Propagating spin excitations along skyrmion strings

TL;DR

This study demonstrates the experimental and theoretical analysis of coherent, non-reciprocal spin excitation propagation along skyrmion strings in a chiral magnet, revealing long-range transfer potential for topological information processing.

Contribution

It provides the first experimental evidence and theoretical understanding of directional spin excitation propagation along three-dimensional skyrmion strings.

Findings

Spin excitations propagate over distances exceeding 10^3 times the skyrmion diameter.

Propagation is directionally non-reciprocal and mode-dependent.

Theoretical dispersion matches experimental observations.

Abstract

Magnetic skyrmions, topological solitons characterized by a two-dimensional swirling spin texture, have recently attracted attention as stable particle-like objects. In a three-dimensional 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 chiral-lattice magnet Cu2OSeO3. We find that this propagation is directionally non-reciprocal, and the degree of non-reciprocity, as well as the associated group velocity and decay length, are strongly dependent on the character of the excitation modes. Our theoretical calculation establishes the corresponding dispersion relationship, which well reproduces the experimentally observed features. Notably, these spin excitations can propagate over a distance exceeding 10^3 times the skyrmion diameter, demonstrating the excellent long-range nature of the excitation propagation on the skyrmion strings. Our combined experimental and theoretical results offer a comprehensive account of the propagation dynamics of skyrmion-string excitations, and suggest the possibility of unidirectional information transfer along such topologically-protected strings.

Equations

$$\Delta {\nu }_{\mathrm{DM}}\propto D|k^{\mathrm{SW}}|{\int }_{u.c.}{d}^2\mathbf{r}\hat{z}i(\delta \mathbf{m}(\mathbf{r})\times \delta {\mathbf{m}}^{\ast }(\mathbf{r}))\propto D|k^{\mathrm{SW}}|{\int }_{u.c.}{d}^2\mathbf{r}{m}_0^z(\mathbf{r})\mathcal{A}(\mathbf{r}),$$

Full text

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 attentionNeutronMnSi ; TEMFeCoSi ; SkXReviewFertTwo ; SkXReviewTokura . In a magnetically two-dimensional system, a skyrmion appears as a vortex-like swirling spin texture with particle nature as shown in Fig. 1j. The stable particle nature of the skyrmion suggests its potential application as a novel magnetic bit for future data storage devices, and the nontrivial topology and symmetry of the skyrmion also cause exceptional electromagnetic responsesSkXReviewFertTwo ; SkXReviewTokura .

In three dimensional systems, the skyrmion can form a string structure by extending in the third dimension, which consists of the uniform stacking of two-dimensional skyrmions along the string direction (Fig. 1a)Monopole ; Holography . Skyrmion strings can be considered as an analogue of the vortex-line in superfluidsRMP_superfluids ; PNAS_Kelvinwave , type-II superconductorsRMP_superconductor and trapped dilute-gas Bose-Einstein condensatesScience_BEC or the cosmic string in the universecosmic_string . 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, while 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 B20NeutronMnSi ; TEMFeCoSi or $\beta$-Mn type Co-Zn-MnCoZnMn_First alloys and insulating Cu2OSeO3Cu2OSeO3_Seki ; Cu2OSeO3_SANS_Pfleiderer , 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 modesMochizuki12 ; B20_FMR ; PRB_SkX_spinwave (central panels in Figs. 1f-h), i.e. the counter-clockwise (CCW) and clockwise (CW) rotational modes both excited by an oscillating magnetic field $H^{\nu}\perp H$, as well as the breathing (B) mode excited by $H^{\nu}\parallel H$, which have recently been identified by magnetic resonance experimentsB20_FMR ; Onose . However, these previous works mostly focused on the character of non-propagating uniform excitations with wave number $k^{\rm SW}=0$. To understand their propagation character, the employment of a different experimental approach sensitive to the $k^{\rm SW}\neq 0$ 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 counter-propagating spin excitations show different propagation behavior, and the degree of non-reciprocity, 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. Moreover, the observed decay lengths exceed 103 times the skyrmion diameter, reflecting the excellent long-range order of the skyrmion string structure. The present results revealed the peculiar propagation dynamics of skyrmion string excitations, and suggest that skyrmion string can be a good medium for magnon transport with unique functionalities.

The detailed operating principle of the present measurement technique (propagating spin wave spectroscopy) is provided in Refs. AESWS_doppler_B ; AESWS_doppler_Science . Figures 1b and c indicate the device structure employed in this study, where a plate-shaped Cu2OSeO3 single crystal with thickness $b\approx 2\mu$m is placed on top of a pair of Au coplanar waveguides (CPW) fabricated on a Si substrate. When the oscillating electric current of frequency $\nu$ is injected into a CPW, the generated oscillating magnetic field $H^{\nu}$ induces resonant spin excitations in the neighboring Cu2OSeO3 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 self-inductance $\Delta L_{11}$ and mutual inductance $\Delta L_{nm}$ 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 $\lambda^{\rm SW}=12\mu$m and $d=20\mu$m for all the measurements; here, $\lambda^{\rm SW}$ and $d$ are the period of a CPW pattern and the gap distance between two CPWs as shown in Figs. 1c and S3a, respectively. The wave number distribution of the induced spin excitations is determined by the Fourier transform of CPW pattern (Fig. S2)AESWS_doppler_B , whose maximum intensity is always located at $k^{\rm SW}=2\pi/\lambda^{\rm SW}\approx 0.5\mu m^{-1}$. The propagation direction of $k^{\rm SW}$ and $H$ are set parallel to the [110] direction of the Cu2OSeO3 crystal.

Figure 1d indicates the $H$-$T$ magnetic phase diagram for Cu2OSeO3, obtained for the field cooling path shown by the arrows. While the SkX phase usually appears as the thermodynamic equilibrium state only for the narrow temperature range just below $T_{c}$ (Fig. S1)Cu2OSeO3_Seki ; Cu2OSeO3_SANS_Pfleiderer , the rapid cooling ($\approx 5$K/min) of the sample at 17.5 mT enabled us to keep the SkX phase as a metastable stateQuench even down to 10 K. The corresponding $H$-dependence of local magnetic resonance spectra $\Delta L_{11}$ measured at 25 K is plotted in Fig. 1e, where the pure SkX phase characterized by the three magnetic resonance modesMochizuki12 ; B20_FMR is clearly identified between 20 mT and 47.5 mT.

Next, we investigated the propagation characteristics of these spin excitations in the SkX phase. Figures 2a-c indicate the spectra of mutual inductance $\Delta L_{21}$ and $\Delta L_{12}$ measured at 25 K and +25 mT, which represent the propagation character of spin excitations with the wavevector $+k^{\rm SW}$ and $-k^{\rm SW}$ (i.e., parallel and anti-parallel to the external magnetic field as shown in Fig. 2d), respectively. Here, Figs. 2a-c correspond to the CCW, breathing and CW modes as schematically illustrated in Figs. 1f-h, respectively. For all modes, a propagating signal of coherent spin excitations is observed, implying that the skyrmion strings are well-defined and free of defects over an extended distance. Importantly, for the CCW mode, we can identify a clear frequency shift $\Delta\nu$ between $\Delta L_{21}$ and $\Delta L_{12}$, 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 $\Delta\nu$ is relatively small for the breathing and CW modes. When the direction of external $H$ is reversed, the sign of $\Delta\nu$ is reversed as shown in Figs. 2e-h. Such a nonreciprocal behavior and its mode dependence can be interpreted in terms of the dispersion relation as discussed below.

Figure 2i shows the spin excitation dispersion in the SkX phase, theoretically calculated for the present $k^{\rm SW}\parallel H$ configuration (see supplementary information for details). The dispersion relations are asymmetric (i.e. the resonance frequency is not an even function of $k^{\rm SW}$) for all three modes, and this asymmetry is most pronounced for the CCW mode. Here, the experimentally measured $\Delta\nu$ corresponds to the difference of eigen frequencies between $k^{\rm SW}=\pm 2\pi/\lambda^{\rm SW}$, which directly reflects the degree of asymmetry in the dispersion relation. Therefore, the observation of the largest magnitude of $\Delta\nu$ 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 $\Delta\nu$: the first is attributed to the DM interaction, $\Delta\nu_{\rm DM}$, and the second arises from the dynamic dipolar interaction, $\Delta\nu_{\rm dip}$. We refer for a discussion of the latter to the supplementary information, and focus here on the contribution from the DM interactionKataoka ; SW_DM_BLS ; LiFe5O8 ; SW_DM_COSO ; SW_DM_MnSi . 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 the general analytic expression of $\Delta\nu$, which reveals that the spatial distribution of local precession amplitude, as well as the static spin texture in the ground state, plays an important role to define the degree of dispersion asymmetry in each mode.

Figures 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 $H\parallel z$ direction) with the wave vector $+k^{\rm SW}$ and $-k^{\rm SW}$. In the right panel, the corresponding cross-sectional images for selected $z$-planes are also indicated. Such a propagating spin excitation induces a change of the DM energy density $\varepsilon_{\rm DM}={\bf D}_{z}({\bf m}_{i}\times{\bf m}_{j})$, which is defined between the local moments ${\bf m}_{i}$ and ${\bf m}_{j}$ 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 ${\bf H}$ in the ground state, and precesses counter-clockwise around ${\bf H}$ in the excited state as shown in Fig. 1i. In this case, the $+k^{\rm SW}$ and $-k^{\rm SW}$ excitation modes locally induce a conical spin arrangement along the ${\hat{z}}$-direction but with opposite sign of the spin helicity $({\bf m}_{i}\times{\bf m}_{j})$ and thus of $\varepsilon_{\rm DM}$, which leads to a different excitation energy for modes with $\pm k^{\rm SW}$ and causes an asymmetric dispersion. At the core region of the skyrmion, on the other hand, the local moment is antiparallel to ${\bf H}$ and precesses clockwise around ${\bf H}$; therefore, the sign of $\varepsilon_{\rm DM}$ contribution is reversed as compared with the skyrmion edge position (Fig. 1i). As a consequence, the magnitude of non-reciprocity 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, $\Delta\nu_{\rm DM}$ due to the DM interaction in the limit of small $k^{\rm SW}$ is given by

[TABLE]

with the integration range defined by the two-dimensional magnetic unit cell of the SkX. Here, we assume the local magnetization dynamics ${\bf m}({\bf r})={\bf m}_{0}({\bf r})+(\delta{\bf m}({\bf r})\exp[i2\pi\nu t]+{\rm c.c.})$, with ${\bf m}_{0}({\bf r})$ and ${\bf\delta m}({\bf r})$ representing the static and dynamical parts of local magnetization component, respectively; $m^{z}_{0}({\bf r})$ is ${\hat{z}}$-component of ${\bf m}_{0}({\bf r})$. 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, that quantifies the probability density of the magnon). This allows for an intuitive geometric interpretation of $\Delta\nu_{\rm DM}$ as given by the last term in Eq. (1). Its integrand is determined by the product of $m^{z}_{0}({\bf r})$ and precession density $\mathcal{A}({\bf r})$ for each excitation mode, whose spatial distribution are presented in Fig. 1j and Figs. 1k-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 $\Delta\nu_{\rm DM}$ according to Eq. (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 $\Delta\nu_{\rm DM}$. 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 $\Delta\nu_{\rm DM}$. This analysis accounts already qualitatively for the large $\Delta\nu$ 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 non-reciprocity. As we explain in the supplementary information, for a quantitative comparison an additional contribution $\Delta\nu_{\rm dip}$ attributed to the dynamical stray field must be taken into account.

The preceding discussion focused on the dispersion relation in the bulk limit $k^{\rm SW}\gg 1/b$. For $k^{\rm SW}\lesssim 1/b$, the magnetostatic (i.e. magnetic dipolar) interaction results in an additional peak structure in the dispersion whose sharpness scales with the sample thickness $b$MSW_Text . Our present experimental setup corresponds to $k^{\rm SW}\approx 1/b$. As sketched in Fig. 2j and k, this magnetostatic dispersion (solid lines) develops on the background of the non-reciprocal bulk spectrum (dashed lines) so that $\Delta\nu$ is still dominated by the latter. Nevertheless, the group velocity $v_{g}=2\pi(\partial\nu/\partial k^{\rm SW})$ is strongly influenced by the magnetostatic interactions for $k^{\rm SW}\lesssim 1/b$. Experimentally, the magnitude of group velocity can be deduced based on the relationship $|v_{g}|=\nu_{pp}d$AESWS_doppler_B , with $\nu_{pp}$ representing the oscillation period of $\Delta L_{nm}$ signals as shown in Fig. 2b.

Figures 3a-c indicate the magnetic field dependence of magnetic resonance frequency $\nu_{0}$, frequency shift $\Delta\nu$ between $\pm k^{\rm SW}$, and magnitude of group velocity $|v_{g}|$, experimentally deduced at 25 K from the similar data sets as shown in Figs. 2a-c. Theory provides parameter-free predictions for these quantities that are also plotted in Figs. 3d-f. In both cases, the non-reciprocal frequency shift $\Delta\nu$ 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 $\Delta\nu$ 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 skyrmion string excitations.

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 self inductance $|\Delta L_{11}|$ and mutual inductance $|\Delta L_{21}|$ 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 $\alpha$ in the form of $l=v_{g}/(2\pi\nu_{0}\alpha)$AESWS_doppler_B . In Figs. 4b and c, the magnetic field dependence of $l$ and $\alpha$ at 25 K are plotted. While the decay length $l$ largely depends on the character of spin excitation modes, basically the same value of $\alpha$ 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 $\nu_{0}$ and small $v_{g}$ values (Figs. 3a and c). In Figs. 4d and e, temperature dependence of $l$ and $\alpha$ for various spin excitation modes are plotted. For all temperatures, the SkX phase is characterized by a damping parameter $\alpha$ that is slightly larger, but less than a factor of two, than the one in the field-induced ferromagnetic phase. This slight enhancement might arise from the elliptical nature of local magnetization precessiondampingEnhance 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 monopolesMonopole , which act like the slider of a zipper connecting two skyrmion strings. The existence of such monopole-like defects diminishes the spatial long-range 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 $\mu$m at low temperatures, proving that the spin excitations on the skyrmion strings can propagate a distance exceeding 103 times the diameter of single skyrmion string ($\approx 50$ nmCu2OSeO3_Seki ; Cu2OSeO3_SANS_Pfleiderer ). Such a small damping and long propagation distance of the excitation are rather unexpected for the SkX phase with its intricate three-dimensional spin texture, and the above results suggest that the skyrmion string with excellent long-range order can be a good medium for magnon transport.

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 mode-dependent in full agreement with our calculated dispersion relations. These results establish the comprehensive picture of the propagation dynamics of skyrmion string excitations. Moreover, we developed a general theoretical framework to evaluate the dispersion asymmetry for non-collinear 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. Since 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 skyrmion string dynamics in 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 recentlyisolatedStringOne ; isolatedStringTwo ; isolatedStringThree . Skyrmion strings can be bent into curved shapes due to their topological protectionMonopole , and the associated spin excitations are expected to propagate along the string path in analogy with the Kelvin mode of vortex lines in superfluidsRMP_superfluids ; PNAS_Kelvinwave ; isolatedStringTwo . These features suggest that skyrmion strings may serve as robust and flexible information transmission lines without Joule heat loss. The nonlinear dynamics and networking of these strings are another fascinating issues. Our results highlight that not only the skyrmion particle in two-dimensional systems, but also the skyrmion string in three-dimensional systems is an attractive topological object, and further investigation of its fundamental and functional properties will be a promising endeavor.

Methods

Single crystals of Cu2OSeO3 were grown by the chemical vapor transport method. A pair of Au coplanar wave guides (CPW: 200 nm thickness) were fabricated on the oxidized silicon substrate through the standard photolithography technique, and a plate-shaped single crystal of Cu2OSeO3 ($\sim 2\mu$m thickness) was placed across them with W (tungsten) deposition at an edge of the crystal using the focused ion beam (FIB) micro fabrication technique (Fig. 1b). By measuring the magnetic contribution to the complex spectra of self-inductance $\Delta L_{11}$ and mutual inductance $\Delta L_{nm}$ (with $m$ and $n$ being the port number used for the excitation and detection, respectively) for these CPWs with the vector network analyzer, the local excitation character and propagation character of spin excitation can be directly evaluated, respectivelyAESWS_doppler_B ; AESWS_doppler_Science . The spin excitation contribution to the inductance spectrum $\Delta L_{nm}(\nu)=L_{nm}(\nu)-L_{nm}^{\rm ref}(\nu)$ is derived by the subtraction of the common background $L_{nm}^{\rm ref}(\nu)$ from the raw data $L_{nm}(\nu)$. Here, $L_{nm}(\nu)$ taken at $\mu_{0}H=250$ mT is adopted as the reference spectrum $L_{nm}^{\rm ref}(\nu)$, where the magnetic resonance is absent within our target frequency range of 0.2 GHz $\leq\nu\leq$ 7.0 GHz. Unless specified, we employed the device with the central wave length $\lambda^{\rm SW}=12\mu$m and the gap distance $d=20\mu$m.

Author contributions

S.S. performed the measurements. S.S., R.T., Y.O., K.K., F.K., 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.

Acknowledge

The authors thank J. Iwasaki, S. Hoshino, N. Nagaosa, and H. Hosono for enlightening discussions and experimental helps. This work was partly supported by Grants-In-Aid for Scientific Research (A) (Grant No. 18H03685) and Grant-in-Aid for Scientific Research on Innovative Area, ”Nano Spin Conversion Science” (Grant No.17H05186) from JSPS, and PRESTO (Grant No. JPMJPR18L5) from JST. M. G. was supported by the DFG via SFB 1143 ”Correlated magnetism: from frustration to topology” and grant GA 1072/6-1.