Magnonic Activity of Circularly Magnetized Ferromagnetic Nanotubes Induced by Dzyalonshinskii-Moriya Interaction

Abstract

Magnonic activity, a chiral effect in magnetization dynamics, was recently reported in ferromagnetic nanotubes. Being a perfect analogy to the optical activity, it refers to the continuous rotation of a standing-waves pattern formed in the circumferential direction during the wave propagation along the tube. This effect only occurs when the tube is longitudinally magnetized. Here we report that a similar phenomenon can also take place in circularly magnetized nanotubes with the presence of Dzyalonshinskii-Moriya interaction (DMI). While in the former case, the chiral-symmetry breaking is caused by the curvilinear shape of the tube, it is attributed to the intrinsic asymmetry of the DMI in the latter one. We present the results obtained in both numerical simulations and semi-analytical calculations, which are in great agreement. This work provides new aspects for the manipulation of spin waves, which may bear potential applications in the development of novel spintronic devices.

1. Introduction

Historically, the discovery and exploration of optical activity played an important role in the recognition of chirality in nature [1,2,3,4,5]. It reveals the dependence of the propagation of light on the chiral stereo-structure of molecules composing the medium [1]. Recently, an analogous wave effect, referred to as magnonic activity, was reported for spin waves (SWs) traveling in ferromagnetic nanostructures [6]. While an optically-active substance causes the continuous rotation of the polarization of electromagnetic waves passing through, a longitudinally magnetized nanocylinder features SW modes with rotating circumferential standing-wave patterns during the propagation. Such a phenomenon, being a direct consequence of chiral-symmetry breaking attributed to the curvilinear shape of the nanocylinder acting as the SW wave guide, is thus a pure geometric effect [6].

In this work, we illustrate another mechanism that can lead to the magnonic activity, the Dzyalonshinskii-Moriya interaction (DMI). DMI is an intrinsically asymmetric exchange interaction, existing in systems that lack inversion-symmetry [7,8,9]. With the presence of DMI, various magnetic chiral structures can form. A particularly interesting example is the magnetic skyrmions, which have been extensively studied in the last decade [7,10,11,12,13,14,15,16,17]. DMI also exerts influences on magnetization dynamics, such as the domain-wall motion [18,19] and SW propagations [20,21,22,23,24,25]. In ferromagnetic thin-film strips, for instance, DMI can cause the nonreciprocal dispersion relation of SWs [22,25]. Here we consider ferromagnetic cylindrical nanotubes with bulk DMI and study the SW modes propagating along the tubes using micromagnetic simulations. Our results show that the effect of magnonic activity also occurs in a circularly magnetized tube. Similar to the case of a longitudinally magnetized tube without DMI, the SW modes characterized by the rotation of the wave front with a circular standing-wave pattern can be decomposed into two eigen states, the L- and R-state (corresponding to the left- and right-handed circularly polarized lights in optics, respectively) with different dispersion relations. An excellent agreement is achieved between two approaches for the calculation of the dispersion relations of the L- and R-state, either by directly measuring the independently excited eigen modes or extracting from the profiles of the composite modes. By both numerical and analytical analysis, we argue that it is the DMI in this case that is responsible for this dynamic chiral effect. As a hot topic in spintronics, DMI has been intensively studied in the last decade. However, DMI-induced magnonic activity in magnetic nanostructures was never reported before. Our work enriches chiral magnetization dynamics in magnetic cylindrical nanowires and provides in principle new aspects for the manipulation of SWs.

2. Materials and Methods

We use the open source package MuMax3 [26] to perform micromagnetic simulations. The magnetization dynamics are governed by the Landau-Lifshitz-Gilbert (LLG)equation, as shown in Equation (1), which can be numerically solved to analyze the magnetization evolution in magnetic systems.

$$\frac{d\overrightarrow{m}}{dt}=-\mathsf{\gamma }\overrightarrow{m}\times {\overrightarrow{H}}_{eff}+\alpha \left[\overrightarrow{m}\times \frac{d\overrightarrow{m}}{dt}\right]$$

(1)

where $\overrightarrow{m}=\frac{\overrightarrow{M}}{M_s}$ is the normalized magnetization vector, γ the gyromagnetic ratio, ${\overrightarrow{H}}_{eff}$ the effective field which takes into account the DMI, α the Gilbert damping parameter, and $M_s$ the saturation magnetization. We consider a cylindrical tube with a length of 4000 nm and internal and external radii of 25 nm and 30 nm, respectively. In the simulation, the tube is discretized into 1 nm³ cubic cells. We use Permalloy as a modeling system. The parameters of the material are set as ${\mu }_0{M}_s=1\mathrm{T}$, the exchange constant $A=1.3\times {10}^{-11}$ J/m and $\alpha =0.01$. The magnetocrystalline anisotropy is zero. A circumferential static magnetic field is applied to magnetize the tube around its axis in equilibrium [27,28]. Consequently, the effective field ${\overrightarrow{H}}_{eff}$ includes four terms, the exchange field, the external field, the dipole field and the DMI equivalent field, which can be written as

$${\overrightarrow{H}}_{eff}={\overrightarrow{H}}_{ex}+{\overrightarrow{H}}_0+{\overrightarrow{H}}_d+{\overrightarrow{H}}_{DM}$$

(2)

The exchange field is given by ${\overrightarrow{H}}_{ex}=A^{*}{\nabla }^2\overrightarrow{m}$, $A^{*}=\frac{2A}{{\mu }_0{M}_s}$. The static field ${\overrightarrow{H}}_0=H_o\widehat{\phi }$ (φ is the azimuthal angle in a cylindrical coordinate system). The dipolar field ${\overrightarrow{H}}_d=-\nabla \varnothing$ (∅ is the dynamic dipolar potential). The DMI energy density is ${\epsilon }_{DM}=D\overrightarrow{m}·\left(\nabla \times \overrightarrow{m}\right)$, where D is the DMI constant. The corresponding DMI field term is given by

$${\overrightarrow{H}}_{DM}=-\frac{1}{{\mu }_0{M}_s}\frac{\partial {\epsilon }_{DM}}{\partial \overrightarrow{m}}=-D^{*}\nabla \times \overrightarrow{m}$$

(3)

where $D^{*}=\frac{2D}{{\mu }_0{M}_s}$. The total magnetization vector in the excited system can be written as $\overrightarrow{m}={\overrightarrow{m}}_0+\delta \overrightarrow{m}$, where ${\overrightarrow{m}}_0=\widehat{\phi }$ is the static part and $\delta \overrightarrow{m}$ the dynamic one.

The equilibrium state of the tube is shown in Figure 1, as the direction of the local magnetization represented by gray arrows. In order to stimulate the system, a rf field oscillating with a particular frequency is applied in the middle region of the tube, indicated by the yellow ring. Two SW branches are thus excited propagating along the tube to the left and right directions, respectively. We consider those two SWs traveling in opposite directions separately and define two cylindrical coordinates correspondingly. For each of them, the positive direction of the polar axis is parallel to the wave vectors ${\overrightarrow{k}}_z$ of the SW. Note that one can define a chirality by combining the magnetization direction of the tube with the propagation direction of the SW. For the left (right) half of the tube, the chirality defined here is left (right)-handed.

In the following, we discuss the properties of SW modes obtained in our simulations. For comparison, we first demonstrate a typical mode pattern obtained in a circularly magnetized tube without DMI, as shown in Figure 2a. The upper panel is the side view of the original tube, showing a snapshot of the dynamic radial component ($\delta m_{\rho }$) of magnetization when excited by a 52 GHz oscillating field. The lower panel shows the same mode profile when the tube is “rolled-out”. Clearly, a normal circular standing-wave pattern with an order number n = 2 ($k_{\phi }=\frac{2n\pi }{L}$, where L is the outer circumference of the tube) is formed in the azimuthal direction. Note that during the wave propagation along the tube, the azimuthal locations of the nodes (antinodes) remain unchanged, just like regular electromagnetic waves traveling in waveguides.

However, an obviously different wave pattern is obtained in the same tube with the presence of bulk DMI, as shown in Figure 2b. The mode profile in this case is characterized by a continuous and uniform rotation of the azimuthal positions of the nodes (antinodes) of the circular standing-wave. The sense of rotation depends on the propagating direction of the SW branches. For the (left) right-running wave, the rotation appears to be (clockwise) counterclockwise. We point out that this peculiar SW behavior is essentially the same as that observed in a longitudinally magnetized nanocylinder without DMI, referred to as magnonic activity [6]. In the next section, we will provide a physical explanation for this chiral SW effect.

3. Analysis and Discussions

In Ref. [6], a physical picture of the magnonic activity of a longitudinally magnetized tube was provided, based on Fresnel’s model for optical activity [1]. In this picture, the SW modes possessing a circumferential standing-wave pattern can be decomposed into two circular eigen modes (L- and R-state). This is analogous to a linearly polarized light beam consisting of two circularly polarized lights. If chiral symmetry is not preserved in the magnetic system, the L- and R-state will acquire different dispersion relations, resulting in the magnonic activity effect for the composite SW mode. In a longitudinally magnetized tube without DMI, the chiral-symmetry breaking responsible for the magnonic activity is attributed to the curvilinear shape of the cylinder. In this case, as we will show below, it is the DMI that violates the chiral symmetry and causes the discrepancy between the L- and R-state.

For a circularly magnetized tube, the dynamic magnetization after excitation can be written as $\delta \overrightarrow{m}=\delta {\overrightarrow{m}}_{\rho }+\delta {\overrightarrow{m}}_z$ in a cylindrical coordinate system. Following the spirit of Ref. [6], each dynamic component of the SW mode described above can be written as

$$\begin{array}{l}\delta {\overrightarrow{m}}_j\left(\phi ,z,t\right)=\delta {\overrightarrow{m}}_j\left(e^{i\left(\omega t+n\phi -k_{z1}z\right)}+e^{i\left(\omega t-n\phi -k_{z2}z\right)}\right)\\ =2\delta {\overrightarrow{m}}_j\cos \left[n\left(\phi -\frac{k_z^{\prime }}{n}z\right)\right]e^{i\left(\omega t-k_z^{″}z\right)},j=\rho ,z\end{array}$$

(4)

where $e^{i\left(\omega t+n\phi -k_{z1}z\right)}$ and $e^{i\left(\omega t-n\phi -k_{z2}z\right)}$ are the exponential parts of the so-called R-state and L-state, respectively. Those two states are assumed to have the same frequency $\omega$, yet different wave vectors $k_{z1}$ and $k_{z2}$. The superposition of the two states yields a mathematical expression (Equation (4)), which gives a precise description of the SW mode profile obtained in simulations shown in Figure 2b. In Equation (4), $n$ indicates the order of the circumferential standing wave, $k_z^{\prime }=\frac{k_{z1}-k_{z2}}{2}$ defines the rotational rate (angle rotated per length during propagation) of the wave front and $k_z^{″}=\frac{k_{z1}+k_{z2}}{2}$ the propagation speed of the wave.

Next, we demonstrate that in the system studied here, the two eigen states, L- and R-state, indeed possess different dispersion relations. In fact, those two states can be independently excited in simulations by using specially-designed local fields [6]. Figure 3 shows the comparison of the mode profiles of the two states with a degenerate frequency. In Figure 3a, an oscillating field is applied in the middle of the tube (yellow region), which is set up to excite a right-running R-state (right-half of the tube), while in Figure 3b, a right-running L-state. Clearly, those two modes have distinct wave lengths, and thus propagating speeds. We emphasize that without DMI, the L- and R-state propagating to the same direction would have the same dispersion relation (not shown). Note that a left-running L-state (R-state) is simultaneously excited by the local field in Figure 3a,b. We point out that the L- and R-state propagating to the opposite direction also acquire different dispersion relations, which can be seen by comparing the left-half with the right-half of the tube in Figure 3a or Figure 3b. This difference is also attributed to the curvature-induced asymmetric propagation of SWs in a circularly magnetized tube, as reported previously [28,29,30]. In this case, we focus on the different dispersion of the L- and R-state propagating in the same direction, whose superposition yields the SW mode showing the magnonic activity effect.

Figure 3. Profiles of independently excited L- and R-state with f = 52 GHz in a circularly magnetized tube with a DMI constant D = 0.4 mJ/m², shown by snapshots of the dynamics radial component ($\delta m_{\rho }$). (a) A right-running R-state together with a left-running L-state. (b) A right-running L-state together with a left-running R-state. Upper panel of each is the side view of the original tube and lower panel a “rolled-out” one. Note that the L-state and R-state propagating in the same direction have obviously different wave length.

Figure 3. Profiles of independently excited L- and R-state with f = 52 GHz in a circularly magnetized tube with a DMI constant D = 0.4 mJ/m², shown by snapshots of the dynamics radial component ($\delta m_{\rho }$). (a) A right-running R-state together with a left-running L-state. (b) A right-running L-state together with a left-running R-state. Upper panel of each is the side view of the original tube and lower panel a “rolled-out” one. Note that the L-state and R-state propagating in the same direction have obviously different wave length.

By varying the excitation fields, one can numerically calculate the dispersion relations of the L- and R-state. The obtained results for n = 1 and n = 2 are shown in Figure 4. Note that one has to consider the left- and right-running modes separately, as illustrated in Figure 4a,b, respectively. On the other hand, the dispersion relation can also be extracted from the mode profile shown in Figure 2 by calculating $k_{z1}$ and $k_{z2}$ from $k_z^{\prime }$ and $k_z^{″}$ in Equation (4). As shown in Figure 4, an almost perfect agreement is achieved between those two approaches, providing quantitative proof for the physical explanation of the magnonic activity effect observed in this system.

Based on the analysis above, it can be easily deduced that the rotational rate of the SW modes should rely on the strength of the DMI in the system. In Figure 5, $k_z^{\prime }$, which determines the rotational rate, is plotted as a function of the DMI constant D for SW modes with various frequencies and order numbers. As expected, a clear positive dependence of the discrepancy between the dispersion relations of the L- and R-state and thus the rotational rate on the DMI strength is observed.

Finally, we explain mathematically why the DMI distinguishes the L- and R-state in our studied system. The contribution of the DMI to the total effective field ${\overrightarrow{H}}_{eff}$ in LLG equation is given in Equation (3). Considering the L- and R-state defined in Equation (4), the corresponding DMI field can be calculated in a cylindrical coordinate system, given by

$${\overrightarrow{H}}_{DM}=-\frac{i\left(\mp n\right)}{\rho }{D}^{*}\delta m_z\widehat{\rho }+i{k}_z{D}^{*}\delta m_{\rho }\widehat{\phi }+\frac{D^{*}}{\rho }\left(i\left(\mp n\right)\delta m_{\rho }\right)\widehat{z},$$

(5)

where the −(+) sign in front of $n$ corresponds to the L- (R-)state. Obviously, the DMI field and thus the dispersion relation for L- and R-state are different. Mathematically, this originated from the first-order spatial derivative of the magnetization introduced in the DMI field term. We point out that for a circularly magnetized tube without DMI, the dispersion relations of the SW eigen modes have been calculated analytically in previous works [28]. A full analytical calculation including DMI is beyond the scope of this work.

4. Summary

In summary, we study propagating SW modes in a circularly magnetized nanotube with the presence of DMI. The intrinsic asymmetry of the DMI results in a dynamic chiral effect in the system, i.e., a discrepancy between two eigen states, the L- and R-state, which are otherwise indistinguishable in terms of dispersion relation. Consequently, the magnonic activity effect is observed for a SW mode, which can be considered as the superposition of the L- and R-state. Besides the aesthetics lying in its apparent analogy to the optical activity in optics, the magnonic activity may also bear application potential in the development of new-concept magnonic and spintronic devices. For instance, SWs can not only serve as high-efficiency spin current sources but also influence tuning of fundamental electron-magnon interaction [31,32]. The experimental fabrications of ferromagnetic nanocylinders underway will pave the way for the explorations of magnetization in such systems.