Electrostatic resonances of heterostructures with negative permittivity: Homogenization formalisms versus finite-element modeling

We investigate the effective permittivity epsilon of two-phase two-dimensional heterostructures, consisting of an inclusion (or cross section of infinite parallel, infinitely long, identical, cylinders, where the properties and characteristics are invariant along the perpendicular cross sectional plane), of permittivity epsilon(2)=epsilon(2)'+epsilon(2)"i with epsilon(2)' being positive or negative, in a matrix of permittivity epsilon(1) (hereafter, assumed to be real valued and positive). Our method for computing epsilon=epsilon'+epsilon"i is based on formulating the conservation of electric displacement flux through the interface separating the two media on systems with periodic boundary conditions in one direction. We identify two distinct behaviors in the surface fraction phi(2) dependence of the effective permittivity according the value of |epsilon(2)'|/epsilon(1) relative to 1, which is a consequence of the duality symmetry. The incorporation of negative values of epsilon(2)' into our calculations leads to a peak in epsilon"(phi(2)) whereas epsilon'(phi(2)) decreases to zero, which are both results of an electrostatic resonance (ER) phenomenon. We demonstrate that one can generate heterostructures characterized by an upward shift in the ER position as epsilon(2)" is increased. This suggests that, in principle, this property can be used to provide a wide range of innovative structures from specially designed composite materials, e.g., reconfigurable composite device. The comparison of our data with Maxwell Garnett (MG) and Bruggeman (SBG) homogenization formalisms permits a quantitative assessment of the ability of the two methods to capture the effects of surface fraction on epsilon. These methods have severe inadequacies, which arise physically from an incorrect treatment of the higher multipoles than dipole moments. We argue that the inappropriateness of SBG formula can originate from its prerogative that phase 1 and phase 2 are treated symmetrically. Our calculations show that MG formula may provide reasonable estimates for epsilon, even close to the ER position, of homogenized two-phase heterostructures with the real part of the complex-valued permittivities of phases having opposite signs and provided that |epsilon(2)'|<=epsilon(2)".