Generation of Magnon Orbital Angular Momentum by a Skyrmion-Textured Domain Wall in a Ferromagnetic Nanotube

Abstract

We develop a theory for the dynamics of a magnon on top of a domain wall in a ferromagnetic nanotube. Due to the geometry of the sample, domain walls are classified by the Skyrmion charge which counts the winding number of magnetic textures. The domain wall with a non-zero Skyrmion charge generates an emergent magnetic field for magnons, which exerts the Lorentz force on moving magnons and thereby deflects their trajectories. This deflection is manifested as the generation of the finite orbital angular momentum of the magnon that traverses the domain wall. We obtain exact solutions for the magnon on top of the Skyrmion-textured domain wall and also their scattering properties with the domain wall with the aid of supersymmetric quantum mechanics. We show that there is a critical wavenumber for the total reflection of magnons and it is discretized by the Skyrmion charge of the domain wall. Our results show that the orbital angular momenta of magnetic textures and magnons can be intertwined in a curved geometry.

1 Introduction

Magnons are quanta of spin waves which are low-energy collective excitations of ordered magnets [1–3]. Due to fundamental curiosity and potential technical applicability, magnons have received significant attention both theoretically and experimentally [4–10]. A magnon carries a spin angular momentum and thus it can apply torque to magnetic textures such as a domain wall and a Skyrmion, as an electron does through spin-transfer torque [11–14]. Since a magnon current does not involve electron transport, magnons can be used to transport information even in insulators, enabling magnon-based information technology free from Joule heating [15]. Because of this potential utility for low-power devices, it is important to investigate the dynamics of magnons, spawning the rising fields of magnonics or magnon spintronics in condensed matter physics [16–19].

Over the past decades, magnonics has been mainly focused on flat geometry. Recently, magnonics in curved geometry has emerged [20–24]. One of the simplest curved geometries is a nanotube. In the magnetic nanotube, orbital angular momenta, as well as spin angular momenta, are keys to understanding the dynamics of magnetic textures and particles in the system [25, 26]. In particular, in a ferromagnetic nanotube, a magnon can carry orbital angular momentum, which can be exploited to magnonic computing [20, 27, 28]. A domain wall with non-trivial magnetic texture has the Skyrmion charge and has been recently shown to possess an orbital angular momentum [25, 29]. An electron also can carry orbital angular momentum and the exchange of orbital angular momenta between electrons and the domain wall can be interpreted as a current-induced torque. Current-induced domain wall motion and interaction between electrons and the Skyrmion-textured domain wall in a nanotube have been studied [25, 30].

Waves carrying orbital angular momentum have been intriguing physicists throughout various fields. For example, orbital angular momenta of photons [31–36], phonons [37–39], neutrons [40, 41], electrons [42, 43], and gravitational waves [44, 45] have been investigated. Orbital angular momentum exchange between photons and magnetic Skyrmions has also been studied [46]. These waves and particles with orbital angular momentum can offer novel functionalities. In particular, it has been proposed that a magnon carrying an orbital angular momentum can be used to construct a topology-protected logic gate [27].

In this paper, we propose a way to generate a twisted magnon current that carries orbital angular momentum by using a domain wall. In a two-dimensional magnet, a domain wall on the magnet is a line and spin-textures on the domain wall can spatially vary [47]. When the magnet is curved, the domain wall line can also be curved. In particular, when the magnet has the cylindrical geometry, the domain wall become a circle. Figure 1 for the illustration of a domain wall with the nontrivial texture in a nanotube. Due to the periodicity of the nanotube along the azimuthal direction, spin textures through the domain wall circle can have integer winding number and this winding number determines the system’s Skyrmion charge . The Skyrmion-textured domain wall can be classified by its Skyrmion charge and the Skyrmion charge characterizes the emergent magnetic field of the domain wall. We find that the emergent magnetic field exerts the Lorentz force for the magnon beam and thereby the beam is bent, one can tune the orbital motion of the magnon by the domain wall. In other words, one can generate the magnon orbital angular momentum by a domain wall. In order to solve the problem analytically, we use a method of supersymmetric quantum mechanics (SUSY QM) [48, 49]. We obtain the exact solutions for the magnon on the domain wall possessing a finite Skyrmion charge. We solve the scattering problem and also find the bound states. Our work deals with the exchange of the angular momenta of magnets and their fluctuations, similar to that the Einstein-de Haas effect [50] deals with the exchange of angular momenta of the magnets and the lattice. We hope that our study paves the way for the understanding of angular momentum conservation.

FIGURE 1

Our paper is organized as follows. In Section 2.1, we introduce the general formalism to obtain the magnon on top of the Skyrmion-textured domain wall in a ferromagnetic nanotube. The equation of motion for the magnon which we obtain can be interpreted as “Schrödinger” equation with the “Hamiltonian” for the charged particle in an electromagnetic field. In Section 2.2, orbital angular momentum exchange between the magnon and the domain wall is calculated by the Lorentz force. In Section 2.3, we define the SUSY partner Hamiltonian and potential. In Section 2.4, we discuss magnon-bound states. In Section 2.5, precise reflection probability is obtained. Also, the result from Section 2.2 is reproduced. Section 3 is devoted to the dynamics of the domain wall. Due to the Skyrmion charge, critical behavior of the domain wall velocity occurs. In Section 4, we summarized main results and discuss potential future outlook.

2 Exact Solution of a Spin Wave on Top of the Domain Wall

In this section, we construct the equation of motion for the magnon on top of the Skyrmion-textured domain wall and solve the equation analytically with the aid of SUSY QM. To this end, we introduce the general formalism of the dynamics of the magnetization and obtain the “Schrödinger” equation and the “electromagnetic” gauge field for the magnon.

2.1 General Formalism

To investigate the low-energy dynamics of magnetizations in a ferromagnetic nanotube, we take the continuum limit. The normalized magnetization m is considered as a field with two spatial variables z and φ, which are the axial coordinate and the azimuth, and a temporal variable t. The axis of the nanotube is chosen to be the z-axis. For theoretical description of a ferromagnetic nanotube, we use the continuum Heisenberg model with easy-axis anisotropy, whose potential energy density is given bywhere ρ is the radius of the nanotube, A is the exchange constant, and K > 0 is the anisotropy constant [25]. The dynamics of the magnetization stems from the kinetic part of the Lagrangian density , which is given by , and the Rayleigh dissipation , where α is the Gilbert damping coefficient. The variational principle with respect to m yields the Landau-Lifshitz-Gilbert equation [51]where s is the spin density and h_eff is the effective magnetic field conjugate to m given bywhere . Solving Eq. 2 for the case of ∂_tm = 0, we can obtain static solutions. We will consider a set of solutions of the Skyrmion-textured domain wall as a family of the slow modes with an angle representation m₀ = (sin θ₀ cos ϕ₀, sin θ₀ sin ϕ₀, cos θ₀),where and [25]. Here, Z, ϒ, Φ and Q are the position of the domain wall, the angle of the Skyrmionic texture, the global rotation angle, and the Skyrmion charge, respectively. Since the system is invariant under translations along the z axis, orbital rotations about the axis of the nanotube, and spin rotations about the easy-axis, linear momentum, orbital angular momentum, and spin angular momentum are conserved. The domain-wall solution (4) breaks these three symmetries and the three collective coordinates Z, ϒ, Φ represent corresponding zero-energy modes. We decompose the magnetization into a slow mode m₀ (z, φ) and a perturbative fast mode δm (z, φ, t) which are orthogonal. The magnetization can be written asWith this decomposition, the equation of motion for the fast mode can be derived from the expansion of the Landau-Lifshitz equation up to linear order of the fast mode δm,where . To obtain magnon modes on top of the domain wall, we neglect the Gilbert damping by following previous literature on the interaction of magnons with magnetic textures, e.g., Refs. [52–54], which allows us to invoke the conservation of angular momentum to understand the interaction with transparent physical picture. Note that the effect of the damping on magnons can be captured simply by introducing a finite lifetime, τ ∝ (αω)⁻¹ with ω the magnon frequency as has been done in, e.g., Refs. [13, 52, 55]. Plugging the domain-wall solution (4) into Eq. 6, the equation of motion for the fast mode on top of the Skyrmion-textured domain wall becomes

It is convenient to use the following as an orthonormal basis

The magnetic texture of the spin-wave excitation with the solution (4) is illustrated in Figure 1. For the convenience, we define a complex field Ψ = δm₁ − iδm₂, where . The complex filed Ψ contains full information of the spin wave. Reassembling the equations of motion for each component δm_j yields the equation of motion for the complex filed Ψ. The complex field Ψ obeys the “Schrödinger” equation ¹with the “Hamiltonian”with vector potential and scalar potential . The independency of the Hamiltonian (Eq. 10) on the spin azimuthal angle ϕ₀ is due to the spin rotational symmetry. The independency on the spatial azimuthal angle φ is due to the orbital rotational symmetry. Conservation of spin and orbital angular momenta of the system stems from these two rotational symmetries.

2.2 Emergent Magnetic Field

Before solving the Hamiltonian (Eq. 10) exactly, we can invoke the following simple argument to compute orbital angular momentum change of magnons. Magnons on the Skyrmion-textured domain wall experience an emergent magnetic field [56] . For the magnetic field which is a curl of a, we only take its surface-normal component, since the nanotube is considered as a two-dimensional surface ². The orbital angular momentum change of each magnon can be evaluated from the Lorentz force, , where v is the magnon velocity and p_φ is momentum in the azimuthal direction. Since the total orbital angular momentum of the domain wall and magnons is a conserved quantity, an orbital transfer torque to the domain wall per magnon is 2Qℏ. Figure 2 depicts the bent trajectory of the magnon due to the surface-normal emergent magnetic field. In this classical viewpoint, we can calculate the orbital angular momentum transfer, but it does not offer concise reflection property. In the section below we solve the reflection problem exactly.

FIGURE 2

2.3 SUSY QM

Since the field θ₀ of the domain-wall solution (4) has z-dependence only, the “Hamiltonian” H(Eq. 10) is invariant under global translation of the azimuthal angle φ → φ + δφ. Therefore we can write the wavefunction as an eigenmode Ψ (z, φ, t) = ψ (z)e^i(lφ−ωt), l = 0, ±1, ±2, ⋯. The equation of motion for ψ (z) is written as a time independent Schrödinger equation sωψ (z) = H_effψ (z) with the effective Hamiltonian , where the effective potential is

Then the original two-dimensional problem is simplified as a one-dimensional quantum mechanics. The effective potential is also known as the Rosen-Morse potential [57]. The tanh term plays role of an energy barrier which reduce longitudinal momentum of incoming magnon. The sech² term represents a potential well, which gives us possibility of existence of bound modes.

The problem can be solved by using a method of SUSY QM [48, 49]. In terms of creation and annihilation operators , the effective Hamiltonian and its SUSY partner Hamiltonian can be written aswhere β = −Qlλ/ρ² and

Note that, in the case of Q = 0, the partner potential is a constant potential so that there is no reflection of magnons and the linear momentum is preserved. Also, the original potential V_eff becomes the Pöschl-Teller potential [58, 59] which is a well known reflectionless potential as shown in Figure 3. Here, we can conclude that, for moving magnons, the Skyrmionic texture of the domain wall generates the potential barrier. Since , the commutation relationeliminate the term in Eq. 11 and yields Eq. 14. The problem with the potential V_eff is simplified to the problem with . Eigenfunctions of the two Hamiltonians are easily transformed into each other’s ones by annihilation and creation operators:where . Since the a (a^†) operation on the eigenfunction of is the eigenfunction of , we can switch the problem to its SUSY partner which turn out to be easier to tackle than the original one.

FIGURE 3

2.4 Bound Magnon Modes

One can obtain the bound state solution by solving aψ (z) = 0, which is given bywhere ω_b is the frequency of the bound solution. The condition of normalizability is |l| < ρ²/(|Q|λ²). This mode is local [60, 61] and the magnon position δ obeyswhere the position is defined as the maximum position of wavefunction’s amplitude. Figure 4 shows the positions of bound magnons schematically, the white lines are trajectories of bound magnon. Note that, for the non-zero Skyrmion charge, the bound modes with different l are separated [47, 53, 62–65] unlike the bound modes with Q = 0 [66, 67].

FIGURE 4

2.5 Propagating Magnon Modes

Let us consider a scattering problem of the spin wave. We now can write the wavefunction ψ (z) aswhereand k₋ is the wavenumber of the incoming magnon. Because of the asymmetric potential, wavenumbers of incoming and outgoing magnons are different. Using the relations (Eqs 16, 17), we can obtain asymptotic behaviors of the corresponding partner wavefunction:

Then we can check the fact that the reflection probabilities of original and partner potentials are equivalent:where is the reflection probability of the original (partner) potential. Now the problem is simplified as a hyperbolic-tangent-potential-barrier problem which is exactly solvable. The reflection probability of the SUSY partner is given by

[68–71]. Here, we use the Euler’s reflection formula [72] to simplify the reflection probability. We can assume that l is negative and Q is positive, without loss of generality, since the reflection probability is symmetric under the exchange of k₋, k₊. The transmission probability is determined by T_k = 1−R_k for both Hamiltonians. Figure 5 illustrates trajectories of reflected, transmitted, and incoming spin waves.

FIGURE 5

The wavefuntion Ψ is defined in the spatially varying local frame (Eq. 8). Thus, to see scattering properties of the magnon in the laboratory frame, we obtain wavefunction in the laboratory frame. Since there are two domains, local axis of a moving magnon is flipped passing by the domain wall and it makes difference between functional forms of incoming and outgoing wavefunctions:up to constant phase factors ³. Since Φ₀ = Q (φ−ϒ) + Φ, orbital wavenumbers of incoming and outgoing magnons are different. An orbital angular momentum of a transmitted magnon is changed from ℏ (l + Q) to ℏ (l−Q). Hence the change of magnon’s orbital angular momentum after passing the domain wall is Δl_z = −2ℏQ and orbital-transfer torque applying to the domain wall per magnon is 2ℏQ. The resultant torque derived from the calculation by SUSY QM is consistent with the result from the emergent field dynamics of Section 2.2. This is one of our main results: By using a Skyrmion-textured domain wall, we can generate an orbital angular momentum of a magnon.

3 Magnon-Driven Domain Wall Dynamics

Here, we study the dynamics of a skyrmion-textured domain wall driven by a magnon current. As shown in Section 2.5, the orbital angular momentum of a magnon on the domain wall varies due to the interaction of the magnon and the background domain wall. Analogous to Newton’s third law, which is also known as the action-reaction law, the domain-wall position is shifted to compensate the change of angular momentum of magnon traversing the domain wall. In order to describe the low energy dynamics of the domain wall, we introduce the two variables Z, Ξ ≡Φ−Qϒ as collective coordinates. The linear momentum, spin angular momentum and orbital angular momentum are expressed as linear functions of the collective coordinates [25, 73].

Note that angular momenta are linearly dependent on the domain wall position Z and the orbital angular momentum is proportional to the Skyrimon charge Q.

The continuity equation of the magnon wavefunction can be written aswhere j is the corresponding current. From the Eq. 9 and Eq. (10), we obtain the z-component of jwhere and k are the amplitude and wavenumber of a plane wave Ψ. To connect |Ψ|² to the magnon number density, we see the longitudinal fluctuation of m(Eq. 5), the reduction of the longitudinal component due to the magnon is given by

Then the reduction of spin density is sΔm_l and it must be the ℏ times magnon number density. Consequently, plugging the relation sΔm_l = ℏσ into the Eq. 34, the magnon number density σ_m is given by

Finally, we obtain the z-component of the number density current

Since we know the angular momentum exchange per magnon, we can obtain the torque using the magnon number density current (Eq. 36). The magnonic orbital-transfer torque for the given number density current j_m and magnon wavenumber k is given bywhere T_k is the transmission probability and 2ℏQ is the orbital-transfer torque per magnon. Here, the factor T_k is present since only the magnons passing through the domain wall exert the torque. Similarly, the magnonic force and the magnonic spin-transfer torque are given bywhere k′ is the wavenumber of the transmitted magnon. Including the Rayleigh dissipation, Euler-Lagrange equations for the collective coordinates are given by [25].

Solving these equations, the magnon-driven domain-wall velocity is expressed aswhere k and k′ are wavenumbers of incoming and outgoing magnons, respectively.

In Figure 6, we consider the velocity of the domain wall driven by the magnon beam which is coming from the left end (z = −∞) and moving parallel to the z-axis. Here, we assume α = 0.1, and use material constants of Co: A = 31 pJ/m, K = 410 kJ/m³, and s = 8.23 × 10⁻⁶ J⋅s/m³ [74]. Note that the magnon whose wavenumber is smaller than is totally reflected. The orbital angular momentum and spin angular momentum of the totally reflected magnon are unchanged by the scattering and thus the domain wall maintains its position.

FIGURE 6

4 Summary and Discussion

We have shown that a magnon moving in a curved geometry can exchange its orbital angular momentum with the background magnetic texture. In particular, the Skyrmion-textured domain wall in the ferromagnetic nanotube generates the Lorentz force on the moving magnon, whereby we have verified the orbital-angular-momentum exchange between the magnon and the domain wall by solving the Hamiltonian with the aid of SUSY QM. We have investigated the reflection property of the magnon-domain-wall scattering. An analytic form of the reflection probability is obtained. We have also shown that a magnon whose wavenumber is lower than the critical wavenumber is totally reflected. The critical wavenumber is discretized by the Skyrmion charge of the domain wall.

An orbital angular momentum of a magnon can be used to carry additional information. Since the orbital angular momentum of the magnon can be generated and tuned by the domain wall with the Skyrmion-charge, the Skyrmion-textured domain wall may find its role in magnon-based devices. Due to the orbital symmetry, the domain wall with a non-zero Skyrmion charge interacts weakly with an external magnetic field in comparison to an ordinary domain wall [25]. Moreover, the domain wall with the Skyrmion charge is topologically stable in the cylindrical geometry. These properties support the technological utility of the Skyrmion-textured domain wall.

Magnons have been known to pass through the domain wall without reflection in a quasi-one-dimensional wire. We have shown that this does not hold for a domain wall with the Skyrmion charge. The Skyrmion-textured domain wall induces the effective potential for magnons and it is reflective. The magnon whose wavenumber is below the critical value is totally reflected and the critical wavenumber is proportional to the Skyrmion charge. Therefore the domain wall can play a role of a magnon filter.

In this work, we have neglected the effects of the thermal noise that are present at finite temperatures [75–78] and the effects of the Dzyaloshinskii-Moriya interaction that can exist if the inversion symmetry of the system is broken [79, 80]. The former can affect the stabilization of the Skyrmion-textured domain wall at elevated temperatures and the latter can also modify our results appreciably. The investigation of these effect is beyond the scope of the current work. We would also like to mention that we neglect the spin-inertia effects that add the second-order time derivative term to the LLG equation [81–85], which has been generally thought to be important only at the ultrafast time scales of the order of THz and beyond, but has recently been shown to be able to significantly affect the GHz-scale switching behavior of nanoscale ferromagnets via the long-term nutation dynamics [86]. We leave the investigation of the effects of the spin inertia on our results as a future research topic.

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