Metamaterials 6 (2012) e127–e138 Available online at www.sciencedirect.com Metamaterial properties of ferromagnetic antidot lattices R. Zivieri ∗ , L. Giovannini Dipartimento di Fisica and Scienze della Terra and CNISM Unità di Ferrara, Via Saragat 1, I-44122 Ferrara, Italy Received 1 October 2012; received in revised form 6 November 2012; accepted 8 November 2012 Available online 13 December 2012 Abstract In this paper the metamaterial properties of two-dimensional arrays of circular antidots (holes) embedded into a ferromagnetic medium of Permalloy are studied according to both micromagnetic and analytical calculations. The periodicity of the arrays and the diameters of the antidots are in the nanometric range. The collective mode dynamics is described by means of effective physical quantities for the scattering geometry with the external magnetic field applied perpendicularly to the Bloch wave vector in the antidot plane. As an example, the definition of an effective field, incorporating the demagnetizing effects due to the holes, permits to describe the dynamical properties of collective modes in terms of effective properties in the travelling regime. An effective wavelength and a small wave vector are introduced both for extended and localized magnonic modes. By means of these effective quantities it is shown that holes play the role of point defects affecting the spin dynamics in the microwave range. Relations between the effective wavelength and the Bloch wavelength and between the corresponding small wave vector and the Bloch wave vector are found. Some effective rules on the dynamic magnetization, based upon the effective wavelength and the corresponding small wave vector, are derived. An application that exploits the definition of the small wave vector is proposed and an experiment based upon the notion of effective wavelength and small wave vector is suggested. © 2012 Elsevier B.V. All rights reserved. PACS: 81.05.Xj; 78.67.Pt; 75.30.Ds Keywords: Metamaterials; Low-dimensional magnetic systems; Magnonic crystals; Effective properties 1. Introduction In these last years great attention has been devoted to the study of metamaterials both for their interest- ing properties and for the high number of potential applications. Among them the most studied are the elec- tromagnetic metamaterials, because of their negative refractive index able to permit the creation of negative superlenses overcoming the diffraction limit [1,2]. Due to the high ratio between the vacuum electromagnetic ∗ Corresponding author. Tel.: +39 0532 974232; fax: +39 0532 974210. E-mail address: zivieri@fe.infn.it (R. Zivieri). wavelength and the periodicity of the structure, electro- magnetic metamaterials [3] can be distinguished from the well-known photonic crystals [4–6]. An important step forward was done showing that it is possible to build a material exhibiting simultaneously negative effective permeability and negative effective permittivity in the microwave regime [7] confirming the prediction of Veselago [8]. In the last decade great efforts have been devoted also to the investigation of plasmonic metama- terials for their interesting optical properties. Since the wavelength of electromagnetic waves and of plasmonic waves is much larger than the periodicity of the artificial arrays, the description of the most important physical quantities can be made in terms of effective properties [9–11]. Other classes of widely investigated special 1873-1988/$ – see front matter © 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.metmat.2012.11.003