Direct Observation of Attractive Skyrmions and Skyrmion Clusters in the Cubic Helimagnet Cu$_2$OSeO$_3$

TL;DR

This paper reports the direct observation of attractive skyrmions and their clusters in the cubic helimagnet Cu$_2$OSeO$_3$, revealing a new type of three-dimensional magnetic solitons with potential implications for magnetic materials.

Contribution

It provides the first real-space images of attractive skyrmion clusters in Cu$_2$OSeO$_3$ and introduces a phenomenological theory describing these non-standard skyrmions.

Findings

Observation of attractive skyrmions in Cu$_2$OSeO$_3$

Visualization of skyrmion clusters via transmission electron microscopy

Development of a theoretical model for these skyrmions

Abstract

We report the discovery of attractive magnetic skyrmions and their clusters in non-centrosymmetric ferromagnets. These three-dimensional solitons have been predicted to exist in the cone phase of chiral ferromagnets (J. Phys: Condens. Matter 28 (2016) 35LT01) and are fundamentally different from the more common repulsive axisymmetric skyrmions that occur in the magnetically saturated state. We present real-space images of these skyrmion clusters in thin (~70 nm) single-crystal samples of Cu$_2$OSeO$_3$ taken using transmission electron microscopy and develop a phenomenological theory describing this type of skyrmion.

Equations

$$w=A(\mathbf{grad}\mathbf{M}{)}^2+K(\mathbf{M}\cdot \mathbf{n}{)}^2-{\mu }_0\mathbf{M}\cdot \mathbf{H}+w_D,$$

$${\theta }_{\rho \to \infty }={\theta }_c=\arccos \left(H/H_C\right),{\psi }_{\infty }={\psi }_c=2\pi z/L_D,$$

$$e(\rho )=(2\pi L_D{)}^{-1}{\int }_0^{L_D}dz{\int }_0^{2\pi }d\phi w_s(\theta ,\psi )$$

Full text

I Introduction

In magnetic compounds lacking inversion symmetry, the handedness of the underlying crystal structure induces an asymmetric exchange coupling called the Dzyaloshinskii-Moriya (DM) interaction Dzyaloshinskii (1964) which stabilizes long-period spatial modulations of the magnetization with a fixed rotation sense Dzyaloshinskii (1964); Bak and Jensen (1980). There has been a renewed interest in chiral helimagnetism in the last few years since the discovery of two-dimensional localized modulations called chiral skyrmions Bogdanov and Yablonsky (1989); Bogdanov and Hubert (1994); Romming et al. (2013); Leonov et al. (2016a).

In most nonlinear physical systems, similar two-dimensional localized states are radially unstable and collapse spontaneously into linear singularities. Chiral interactions provide a unique stabilization mechanism, protecting two-dimensional localized states from this instability Bogdanov and Hubert (1994); Leonov et al. (2016a). That is why non-centrosymmetric magnets and other chiral condensed matter systems are of special interest in fundamental physics and mathematics as a particular class of materials where skyrmions can exist Rößler et al. (2006); Melcher (2014). Chiral magnetic skyrmions are also considered promising objects for various spintronic applications, notably racetrack computer memories Kiselev et al. (2011); Fert et al. (2013); Parkin and Yang (2015).

The generic phase diagram for the magnetic states which can occur in in non-centrosymmetric magnets is shown in Fig. 1(d). Skyrmions occur as lattices or as isolated particles within a different magnetic phase. The chiral skyrmions that were theoretically introduced in Refs. Bogdanov and Yablonsky, 1989; Bogdanov and Hubert, 1994; Bogdanov, 1995 and experimentally investigated in nanolayers of chiral ferromagnets Romming et al. (2013); Leonov et al. (2016a) were embedded in the magnetically saturated phase in which all the atomic spins are parallel. Recently, fundamentally different solitonic states embedded in the cone phase of chiral ferromagnets have been investigated mathematically Leonov et al. (2016b). Unlike the axisymmetric skyrmions, these three-dimensional chiral solitons are inhomogeneous along their axes and asymmetric in the basal planes as shown in Fig. 1(a-c)Leonov et al. (2016b). Whereas skyrmions embedded in the saturated phase repel one another, these skyrmions are mutually attractive and so tend to produce clusters of skyrmions Leonov et al. (2016b, c).

Non-axisymmetric skyrmions have been investigated theoretically in bulk Leonov et al. (2016b) and confined chiral helimagnets Leonov et al. (2016c) but no experimental observations of these states have been reported to date. Here we show the first observation of attractive skyrmions in the cone phase of the cubic helimagnet Cu2OSeO3 by presenting transmission electron micrographs of skyrmion clusters and provide a theoretical description of these results.

II Theory

Chiral solitons and modulated phases are described mathematically by equations which minimize the energy functional for a chiral ferromagnet Dzyaloshinskii (1964); Leonov et al. (2016a):

[TABLE]

where $\mathbf{M}$ is the magnetization $\mathbf{M}=M(\sin\theta\cos\psi;\sin\theta\sin\psi;\cos\theta)$ with $\theta$ being the polar angle and $\psi$ being the azimuthal angle between each magnetic moment and the applied magnetic field $\mathbf{H}$ which points in the $z$ direction. $A$ is the exchange stiffness constant, $K$ is the uniaxial anisotropy constant and $\mathbf{n}$ is the unity vector along uniaxial anisotropy axis. The DM energy functionals $w_{D}$ are composed of Lifshitz invariants $\mathcal{L}^{(k)}_{i,j}=M_{i}\partial M_{j}/\partial x_{k}-M_{j}\partial M_{i}/\partial x_{k}$ Dzyaloshinskii (1964); Bogdanov and Yablonsky (1989) where $x$ is the spatial coordinate.

Denoting the distance from the skyrmion axis as $\rho$, skyrmions in the cone phase approach the solutions for the cone phase Butenko et al. (2010); Wilson et al. (2014) ($\theta_{c},\psi_{c}$) at large distances from the axis Leonov et al. (2016a):

[TABLE]

where $H_{C}=H_{D}\left(1-K/K_{0}\right)$ is the magnetic field above which the saturated state forms and $H_{D}=D^{2}M/(2A)$. The pitch of the cone phase is $L_{D}=4\pi A/|D|$ and $K_{0}=D^{2}/(4A)$ where $D$ is the Dzyaloshinskii-Moriya (DM) coupling energy.

For the important case of non-centrosymmetric cubic ferromagnets (with space group P213), the DM energy functional is reduced to the isotropic form Bak and Jensen (1980): $w_{D}=D(\mathcal{L}_{yx}^{(z)}+\mathcal{L}_{xz}^{(y)}+\mathcal{L}_{zy}^{(x)})=D\,\mathbf{M}\cdot\mathrm{rot}\mathbf{M}$. The calculated phase diagram for this case is representative for a whole class of non-centrosymmetric ferromagnets and shown in Fig. 1(d) Butenko et al. (2010). In this phase diagram, axisymmetric skyrmions exist as ensembles of weakly repulsive strings within the magnetically saturated phase and non-axisymmetric skyrmions inhabit the cone phase. During the phase transition from the saturated to the cone phase through the critical line $H_{C}$ (Fig. 1 (d)), axisymmetric skyrmions continuously transform into non-axisymmetric skyrmions. The structure of these non-axisymmetric skyrmions is discussed in detail in Supplemental Information I sup and unlike the axisymmetric skyrmions, the point at which $\theta=\pi$ is no longer lies on the central axis of the skyrmion but its position oscillates on moving in $z$ from one layer to the next. Experimentally, we produced individual non-axisymmetric skyrmions and their clusters by starting in the skyrmion lattice phase and increasing the magnetic field so the lattice fragments during the first-order transition to the cone phase.

III Experimental Methods

Copper oxy-selenite (Cu2OSeO3) is a cubic helimagnet with a characteristic helical period Seki et al. (2012) of 63 nm. Ab-initio calculations show that this helimagnetism is induced by the DM interaction Yang et al. (2012). To acquire images of skyrmions, a single crystal was thinned to electron transparency (about 70 nm) by the conventional process of mechanical grinding and argon-ion polishing on the (110) face. Lorentz transmission electron microscopy (LTEM) was conducted using an FEI Tecnai F20 electron microscope and the sample cooled in-situ using a liquid-helium cooled holder with a base temperature of 10 K. Skyrmions appear as black or white circles in the images produced using this technique. The images also show white linear features which are not magnetic but are parts of the specimen surface which had peeled off and rolled into tiny tubes. We have not encountered this with the preparation of other materials. (See Supplemental Information II for full experimental methods sup .)

IV Experimental Results

Fig. 2 shows frames from one of several videos we acquired (see Supplemental Information III sup ) showing two skyrmion clusters (outlined in red) embedded in a host phase that produced no contrast in the image. At lower fields, the skyrmion lattice filled the whole sample. The smaller cluster contained 7 skyrmions and the larger, 13. There does not appear to be a preferred number of skyrmions in a cluster and in other videos we observed a single skyrmion as well as clusters with 6, 18 and 21 skyrmions. The fact that the skyrmions form clusters demonstrates that the interaction between them must be attractive and later we show theoretically that skyrmions embedded in the cone phase can be attractive whereas skyrmions embedded in the saturated state are purely repulsive.

Figs. 3(a)–(c) were taken under the same conditions from a different region of the sample and show a cluster of 30 skyrmions moving across the host phase and merging with the skyrmion lattice phase over 21 s. The boundaries of the cluster and the edge of the skyrmion lattice phase are delineated by red lines and by comparing panels (b) and (c), it can be seen that the phase boundary advances after merging and that the skyrmions in the cluster have spread evenly across the boundary.

The host phase appears featureless in the images so it is either the cone or saturated state since neither produce any contrast variation. Whilst the image alone does not allow us to identify the host phase, it is almost certainly the cone phase. It is clear from Figs. 3(a)–(c) that the experimental conditions are close to the phase boundary of the skyrmion lattice. Both the phase diagram for bulk Cu2OSeO3 derived by neutron diffraction Adams et al. (2012) and the theoretical phase diagram for thin cubic helimagnets in Fig. 3(d) Leonov et al. (2016d) indicate that if the material is initially in the skyrmion lattice phase, an increase in magnetic field will first create the cone phase and any coexistence should be between these two phases. Furthermore, coexisting phases like those we see here can only result from a first order transition whereas a transition from the skyrmion lattice to the saturated state should be a second order process which occurs via the gradual expansion of the period of the skyrmion lattice Bogdanov and Hubert (1994); Leonov et al. (2016d).

In all of the videos we recorded, the skyrmions were in constant motion as shown in Fig. 2(b)–(d). Similar motion has been reported by Mochizuki et al. Mochizuki et al. (2014) who attribute this to the heating of the specimen by the electron beam. We suggest instead that it may be caused by the specimen charging under the electron beam as coating the sample in a thin layer of carbon to improve its electrical conductivity slowed the movement of the skyrmions. Such charging can occur if the sample is insulating, like Cu2OSeO3, or not well grounded. Mochizuki et al. calculated that a steady flow of electrons through the sample from the electron beam was three orders of magnitude too low to move skyrmions via the spin-torque effect but it is possible that bursts of current caused by the specimen charging and discharging may be sufficient to cause this movement.

Areas with the cone and skyrmion lattice phases are separated by domain walls. The calculated contour plots of such a domain wall is presented in Fig. 3(e). The frontal parts of skyrmion cores in the wall have a similar structure as those in non-axisymmetric skyrmions and play the role of nuclei for individual non-axisymmetric skyrmions. The attractive interaction between such skyrmions Leonov et al. (2016b) explains the formation of skyrmion clusters observed in our experiments.

V Discussion

A non-axisymmetric skyrmion can be thought as the result of a continuous transformation of an axisymmetric skyrmion during the phase transition from the saturated to the cone phase (Fig. 1(d)). The equilibrium structure of a non-axisymmetric skyrmion is reached by the compatibility of the axisymmetric skyrmion core with the transversely modulated cone phase (Fig. 1(a-c))  Leonov et al. (2016b, c). The skyrmion core is separated from the host phase by a broad asymmetric “shell”.

The numerical calculations presented in Fig. 4 elucidate the main features of non-axisymmetric skyrmions and their bound states. The calculated radial skyrmion energy densities

[TABLE]

are plotted as functions $\Delta e(\rho)=(e(\rho)-e_{\mathrm{cone}})$ for different values of the applied field (Fig. 4) where $w_{s}(\theta,\psi)$ is the energy density (Eqn. 1) for an isolated non-axisymmetric skyrmion and $e_{\mathrm{cone}}$ is energy density (Eqn. 3) calculated for the cone phase (Eqn. 2). It should be noted that as the host cone phase and embedded non-axisymmetric skyrmions are modulated along the film thickness $L$, the skyrmion energy density (Eqn. 3 and Fig. 4(e)) and the inter-skyrmion coupling depend on the confinement ratio, $\nu=L/L_{D}$. These subtle effects could be a subject of future theoretical and experimental investigations.

The characteristic lengths $R_{1}$, $R_{2}$, $R_{3}$ indicate several distinct regions in the radial energy density profiles $\Delta e(\rho)$ (Fig. 4(c)–(e)). The functions $R_{i}(h)$ are plotted in Fig. 4(d). For axisymmetric skyrmions (Fig. 4(c)), the energy densities $\Delta e(\rho)$ consist of a positive energy “bag” located in the skyrmion center ($\rho<R_{2}$) surrounded by extended areas with negative energy density, where the DM coupling dominates Bogdanov and Hubert (1994). Negative asymptotics of the radial energy densities ($\Delta e(\rho)<0$ for $\rho>>1$ predetermines the repulsive inter-soliton potential for axisymmetric skyrmions Bogdanov and Hubert (1994). For non-axisymmetric skyrmions (Fig. 4(a) and (b)) the energy densities $\Delta e(\rho)$ are positive at large distances from the skyrmion center ($\rho>R_{3}$). These areas correspond to “shells” separating the skyrmion core from the cone phase. The positive energy density of the shell leads to attractive interactions between non-axisymmetric skyrmions Leonov et al. (2016b). In other words, the attractive interaction between skyrmions in the cone phase is explained by the excessive energy density of the asymmetric shell compared with the skyrmion core and the cone phase Leonov et al. (2016b). Importantly, the equilibrium distribution of the magnetization in the shell determines the material parameters of the cluster: the bound energy and the distance between the constituent skyrmions.

Another type of attractive chiral skyrmion has been investigated theoretically in the precursor region of cubic helimagnets Wilhelm et al. (2011, 2012). It was shown that due to the “softening” of the magnetization near the ordering temperature, the skyrmion-skyrmion coupling acquires an oscillatory character and promotes the formation of skyrmion clusters in this region.

The solutions for two-dimensional “baby” skyrmions with an oscillatory inter-particle potential have also been derived within the canonical Skyrme model Leonov and Mostovoy (2015). In magnetism the Skyrme model is applied to describe a group of magnetic compounds with competing exchange interactions. Ab-initio calculations of attractive two-dimensional localized states in these magnetic systems have been carried out by Rozsa et al. Rózsa et al. (2016). It was also found that the solutions for attractive baby skyrmions exist in modified Faddeev-Skyrme models with a Lennard-Jones type potential term describing a short-range repulsion and a long-range attraction Salmi and Sutcliffe (2015).

The oscillatory vortex-vortex interaction attributed to type-II superconductors with small values of the Ginzburg-Landau parameter Hubert (1971) leads to the first-order transition from the superconducting state into the Abrikosov vortex phase accompanied with the multidomain states of the competing phases Essmann (1971). Vortex clusters stabilized by the attractive inter-vortex coupling have been also observed in MgB2 and Sr2RuO4 Moshchalkov et al. (2009); Gutierrez et al. (2012); Curran et al. (2011); Garaud et al. (2014). The attractive skyrmions in the cone phase of non-centrosymmetric ferromagnets represent an alternative to solitons with the oscillatory inter-particle potential investigated in Refs. Wilhelm et al., 2011, 2012; Leonov and Mostovoy, 2015; Rózsa et al., 2016; Hubert, 1971; Essmann, 1971.

In conclusion, we report the first direct observations of clusters of attractive skyrmions embedded in the cone phase of a non-centrosymmetric ferromagnet. The clusters were generated by the magnetic-field-induced fragmentation of the skyrmion lattice during the first-order transition to the cone phase. This investigation used Cu2OSeO3 but the same method could be used to investigate skyrmion clusters in the cone phases of other non-centrosymmetric ferromagnets.

Acknowledgements.

The authors are grateful to E. Babaev, K. Inoue, D. McGrouther, T. Monchesky and Y. Togawa for useful discussions. This work was funded by the Royal Society (United Kingdom) and the United Kingdom Engineering and Physical Sciences Research Council (EPSRC), grant number EP/N032128/1. M.C.H. and G.B. also acknowledge financial support from the EPSRC grant number EP/M028771/1. A.O.L. acknowledges the Japan Society for the Promotion of Science (JSPS) Core-to-Core Program, Advanced Research Networks and the JSPS Grant-in-Aid for Research Activity Start-up 17H06889. A.O.L. thanks Ulrike Nitzsche for technical assistance.