Kelvin modes of a skyrmion line in chiral magnets and the associated magnon transport

TL;DR

This paper investigates Kelvin modes of skyrmion lines in chiral magnets, revealing symmetric low-energy modes and asymmetric higher modes, suggesting skyrmion lines can act as one-way magnon waveguides.

Contribution

It provides the first detailed analysis of Kelvin mode dispersion in skyrmion lines, highlighting symmetry properties and potential for unidirectional magnon transport.

Findings

Lowest Kelvin mode is symmetric at low energy despite inversion symmetry breaking.

Higher Kelvin modes are asymmetric and mostly have positive group velocity.

Skyrmion lines can serve as one-way waveguides for magnons.

Abstract

Magnetic skyrmions in bulk crystals are line-like topologically protected spin textures. They allow for the propagation of magnons along the skyrmion line but are localized inside the skyrmion line. Analogous to the vortex line, these propagating modes are the Kelvin modes of a skyrmion line. In crystals without an inversion center, it is known that the magnon dispersion in the ferromagnetic state is asymmetric in the wavevector. It is natural to expect that the dispersion of the Kelvin modes is also asymmetric with respect to the wavevector. We study the Kelvin modes of a skyrmion line in the ferromagnetic background. In contrast, we find that the lowest Kelvin mode is symmetric in the wavevector in the low energy region despite the inversion symmetry breaking. Other Kelvin modes below the magnon continuum are asymmetric, and most of them have a positive group velocity. Our results suggest that a skyrmion line can function as a one-way waveguide for magnons.

Figures

Equations

$$\mathcal{H}=\int d{r}^3\left[\frac{J}{2}\sum _{\mu =x,y}{\left({\partial }_{\mu }\mathbf{n}\right)}^2+D\mathbf{n}\cdot \nabla \times \mathbf{n}-B_z{n}_z-\frac{A}{2}{n}_z^2\right],$$

$$\frac{S\hslash }{a^3}{\omega }_{\mathrm{FM}}=J{\mathbf{k}}^2+2D{k}_z+B_z+A.$$

$$E_z=\frac{1}{2}J({\partial }_z\mathbf{n}{)}^2+D\left(n_y{\partial }_z{n}_x-n_x{\partial }_z{n}_y\right).$$

$$E_z=\frac{1}{2}\eta J\int dz\left[({\partial }_z{u}_x{)}^2+({\partial }_z{u}_y{)}^2\right],$$

$$S_B=S_B(\mathbf{u}=0)+\frac{S\hslash \pi }{a^3}\left(u_x{\partial }_t{u}_y-u_y{\partial }_t{u}_x\right).$$

$$\frac{2\pi \hslash S}{a^3}{\partial }_t{u}_y-J\eta {\partial }_z^2{u}_x=0,$$

$$-\frac{2\pi \hslash S}{a^3}{\partial }_t{u}_x-J\eta {\partial }_z^2{u}_y=0.$$

$$\omega =\frac{a^3\eta J}{2\pi \hslash S}{k}_z^2.$$

$$\frac{J}{2r}\sin \left(2\theta \right)+D\cos \left(2\theta \right)+B_{z}r\sin \theta +\frac{A}{2}r\sin \left(2\theta \right)$$

$$-\left(J{\partial }_r\theta +D\right)-rJ{\partial }_r^2\theta =0,$$

$$\hat{O}=\left(\begin{array}[]{ccc}-\text{sin$\varphi$}&-\text{cos$\varphi$}\text{cos$\theta$}&\text{cos$\varphi$}\text{sin$\theta$}\\ \text{cos$\varphi$}&-\text{sin$\varphi$}\text{cos$\theta$}&\text{sin$\varphi$}\text{sin$\theta$}\\ 0&\text{sin$\theta$}&\text{cos$\theta$}\\ \end{array}\right).$$

$$\psi =\frac{L_X+i{L}_Y}{\sqrt{2}},{\psi }^{\ast }=\frac{L_X-i{L}_Y}{\sqrt{2}},$$

$${\mathcal{H}}_{\psi }=\frac{1}{2}{\hat{\psi }}^{†}{H}_{\psi }\hat{\psi },{\hat{\psi }}^{†}=({\psi }^{\ast },\psi ),$$

$$H_{\psi }=(-J{\nabla }^2+V_0){\sigma }_0+V_1{\sigma }_x$$

$$-2{\sigma }_z\left[\left(J\frac{\cos \theta }{r^2}-D\frac{\sin \theta }{r}\right)i{\partial }_{\varphi }-iD\cos \theta {\partial }_z\right],$$

$$V_0=J\frac{1+3\cos \left(2\theta \right)}{4r^2}-D\frac{3\sin \left(2\theta \right)}{2r}+B_z\cos \theta -D{\partial }_r\theta$$

$$-\frac{J}{2}{\left({\partial }_r\theta \right)}^2-\frac{A}{2}\left(1-3{\cos }^2\theta \right),$$

$$V_1=J\frac{{\sin }^2\theta }{2r^2}+D\frac{\sin \left(2\theta \right)}{2r}-D{\partial }_r\theta -\frac{J}{2}{\left({\partial }_r\theta \right)}^2+\frac{A}{2}{\sin }^2\theta .$$

$$\mathbf{a}=-\hat{\varphi }\left(\frac{\cos \theta }{r}-\frac{D\sin \theta }{J}\right)+\frac{D\cos \theta }{J}\hat{z},$$

$$H_{\psi }=J{\left(-i\nabla -{\sigma }_3\mathbf{a}\right)}^2+{\sigma }_0\left(V_0-J{\mathbf{a}}^2\right)+{\sigma }_1{V}_1.$$

$$-i\frac{S\hslash }{a^3}{\sigma }_z{\partial }_t\hat{\psi }=H_{\psi }\hat{\psi }.$$

$${\hat{\psi }}_m=b\exp [i(\omega t+m\varphi +k_{z}z)]{\hat{\eta }}_m+b^{\ast }\exp [-i(\omega t+m\varphi +k_{z}z)]{\sigma }_{x}K{\hat{\eta }}_m.$$

$$\frac{S\hslash }{a^3}{\omega }_m{\sigma }_z{\hat{\eta }}_m=H_{\psi }{\hat{\eta }}_m.$$

$$\displaystyle|{p_{m,i}}\rangle=\frac{{\sqrt{2}}}{{{R_{c}}{J_{m}}\left({{k_{m-1,i}}}\right)}}{J_{m-1}}\left({{k_{m-1,i}}\frac{r}{{{R_{c}}}}}\right)\exp\left({{{i}}m\phi+ik_{z}z}\right)\left({\begin{array}[]{*{20}{c}}1\\ 0\end{array}}\right),$$

$$\displaystyle|{h_{m,i}}\rangle=\frac{{\sqrt{2}}}{{{R_{c}}{J_{m+2}}\left({{k_{m+1,i}}}\right)}}{J_{m+1}}\left({{k_{m+1,i}}\frac{r}{{{R_{c}}}}}\right)\exp\left({{{i}}m\phi+ik_{z}z}\right)\left({\begin{array}[]{*{20}{c}}0\\ 1\end{array}}\right),$$

$${\hat{H}}_{11;ij}^{(m)}=⟨p_{m,i}|H_{\psi }|p_{m,j}⟩,{\hat{H}}_{12;ij}^{(m)}=⟨p_{m,i}|H_{\psi }|h_{m,j}⟩,$$

$${\hat{H}}_{21;ij}^{(m)}=⟨h_{m,i}|H_{\psi }|p_{m,j}⟩,{\hat{H}}_{22;ij}^{(m)}=⟨h_{m,i}|H_{\psi }|h_{m,j}⟩.$$

Full text

††preprint: S.-Z. Lin, J.-X. Zhu and A. Saxena, Phys. Rev. B 99, 140408(R) (2019).

Lord Kelvin calculated stable propagating wave modes along a straight vortex tube of uniform vorticity in a classical fluid about 130 years ago. Kelvin1880 These modes were later called Kelvin modes. The Kelvin modes in quantized vortex lines were subsequently studied Pitaevskii1961; PhysRev.162.143 and observed experimentally RevModPhys.59.87 in quantum superfluids. The spectrum of the Kelvin modes in the long wavelength limit is $\hbar\omega=\frac{\hbar^{2}k^{2}}{2m}\ln(1/k\xi)$, where $m$ is the mass of a particle in the superfluid, $k$ is the momentum, and $\xi$ is the healing length.

Recently, a vortex-line like topological spin texture, known as skyrmion, has been observed in magnets by experiments.Muhlbauer2009; Yu2010a A large family of skyrmion-hosting materials has been identified. A skyrmion has several intrinsic properties, such as polarization, vorticity and helicity, and these properties are determined by the symmetries of the material and/or external magnetic fields. For skyrmions in materials without inversion symmetry, the Dzyaloshinskii-Moriya interaction Dzyaloshinsky1958; Moriya60; Moriya60b (DMI) is responsible for the stabilization of a skyrmion lattice. Bogdanov89 Skyrmions can also exist in systems with inversion symmetry, where competing magnetic interactions stabilize the skyrmions. PhysRevLett.108.017206; leonov_multiply_2015; PhysRevB.93.064430; PhysRevB.93.184413; hirschberger_skyrmion_2018; kurumaji_skyrmion_2018 In thin films, skyrmions appear as disk-like excitations and in bulk materials, skyrmions are line-like excitations. Skyrmions can be manipulated by various external drives, such as electric current, electric field, thermal gradient, etc. Jonietz2010; Yu2012; Schulz2012; White2012; PhysRevLett.113.107203; Kong2013; Lin2014PRL; Mochizuki2014 Remarkably, skyrmions can be driven into motion by a small current density of the order of $10^{6}\ \mathrm{A/m^{2}}$, which is $5$ to $6$ orders of magnitude smaller than that for magnetic domain walls. Jonietz2010; Yu2012; Schulz2012 For their superior properties including compact size, high mobility and stability, skyrmions have attracted tremendous attention recently and are deemed as promising candidates for applications in the next generation spintronic devices. Fert2013; nagaosa_topological_2013

A skyrmion line can also support propagating Kelvin modes inside the line, see Fig. 1 (a). In centrosymmetric systems, the Kelvin mode is symmetric in the propagating wavevector. PhysRevD.90.025010 In systems without inversion symmetry, the question is whether the dispersion of the Kelvin modes is asymmetric with respect to the wavevector. If the answer is positive, this would imply a one-way propagation of magnons inside a skyrmion line, and therefore the skyrmion line can work as a one-way magnon waveguide. This is the question we will address in this work.

First, we consider the magnon dispersion in the fully spin polarized state. We begin with a phenomenological description of the magnetization in a chiral magnet. The Hamiltonian of the system in terms of the magnetization field $\mathbf{n}(\mathbf{r})$ with $|\mathbf{n}|=1$ is bak_theory_1980; Bogdanov89

[TABLE]

which successfully captures many experimental observations in chiral magnets. Here $J$ is the exchange interaction, $D$ is the DMI Dzyaloshinsky1958; Moriya60; Moriya60b and $B_{z}$ is the external magnetic field. We have introduced an easy axis anisotropy $A>0$. For B20 compounds with cubic symmetry, this term is not allowed, bak_theory_1980 but this anisotropy can be generated by uniaxial stress. It can also exist in other crystals with a layered structure. We have neglected the weak dipolar interaction. Note that the skyrmion size is much bigger than the spin lattice constant, and this justifies the continuum approximation in Eq. (1). For the field value above the saturation field, a ferromagnetic state is stabilized, where $\mathbf{n}=\hat{z}$ with $\hat{z}$ being a unit vector in the $z$ direction. Butenko2010; leonov_properties_2016 The dynamics of $\mathbf{n}$ is determined by the Berry phase contribution to the action $S_{B}=\frac{S\hbar}{2a^{3}}\int dr^{3}dt\partial_{t}\varphi(\cos\theta+1)$, where $S$ is the total spin of the ion (in the material) and $a$ is the crystal lattice parameter. Here $\varphi$ and $\theta$ are the spherical angles of $\mathbf{n}$, i.e. $\mathbf{n}=(\sin\theta\cos\varphi,\ \sin\theta\sin\varphi,\ \cos\theta)$. The magnon dispersion is

[TABLE]

The magnon dispersion is asymmetric with respect to $k_{z}$ consistent with the inversion symmetry breaking. This asymmetric magnon dispersion has been observed in experiments, PhysRevB.97.224403; PhysRevLett.120.037203 and provides a useful way to determine the strength of the DMI. Here the ferromagnetic (FM) state is stable for the field above $B_{c}=D^{2}/J-A$. The asymmetry gives rise to unconventional magnon propagation such as the modified Snell’s law. PhysRevB.94.140410 Note that the asymmetry only appears in $k_{z}$ along the field direction, while the dispersion with respect to $k_{x}$ and $k_{y}$ remain symmetric. Therefore for a thin film with a normal magnetic field, the magnon dispersion is symmetric with respect to the in-plane wave vectors, even though the inversion symmetry is broken.

A skyrmion line can exist as a metastable state in the background of the FM state. The skyrmion line provides a centrosymmetric potential for magnon excitations, and it allows for the existence of localized magnons. This was calculated in thin films. PhysRevB.90.094423; Lin_internal_2014 In clean systems, the magnon modes can be labeled by angular momentum $m$ and wavevector $k_{z}$. In the following discussion, we call these modes the Kelvin modes with quantum numbers $m$ and $k_{z}$. Let us consider the lowest mode associated with the translation of the skyrmion line. The translation of the whole straight skyrmion line does not cost any energy and it is a Goldstone mode of the system. The bending of the skyrmion line costs energy and results in the dispersion of the corresponding Kelvin mode. The displacement of a rigid skyrmion line can be described by $\mathbf{n}_{s}[\mathbf{r}-\mathbf{u}(z)]$, where $\mathbf{u}(z)=[u_{x}(z),\ u_{y}(z)]$ is the displacement vector. The $z$ independent displacement of a skyrmion line does not cost energy, i.e. $\mathcal{H}[\mathbf{n}_{s}(\mathbf{r}-\mathbf{u}_{0})]=\mathcal{H}[\mathbf{n}_{s}(\mathbf{r})]$. For a long wavelength distortion, the energy functional can be expanded in the basis of $\partial_{z}\mathbf{u}$, $\mathcal{F}(\mathbf{u})\propto\int dz(\partial_{z}\mathbf{u})^{2}+\cdots$. The first order term $\partial_{z}\mathbf{u}$ appears as a surface term upon integration. It vanishes when the two ends of the skyrmion line are fixed.

The energy cost to distort a skyrmion line can also be obtained directly from Eq. (1) and is

[TABLE]

The contribution from the DMI vanishes as obtained by straightforward calculations. Therefore the distorted skyrmion line has an energy cost

[TABLE]

with $\eta\equiv\int dr^{2}(\partial_{x}\mathbf{n}_{s})=\int dr^{2}(\partial_{y}\mathbf{n}_{s})$. The stiffness of the skyrmion line is $\eta J$ and is independent of the DMI, which is consistent with the fact that $J$ is the largest energy scale of the problem.

In terms of $\mathbf{u}(z)$, the Berrry phase part of the action becomes

[TABLE]

The total action associated with the distortion of a skyrmion line is $S_{T}=S_{B}-E_{z}$. The equation of motion for $\mathbf{u}(z)$ is

[TABLE]

Therefore the dispersion of this Kelvin mode is (as will be shown below, this Kelvin mode has $m=\pm 1$)

[TABLE]

This Kelvin mode is symmetric with respect to $k_{z}$. This is different from the magnon mode in the ferromagnetic state, where the magnon dispersion is asymmetric due to the presence of DMI. The mode is gapless. A gap exists when there is a local pinning potential or geometric confinement in a small system. The gap can be introduced into the action $S_{T}$ by adding a mass term $M\mathbf{u}^{2}/2$ for a straight skyrmion line. Nevertheless, the dispersion of other Kelvin modes is asymmetric with respect to $k_{z}$ as will be shown below.

To go beyond the analysis of the skyrmion displacement field, we calculate the magnon spectrum in the presence of a straight skyrmion line embedded in the ferromagnetic background. For a disk-like skyrmion in thin films, the spectrum was calculated in in Refs. PhysRevB.90.094423; Lin_internal_2014. Here we extend the method used in Ref. PhysRevB.90.094423 to three dimensions. First, we find the stationary solution of a straight skyrmion line. The symmetry of the problem allows us to use cylindrical coordinates $\mathbf{r}=(r,\ \phi,\ z)$. The skyrmion line solution in Eq. (1) has the form $\varphi=\phi+\pi/2$ (skyrmion helicity is $\pi/2$ determined by the DMI) and $\theta(r)$ with $\theta$ changing from $\theta=-\pi$ at the skyrmion center $r=0$ to $\theta=0$ at $r=\infty$. We obtain the equation for $\theta(r)$ by minimizing $\mathcal{H}$

[TABLE]

from which $\theta(r)$ can be found numerically.

We then introduce a local coordinate system with the local $z$ axis along the spin direction $\mathbf{n}_{s}(\mathbf{r})$. The spin representation in the lab coordinate and the local coordinate is sketched in Fig. 1 (b). The local coordinate is obtained by the subsequent rotation operations in the lab frame: rotation along the $z$ axis by $\phi_{0}=\pi/2$, rotation along the $y$ axis by $\theta$ and rotation along the $z$ axis by $\varphi$. Then the spin in the lab frame $\mathbf{n}$ can be obtained from the local coordinate $\mathbf{L}=(L_{X},\ L_{Y},\ L_{Z})$ according to $\mathbf{n}=\hat{O}\mathbf{L}$, with

[TABLE]

The small deviations $\mathbf{L}$ from the skyrmion line solution $\bar{L}_{X}=\bar{L}_{Y}=0$ and $\bar{L}_{Z}=1$ are described by the complex magnon fields

[TABLE]

and $L_{Z}=1-\psi\psi^{*}$ with $|\psi|\ll 1$. Expanding the Hamiltonian to second order in $\psi$, we obtain

[TABLE]

[TABLE]

with $\sigma_{i}$ ($i=x,\ y,\ z$) being the Pauli matrices and $\sigma_{0}$ is the unit matrix. Here

[TABLE]

[TABLE]

The presence of skyrmion gives rise to an emergent magnetic field acting on the magnons. This can be seen explicitly by introducing an effective vector potential

[TABLE]

with $\hat{\phi}$ being the unit vector in the $\phi$ direction. Using $\nabla\cdot\mathbf{a}=0$, $H_{\psi}$ can be written in a compact form

[TABLE]

The emergent vector potential $\mathbf{a}$ couples to the magnons and induces a screw scattering of the extended magnons by skyrmions.PhysRevB.89.064412; PhysRevB.90.094423

The eigenmodes are determined by the equation

[TABLE]

This equation has the form of the Schrödinger equation describing the magnon wave function in a centrosymmetric potential. We can introduce an angular momentum $m$ and wavevector $k_{z}$ with $\psi=\psi_{m}(r,t)\exp(im\phi+ik_{z}z)$ to label the eigenmodes. The two components of $\hat{\psi}$ are related by complex conjugation because the magnetic moment $\mathbf{n}$ is real. This indicates that the matrix equation, Eq. (17), is redundant. Indeed ${H}_{\psi}$ has particle-hole symmetry, ${H}_{\psi}=\sigma_{x}K{H}_{\psi}K\sigma_{x}$ with $K$ being the complex conjugate operator. This means that if $\exp[i(\omega t+m\phi+k_{z}z)]\hat{\eta}_{m}$, with $\hat{\eta}_{m}^{\dagger}\equiv(\eta_{1}^{*},\ \ \eta_{2}^{*})$, solves Eq. (17), then $\exp[-i(\omega t+m\phi+k_{z}z)]\sigma_{x}K\hat{\eta}_{m}$ also solves Eq. (17). We therefore only take the magnon branch with $\omega\geq 0$. Then $\hat{\psi}$ can be obtained by a linear superposition of the two symmetry-related solutions

[TABLE]

The two components of $\hat{\psi}_{m}$ are complex conjugate to each other. Here $\hat{\eta}$ is determined by the eigenvalue problem

[TABLE]

When the frequency is much larger than the magnon gap of the FM state, $\frac{S\hbar}{a^{3}}\omega_{g}=B_{z}+A-D^{2}/J$, i.e. $\omega\gg\omega_{g}$, the magnon dispersion reduces to that in Eq. (2) and the eigenmodes are $\hat{\eta}_{m}^{\dagger}=(1,\ 0)J_{m}(kr)$. We represent the matrix ${H}_{\psi}$ using the Bessel function $J_{m}(kr)$ as an orthogonal basis. PhysRevB.96.014407 The basis functions are

[TABLE]

[TABLE]

where we have used the box normalization with $R_{c}$ being the radius of the box and $k_{m,i}$ is the $i$-th zero of the Bessel function $J_{m}(kr)$. Then the matrix elements of ${H}_{\psi}$ are

[TABLE]

By diagonalizing the matrix $\sigma_{z}{H}_{\psi}$, we obtain the eigenfrequencies and eigenmodes. We take $R_{c}=20$ and truncate the Bessel series at $i_{\mathrm{max}}=20$.

The calculated dispersion of the Kelvin modes with different $m$ is shown in Fig. 2. There are only two Kelvin modes below the magnon continuum when $B_{z}=1.4D^{2}/J$ in Fig. 2 (a), and as a consequence, these modes are radially localized inside the skyrmion. The other modes mix with the magnon continuum and can easily decay into the extended magnon modes. The Kelvin mode with $m=-1$ corresponds to the distortion of a rigid skyrmion line discussed above. It is symmetric with respect to $k_{z}$ in the low energy region consistent with that in Eq. (8). The mode with $m=0$ corresponds to the uniform radial breathing of the skyrmion line. The group velocity $v_{g}=d\omega/dk_{z}$ is always positive, indicating a one-way propagation of this Kelvin mode. At a lower field, $B=1.05D^{2}/J$ in Fig. 2 (b), the Kelvin mode with $m=-2$ also appears below the magnon continuum. In the presence of an easy axis anisotropy, there appear more Kelvin modes below the magnon continuum, see Fig. 3. The Kelvin mode with $m=-1$ in Figs. 2 and 3 has a very small gap originating from the numerical discretization in the calculations, which breaks the translation symmetry. The left branch of the Kelvin mode with $m=-1$ at high energy merges into the magnon continuum, and therefore is strongly damped. It only allows magnons with a positive group velocity to propagate in this region. Here all the Kelvin modes with $m\neq-1$ are gapped, which guarantee the meta-stability of the skyrmion line in the ferromagnetic background.

To use the skyrmion line as a one-way magnonic waveguide, it is required to excite the Kelvin mode with $m\neq-1$. This can be achieved by choosing the angular momentum of the source field. Recently, the propagation of magnons both with symmetric and asymmetric dispersion in a skyrmion line in prismatic geometry was demonstrated using micromagnetic simulations. xing_skyrmion_2019 The propagation of linear magnon wave and nonlinear solitary wave excitations along a skyrmion line in chiral magnets was considered in Ref. kravchuk_solitary_2019. The results on the linear magnon wave are consistent with ours. The unidirectional propagation of magnon in the skyrmion line crystal in $\mathrm{Cu_{2}OSeO_{3}}$ was investigated both experimentally and theoretically. seki_propagating_2019

We have focused on a system with DMI form in Eq. (1), which can be realized in crystals having $D_{n}$ or $C_{n}$ symmetry, Bogdanov89 for example B20 chiral magnets including FeGe and MnSi. For crystals with $C_{nv}$ or $D_{2d}$ symmetry, such as Mn-Pt-Sn Heusler materials, nayak_magnetic_2017 $\mathrm{GaV_{4}S_{8}}$, kezsmarki_neel-type_2015 $\mathrm{GaV_{4}Se_{8}}$, PhysRevB.95.180410 and $\mathrm{VOSe_{2}O_{5}}$ PhysRevLett.119.237201 no spatial derivative along the crystal $c$ axis is allowed in the DMI. PhysRevB.96.214413 In these systems, all the Kelvin modes are symmetric with respect to $k_{z}$.

It is possible that skyrmion lines that do not percolate the whole system are stabilized. Milde2013; yokouchi_current-induced_2018; SZLinMonopole2016; PhysRevB.94.174428; PhysRevB.98.054404 At the ends of the lines, there appear emergent magnetic monopoles or antimonopoles. When both ends of a skyrmion line are terminated by a monopole and an antimonopole, the skyrmion line serves as a magnonic cavity for the Kelvin modes because these localized modes cannot penetrate into the ferromagnetic state. In imperfect systems, the skyrmion line is distorted in order to accommodate the pinning potential. Blatter94 The pinning opens a gap for the symmetric lowest Kelvin mode. The bent skyrmion line can still guide the Kelvin modes to propagate along the line.

To summarize, we have studied the Kelvin modes of a straight skyrmion line in chiral magnets. There exist several Kelvin modes below the magnon continuum, and these modes are radially localized in the skyrmion line. The Kelvin mode with angular momentum $m=-1$ is symmetric with respect to the wavevector along the skyrmion line in the low energy region. The Kelvin modes with other $m$ are asymmetric. Our results suggest that the skyrmion lines can function as a one-way magnonic waveguide.