Nonuniform Spin-Wave Softening in Two-Dimensional Magnonic Crystals as a Tool for Opening Omnidirectional Magnonic Band Gaps

By means of the plane-wave method we study spin-wave dynamics in two-dimensional bicomponent magnonic crystals based on a squeezed hexagonal lattice and consist of a permalloy thin film with cobalt inclusions. We explore the dependence of a spin-wave frequency on the external magnetic field, especially in weak fields where the mode softening takes place. For considered structures, the mode softening proves to be highly nonuniform on both the mode number and the wave vector. We find this effect to be responsible for the omnidirectional band gap opening. Moreover, we show that the enhancement of the demagnetizing field caused by the squeezing of the structure is of crucial importance for the nonuniform mode softening. This allows us to employ this mechanism to design magnonic gaps with different sensitivity for the tiny change of the external field. The effects we find should be useful in designing and optimization of spin-wave filters highly tunable by an external magnetic field.

Figure 1

Thin-film MC based on the 2D hexagonal lattice composed of cobalt rods (blue) in a permalloy matrix (orange). (a) The base structure with the lattice constant $a$ and the radius of $\mathrm{Co}$ rods $R$. (b) The structure squeezed in the $x$ direction by the structure ratio $s$. (c) First Brillouin zone for the base structure shown in (a) with high-symmetry paths indicated (blue lines).

Figure 2

Spin-wave spectra of $\mathrm{Co}$/$\mathrm{Py}$ 2D MCs along the high-symmetry paths in the FBZ [compare Fig. 1] for (a) the base structure at 50 mT, and the squeezed structure with a structure ratio $s=0.6$ at (b) 150, (c) 100, and (d) 50 mT. (e) The spectrum for the squeezed structure with $s=0.3$ at 52 mT. Line colors depict the concentration factor calculated from Eq. (1) according to the color scale shown in the inset of (a). Dotted horizontal lines represent the upper and lower band edges of complete magnonic gaps.

Figure 3

(a) Demagnetizing field $H_d$ for the squeezed $\mathrm{Co}$/$\mathrm{Py}$ MC ($s=0.6$) along with its cross-sectional profiles parallel (bottom) and perpendicular (left) to the external field. (b),(c) Spin-wave profiles for the MC in question in an external field of 50 mT for the lowest ($n=1$) and the second ($n=2$) mode in the spin-wave spectrum at the center of the FBZ [compare Fig. 2]. Colors represent argument (phase) and their intensity the modulus of the dynamic magnetization, as shown in the inset.

Figure 4

Width of magnonic band gaps vs structure ratio $s$ and external field magnitude $H$ for (a) the first and (b) the second gap. Colors depict the gap width.

Figure 5

(a) Magnonic band gaps vs the external field magnitude $H$ for two structure ratios: $s=0.5$ and $s=0.34$. (b) The dependence of the gap width on $H$ for both gaps shown in (a).