## Abstract

The effective plasma frequency *f _{p}* of periodic metallic wires whose characteristic dimensions are comparable to their skin depth has been analyzed. And a relevant analytic model is constructed by considering the skin effect and making a reasonable shape approximation, which is suitable for the case that the cross section of the wire is noncircular. To verify this model, a wires array with rectangle cross section is designed and the corresponding stacked Au-SiO

_{2}nanostrips are fabricated. The experimental and simulational transmittances of the metamaterial have been evaluated with a good agreement, although the presence of quartz substrate and structural imperfections in experiment will have an impact, which validates that the multilayer Au-SiO

_{2}nanostrips could function similarly to a natural bulk metal with discrepancies of

*f*values less than 8%. It could be confirmed that the theoretic formula is trustworthy in predicting

_{p}*f*for designing and realizing a controllable artificial metal in optical region.

_{p}© 2010 OSA

## 1. Introduction

Early in 1962, periodic arrangements of thin metallic wires were known to use as a negative permittivity medium [1]. It was therefore important to understand how the electromagnetic response (especially *f _{p}*) depends on the structural parameters of the wire system. The early relevant work was proposed by Pendry

*et al*., they demonstrated that the metallic wire-mesh structures have a low frequency stop band from zero frequency up to a cut-off frequency, which was attributed to the motion of electrons in the metal wires [2,3]. What is more, this artificial material could exhibit novel electromagnetic properties being similar to bulk metals. And

*f*was depressed into GHz region. The metamaterial of Pendry was composed of very thin wires (namely r<<

_{p}*δ*,

*r*and

*δ*is the radius and skin depth of the metal wire, respectively), and the cross section of wire is specifically required to be circular. All electrons would equally participate in the modulation of

*f*. Thus the skin effect could be ignored, and Pendry’s model was given by

_{p}*c*

_{0}is the velocity of light in vacuum, and

*a*is the lattice constant of the wire array. Since then several alternative theories have been proposed [4–6]. However, the skin effect which influences to the interaction of the electromagnetic wave with metal is insufficiently considered. With regard to the condition that the wire is thick, namely the radius is relatively larger than the skin depth

*δ*, it has been verified that only the effective active electrons near the wire surface will work and take part in the modulation of

*f*[7]. In addition, the effects of skin depth on

_{p}*f*were studied for the case of rods with circular cross-section [8,9].

_{p}However, for creating artificial structured material in the optical region, besides the lattice period is reduced to the nanoscale level, the dimension of the metal wire is very close to the skin depth and the cross section of the metal wire can normally not be formed in a circular shape by present nano-fabrication means. It is very important to develop a relevant method for predicting *f _{p}* of the optical structured material with above requirement.

In this paper, we demonstrate a design for predicting *f _{p}* of the structured material in optical region with a noncircular cross section being comparable to the skin depth

*δ*. Accordingly, an improved model is deduced by considering the skin effect and making a reasonable shape approximation. The noncircular wires could be approximated into circular ones by keeping the total amount of the electrons invariable which determines

*f*. For instance, the design is given for wires array with rectangle cross section. According to the theoretic investigation of

_{p}*f*, we report a successful fabrication of an elementary artificial metal composed of multilayer Au-SiO2 nanostrips whose

_{p}*f*is in optical region. The experimental and simulational transmittance results have been evaluated with a good agreement. It is validated that the multilayer Au-SiO

_{p}_{2}nanostrips could function similarly to a natural bulk metal based on the model. This metamaterial may lead to various plasmonics-based applications such as subwavelength waveguides and antennas [10,11], spectral selective filters [12], superlens [13,14] and negative refraction [15].

## 2. Modeling of effective plasma frequency

In Fig. 1
, an array of infinitely long, parallel metal wires with noncircular cross section is placed periodically in a square lattice with distance *a.* The electric field is applied parallel to the wires (along the *y* axis). When the waves impinge this system, the electrons are confined to move within the wires. Thus, two vital consequences are brought: the first is the decrease of the average electrons density due to diluting of the metal by the air space; the second is the distinct enhancement of the effective mass of the electrons caused by magnetic effects [2].

As is well known, the plasma angular frequency *ω _{p}* is given in terms of the effective electron density $n{\text{'}}_{eff}$, the effective electron mass ${m}_{eff}$ and charge

*e*.

*n*', then the total amount of the electrons participating in the modulation of

*f*, can be expressed approximately aswhere $\rho (x,z)$ represents the density factor of the electrons, which can be approximately expressed as ${\mathrm{exp}}^{-\mathrm{\Delta}d/\delta}$ due to skin effect, and Δ

_{p}*d*is the decaying distance. By keeping

*N*' unchangeable, we make an approximation as shown in Fig. 1. The noncircular wire has been replaced by a newer wire with an effective radius

*r*, in which the active electrons can uniformly participate in the

_{eff}*f*modulation process.The effective electrons density in the effective structure as a whole is given by the fraction of space occupied by the effective wire,In addition, the distinct enhancement of the effective mass of the electrons could be well expressed through the above approximation. The effective mass of the electrons can be achieved with the same principle as in Pendry’ work [2]Thus, we conclude that

_{p}*f*of the arranged metal wires, whose characteristic dimensions are comparable to

_{p}*δ*, could be expressed as

_{2}is used to act as the filling layer, a modulation factor

*ε*called relative permittivity of the relevant dielectric must be added which have been verified in our previous report [7]. Then, the ultimate expression of

_{r}*f*can be expressed as

_{p}*f*in Eq. (8) is independent of the microscopic quantities such as the electron density and the mean drift velocity. It only depends on the spacing

_{p}*a*and the effective radius of the wire

*r*. And

_{eff}*r*could be obtained through solving Eqs. (3) and (4), which is determined by the structural parameters of the wire and the skin depth

_{eff}*δ*.

In order to test the improved model, we first carried out 3D Finite-Difference Time-Domain (FDTD) calculations for Au wire with a circular cross section. Au could be well described by the free-electron Drude model [16] with the parameters plasma frequency *ω _{pl}* = 1.32 × 10

^{16}s

^{−1}and collision frequency

*ω*= 1.2 × 10

_{col}^{14}s

^{−1}. Note that the skin depth of Au is frequency dependent, whereas the change is not obvious. Therefore, we consider it as an invariable parameter with the value of 20 nm in the visible and near-infrared region [17]. As the cross section is circular, Eq. (3) can be specifically expressed as

*r*is the radius of the original circular wire, which is comparable to

*δ*. When

*r*= 25 nm,

*r*can be obtained to be 20.7 nm through solving Eqs. (4) and (9). To obtain

_{eff}*f*through simulation, we applied the well known retrieval procedure [18] to calculate the effective permittivity

_{p}*ε*from the simulated reflection and transmission data. And

_{eff}*f*can be determined at the frequency where

_{p}*ε*= 0. Figure 2 shows the comparisons of

_{eff}*f*values, which are derived by the FDTD simulation and our analytic model, separately. The discrepancies between the simulation and our model are small, and the maximum value is less than 8%.

_{p}The rectangular cross section of wire is considered as shown in Fig. 3
, where the width and height of the rectangle is *w* and *t*, respectively. For this case, *N*' can be formulated as the following expression corresponding to the Eq. (3). Here *t* is comparable to *δ* and much smaller than *w*,

*w*= 70 nm and

*t*= 30 nm can be replaced by an equivalent circle with

*r*= 21.7 nm. In Fig. 3, the simulation results by FDTD as the criterion are compared with the predicting ones by model which indicates the validity of the improved model for the rectangle wire. Similarly, this model could be suitable for many other shapes, such as ellipse, square and irregular shape through above reasonable approximation.

_{eff}## 3. Experiment

In what follows, the fabrication is described for the artificial metal whose *f _{p}* is in optical region by using Au strip surrounded by SiO2. An elementary bulk metamaterial composed of triple Au strips has been designed, as shown in Fig. 4(a)
. We have actualized an etching-based procedure [19] to fabricate these multilayer materials. Figure 4(b) shows the major fabrication processes.

The first approach was to evaporate Au and SiO_{2} layer by layer (Au-SiO_{2}-Au-SiO_{2}-Au) onto the SiO_{2} substrate at pressures about 4 mTorr by magnetron sputtering (LAB-18, Kutt. J. Lesker). The depths of Au and SiO_{2} were 20 nm and 40 nm, respectively. Subsequently, 200 nm thick poly(methy lmethacrylate) (PMMA) was spun on the top of the upper Au film. And then, lithography was performed by using standard electron beam exposure apparatus (JBX5500ZA), followed by an appropriate development for 90 s in a solution of methyl isobutyl ketone (MIBK) diluted 1:3 by volume with isopropyl alcohol (IPA) at 21°C. For achieving the artificial metal with excellent performance, the results of electron beam lithography were examined using field emission scanning electron microscope (Quanta 400 FEG) to confirm the fine quality. Although the sample of 40nm minimum lateral feature size and 200 nm thickness were fabricated, some parts of the PMMA patterns collapsed as shown in Fig. 5(a)
because of the high aspect ratio (i.e., height/width). Obviously, this result would not meet the requirement for the following etching step. However, the unwanted effect could be avoided by extending the strip width *w* to 70 nm. Next, the PMMA pattern was transferred into the multilayer Au and SiO_{2} film using deep anisotropic etching in LKJ-1C-150 IBE system with Ar at pressures below 2 × 10^{−2} Pa. The samples were rotated to achieve uniform etching rate in various directions. The cooling system of etching device could help to control working temperature to avoid the PMMA deformation caused by ion bombardment. The etching rates of Au, SiO_{2}, and PMMA were beforehand stabilized at 30 nm/min, 15 nm/min, and 23 nm/min, respectively, with 80 mA ion beam, 300 eV ion energy, 180 V acceleration voltage, and 2 mA neutralization current. After 8 min etching, the pre-patterned PMMA topography could be perfectly transferred into the alternating Au-SiO2 layers. The electron micrograph of the best sample (500 μm × 500 μm footprint) shown in Fig. 5(b) reveals a good large-scale homogeneity with the feature dimension about 70 nm line width and 250 nm period, and the sidewall roughness is about 10 nm. With these experimental parameters, the effective plasma wavelength can be estimated by our model. We can obtain *r _{eff}* = 18.7 nm through solving Eqs. (4) and (10) and

*a*can be appropriately decided to be $\sqrt{p\cdot (s+t)}\approx 123$nm. Since the dielectric material is composed by air and SiO

_{2}, the modulation factor could be effectively decided to be ${\epsilon}_{r}({\text{SiO}}_{\text{2}})g+{\epsilon}_{r}(\text{a}ir)(1-g)\approx 1.3$by effective medium theory, where

*g*represents the ratio of the area of SiO

_{2}to the whole filling layer, ${\epsilon}_{r}({\text{SiO}}_{\text{2}})\approx 2.31$ and ${\epsilon}_{r}(\text{a}ir)\approx 1$. Then, the effective plasma wavelength of ${\lambda}_{p}=c/{f}_{p}=483$nm can be calculated by using Eq. (8).

In order to test the effective plasma wavelength and the properties of the artificial structure, we measured the transmission spectra of the nanostrips sample for two orthogonal polarizations (parallel and perpendicular to the strips) over a broad spectral range. For the optical characteristic evaluation, a commercial ellipsometer (MD2000D, J. A. Woollam) which could provide and collect a linearly polarized broadband light with the range from visible to near-infrared were employed. By complementing with a home-built setup (Diaphragm and travel translation stage), only the lights that passed through the sample were collected. Normalization was performed with respect to the bare quartz substrate.

## 4. Results and discussion

These measured spectra [dash lines in Fig. 6(a)
] are directly compared with numerical calculations [solid lines in Fig. 6(a)]. For the perpendicular polarization, this metamaterial can be regarded as an effective medium [20]. A transmission coefficient with the value of 100% at 725 nm is presented in the simulation curve, as the Fabry-Perot resonance condition is well fulfilled. However, this condition is destroyed due to the existence of quartz substrate in the experiment, so the transmittance peak is not exhibited in the measured spectrum. We can also find that two minima occur at around 570 nm in the transmittance spectrums of the experiment and simulation. This phenomenon results from an anti-symmetric current from in the upper and lower strips, which forms a circular current and gives rise to a magnetic response [21]. The magnetic resonance is greatly sensitive to the geometrical parameters [22]. So, the minima positions which have a minor discrepancy (about 23 nm) between experiment and simulation are responsible for the sidewall roughness of the structure. Anyway the experimental and simulational transmittances reveal a good agreement. For the polarization parallel to the strips, the spectrum displays a diluted metal with *λ _{p}* = 524 nm as shown in Fig. 6(b). And the discrepancy of the effective plasma wavelength between above-mentioned prediction by model and the experiment is below 8%. Above the interested plasma wavelength, the real part of effective permittivity is regularly negative. Note that only the modes whose electric fields are parallel to the strips will play a great role in the modulation of

*f*.

_{p}## 5. Conclusion

In summary, we have demonstrated a metamaterial composed of Au-SiO2 nanostrips numerically and experimentally to create diluted plasma responses in the visible spectrum. Also, we have proposed an improved analytical model to predict *f _{p}*, which is suitable for the case that the characteristic dimension of the metal wire is comparable to

*δ*. The dependence of

*f*on the geometric parameters, which can be predicted by the model, provides us with a general recipe for designing and fabricating such artificial metal at desired frequency. This new artificial material may open new possibilities for many plasmonics-based applications in much wider regime, and the fascinating electrodynamic effects of such metamaterials are expected to be investigated further.

_{p}## Acknowledgment

This work was supported by the National Basic Research Program (2006CB302900), High Tech. Program of China (2007AA03Z332) and the Chinese Nature Science Grant (10904118, 60727006). We thank Wenhua Shi, Qiang Zha, Baoshun Zhang, Yongxin Qiu, Kai Huang and Xionghui Zeng of Suzhou Institute of Nano-tech and Nano-bionics, Chinese Academy of Sciences, for assisting in experiment.

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