Abstract
Magnetic skyrmions, topological solitons characterized by a twodimensional swirling spin texture, have recently attracted attention as stable particlelike objects. In a threedimensional system, a skyrmion can extend in the third dimension forming a robust and flexible string structure, whose unique topology and symmetry are anticipated to host nontrivial functional responses. Here we experimentally demonstrate the coherent propagation of spin excitations along skyrmion strings for the chirallattice magnet Cu_{2}OSeO_{3}. We find that this propagation is directionally nonreciprocal and the degree of nonreciprocity, as well as group velocity and decay length, are strongly dependent on the character of the excitation modes. These spin excitations can propagate over a distance exceeding 50 μm, demonstrating the excellent longrange ordered nature of the skyrmionstring structure. Our combined experimental and theoretical analyses offer a comprehensive account of the propagation dynamics of skyrmionstring excitations and suggest the possibility of unidirectional information transfer along such topologically protected strings.
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Introduction
Recently, the concept of a magnetic skyrmion, i.e., topologically stable spin configuration whose spins point in all of the directions wrapping a sphere, has attracted enormous attention^{1,2,3,4}. In a magnetically twodimensional system, a skyrmion appears as a vortexlike swirling spin texture with particle nature as shown in Fig. 1j. The stable particle nature of the skyrmion suggests its potential application as a magnetic bit for future data storage devices, and the nontrivial topology and symmetry of the skyrmion also cause exceptional electromagnetic responses^{3,4}.
In threedimensional systems, the skyrmion can form a string structure by extending in the third dimension, which consists of the uniform stacking of twodimensional skyrmions along the string direction (Fig. 1a)^{5,6}. Skyrmion strings can be considered as an analog of the vortex line in superfluids^{7,8}, typeII superconductors^{9}, and trapped dilutegas Bose–Einstein condensates^{10} or the cosmic string in the universe^{11}. They are all flexible and some of these strings are proposed to host a resonant oscillation mode propagating through the string path. This implies the possible coherent signal transfer along skyrmion strings, whereas the propagation character of their excitations has rarely been investigated before.
Experimentally, such skyrmion strings appear in a series of bulk magnets with chiral cubic atomic lattice. The examples are metallic B20^{1,2} or βMntype CoZnMn^{12} alloys and insulating Cu_{2}OSeO_{3}^{13,14}, the latter of which is the target of this work. In these materials, the Dzyaloshinskii–Moriya (DM) interaction is the key for the skyrmion formation. For a limited temperature (T) range, these compounds host a hexagonal lattice of skyrmion strings aligned along the static magnetic field (H) direction (Fig. 1a). This skyrmion crystal (SkX) phase is predicted to host three distinctive magnetic resonance modes^{15,16,17} (central panels in Fig. 1f–h), i.e., the counterclockwise (CCW) and clockwise (CW) rotational modes both excited by an oscillating magnetic field H^{ν} ⊥ H, as well as the breathing (B) mode excited by H^{ν}∥H, which have recently been identified by magnetic resonance experiments^{16,18}. However, these previous works mostly focused on the character of nonpropagating uniform excitations with wave number k^{SW} = 0. To understand their propagation character, the employment of a different experimental approach sensitive to the k^{SW} ≠ 0 regime, as well as the theoretical identification of their dispersion relation, is essential.
In this work, we investigated the propagation characteristics of such spin excitations along skyrmion strings. We find that the counterpropagating spin excitations show different propagation behavior and the degree of nonreciprocity, as well as the associated group velocity and decay length, are strongly dependent on the character of the excitation modes. These experimental features are well reproduced by our theoretically calculated dispersion relations. The observed decay lengths exceed 50 μm, reflecting the excellent longrange order of the skyrmionstring structure. The present results revealed the peculiar propagation dynamics of skyrmionstring excitations and suggest that skyrmion string can be a good medium for magnon transport with unique functionalities.
Results
Propagation dynamics of spin excitations
The detailed operating principle of the present measurement technique (propagating spinwave spectroscopy) is provided in refs. ^{19,20}. Figure 1b, c indicate the device structure employed in this study, where a plateshaped Cu_{2}OSeO_{3} single crystal with thickness b ≈ 2 μm is placed on top of a pair of Au coplanar waveguides (CPWs) fabricated on a Si substrate. When the oscillating electric current of frequency ν is injected into a CPW, the generated oscillating magnetic field H^{ν} induces resonant spin excitations in the neighboring Cu_{2}OSeO_{3} sample as shown in Fig. 1c. These spin excitations propagate along the sample and induce an oscillating electric voltage in the CPWs through the inverse process. By measuring the magnetic contribution to the selfinductance ΔL_{11} and mutual inductance ΔL_{nm} spectra (with m and n representing the port number used for the excitation and detection, respectively) for CPWs using a vector network analyzer (NA), we can directly evaluate the local excitation character and propagation character of spin excitations, respectively. Unless specified, we employed a device structure with λ^{SW} = 12 μm and d = 20 μm for all the measurements; here, λ^{SW} and d are the period of a CPW pattern and the gap distance between two CPWs as shown in Fig. 1c and Supplementary Fig. 3a, respectively. The wave number distribution of the induced spin excitations is determined by the Fourier transform of CPW pattern (Supplementary Fig. 2)^{19}, whose maximum intensity is always located at k^{SW} = 2π∕λ^{SW} ≈ 0.5 μm^{−1}. The propagation direction of k^{SW} and H are set parallel to the [110] direction of the Cu_{2}OSeO_{3} crystal.
Figure 1d indicates the H–T magnetic phase diagram for Cu_{2}OSeO_{3}, obtained for the field cooling path shown by the arrows. Although the SkX phase usually appears as the thermodynamic equilibrium state only for the narrow temperature range just below T_{c} (Supplementary Fig. 1)^{13,14}, the rapid cooling (≈5 K min^{−1}) of the sample at 17.5 mT enabled us to keep the SkX phase as a metastable state^{21} even down to 10 K. The corresponding Hdependence of local magnetic resonance spectra ΔL_{11} measured at 25 K is plotted in Fig. 1e, where the pure SkX phase characterized by the three magnetic resonance modes^{15,16} is clearly identified between 20 mT and 47.5 mT.
Next, we investigated the propagation characteristics of these spin excitations in the SkX phase. Figure 2a–c indicate the spectra of mutual inductance ΔL_{21} and ΔL_{12} measured at 25 K and +25 mT, which represent the propagation character of spin excitations with the wavevector +k^{SW} and −k^{SW} (i.e., parallel and antiparallel to the external magnetic field as shown in Fig. 2d), respectively. Here, Fig. 2a–c correspond to the CCW, breathing and CW modes as schematically illustrated in Fig. 1f–h, respectively. For all modes, a propagating signal of coherent spin excitations is observed, implying that the skyrmion strings are welldefined and free of defects over an extended distance. Importantly, for the CCW mode, we can identify a clear frequency shift Δν between ΔL_{21} and ΔL_{12}, which demonstrates that the spin excitations propagating along the positive and negative direction on the skyrmion strings are not equivalent. On the other hand, the magnitude of Δν is relatively small for the breathing and CW modes. When the direction of external H is reversed, the sign of Δν is reversed as shown in Fig. 2e–h. Such a nonreciprocal behavior and its mode dependence can be interpreted in terms of the dispersion relation as discussed below.
Dispersion relation in the SkX state
Figure 2i shows the spin excitation dispersion in the SkX phase, theoretically calculated for the present k^{SW}∥H configuration (see Supplementary Note 2 for details). The dispersion relations are asymmetric (i.e., the resonance frequency is not an even function of k^{SW}) for all three modes and this asymmetry is most pronounced for the CCW mode. Here, the experimentally measured Δν corresponds to the difference of eigen frequencies between k^{SW} = ±2π∕λ^{SW}, which directly reflects the degree of asymmetry in the dispersion relation. Therefore, the observation of the largest magnitude of Δν in the CCW mode is in accord with the predicted dispersions in Fig. 2i.
In the following, we discuss the microscopic origin of the observed nonreciprocity and its mode dependence. For skyrmion strings, we find that there are two contributions to Δν: the first is attributed to the DM interaction, Δν_{DM}, and the second arises from the dynamic dipolar interaction, Δν_{dip}. We refer for a discussion of the latter to Supplementary Note 2, and focus here on the contribution from the DM interaction^{22,23,24,25,26}. Previously, it has not systematically been understood how DM interaction affects the dispersion relation in the noncollinear magnets with arbitrary excitation modes. As detailed below, we derive a general analytic expression for Δν, which reveals that the spatial distribution of the local precession amplitude, as well as the static spin texture in the ground state, has an important role to define the degree of dispersion asymmetry in each mode.
Figure 1f–h represent snapshot images describing how the local spin excitations launched at the z = 0 plane propagates on a skyrmion string (aligned along the H∥z direction) with the wave vector +k^{SW} and −k^{SW}. In the right panel, the corresponding crosssectional images for selected zplanes are also indicated. Such a propagating spin excitation induces a change of the DM energy density ε_{DM} = D_{z}(m_{i} × m_{j}), which is defined between the local moments m_{i} and m_{j} on adjacent sites i and j along the \(\hat{z}\)direction with the DM vector \({{\bf{D}}}_{z}=D\hat{z}\). At the edge region of the skyrmion, the local moment is pointing parallel to H in the ground state, and precesses CCW around H in the excited state as shown in Fig. 1i. In this case, the +k^{SW} and −k^{SW} excitation modes locally induce a conical spin arrangement along the \(\hat{z}\)direction but with opposite sign of the spin helicity (m_{i} × m_{j}) and thus of ε_{DM}, which leads to a different excitation energy for modes with ±k^{SW} and causes an asymmetric dispersion. At the core region of the skyrmion, on the other hand, the local moment is antiparallel to H and precesses CW around H; therefore, the sign of ε_{DM} contribution is reversed as compared with the skyrmion edge position (Fig. 1i). As a consequence, the magnitude of nonreciprocity deriving from the DM interaction is basically determined by the imbalance of local precession amplitudes between the edge and core positions with local moments pointing along \(+\hat{z}\) and \(\hat{z}\) directions, respectively.
More precisely, Δν_{DM} due to the DM interaction in the limit of small k^{SW} is up to a normalization given by
with the integration range defined by the twodimensional magnetic unit cell of the SkX. Here, we assume the local magnetization dynamics \({\bf{m}}({\bf{r}},t)={{\bf{m}}}_{0}({\bf{r}})+(\delta {\bf{m}}({\bf{r}})\exp [i2\pi \nu t]+{\rm{c.c.}})\), with m_{0}(r) and δm(r) representing the static and dynamical parts of local magnetization component, respectively; \({m}_{0}^{z}({\bf{r}})\) is \(\hat{z}\)component of m_{0}(r). The cross product in the integrand is proportional to the local precession intensity \({\mathcal{A}}({\bf{r}})\) as defined in Fig. 1n, i.e., the area enclosed by the precessing local magnetization during a single oscillation period. This allows for an intuitive geometric interpretation of Δν_{DM} as given by the last term in equation (1). Its integrand is determined by the product of \({m}_{0}^{z}({\bf{r}})\) and precession density \({\mathcal{A}}({\bf{r}})\) for each excitation mode, whose spatial distribution are presented in Fig. 1j, k–m, respectively. In case of the CCW mode (Fig. 1k), the precession density is confined to the edge of the unit cell, resulting in a large magnitude of Δν_{DM} according to equation (1). For the CW mode (Fig. 1m), in contrast, a finite precession density is present both at the core and the edge regions, which leads to a cancelation for Δν_{DM}. The breathing mode (Fig. 1l) is characterized by a small precession density at the core and the edge region and therefore hosts only a tiny Δν_{DM}. This analysis accounts already qualitatively for the large Δν observed for the CCW mode, and demonstrates that the spatial details of the internal magnetic texture, as well as its excitation manner plays a crucial role for the magnitude of nonreciprocity. As we explain in Supplementary Note 2, for a quantitative comparison an additional contribution Δν_{dip} attributed to the dynamical stray field must be taken into account.
The preceding discussion focused on the dispersion relation in the bulk limit k^{SW} ≫ 1∕b. For k^{SW} ≲ 1∕b, the magnetostatic (i.e., magnetic dipolar) interaction results in an additional peak structure in the dispersion whose sharpness scales with the sample thickness b^{27}. Our present experimental setup corresponds to k^{SW} ≈ 1∕b. As sketched in Fig. 2j, k, this magnetostatic dispersion (solid lines) develops on the background of the nonreciprocal bulk spectrum (dashed lines) so that Δν is still dominated by the latter. Nevertheless, the group velocity v_{g} = 2π(∂ν∕∂k^{SW}) is strongly influenced by the magnetostatic interactions for k^{SW} ≲ 1∕b. Experimentally, the magnitude of group velocity can be deduced based on the relationship ∣v_{g}∣ = ν_{pp}d^{19}, with ν_{pp} representing the oscillation period of ΔL_{nm} signals as shown in Fig. 2b.
Figure 3a–c indicate the magnetic field dependence of magnetic resonance frequency ν_{0}, frequency shift Δν between ±k^{SW}, and magnitude of group velocity ∣v_{g}∣, experimentally deduced at 25 K from the similar data sets as shown in Fig. 2a–c. Theory provides parameterfree predictions for these quantities that are also plotted in Fig. 3d–f. In both cases, the nonreciprocal frequency shift Δν is much larger for the CCW mode than for the other modes, and the magnitude of ∣v_{g}∣ is the largest and smallest for the breathing and CW modes, respectively. The order of Δν and ∣v_{g}∣ are also roughly in accord with each other. Such a good agreement between the experimental and theoretical results firmly establishes the overall picture of dispersion relations for skyrmionstring excitations.
Decay length and damping parameter
On the basis of the present experimental data, we can further deduce the decay length l of propagating spin excitations. Figure 4a indicates the amplitude spectra of selfinductance ∣ΔL_{11}∣ and mutual inductance ∣ΔL_{21}∣ measured at 25 K and +25 mT, whose ratio provides the decay rate of spin excitation amplitude during the propagation over the distance d. The decay length l can be estimated from the relationship \(2 \Delta {L}_{21} / \Delta {L}_{11} =\exp (d/l)\), which is associated with the damping parameter α in the form of l = v_{g}∕(2πν_{0}α)^{19}. In Fig. 4b, c, the magnetic field dependence of l and α at 25 K are plotted. Although the decay length l largely depends on the character of spin excitation modes, basically the same value of α is shared by all three modes in the SkX phase. Here, the CW mode hosts considerably shorter l than the other two modes, reflecting its large ν_{0} and small v_{g} values (Fig. 3a, c). In Fig. 4d, e, temperature dependence of l and α for various spin excitation modes are plotted. For all temperatures, the SkX phase is characterized by a damping parameter α that is slightly larger, but less than a factor of two, than the one in the fieldinduced ferromagnetic phase. Here, the effective damping parameter α reflects not only the intrinsic Gilbert damping, but also the additional contribution of extrinsic origin. In the present case, this slight enhancement of α might arise from the elliptical nature of local magnetization precession^{28} and/or the additional scattering, e.g., from defects of the SkX order. Recently, it was suggested that the SkX contains a considerable amount of singular point defects interpreted as emergent magnetic monopoles^{5}, which act like the slider of a zipper connecting two skyrmion strings. The existence of such monopolelike defects diminishes the spatial longrange order of the SkX and thus also the coherent propagation of spin excitation along the skyrmion strings. Nevertheless, the decay length l in the SkX phase still exceeds 50 μm at low temperatures, proving that the spin excitations on the skyrmion strings can propagate a distance exceeding 10^{3} times the diameter of single skyrmion string (a_{sk} ≈ 50 nm^{13,14}). This large l∕a_{sk} ratio yields a lower bound characterizing the distance between defects, and demonstrates the excellent static longrange order of the magnetic skyrmion crystal. Such a small damping and long propagation distance of the excitation are rather unexpected for the SkX phase with its intricate threedimensional spin texture and the above results suggest that the skyrmion string with excellent longrange order can be a good medium for magnon transport.
Discussion
In this study, we experimentally demonstrated the coherent and nonreciprocal propagation of spin excitations along skyrmion strings. Unlike conventional ferromagnets, skyrmion strings host multiple excitation modes, and their group velocity, decay length, and degree of nonreciprocity turned out to be strongly modedependent in full agreement with our calculated dispersion relations. These results establish the comprehensive picture of the propagation dynamics of skyrmionstring excitations. Moreover, we developed a general theoretical framework to evaluate the dispersion asymmetry for noncollinear magnets, which reveals that the underlying static spin texture and spatial distribution of local precession amplitude are key factors that determine the degree of nonreciprocity for each mode. As the propagating spin excitations carry energy and momentum, the above findings suggest the possibility of unidirectional information transfer along skyrmion strings.
On a broader perspective, the interesting extension of the present work includes the study of the skyrmionstring dynamics under more general conditions. While the current experiments focused on straight skyrmion strings embedded in the skyrmion crystals, the dynamics of a single isolated string has also been discussed theoretically very recently^{29,30,31}. Such skyrmion strings can be bent into curved shapes due to their topological protection^{5} and the spin excitations are expected to propagate along the same string path in analogy with the Kelvin mode of vortex lines in superfluids^{7,8,30}. These features suggest that skyrmion strings may serve as robust and flexible information transmission lines potentially enabling the design of associated network or circuits without Joule heat loss, in a similar manner as the recently proposed reconfigurable spinwave nanochannels^{32}. The nonlinear dynamics of these strings is another fascinating issue. Our results highlight that not only the skyrmion particle in twodimensional systems, but also the skyrmion string in threedimensional systems is an attractive topological object as a building block for energyefficient spintronic devices and further investigation of its fundamental and functional properties will be a promising endeavor.
Methods
Crystal growth and device fabrication
Single crystals of Cu_{2}OSeO_{3} were grown by the chemical vapor transport method. A pair of Au coplanar waveguides (CPW: 200 nm thickness) were fabricated on the oxidized silicon substrate through the standard photolithography technique, and a plateshaped single crystal of Cu_{2}OSeO_{3} (~2 μm thickness) was placed across them with W (tungsten) deposition at an edge of the crystal using the focused ion beam micro fabrication technique (Fig. 1b).
Propagating spinwave spectroscopy
By measuring the magnetic contribution to the complex spectra of selfinductance ΔL_{11} and mutual inductance ΔL_{nm} (with m and n being the port number used for the excitation and detection, respectively) for these CPWs with the vector NA, the local excitation character and propagation character of spin excitation can be directly evaluated, respectively^{19,20}. The spin excitation contribution to the inductance spectrum \(\Delta {L}_{\mathrm{{nm}}}(\nu )={L}_{\mathrm{{nm}}}(\nu ){L}_{\mathrm{{nm}}}^{{\rm{ref}}}(\nu )\) is derived by the subtraction of the common background \({L}_{\mathrm{{nm}}}^{{\rm{ref}}}(\nu )\) from the raw data L_{nm}(ν). Here, L_{nm}(ν) taken at μ_{0}H = 250 mT is adopted as the reference spectrum \({L}_{\mathrm{{nm}}}^{{\rm{ref}}}(\nu )\), where the magnetic resonance is absent within our target frequency range of 0.2 GHz ≤ ν ≤ 7.0 GHz. Through this background subtraction process, the possible contribution of crosstalk can be safely excluded. Unless specified, we employed the device with the central wave length λ^{SW} = 12 μm and the gap distance d = 20 μm.
Data availability
The data presented in the current study are available from the corresponding authors on reasonable request.
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Acknowledgements
We thank J. Iwasaki, S. Hoshino, N. Nagaosa, and H. Hosono for enlightening discussions and experimental helps. This work was partly supported by GrantsInAid for Scientific Research (A) (Grant Number 18H03685) and GrantinAid for Scientific Research on Innovative Area, “Nano Spin Conversion Science” (Grant Number 17H05186) from JSPS, and PRESTO (Grant Number JPMJPR18L5) from JST. M.G. was supported by the DFG via SFB 1143 “Correlated magnetism: from frustration to topology” and grant GA 1072/61.
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S.S. performed the measurements. S.S., R.T., N.D.K., Y.O., K.K., F.K., and Y.O. contributed to the crystal growth and device fabrication. M.G. and J.W. performed theoretical calculation. S.S., M.G., and Y.T. planned the project and wrote the manuscript. All authors discussed the results and commented on the manuscript.
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Seki, S., Garst, M., Waizner, J. et al. Propagation dynamics of spin excitations along skyrmion strings. Nat Commun 11, 256 (2020). https://doi.org/10.1038/s41467019140950
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DOI: https://doi.org/10.1038/s41467019140950
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