The properties of isolated chiral skyrmions in thin magnetic films

In this paper we investigate isolated skyrmions in a thin layer of a noncentrosymmetric ferromagnet. As a model we consider a thin plate infinite along the $x-$ and $y-$ axes and of thickness L along the $z-$ axis. In the following sections we specify the model and discuss its limitations. For a film of a noncentrosymmetric uniaxial ferromagnet in the applied magnetic field ${{\bf{H}}}^{(e)}$ perpendicular to the film surface, the micromagnetic energy density written within terms quadratic in the components of the magnetization vector ${\bf{M}}$ has the following standard form [1]:

$$\begin{eqnarray}w & = & A{({\bf{grad}}{\bf{m}})}^{2}+{w}_{D}({\bf{m}})-K{({\bf{m}}\cdot {\bf{n}})}^{2}-{\mu }_{0}{{MH}}^{(e)}{\bf{m}}\cdot {\bf{n}}-{\mu }_{0}M{\bf{m}}\cdot {{\bf{H}}}^{(d)}/2,\ \end{eqnarray} \tag{ 2 }$$

where A is the exchange stiffness constant, K is the uniaxial anisotropy constant, ${{\bf{H}}}^{(d)}$ is the demagnetizing field,

$$\begin{eqnarray}&&{\bf{m}}={\bf{M}}/| {\bf{M}}| =(\mathrm{sin}\theta \mathrm{cos}\psi ,\mathrm{sin}\theta \mathrm{sin}\psi ,\mathrm{cos}\theta )\end{eqnarray} \tag{ 3 }$$

is the reduced magnetization, ${\bf{n}}$ is the unity vector directed perpendicular to the film surface.

The Dzyaloshinskii-Moriya energy density w_D is composed of Lifshitz invariants (1):

$$\begin{eqnarray}&&{{ \mathcal L }}_{{ij}}^{(k)}={m}_{i}\displaystyle \frac{\partial {m}_{j}}{\partial {x}_{k}}-{m}_{j}\displaystyle \frac{\partial {m}_{i}}{\partial {x}_{k}}.\end{eqnarray} \tag{ 4 }$$

The functional forms of energy density w_D are determined by crystallographic symmetry of a noncentrosymmetric magnetic crystal and are listed in equations (A.1), (A.2). Lifshitz invariants (4) favour spatial modulations with a fixed rotation sense along the x_k directions [1]. A competition between the chiral energy w_D and other energy contributions leads to the formation of isolated chiral states [2, 3, 14] and spatially modulated magnetic phases [1, 3].

The Euler equations for energy functional (2) together with Maxwell's equations,

$$\begin{eqnarray}&&\mathrm{rot}{{\bf{H}}}^{(d)}=0,\quad \mathrm{div}[{{\bf{H}}}^{(d)}+{\mu }_{0}{\bf{M}}]=0,\end{eqnarray} \tag{ 5 }$$

yield solutions for different types of chiral modulations (figures 1, 2, 12).

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Generally the equilibrium modulated patterns ${\bf{m}}({\bf{r}})$ in a chiral magnet are derived by numerically solving the above set of nonlinear differential equations including non-local stray-field calculations [14, 34]. Contrary to soft magnetic materials where demagnetization fields sufficiently influence the equilibrium magnetic states [38], in chiral magnetic materials the DM interactions strongly suppress these effects [34]. As a result in many practical cases a magnetostatic problem is reduced to analytical solutions [3, 34, 39], and the stray-field energy can be expressed as local energy contributions in energy functional (2) [3, 34].

It was also found that for one-dimensional modulations and two-dimensional axisymmetric structures, the internal stray-field energy has a local character [3, 38]. Particularly, for ferromagnets with C_nv symmetry the internal stray-field energy can be taken into account by the following redefinition of the anisotropy constant,

$$\begin{eqnarray}&&K\to K+{K}_{d},\quad {K}_{d}={\mu }_{0}{M}^{2}/2.\end{eqnarray} \tag{ 6 }$$

We introduce cylindrical coordinates for the spatial variable ${\bf{r}}=(r\mathrm{cos}\varphi ,r\mathrm{sin}\varphi ,z)$ and consider magnetic patterns homogeneous along the z-axis with the magnetization antiparallel to the applied field in the center ($\theta =\pi$ for r = 0) and approaching the parallel orientation when the distance from the center approaches infinity ($\theta \to 0$ for $r\to \infty$). For $\theta (r,\varphi )$, $\psi (r,\varphi )$ the energy functional (2) is reduced to the following form:

$$\begin{eqnarray}w & = & A\left[{\theta }_{r}^{2}+\displaystyle \frac{1}{{r}^{2}}{\theta }_{\varphi }^{2}+{\mathrm{sin}}^{2}\theta \left({\psi }_{r}^{2}+\displaystyle \frac{1}{{r}^{2}}{\psi }_{\varphi }^{2}\right)\right]\\&&+{w}_{D}-K{\mathrm{cos}}^{2}\theta -{\mu }_{0}{{MH}}^{(e)}\mathrm{cos}\theta -{\mu }_{0}M{\bf{m}}\cdot {{\bf{H}}}^{(d)}/2,\ \end{eqnarray} \tag{ 7 }$$

and the Dzyaloshinskii-Moriya energy functionals ${w}_{D}(\theta ,\psi ,r,\varphi )$ are listed in equations (A.1), (A.2).

The equations minimizing energy (7) include rotationally symmetric solutions,

$$\begin{eqnarray}&&\theta =\theta (r),\quad \psi =\psi (\varphi ),\quad {{\bf{H}}}^{(d)}={{\bf{H}}}^{(d)}(r).\end{eqnarray} \tag{ 8 }$$

Analytical solutions $\psi =\psi (\varphi )$ for uniaxial noncentrosymmetric ferromagnets [2] and cubic helimagnets (figures 1, 2) are listed in equation (A.7).

To date, only two types of skyrmionic states from this list have been identified in chiral ferromagnets by direct experimental observations: skyrmionic patterns with Bloch-type modulations (figure 1(a))

$$\begin{eqnarray}&&{\bf{m}}={\vec{e}}_{\varphi }\mathrm{sin}\theta (r)+{\vec{e}}_{z}\mathrm{cos}\theta (r)\end{eqnarray} \tag{ 9 }$$

have been observed in free standing nanolayers of cubic helimagnets (see e.g. [23, 24, 30]), and skyrmion lattices with ${\rm{N}}\acute{\rm{e}}{{\rm{e}}}$el-type modulations (figure 1(b))

$$\begin{eqnarray}&&{\bf{m}}={\vec{e}}_{r}\mathrm{sin}\theta (r)+{\vec{e}}_{z}\mathrm{cos}\theta (r)\end{eqnarray} \tag{ 10 }$$

have been observed in Fe/Ir(111) and PdFe/Ir(111) nanolayers [27, 31, 32, 41] and in the rhombohedral ferromagnet GaV₄O₈ with ${{\rm{C}}}_{3v}$ symmetry [28].

The first direct observations of isolated skyrmions have been reported in PdFe/Ir(111) nanolayers [31]. These chiral solitonic structures have been investigated in a broad range of applied fields [31, 32].

After integration with respect to ϕ, the total energy ${ \mathcal F }$ for an isolated skyrmion of Bloch- and ${\rm{N}}\acute{\rm{e}}{{\rm{e}}}$el-type in an applied magnetic field perpendicular to the film surface can be reduced to the following form:

$$\begin{eqnarray}&&{ \mathcal F }=2\pi {\int }_{0}^{\infty }f(\theta ,r){r}{\rm{d}}{r}.\end{eqnarray} \tag{ 11 }$$

Here $f(\theta ,r)=w(\theta ,r)-w(0)$ is the difference between the skyrmion energy density and that of the saturated state, $w(0)=-K-{\mu }_{0}{MH}$:

$$\begin{eqnarray}f(\theta ,r) & = & A\left({\theta }_{r}^{2}+\displaystyle \frac{1}{{r}^{2}}{\mathrm{sin}}^{2}\theta \right)-D\left({\theta }_{r}+\displaystyle \frac{1}{r}\mathrm{sin}\theta \mathrm{cos}\theta \right)+K{\mathrm{sin}}^{2}\theta +{\mu }_{0}H(1-\mathrm{cos}\theta ).\end{eqnarray} \tag{ 12 }$$

In equation (12) the perpendicular component of the internal magnetic field $H\equiv {H}_{z}$ equals the difference between the value of the applied magnetic field ${H}^{(e)}\equiv {H}_{z}^{(e)}$ and the perpendicular component of the demagnetization field imposed by the film with an isolated axisymmetric skyrmion (${H}_{d}^{(\mathrm{sk})}$): $H={H}^{(e)}-{H}_{d}^{(\mathrm{sk})}$. The analytical solution for the stray-field ${H}_{d}^{(\mathrm{sk})}$ has been derived by Y. Tu [34, 39]. In most nanolayers of chiral magnets investigated so far, skyrmion sizes are smaller than the layer thickness (${r}_{s}\lt d$). For such 'thick' films the stray-field ${H}_{d}^{(\mathrm{sk})}$ can be simplified and yields the following relation between the applied and internal fields: $H={H}^{(e)}-{\mu }_{0}M$. In PdFe/Ir (111) films investigated in this paper, isolated skyrmion cores are much larger than the film thickness (${r}_{s}\gg d$). In this limiting case of 'ultrathin' films, the stray-field ${H}_{d}^{(\mathrm{sk})}$ becomes exponentially small due to dipole-dipole interactions between the upper and bottom film surfaces, and the internal field is practically equal to the external field ($H\cong {H}^{(e)}$). A similar drastic reduction of the stray-field occurs in ultrathin magnetic films with isolated bubble and stripe domains [40].

The Euler equation for energy functional (12),

$$\begin{eqnarray}&&A\left({\theta }_{{rr}}+\displaystyle \frac{1}{r}{\theta }_{r}-\displaystyle \frac{1}{{r}^{2}}\mathrm{sin}\theta \mathrm{cos}\theta \right)+\displaystyle \frac{D}{r}{\mathrm{sin}}^{2}\theta -K\mathrm{sin}\theta \mathrm{cos}\theta -{\mu }_{0}{MH}\mathrm{sin}\theta =0,\end{eqnarray} \tag{ 13 }$$

with boundary conditions

$$\begin{eqnarray}&&\theta (0)=\pi ,\quad \theta (\infty )=0,\end{eqnarray} \tag{ 14 }$$

yields the equilibrium structure of isolated axisymmetric skyrmions [2, 3, 14]. Note that for ${\rm{N}}\acute{\rm{e}}{{\rm{e}}}$el-type skyrmions, K includes the stray energy contribution (6).

Dimensionless variables

$$\begin{eqnarray}&&\rho =2\pi r/{L}_{D},\quad h=H/{H}_{D},\quad k=K/{K}_{0},\end{eqnarray} \tag{ 15 }$$

are commonly used in recent papers to describe modulated states in uniaxial chiral ferromagnets and cubic helimagnets (see e.g. [17, 26, 32, 42]). Here we use the characteristic parameters of a uniaxial chiral ferromagnet [3, 42]:

$$\begin{eqnarray}&&{L}_{D}=\displaystyle \frac{4\pi A}{| D| },\quad {\mu }_{0}{H}_{D}=\displaystyle \frac{{D}^{2}}{2{AM}},\quad {K}_{0}=\displaystyle \frac{{D}^{2}}{4A}.\qquad \end{eqnarray} \tag{ 16 }$$

L_D is the period of a helix at zero field and zero anisotropy, H_D is the saturated field and K₀ is the critical anisotropy (A.13).

With variables (15), the equation for axisymmetric skyrmions (13) is reduced to the following form:

$$\begin{eqnarray}&&{\theta }_{\rho \rho }+\displaystyle \frac{{\theta }_{\rho }}{\rho }-\displaystyle \frac{1}{{\rho }^{2}}\mathrm{sin}\theta \mathrm{cos}\theta +\displaystyle \frac{2{\mathrm{sin}}^{2}\theta }{\rho }-k\mathrm{sin}\theta \mathrm{cos}\theta -h\mathrm{sin}\theta =0,\end{eqnarray} \tag{ 17 }$$

with boundary conditions (14).

Solutions $\theta (r)$ of the boundary value problem (14) can be derived by solving the auxiliary Cauchy initial value problem for equation (13),

$$\begin{eqnarray}&&\theta (0)=\pi ,\quad {\theta }_{r}(0)=-a.\end{eqnarray} \tag{ 19 }$$

For illustration we consider the Cauchy problem given by (13) and (19) for H = 0 and $\varkappa =\pi D/(4\sqrt{{AK}})=0.8$ (A.16). The calculated profiles $\theta (r,a)$ and the corresponding curves ${\theta }_{r}(\theta )$ in the interval ${\rm{[}}0.4\lt a\lt 4.0{\rm{]}}$ are plotted in figures 3(c), (d). Most of curves $\theta (r,a)$ oscillate near lines ${\theta }_{\mathrm{1,2}}=\pm \pi /2$, the maximum values of ${w}_{a}=K{\mathrm{sin}}^{2}\theta$, and the corresponding profiles ${\theta }_{r}(\theta )$ spiral around the attractors, points ($\pm \pi /2,0$). Among these curves there is a singular line (with $a=1.62471$) which ends in the saddle point ($0,0$) and, thus, represents a solution of the boundary value problem for isolated skyrmions.

The visual representation of the solutions for the auxiliary Cauchy problem (13), (19) as parametrized profiles $\theta (r,a)$ (figure 3(c)) and ${\theta }_{r}(\theta )$ curves in ($\theta ,{\theta }_{r}$) phase plane (figure 3(d)) reveal mathematical regularities in the formation of the localized states.

To demonstrate a crucial role of the DM interactions in the stabilization of chiral skyrmions, in the following we compare the phase portrait in figure 3(d) with special cases of model (12) with D = 0.

Isotropic ferromagnets ($D=K=H=0$). The Euler equations for energy functional of an isotropic ferromagnet $w=A{({\bf{grad}}{\bf{m}})}^{2}$ yield rigorous analytical solutions for axisymmetric skyrmions $\theta (r)$, $\psi (\varphi )$ derived by Belavin and Polyakov [52]

$$\begin{eqnarray}&&\psi =N\varphi +\alpha ,\quad \mathrm{tan}(\theta /2)={(\delta /r)}^{N},\end{eqnarray} \tag{ 20 }$$

where α and $\delta \gt 0$ are arbitrary values and N are positive integers. The energy (11) for solutions (20) ${{ \mathcal F }}_{0}=8\pi {AN}$, does not depend on values δ and α [52]. For N = 1 a set of magnetization profiles $\theta (r/\delta )$ (20) and phase portrait trajectories ${\theta }_{r}(\theta )$

$$\begin{eqnarray}&&\theta =2\mathrm{arctan}(\delta /r),\quad \delta {\theta }_{r}=-2{\mathrm{sin}}^{2}(\theta /2),\end{eqnarray} \tag{ 21 }$$

are plotted in figure 5. For $\delta \gt 0$, the curves ${\theta }_{r}(\theta )$ start in points ($\pi ,-2/\delta$) and end in the saddle point ($0,0$). However, any anisotropy or magnetic field will destabilize this solution.

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Uniaxial centrosymmetric ferromagnets (D = 0). In this case equation (13) has no stable solutions for isolated skyrmions. For $H\gt 0$ all phase trajectories ${\theta }_{r}(\theta )$ spiral around attractor ($-\pi /2,0$). For $H\lt 0$ equation (13) has radially unstable solutions for isolated skyrmions as proved by Derrick-Hobart theorem (For details see [3, 10]).

Analysis of skyrmion energy ${ \mathcal F }$ (11) under scaling transformations offers further important insight into the physics of chiral skyrmions. We consider a family of functions $\vartheta (r)=\vartheta (r/\eta )$ obeying the boundary conditions (14). Here $\eta \gt 0$ is an arbitrary constant describing uniform compressions ($0\lt \eta \lt 1$) or expansions ($\eta \gt 1$) of profile $\vartheta (r)$. For rescaled functions $\vartheta (r)=\vartheta (r/\eta )$, the skyrmion energy $\tilde{{ \mathcal F }}$ (11) can be expressed as a function of η:

$$\begin{eqnarray}&&\tilde{{ \mathcal F }}(\eta )={{ \mathcal E }}_{e}-{{ \mathcal E }}_{D}\eta +{{ \mathcal E }}_{0}{\eta }^{2}.\quad \end{eqnarray} \tag{ 22 }$$

The values of the exchange (${{ \mathcal E }}_{e}$), Dzyaloshinskii-Moriya (${{ \mathcal E }}_{D}$), and potential (${{ \mathcal E }}_{0}$) energy contributions for profile $\vartheta (r)$ (11) are given as follows:

$$\begin{eqnarray*}&&{{ \mathcal E }}_{e}=2\pi A{\int }_{0}^{\infty }\left({\vartheta }_{\xi }^{2}+\displaystyle \frac{1}{{\xi }^{2}}{\mathrm{sin}}^{2}\vartheta \right)\xi {\rm{d}}\xi \equiv A{\alpha }_{1},\end{eqnarray*}$$

$$\begin{eqnarray*}&&{{ \mathcal E }}_{D}=2\pi | D| {\int }_{0}^{\infty }\left({\vartheta }_{\xi }+\displaystyle \frac{1}{\xi }\mathrm{sin}\vartheta \mathrm{cos}\vartheta \right)\xi {\rm{d}}\xi \equiv | D| {\alpha }_{2},\end{eqnarray*}$$

$$\begin{eqnarray}&&{{ \mathcal E }}_{0}=2\pi {\int }_{0}^{\infty }[K{\mathrm{sin}}^{2}\vartheta +{\mu }_{0}{MH}(1-\mathrm{cos}\vartheta )]\xi {\rm{d}}\xi \end{eqnarray} \tag{ 23 }$$

or ${{ \mathcal E }}_{0}=K{\alpha }_{3}+{\mu }_{0}{MH}{\alpha }_{4}$, where ${\alpha }_{i}$ are the numerical coefficients given by the values of the integrals in equation (23).

Equation (22) shows that the DM energy plays a crucial role in stabilizing skyrmions [2, 13]. In centrosymmetric ferromagnets (${{ \mathcal E }}_{D}=0$) isolated skyrmions are unstable with respect to compression and collapse into a singular line ($\eta \to 0$) (Derrick-Hobard theorem [10]). Skyrmion solutions that minimize the free energy (22) only occur for nonzero Dzyaloshinskii-Moriya energy contributions.

Ansatz solutions. Potential $\tilde{{ \mathcal F }}(\eta )$ (22) has a convenient form for analysis of skyrmion solutions with trial functions of type $\vartheta =\vartheta (\rho /\eta )$ that obey the boundary conditions (14). Particularly, a linear ansatz

$$\begin{eqnarray}&&\vartheta =\pi [1-(r/\eta )]\quad (r\lt \eta ),\quad \vartheta =0\quad (r\gt \eta ),\qquad \end{eqnarray} \tag{ 24 }$$

has been used in [2] to introduce the phenomenon of chiral skyrmions. The ansatz,

$$\begin{eqnarray}&&\vartheta (r/\eta )=4\mathrm{arctan}[\mathrm{exp}(-r/\eta )],\end{eqnarray} \tag{ 25 }$$

based on solutions for isolated ${360}^{\circ }$ Bloch walls [38] provides a good fit to the solutions of equation (13). In [32], magnetization profiles for isolated skyrmions have been fitted by a combination of functions of type (25). For the trial function $\vartheta (r/\eta )$ in equation (25), the total energy (22) can be written as

${ \mathcal F }(\eta )/(2\pi )=4.31A+(1.59K+1.39{\mu }_{0}{MH}){\eta }^{2}-3.02D\eta$. For zero anisotropy (k = 0) this ansatz yields the transition field into the skyrmion lattice ${h}_{s}={H}_{s}/{H}_{D}=0.760$ (cf. with the rigorous value ${h}_{s}=0.801$ and ${h}_{s}=0.675$ for the linear ansatz (24) [2]).

For $\tilde{{ \mathcal F }}(\eta )$ (22) the equilibrium skyrmion size is

$$\begin{eqnarray}&&{\eta }_{0}=\displaystyle \frac{{L}_{D}}{2\pi }\ \displaystyle \frac{{\alpha }_{2}}{{\alpha }_{3}k+2{\alpha }_{4}h},\end{eqnarray} \tag{ 26 }$$

expressed as a ratio of the Dzyaloshinskii-Moriya to the potential energy contributions for the trial function.

The virial theorem for isolated axisymmetric skyrmions is derived by integration of the Euler equation (13). Partial integration leads to the following virial relation between the equilibrium values of the potential and DM energies [33] ${\bar{{ \mathcal E }}}_{0}=2{\bar{{ \mathcal E }}}_{D}$ where ${\bar{{ \mathcal E }}}_{0}$, ${\bar{{ \mathcal E }}}_{D}$ are the integrals (23) calculated for the solutions of equation (13), $\vartheta =\theta (\rho )$.

The stability of the solutions $\theta (\rho )$ of the boundary value problem (13), (14) under small radial distortions $\zeta (\rho )$ ($\zeta (0)=\zeta (\infty )=0$) has been investigated in [14]. This problem is reduced to the spectral problem for the perturbation energy functional [14]. By numerically solving the eigenvalue problem for this functional, the radial stability of isolated skyrmions has been established in a broad range of the control parameters $(k,h)$ [3]. Contrary to magnetic bubbles, which collapse with finite radii at certain critical fields [38], the solutions of the boundary value problem (13) and (14) (figure 3) exist at arbitrary high fields. In increasing fields their sizes gradually decrease and asymptotically approach zero.

The continuum model (2), however, becomes invalid for localized solutions with sizes of few lattice constants. In this region we investigate solutions for chiral skyrmions within the discrete models. We consider classical spins, ${{\bf{S}}}_{i}$, of unit length on a two-dimensional square lattice with the following energy functional [47] $E={E}_{0}+{E}_{D}$ where

$$\begin{eqnarray}&&{E}_{0}=-J\;\displaystyle \sum _{\langle i,j\rangle }({{\bf{S}}}_{i}\cdot {{\bf{S}}}_{j})-\displaystyle \sum _{i}[{\bf{H}}\cdot {{\bf{S}}}_{i}+K{({{\bf{S}}}_{i}\cdot {\bf{n}})}^{2}],\quad \end{eqnarray} \tag{ 27 }$$

and the Dzyaloshinskii-Moriya energy equals

$$\begin{eqnarray}&&{E}_{D}=-D\;\displaystyle \sum _{i}({{\bf{S}}}_{i}\times {{\bf{S}}}_{i+\hat{x}}\cdot \hat{x}+{{\bf{S}}}_{i}\times {{\bf{S}}}_{i+\hat{y}}\cdot \hat{y})\end{eqnarray} \tag{ 28 }$$

for Bloch-type modulations, and

$$\begin{eqnarray}&&{E}_{D}=-D\;\displaystyle \sum _{i}({{\bf{S}}}_{i}\times {{\bf{S}}}_{i+\hat{x}}\cdot \hat{y}-{{\bf{S}}}_{i}\times {{\bf{S}}}_{i+\hat{y}}\cdot \hat{x})\end{eqnarray} \tag{ 29 }$$

for Néel-type modulations ($\langle i,j\rangle$ denotes pairs of nearest-neighbor spins).

For a helix ${{\bf{S}}}_{i}=(\mathrm{cos}{\theta }_{i},\mathrm{sin}{\theta }_{i},0)$ propagating along the x-axis at field and anisotropy (${\bf{H}}=K=0$), model (27) is reduced to

$$\begin{eqnarray}&&E=\displaystyle \sum _{i}[-J\mathrm{cos}({\theta }_{i}-{\theta }_{i+\hat{x}})-D\mathrm{sin}({\theta }_{i}-{\theta }_{i+\hat{x}})],\end{eqnarray} \tag{ 30 }$$

and yields the equilibrium period ${p}_{0}=2\pi /\mathrm{arctan}(D/J)$ (p₀ is the number of magnetic ions corresponding to ${\rm{\Delta }}\theta =2\pi$).

The equilibrium solutions for isolated skyrmions are derived by numerically solving the equations minimizing the energy functional E (27). Typical solutions for the equilibrium skyrmion size p as a function of the applied field are plotted in figure 6(c). The solutions with finite sizes exist only below a certain critical (collapse) field ${h}_{c}({p}_{0})$ (figure 6(a)). The critical field ${h}_{c}({p}_{0})$ increases without limit with increasing p₀ (figure 6(b)) and, thus, signifies a transition from the discrete model to the continuous model.

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Isolated skyrmions exist as metastable states above the critical field ${h}_{s}(k)$ (figures 4(a), (b)). Below this line the energy ${ \mathcal F }$ (11) becomes negative and skyrmions tend to condense into a hexagonal lattice [3]. However, if the formation of skyrmion lattices is suppressed (as in PdFe/Ir (111) films [32]) isolated skyrmions continue to exist below the critical line ${h}_{s}(k)$ (with the skyrmion core energy density lower than that of the surrounding saturated state). At the same time isolated skyrmions have a tendency to elongate and expand into a band with helicoidal or cycloidal modulations and eventually to fill the whole space, since the spiral state represents the minimum with lower energy as compared to the local minima with the metastable isolated skyrmions. These (elliptic) instabilities are similar to 'strip-out' instabilities of isolated magnetic bubbles at a certain critical field [53] observed in common 'bubble-domain' films [38] and in magnetic nanolayers with perpendicular anisotropy [54]. A theory of elliptic instabilities for chiral skyrmions have been developed in [33] where a strip-out field has been derived from the stability analysis of the skyrmion energy (11) with respect to (elliptic) perturbations of type

$$\begin{eqnarray}&&\tilde{\rho }=\rho +\varepsilon \eta (\rho )\mathrm{cos}2\varphi ,\quad \tilde{\psi }=\psi +\zeta (\rho ,\varphi ),\end{eqnarray} \tag{ 31 }$$

where ($\varepsilon \ll 1$)[33] . In [33], the strip-out field H_el has been calculated for stray-field free elliptic distortions of Bloch-type isolated skyrmions. There, a set of parametrized ansatz functions of type $\eta (\rho )=\mathrm{sin}\theta /(1+a\mathrm{sin}\theta )$ with optimized values of a were used [33]. In this paper the critical line ${h}_{{el}}(k)$ ($0\lt k\lt {k}_{a}$ ) for a Bloch-type-skyrmion (figure 4) has been calculated by direct minimization of the perturbation energy.

Within the discrete model (27) the critical field h_el has been calculated for zero anisotropy (k = 0) and for $6\lt {p}_{0}\lt 30$. Figure 7 (a) shows that the strip-out field h_el essentially decreases with the decreased size of skyrmions what can be beneficial for possible application of such skyrmions. However, the existence region of these isolated skyrmions is restricted by the lower field of collapse h_c.

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In this section we consider the existence area for isolated skyrmions in the magnetic phase diagram (figure 4). The energy functional for uniaxial chiral ferromagnets (2) depends on the two independent control parameters, the reduced values of the applied field, h and uniaxial anisotropy, k (15). The magnetic phase diagram in variables k and h collects all possible solutions for model (2). The calculated phase diagram in the inset of figure 4 shows the existence areas of the cycloids and skyrmion lattices and the transition lines between these modulated phases and the saturated state. The phase diagram indicates the critical fields at zero anisotropy, the bicritical point B (1.90, 0.10), and the critical point A (2.67, 0) [3] (for a detailed description of this phase diagram see [17]). Figure 4 shows critical lines for isolated skyrmions (results of the continuum model (2)). Isolated skyrmions condense into a skyrmion lattice when the applied magnetic field decreases to the critical value ${h}_{s}(k)$. However, isolated skyrmions can exist as localized objects below the critical line ${h}_{s}(k)$ and strip-out into helicoids at the critical line h_el.