Calculation of high-frequency permeability of magnonic metamaterials beyond the macrospin approximation

We present a method of calculation of the effective magnetic permeability of magnonic metamaterials containing arrays of magnetic inclusions of arbitrary shapes. The method fully takes into account the spectrum of spin waves confined in the inclusions. We evaluate the method by considering a particular case of a metamaterial formed by a stack of identical two-dimensional (2D) periodic hexagonal arrays of disk-shaped magnetic inclusions in a nonmagnetic matrix. Two versions of the method are considered. The first approach is based on a simple semianalytical theory that uses the numerically calculated susceptibility tensor of an isolated inclusion as input data for an analytical calculation in which the magnetodipole interaction between inclusions within each 2D array is taken into account. In the second approach, we employ micromagnetic packages with periodic boundary conditions to calculate the susceptibility of the whole 2D periodic array of such inclusions. The comparison of the two approaches reveals the necessity of retaining higher-order terms in the analytical calculation of the magnetodipole interaction via the multipole expansion. Models limited to the dipolar term can lead to remarkable underestimation of the effect of the magnetodipole interaction, in particular, for modes localized near the edge regions of inclusions. To calculate the susceptibility tensor of an isolated inclusion, we have implemented two different methods: (a) a method based on micromagnetic simulations, in which we have compared three different micromagnetic packages: the finite-element package nmag and the two finite differences packages oommf and micromagus; and (b) the modified dynamical matrix method (DMM). The comparison of the different micromagnetic packages and the DMM (based on the calculation of the susceptibility tensor of an isolated inclusion) demonstrate that their results agree to within 3$\%$. Frequency regions in which the metamaterial is characterized by the negative permeability are identified. We speculate that the proposed methodology could be generalized to more complex arrangements of magnetic inclusions, e.g., to those with multiple periods or fractal arrangements, as well as to arrays of inclusions with a distribution of properties.

Figure 1

The geometry of the metamaterial consisting of magnetic disks in a nonmagnetic matrix is shown. The disks are located in nodes of a hexagonal lattice. The disk diameter is $d$ $=$ 195 nm, the in-plane edge-to-edge separation is $b$ $=$ 20 nm, and the distance between the layers is $c$ $=$ 140 nm which is much greater than disk thickness $l$ $=$ 5 nm. The filling factor is ρ $=$ 2.48$\%$.

Figure 2

The real (a) and imaginary (b) parts of four components of effective permeability tensor μ calculated using Eq. (9) are shown as functions of the frequency for the metamaterial depicted in Fig. 1. External magnetic field $H_{\mathrm{bias}}$ $=$ 933 Oe is applied in the plane of the layers along the horizontal line. The blue solid lines, dashed red lines, green open dots, and black filled dots show the permeability calculated using oommf, nmag, micromagus, and DMM, respectively. The insets show the distribution of the normalized magnetization component orthogonal to the applied field in the ground state in a single nanodisk; the spatial profiles of the mode amplitude (top) and phase (bottom) for the two dominant modes of a single nanodisk.

Figure 3

The real (a) and imaginary (b) parts of the $zz$ component of the effective permeability tensor calculated using Eq. (10) with the susceptibility tensor calculated from micromagnetic simulations with PBCs are shown as functions of the frequency for the metamaterial depicted in Fig. 1. External magnetic field $H_{\mathrm{bias}}$ $=$ 933 Oe is applied in the plane of the layers parallel to the $x$ axis. The thin black and thick red dashed lines show the permeability of the metamaterial consisting of noninteracting magnetic disks based on the susceptibility of a single disk calculated by nmag and micromagus, respectively. The thin black and thick red solid lines show the permeability of the metamaterial calculated using PBCs by nmag and micromagus, respectively.