Plasmonics (2014) 9:415–425 DOI 10.1007/s11468-013-9638-3 Effective Permittivity of Metal–Dielectric Plasmonics Nanostructures H. Sadeghi · A. Zolanvar · H. Khalili · M. Goodarzi · J. Nezamdost · S. Nezamdost Received: 10 July 2013 / Accepted: 7 November 2013 / Published online: 27 November 2013 © Springer Science+Business Media New York 2013 Abstract Increasing attention on plasmonics materials has been paid due to their exciting physical behaviors and potential applications. Here, we present the application of the Kubo formula by providing a description of the underlying mechanisms to realize the effective permit- tivity. We obtain plasma frequency and linear relation of frequencies with the effective permittivity. We have proposed an improved method to retrieve the effective parameters (index of refraction, impedance, permittivity, and permeability) of metamaterials from transmission and reflection data. Extents far above the calculated rela- tions are very good match with effective permittivity- related environment, including nanospheres of some met- als and a samples of the high refractive index dielectric material. Keywords Nanostructured materials · Kubo formula · Plasmonics · Effective permittivity Introduction The concept of an effective permittivity is useful to describe the optical response of a metamaterial molded into arbitrary H. Sadeghi () · A. Zolanvar · H. Khalili · M. Goodarzi Department of Physics, Faculty of Science, Arak University, Arak 38156-8-8349, Iran e-mail: H-Sadeghi@araku.ac.ir J. Nezamdost Iranian Academic for Education, Culture and Research, Arak, Iran S. Nezamdost Department of Communication, Islamic Azad University, South Tehran Branch, Tehran, Iran shapes, rather than having to deal with the detailed arrange- ment of particles that form it. Now, the question arises: What is the minimum size for which a piece of metamaterial still behaves as an effective homogeneous medium tailored into the same shape? Theoretically, the permittivity and refractive index have been calculated using effective mod- els by considering the wavelength outside spheres, which is larger compared to sphere diameters [3, 5, 9]. These stud- ies are based on effective medium theories for a composite material consisting of spheres which resonate either in the electric or magnetic mode. Their results show that electri- cal properties of spheres can have a significant effect on the effective permittivity and refractive index of their compos- ite. The size distribution of the spheres causes increasing losses in the frequency band, where backward waves can exist. The mixing of the Clausius–Mossotti relation (CMR), the Kramers-Kronig relations, and the theory of linear response [6] are applied for the formulation of metal– dielectric effective permittivity ε(q, ω) of a homogeneous system near the Mie resonance modes. The structure is assumed to be nonmagnetic metallic and dielectric spheres. Relative to the incident light wavelength, scatter- ing by small spherical particles is based upon the exact Mie solution of the diffraction [4]. All scattering objects can be represented by effective electric polarizability densities. The experimental composites will be made using a vari- ety of fabrication methods, including multilayer thin-film deposition, focused ion beam milling, and self-assembly [7]. Both the past theoretical and experimental studies revealed that the properties of surface plasmons depend on multi- ple parameters, including shape, size, and composition of metal nanostructures, and dielectric constant of the media surrounding metal nanostructures.