Chiral spin-wave edge modes in dipolar magnetic thin films

Based on a linearized Landau-Lifshitz equation, we show that two-dimensional periodic allay of ferromagnetic particles coupled with magnetic dipole-dipole interactions supports chiral spin-wave edge modes, when subjected under the magnetic field applied perpendicular to the plane. The mode propagates along a one-dimensional boundary of the system in a unidirectional way and it always has a chiral dispersion within a band gap for spin-wave volume modes. Contrary to the well-known Damon-Eshbach surface mode, the sense of the rotation depends not only on the direction of the field but also on the strength of the field; its chiral direction is generally determined by the sum of the so-called Chern integers defined for spin-wave volume modes below the band gap. Using simple tight-binding descriptions, we explain how the magnetic dipolar interaction endows spin-wave volume modes with nonzero Chern integers and how their values will be changed by the field.

Figure 1

Periodic array of ferromagnetic islands (gray circles) (a) decorated square-lattice model. (b) decorated honeycomb-lattice model. Each gray point stands for a ferromagnetic island (volume element is $\Delta V$), which we assume to behave as a single big spin whose moment is fully saturated ($M_s$). The spins are coupled via magnetic dipole-dipole interaction. We took $|{\mathit{e}}_x|=|{\mathit{e}}_y|=2.4$, $2r=1.2$, $\Delta V=1.70$, and $M_s=1.0$ for the square-lattice case, while $|{\mathit{e}}_1|=|{\mathit{e}}_2|=|{\mathit{e}}_3|=2.4$, $2r=1.2$, $\Delta V=1.0$, and $M_s=1.0$ for the honeycomb-lattice case. The primitive translational vectors ${\mathit{a}}_{\mu }$ ($\mu =x,y$) are defined as ${\mathit{a}}_x={\mathit{e}}_x$ and ${\mathit{a}}_y={\mathit{e}}_y$ for the square-lattice model and as ${\mathit{a}}_x={\mathit{e}}_3-{\mathit{e}}_2$ and ${\mathit{a}}_y={\mathit{e}}_1-{\mathit{e}}_2$ for the honeycomb-lattice case (see text).

Figure 2

Top view of classical spin configurations for the decorated square-lattice model. The field is lower than the saturation field, so that spins have finite in-plane components, forming a vortex structure.

Figure 3

(a)–(d) Side view of wavelength-frequency dispersions of four spin-wave volume-mode bands in the square lattice model under fields ($H$) normal to the two-dimensional plane; (a) $H=0.0$, (b) $H=0.47H_c$, (c) $H=0.76H_c$ (d) $H=0.82H_c$, (e) $H=1.01H_c$, (f) $H=1.1H_c$, (g) $H=1.4H_c$, and (h) $H=2.35H_c$ where $H_c=1.71$. The Chern integer for red/blue-colored spin-wave bands is $-1/+1$, while 0 otherwise.

Figure 4

(a)–(d) Wavelength-frequency dispersions calculated with open boundary condition along one direction ($y$ direction) and periodic boundary condition along the other (decorated square-lattice model); (a) $H=0.0$; (b), (b-1) $H=0.47H_c$; (c), (c-1), (c-2) $H=0.76H_c$; (d), (d-1) $H=0.82H_c$ with $H_c=1.71$. The system along the $y$ direction includes 40 unit cells ($L=40$). More than $80\%$ of wave functions for red-colored points are localized from $y=L-3$ to $y=L$, while those for blue-colored points are localized from $y=1$ to $y=4$. Compared with Figs. 3a, 3b, 3c, 3d, spectra are comprised also of edge-mode bands, whose chiral dispersions runs across band gaps for spin wave volume modes.

Figure 5

(a)–(d) Wavelength-frequency dispersions calculated with open boundary condition along one direction ($y$ direction) and periodic boundary condition along the other (decorated square-lattice model). (a), (a-1) $H=1.01H_c$; (b), (b-1) $H=1.1H_c$; (c), (c-1) $H=1.4H_c$; (d), (d-1) $H=2.35H_c$ with $H_c=1.71$.

Figure 6

(a)–(b) Wavelength-frequency dispersions calculated with open boundary condition along one direction ($y$ direction) with the zigzag boundary (decorated honeycomb-lattice model). The system along the $y$ direction includes 30 unit cells ($L=30$). Wave functions for red-colored points are localized from $y=L-2$ to $y=L$ ($>80\%$), while those for blue-colored points are localized from $y=1$ to $y=3$. (a), (a-1) $H=1.05H_c$; (b), (b-1) $H=3.5H_c$, where $H_c=0.57$.

Figure 7

A single $M$-spin cluster for $M=4,3,6$. $M$ spins (gray circle) align along a circle (radius $r$), taking an equal distance between their nearest-neighbor spins. Each spin has a saturation magnetization ($M_s$) and a finite volume element ($\Delta V$).

Figure 8

(a) Atomic-orbital levels as a function of the field in a single four-spin cluster case. Equation (13) is diagonalized, where the demagnetization field for each site ($\alpha$) comes from the spins in the same cluster. The saturation field at which $n_J=0$ becomes zero is around $H_c=1.26$. The level inversion between $n_J=1$ and $n_J=0$ is around $H/H_c=0.80$, while that between $n_J=3$ and $n_J=2$ is around $H/H_c=0.83$. (a-1) Atomic-orbital levels as a function of the field calculated from Eq. (13) in the decorated square lattice model. Equation (13) is diagonalized, where the demagnetization field for each site includes not only those from the spins in the same cluster but also those from spins in the other cluster. The saturation field is around $H_c=1.71$, where the level of $n_J=0$ goes below the others. (b) Atomic-orbital levels as a function of the field (single three-spin cluster case) with $H_c=0.32$.

Figure 9

Side view of spin-wave band dispersions for decorated square-lattice model calculated from the tight-binding Hamiltonian, Eqs. (15, 16, 17) where only nearest-neighbor hopping integrals are included. As for atomic-orbital levels, Eq. (17), and the respective wave function ${\mathit{T}}_0$ used in Eq. (16), we used those for the single four-cluster. (a) $H=0.23H_c$, (b) $H=0.66H_c$, (c) $H=0.71H_c$, (d) $H=0.79H_c$, (e) $H=0.85H_c$, (f) $H=1.4H_c$, where $H_c$ denotes the saturation field for single four-spin cluster, [$H_c=1.26$; see the caption of Fig. 8a]. At (a), (b), and (f), the lowest spin-wave band and the second lowest one form band touchings at $\Gamma$ point, $X$ points, and $M$ point, respectively. At (c), (d), and (e), the highest spin-wave band and the third lowest one form band touchings at $M$ point, $X$ points, and $\Gamma$ point, respectively.

Figure 10

Chern integers for four spin-wave bands as a function of the field. (a)–(f) depicted in the figure correspond to the fields at which band touchings occur as shown in Figs. 9a, 9b, 9c, 9d, 9e, 9f, respectively. Note also that the third lowest spin-wave band at Fig. 9e is mainly composed of $n_J=0$, while the second lowest and the lowest bands are mainly composed of $n_J=2,3$. From (e) to (f), the former band goes below the latter two until $H=H_c$, while, for $H>H_c$. it increases its frequency again, in the same way as the $s$-wave atomic orbital does in Fig. 8a. For clarity, we call the latter two as second lowest and lowest, even though they are not during (e)–(f).

Figure 11

Shapes of the atomic orbitals in the decorated square lattice model. That of $n_J=0$ is $s$-wave, while those of $n_J=1,2,3$ are $p_x-i{p}_y$-wave, $d_{x^2-y^2}$-wave, and $p_x+i{p}_y$-wave respectively.

Figure 12

(a) Side view of spin-wave band dispersions for the decorated honeycomb-lattice model under a sufficiently strong field ($H=5.0$). Because of a small but finite band gap at two $K$ points, the Chern integers for the lowest two bands are $-1$ and $+1$ respectively. (b) Side view of spin-wave band dispersions calculated from the effective Hamiltonian to the order of $\mathcal{O}\left(1\right)$. For the atomic-orbital levels, we use those for a single three-spin cluster. To evaluate the nearest-neighbor hopping integral within the order of $\mathcal{O}\left(1\right)$, we use Eq. (36). When $H=5.0$ is added, the resonance frequencies of the spin-wave bands in Fig. 12b become comparable to those in Fig. 12a.

Figure 13

Schematic view of a simulated system that comprises ferromagnetic nanograin. Although we took the size of the ferromagnetic nanograin to be $5\times 5\times 5$ nm${}^3$ as a demonstration, the present simulation is scale free. In the beginning of the simulation, we apply a pulse field either at the center (marked by a black triangle) or around the boundary (marked by a black cross).

Figure 14

Fourier power spectra of magnetization dynamics. (a) Frequency dependence of the intensity of spin-wave excitations (see text). (b)–(g) Spatial distribution of the intensity, which is obtained by the application of the pulse field at center (b), (d), (f) and the edge (c), (e), (g); (b), (c) $\omega =10$ GHz, (d), (e) $\omega =29$ GHz, and (f), (g) $\omega =31$ GHz.

Figure 15

Dispersion relations of spin-wave modes. (a) Dispersion relation obtained by the application of the pulse field at the center. (b), (c) Dispersion relation obtained by the pulse at the edge. The Fourier transformation is taken only over the upper side ($Y>L/2$) for (b) and over the lower side ($Y<L/2$) for (c).

Figure 16

Spatial-resolved Fourier power spectra of magnetization dynamics in the presence of stronger dissipation. Spatial distribution of the intensity at $\omega =29$ GHz, which is obtained by the application of the pulse field at the edge. (a) $\alpha =0.001,$ (b) $\alpha =0.01$.