We fabricated silver split-ring resonators (SRRs) using nanosphere lithography and measured the LC resonance wavelength of single isolated SRRs in optical wavelength range. The SRRs’ sizes decreased when smaller polystyrene spheres were used as templates, and their LC resonance wavelength decreased to 721 nm. The LC resonance wavelength corresponding to the observed properties of the SRRs was calculated using the LC circuit model; we confirmed that the observational and calculated results agreed well. The LC resonance frequency of a miniaturized SRR with a constant shape was also calculated. For SRRs of the shape that we fabricated, the estimated short-wavelength limit was 426 nm.
©2012 Optical Society of America
A split-ring resonator (SRR)  causes magnetic resonance near the LC resonance frequency and changes the permeability of its metamaterial. The electric and magnetic fields are expected to be enhanced when LC resonance occurs in SRRs. Thus, it is interesting to use SRRs not only as a unit cell of the metamaterial but also as an antenna. In optical wavelength range, the LC resonance wavelength of an SRR can be explained by the LC resonant circuit theory including its internal impedance [2–6]. In the theory, the inductance L and capacitance C of an SRR depend on its material, shape, and size. Reducing the operating wavelengths of SRRs is a pressing concern in metamaterial applications; several studies have also focused on shortening the LC resonance wavelength [7–9]. One study reported the miniaturization of SRRs to achieve high-frequency operation while maintaining the magnetic characteristics, device structure, and choice of SRR material . However, numerical calculations show that the LC resonance frequency saturates with the miniaturization of the SRR [2, 6, 7]. This saturation is caused by the existence of the so-called kinetic inductance phenomenon .
The size of the SRR must be reduced to around 100 nm for operation in the visible/near-infrared region; however, it is technically difficult to make an SRR this small with high accuracy. Electron beam lithography was used to fabricate SRRs that operated in this region [7, 9–12]. However, this method is unsuitable for mass production because the process is complex and the system is expensive. Shumaker-Parry et al. produced a crescent-shaped gold nanorod by nanosphere lithography (NSL) . We recently fabricated a silver SRR approximately 130 nm in diameter in a similar manner. This method will make it possible to produce a large number of SRRs at a comparatively low cost. In addition, we successfully measured the scattered light spectrum of a single isolated SRR in the visible/near-infrared region, and we clarified that the long-wavelength peaks in the light-scattering spectra corresponded to a fundamental LC resonance mode excited by an incident electric field . However, to our knowledge, there have been no reports on the dependence of the LC resonance wavelength on the size of a silver SRR fabricated by the NSL method. Moreover, it has not been clarified yet whether the counting approach based on the LC circuit theory is appropriate for evaluating this dependence for our SRR.
In this study, we aimed to develop an LC circuit model corresponding to the silver SRR fabricated by NSL to clarify its size parameters and the relationship of these parameters with the LC resonance wavelength. In addition, the short-wavelength limit of the LC resonance in an SRR fabricated by NSL was considered using LC circuit theory. In SRR fabrication using NSL, the size of the SRR depends mainly on the diameter of the polystyrene spheres used as a template. Therefore, we reduced the size of the spheres and miniaturized the SRRs in order to further shorten their LC resonance wavelength. Most current reports on the experimental investigation of the SRR LC resonance wavelength are concerned with the study of an array of SRRs. However, in such a case, the measurement results represent ensemble data for numerous SRRs. Therefore, a strict evaluation concerning the relationship between the SRR size and the LC resonance wavelength cannot be performed because of the slight variations in the sizes of the SRRs in the array. Hence, in our research, the scattering spectrum of individual SRRs was observed. In addition, the size parameters of the SRRs were determined from the electron microscopy image of the corresponding SRR.
2. Experimental and calculation methods
SRR fabrication method
The SRR structure was produced by NSL . Silver, which has a small optical loss, was chosen as the SRR material. The fabrication procedure is as follows. A colloidal solution of polystyrene spheres 100 nm or 60 nm in diameter was deposited on a glass substrate. The spheres were then coated with a thin silver film 40 nm thick via thermal evaporation deposition. The substrate was tilted by θd during deposition. Next, the silver was removed by argon ion sputtering. For this step, the substrate was tilted by -θs. The polystyrene spheres were melted and removed with acetone, leaving SRRs distributed on the glass substrate. Refer to  for details. The SRRs were observed by electron microscopy, and their sizes were measured.
Method of measuring scattered light spectrum of single isolated SRR
An optical system for dark-field microscopy/spectroscopy similar to that in  was used to observe the scattered light spectrum of a single SRR. Plane polarized white light was input to the SRR. The z direction was taken as the direction of the optical axis of the incident light perpendicular to the glass substrate. The y direction was the direction of the gap from the center of the SRR, and the x direction was the direction perpendicular to the y axis with respect to the substrate surface. The object lens used for the incident light had a magnification of 50 × and a numerical aperture (NA) of 0.55. The transmitted and scattered light from the SRR passed a 100 × object lens (NA = 0.95) and reached the pupil plane, where a filtering mask was located. The filter blocked the transmitted light of small NA and passed the scattered light of large NA, yielding a dark-field image from the scattered light. A light beam 100 μm in diameter obtained from the projected image at confocal plane was routed to a multichannel spectrometer via an optical fiber.
The dark-field spectrum from the SRR is denoted by IS. Away from the SRR, the dark-field and bright-field spectra are denoted by IBD and IBB, respectively. The bright-field spectrum was obtained when the filtering mask was removed from the pupil plane. Thus, the light scattering can be expressed as
LC circuit model
The theoretical value of the LC resonance wavelength λLC-cal was calculated using the LC circuit model [2, 4–6]. The SRR is assumed to have the shape shown in Fig. 1(a) , and the LC circuit shown in Fig. 1(b) is considered. The incident light is vertical to the substrate. Here, 2R, 2r, d, wg, and wr are the outside diameter, inside diameter, gap distance, width of the gap, and width of the ring body, respectively. These values were measured from scanning electron microscopy (SEM) images of an SRR. The thickness h of the SRR was assumed to be uniform to simplify the calculations.
The capacitance C of the SRR depends on the shape of the gap and is generally calculated as15]. It was assumed that the gap in our SRR can be treated similarly, and hence, we assumed that εg = (neff)2.
The inductance of the SRR consists only of the self-inductance caused by the electric current in the metallic ring. Because the distance between SRRs is sufficiently large compared with the diameter of the ring, mutual inductance can be disregarded. The electric current in the metal is induced by the electric field of the incident light. Assuming a skin depth of δ, the distribution of the electric field that penetrates the metallic surface in the z direction is proportional to exp(–z/δ). Thus, if the electric current density on the surface is J0, the total current that flows in an SRR of thickness h is
The self-inductance Ls of the ring is generally determined by the geometric shape of the ring. We assume that Ls  is14]), the electric charge in the SRR had concentrated in the top region of the gap of the SRR, where the electric field was especially strong. Therefore, the effective path of the electric current in the SRR was considered to be a circle passing through the top of the gap and the center of wr, with approximated radius a = (2R – wr/2)/2. Incidentally, in optical wavelength range, the kinetic energy of free electrons in metal cannot be ignored as another magnetic energy. The inductance Lkin arising from the kinetic energy of the electrons is derived from the motion equation of free electrons . Alternatively, Lkin can be derived from Ohm’s law by considering the SRR to be a current circuit . In the optical frequency region, the conductivity of a metal is described as 
3. Results and discussion
Size of fabricated SRRs
Figure 3 shows the relationship between the diameter of the polystyrene spheres and the size of the SRR. The data confirm that SRRs can be miniaturized if smaller polystyrene spheres are used as a template. The outside diameter of the fabricated SRR was several tens of nanometers larger than the diameter of the polystyrene spheres. This is attributed to redeposition by sputtering during argon ion etching. Moreover, the diameter varied although the SRRs were fabricated under the same conditions. The ratio 2R:2r:d of the fabricated SRRs also varied. The reason is thought to be that the size of the polystyrene spheres used for the template varied. In addition, local variations in the thickness of the silver film and nonuniform argon ion etching are thought to contribute.
Scattered light spectra of single isolated SRR
The scattered light spectrum of an isolated single SRR was observed and measured across each LC resonance wavelength. Figure 4 shows an example of the results.
For Ex polarization, the peak on the long-wavelength side (LC1) indicates excitation of the fundamental LC resonance, and the peak on the short-wavelength side indicates excitation of the higher-order quadrupole-type LC resonance (LC2) . For Ey polarization, the peak on the long-wavelength side indicates excitation of the electric-dipole-type localized surface plasmon excitation (LSP1). The fundamental LC resonance wavelength is the focus of this paper. The resonance wavelength obtained in this experiment is denoted as λLC-exp. In Fig. 4(a) and 4(b), the fundamental LC resonances are 950 nm and 760 nm, respectively.
Figure 5 shows the relationship between the diameter of the polystyrene spheres used to fabricate the SRRs and λLC-exp. The LC resonance wavelength of SRRs fabricated using 60 nm polystyrene spheres was shorter than that of SRRs fabricated using 100 nm polystyrene spheres. The shortest LC resonance wavelength that we obtained was 721 nm. The difference in the LC resonance wavelengths is attributed to differences in the outside diameter of the SRRs due to differences in the size of the spheres, as described above. However, λLC-exp was not necessarily the same for SRRs having the same outside diameter. The reason is that λLC-exp depends on not only the outside diameter of the SRR but also the gap distance and width and the ring width .
Comparison of experimental and theoretical values of LC resonance wavelength
The size parameters of the fabricated SRRs were measured from the SEM images. Their thickness t cannot be confirmed in the SEM images. However, on the basis of , we estimated that t is about 22 nm. The skin depth of silver at visible/near-infrared wavelengths is about 22 nm . Therefore, we assumed that h = δ = 22 nm. For the plasma frequency ωp of silver, we used ωp = 14.0 × 1015 rad/s . The capacitance of the SRRs was influenced by not only air but also the glass substrate on which the SRRs were fabricated. When the refractive indices of the gap medium and the substrate are assumed to be 1.00 and 1.51, respectively, and α is taken as 0.58, neff becomes 1.21. Thus, εg was assumed to be (1.21)2. Figure 6 shows the relationship between λLC-cal, obtained by the LC circuit theory, and λLC-exp, obtained from the scattered light spectrum; the λLC-cal and λLC-exp values are relatively close. The good agreement between the measured and calculated LC resonance wavelengths leads us to conclude that the LC circuit theory accurately describes the optical response of the SRRs.
Estimation of short-wavelength limit in LC resonance wavelength
The dependence of the LC resonance frequency on the SRR size was calculated using the LC circuit model. The results are shown in Fig. 7 . The scaling factor F is a coefficient indicating the diameter of the SRRs. For a typical SRR fabricated by NSL, we assumed that 2R = 100 nm, 2r = 44 nm, d = 36 nm, wg = 18 nm, wr = 33 nm, and h = 20 nm. The F value of an SRR having these dimensions was defined as 1. The LC resonance frequency of SRRs that vary in size but retain this shape is indicated by a solid red line in Fig. 7. The LC resonance frequency increased with increasing 1/F, that is, with the miniaturization of the SRRs. However, the rate of increase became small at about 1/F > 1. When all the size parameters of the SRR were reduced, fLC became 704 THz (λLC = 426 nm) in the limit 1/F → ∞. This value can be considered the theoretical limit value in the silver SRR shown in Fig. 1.
We consider why the rate of increase of fLC decreases when 1/F > 1. According to Eqs. (2), (4), and (6), Lkin increases, although C and Ls decrease, with an increase in 1/F. In Fig. 7, the inductance of the SRRs was Lkin < Ls at 1/F < 1. In contrast, it became Lkin > Ls at 1/F > 1, and Lkin became predominant. It is thought that the variation in the LC resonance frequency became small at 1/F > 1 because the change in Lkin with an increase in 1/F is opposite to the change in C.
Note that a silver film deposited by thermal evaporation is known to become discontinuous and exhibit island structure even if deposition to a thickness of less than 10 nm is attempted, and the characteristics of the metal are lost . Therefore, fabrication of SRRs having a thickness of less than 10 nm is expected to be difficult. The blue line in Fig. 7 indicates the result of fixing the thickness h of the SRR at 20 nm and varying the size while retaining the shape. When all the size parameters of the SRR were reduced, fLC became 570 THz (λLC = 526 nm) in the limit of 1/F → ∞.
Silver SRRs fabricated by NSL were miniaturized by reducing the size of the polystyrene spheres used for a template. Their LC resonance wavelength and the scattered light spectra of single isolated SRRs were observed. Consequently, we confirmed that the LC resonance wavelength could be shortened by reducing the size of the SRRs. The LC resonance wavelength of the fabricated SRRs was calculated using the LC circuit theory and SRR dimensions estimated from SEM images. The experimental and calculated LC resonance wavelengths agreed well. In addition, we calculated the LC resonance frequency and wavelength when an SRR of constant shape was miniaturized. We found that for an SRR having the shape of those we fabricated, the short-wavelength limit was 426 nm. We believe that there is a limit to the degree of miniaturization because of limits on the fabrication accuracy.
The shortest LC resonance wavelength confirmed by our experiment was 721 nm, which is longer than the theoretical short-wavelength limit. Obtaining a shorter LC resonance wavelength will become possible if SRR miniaturization is advanced by improving the fabrication technology.
This study was supported by a Grant-in-Aid for Scientific Research (C) from the Japan Society for the Promotion of Science (No. 21510134) and a Grant-in-Aid for Scientific Research on Innovative Areas from The Ministry of Education, Culture, Sports, Science and Technology of Japan (No. 22109007).
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