High frequency permeability of magnonic metamaterials with magnetic   inclusions of complex shape
  We present a method of calculation of the effective magnetic permeability of magnonic metamaterials containing periodically arranged magnetic inclusions of arbitrary shapes. The spectrum of spin wave modes confined in the inclusions is fully taken into account. Within the scope of the proposed method, we compare two approaches. The first approach is based on a simple semi-analytical theory that uses the numerically calculated susceptibility tensor of an isolated inclusion as input data. Within the second approach, micromagnetic packages with periodic boundary conditions (PBC) are used to calculate the susceptibility of a single 2D periodic array of such inclusions, with the whole 3D metamaterial consisting of a stack of such arrays. To calculate the susceptibility tensor of an isolated inclusion, we have implemented and compared two different methods: (a) a micromagnetic method, in which we have employed 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. To illustrate the methodology, we have calculated the effective permeability of a metamaterial consisting of a stack of hexagonal arrays of magnetic nanodisks in a non-magnetic matrix. The range of geometrical parameters for which such a metamaterial is characterized by the negative permeability has been identified. The critical comparison of the different micromagnetic packages and the dynamical matrix method (based on the calculation of the susceptibility tensor of an isolated inclusion) has demonstrated that their results agree to within 3 %. 
O. Dmytriiev, M. Dvornik, R. V. Mikhaylovskiy, M. Franchin, H.   Fangohr, L. Giovannini, F. Montoncello, D. V. Berkov, E. K....
cond-mat.mes-hall