Anti-helical edge magnons in patterned antiferromagnetic thin film

Chiral or helical edge states are a hallmark feature of two-dimensional topological insulators MZHasan ; XLQi . Chiral edge states emerge in quantum Hall or Chern insulators with time-reversal symmetry breaking and propagate along the edges clockwise or counterclockwise. As a consequence, a typical strip geometry supports opposite propagating edge states along the two parallel edges. Helical edge states can be treated as two copies of chiral edge states, protected by the time-reversal symmetry, holding great potential applications in spintronics. The edge states for opposite spin hold opposite chirality, leading to a vanished charge current and a net spin current, which flow in opposite direction at the two parallel edges in the strip geometry.

Recent theory predicted the so-called antichiral edge states EColomes ; DBhowmick ; XCheng ; YangY ; PZhou ; MMDenner ; SMandal ; JChen , which copropagate in the same direction at opposite edges, compensated by counterpropagating bulk modes, distinct from the chiral edge states. The original work is based on the modified Haldane model in electronic systems EColomes , quite hard to realize experimentally. Then a series of the theoretical even experimental works addressed the realization in many other systems, including the electron system based on graphene MMDenner ; XCheng , exciton polariton SMandal , magnon DBhowmick , photon PZhou ; JChen and classical electric circuit YangY systems. The experimental works in photonic PZhou and electric circuit YangY systems observed the copropagating behavior of antichiral edge states based on the modified Haldane model. All these works require the broken time-reversal symmetry. A natural question arises. In the time-reversal symmetric systems, is it possible to realize “anti-helical” edge states, where the spin current on the parallel edges propagates in the same direction? For now, quite few works made the explorations theoretically or experimentally.

In recent yeas, antiferromagnets have attracted significant attention in spintronics due to the ultrafast spin dynamics and lack of stray fields TJungwirth ; VBaltz . Importantly, antiferromagnets support both left-handed and right-handed polarized magnon modes, related by the pseudo-time reversal symmetry YMLi1 , analogous to the electron spin. This coexistence of both polarizations give rise to many novel spin-related physical phenomena, such as the magnon spin Nernst effect RCheng ; VAZyuzin , Stern-Gerlach effect ZWang and Hanle effect TWimmer . Manipulating the polarization may facilitate the chirality-based computing CJia ; MWDaniels and logic devices WYu1 . Therefore, the exploration on topologically protected helical or even anti-helical edge state could be greatly helpful in the field of magnon spintronics based on antiferromagnets ABarman .

In this paper, we theoretically propose the realization of anti-helical edge states of magnons in patterned antiferromagnetic thin film, in which the spin current at parallel edges flow in the same direction, compensated by the counterpropagating bulk confined spin currents, as illustrated in Fig. 1 (a). Here the degenerate two magnon modes with opposite chirality is treated as the spin degree of freedom, as widely discussed in previous works above. Heavy metal dot array is embedded into the thin film to fold the free dispersion of magnons into bands. Moreover, the inversion symmetry breaking at the interfaces between metal dots and the thin film will generate interfacial Dzyaloshinskii-Moriya interactions (iDMIs), which modify the exchange boundary condition and induce nontrivial topological phase for the magnon bands, characterized by the spin Chern number. As a result, helical edge magnons arise in a finite width strip geometry. The left-handed and right-handed edge magnons flow towards opposite directions at the same edge, leading to a pure spin current, revealed by the micromagnetic simulations. The direction of the spin current is determined by the sign of iDMI parameter $D$. In a finite width ribbon geometry, we combine two subsystems embedded with different metal dot arrays, which induce iDMIs with opposite sign of $D$, as shown in Fig. 1 (a). Interesting, the spin currents along the two edges flow in the same direction, while the spin currents along the domain wall flow in opposite direction. The emergence of the anti-helical edge states can also be verified by the micromagnetic simulations.

This paper is organized as follows. In Sec. II, we derive the eigenvalue problem for the magnons in the patterned antiferromagnetic thin film in the presence of metal dot array and the iDMIs. In Sec. III, we discuss the band topological properties due to the iDMIs, the topological phase transition and the construction of strip geometry for anti-helical edge states. We also performed the micromagnetic simulations for verifications. In Sec. IV, we summarize our results. In Sec. V, we present the methods and the parameters we adopted for the calculations.

We consider an antiferromagnetic thin film with periodically embedded metal dots. The metal dots are arranged in a triangular lattice configuration. We construct a domain along two sides with different embedded metal dot array to get the anti-helical edge states in a strip geometry, as illustrated in Fig. 1 (a). An infinite system with only one type embedded metal dot array is a two-dimensional artificial magnonic crystal. The corresponding unit cell is shown in Fig. 1 (b). The strip geometry with one or two embedded metal dot array is one-dimensional artificial magnonic crystal. The magnetization dynamics in antiferromagnets can be described by two coupled Landau-Lifshitz-Gilbert (LLG) equations for each sublattice,

$$\mathbf{\dot{m}}_{i}=-\gamma\mathbf{m}_{i}\times\mathbf{h}_{i}+\alpha\mathbf{m}_{i}\times\mathbf{\dot{m}}_{i},$$

(1)

where $i=1,2$ denote the two sublattices. $\gamma$ is the gyromagnetic ratio, $\alpha$ is the Gilbert damping constant. Here $\gamma\mathbf{h}_{i}=K_{z}m_{i}^{z}\mathbf{z}+A\nabla^{2}\mathbf{m}_{i}-J_{ex}\mathbf{m}_{\bar{i}}$ (with $\bar{1}=2$ and $\bar{2}=1$) is the effective magnetic field acting locally on sublattice $\mathbf{m}_{i}$, where $K_{z}$ is the easy-axis anisotropy along $\mathbf{z}$ direction $A$ and $J_{ex}$ characterize the Heisenberg exchange coupling constant of intra-sublattice and inter-sublattice, respectively. The equilibrium magnetization is in collinear order for the two sublattices and parallel to the $z$-direction. To get the magnon dispersions, we can divide the magnetization and effective magnetic fields on each sublattice into the static and dynamical ones. Let $\mathbf{m}_{i}=\mathbf{m}_{i}^{0}+\delta\mathbf{m}_{i}$ and $\mathbf{h}_{i}=\mathbf{h}_{i,0}+\delta\mathbf{h}_{i}$ with $\mathbf{m}_{i}^{0}$ the equilibrium magnetization and $|\delta\mathbf{m}_{i}|\ll\mathbf{m}_{i}^{0}$. $\mathbf{h}_{i,0}$ is the $z$-component of the effective magnetic field. By neglecting the damping first, we have $\partial\delta\mathbf{m}_{i}/\partial t=-\gamma(\mathbf{m}_{i}^{0}\times\delta\mathbf{h}_{i}+\delta\mathbf{m}_{i}\times\mathbf{h}_{i,0})$. To solve the dispersion in the artificial crystals, we employ the ansatz $\delta\mathbf{m}_{i}=\delta\mathbf{m}_{i}^{\mathbf{k}}(\mathbf{r})e^{i(\mathbf{k}\cdot\mathbf{r}-\omega t)}$, and after a transformation, we can get the eigenvalue equations, given by

$$\omega\psi_{\pm,\mathbf{k}}=\mp\sigma_{z}\hat{H}_{\mathbf{k}}\psi_{\pm,\mathbf{k}},$$

(2)

where $\psi_{\pm,\mathbf{k}}=(\delta m_{1,\pm}^{\mathbf{k}},\delta m_{2,\pm}^{\mathbf{k}})^{T}$, $\delta m_{i,\pm}^{\mathbf{k}}=\delta m_{i,x}^{\mathbf{k}}\pm i\delta m_{i,y}^{\mathbf{k}}$, $\hat{H}_{\mathbf{k}}=[K_{z}+J_{ex}+(\nabla+i\mathbf{k})^{2}]\sigma_{0}+J_{ex}\sigma_{x}$. $\sigma_{0}$ is $2\times 2$ identity matrix, $\sigma_{x,y,z}$ are the Pauli matrices. The solution for $\psi_{+,\mathbf{k}}$ denotes left-handed modes while $\psi_{-,\mathbf{k}}$ is solution for right-handed modes. As discussed in our recent work YMLi1 , the two polarized modes are related by a pseudo-time reversal symmetry. Note that the two polarized modes are totally decoupled.

We turn to the role of the embedded heavy metal dots. The first effect is to fold the free dispersion into magnon bands due to the scattering by the periodic interfaces from the metals. More importantly, as discussed in many theoretical LUdvardi ; JHMoon ; MdRKAkanda ; KYamamoto ; MKostylev and verified in experimental works JCho ; NembachH ; KZakeri ; PFerriani ; KDi , the heavy metals (such as Pt, W, Au, Re, Ir, etc) with strong spin-orbit coupling can generate chiral interfacial Dzyaloshinskii-Moriya interactions (iDMI) due to the inversion symmetry breaking at the magnets/heavy metal interfaces. In our system, the iDMI $\mathbf{h}_{\mathrm{DM}}$ is given by $\gamma\mathbf{h}_{\mathrm{DM}}=D\mathbf{z}\times(\mathbf{e}_{t}\cdot\nabla)\mathbf{m}_{i}$ by comparing our bent surface to the previous plane surface JHMoon ; KDi . $\mathbf{e}_{t}=\mathbf{n}\times\mathbf{z}$ is the in-plane tangential vector of the interface and $\mathbf{n}$ the normal vector of the interface, both depicted in Fig. 1 (b). $D$ may be either positive or negative, depending on the metal materials MdRKAkanda ; KYamamoto . Without loss of generality, we assume $D$ is uniform at the bent surfaces. The iDMI will give an interface torque MKostylev , change the exchange boundary condition into the formalism below,

$$\mathbf{n}\cdot\nabla\mathbf{m}_{i}+\frac{Dd}{A}\mathbf{z}\times(\mathbf{e}_{t}\cdot\nabla)\mathbf{m}_{i}=0,$$

(3)

with $d$ the thickness of interface atomic layer and $D_{0}=Dd$ is a constant, only depending the materials. This constant $D_{0}$ value accounts for the experimentally observed approximately inverse thickness dependence of spin wave frequency shift in ferromagnets/heavy metal heterostructures JCho ; NembachH .

By solving the eigenvalue equation in Eq. (2) with the boundary condition in Eq. (3), we can obtain the magnon dispersions in the $\mathbf{k}$-space. The strip geometry illustrated in Fig. 1 (a) can also be calculated. The results are shown in the subsequent section.

In this work, we proposed a theoretical scheme to realize the anti-helical edge magnons in antiferromagnetic thin film with embedded heavy metal dot array. The iDMIs induce helical edge states with the direction of spin current dependent on the sign of iDMI parameter. By constructing an interface with embedded different metal dots along the two sides in a finite width ribbon, edge spin current on the parallel two edges propagate in the same direction, compensated by conterpropagating spin current along the interface. Our proposal is experimentally feasible and the anti-helical edge states can be probed by the state-of-the-art techniques. Our work provides a new method controlling the magnon spin in antiferromagnets and should be quite helpful in the field of magnon spintronics based on antiferromagnets.

The nearest distance between the metal dot center (lattice constant) is $a=50$ nm. The radius of the metal dot is $r=15$ nm. The parameters in the coupled LLG equation are as follows: WYu2 $K_{z}=8.55$ GHz, $A=7.25\times 10^{-6}$ Hz$\cdot$m², $J_{ex}=1.105\times 10^{11}$ Hz. The saturation magnetization $M_{s}=1.94\times 10^{5}$ A/m, the gyromagnetic ratio $\gamma=2.21\times 10^{5}$ Hz$\cdot$m/A. The values of $D_{0}$ in the unit of $A$ are given in the main text according to our need for the discussions. The micromagnetic simulation is based on the Eq. (1). The damping constant is adopted as $\alpha=2\times 10^{-4}$ to get a long decay length. We did not take the dipolar fields into calculation due to the antiferromagnet environment.

We can selectively excite either one or both of the polarized modes simultaneously by setting the microwave fields. The effective field of the radio wave is given by $\mathbf{h}_{rf}(t)=h_{0}[\cos(\omega t)\mathbf{e}_{x}\pm\sin(\omega t)\mathbf{e}_{y}]$, where $+$ denotes the right-handed source and $-$ denotes the left-handed source. Only one component above along the $x$ or $y$ direction give a linearly polarized source. In our micromagnetic simulation, $h_{0}=2\times 10^{9}$ Hz, $\omega=2\pi\times 16.2$ GHz.

This work is supported by the startup funding from Xiamen University.