Topologically stable magnetization states on a spherical shell: curvature stabilized skyrmions
Topologically stable structures include vortices in a wide variety of matter, such as skyrmions in ferro- and antiferromagnets, and hedgehog point defects in liquid crystals and ferromagnets. These are characterized by integer-valued topological quantum numbers. In this context, closed surfaces are a prominent subject of study as they form a link between fundamental mathematical theorems and real physical systems. Here we perform an analysis on the topology and stability of equilibrium magnetization states for a thin spherical shell with easy-axis anisotropy in normal directions. Skyrmion solutions are found for a range of parameters. These magnetic skyrmions on a spherical shell have two distinct differences compared to their planar counterpart: (i) they are topologically trivial, and (ii) can be stabilized by curvature effects, even when Dzyaloshinskii-Moriya interactions are absent. Due to its specific topological nature a skyrmion on a spherical shell can be simply induced by a uniform external magnetic field.
pacs:75.10.Hk, 75.10.Pq, 75.40.Mg, 75.60.Ch, 75.78.Cd, 75.78.Fg
Topological methods are increasingly used to describe observed states in condensed matter systems. Prominent examples are the description of vortex textures in superfluid helium;Anderson and Toulouse (1977); Volovik (2003) band theory for topological insulators;Hasan and Kane (2010); Moore (2010); Hsieh et al. (2008) topological superconductivity in a helical Dirac gasXu et al. (2014) and in Dirac semimetals;Kobayashi and Sato (2015) and topological defects in liquid crystals,Alexander et al. (2012); Kleman and Lavrentovich (2006) ferromagnets,Thiele (1973); Belavin and Polyakov (1975); Malozemoff and Slonzewski (1979); Papanicolaou and Tomaras (1991); Komineas and Papanicolaou (1996)and antiferromagnets.Barker and Tretiakov (2016) In this context, thin curvilinear films of ordered matter are in the focus of strongly growing interest, because in these systems a nontrivial geometry can induce topological defects in the order parameter fieldBowick and Giomi (2009); Vitelli and Turner (2004); Turner et al. (2010) and can result in new effective interactions.Napoli and Vergori (2012, 2013); Gaididei et al. (2014); Sheka et al. (2015) Among curvilinear films the most promising candidates for new physical effects are closed surfaces due to the natural appearance of topological invariants in the system. In this case the normalized vector field defined on the surface realizes a map of the surface into a sphere . The degree of this map is an integer topological invariant,Mermin (1979); Thouless (1998); Dubrovin et al. (1985) i.e. each given distribution of the vector field on a closed oriented surface is characterized by an integer number which is conserved for any continuous deformation (homotopy) of the field . Moreover, any two distributions of the field are topologically equivalent (homotopic), i.e. they can be matched by means of a continuous deformation provided they have the same .Dubrovin et al. (1985); Kosevich et al. (1990); Manton and Sutcliffe (2004) Since a discontinuity in the physical field is usually energetically non-favorable, two solutions with different are separated by a high energy barrier. This causes topological stability. For example, an isolated magnetic skyrmionBogdanov and Yablonskiĭ (1989); Bogdanov and Hubert (1994a, 1999); Bogdanov and Rößler (2001); Romming et al. (2013, 2015); Büttner et al. (2015); Leonov et al. (2016) in a planar film with Dzyaloshinskii-Moriya interaction (DMI) is an excited state of the system (for the case of low temperature and absence of external magnetic fields). However, this excitation is topologically stable, because the invariant is for the skyrmion,Papanicolaou and Tomaras (1991); Komineas and Papanicolaou (2015) while for the ground state. Topological stability occurs for a variety of defects in ordered matter, such as disclination loops, hedgehog point defects and knots in nematic liquid crystals;Alexander et al. (2012); Kleman and Lavrentovich (2006); Machon and Alexander (2013); Jampani et al. (2011); Senyuk et al. (2012) and vorticesMertens and Bishop (2000) and Bloch pointsFeldtkeller (1965); Malozemoff and Slonzewski (1979) in ferromagnets.
Conservation of the topological index for a closed surface raises two fundamental questions: (i) what is the lowest energy equilibrium solution for a given , which is not necessarily the ground state owing to the topological constraint. And (ii) which corresponds to the ground state for a given surface? The answers can lead to new phenomena, specific to the physical system under consideration. In this paper we answer these questions for the case of thin ferromagnetic spherical shells. Even such a relatively simple model brings a number of surprising results.
We show that for a spherical shell a skyrmion solution exists as a topologically stable excitation above the hedgehog ground state. An important feature is that the skyrmion may be stabilized by curvature effects only, specifically by the curvature-induced, exchange-driven effective DMI.Gaididei et al. (2014); Sheka et al. (2015) This is in contrast to the planar case, where the intrinsic DMI is required for the skyrmion stabilization.Bogdanov and Hubert (1994a); Sampaio et al. (2013); Rohart and Thiaville (2013)
The case of the spherical shell is topologically opposite to that of the planar film: the skyrmion has the index , in other words it is topologically trivial, while the ground state is characterized by . This is due to a shift of the topological index of the vector field, caused by topology of the surface itself. Since the skyrmion solution on a spherical shell is homotopic to a uniform state, it can be induced by means of a uniform external magnetic field, similarly to the excitation of onion magnetic states in nanorings.Rothman et al. (2001) In a continuous medium the switching between states with different is topologically forbidden. However, in discrete spin lattices such a transition is possible, though it requires a strong external influence.
Ii General case of an arbitrary curvilinear shell
We first present a set of general results valid for an arbitrary thin curvilinear shell. In the following we apply these results to calculate the magnetic energy and topological properties of magnetization states of spherical shells.
ii.1 The mapping Jacobian
The degree of a map, realized by a normalized three-dimensional vector field defined on a two-dimensional closed surface , readsDubrovin et al. (1985) . In this particular case the mapping Jacobian can be presented in the form of the triple productDubrovin et al. (1985) , where the minus sign is introduced solely to conform with the traditional notation used in ferromagnetic research. Here and everywhere below the Greek indices numerate the curvilinear coordinates , introduced on the surface, and the vector components defined in the corresponding curvilinear local basis ; while the Latin indices numerate coordinates and vector components in the Cartesian basis . The summation over repeated dummy indices is implied, unless stated otherwise. The local basis is assumed to be orthonormal , therefore the metric tensor is diagonal. Details on the definition of the orthonormal basis for a given surface are presented in the Appendix A. The operator , where the summation over is not implied and , denotes the corresponding component of the surface del operator . The surface element reads , where .
Since using Cartesian components of the vector field is not convenient for curvilinear systems, we will switch to curvilinear coordinates , where is the surface normal. Moreover, it is useful to incorporate the constraint by means of the angular parameterization , , where and represent colatitude and longitude, the spherical angles of the local curvilinear basis, respectively. In this case one can show (see Appendix B) that
Here, , where is the normalized projection of the vector on the tangential plane and is a tensor known as the Weingarten map or modified second fundamental form.Kamien (2002) Vector denotes the spin connection and is the Gauß curvature. The corresponding definitions are presented in the Appendix A.
One can easily check that for the case of a plane with a Cartesian frame of reference the expression (1) results in the well knownThiele (1973); Belavin and Polyakov (1975); Malozemoff and Slonzewski (1979); Papanicolaou and Tomaras (1991); Komineas and Papanicolaou (1996) formula .
Remarkably, for a strictly normal distribution of the vector field (normal Gauß map) one obtains the well knownKamien (2002); Dubrovin et al. (1985) result . Applying the Gauß-Bonnet theorem we obtain the famous relation between degree of the normal Gauß map and genus of the surface. Thus, for a normally magnetized sphere (hedgehog), for a normally magnetized torus, etc. In a topological classification of the solutions the value should be taken into account as a topological charge shift, which originates from the topology of the surface itself. To establish a link with the well-known skyrmions in the planar geometryBogdanov and Yablonskiĭ (1989); Bogdanov and Hubert (1994a, 1999); Bogdanov and Rößler (2001); Romming et al. (2013, 2015); Büttner et al. (2015); Leonov et al. (2016) one has to introduce the skyrmion number .
Note that the term “skyrmion” is used rather broadly: any localized two-dimensional structure with unit (integer) mapping degree may be considered a skyrmion. However, in addition to chiral skyrmionsBogdanov and Yablonskiĭ (1989); Bogdanov and Hubert (1994a, 1999); Bogdanov and Rößler (2001); Romming et al. (2013, 2015); Büttner et al. (2015); Leonov et al. (2016) and bubbles,Malozemoff and Slonzewski (1979) this definition includes a variety of objects with very different physical properties, such as vortex domain walls on tubes;Landeros and Núñez (2010); Yan et al. (2011); Villain-Guillot et al. (1995) hedgehog states and some vortex states on a spherical shell;Milagre and Moura-Melo (2007); Kravchuk et al. (2012) and rotating vortex dipoles.Komineas (2007)
It is instructive to introduce a narrower definition which considers skyrmions as localized solutions with the structure of a vortex.
The vector is the limit for the two-dimensional case for the gyrocoupling vectorThiele (1973); Malozemoff and Slonzewski (1979); Papanicolaou and Tomaras (1991); Komineas and Papanicolaou (1996)(topological density, topological current, vorticity) , whose Cartesian components read . The gyrocoupling vector is widely used for the topological description of a unit vector field defined in a three-dimensional domain. If the shell thickness is small enough to ensure the uniformity of along the normal direction: , then , see Appendix B. In magnetism, the gyrocoupling vector is the key quantity to describe the dynamics of topologically nontrivial solutions, such as domain walls,Thiele (1973); Malozemoff and Slonzewski (1979) vortices,Huber (1982); Papanicolaou and Tomaras (1991); Mertens and Bishop (2000) skyrmions,Komineas and Papanicolaou (2015); Lin et al. (2013a, b); Lin and Saxena (2015) skyrmion linesMilde et al. (2013); Lin and Saxena (2016) and Bloch points.Malozemoff and Slonzewski (1979); Pylypovskyi et al. (2012); Milde et al. (2013); Lin and Saxena (2016) It determines important integrals of motion in the dynamics of ferromagnetic media.Papanicolaou and Tomaras (1991); Komineas and Papanicolaou (1996) Recently it was shownSchulz et al. (2012); Lin and Saxena (2016) that is proportional to the emergent magnetic field, which appears due to the Hund’s coupling between spins of the conducting electrons and localized magnetic moments. This gives rise to the topological Hall effect.Neubauer et al. (2009); Li et al. (2013); Kanazawa et al. (2011)
Let us provide physically illustrative explanations why the topological charge or index is an integer number and a conserved quantity. A direct consequence of the definition of with the constraint is , where is the Dirac delta-function and the vector determines the position of a Bloch point (monopole), , whose infinitesimal neighborhood of the center has the structureMalozemoff and Slonzewski (1979) . Here, is an arbitrary matrix of three-dimensional rotations and is the monopole charge. Thus, the monopoles are sources and sinks of the gyrovector field.Malozemoff and Slonzewski (1979) Likewise, electrical charges are sources and sinks of the electrical field. For any closed surface enclosing the volume , the integral yields an integer number equal to the difference of negatively and positively charged monopoles inside . Since two monopoles with opposite charges are connected by the Dirac string, which may be considered a skyrmion line,Milde et al. (2013); Lin and Saxena (2016) one can also say that is the difference of outgoing and incoming skyrmion lines.Milde et al. (2013) Thus the only way to change for a given closed is to replace a monopole across . When the monopole center crosses the surface, i.e. it is located exactly on , the vector field is discontinuous at the point . Thus, one can conclude that a continuous deformation of the continuous field can not change the mapping degree of . The rigorous proof of the latter statement can be found elsewhere, for instance see Ref. Dubrovin et al., 1985.
The expression for the mapping Jacobian (1) is general: it is the key formula for topological analysis of a normalized vector field of an arbitrary physical nature on an arbitrary curvilinear surface.
ii.2 Magnetic energy of a curvilinear shell
The topological analysis is independent of the physical nature of the vector field . In the following, we focus on possible equilibrium magnetization states of thin ferromagnetic curvilinear shells. To this end, we introduce the energy functional . Here, we take into account three magnetic interactions. The first term of the integrand represents the exchange energy with the energy density and the exchange constant, . The second term is a uniaxial anisotropy: easy-normal for or easy-surface for . The presence of this anisotropy, which conforms to the geometry, is crucial for our model. The anisotropy forces spins to follow the geometry which is why the spin subsystem ultimately “fills” the geometry. This is a fundamental difference between our approach and a number of previous studies, where soliton solutions were found on curvilinear shells, yet anisotropy was either neglected,Villain-Guillot et al. (1995); Carvalho-Santos and Dandoloff (2012); Priscila S.C. Vilas-Boas (2015); Carvalho-Santos et al. (2015) or it was spatially uniform lacking any correlation with the geometry.Milagre and Moura-Melo (2007); Carvalho-Santos et al. (2008, 2013) Our approach is based on the fundamental behavior of magnetically ordered media, where spin-orbit couplings provide the vital link between nontrivial curved geometry and the spin-system. Therefore, any realistic assessment of possible magnetization states in curved geometries must include the geometrically allowed anisotropic couplings.
The last term in is the DMI with the energy densityCrépieux and Lacroix (1998); Bogdanov and Rößler (2001); Thiaville et al. (2012) and the DMI constant, . This kind of DMI originates from the spin-orbit coupling and is related to the inversion symmetry breaking on the film interface; it is typical for ultrathin filmsCrépieux and Lacroix (1998); Bogdanov and Rößler (2001); Thiaville et al. (2012) or bilayers.Yang et al. (2015) In the curvilinear basis one can represent the DMI density as follows
In our model we assume that the magnetostatic interaction, which is always present in the system, can be reduced to the easy-surface anisotropy, resulting in the shift of the anisotropy coefficient . This was rigorously demonstratedGioia and James (1997); Kohn and Slastikov (2005a, b) for plane films, when thickness is substantially smaller than the size of the system and . Here, we assume that the same model is sufficient for smoothly curved shells, if is much smaller than the curvature radius.Slastikov (2005)
Iii Case of a spherical shell
As the simplest example we consider a thin spherical shell with radius . For the case of easy-normal anisotropy () there exists a class of azimuthally symmetric solutions , see Appendix D. The basis vector points, tangential to the surface, towards the direction of increasing polar angle and is the outward normal. The function satisfies the following equation
Here, is the characteristic magnetic length and is the strength of the curvature-induced effective DMI that is solely exchange-driven.Sheka et al. (2015) This geometrical DMI contribution competes with the intrinsic spin-orbit-driven DMI. Full compensation takes place when .
In the limit the equation (3) is transformed
There are two kinds of boundary conditions (BC) possible for Eq. (3), namely (i) , , and (ii) , . Here, , the helicity number, is formally a winding number of the magnetization along a circle loop passing through both pole points and . Using the helicity number one can introduce the chirality of the structure: . Thus, the skyrmions shown in Fig. 1A and A have the chiralities and , respectively.
From the general expression for the gyrocoupling vector (1), it follows that the mapping index for an azimuthally symmetrical solution is
which implies that for the mentioned class of solutions. It is interesting to note that a one-dimensional magnetization in the planar case, , results in . However, in the case of a spherical shell a solution with is possible even if depends on one coordinate only. According to (4) an even results in for both kinds of BC and an odd results in and for BC of type (i) and (ii), respectively. Note that, in contrast to the mapping degree , the helicity number is not a topological invariant: we merely use it for the classification of solutions. Any two solutions with different but with the same belong to the same homotopy class and they can be transformed into each other by means of a continuous deformation of the vector field .
Any two solutions of (3), obtained under different kinds of BC but for the same , differ by sign only: . The latter transformation does not change the energy of the system, as the energy functional , in the absence of external fields, is quadratic with respect to components of vector . However, it changes the sign of because the mapping Jacobian is cubic in the magnetization. Thus a state with given is doubly degenerate with respect to replacements and .
iii.1 Topologically trivial case : skyrmion solutions
Equation (3) can have skyrmion solutions for the cases and , see Fig. 1A and Fig. 1A, respectively. In contrast to the planar case, where the skyrmion solution has , on a spherical shell the skyrmion is topologically trivial (). Let us define the skyrmion radius as , where . For the planar case with being distance to the skyrmion center and can be defined analogously: . In planar films skyrmions are widely studied; it is well knownBogdanov and Hubert (1994a); Kiselev et al. (2011); Rohart and Thiaville (2013) that the skyrmion radius strongly depends on the DMI constant : the skyrmion collapses, , when ; and when , see dashed line in Fig. 1. For this type of DMI, the so called hedgehog (Néel) skyrmions appear with zero azimuthal magnetization component, see Fig. 1B,B. In planar films such type of skyrmions have been predicted theoreticallyRohart and Thiaville (2013); Sampaio et al. (2013) and were observed experimentally.Romming et al. (2013, 2015); Büttner et al. (2015) The same type of skyrmions appear on a spherical shell with an analogous dependence , see the solid line in Fig. 1. There are, however, a number of new, important features:
(i) Skyrmions collapse for a finite value of the DMI constant, , and as a consequence a skyrmion of finite radius exists for the case , see point A in Fig. 1 and the corresponding inset. The shift along the axis is due to the additional curvature-induced DMI (7c), which appears as an effective term in the exchange interaction.Sheka et al. (2015)
(ii) For a given radius of the spherical shell the skyrmion exists for a certain range of the DMI constant, , see Fig. 1 and Fig. 2. Beyond this range, at , see Fig. 1, the skyrmion transforms into the 3D-onion state.
(iii) In contrast to the planar case where the function is even, the corresponding curve for spherical shells is highly asymmetrical.
Fig. 1 shows possible equilibrium states for the case , answering our introductory question (i) about the physically stable magnetization structures in topologically different sectors.
iii.2 Diagram of ground states
Though a continuous transition between solutions with and is not possible (topological stability), a solution with can have lower energy than the corresponding solution with for some range of parameters. In order to clarify this picture and answer question (ii) about the globally stable magnetization configurations, we build the diagram of the ground states for the class of azimuthally symmetrical solutions determined by Eq. (3), see Fig. 2. One can distinguish two main states: the hedgehog states with () and 3D-onion state with (). However, for large enough sphere radii and magnitudes of the DMI constant a variety of states with higher helicity numbers appear. These states can be interpreted as helical structures on a spherical shell. Similar skyrmionic structures were recently observed in disk-shaped chiral nanomagnets.Streubel et al. (2015); Beg et al. (2015)
Like the solitary skyrmion on a planar film, the skyrmion on a spherical shell does not form the magnetic ground state, yet skyrmions with and can exist as topologically stable excitations, see the dashed areas and , respectively.
The diagram of the ground states (Fig. 2) was built for the class of azimuthally symmetrical solutions . Hence, we address the question about azimuthal stability of these solutions. Performing a standard stability analysis, see Appendix E, we found a number of narrow regions, where elliptical instabilityBogdanov and Hubert (1994b) is possible. Remarkably, the instability regions are in the vicinity of boundaries which separate different magnetization states, see Fig. 2.
Iv Skyrmion formation without DMI
The results on static skyrmion state configurations on spherical shells immediately pose the problem whether and how these states can be realized. In the following, we discuss in more detail an intriguing case of the skyrmion formation, when . For this purpose, we will move along the vertical axis of the ground states diagram (Fig. 2) starting from small sphere radii . When the sphere’s radius is small enough (), the ground state of the system is topologically trivial () and close to the uniform state, we call it a 3D-onion state, see Fig. 2 and Fig. 3a. The ground state of the spheres with is one of the hedgehog states with . Due to the topological stability the 3D-onion state survives when the sphere radii . At the point the 3D-onion state becomes unstable resulting in a fork-like bifurcation, see Fig. 3b, where the lines are obtained by solving Eq. 3 with the boundary conditions , and dots correspond to the micromagnetic simulations. As a result of the bifurcation a skyrmion is formed either on the north or on the south pole of the sphere, see Fig. 3. The skyrmion exists as a topologically stable excitation of a hedgehog state for the case , see region in Figs. 2, 3a. However the radius of the skyrmion decreases rapidly when the radius of the sphere further increases, see Fig. 3b.
The observed behavior of this system can be explained as follows. The 3D-onion state can be interpreted as a skyrmion solution with the angular radius , or with lateral radius . However, it is well knownBogdanov and Hubert (1994a); Kiselev et al. (2011); Rohart and Thiaville (2013) that skyrmions on a planar film collapse when the intrinsic spin-orbit-driven DMI vanishes, i.e., in the limit . Thus, it is natural to expect the skyrmion to collapse for the case when the curvature effect vanishes. That is why the instability of the 3D-onion state appears for a certain value of and with a further increase of the radius of the sphere the skyrmion collapses to either the north or the south pole, see Fig. 3b. However, for a sphere radius skyrmions with finite radii exist, see Fig. 3b. This can be interpreted as a skyrmion stabilization due to the curvature-induced, exchange-driven effective DMI.Gaididei et al. (2014); Sheka et al. (2015) The obtained results agree very well with micromagnetic simulations based on a model where the magnetostatic interaction was reduced to the easy-surface anisotropy, see disk- and square-shaped symbols in Fig. 3b. It is important that taking into account the magnetostatic interaction does not change the physical picture, but results in an increase of the skyrmion radius, see triangle-shaped markers in Fig. 3b. The increase of the skyrmion radius appears due to the volume magnetostatic charges. Thus, the full micromagnetic simulation results vindicate our analytical approach that neglect the dipolar stray-fields to construct the diagrams of equilibrium states.
It is noteworthy that the energy difference of the skyrmion (3D-onion) and the hedgehog solutions varies in the region , where is the energy of the hedgehog state sphere with , and is the energy of the Belavin-Polyakov soliton,Belavin and Polyakov (1975) which is equal to the energy of a skyrmion with infinitesimal radius, see Fig. 3a. It is also remarkable that , which matches the energy of the vortex-antivortex pair with the opposite polarities just before their annihilation.Tretiakov and Tchernyshyov (2007); Hertel and Schneider (2006)
Finally, we demonstrate how the skyrmion configuration can be created by means of a uniform magnetic field. We consider a spherical shell, whose radius corresponds to the hedgehog ground state. By means of micromagnetic simulations (see Appendix F) we find that the adiabatically slow increase of the external uniform magnetic field results in the transition from the hedgehog state to the 3D-onion state, see Fig. 4. A subsequent decrease of the field leads to a skyrmion state formation. This transition from to is topologically forbidden for a continuous system and appears here merely due to the discretization. However, since real magnetic crystals have a discrete structure, one can expect this behavior in strong enough external magnetic fields. This mechanism of the skyrmion formation is similar to the formation of onion magnetic states in nanorings.Rothman et al. (2001)
In conclusion, we demonstrate that different types of axially symmetrical solutions of the magnetization exist for a thin ferromagnetic spherical shell. These solutions can be divided into three homotopic classes with topological index . To calculate we developed the general expression for the mapping Jacobian (1) valid for an arbitrary curvilinear shell. Skyrmion solutions are found in the topologically trivial class with . Remarkably, a skyrmion solution on a spherical shell can be stabilized by curvature effects only, namely by the curvature-induced, exchange-driven effective DMI.Gaididei et al. (2014); Sheka et al. (2015) This is in contrast to the planar case, where the spin-orbit-driven intrinsic DMI is required for the skyrmion stabilization.Bogdanov and Hubert (1994a); Kiselev et al. (2011); Rohart and Thiaville (2013) Since a skyrmion on a spherical shell is homotopic to a uniformly magnetized sphere, it can be induced by a strong uniform external magnetic field.
Experimental advances in fabrications of curvilinear nanomagnetsStreubel et al. (2016) make us optimistic in forthcoming experimental confirmation of the curvature stabilized skyrmions. Indeed, magnetic spherical nanoshells can be preparedZhang et al. (2009); Cabot et al. (2009); Gong et al. (2014) by coating of a nonmagnetic spherical core with a ferromagnetic material. The small size (10–20 nm) of the obtained particlesZhang et al. (2009); Cabot et al. (2009); Gong et al. (2014) enable one to expect discernible curvature effects. Note that spherical magnetic nanocaps with normally oriented anisotropy axis can also be created experimentally.Albrecht et al. (2005); Ulbrich et al. (2006); Makarov et al. (2008, 2009)
V.P.K. acknowledges the Alexander von Humboldt Foundation for the support and IFW Dresden for kind hospitality. This work was funded in part by the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013)/ERC grant agreement No. 306277 and the European Union Future and Emerging Technologies Programme (FET-Open Grant No. 618083). D.D.S. acknowledges Prof. Avadh Saxena for fruitful discussions.
Appendix A Introduction of the curvilinear basis
In order to formalize the geometry of the shell we use the parametric representation . Here, is the 3D position vector, which determines a 2D surface embedded in with being local curvilinear coordinates on . The unit vector denotes the surface normal and the parameter is the corresponding curvilinear coordinate along the normal direction. We restrict ourselves to the limiting case . Specifically, we assume that the thickness is much smaller than the curvature radius as well as the characteristic magnetic length . As a consequence, we assume that the magnetization is uniform along the normal direction: .
The parameterization induces the natural tangential basis with the corresponding metric tensor . Assuming that vectors are orthogonal, one can introduce the orthonormal basis , where and . Using the Gauß-Codazzi formula and Weingarten’s equationsDubrovin et al. (1984) one can obtain the following differential properties of the basis vectors
Here, matrix is a tensor, known as the Weingarten map or modified second fundamental form.Kamien (2002) The formula with being components of the second fundamental form, is practically useful. The Weingarten map determines the Gauß curvature and the mean curvature . Components of the spin connection vector are determined by the relation .
In general, the basis need not be orthogonal. However, if the vectors and are not collinear then one can always introduce the orthonormal basis in the following way: , , and , where is dual tangential basis with . However, in this case the relations (5) should be revised.
Appendix B Gyrocoupling vector in a curvilinear reference frame
We start from the Cartesian representation . Taking into account that the magnetization is uniform along the shell thickness, i.e. the normal coordinate, one can replace the del operator by its surface analogue and represent the Cartesian derivatives via its curvilinear counterparts . Simple calculations result in , where is the mapping Jacobian defined in the main text. Introducing the curvilinear magnetization components and taking into account differential properties of the curvilinear basis (5) one obtains
Notice the difference between volume and surface gyrocoupling vectors. can be introduced for a vector field defined in a 3D region . One can cut out from a curvilinear shell with small but finite thickness . If is uniform along the shell thickness (otherwise can not be introduced), then . However, if we build a 2D surface () in , then on the surface.
Appendix C Magnetic interactions on a curvilinear shell
Let us first consider the exchange interaction. Under the main assumption of magnetization uniformity along the shell thickness, one can represent the exchange energy density in its curvilinear form,
Here, is the “common” isotropic exchange, and can be treated as an effective curvature-induced DMI and anisotropy, respectively. In the angular representation the exchange energy readsGaididei et al. (2014)
Applying the same procedure for the DMI energy density one obtains
In the angular representation the density of the anisotropy energy looks particularly simple , because the anisotropy has the symmetry of the surface.
Appendix D Case of a spherical shell
In order to describe a spherical shell of radius we use the parameterization . Here, and are polar and azimuthal spherical angles, respectively. Basis vectors are , and the normal vector is directed outward of the sphere. In this case the Weingarten map is the diagonal matrix with components , and consequently , and . The spin-connection vector has only one (azimuthal) component: .
Taking into account (8) and (2) one can show that in the case of easy-normal anisotropy () the functions and , which minimize the energy functional , are a solution of the following Euler-Lagrange equations
When deriving (10) we used that for a spherical shell and . In addition, we use the following general properties of vector
Taking into account that the vector of spin connection is one can see that Eqs. (10) have a solution , . In this case Eq. (10b) turns to identity and the function can be determined as a solution of Eq. (10a). Since for the considered class of solutions, one can consider the colatitude angle as the only parameter: . In this case the function is determined by equation (3). For the axially symmetric solutions and the gyrocoupling vector (1) for the spherical shell can be written as follows
The integration over the sphere results in the topological index (4).
Appendix E Stability analysis
Our goal is to analyze the stability of azimuthally symmetrical solutions, shown in Fig. 2. For this purpose we use the parameterization . Taking into account the exchange (7), DMI (9) and anisotropy, , contributions we can construct the energy functional . This functional has an extremal for , , where the function is a solution of Eq. (3). We need to check whether the solution and corresponds to the energy minimum. For this purpose, we consider small deviations and . Now the harmonic approximation of the energy reads
Here, is stationary value of the energy functional, and the operator reads
where is the angular part of the Laplacian in the spherical reference frame. The potentials are as follows
where . One can easily check that the Euler equations with respect to small deviations and (Jacobi equation) have the solutions and , with . Introducing one can present the energy (13) in the form
is a Hermitian operator in the space of functions with the scalar product . Here, .
The solution and minimizes the energy functional iff all eigenvalues of the operator are positive for all . Note that sign of does not effect the eigenvalues of the operator (17). For a given pair of parameters we found numerically a set of eigenvalues of operator (17) for the range and for the fixed boundary conditions . Using the mentioned criterion, we found some narrow instability regions in the diagram Fig. 2 in the vicinity of boundaries between states with different helicity numbers . These instability are found there only for modes , thus following Ref. Bogdanov and Hubert, 1994b, we call it elliptical instability.
Appendix F Micromagnetic simulations
In order to verify our analytical results we performed micromagnetic simulations with the FinMag code, which is the successor to the Nmag tool.Fischbacher et al. (2007) We used the material parameters of cobalt: J/m, A/m and J/m, which are typical for Pt/Co/AlO layer structures.Rohart and Thiaville (2013) For comparison, we also performed a simulation neglecting the magnetostatic interaction, assuming that it can be reduced to an effective easy-surface anisotropy, we used J/m. These parameters correspond to nm. In all simulations the ratio is kept constant. The size of the discretization mesh is .
To verify the dependence shown in Fig. 1 we simulate the shell with radius nm, thickness nm and an average mesh size of 0.42 nm. Since the mesh discreteness breaks the topological stability, we are not able to obtain the skyrmions with small radii . The same geometrical parameters are used in simulations for Fig. 2.
To simulate the formation of skyrmions by means of a uniform field, see Fig. 4, we consider a spherical shell with radius nm and thickness nm. The applied magnetic field is increased from zero up to the value 1.5 T with a rate of 330 mT/ns. The field is then decreased back to zero with rate 82 mT/ns. In order to break the symmetry and avoid the unstable equilibrium state we introduce a radial field with a small amplitude mT. Fields in Fig. 4 are normalized to the value T. In this numerical experiment we reduce magnetostatics to the effective easy-surface anisotropy.
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