---
abstract: 'We study the geometrical conditions for stabilizing magnetic skyrmions in cylindrical nanostrips and nanotubes of ferromagnetic materials with Dzyaloshinskii-Moriya interactions. We obtain the ground state of the system implementing a simulation annealing technique for a classical spin Hamiltonian with competing isotropic exchange and chiral interactions, radial anisotropy and an external field. We address the impact of surface curvature on the formation, the shape and the size of magnetic skyrmions. We demonstrate that the evolution of the skyrmion phase with the curvature of the nanoshell is controlled by the competition between two characteristic lengths, namely the curvature radius, $R$ (geometrical length) and the skyrmion radius, $R_{Sk}$ (physical length). In narrow nanotubes ($R<R_{Sk}$) the skyrmion phase evolves to a stripe phase, while in wide nanotubes ($R>R_{Sk}$) a mixed skyrmion-stripe phase emerges. Most interestingly, the mixed phase is characterized by spatially separated skyrmions from stripes owing to the direction of the applied field relative to the surface normal. Below the instability region ($R \lesssim R_{Sk}$) skyrmions remain circular and preserve their size as a consequence of their topological protection. Zero-field skyrmions are shown to be stable on curved nanoelements with free boundaries within the same stability region ($R\gtrsim R_{Sk}$). The experimental and technological perspectives from the stability of skyrmions on cylindrical surfaces are discussed.'
author:
- 'D. Kechrakos'
- 'A. Patsopoulos'
- 'L. Tzannetou'
title: Magnetic skyrmions in cylindrical ferromagnetic nanostructures with chiral interactions
---

Introduction
============

Magnetic skyrmions are self-localized vortex-like spin structures with axial symmetry [@bog94a]. They have been mainly studied in noncentrosymmetric bulk crystals and their thin films[@muhl09; @pap09; @yux10], as well as, in ultrathin ferromagnetic (FM) films on heavy metal (HM) substrates [@hein11; @rom13], in which a sizable Dzyaloshinskii-Moriya interaction (DMI) [@dzi58; @mor60] leads to their formation. From the technological point of view, two-dimensional magnetic skyrmions formed in ferromagnetic-heavy metal interfaces have potentials for a variety of innovative robust and high-density spintronics applications due to their protected topology and nanoscale size [@fer13]. In particular, they can be driven by lateral spin currents [@fer13; @samp13; @nag13], produced by electrical currents with five to six orders of magnitude smaller current density than those needed for domain wall motion [@rom13], thus pointing to energy efficient [@fer13] skyrmion-based racetrack-type memory devices[@par08]. However,current-driven skyrmions will drift from the racetrack direction due to the presence of Magnus force [@iwa13; @yux12], if the velocity is high enough. This phenomenon known as the Skyrmion Hall effect (SkHE) leads to their annihilation at the racetrack edge and the loss of stored information. An approach for limiting the SkHE effect is through spin-wave driven skyrmion motion [@zha15; @sch15]. Skyrmions can be displaced by magnons induced by thermal gradients in insulating chiral ferromagnets [@kon13], while the SkHE deviation vanishes for high energy magnons [@gar15]. However, compared with the current-driven skyrmion motion, it is difficult to generate spin waves in a nanometre-size nanotrack with appropriate spectral properties for driving the motion of a skyrmion. It is also difficult to realize a skyrmion nanocircuit based on thermal gradients. Consequently, the current-driven skyrmion motion is the most promising method and as such it attracts a great deal of research effort. To this end, various potential barriers have been proposed to confine skyrmions in the central region of the racetrack so that the annihilation at the racetrack edge is avoided [@zha16; @bar16; @pur16; @lai17; @foo15]. A suggested method is by tuning the perpendicular [@foo15] or the crystalline [@lai17] magnetic anisotropy. As a result, a path of lower resistance is created at the racetrack center, allowing the skyrmions to pass the racetrack without annihilation. Another approach, is by tuning the height of the ferromagnetic layers, creating a rectangular groove on the center of the racetrack. As a result, a curb structure is formed, which functions to confine the skyrmion within the groove [@pur16]. Furthermore, the damping constant of the racetrack can be tuned in either the transverse or the longitudinal direction in different regions of the racetrack [@liu16], so that the deviations of the skyrmions are in opposite directions and cancel each other out. Therefore, the skyrmions can be efficiently confined in the racetrack center and the SkHE is avoided. Another aspect hampering the use of magnetic skyrmions in racetrack memory applications, is their uncontrollable excitation realized at the edges of magnetic nanostrips and thin films [@ran17] leading to error writing events.. This phenomenon is known as the edge effect. In addition, skyrmion motion, even including the oscillating motion and the gyration [@gar16], is affected by the edges in confined geometries due to their potential force [@nav16; @gar16] acting on skyrmions. From the aforementioned works, it appears that the possibility of magnetic skyrmions generation and manipulation on boundary-free samples offers be a desirable direction of research and curved nanostructures, as for example, magnetic nanotubes, constitute a promising option.

The study of magnetic structure and solitonic excitation on curved surfaces has recently attracted intensive interest as curvature was shown to control physical properties of the system [@streu16]. The curvilinear geometry of bent and curved ferromagnetic wires and surfaces [@pyl15; @gai15; @car15] introduces effective chiral interactions and curvature-induced anisotropy.[@streu16] As a consequence, curvature-driven effects emerge, such as magnetochiral effects [@kra12; @ota12] and topologically induced magnetization patterning,[@kra12; @pyl15] resulting in high domain wall velocities [@yan12] and chirality symmetry breaking [@pyl15]. Despite the fact that recent works have focused on the impact of surface curvature on the emerging chiral properties and related magnetic order of otherwise achiral ferromagnetic materials [@streu16], to the best of our knowledge, the conditions for skyrmion formation on chiral curved surfaces has not been addressed yet. We anticipate on physical grounds, that the skyrmion phase supported on a planar nanostructure, such as a FM/HM interface, will be driven to instability under curving.

It is the main aim of the present work, to investigate the ground state properties of curved ferromagnetic nanostructures with chiral interactions (DMI) and examine the conditions under which curvature-driven skyrmion instability occurs. Our structural model accounts for direction modulation of the DMI vector induced by the curvature of the nanostructure under consideration, thus providing a more realistic description of the interplay between isotropic exchange (Heisenberg) and chiral interactions on curved surfaces. We focus on cylindrical nanoelements and nanotubes. Our results demonstrate the feasibility of skyrmion formation on the ridge of a nanotube, where the external field remains almost normal to the surface, provided that the radius of the nanotube remains at least comparable to the skyrmion radius $(R_{tube} \ge R_{Sk} )$. The same geometrical criterion ensures the stability of skyrmions without an external magnetic field on curved nanoelements.

Micromagnetic Model and Simulation Method
=========================================

We consider a thin ferromagnetic cylindrical nanostrip along the z-axis with length $L_z$, width $L_y$, inner radius $R$ and thickness $t\ll R$ (Fig.\[fig:sketch\]).

![(Color online) Cylindrical nanostrip extended along the $z$-axis with width $L_y$, thickness $t$, curvature radius $R$ and curvature angle $\phi_0$, used as our model system. []{data-label="fig:sketch"}](fig1.jpg){width="0.40\linewidth"}

The central angle of the curved nanostrip is defined as $\phi_0=L_y/R$. A planar nanostrip ($R\rightarrow\infty,\phi_0=0 $) and a cylindrical nanotube ($R\ne0,\phi_0=360^0$) naturally occur as limiting cases of the curved nanostrip.

The micromagnetic energy of the system as a functional of the continuous magnetization field $\textbf{m}(\textbf{r})=\textbf{M}(\textbf{r})/M_s$ reads $$\begin{aligned}
    E[\textbf{m}]=\int d^3\textbf{r}~
    \{ 
    A |\nabla\textbf{m}|^2
    -K_u (\textbf{m}\cdot \textbf{e}_\rho)^2 
    \nonumber \\
    -M_s\textbf{m}\cdot\textbf{B}
    +w_{DM}
    \} 
    \label{eq:microm}\end{aligned}$$ where the integral runs over the volume of the nanostructure, $A$ is the exchange constant and $K_u$ is the radial anisotropy density, which we adopt here as a generalization of the perpendicular anisotropy observed in thin ferromagnetic films on a heavy metal substrate[@fer13; @hag15]. $w_{DM}$ is the Dzyaloshinskii-Moriya energy density and $\textbf{e}_\rho(\textbf{r})$ is the radial unit vector. The DMI energy is considered here as arising from the interface coupling between the ferromagnetic nanostrip and a heavy metal layer, which is assumed to be coupled to the nanostrip. Generalizing the expression for the interface DMI energy density[@cor18], we write $$\begin{aligned}
w_{DM}=D[(\textbf{m}\cdot\nabla)m_\rho-m_\rho(\nabla\cdot\textbf{m})]
\label{eq:dmi}\end{aligned}$$ where $\textbf{m}(\textbf{r})=(m_\rho,m_\phi,m_z)$ are the components of the magnetization field with respect to the local cylindrical coordinate system $(\textbf{e}_\rho,\textbf{e}_\phi,\textbf{z})$ (Fig.\[fig:sketch\]). Magnetostatic terms are neglected in Eq.(\[eq:microm\]), because in the limit of a very long cylinder $(L_z\gg R)$ and a mean field approximation, they can be approximated by a uniaxial anisotropy term along the z-axis leading to reduction the radial anisotropy term as[@roh13] $K_u^{'}=K_u-\frac{1}{2}\mu_0 M_s^2$. Upon discretization of Eq.(\[eq:microm\]) on a cylindrical grid , we obtain for the total energy $$\begin{aligned}
E = 
-\frac{1}{2} J \sum_{<ij>} \textbf{m}_i \cdot \textbf{m}_j 
\nonumber \\
-\frac{1}{2} d \sum_{<ij>} \textbf{D}_{ij} \cdot 
(\textbf{m}_i \times \textbf{m}_{j})
\nonumber \\
-k \sum_i (\textbf{m}_i \cdot \textbf{e}_{\rho,i} )^2  
-h \sum_i \textbf{m}_i \cdot \textbf{h}_i
\label{eq:energy}\end{aligned}$$ with bold characters indicating unit vectors. $\textbf{m}_i$ is the unit vector (spin) along the magnetic moment of the $i$-th cell. The $1/2$ prefactor of the first and second terms accounts for the double-counting of energy contribution from pairs of nearest neighboring sites. The DMI vector takes the form $\textbf{D}_{ij}=\textbf{e}_{\rho,i} \times \textbf{r}_{ij}$, which is a generalization of the expression $\textbf{D}_{ij}=\textbf{x} \times \textbf{r}_{ij}$, that describes the DMI coupling at planar interfaces in the $yz$-plane.[@hag15; @yin16] Note that a major physical difference compared to the flat interface is that for a curved interface the vector $\textbf{D}_{ij}$ becomes site-dependent owing to the variation of the radial direction across the surface. The applied field, is assumed either homogeneous along the $x$-axis ($\textbf{h}_i=\textbf{x}$) or radial ($\textbf{h}_i=\textbf{e}_{n,i}$), as explicitly mentioned below.

Under the assumption of a grid cell with equal sizes along the azimuthal and $z$ axes ($a_\phi=a_z$) and in the limit of a very thin FM nanostrip ($t=a_z$), the energy parameters entering Eq.(\[eq:energy\]) are related to the micromagnetic material parameters of Eq.(\[eq:microm\]) through the relations $J\approx 2Aa, 
d \approx Da^2$ $k \approx K_ua^3$, $h\approx M_sBa^3$. We use material parameters typical of a transition metal thin film on a heavy metal substrate[@hag15; @lel19], namely, $
M_s=580kA/m, 
A=10pJ/m, 
D=4mJ/m^2, 
K_u=500kJ/m^3
$ and a cell size $a=2nm$, which is well below the exchange length $l_{ex}=\sqrt{2A/\mu_0 M_s^2}\sim 7nm$. An applied field $B=0.9~T$ is considered. Then the rationalized (dimensionless) parameters $d/J=0.4, 
k/J=0.1$ and $h/J=0.1$, consist a complete set of parameters that determine the magnetic configuration at the ground state.

Furthermore, the pitch length of the helical phase is determined by the rationalized parameter $d/J$ through the relation[@kee15; @sek16] $$\begin{aligned}
p=\frac{2\pi a}{tan^{-1}(d/J)}. 
\label{eq:pitch}\end{aligned}$$ For the material parameters mentioned above we obtain $p\approx 16.5a=32nm$. This is a characteristic length scale of the skyrmion phase as it is approximately equal to the skyrmion radius.[@kee15; @sek16]

To obtain the ground state we perform simulation annealing using the Metropolis Monte Carlo algorithm with single spin updates and temperature-dependent spin aperture that accelerates the approach to equilibrium. In particular, a field-cooling procedure under a field $h/J=0.1$ is performed from a high temperature $k_BT/J=20$ ($k_BT_C/J\approx 1$) to a low temperature $k_BT/J=0.001$, with a variable step $dT/T=5\%$ that produces an exponential decrease of temperature. At each temperature value we perform $5000$ Monte Carlo steps per spin (MCSS) for thermalization followed by $5000$ MCSS for calculations of thermal averages. The latter are calculated from sampling every $\tau=10$ MCSS, in order to minimize statistical correlations between sampling points. The thermodynamic quantities at each temperature are averaged over many ($\approx 20-40$) independent relaxation sequences to obtain an estimate of the statistical errors.

Results and Discussion
======================

Skyrmion phase
--------------

\[htb!\] ![ (Color online) Ground state configuration showing skyrmion formation in cylindrical nanostructures under application of a uniform magnetic field along the $x$-axis. Cylindrical surfaces are constructed by gradually wrapping an initial square sample ($L_y=L_z\equiv L$) around the $z$-axis. Spin configurations are color coded with the values of magnetization along the applied field direction ($x$-axis). (a), (d) planar surfaces, (b) $L=50a,  \phi_0=150^0, R=19.1a$, (c) $L=50a,  \phi_0=360^0, R=8.0a$, (e) $L=100a, \phi_0=150^0, R=38.2a$, and (f) $L=100a, \phi_0=360^0, R=15.9a$, with $a=2nm$. With increasing angle of curvature ($\phi_0$) the skyrmion phase transforms to either a spiral phase, as in (c), or to a mixed skyrmion-spiral phase, as in (f), depending on the value of the curvature radius ($R$). []{data-label="fig:Skphase"}](fig2.jpg "fig:"){width="0.95\linewidth"}

We consider first the evolution of a skyrmion ground state as the curvature of the nanostructure increases. We start from a planar surface (PS) in the yz-plane and wrap it gradually along the z-axis to form an open cylindrical surface (CS) and eventually, a closed cylindrical surface corresponding to a nanotube (NT) (Fig.\[fig:Skphase\]). When we curve the 2D sample, we preserve the dimensions $(L_y,L_z)$ of the initial planar system in order to emphasize the role of curvature and exclude finite size effects. Periodic boundary conditions are used solely along the z-axis of our curved samples, except for nanotubes, when the lateral free boundaries couple among themselves, naturally.

In planar systems, we observe the well-known skyrmion lattice[@yis09] consisting of a hexagonal arrangement of skyrmions. Obviously, the number of skyrmions increases with the area of the planar sample, however their spatial density remains almost unchanged.

As the angle of curvature increases, skyrmions close to the free edges of the curved surface become elongated and finally transform into spirals. This effect becomes more evident in smaller samples, which are characterized by smaller values of the curvature radius (Fig.\[fig:Skphase\]b,c). In a small nanotube with radius $R=8a$ (Fig.\[fig:Skphase\]c) stripes form almost all around the surface. On the contrary, in a larger nanotube with radius $R=15.9a$ isolated skyrmions are observed along the front and the back ridge of the cylinder, where the external field is almost normal to the surface, but spiral structures form along the left and right sides of the large tube (Fig.\[fig:Skphase\]f) where the applied field is almost tangential to the surface.

Thus, skyrmion formation on nanotubes is strongly dependent on the nanotube radius, with large radius nanotubes supporting the coexistence of both skyrmion and stripe phases. We underline the fact that the two phases are spatially separated with skyrmions forming along the ridge and stripes forming on the sides of the nanotube. The width of the region supporting skyrmions is determined by the size of the skyrmion radius ($R_{Sk}$) relative to the curvature radius ($R$). This point is discussed further below.

To quantify the evolution of the skyrmion phase with sample curvature, as depicted in Fig.\[fig:Skphase\], we calculate the topological charge ($Q$) of the ground state. For a three component spin field $\textbf{m}(\phi,z)$ on a cylindrical surface described by the coordinates ($\phi,z$), the topological charge is given as $$\begin{aligned}
Q=\frac{1}{4\pi} \iint d\phi~dz~
\textbf{m}\cdot
(\frac{\partial\textbf{m}}{\partial\phi}
\times
\frac{\partial\textbf{m}}{\partial z}).
\label{eq:topol_charge}\end{aligned}$$ For the numerical computations we implement a lattice expression of the topological charge[@ber81], appropriate to a square lattice wrapped around a cylindrical surface. Skyrmions have a topological charge $Q=\pm1$, depending on the direction of the applied field. Thus the absolute value of $Q$ for a nanostrip in the skyrmion phase equals to the number of skyrmions supported.

The dependence of the topological charge on the curvature angle is shown in Fig.\[fig:Q\_vs\_a\] for nanostrips with different widths $L_y$.

\[htb!\] ![ (Color online) Dependence of the ground state topological charge (Q) on the angle of curvature ($\phi_0$) for magnetic nanostructures with different sizes ($L\times L$) and $a=2nm$. The applied magnetic field is uniform along the x-axis (circles, closed triangles) or radial (open triangles). []{data-label="fig:Q_vs_a"}](fig3.jpg "fig:"){width="0.95\linewidth"}

We notice that $Q$ remains almost constant up to an angle $\phi_0\approx 100^0$ and further on it decreases smoothly from the initial value for planar nanostrips (skyrmion phase), to a smaller value close to zero for nanotubes, indicating that only a small fraction of the initial number of skyrmions are stabilized (mixed skyrmion-stripe phase). In contrast to this trend, when the applied field is radial, the topological charge is only weakly dependent on the curvature angle. This weak decay of $Q$ with curvature angle is due to the gradual reduction of the co-planar condition between the DMI vectors on each lattice site. A radial field for a cylindrical nanostrip is geometrically analogous to the case of a uniform field normal to a planar nanostrip and the conservation of the topological charge with curvature underlines the importance of the normality condition for the applied filed in the stabilization of skyrmions on any surface.

For planar nanostrips shown in Fig.\[fig:Skphase\]a ($Q=7.2$) and Fig.\[fig:Skphase\]d ($Q=37.8$), the values of $Q$ deviate weakly from integer values due to the misalignment of the moments located on the free boundaries of the sample[@roh13] and the thermal fluctuations inherent to the Monte Carlo method. For curved surfaces, however, the shape distortion of the skyrmions and their evolution to stripe-like structures is not characterized by integer values of $Q$, thus the calculation of $Q$ based on Eq.(\[eq:topol\_charge\]) assumes non-integer values, as in Fig.\[fig:Q\_vs\_a\], and is only indicative of the number of skyrmions observed in the mixed phase.

\[htb!\] ![ Dependence of the ground state topological charge ($Q$) on the radius ($R$) of magnetic nanotubes ($\phi_0=360^0$) with length $L_z=220a$ and $a=2nm$. []{data-label="fig:Q_vs_R"}](fig4.jpg "fig:"){width="0.95\linewidth"}

The magnetization landscape at the ground state of a curved nanostrip depends on both the curvature radius ($R$) and the curvature angle ($\phi_0$). Next, we fix the curvature angle by choosing to consider nanotubes ($\phi_0=360^0$) of various radii and fixed length. We show in Fig.\[fig:Q\_vs\_R\] the dependence of the topological charge on the nanotube radius. For radius $R\lesssim 15a$ the topological charge assumes low values indicating stripe formation around the tube, while in nanotubes with larger radius ($R\gtrsim18a$) a sharp increase of $Q$ is observed signifying skyrmion formation. This behavior of $Q$ is in accordance with the magnetization configuration seen in Fig.\[fig:Skphase\]f ($R=15.9a$), that indicates skyrmion formation along the ridge of the nanotube.

In conclusion, the numerical data so far demonstrate that the skyrmion phase of a planar nanostrip with material parameters typical of a ferromagnetic/heavy metal interface ($p\approx16a$), transforms to a stripe phase when the curvature angle or the curvature radius exceeds some characteristic values ($\phi_0\gtrsim100^0$, $R/a \lesssim 15$). It is important to notice in Fig.\[fig:Q\_vs\_R\] that the skyrmion phase disappears when the curvature radius becomes comparable to the pitch length ($R\sim p\sim 16a$). We elaborate further on this point in the next section, by focusing on a nanostrip with a single skyrmion.

Skyrmion shape and size
-----------------------

The analysis of the skyrmion shape and size in the ground state is a numerically intricate task[@ziv19] especially when the system is in a mixed phase as it occurs in curved nanostrips (Fig.\[fig:Skphase\]). For this reason we increase the discretization level in order to stabilize a single skyrmion in the simulation cell and facilitate the analysis. In particular, we use further on a cell size $a=1nm$ and keep the material parameters $(A_{ex},D,K_u,M_s)$ and the applied field $B$ unchanged. This leads to new rationalized parameters $d/J=0.2$, $k/J=0.025$, $h/J=0.025$, and a pitch length $p\approx33.0a=33nm$. Notice that increasing the discretization level does not affect substantially the physical value of the pitch length, because the latter is a slowly varying function of the grid cell size for $D/A_{ex} \lesssim 1nm^{-1}$ (see Eq.(\[eq:pitch\])).

\[htb!\] ![ (Color online) Snapshots of ground state magnetic configurations in curved nanostrips with size $50a\times50a$ $(a=1nm)$. Curvature angles and topological charge are (a) $\phi_0=  0^0$, $Q=0.84$, (b) $\phi_0= 50^0$, $Q=0.83$, (c) $\phi_0=100^0$, $Q=0.84$, (d) $\phi_0=150^0$, $Q=0.99$, (e) $\phi_0=160^0$, $Q=0.52$, and (f) $\phi_0=200^0$, $Q=0.10$. The curved nanostrips (b)-(f) are unwrapped on the $yz$-plane for visual clarity. A uniform field ($h/J=0.025$) is applied in all cases along the $x$-axis. The color code indicates the values of magnetization along the field axis. A transformation from purely skyrmion phase (a), to a mixed skyrmion-stripe phase (e,f), due to increasing curvature, is seen. []{data-label="fig:Single_Sk"}](fig5.jpg "fig:"){width="0.95\linewidth"}

In Fig.\[fig:Single\_Sk\], we show the evolution of a single skyrmion that is stable on a planar nanostrip as the nanostrip is curved gradually. For small angles ($\phi\lesssim 100^0$), the skyrmion retains its basic geometrical features, such as its size and axially symmetric shape. The robustness of the skyrmion at small curvature angles is consistent with the constant value of the topological charge at small curvature angles, seen in Fig.\[fig:Q\_vs\_a\]. As the curvature increases, the skyrmion obtains a more elliptical shape while its size decreases. Finally, for larger angles ($\phi\gtrsim160^0$) skyrmion formation is not stable anymore.

\[htb!\] ![ (Color online) Dependence of skyrmion circularity ($M_{circ}$) and linearity ($M_{lin}$) on curvature angle of a cylindrical nanostrip with size $50a\times50a~ (a=1nm)$. Error bars are obtained from an average over $30$ independent configurations of the ground state. []{data-label="fig:Sk_MM"}](fig6.jpg "fig:"){width="0.95\linewidth"}

To quantify our observations on the magnetic configurations of Fig.\[fig:Single\_Sk\], we perform shape analysis of the skyrmion core ($S$), which is is defined as the compact region of the nanostrip with negative local magnetization ($m_{i,x}<0$). We compute two shape measures of $S$, namely the invariant moments $M_{circ}$ and $M_{lin} $ that measure the degree of circularity[@zun14] and linearity [@sto08], respectively. These are defined as $$\begin{aligned}
    M_{circ}=\frac{\mu_{00}}{\mu_{20}+\mu_{02}} 
    \label{eq:hue1}\end{aligned}$$ and $$\begin{aligned}
    M_{lin}=\frac{ \sqrt{(\mu_{20}-\mu_{02})^2+4\mu_{11}^2} }{\mu_{20}+\mu_{02}},
    \label{eq:hue2}\end{aligned}$$ where the second order geometric moments are $$\begin{aligned}
    \mu_{pq}=\frac{1}{N_S}\sum_{i\in S}(y_i-y_c)^p(z_i-z_c)^q
    \label{eq:moms}\end{aligned}$$ with $p,q$ positive integers satisfying $p+q\le 2$, $N_S$ the number of cells in $S$ and $(y_c,z_c)$ the centroid coordinates $y_c=\sum_i y_i/N_S$ and $z_c=\sum_i z_i /N_S$. In the limiting case of a circle $M_{circ}=1, M_{lin}=0$ and in the case of straight line $M_{circ}=0, M_{lin}=1$.

In Fig.\[fig:Sk\_MM\] we show the evolution of the shape measures of a skyrmion with curvature angle. The sudden drop of $M_{circ}$ above $\phi_0 \simeq 100^0$ signifies the skyrmion elongation and eventual annihilation. Below this characteristic angle, the skyrmion retains the circular shape ($M_{circ}\simeq 1$ and $M_{lin}\simeq 0$). Before annihilation, a weak hump in the curve of $M_{lin}$ indicates a weak elongation the skyrmion shape.

\[htb!\] ![ (Color online) Dependence of the effective skyrmion radius ($R_{sk}$) on curvature radius ($R$) for curved nanostrips with size $50a\times50a~(a=1nm)$. Dashed line is the $R_{sk}=R$ plot that serves as guide to the eye. Skyrmion annihilation is observed when $R_{sk} \simeq R$ ($\phi_0\simeq 150^o$). The critical curvature radius for skyrmion stability decreases with increasing value of the applied field. []{data-label="fig:Rsk"}](fig7.jpg "fig:"){width="0.95\linewidth"}

To study the evolution of skyrmion size with curvature we compute the effective skyrmion radius , with $R_g=\sqrt{\mu_{20}+\mu_{02}}$ the radius of gyration of the skyrmion region $S'$. We define $S'$ as the compact region of the nanostrip with local magnetization less than the saturation value ($m_{i,x} < 0.98$ for $B_x > 0$) and topological charge $Q>0.5$. For a circular region (disk), obviously $R_{eff}$ equals the disk radius, while for an ellipsoidal region $R_{min}<R_{eff}<R_{max}$. In Fig.\[fig:Rsk\] we show the dependence of skyrmion radius on curvature radius for the same nanostrips as in Fig.\[fig:Sk\_MM\]. Starting from the planar limit ($R\gg a$), we notice that $R_{sk}$ remains constant as $R$ decreases up to the point that the two radii become approximately equal. Then a sudden drop of $R_{sk}$ indicates the skyrmion instability and its annihilation. This behavior is also observed for higher field values ($h/J=0.030, 0.035$), where the skyrmion radius is slightly reduced. The stability of the skyrmion phase when $R \gtrsim R_{eff}$ is consistent with what is shown in Fig.\[fig:Q\_vs\_R\] regarding the evolution of the topological charge with nanotube radius. Again in that case, the sudden increase of the curve $Q(R)$, indicating the appearance of the skyrmion phase, occurs when $R\simeq p \simeq ~ R_{sk}$ Seen from a general point of view the curvature radius is a geometrical length scale and the skyrmion radius a physical length scale. Skyrmion stability is established in planar nanostrips where $R/R_{sk}\gg1$ and the stability condition is violated when the two length scales become comparable, in other words when $R/R_{sk} \simeq 1$. This geometrical argument summarizes the stability of skyrmions on curved surfaces as a matter of competition between length scales.

Zero-field skyrmions
--------------------

\[htb!\] ![ (Color online) (a)-(c) Time evolution of zero-field skyrmions in a nanoelement with size $50a\times50a~(a=1nm)$ and different curvature angles ($\phi_0=50^0,100^0,150^0$). The nanostrips are unwrapped on the $yz$-plane for visual clarity. A uniform field is applied in all cases along the $x$-axis. The color code indicates the values of magnetization along the field direction. (d) Time evolution of the topological charge ($Q$) after switching off the applied field ($t=0$) []{data-label="fig:Sk_t"}](fig8a.jpg "fig:"){width="0.95\linewidth"} ![ (Color online) (a)-(c) Time evolution of zero-field skyrmions in a nanoelement with size $50a\times50a~(a=1nm)$ and different curvature angles ($\phi_0=50^0,100^0,150^0$). The nanostrips are unwrapped on the $yz$-plane for visual clarity. A uniform field is applied in all cases along the $x$-axis. The color code indicates the values of magnetization along the field direction. (d) Time evolution of the topological charge ($Q$) after switching off the applied field ($t=0$) []{data-label="fig:Sk_t"}](fig8b.jpg "fig:"){width="0.95\linewidth"}

It is well established that magnetic skyrmions can be stabilized in planar nanoelements of circular shape (dots) in absence of an applied field. These are referred to as zero-field skyrmions. We examine here the possibility of stabilizing zero-field skyrmions in nanoelements that deviate from the planar shape. The size of the nanoelement and the skyrmion pitch are chosen, as in the previous section, such that a single skyrmion is stabilized in the nanoelement. We field cool the system to a very low temperature and at the end of the cooling process we switch off the magnetic field and study the time evolution of the system by recording the magnetization configuration and the topological charge values. Results of the zero-field relaxation of the topological charge are shown in Fig.\[fig:Sk\_t\], where the observation time after reaching the ground state and switching off the field has been about 10 times longer (MCSS=$10^5$) than the relaxation time used during the field-cooling process (MCSS=$10^4$).

Distinct behaviors are recored for systems with different degree of curvature. In case of planar nanoelements the topological charge remains almost constant in time indicating the stability of skyrmion at zero field. In systems with small curvature angle ($\phi_0 \simeq 100^0$), the skyrmion is still stable, however, its size increases slightly in the absence of a magnetic field. Increase of skyrmion radius at zero field relative to the non-zero field case is expected on physical grounds, because the Zeeman energy favors ferromagnetic order in expense of moments misalignement within the skyrmion region. However, as seen in Fig.\[fig:Sk\_t\]a, the curvature of the nanoelement enhances this effect. The weak increase of the topological charge from $Q\simeq 0.8$ to $Q\simeq 1.2$ that accompanies the increase in size of the zero-field skyrmion ($\phi_0 \lesssim 100^0$) is understood as an outcome of thermal fluctuations and misalignment of the moments along the free boundaries.[@roh13] For larger curvature angles ($\phi_0 \simeq 150^0$) the skyrmion becomes unstable at zero field and it gradually transforms to a stripe-like structure. This behavior is characterized by decreasing values of the topological charge with time. Interestingly, in case of planar nanoelemets the stabilization of zero-field skyrmions is attributed to the presence of free boundaries that repel the skyrmion. It becomes clear form Fig.\[fig:Sk\_t\] that the same argument holds in the case of curved nanoelements provided the curvature angle remains below a characteristic angle ($\phi_0~100^0$) that corresponds to a curvature radius ($R=L/\phi_0$) comparable to the skyrmion radius($R\sim R_{sk})$.

Conclusion and Discussion
=========================

We have studied the influence of curvature on the stabilization of Néel skyrmions in thin nanostructures with cylindrical shape and competing exchange and Dzyaloshinskii-Moriya interactions. We showed that application of a uniform magnetic field normal to the cylinder axis is adequate to stabilize the skyrmions. A geometrical criterion for the stabilization of skyrmions is shown to be the curvature radius of the surface to be at least of the size of the skyrmion radius ($R\gtrsim R_{Sk}$). Similarly, zero-field skyrmions can also be stabilized on cylindrical nanoelements, provided the above geometrical criterion is satisfied. With increasing curvature of the magnetic surface a transformation from a purely skyrmion phase to a mixed skyrmion-stripe phase occurs. The appealing fact is that the two phases are spatially separated. Skyrmions form on the ridge of the curved surface, namely a zone parallel to the cylinder axis where the external field is normal or almost normal to the surface and stripes form on the lateral side of the surface, where the magnetic field is parallel or almost parallel to the surface. Our study showed the feasibility of stabilizing skyrmions on nanotubes. In particular, a core/shell magnetic nanowire with heavy metal core and thin transition metal shell could be candidate physical systems to support interface skyrmions in the shell layer. Alternatively, nanotubes of a B20 material are also expected to support Bloch skyrmions in the surface with a similar physical behavior as the Néel skyrmions studied here. The spatial separation of skyrmions from stripes in the thin ferromagnetic cylindrical shell layer is anticipated to bring new perspectives in current-driven dynamics of skyrmions in curved nanostructures, since the applied magnetic field provides the required confining energy barrier that keeps skyrmions along the ridge of the nanotube and prohibits boundary annihilation due to the Skyrmion Hall effect. We hope that our results will stimulate further experimental work in the field of spintronics with magnetic skyrmions in nanowires and nanotubes.

Acknowledgments {#acknowledgments .unnumbered}
===============

The authors (DK and LT) acknowledge financial support by the Special Account for Research of ASPETE through project *NanoSky* (No 80146). AP is co-financed by Greece and the European Union (European Social Fund- ESF) through the Operational Programme «Human Resources Development, Education and Lifelong Learning» in the context of the project “Strengthening Human Resources Research Potential via Doctorate Research” (MIS-5000432), implemented by the State Scholarships Foundation (IKY).

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