Behavior of Vortex-Like Inhomogeneities Originating in Magnetic Films with Modulated Uniaxial Anisotropy in a Planar Magnetic Field

Abstract

This paper investigates the processes of magnetization reversal of a uniaxial ferromagnetic disk containing a columnar defect of the potential well type in perpendicular and planar magnetic fields. The characteristic stages of magnetization reversal of the domain structure of the disk and vortex-like inhomogeneities forming on the defect are determined. The critical fields of their existence are found and an explanation is given for the presence of a significant difference in their values for the perpendicular and planar fields of the defect magnetization reversal. The role of chirality in the behavior of a Bloch-type magnetic skyrmion during the magnetization reversal of a defect in a planar field is shown.

1. Introduction

Currently, research in skyrmionics is being actively conducted [1] in relation to the structure and properties of magnetic skyrmions formed in non-chiral magnets, as well as the search for their possible applications [2,3]. This is due to the fact that the magnetic skyrmions first discovered in chiral magnets [4] are stabilized in them due to the Dzyaloshinskii–Moriya interaction (DMI) [5]. Their topological protection, nanoscale dimensions, ease of manipulating them with a low-density electric current [5,6,7], and other unique spin-electronic properties [8,9,10,11] have caused increased attention to them. In addition, after the identification of real prospects for their use in various spintronics devices, including new-generation magnetic memory, interest in them has significantly increased [1,12,13]. More recently, however, it has emerged that the DMIs in these materials (the so-called bulk Dzyaloshinskii–Moriya interactions) make a significant contribution to the stabilization of skyrmions only at low temperatures [14,15]. At the same time, in the case of interfacial DMIs realized in multilayer nanoscale films [16,17], skyrmions form stable states even at room temperature. Nevertheless, there are certain difficulties associated with the control of the interaction parameters responsible for their existence at such small film thicknesses ($\mathrm{D}~1$nm) [3]. In addition, fields greater than H ≥ 10 mT are also required to stabilize such skyrmions [18]. As a result, the possibility of their use as a data carrier in spintronics devices becomes problematic. Therefore, there is a demand for alternative ways to stabilize magnetic skyrmions in materials without DMI, i.e., in non-chiral magnets [3,19]. One of the options for implementing such an approach was the idea of using uniaxial ferromagnetic films with spatially modulated material parameters as such magnets [20]. It should be noted that regardless of the studies conducted in [20], similar conclusions were reached in [21,22], in which magnetic inhomogeneities originating in the defects of the ”potential well” type in magnetically uniaxial films were studied. Subsequently, this idea was implemented in [23] and a multilayer Co/Pt film with locally modified sections was experimentally obtained, in which the perpendicular anisotropy constant had a reduced value (due to focused irradiation of the material surface with a ${\mathrm{He}}^{+}$ ion beam). By adjusting the radiation dose, it was possible to ensure that the anisotropy constant in the area of the irradiated areas assumed the required values [24] up to negative values (light-plane anisotropy). Thus, in such a film, as shown in [25], it would be possible to observe at room temperature a lattice of magnetic skyrmions, the stability of which would not be disturbed in a magnetic field either (with the exception of fields exceeding the critical field of their stability). The latter property has an important application value, because with the help of an external magnetic field (homogeneous [26] or inhomogeneous [27,28]), the skyrmion states of ferromagnetic multilayer films can be effectively controlled. Therefore, the need to study the effect of the Zeeman interaction on the structure and stability of magnetic skyrmions formed on columnar defects with modulated uniaxial anisotropy is obvious, and this suggests continued research in this direction. Despite the fact that some aspects of the problem under consideration have already been touched upon in [26,27,28], these studies nevertheless do not create a complete picture of the processes of magnetization reversal of non-chiral magnets of the above type. In particular, it is of interest to study the behavior of magnetic skyrmions in a magnetically uniaxial film containing a columnar defect of the ”potential well” type in a planar magnetic field.

2. Basic Ratios

The paper considers a uniaxial ferromagnetic in the form of a disk (thickness $\mathrm{D}$) in which there is a defect in the form of a cylinder of radius ${\mathrm{R}}_0$ (columnar defect [29]) with an axis of symmetry coinciding with a similar axis of the disk and directed along Oz (Figure 1).

It is assumed that the easy axis of uniaxial anisotropy also coincides with the Oz axis and, in addition, the ratio $\mathrm{R}\gg {\mathrm{R}}_0$ holds, where $\mathrm{R}$ is the radius of the disk. The expression for the energy density Ԑ takes into account the exchange interaction (characterized by the exchange parameter $\mathrm{A}$), uniaxial anisotropy ${\mathrm{K}}_{\mathrm{u}}$, the Zeeman interaction and the demagnetizing fields of the disk due to its finiteness. Accordingly, Ԑ has the form

$$\mathsf{\epsilon }={\mathsf{\epsilon }}_{\mathrm{ex}}+{\mathsf{\epsilon }}_{\mathrm{u}}+{\mathsf{\epsilon }}_{\mathrm{H}}+{\mathsf{\epsilon }}_{\mathrm{ms}}.$$

(1)

Here

$${\mathsf{\epsilon }}_{\mathrm{ex}}=\mathrm{A}\left(\frac{\partial {\mathrm{m}}_{\mathrm{i}}}{\partial {\mathrm{x}}_{\mathrm{j}}}\frac{\partial {\mathrm{m}}_{\mathrm{i}}}{\partial {\mathrm{x}}_{\mathrm{j}}}\right), {\mathsf{\epsilon }}_{\mathrm{u}}={\mathrm{K}}_{\mathrm{u}}\left({\mathrm{m}}_{\mathrm{x}}^2+{\mathrm{m}}_{\mathrm{y}}^2\right), {\mathsf{\epsilon }}_{\mathrm{H}}=-{\mathrm{M}}_{\mathrm{s}}\left(\mathrm{mH}\right), {\mathsf{\epsilon }}_{\mathrm{ms}}=-\frac{1}{2}{\mathrm{M}}_{\mathrm{s}}\left({\mathrm{mH}}_{\mathrm{m}}\right),$$

(2)

where ${\mathrm{m}}_{\mathrm{i}}$ are the components of the unit magnetization vector $m=M/{\mathrm{M}}_{\mathrm{s}}$ ($\mathrm{i},\mathrm{j}=1,2,3-$summation is carried out according to the indices $\mathrm{i}$ and $\mathrm{j}$ in the expression for ${\mathsf{\epsilon }}_{\mathrm{ex}}$), ${\mathrm{M}}_{\mathrm{s}}$ is the saturation magnetization, $M$ is the magnetization vector, $H$ is the external magnetic field, $H_{\mathrm{m}}$ is the demagnetizing field, which is found from the magnetostatics equations

$$\mathrm{div}\left(H_{\mathrm{m}}+4{\mathsf{\pi }\mathrm{M}}_{\mathrm{s}}m\right)=0, \mathrm{rot}{H}_{\mathrm{m}}=0$$

(3)

taking into account the boundary conditions imposed on $H_{\mathrm{m}}$ [30].

It is assumed that in the defect region, the material parameters of the sample $\mathrm{P}=\left(\mathrm{A},{\mathrm{K}}_{\mathrm{u}},{\mathrm{M}}_{\mathrm{s}}\right)$ change abruptly according to the formula

$$\mathrm{P}=\{\begin{array}{c}{\mathrm{P}}_1, r>{\mathrm{R}}_0 \\ {\mathrm{P}}_2, 0<r<{\mathrm{R}}_0,\end{array}$$

(4)

where ${\mathrm{P}}_{\mathrm{i}}$ represents the values of the material parameters outside $\left(\mathrm{i}=1\right)$ and inside $\left(\mathrm{i}=2\right)$ the defect. In addition, we will assume that ${\mathrm{K}}_{\mathrm{u}2}<0$, i.e., the defect is a potential well [31]. Thus, the defect under consideration has a cylindrical shape, on which, according to [29], magnetic inhomogeneities with a magnetization distribution determined by the symmetry of this system can form. In the absence of a magnetic field $\left(H=0\right)$ the symmetry of the system under consideration is axial. In addition, the symmetry group of the system also includes the reflection plane ${\mathsf{\sigma }}_{\mathrm{h}}$ coplanar to the surface of the film. Then by solving the Euler–Lagrange equations corresponding to the minimum energy of an unlimited magnetically uniaxial film $\left(\mathrm{R}\to \infty \right)$

$$\mathrm{E}=2\mathsf{\pi }\mathrm{D}\underset{0}{\overset{\infty }{{\displaystyle \int }}}\mathsf{\epsilon }\mathrm{rdr}$$

(5)

there will be vortex-like inhomogeneities (VLIs) of four types [26,32] (Figure 2). They have a Bloch magnetization distribution and differ in the polarity of the core and the orientation of the unit magnetization vector $m_0$ at the film periphery: $m_0=m(\infty )$. Two of them (the first one with ${\mathrm{m}}_{\mathrm{z}}\left(0\right)=1,{\mathrm{m}}_{\mathrm{z}}(\infty )=1$, and the second one with ${\mathrm{m}}_{\mathrm{z}}\left(0\right)=-1,{ \mathrm{m}}_{\mathrm{z}}(\infty )=-1$) represent degenerate states of a non-topological soliton [2,26], while the other two (the third one with ${\mathrm{m}}_{\mathrm{z}}\left(0\right)=-1,{\mathrm{m}}_{\mathrm{z}}(\infty )=1$ and the fourth one with ${\mathrm{m}}_{\mathrm{z}}\left(0\right)=1,{ \mathrm{m}}_{\mathrm{z}}(\infty )=-1$) are degenerate states of the magnetic skyrmion (Figure 2). The twofold degeneracy of vortex-like states with respect to the polarity of the core is due to the presence of the element ${\mathsf{\sigma }}_{\mathrm{h}}$ (reflection plane) in the symmetry group of the system under consideration. The VLI stability significantly depends on the parameters of the defect (${\mathrm{P}}_2$ and ${\mathrm{R}}_0$), as well as on the quality factor of the material $\mathrm{Q}={\mathrm{K}}_{\mathrm{u}1}/2\mathsf{\pi }{M}_{s1}^2$ [26,29]; there are always limiting values ${\mathrm{R}}_0$ and $\mathrm{Q}$, below which these inhomogeneities are not formed, as well as the limiting value $\mathrm{k}={\mathrm{K}}_{\mathrm{u}2}/{\mathrm{K}}_{\mathrm{u}1}$, above which they do not exist. In what follows, we will assume that ${\mathrm{M}}_{\mathrm{S}1}={\mathrm{M}}_{\mathrm{S}2}$, ${\mathrm{A}}_1={\mathrm{A}}_2$. It should be noted that skyrmion states are energetically more favorable than the states corresponding to a non-topological soliton. This is due to the absence of magnetic flux closures in the latter, and therefore they represent metastable micromagnetic structures [26]. In addition, VLIs differ in topological charge $\mathrm{q}$ and configuration: for example, in the case of magnetic skyrmions $-\mathrm{q}=1$ and the magnetization distribution profile has three inflection points, and in the case of non-topological solitons $-\mathrm{q}=0$ and the profile contains only two inflection points. At the same time (as can be seen from Figure 2) in all the four types of VLI in their structure, one can conditionally distinguish three sections of rotation of magnetic moments: the core (central region), the intermediate and the boundary sections. If there is a sharp change in the value of ${\mathrm{m}}_{\mathrm{z}}\left(\mathrm{r}\right)$ per unit length in the core and in the boundary section, then the intermediate section is characterized by a smoother change in ${\mathrm{m}}_{\mathrm{z}}\left(\mathrm{r}\right)$. The appearance of the intermediate region (as of the VLIs) is completely due to the presence in the film of a columnar defect of this type. With an increase in ${\mathrm{R}}_0$, its configuration becomes more prominent [26], which is expressed in an increase in its size and in a decrease in the slope of the ${\mathrm{m}}_{\mathrm{z}}\left(\mathrm{r}\right)$ curve to the radial axis Or in its region. Hence, it follows that at large ${\mathrm{R}}_0,$ the magnetic moments in the intermediate region will practically lie in the plane of the film.

3. Vortex-Like Inhomogeneities in a Magnetic Field: $H\parallel Oz$

Obviously, the inclusion of a magnetic field will lead to the transformation of the structure of the Bloch-type VLI [26]. The nature of its changes depends on the orientation of the field relative to the axis of symmetry of the magnetic system. In the case of $H\parallel \mathrm{Oz}$, the symmetry of the system decreases (the element ${\mathsf{\sigma }}_{\mathrm{h}}$ disappears), which leads to the removal of the degeneracy of vortex-like states, and then each type of VLI has its own scenario of its behavior in a magnetic field. Let us consider the behavior of two types of VLI, in which the magnetization at the periphery of $m_0$ coincides with the direction of the field $H$: $m_0\uparrow \uparrow H$. Consideration of the behavior of the other two VLIs with $m_0\uparrow \downarrow H$ is described in sufficient detail in [26] and it does not seem appropriate to re-analyze them here. For the first type of VLI (non-topological soliton), the polarity of the core and the direction $m_0$ coincide with the field H. In this case, as $\mathrm{H}$ increases, the size of its core increases, and the size of the intermediate section decreases. In addition, the maximum deviation angle ${Ɵ}_{\mathrm{m}}$ of the magnetization $m$ from the homogeneous state $m_0$ also decreases. In general, the size of this inhomogeneity also decreases, which is a consequence of the narrowing of the region of unfavorably directed magnetic moments with respect to $H$. At the same time, its energy as a result of these processes increases, and when the field H reaches a certain limit value ${\mathrm{H}}_l$, the energy $\mathrm{E}$ of the first type of vortex-like inhomogeneities becomes zero. As a result, it decreases in size and disappears, which is similar to the process of evaporation of a drop of water located on the surface of a solid when it is heated [22].

For a magnetic skyrmion, there will be a slightly different scenario for its behavior in the magnetic field $H$. According to calculations [26,32], with an increase in the magnitude of H, the core, representing the region of unfavorably directed magnetic moments with respect to the field $H$, will decrease in size, and the size of the intermediate section of the skyrmion, on the contrary, will increase. At the same time, the radius of the skyrmion will decrease. When the value of the field H exceeds ${\mathrm{H}}_l$, an abrupt transformation of the structure of the second type of VLI occurs, accompanied by a sudden disappearance of the intermediate section and an equally sharp decrease in the radius of the skyrmion. Mathematically, this is explained by the fact that at $\mathrm{H}={\mathrm{H}}_l$, the number of inflection points in the graph of the function ${\mathrm{m}}_{\mathrm{z}}={\mathrm{m}}_{\mathrm{z}}\left(\mathrm{r}\right)$ changes abruptly from three to one. In fact, the transformation process is continuous, just the border between the intermediate and boundary sections becomes blurred and the intermediate section disappears. At the same time, near the value $\mathrm{H}={\mathrm{H}}_l$, the energy of the skyrmion becomes zero, and in the fields $\mathrm{H}>{\mathrm{H}}_l$, its energy becomes positive. Accordingly, the homogeneous state of the film becomes more advantageous. In this case, an arbitrary fluctuation of the magnetization with a Neel-type component will lead to the destruction of its structure and the film will become uniformly magnetized [32,33]. However, the process of restructuring the structure of the magnetic skyrmion may occur somewhat earlier (in smaller fields). This is due to the fact that in the zero field, the magnetic skyrmion is an energetically more advantageous formation than the first type of VLI. However, in the presence of an external magnetic field, the situation with the status of these states changes as the field increases. In small fields, the skyrmion is still a more advantageous formation, at the same time in larger fields, when the value of H reaches some characteristic value ${\mathrm{H}}_{\mathrm{t}}({\mathrm{H}}_{\mathrm{t}}<{\mathrm{H}}_l)$, their energies are compared. Accordingly, at $\mathrm{H}>{\mathrm{H}}_{\mathrm{t}}$, the situation with their stability reverses: the non-topological soliton becomes stable while the skyrmion becomes a metastable state. In this case, also, the fluctuation mechanism of transformation of the structure of the VLI is triggered, and the skyrmion, by switching the polarity of the core, turns into the VLI of the first type. Its further behavior in the status of a non-topological soliton with an increase in the field H has already been considered previously.

To identify the final version of the magnetization reversal scenario of a non-chiral magnetic with modulated uniaxial anisotropy, we investigated this process using the open-source software package OOMMF [34,35]. The parameters of the sample, representing a uniaxial ferromagnetic taken in the form of a disc containing a columnar defect with a reduced value of ${\mathrm{K}}_{\mathrm{u}}$ constant, were taken with the following values [23]: $\mathrm{R}=300$ nm, $\mathrm{D}=30$ nm, ${\mathrm{R}}_0=30$ nm, ${\mathrm{A}}_1={\mathrm{A}}_2=2.5\times {10}^{-13}$ J/m, ${\mathrm{K}}_{\mathrm{u}1}=3\times {10}^4$ J/${\mathrm{m}}^3$, ${\mathrm{K}}_{\mathrm{u}2}=-0.5\times {10}^4$ J/${\mathrm{m}}^3$, ${\mathrm{M}}_{\mathrm{s}1}={\mathrm{M}}_{\mathrm{s}2}=2\times {10}^5$ A/m. The dimensions of the partition cell were taken as $2.5\times 2.5\times 2.5$ nm. The results of micromagnetic modeling of the process of magnetization reversal of the disk under study with a defect in a perpendicular magnetic field are shown in Figure 3a–e. According to calculations, with the exception of the central region (in the absence of a field), a labyrinth domain structure is formed on the disk (Figure 3a), typical for uniaxial ferromagnetic films [25]. It does not have any particular symmetry, although there is some ordering of the domain structure in Figure 3a is available. Obviously, the configuration of the domain structure of the sample calculated using the OOMMF software package is influenced by many factors: the geometry of the sample itself, the ratio of the sizes of the domains, the defect and the disk, the choice of partition cells in the form of cubes when numerically integrating the Landau–Lifshitz equations, and the error of calculations, etc. In particular, the dimensions of the defect and of the domains are approximately equal, which does not contribute to the manifestation of a certain symmetry. In reality, the domain structure of such a sample, as a rule, also does not have any symmetry, except for the central region, where a vortex-like inhomogeneity with axial symmetry is observed [23]. In addition, it may so happen in the course of calculations that a significant asymmetry appears in the magnetization distribution in the region located outside the defect. Then the formation of a VLI with a core shifted relative to its center is possible on the defect.

When a magnetic field normal to the film surface ($H\Vert \mathrm{Oz}$) is applied, the initial configuration of the magnetic system is disrupted (Figure 3b); at low values ($\mathrm{H}=25 \mathrm{mT}$), some of the stripe domains magnetized along the field expand, while some of those magnetized opposite to the field contract. The latter (or rather, some of them) in the field $\mathrm{H}=61 \mathrm{mT}$ turn into cylindrical magnetic domains (Figure 3c). At the same time, a mixed domain structure consisting of both bubbles and stripe domains is formed outside the defect. As the field increases, the area occupied by the labyrinth domain structure continues to decrease, and finally, there comes a moment (at $\mathrm{H} =71 \mathrm{mT}$) when it completely disappears, with its simultaneous transformation into a certain lattice of bubbles with a certain order of arrangement, with the axis of symmetry of the sixth order (${\mathrm{C}}_6$) (Figure 3d). A further increase in the field leads to the fact that the density of bubbles decreases and the symmetry of the system decreases (Figure 3e). In some other critical field $\mathrm{H}={\mathrm{H}}_{{\mathrm{C}}_1} \left({\mathrm{H}}_{{\mathrm{C}}_1}=100 \mathrm{mT}\right)$, only a magnetic skyrmion with a completely magnetic reversal disk periphery remains in the sample (Figure 3f). At the same time, the dimensions of the magnetic skyrmion core decrease and at a certain field $\mathrm{H}={\mathrm{H}}_{\mathrm{t}} ({\mathrm{H}}_{\mathrm{t}}=123 \mathrm{mT}$), the polarity of the core reverses (the core switches), i.e., the magnetic skyrmion turns into a non-topological soliton (Figure 3g). Finally, when the field exceeds the value $\mathrm{H}={\mathrm{H}}_{{\mathrm{C}}_2}$ $({\mathrm{H}}_{{\mathrm{C}}_2}=254 \mathrm{mT}$), the magnetic inhomogeneity corresponding to the non-topological soliton is completely remagnetized and the sample becomes uniformly magnetized (Figure 3h).

It should be noted that the presence of a domain structure on the periphery of the disk reduces the magnitude of its magnetostatic energy [26]. At the same time, as the field H increases, the area occupied by those domains in which the magnetization is directed along the field increases. Accordingly, the energy of the demagnetizing fields increases, which in turn affects the structure and characteristics of the magnetic skyrmion. Therefore, the behavior of a magnetic skyrmion in an unlimited film, in which the presence of a domain structure was not taken into account in its simulation [26], and the behavior in a disk of finite dimensions, in which the domain structure is present (according to calculations using the OOMMF package), may differ slightly, but not significantly. This statement follows from the fact that the magnetic skyrmion in the case under consideration is stabilized by the presence of a columnar defect of the potential well type [26]. Consequently, its structure and characteristics depend to the greatest extent on the radius ${\mathrm{R}}_0$ and the depth of the potential well ${\mathrm{K}}_{\mathrm{u}2}$, and to a much lesser extent on demagnetizing fields. Thus, it can be seen from (Figure 4) that the size of the ${\mathrm{R}}_{\mathsf{\upsilon }}$ skyrmion stabilized in a disk of finite dimensions increases with small ${\mathrm{R}}_0$, and with its further increase, the dependence of ${\mathrm{R}}_{\mathsf{\upsilon }}$ on ${\mathrm{R}}_0$ becomes more flat. This dependence correlates well with the similar dependence ${\mathrm{R}}_{\mathsf{\upsilon }}({\mathrm{R}}_0)$ obtained for a film of unlimited dimensions in which there is no domain structure [29]. In addition, the analysis of the obtained results shows (Figure 4) that the formation of a skyrmion on a defect of the “potential well” type in a disk of finite dimensions is also of a threshold nature, as in the case of an infinite film. In this case, at ${\mathrm{R}}_0<21$ nm, a skyrmion is not formed on the defect. At the same time, the threshold value of the defect radius (${\mathrm{R}}_0=21$ nm) practically does not depend on the quality factor, which once again confirms the thesis that the characteristics of the skyrmion are clearly dependent mainly on the defect parameters.

It is also possible to note the dependence of the critical switching fields of the core ${\mathrm{H}}_{\mathrm{t}}$ (Figure 4) and the magnetization reversal of the defect ${\mathrm{H}}_{{\mathrm{C}}_1}$ (Figure 4) on the radius of the defect ${\mathrm{R}}_0$: the field ${\mathrm{H}}_{\mathrm{t}}$ significantly depends on the size of the defect—with increasing ${\mathrm{R}}_0$, the value of decreases significantly (Figure 4), and the value of ${\mathrm{H}}_{{\mathrm{C}}_2}$, on the contrary, increases noticeably with small ${\mathrm{R}}_0$, whereas with large ${\mathrm{R}}_0$, it increases only to a very small extent (practically goes to a plateau).

4. Behavior of Vortex-Like Inhomogeneities in a Planar Magnetic Field

Let us now consider the process of magnetization and magnetization reversal of the sample under study in a magnetic field lying in the plane of the disk. The initial equilibrium domain structure in the disk (both on the periphery of the disk and on the defect) corresponds to the configuration shown in Figure 3a. When the planar field is turned on, the symmetry of the magnetic system decreases [36]. There remains only one element of symmetry—the plane of reflection ${\mathsf{\sigma }}_{\mathrm{h}}$ orthogonal to the Oz axis. In this case, the two-fold degeneration that took place in the absence of a field in both types of VLI remains unchanged, but their structure transforms according to approximately the same scenario. Thus, in a magnetic skyrmion, when the $H$ field is turned on, an asymmetry of the Bloch distribution of magnetic moments in the defect region occurs (Figure 5a). This is expressed in the fact that in the section (if you move from the center of the defect perpendicular to the field upwards), where the chirality of the magnetic moments coincides with the direction $H$, there is an increase in its transverse dimensions, whereas in the section from the bottom of the center, in which the magnetic moments are directed opposite to the field, on the contrary, there is a decrease in their sizes. As a result of these processes, the core shifts downwards, and the larger the field value, the further away from the center. Finally, at the field $\mathrm{H}=19.25$mT, the core gradually reaches the edge of the defect (Figure 5b). When the field reaches the value $\mathrm{H}={\mathrm{H}}_{\mathrm{C}} ({\mathrm{H}}_{\mathrm{C}}=19.4$mT) the core is completely displaced from the defect (Figure 5c) and the defect region becomes permanently magnetized along the field $H$. At the same time, the labyrinth domain structure located outside the defect does not disappear, but transforms: the planes of the domain walls in which the magnetic moments rotate begin to orient themselves along the field. This leads to the gradual alignment of the stripe domains parallel to $H$.

It should be noted that the critical value of the in-plane defect remagnetization field ${\mathrm{H}}_{\mathrm{C}}$ depends only minimally on its size. The observed slight decrease and weak oscillation of the dependence of ${\mathrm{H}}_{\mathrm{C}}$on the defect size are not significant, in particular, the amplitude of the latter is about 1 mT.

Two circumstances should be noted here: the first one concerns the value of the planar field at which the magnetization reversal of the defect area occurs. It is more than an order of magnitude smaller than a similar field (the field of magnetization reversal), only directed perpendicular to the plane of the disk. Such a significant difference is due to the fact that the magnetization reversal of magnetic moments on a defect in a planar field is facilitated by uniaxial anisotropy of the ”easy plane” type, which takes place in the defect region. Its value is 6 times smaller than the value of uniaxial anisotropy of the ”easy axis” type acting outside the defect.

The second circumstance is related to the chirality of the magnetic skyrmion of the Bloch type: if the skyrmion localized on the defect has a different direction of chirality, then the core ejection from the defect area will go in the opposite direction (from the bottom upwards). However, its behavior in a planar magnetic field will be exactly the same as in the first case of the chirality of the magnetic skyrmion.

It should also be noted that a similar movement of skyrmions in a planar field (in a chiral magnet) was noted in [36], while also noted in the case of magnetic vortices in [37,38]. In [39], also investigated was the effect of a planar magnetic field on the structure of a skyrmion, which was stabilized in a multilayer Co/Pt film with interfacial DMI. However, the magnetic skyrmion in such a film had a Neel-type structure, and under the action of the field, it also transformed, but according to a different scenario: the skyrmion was deformed and stretched along the direction of the field.

5. Discussion of the Results

These studies show that the external magnetic field significantly affects the structure of vortex-like inhomogeneities of both types. The scenario of transformation of the VLI topology depends on the direction of the magnetic field and the type of VLI, more precisely, on their characteristics: on the topological charge, on chirality, on polarity and on the type of magnetization distribution: (Bloch-type or Neel-type) [39]. Under the action of a magnetic field, the direction of which coincides with the normal to the surface of the film and with the polarity of the core of a non-topological soliton, the structure of the latter is transformed according to the scenario of the evaporation process of a drop of water located on the surface of a solid when it is heated. In this case, as the field increases, it decreases in size, gradually remagnetizes, and then disappears as an object.

If a magnetic field acts on a magnetic skyrmion formed on a defect with a polarity opposite to $H$, then its transformation occurs in several stages. As H increases, the size of the core decreases as well as that of the skyrmion (to a lesser extent). Then, when the field reaches the value ${\mathrm{H}}_{\mathrm{t}}$ ($\mathrm{H}={\mathrm{H}}_{\mathrm{t}}$), the polarity of the core is switched; the skyrmion turns into a non-topological soliton. Further, its transformation in an increasing field repeats all the stages of changing the structure of a non-topological soliton until its disappearance.

It should be noted that the analysis of the Euler–Lagrange equations for chiral magnets shows that they lack solutions corresponding to a non-topological soliton. At the same time, the existence of $k\pi -$skyrmions is possible in them, where $k\in \left\{1,2,3,..\right\}$ [40,41]. Here the $\pi -$skyrmion is the VLI in which the magnetization $m$ rotates by the angle $\pi$ between the core and the periphery of the film. It is an ordinary skyrmion, studied in most studies on skyrmionics, including this paper. However, the 2$\pi -$skyrmion is not an analogue of a non-topological soliton, despite the fact that in both cases the orientations of the vector $m$ in the center and on the periphery of the film coincide. While for a 2$\pi -$skyrmion the chirality $m=1$ is constant, for a non-topological soliton, the direction of rotation of the vector $m$ (when it reaches a rotation by the maximum ${\theta }_m$) changes to the opposite. The prototype of a non-topological soliton in a one-dimensional model is a 0-degree DW [26,29], and a $2\pi -$-skyrmion is a 360° DW. In this sense, a non-topological soliton can be called a 0° skyrmion or a 0-skyrmion (a skyrmion with an inverted core [2].) According to calculations [29], a 0-skyrmion is a metastable formation, but it can originate as a stable formation at an intermediate stage of the process of magnetization reversal of a $\pi -$skyrmion localized on a columnar defect in a perpendicular field.

When a planar magnetic field acts on a Bloch-type skyrmion, the symmetry of the magnetic system is broken; it ceases to be axial. Due to the fact that part of the magnetic moments of the skyrmion located in the region lying on one side of the core and directed along the field $H$, or forming an acute angle with it ($0\le \psi \le \pi /2$), occupy energetically a more advantageous position compared to the other part located on the other side of the core, in which the magnetic moments form an obtuse angle with the field ($\pi /2<\psi \le \pi$), the core moves perpendicular to $H$ towards the latter part (from top to bottom). As a result, indeed, the symmetry of the system ceases to be axial. With a field exceeding $\mathrm{H}={\mathrm{H}}_{\mathrm{C}}$ the process of moving the core is completed with its displacement outside the defect region and with its complete magnetization reversal along the field. Thus, with this orientation of the field, the chirality of the skyrmion plays an essential role in the process of magnetization reversal of the defect region. The latter means that by the action of a planar field on a Bloch-type skyrmion, its chirality can be determined.

In addition, it follows from the above calculations that the methods used for numerical analysis of the behavior of VLI in non-chiral magnets in magnetic fields of different orientations, which are the direct numerical integration of the Euler–Lagrange equations and the OOMMF open access software package, generally give different information about the objects under study. If in the first case it is possible to clearly determine what role each type of interaction plays (exchange interaction, uniaxial anisotropy, the presence of defects, etc.) on the process under study, then in the second case it is not always possible to clearly identify this, because two more factors are added to the multifactorial analysis of the process: the finiteness of the sample and the presence of a domain structure formed in the disk outside the defect. It follows from the above results that the descriptions of the main stages of the evolution of the magnetic skyrmion obtained for the perpendicular field by both methods coincide, because the decisive factor stabilizing the VLI in these materials is the presence of a defect of the “potential well” type. This agreement of the results made it possible to study the behavior of the skyrmion in a planar field using the OOMMF micromagnetic modeling package.