it reaches 1 and magnitudes of the rest components decrease to zero, indicating that the *r*-phase polar state gradually transforms into the *c*-phase polar state. During these two stages, the toroidal moment remains zero. Finally, in the third stage, the nanodot gradually changes its domain state from the *c*-phase polar state to a vortex state. For this stage, the *z*-component of the toroidal moment of the domain pattern |*g*_{3}| rapidly increases and the other two components keep near zero. Meanwhile, $ m 3 \xaf $ decreases by a small amount. By plotting the evolution of domain morphologies at different stages as shown in Fig. 3(c), we can clearly see that the nanodot ultimately adopts a vortex domain pattern with its vortex axis along the *z*-axis. We would like emphasize that the process of the transformation from the polar state (*a*_{2}-phase) into vortex state under -0.5 GPa in-plane stress undergoes two pre-stages which involve two polar-polar transformations. As has been pointed out, the vortex state of the nanodots has a net vortex core magnetization along the *z*-axis. This *z*-directed vortex core magnetization links a transformation path (acting as a nucleation region) between the vortex state and the *z*-directed polar state. Therefore, the nanodot needs to transform into a *z*-directed polar state before it can undergo the polar to vortex transformation.

In addition to the polar to vortex transformation that has been shown can be induced by the compressive in-plane stress load, the reverse vortex to polar transformation has also been simulated as shown in Fig. 3(d), 3(e) and 3(f),. The nanodot initially adopts a vortex state and then undergoes an in-plane compressive stress of -1.0GPa. One can see that a vortex to polar transformation indeed happens. During this transformation, both the average magnetization and the toroidal moment keep along the *z*-axis, and they vary smoothly as shown in Fig. 3(d) and 3(e). Specifically, $ m 3 \xaf $ gradually increases to 1 and |*g*_{3}| gradually decreases to zero, indicating that the vortex state transforms into the polar state (*c*-phase). The much simpler process of vortex to polar transformation again reflects that the energy barrier between the vortex state and the *z*-directed polar state is small due to *z*-directed vortex core magnetization of the vortex state. Comparing to the experimental studies of the polar-vortex transformation, we find that a similar transitional region in the phase diagram has been proved in the experimental work.^{9} Nevertheless, they only studied the size effect and transformation from single domain state to vortex state with annealing. Another experimental work investigated transformation between vortex and collinear magnetic states by thermal fluctuations and magnetic field. A similar transitional region was found to exist as function of temperature and magnetic field. The magnetic field changes the relative value of the two energy minimum, corresponding to vortex and collinear magnetic states.^{15} Our work demonstrates that the stress also has an effect on the corresponding energies, and similar transition can be achieved.

To gain a further insight into the effect of mechanical load on the control of polar-vortex transformation, we further simulate the domain evolution of the nanodot in size of 15 × 15 × 6 nm^{3} under a series of in-plane stress ranging from -1.0Gpa to 1.5Gpa. Initially, the nanodot adopts an existent polar state or vortex state as that shown in Fig. 2(b). The equilibrium average magnetization and the toroidal moment of the nanodot as functions of the in-plane stress are shown Fig. 4, with Fig. 4(a) and 4(b) corresponding to the case of an existent vortex state and the case of an existent polar state, respectively. From Fig. 4(a), the vortex state is stable under the tensile stress. Meanwhile, the vortex state would become unstable and transform into the polar state (*c*-phase) under compressive stress as the stress exceeds 0.8Gpa. Furthermore, from Fig. 4(b), the initial polar state (*a*_{2}-phase) also remains stable under the tensile stress. In contrast, a small compressive stress (0.05Gpa to 0.8Gpa) can make the polar state (*a*_{2}-phase) transform into the vortex state. Meanwhile, at larger compressive stress (>0.8GPa), the initial polar state (*a*_{2}-phase) would transform into another polar state (*c*-phase). Our result therefore demonstrates that we can achieve fruitful controllability of mechanical load on the domain state stability and evolution in Fe_{81.3}Ga_{18.7} nanodot with a moderate size.

In conclusion, phase field simulations have been performed to investigate the stability of polar and vortex domain states and their transformations in soft ferromagnetic nanodots. We focus on the combining effects of size, geometry and mechanical loads. The phase diagram of the nanodot as a function of the lateral dimensions under various mechanical loads has been calculated. Results show that the stability of the domain state is sensitive to its lateral dimensions. With increase of the lateral dimensions, the stable domain state gradually changes from polar state to vortex state, with a transitional region where both the two ordered states are stable. More significantly, it is found that isotropic biaxial in-plane compressive stress can remarkably changes the phase diagram. As a consequence, by applying isotropic biaxial in-plane compressive stress, transformations between polar state and vortex states can be induced in the nanodot with its size lying in the transitional region of the phase diagram. Our result sheds light on the possibility of designing ferromagnetic nanostructures with controllable responses to mechanical loads, which can be used in applications such as non-volatile memories, energy storages and sensors, etc.

The authors acknowledge support of NSFC (Nos. 11474363, 51172291), RF-SZ Science, T&I Commission (20140606094908124 and 20150513151706572). Y. Zheng also thanks the FRFCU, FYTF, GNSF-DYS.