Magnetostatic wave analog of integer quantum Hall state in patterned magnetic films

A magnetostatic spin-wave analog of integer quantum Hall (IQH) state is proposed in realistic patterned ferromagnetic thin films. Due to magnetic shape anisotropy, magnetic moments in a thin film lie within the plane, while all spin-wave excitations are fully gapped. Under an out-of-plane magnetic field, the film acquires a finite magnetization, where some of the gapped magnons become significantly softened near a saturation field. It is shown that, owing to a spin-orbit locking nature of the magnetic dipolar interaction, these soft spin-wave volume mode bands become chiral volume mode bands with finite topological Chern integers. A bulk-edge correspondence in IQH physics suggests that such volume mode bands are accompanied by a chiral magnetostatic spin-wave edge mode. The existence of the edge mode is justified both by micromagnetic simulations and by band calculations based on a linearized Landau-Lifshitz equation. Employing intuitive physical arguments, we introduce proper tight-binding models for these soft volume mode bands. Based on the tight-binding models, we further discuss possible applications to other systems such as magnetic ultrathin films with perpendicular magnetic anisotropy.

Figure 1

Schematic top view of two-dimensional patterned magnetic thin films (blue region represents magnetic media, while the other stands for the vacuum). An external magnetic field is applied perpendicular to the plane with out-of-plane moments. Either circular rings (a) or disks (b) form a square lattice. We assume that the film is sufficiently thin, so that there is no texture along the direction perpendicular to the plane.

Figure 2

Resonance frequency levels in a magnetic ring (a $M$-spins chain with $M=60$) as a function of the angular momentum $q_J\equiv \frac{2\pi n_J}{M}$ with $n_J=-M/2,-M/2+1,\cdots ,M/2$. (a) $H=0.7H_c$ and (b) $H=0.9H_c$. (c) When demagnetization fields from the surrounding magnetic rings are included, the angular momentum $n_J$ is defined mod 4. (d) Wave functions with $n_J\equiv$ $-1$, 0, $+1$, $+2$ (mod 4) are referred to as $P_{-}$, $S$, $P_{+}$, $D_{x^2-y^2}$ wave respectively, since they acquire $-i$, $+1$, $+i$, and $-1$ phase under the $\frac{\pi }{2}$ spatial rotation [Eqs. (2, 3, 4)].

Figure 3

Wavelength-frequency dispersions for spin-wave excitations for $H_d<H<H_c$ $(H=0.94H_c)$. (a) A side-view of lowest eight volume mode bands with the Chern integer. The red bands have $-1$ Chern integer, while blue bands have $+1$. The dispersions are calculated with periodic boundary conditions for both $x$ and $y$ directions. Since the seventh and eighth lowest bands have frequency degeneracies around $M$ points, only the sum of their integers is quantized to $+1$. (b) Spin-wave excitations calculated with an open/periodic boundary condition along the $y$/$x$ direction respectively. The resonance frequencies are given as a function of the wave vector along the $x$ direction. The system along the $y$ direction includes 18 square-lattice unit cells ($L=18$). More than $75\%$ of amplitudes of eigenwave functions with red points are localized within $y=1$ and $y=2$, while those with blue points are localized within $y=L-1$ and $y=L$ (edge modes). Compared with (a), the calculated spectra have additional spin-wave modes which are localized along the edges. (c) With the out-of-plane field up-headed, the chiral edge modes rotate in the counterclockwise way.

Figure 4

(a) Four corners in a ring (red regions; $\theta =0,\frac{\pi }{2},\pi ,\frac{3\pi }{2}$) feel larger demagnetization field than other regions. $\mathit{b}$ denotes a coordinate of the center of the ring, while $r$ is a radius of the ring. (b) Two-orbital tight-binding model with nearest-neighbor (NN in the figure) intercluster transfer integral (${\mathit{H}}_0$), next-nearest-neighbor $\sigma$-$\sigma$ coupling (NNN,$\sigma \sigma$) intercluster transfer integral ($c$ in ${\mathit{H}}_1$), and next-nearest-neighbor $\pi$-$\pi$ coupling (NNN,$\pi \pi$) intercluster transfer integral ($c^{\prime }$ in ${\mathit{H}}_1$). The in-phase orbital (red peanut-shape item) at the $x$ link is extended along the $x$ direction, while that at the $y$ link is along the $y$ direction. ${\mathit{e}}_x$ and ${\mathit{e}}_y$ denote the primitive translation vectors of the square lattice.

Figure 5

Wavelength-frequency dispersions for lowest four volume mode bands and chiral edge modes in $H>H_c$ (a) $H=1.09H_c$, (b) $H=1.17H_c$. The dispersion are obtained with an open/periodic boundary condition along the $y$/$x$ direction, where the resonance frequencies for spin-wave excitations are given as a function of the wave vector along the $x$ direction. We used the same system size along the $y$ direction as in Fig. 3 and the same definition of red and blue points as in Fig. 3. In both (a) and (b), the lowest two volume modes (black points) consist of the in-phase atomic orbitals on $x$ link and $y$ link, while the upper two volume modes mainly consist of out-of-phase orbitals on these two links. The spectra clearly contain a chiral edge mode connecting the lowest two volume mode bands. Compared to the lowest two bands, the third and fourth lowest bands have smaller bandwidth and no band gap in between. This is because, contrary to the in-phase atomic orbital, the out-of-phase atomic orbital has a node at the center of each link, which results in smaller transfer integrals. Comparing (a) and (b), note also that a frequency spacing between the in-phase atomic-orbital level and the out-of-phase atomic-orbital level increases on increasing the field (see Appendix pp2 for the reasoning).

Figure 6

(a),(b) Distribution of resonance frequency levels of magnon modes in a single circular magnetic disk as a function of the out-of-plane field. The fourfold-rotational demagnetization field from other disks are also included in the calculation. The red curve plots the lowest resonance frequency level as a function of the out-of-plane field, while the blue curve represents the highest resonance frequency level. We simulate the circular magnetic disk by a cluster of spins, respecting the circular symmetry as much as possible. For a given radius $R$, we discretize $R$ into $n$ pieces. For the radial coordinate ranging from $[\frac{Rj}{n},\frac{R(j+1)}{n}]$ ($j=0,1,\cdots ,n-1$), we discretize the azimuth coordinate into $4(2j+1)$ pieces, so that area of each element is same, $\frac{\pi R^2}{4n^2}$. We put a spin at the center of each element specified by $(x,y)=r_j(cos{\theta }_{j,m},sin{\theta }_{j,m})$ with $r_j=\frac{R(2j+1)}{2n}$ and ${\theta }_{j,m}=\frac{\pi (2m+1)}{4(2j+1)}$ [$m=0,\cdots ,4(2j-1)$]. In the calculation, we take $n=8$, so that a cluster has 256 spins.

Figure 7

Wavelength-frequency dispersions for spin-wave excitations for $H<H_{c,1}$ [see Fig. 6]. (a) Wavelength-frequency dispersion for volume mode bands with the Chern integer ($H=0.94H_{c,1})$. The dispersions are calculated with periodic boundary conditions for both $x$ and $y$ directions. (b) Wavelength-frequency dispersion for volume mode bands and edge mode bands ($H=0.94H_{c,1}$) calculated with an open/periodic boundary condition along the $y$/$x$ direction. Resonance frequencies are given as a function of the wave vector along the $x$ direction. The system along the $y$ direction includes nine unit cells ($L=9$). More than $50\%$ of amplitudes of eigenwave functions with red points are localized only within $y=1$, while those for blue points are localized from $y=L$ (edge modes). Compared with (a), the spectra have additional spin-wave modes which are localized along the edges, whose chiral dispersion connect spin-wave volume modes bands with opposite Chern integers.

Figure 8

magnetic superlattice with 14 $\times$ 14 unit cell. Right: A unit cell contains 12 ferromagnetic grain forming a square ring. Each grain is cubic shaped with its linear dimension 5 nm. Left: To excite volume mode/edge mode excitations, we apply a pulse field at the center/boundary of the superlattice (blue/red crossed point) respectively.

Figure 9

(a) Contour plot of the integrated power spectrum $A\left(\omega \right)$ as a function of the static out-of-plane field $H$, where the initial pulse field is applied at the center of the magnetic superlattice. The out-of-plane field is greater than the saturation field ($\simeq$620 Oe). (b) Contour plot of the density of state for volume mode bands obtained from spin-wave calculations on the same magnetic superlattice. The horizontal axis is the static out-of-plane field, where the unit is taken to be the saturation field. In both figures, darker regions have higher intensities.

Figure 10

(a) Integrated power spectra calculated with the pulse field at the center (blue) and at the boundary (red). The static out-of-plane field is set to 800 Oe. (b)–(d) Spatial-resolved power spectra $|s_{+}(X,Y,\omega )|$ calculated with the pulse field at the center (blue crossed point); (b) $\omega =$ 6.25 GHz, (c) 6.69 GHz, (d) 6.54 GHz. (e) Spatial-resolved power spectrum calculated with the pulse field at the boundary (red crossed point) with $\omega =6.54$ GHz.

Figure 11

Snapshots of a transverse magnetic moment after the ac field is applied at $t=0$. (a) $t=100$ ns, (b) 200 ns, (c) 300 ns, (d) $t=350$, 352, 354 ns. The frequency of the ac field and the static out-of-plane field is set to 654 GHz and 800 Oe respectively. Colors specify the sign of the transverse moment (red is for positive and blue is for negative). In (a)–(c), the spin density propagates in the counterclockwise direction, while, in (d), the node of the transverse moment (indicated by black arrows) moves in the clockwise direction; the phase velocity is opposite to the group velocity.

Figure 12

Patterned magnetic films with periodically aligned gutters (a) or cambers (b). Without magnetic crystalline anisotropy (MCA), spins at the thinner regions feel stronger easy-plane anisotropy than those at that thicker regions.

Figure 13

(a) Ring is decomposed into four quadrants (grey shadow regions), which are ranged as $-\frac{\pi }{4}\le {\theta }_j\le \frac{\pi }{4}$ ($n=0$), $\frac{\pi }{4}\le {\theta }_j\le \frac{3\pi }{4}$ ($n=1$), $\frac{3\pi }{4}\le {\theta }_j\le \frac{5\pi }{4}$ ($n=2$), $\frac{5\pi }{4}\le {\theta }_j\le \frac{7\pi }{4}$ ($n=3$) respectively with ${\mathit{r}}_j={\mathit{b}}_j+r(cos{\theta }_j,sin{\theta }_j)$. Here ${\mathit{b}}_j$ denotes a center of the ring and $r$ is a radius of the ring. (b) One quadrant in a ring and its closest quadrant in the nearest-neighbor ring are combined together, to form a cluster (grey shadow region encompassed by a black dotted line). The cluster thus defined is centered at ${\mathit{b}}_j+\frac{{\mathit{e}}_x}{2}$ (a midpoint of the nearest-neighbor $x$ link) or ${\mathit{b}}_j+\frac{{\mathit{e}}_y}{2}$ (a midpoint of the $y$ link), where ${\mathit{e}}_x$ and ${\mathit{e}}_y$ are the basic translational vectors.