Reversible tuning of omnidirectional band gaps in two-dimensional magnonic crystals by magnetic field and in-plane squeezing

By means of the plane-wave method, we study nonuniform, i.e., mode- and $\mathit{k}$-dependent, effects in the spin-wave spectrum of a two-dimensional bicomponent magnonic crystal. We use the crystal based on a hexagonal lattice squeezed in the direction of the external magnetic field wherein the squeezing applies to the lattice and the shape of inclusions. The squeezing changes both the demagnetizing field and the spatial confinement of the excitation, which may lead to the occurrence of an omnidirectional magnonic band gap. In particular, we study the role played by propagational effects, which allows us to explain the $\mathit{k}$-dependent softening of modes. The effects we found enabled us not only to design the width and position of magnonic band gaps, but also to plan their response to a change in the external magnetic field magnitude. This allows the reversible tuning of magnonic band gaps, and it shows that the studied structures are promising candidates for designing magnonic devices that are tunable during operation.

Figure 1

2D MC based on the hexagonal lattice: cobalt rods (blue) in a permalloy matrix (gray). (a) The base structure: $a$ is the lattice constant, $R$ is the radius of Co rods. (b) The structure squeezed in the $x$ direction by the structure ratio $s$. (c,d) First Brillouin zone for the base and squeezed structures, respectively. Squeezing of the structure results in the proportional stretching of the FBZ. High-symmetry paths are marked by blue lines.

Figure 2

Ten lowest modes in spin-wave spectra of Co/Py 2D MCs along the high-symmetry path in the FBZ [Figs. 1 and 1] for two external magnetic field magnitudes: (a)–(c) 50 mT and (d)–(f) 100 mT, and for three squeezes: (a,d) the base structure (the structure ratio $s=1)$, (b,e) $s=0.6$, and (c,f) $s=0.4$. Line colors depict the concentration factor calculated from Eq. (3) according to the color scale shown in the inset of (a). Dotted horizontal lines represent the upper and lower band edges of complete band gaps. All graphs are accompanied by the density of states calculated over the whole FBZ and plotted to the left—all on the same scale.

Figure 3

Spin-wave profiles of three modes with the lowest frequency at the FBZ center for the structure ratio (a) $s=0.6$ and (b) $s=0.4$ at the external field 50 mT calculated at three high-symmetry points in FBZ: $M^{\prime }$ (left column), $\mathrm{\Gamma }$ (central column), and $K$ (right column). Profiles in the same row correspond to each other and they are not always ordered according to their frequency in the spectrum (see mode number $n)$. Ellipses mark Co rod borders. Colors represent argument (phase), and their intensity indicates the modulus of the dynamic magnetization, as is shown in the inset.

Figure 4

(a) Two lowest band gaps vs squeezing (structure ration, $s)$ for Co/Py 2D MC at three magnitudes of the external magnetic field. (b)–(d) The gap width evolution with $s$ for band gaps presented in (a). Red bars depict the maximal blurring of band gaps caused by the Gilbert damping.

Figure 5

Two lowest band gaps vs the external field magnitude $H$ for four structure ratios: (a) $s=0.34$, (b) $s=0.42$, (c) $s=0.6$, and (d) $s=0.85$. Below each graph, the dependence of the gap width on $H$ is shown (red bars depict the maximal blurring of band gaps caused by the Gilbert damping). The horizontal dashed line stands for the frequency of 7 GHz (see the text).