# Magnetic response of nanoscale left-handed metamaterials 

R. S. Penciu $^{1}$, M. Kafesaki1 , 2 , $^{∗}$, Th. Koschny 1 , $^{3}$, E. N. Economou1 , $^{4}$, and C. M. Soukoulis1 , 2 , 3 

$^{1}$Foundation for Research and Technology Hellas (FORTH), Institute of Electronic Structure and Laser (IESL), P.O. Box 1385, 71110 Heraklion, Crete, Greece 

$^{2}$Dept. Materials Science and Technology, University of Crete, 71003 Heraklion, Greece 

$^{3}$Ames Laboratory and Dept. Physics and Astronomy, Iowa State University $^{†}$, Ames, Iowa 50011, USA and 

$^{4}$Dept. of Physics, University of Crete, 71003 Heraklion, Greece ∗ 

Using detailed simulations we investigate the magnetic response of metamaterials consisting of pairs of parallel slabs or combinations of slabs with wires (including the fishnet design) as the length-scale of the structures is reduced from mm to nm. We observe the expected saturation of the magnetic resonance frequency when the structure length-scale goes to the sub-micron regime, as well as weakening of the effective permeability resonance and reduction of the spectral width of the negative permeability region. All these results are explained by using an equivalent resistor- inductor-capacitor (RLC) circuit model, taking into account the current-connected kinetic energy of the electrons inside the metallic parts through an equivalent inductance, added to the magnetic field inductance in the unit-cell. Using this model we derive simple optimization rules for achieving optical negative permeability metamaterials of improved performance. Finally, we analyze the magnetic response of the fishnet design and we explain its superior performance regarding the high attainable magnetic resonance frequency, as well as its inferior performance regarding the width of the negative permeability region. 

Left-handed metamaterials (LHMs), i.e. artificial com- posite structures with overlapping negative permittivity and permeability frequency bands giving rise to negative index of refraction $^{1,2}$, have attracted recently an expo- nentially increasing attention. The main reason behind this attention is mainly the novel physical phenomena associated with those materials (negative refraction, op- posite phase and energy velocity, reversed Doppler effect etc), which result to new capabilities in the manipulation of electromagnetic waves. Such an important capability is the superlensing capability of LHMs $^{3}$, i.e. the abil- ity to offer subwavelength resolution imaging, which can have important implications in many scientific, techno- logical and every-day life areas, like imaging, microscopy, lithography, ultra-compact data storage, etc. 

Since the demonstration of the first left-handed material $^{4}$, in 2000, operating in the microwave regime, many left-handed (LH) structures have been created $^{5–11}$, and important efforts for the better understanding and the optimization of those structures have taken place. Among the various efforts within the LHM research, a large part has been devoted to the extension of the fre- quency of operation of LHMs from the microwaves to the optical regime, where the superlensing-based applications can find an important ground for manifestation. These efforts led to metamaterials with negative permeability operating in the few THz regime already in 2004 $^{12,13}$, which soon were followed by the first structures of neg- ative permeability and/or negative index of refraction in the telecommunications regime and more recently in the lower visible regime. (For reviews of the existing research 

PACS numbers: 41.20.Jb, 42.70.Qs, 81.05.Xj, 78.67.Pt 

## I. INTRODUCTION 

efforts on infrared (IR) and optical metamaterials see Refs. $^{14,15}$.) 

While the first and most of the existing microwave LHMs are systems made of split-ring resonators 16 (SRRs, i.e. interrupted metallic rings, giving rise to resonant loop-like currents, and thus to resonant permeability involving negative permeability values) and continuous wires (leading to the negative permittivity response $^{17}$), in most of todays high frequency LHMs the SRRs have been replaced by pairs of slabs (or stripes, or wires) 18–21 - see Fig. 1(a). Like the SRR, the slab-pair also be- haves as a resonant magnetic moment element, where the magnetic moment is created by resonant currents, antiparallel in the two slabs of the pair, forming a loop- like current. In most of the experimentally realized opti- cal slab-pair&wire structures the slabs are as wide as the corresponding unit cell side and are physically connected with the wires, leading to a design known as fishnet 22–25 (see Fig. 1(d)). Fishnet design was able to give the high- est in frequency LHMs up to now $^{26,27}$. 

The main reason behind the replacement of SRR by the slab-pair for the high frequency metamaterials, apart of slab-pair simplicity in fabrication (which is also a crucial parameter), is its ability to exhibit negative permeability response for incidence normal to the plain of the pair; this makes possible the demonstration of the negative permeability response with just a monolayer of slab-pairs. Indeed, up to now most of the demonstrated “magnetic” metamaterials (i.e. materials of resonant and negative permeability) and LHMs are single layers, while only few multilayer samples have been fabricated $^{13,28}$. (Note that in the optical regime what is difficult to be achieved is the negative permeability component of a LHM, since 

