Eel-type Skyrmion Research Articles

Giant resonances in topological spin Hall effect due to electron-skyrmion scattering in two-dimensional Rashba spin-orbit ferromagnets

We study a topological spin Hall effect where conduction electrons are scattered by N\'eel-type skyrmions in two-dimensional Rashba spin-orbit ferromagnets. We find the resonance structures in direct and Hall electric conductivities. The resonances strongly depend on the skyrmion size, the Rashba spin-orbit coupling vector value, and the relative direction between magnetization and Rashba vectors. The antiparallel arrangement of the Rashba spin-orbit coupling vector with respect to the magnetization determines the resonance structure and increases the direct conductivity by about two orders of magnitude. For the parallel arrangement the electric conductivities decrease by one to three orders of magnitude and the Hall conductivity changes the sign. The Rashba spin-orbit coupling also modifies electron bands depending on the ratio ${\ensuremath{\varepsilon}}_{R}/J$, where ${\ensuremath{\varepsilon}}_{R}$ is Rashba energy and $J$ is an exchange integral between conduction and localized electrons. If ${\ensuremath{\varepsilon}}_{R}/J\ensuremath{\le}1$, each energy band has the single minimum while for ${\ensuremath{\varepsilon}}_{R}/J>1$ the lower band has the ``Mexican-hat'' shape with the maximum at $k=0$. We focus on the dependencies of direct and Hall electric conductivities on Fermi energy, ${\ensuremath{\varepsilon}}_{F}$, skyrmion sizes, and Rashba spin-orbit coupling constant values and its relative direction with the magnetization. Analyzing both types of the energy bands, in general, we find two types of resonances: at the minimum of the upper band and behaviors, we employ the scattering pattern analysis. The resonance dependencies on skyrmion sizes and Rashba, a spin-orbit coupling contestant, can be qualitatively explained in terms of the Ramsauer-Townsend scattering of the upper band electrons by the skyrmion quantum well. The resonance properties can be used in spin transistors. To discover the resonances it is necessary to know the specific range of the parameters. At some values of the parameters the electric conductivity changes by about two orders of magnitude in the narrow range of ${\ensuremath{\varepsilon}}_{F}$ ($<0.01$ eV). To detect the resonances it is also important to identify the relative direction of the Rashba spin-orbit coupling vector with respect to the magnetization.

Manipulation of topological spin configuration via tailoring thickness in van der Waals ferromagnetic Fe5−xGeTe2

Topological spin textures with versatile configurations exhibit fascinating physical behavior as high-efficiency information units. Recent observations of nontrivial spin textures in two-dimensional (2D) van der Waals (vdW) ferromagnets have shed light on their functionality as a paradigm for spintronic devices. Here, the spin configuration of single-crystal 2D vdW ${\mathrm{Fe}}_{5\text{\ensuremath{-}}x}\mathrm{Ge}{\mathrm{Te}}_{2}$ is investigated by Lorentz transmission electron microscopy, where the conversion from Bloch-type bubbles to N\'eel-type skyrmions can be observed by tuning sample thickness. Remarkably, high-density N\'eel-type skyrmions can be obtained via field-cooling manipulation. We further demonstrate the underlying mechanism by varying the sample thickness and Dzyaloshinskii-Moriya interaction (DMI) parameters in a series of micromagnetic simulations, manifesting the presence of DMI at the ${\mathrm{Fe}}_{5\text{\ensuremath{-}}x}\mathrm{Ge}{\mathrm{Te}}_{2}$ surface by first-principles calculation. Our findings disclose the rich variety of topological spin textures in 2D vdW ${\mathrm{Fe}}_{5\text{\ensuremath{-}}x}\mathrm{Ge}{\mathrm{Te}}_{2}$, possessing great promise for future nonvolatile memories based on spin topology.

Manipulating density of magnetic skyrmions via multilayer repetition and thermal annealing

The magnetic skyrmion, a tiny magnetic texture that holds promise as the next-generation information carrier, has been widely studied in recent years. A fine tunability of skyrmion density is required for its real applications in novel memory and logic devices. Here, we report on the manipulation of skyrmion density at room temperature in a Pt/Co/Ta/MgO system composed of multiple repetitions with perpendicular magnetic anisotropy. N\'eel-type skyrmions are observed by performing Lorentz transmission electron microscopy and magnetic force microscopy. The influence of structure repetition on skyrmions is experimentally investigated and further understood by micromagnetic simulations. The variation of skyrmion density with repetition number is mainly attributed to magnetic anisotropy. Meanwhile, thermal annealing is utilized to regulate the skyrmion density, where skyrmion-related properties exert a competitive effect. Our findings provide alternative means to manipulate skyrmion density, further allowing for optimized engineering of skyrmion-based devices.

Theory of Magnetic-Texture-Induced Anomalous Hall Effect on the Surface of Topological Insulators

The anomalous Hall effect is caused by magnetic textures such as skyrmions. We derive an analytical formula of the Hall conductivity on the surface of a topological insulator up to third order in magnetization, $\boldsymbol{M}(\boldsymbol{x})$, based on a perturbative approach. We identify the magnetic textures that contribute to the Hall conductivity up to third order in magnetization and second order in spatial differentiation. We treat magnetization as a perturbation to calculate the Hall conductivity for each magnetic texture based on the linear response theory. Furthermore, we estimate the skyrmion-induced Hall conductivity and confirm that it depends on the shape of skyrmions, such as Bloch-type or N\'eel-type skyrmions. The results of this study can be applied not only to conventional skyrmion systems but also to more general magnetic structures.

Noncoplanar multiple- Q spin textures by itinerant frustration: Effects of single-ion anisotropy and bond-dependent anisotropy

We theoretically investigate multiple-$Q$ spin textures, which are composed of superpositions of spin density waves with different wave numbers, for an effective spin model of centrosymmetric itinerant magnets. Our focus is on the interplay between biquadratic interactions arising from the spin-charge coupling and magnetic anisotropy caused by the spin-orbit coupling. Taking into account two types of the magnetic anisotropy, single-ion anisotropy and bond-dependent anisotropy, we elucidate magnetic phase diagrams for an archetypal triangular-lattice system in the absence and presence of an external magnetic field. In the case of the single-ion anisotropy, we find a plethora of multiple-$Q$ instabilities depending on the strength and the sign of the anisotropy (easy plane or easy axis), including a noncoplanar triple-$Q$ state regarded as a skyrmion crystal with topological number of two, and coplanar and noncoplanar double-$Q$ states. In an external magnetic field, we find that another noncoplanar triple-$Q$ state, a skyrmion crystal with topological number of one, is stabilized by the in-plane (out-of-plane) magnetic field under the easy-plane (easy-axis) anisotropy. A part of the results, especially for the relatively large biquadratic interaction, qualitatively reproduce those in the Kondo lattice model, which explicitly includes itinerant electrons [S. Hayami and Y. Motome, Phys. Rev. B 99, 094420 (2019)]. We also examine the stability of the field-induced skyrmion crystal by rotating the field direction. As a by-product, we show that a different triple-$Q$ state with nonzero chirality appears in the presence of the biquadratic interaction and the easy-axis anisotropy. Meanwhile, we find that the bond-dependent anisotropy also stabilizes both types of skyrmion crystals. We show that, however, for the skyrmion crystal with topological number of one, Bloch- and N\'eel-type skyrmion crystals are selectively realized depending on the sign of the bond-dependent anisotropy, since this anisotropy selects a particular set of the helicity and vorticity. Moreover, we find yet another multiple-$Q$ states with nonzero spin scalar chirality, including two types of meron crystals with the skyrmion numbers of one and two. The systematic investigation of multiple-$Q$ instabilities in triangular itinerant magnets will provide a reference to complex magnetic textures in centrosymmetric magnetic metals.

Ultrafast Ratchet Dynamics of Skyrmions by Defect Engineering in Materials with Poor Conductivity Under Gigahertz Magnetic Fields

The ratchet motion of magnetic skyrmions driven by microwave magnetic fields, with the motion direction and speed tunable by field parameters, provides a promising route to drive magnetic skyrmions in materials with poor conductivity. However, as an indirect motion, skyrmion ratchet-motion speed is generally low in comparison with the direct motions driven by currents. Toward practical applications, it is necessary to ask if there are mechanisms to realize ultrafast ratchet motion of magnetic skyrmions and how such a motion can be integrated into racetrack-type skyrmion devices. In this work, we explore the effects of defects and edges on the ratchet motion of magnetic skyrmions under time-varying magnetic fields in GHz. We demonstrate that the ratchet motion of skyrmion is not only guided along the defect tracks or edges, but also with a remarkable speed up (with a factor over 10) compared with that in the bulk region. The skyrmion ratchet-motion speed reaches 100 m/s along a straight defect track and edge and ${10}^{9}$ rad/s along a circular edge under a field of approximately 50 mT, comparable to those direct motions driven by currents. Moreover, the skyrmion ratchet motion along the defect track or edge can be facilely controlled by the field and defect parameters. Analysis based on the time-averaged Thiele equation of skyrmion verifies that such a speed-up effect is due to the increased time-averaged driving force perpendicular to the skyrmion motion when it approaches the defect track or edge, analogous to that discovered in direct motions driven by currents. The generality of our conclusions has been examined for the ratchet motion of Bloch and N\'eel-type skyrmions driven by a variety of time-varying magnetic fields, and for systems with open edges or defect tracks with modified Dzyaloshinskii-Moriya or exchange interactions and anisotropy.

Field-induced tricritical behavior in the Néel-type skyrmion host GaV4S8

The lacunar spinel compound ${\mathrm{GaV}}_{4}{\mathrm{S}}_{8}$ exhibits a N\'eel-type skyrmion, which holds great promise for future spintronics and ultrahigh-density magnetic memory devices. To gain more insight into the magnetic interactions, the critical behavior of ${\mathrm{GaV}}_{4}{\mathrm{S}}_{8}$ is studied by dc magnetization measurement around the Curie temperature $({T}_{\mathrm{C}})$. A set of reliable critical exponents ($\ensuremath{\beta}=0.220\ifmmode\pm\else\textpm\fi{}0.024$, $\ensuremath{\gamma}=0.909\ifmmode\pm\else\textpm\fi{}0.005$, and $\ensuremath{\delta}=5.161\ifmmode\pm\else\textpm\fi{}0.003$) is obtained by the modified Arrott plot technique, the Kouvel-Fisher method, and critical isothermal analysis. The generated critical exponents fulfill the universality class of tricritical mean-field theory, which suggests a field-induced tricritical phenomenon. Based on the scaling equations, boundaries between the skyrmion and ferromagnetic phases can be distinguished. A tricritical point is revealed at the temperature of ${T}_{\mathrm{Tr}}=12\phantom{\rule{0.16em}{0ex}}\mathrm{K}$ and field of ${H}_{\mathrm{Tr}}=60\phantom{\rule{0.16em}{0ex}}\mathrm{mT}$, which is located at the intersection point among the skyrmion, ferromagnetic, and paramagnetic phases. It is suggested that the origin of the tricritical behavior in ${\mathrm{GaV}}_{4}{\mathrm{S}}_{8}$ is related to the skyrmion state near the magnetic transition temperature ${T}_{\mathrm{C}}$.

Stability of Néel-type skyrmion lattice against oblique magnetic fields in GaV4S8 and GaV4Se8

The orientation of N\'eel-type skyrmions in the lacunar spinels GaV$_4$S$_8$ and GaV$_4$Se$_8$ is tied to the polar axes of their underlying crystal structure through the Dzyaloshinskii-Moriya interaction. In these crystals, the skyrmion lattice phase exists for externally applied magnetic fields parallel to these axes and withstands oblique magnetic fields up to some critical angle. Here, we map out the stability of the skyrmion lattice phase in both crystals as a function of field angle and magnitude using dynamic cantilever magnetometry. The measured phase diagrams reproduce the major features predicted by a recent theoretical model, including a reentrant cycloidal phase in GaV$_4$Se$_8$. Nonetheless, we observe a greater robustness of the skyrmion phase to oblique fields, suggesting possible refinements to the model. Besides identifying transitions between the cycloidal, skyrmion lattice, and ferromagnetic states in the bulk, we measure additional anomalies in GaV$_4$Se$_8$ and assign them to magnetic states confined to polar structural domain walls.

Magnetic phase transitions and spin density distribution in the molecular multiferroic system GaV4S8

We have carried out neutron diffraction and small angle neutron scattering measurements on a high quality single crystal of the cubic lacunar spinel multiferroic, GaV$_4$S$_8$, as a function of magnetic field and temperature to determine the magnetic properties for the single electron that is located on the tetrahedrally coordinated V$_4$ molecular unit. Our results are in good agreement with the structural transition at 44 K from cubic to rhombohedral symmetry where the system becomes a robust ferroelectric, while long range magnetic order develops below 13 K in the form of an incommensurate cycloidal magnetic structure, which can transform into a N\'eel-type skyrmion phase in a modest applied magnetic field. Below 5.9(3) K, the crystal enters a ferromagnetic phase, and we find the magnetic order parameter indicates a long range ordered ground state with an ordered moment of 0.23(1) ${\mu}_\mathrm{B}$ per V ion. Both polarized and unpolarized neutron data in the ferroelectric-paramagnetic phase have been measured to determine the magnetic form factor. The data are consistent with a model of the single spin being uniformly distributed across the V$_4$ molecular unit, rather than residing on the single apical V ion, in substantial agreement with the results of first-principles theory. In the magnetically ordered state, polarized neutron measurements are important since both the cycloidal and ferromagnetic order parameters are clearly coupled to the ferroelectricity, causing the structural peaks to be temperature and field dependent. For the ferromagnetic ground state, the spins are locked along the $[1,1,1]$ direction by a surprisingly large anisotropy.

Validity of magnetotransport detection of skyrmions in epitaxial SrRuO3 heterostructures

A technically simple way of probing the formation of skyrmions is to measure the topological Hall resistivity that should occur in the presence of skyrmions as an additional contribution to the ordinary and anomalous Hall effect. This type of probing, lately intensively used for thin film samples, relies on the assumption that the topological Hall effect contribution can be extracted unambiguously from the measured total Hall resistivity. Ultrathin films and heterostructures of the $4d$ ferromagnet ${\mathrm{SrRuO}}_{3}$ have stirred up a lot of attention after the observation of anomalies in the Hall resistivity, which resembled a topological Hall effect contribution. These anomalies, first reported for bilayers in which the ${\mathrm{SrRuO}}_{3}$ was interfaced with the strong spin-orbit coupled oxide ${\mathrm{SrIrO}}_{3}$, were attributed to the formation of tiny N\'eel-type skyrmions. Here we present the investigation of heterostructures with two magnetically decoupled and electrically parallel connected ${\mathrm{SrRuO}}_{3}$ layers. The two ${\mathrm{SrRuO}}_{3}$ layers deliberately have different thicknesses, which affects the coercive field and ferromagnetic transition temperature of the two layers, and the magnitude and temperature dependence of their anomalous Hall constants. The ${\mathrm{SrRuO}}_{3}$ layers were separated by ultrathin layers of either the strong spin-orbit coupling oxide ${\mathrm{SrIrO}}_{3}$ or of the large band-gap insulator ${\mathrm{SrZrO}}_{3}$. Our magnetic and magnetotransport studies confirm the additivity of the anomalous Hall transverse voltages for the parallel conducting channels originating from the two ferromagnetic ${\mathrm{SrRuO}}_{3}$ layers as well as the possibility to tune the global anomalous Hall resistivity by magnetic field, temperature, or structural modifications at the epitaxial all-oxide interfaces. The Hall voltage loops of these two-layer heterostructures demonstrate the possibility to generate humplike structures in the Hall voltage loops of ${\mathrm{SrRuO}}_{3}$ heterostructures without the formation of skyrmions and emphasize that the detection of skyrmions only by Hall measurements can be misleading.

Tuning the Skyrmion Hall Effect via Engineering of Spin-Orbit Interaction

We demonstrate that the Magnus force acting on magnetic skyrmions can be efficiently tuned via modulation of the spin-orbit interaction strength. We show that the skyrmion Hall effect, which is a direct consequence of the non-vanishing Magnus force on the magnetic structure can be suppressed in certain limits. Our calculations show that the emergent magnetic fields in the presence of spin-orbit coupling (SOC) renormalize the Lorentz force on itinerant electrons and thus influence the topological transport. In particular, we show that for a N\'eel-type skyrmion and Bloch-type antiskyrmion, the skyrmion Hall effect (SkHE) can vanish by tuning appropriately the strength of Rashba and Dresselhaus SOCs, respectively. Our results open up alternative directions to explore in a bid to overcome the parasitic and undesirable SkHE for spintronic applications.

Possible emergence of a skyrmion phase in ferroelectric GaMo4S8

Polar lacunar spinels, such as GaV$_4$S$_8$ and GaV$_4$Se$_8$, were proposed to host skyrmion phases under magnetic field. In this work, we put forward, as a candidate for N\'eel-type skyrmion lattice, the isostructural GaMo$_4$S$_8$, here systematically studied via both first-principles calculations and Monte Carlo simulations of model Hamiltonian. Electric polarization, driven by Jahn-Teller distortion, is predicted to arise in GaMo$_4$S$_8$, showing a comparable size but an opposite sign with respect to that evaluated in V-based counterparts and explained in terms of different electron counting arguments and resulting distortions. Interestingly, a larger spin-orbit coupling of 4d orbitals with respect to 3d orbitals in vanadium-spinels leads to stronger Dzyaloshinskii-Moriya interactions, which are beneficial to stabilize a cycloidal spin texture, as well as smaller-sized skyrmions (radius<10 nm). Furthermore, the possibly large exchange anisotropy of GaMo4S8 may lead to a ferroelectric-ferromagnetic ground state, as an alternative to the ferroelectric-skyrmionic one, calling for further experimental verification.

Optically Driven Collective Spin Excitations and Magnetization Dynamics in the Néel-type Skyrmion Host GaV_{4}S_{8}.

GaV_{4}S_{8} is a multiferroic semiconductor hosting magnetic cycloid (Cyc) and Néel-type skyrmion lattice (SkL) phases with a broad region of thermal and magnetic stability. Here, we use time-resolved magneto-optical Kerr spectroscopy to show the coherent generation of collective spin excitations in the Cyc and SkL phases. Our micromagnetic simulations reveal that these are driven by an optically induced modulation of uniaxial anisotropy. Our results shed light on spin dynamics in anisotropic materials hosting skyrmions and pave a new pathway for the optical manipulation of their magnetic order.

Quantifying chiral exchange interaction for Néel-type skyrmions via Lorentz transmission electron microscopy

Magnetic skyrmions are topological spin textures that have been observed in bulk magnets and magnetic multilayers. For bulk magnetic materials, their noncollinear spin profiles have often been studied by using Lorentz transmission electron microscopy (TEM). We experimentally utilized Lorentz TEM imaging to study an inversion asymmetric ${[\mathrm{Pt}(1.5\phantom{\rule{0.16em}{0ex}}\mathrm{nm})/\mathrm{Co}(1\phantom{\rule{0.16em}{0ex}}\mathrm{nm})/\mathrm{W}(1\phantom{\rule{0.16em}{0ex}}\mathrm{nm})]}_{8}$ heterostructure that exhibits N\'eel-type skyrmions at zero field. By tracking the evolution of skyrmion diameters as a function of magnetic fields, we determined the strength of the interfacial Dzyaloshinskii-Moriya interaction (DMI). Our results suggest that in situ Lorentz TEM imaging combined with simulations can provide valuable quantitative information about the interfacial DMI strengths, which can be helpful for optimizing skyrmion materials. Furthermore, we show that in theory, Lorentz TEM can identify the spin chirality of N\'eel-type skyrmions, although an experimental verification is challenging due to the relatively low signal-to-noise ratio.

Evolution of topological skyrmions across the spin reorientation transition in Pt/Co/Ta multilayers

Magnetic skyrmions in multilayers are particularly appealing as next generation memory devices due to their topological compact size, the robustness against external perturbations, the capability of electrical driving and detection, and the compatibility with the existing spintronic technologies. To date, N\'eel-type skyrmions at room temperature (RT) have been studied mostly in multilayers with easy-axis magnetic anisotropy. Here, we systematically broadened the evolution of magnetic skyrmions with sub-50-nm size in a series of Pt/Co/Ta multilayers where the magnetic anisotropy is tuned continuously from easy axis to easy plane by increasing the ferromagnetic Co layer thickness. The existence of nontrivial skyrmions is identified via the combination of in situ Lorentz transmission electron microscopy (L-TEM) and Hall transport measurements. A high density of magnetic skyrmions over a wide temperature range is observed in the multilayers with easy-plane anisotropy, which will stimulate further exploration for new materials and accelerate the development of skyrmion-based spintronic devices.

Azimuthal spin-wave excitations in magnetic nanodots over the soliton background: Vortex, Bloch, and Néel-like skyrmions

We study azimuthal spin-wave (SW) excitations in a circular ferromagnetic nanodot in different inhomogeneous, topologically non-trivial magnetization states, specifically, vortex, Bloch-type skyrmion and N\'eel-type skyrmion states. Continuous mapping of the SW spectrum between these states is realized with gradual change of the out-of-plane magnetic anisotropy and Dzyaloshinskii-Moriya exchange interaction (DMI). Our study shows lifting of the SW frequencies degeneracy and change in systematics of the frequency levels. The change is induced by a geometrical Berry phase, that is present for the dot-edge localized SWs in a vortex state and vanishes in skyrmion states. Furthermore, channeling of the azimuthal SWs localized at the skyrmion edge is present and induces large frequency splitting. This is attributed to DMI induced nonreciprocity, while coupling of the breathing and gyrotropic modes is related to soliton motion. Finally, an efficient coupling of the dynamic magnetization in the skyrmion state to uniform magnetic field in nanodots with non-circular symmetry is shown.

Direct evidence for cycloidal modulations in the thermal-fluctuation-stabilized spin spiral and skyrmion states of GaV4S8

We report small-angle neutron scattering studies of the lacunar spinel GaV$_4$S$_8$, which reveal the long-wavelength magnetic states to be cycloidally modulated. This provides direct support for the formation of N\'eel-type skyrmions recently claimed to exist in this compound. In striking contrast with all other bulk skyrmion host materials, upon cooling the modulated magnetic states transform into a ferromagnetic state. These results indicate all of the modulated states in GaV$_4$S$_8$, including the skyrmion state, gain their stability from thermal fluctuations, while at lower temperature the ferromagnetic state emerges in accord with the strong easy-axis magnetic anisotropy. In the vicinity of the transition between the ferromagnetic and modulated states, both a phase coexistence and a soliton-like state are also evidenced by our study.

Néel-Type Skyrmion Lattice in the Tetragonal Polar Magnet VOSe_{2}O_{5}.

The formation of the triangular Skyrmion lattice is found in a tetragonal polar magnet VOSe_{2}O_{5}. By magnetization and small-angle neutron scattering measurements on the single crystals, we identify a cycloidal spin state at zero field and a Néel-type Skyrmion-lattice phase under a magnetic field along the polar axis. Adjacent to this phase, another magnetic phase of an incommensurate spin texture is identified at lower temperatures, tentatively assigned to a square Skyrmion-lattice phase. These findings exemplify the versatile features of Néel-type Skyrmions in bulk materials, and provide a further opportunity to explore the physics of topological spin textures in polar magnets.

Relaxation dynamics of modulated magnetic phases in the skyrmion host GaV4S8 : An ac magnetic susceptibility study

We report on the slow magnetization dynamics observed upon the magnetic phase transitions of GaV4S8, a multiferroic compound featuring a long-ranged cycloidal magnetic order and a N\'eel-type skyrmion lattice in a relatively broad temperature range below its Curie temperature. The fundamental difference between GaV4S8 and the chiral helimagnets, wherein the skyrmion phase was first discovered, lies within the polar symmetry of GaV4S8, promoting a cycloidal instead of a helical magnetic order and rendering the magnetic phase diagram essentially different from that in the cubic helimagnets. Our ac magnetic susceptibility study reveals slow relaxation dynamics at the field-driven phase transitions between the cycloidal, skyrmion lattice and field-polarized states. At each phase boundary, the characteristic relaxation times were found to exhibit a strong temperature dependence, starting from the minute range at low temperatures, decreasing to the micro- to millisecond range at higher temperatures.

Spin excitation spectrum in a magnetic nanodot with continuous transitions between the vortex, Bloch-type skyrmion, and Néel-type skyrmion states

We study spin-wave excitations in a circular ferromagnetic nanodot in different inhomogeneous, topologically non-trivial magnetization states, specifically, vortex and skyrmion states. Gradual change in the strength of the out-of-plane magnetic anisotropy and the Dzyaloshinskii-Moriya exchange interaction leads to continuous phase transitions between different stable magnetic configurations and allows for mapping of dynamic spin modes in and between the vortex, Bloch-type skyrmion and N\'eel-type skyrmion states. Our study elucidates the connections between gyrotropic modes, azimuthal spin waves and breathing modes in various stable magnetization states and helps to understand the rich spin excitation spectrum on the skyrmion background.