Spin-wave edge modes in finite arrays of dipolarly coupled magnetic nanopillars

The frequency spectrum of spin-wave edge modes localized near the boundaries of a finite array of dipolarly coupled magnetic nanopillars is calculated theoretically. Two mechanisms of edge mode formation are revealed: inhomogeneity of the internal static magnetic field existing near the array boundaries and time-reversal symmetry breaking of the dipole-dipole interaction. The latter mechanism is analogous to the formation mechanism of a surface Damon-Eschbach mode in continuous in-plane magnetized magnetic films and is responsible for the nonreciprocity of edge modes in finite-width nanopillar arrays. The number of edge modes in nanopillar arrays depends on the spatial profile of the internal static magnetic field near the array boundaries and several edge modes are formed if a substantial field inhomogeneity extends over several rows of nanopillars.

Figure 1

A sketch demonstrating a semi-infinite array of nanopillars with the edge parallel to the primitive lattice vector ${\mathbf{a}}_1$: (a) top view, (b) side view. $l$ and $n$ are the integer pillar indices in the directions parallel and perpendicular to the array's edge, respectively.

Figure 2

(a) Frequency spectrum of the collective spin-wave modes in a finite-width stripe of dipolarly coupled magnetic nanopillars. The number of nanopilars along the width direction of the stripe (which is parallel to the lattice vector $a_2$) is equal to $N=31$. Thick black lines correspond to the edge modes, while thin gray lines correspond to the bulk modes. Yellow region shows the spectrum of bulk modes in an infinitely wide array of nanopillars with the same geometric parameters. (b) Distribution of the time-averaged amplitude of magnetization across the stripe width for the first four edge modes. Corresponding modes are marked by the similar symbols in (a). (c) Distribution of the internal static magnetic field across the stripe width. Parameters of the array: lattice constant $a=4.1R$, pillar aspect ratio $h/R=5$.

Figure 3

A sketch demonstrating formation of edge modes at the external (solid blue line) and internal (dashed green line) boundaries of a nanopillar stripe. (a) A stripe of magnetic pillars with edge pillars experiencing different internal static magnetic field than the pillars in the bulk. (b) Equivalent effective medium consisting of three continuous magnetic films with different parameters. Damon-Eshbach-like edge modes always form at one side (left side in the figure) of the boundaries (external or internal). This results in the nonreciprocity of edge mode profiles localized near the opposite edges of the stripe.

Figure 4

(a) Frequency spectrum of spin-wave modes in a finite-width stripe of magnetic nanopillars with spatially uniform profile of the internal static magnetic field. The black line shows the edge mode, while the gray lines show the bulk modes. (b) Distribution of time-averaged amplitude of magnetization across the stripe width for the edge modes (star and circle symbols denote the edge modes propagating in opposite directions). The array parameters are the same as in Fig. 2, but $h/R=20$.

Figure 5

Dispersion of the spin-wave edge modes in two finite-width stripes (with $N=10$ and $N=31$ nanopillars along the stripe width) of infinitely high nanopillars (black dots). For comparison, the dispersion of the Damon-Eshbach mode propagating in a continuous magnetic film with equivalent parameters is shown (yellow lines).