## Abstract

A symmetric metamaterial superlattice is introduced accommodating a high Q-factor trapped mode. THz time-domain spectroscopy is employed to measure the transmission spectra, identifying the excitation of trapped and open-modes in the meta-resonances. A finite-difference-time-domain calculation showed that the trapped mode excitation is from the cancelation of current densities among the nearest-neighboring meta-particles. A cryogenic temperature THz measurement is carried out to examine the temperature dependence of resonance characteristics of meta-resonances. At low temperatures, the temperature-independent radiative damping is dominant for the open-mode, while the Q-factor of the trapped mode is determined by the temperature-dependent phonon scattering and temperature-independent defect scattering with the radiative damping significantly suppressed. When compared with the room temperature measurement, a 16% increase in Q-factor is observed for the trapped mode, while a 7% increase for the open-mode at the cryogenic temperature.

© 2011 Optical Society of America

## 1. Introduction

Frequency selective meta-surface is one of the important examples of planar metamaterials, where the characteristics of resonance such as the resonance frequency, the quality factor, and the polarization dependence can be tailored by a structural design. The spectral width of a metamaterial resonance in the meta-surface is determined by two damping processes. One is the Joule heating from currents in the metallic film composing the meta-particle. Since the amount of Joule heating is determined by the resistivity of the metallic film, related to the conducting electron mobility, the Joule heating damping is temperature-dependent. The other is the temperature-independent radiative damping coming from an electromagnetic coupling of the current oscillations to the free space.

In order to achieve a high Q-factor in the meta-resonance, the radiative damping can be suppressed by introducing an asymmetric structure of the meta-particle. Upon excitation of the asymmetric meta-particle in a symmetric array by an electromagnetic wave, an asymmetric current flow takes place in the meta-particle, which is a linear sum of co- and counter-flowing currents in the meta-particle. The counter-flowing currents cancel each other through a coherent coupling, which suppresses the electromagnetic coupling to the free space, giving rise to a high Q-factor trapped mode excitation [1]. Rather than adopting an asymmetric meta-particle in a symmetric array, it is shown that a metamaterial superlattice composed of symmetric double-split ring resonators (DSRR) in an asymmetric array can also accommodate a trapped mode excitation with an enhanced Q-factor [2]. Here the trapped mode is excited through a cancelation of counter-flowing currents among the nearest neighboring meta-particles arrayed in an asymmetric way.

Another important means to obtain a high Q-factor resonance is to lower the temperature of metamaterials. In a planar metamaterial possessing an inductive-capacitative (LC) resonance, it is demonstrated that a decreased mobility of conducting electrons at a lowered temperature reduces the Joule heating loss, resulting in the Q-factor enhancement [3]. In this context, a superconducting metamaterial has been introduced to dramatically increase the Q-factor by use of a superconducting phase transition [4].

The purpose of this paper is two-fold. First, we address the question whether the presence of an asymmetry, either in a symmetric array of the asymmetric meta-particles or in the asymmetric array of the symmetric meta-particles, is essential to the excitation of a trapped mode. We propose a novel symmetric metamaterial superlattice composed of symmetric DSRRs in a wavy symmetric array to accommodate a trapped mode excitation. The structure is distinct in the sense that there resides no asymmetry either in the meta-particle or in the array of meta-particles. We demonstrate that a high Q-factor trapped mode is excited in the symmetric meta-material superlattice along with a low Q-factor open-mode. Second, we further investigate the temperature dependence of both trapped and open-modes in the symmetric metamaterial super-lattice to explore the difference in the Q-factor changes upon lowering temperature. The trapped and open-modes show a contrasting temperature dependence, which is understood from the fact that the portion of Joule heating contribution to the spectral width is larger in the trapped mode than in the open-mode, since the radiative damping is already significantly suppressed in the trapped mode.

In section 1, we propose the design of a symmetric metamaterial superlattice accommodating a trapped mode. The room temperature THz transmission spectra is measured, and a finite-difference-time-domain (FDTD) calculation of current densities is performed to unravel the mechanism of the trapped mode excitation. In section 2, the temperature dependence of meta-resonance in the symmetric metamaterial superlattice is investigated by a THz transmission spectra measurement at the cryogenic temperature. Calibrating with the THz absorption spectra of silicon substrate, changes in Q-factors of meta-resonances are examined for both trapped and open-mode as a function of temperature. When compared with the room temperature measurement, a 16% increase in Q-factor is observed for the trapped mode, while a 7% increase for the open-mode. The temperature dependence of trapped and open-mode Q-factors are examined by taking into account the electron scattering with defects and phonons.

## 2. Symmetric metamaterial superlattice

Design and fabrication of a symmetric metamaterial superlattice are introduced. A THz time domain spectroscopy (TDS) measurement is performed to identify the trapped mode excitation. FDTD numerical analysis is carried out to find the current densities at each meta-resonance.

#### 2.1. Sample fabrication

In addition to an asymmetric meta-particle in a symmetric array, a spatially symmetric grating structure composed of wavy strips has been introduced and analyzed as a structure accommodating the trapped mode excitation in Ref. [5], where the current flows cancel each other in the one single wavy strip giving rise to the trapped mode. Upon excitation of the grating wavy strip in Fig. 1(a) by a *y*-polarized incident electro-magnetic wave, counter-flowing currents take place along *x*-axis in the wavy strips. Since the occurrence of the trapped mode in the single grating wavy strip is owing to the two counter-flowing currents, the grating wavy strip can be viewed as two-element array of curvilinear dipoles as far as the trapped mode excitation is concerned [5]. Based on this interpretation, we replace the wavy strip with two separate semicircular dipoles in a wavy array. Namely, we introduce a novel design of a symmetric metamaterial superlattice consisting of symmetric DSRRS oriented symmetrically with a non-zero inclination angle with respect to *x*-axis. See Fig. 1(b).

The superlattice structure is different from that shown in Fig. 1(b) of Ref. [2] in that the metamaterial superlattice is mirror-symmetric with respect to the *x*–*z* and *y*–*z* planes. That is, we removed the asymmetry present in the array DSRRs of metamaterial superlattice of Ref. [2]. However, we still expect the occurrence of a trapped mode, since the alternating orientation of meta-particles will allow for a cancelation of current flows among the nearest neighboring DSRRs, similar to what takes place in the wavy strip of Fig. 1(a).

The meta-particle is a DSRR with the size of 36*μ*m×36*μ*m, and a symmetric metamaterial superlattice of the lattice constant of 50*μ*m is constructed by rotating the gap orientation ±22.5° with respect to the *x*-axis. The substrate is a *p*-doped silicon wafer, and a 10nm-thick Ti was deposited as adhesion layer, and a 200nm-thick gold was deposited on top. After a photolithography by use of mask aligner, the lift-off process yielded a symmetric metamaterial superlattice.

#### 2.2. THz Time Domain Spectroscopy Measurement

Figure 2(a) shows the optical microscope picture of a symmetric metamaterial superlattice. Terahertz TDS measurements were carried out with a TeraView TPS Spectra 3000 Spectrometer [6] at a resolution of 1.0 cm^{−}^{1} at room temperature. The time-domain pulse duration is about 2 ps, leading to the accessible spectral range of 0.1–3 THz (3–100cm^{−}^{1}). Figure 2(b) shows the THz transmission spectra of the symmetric metamaterial superlattice for the electric fields with various polarization angles. In Fig. 2(c) we plot the absorbance, namely, a log plot of the absorption, for the *x*- and *y*-polarizations. For the *x*-polarization, a single low frequency resonance is excited. For the *y*-polarization, however, there appear two distinct resonances, one high Q-factor low frequency resonance at 1.2THz and the other low Q-factor high frequency resonance at ≈2.4THz. The sharp resonance occurring at 1.2THz (40cm^{−}^{1}) corresponds to the trapped mode excitation coming from the cancelation of counter-flowing currents among the nearest-neighbors, similar to that observed in Ref. [2].

#### 2.3. Finite Difference Time Domain Calculation of Current Density

In order to understand the mechanism of trapped mode excitation in the symmetric metamaterial superlattice, we carried out an FDTD simulation of the current densities by use of a commercial software Lumerical [7]. Upon excitation with an electro-magnetic wave with the polarization along *y*-axis, the current densities of the trapped and open-modes are plotted.

Figure 3(a)and 3(b) show the current density plots *J _{x}* and

*J*of trapped mode resonance, respectively. As can be seen in the schematic drawing of current flows in Fig. 3(c), the main contribution is from the current flows along the entire arc, corresponding to the low resonance in Fig. 2 of Ref. [8]. In other words, the

_{y}*J*is much larger than

_{x}*J*, and the cancelation takes place mainly among the oppositely-directed

_{y}*J*s of nearest-neighboring DSRRs, suppressing an electromagnetic coupling to the free space. We find that the mechanism of trapped mode excitation is similar to that observed in the wavy strip grating of Ref. [5].

_{x}Figure 4(a) and 4(b) show the current density plots *J _{x}* and

*J*of open-mode resonance, respectively. Figure 4(c) shows the schematic drawing of current flows. Differently from the trapped mode, the main contribution is from the current flows along the halves of the arc, corresponding to the high resonance in Fig. 2 of Ref. [8]. With the given geometry of wavy arrangement, the

_{y}*J*is more significant than

_{y}*J*for the open-mode excitation. According to the analysis of wavy strip grating shown in Fig. 7 of Ref. [5], as the

_{x}*y*-dimension of wavy strip is elongated, the co-flowing, uncanceled, remnant current densities

*J*starts to deteriorate the high Q-factor feature of trapped mode excitation. The excitation of open-mode in the symmetric metamaterial superlattice is similar to what takes place for a

_{y}*y*-dimension elongated wavy strip grating. That is, the directions of main contribution

*J*s are in the same direction, or

_{y}*J*s in the nearest neighboring DSRRs are in-phase leading to a high radiative damping, which is characteristic of an open-mode excitation.

_{y}The relative amount of coherent coupling between the top and bottom DSRRs and the left and right DSRRs has been examined by varying the distance between the DSRRs along the *x*- and *y*-axis in the FDTD simulation. It is found that the coherent coupling between the top and bottom DSRRs is more efficient.

## 3. Low temperature measurement

Now that the spectral response of metamaterial superlattice is identified at room temperature, we investigate how the spectral responses of the trapped and open-modes change as the temperature is lowered. In Ref. [3], a spectral response at cryogenic temperature has been investigated in a planar metamaterial of an LC structure. Differently from the metamaterial in Ref. [3], the symmetric metamaterial superlattice possesses both the trapped and open-mode excitations. Since the trapped mode is excited from the cancelation of current flows by suppressing the temperature-independent radiative loss, we expect that the Q-factor of the trapped mode will experience a more rapid change than that of the open-mode as cooled down to the cryogenic temperature, while the radiative damping is still significant even at low temperature in the open-mode.

#### 3.1. Metametarial superlattice

In THz regime, the temperature-dependent TDS transmission spectra of moderately doped silicon has been measured and interpreted in terms of carrier density and scattering time [9–12]. Since we employed *p*-doped silicon as the substrate for the symmetric metamaterial superlattice, THz-TDS of both bare silicon and symmetric metamaterial superlattice are measured at temperature range 4.0K≤T≤300K in vacuum by use of a temperature-controllable cryostat. Figure 5(a)–5(d) show 3-D plot of THz amplitude transmission and phase spectra of the bare silicon substrate and the symmetric metamaterial superlattice, respectively.

In order to extract the Q-factor by Lorentzian resonance curve fitting, we plot the temperature dependence of THz transmission spectra of the symmetric metamaterial superlattice in Fig. 6.

#### 3.2. Temperature dependence of Q-factors

From the Lorentzian resonance fitting of meta-resonances of the symmetric metamaterial superlattice THz spectra, we obtain the Q-factor of trapped and open-modes as a function of temperature, which are summarized in Fig. 7. A 16% and 7% increases in the Q-factor are observed for the trapped and open-mode resonances, respectively, at the cryogenic temperature when compared with the values at the room temperature.

In understanding the temperature dependence of the spectral width, the Mathiessen’s rule of the conducting electron mobility Γ = Γ* _{D}* + Γ

*is normally adopted [3, 13]. For Γ*

_{P}*originating from the electron-phonon scattering, the temperature dependence is determined from the fact that the number of scatters, namely phonons, increases linearly with the absolute temperature*

_{P}*T*, Γ

*(*

_{P}*T*) ∝

*T*[14]. Γ

*denotes the temperature-independent scattering such as from impurities, defects, and grain-boundaries. In case of thin film, the scattering from the surface acts as an additional important scattering source. Hence, the increase in the conductivity of thin film is slower than that of the bulk both at DC and THz as the temperature gets lowered. See Table 1 of Ref. [13].*

_{D}By taking into account that the contribution of radiative damping Γ* _{R}* is different for the close-and open-modes in the symmetric metamaterial superlattice, the temperature dependence of the spectral width is expressed as a sum of 3 terms,

Since the open- and trapped mode resonances take place at different frequencies, namely at ≈2.4THz and at 1.2THz respectively, we take into account the frequency dependence of conductivity by resorting to the the Drude model. According to the Drude model, as the considered frequency approaches from DC to the electron scattering rate, ac conductivity is reduced [13]. For the Au films, the electron scattering rate is in the range of 3–10THz. Hence, the phonon-scattering loss is larger at the open-mode resonance frequency (≈2.4THz) than at the trapped mode resonance frequency (1.2THz), which is more pronounced at low temperature range where the electron scattering rate gets lowered.

Assuming a linear dependence of the phonon scattering on the absolute temperature *T* and allowing for the frequency-dependent ac conductivity, the least-squared fit of the measured damping Γ (in THz) to Eq. (1) is carried out. The fit yields the following relations with the fitting shown as solid curves in Fig. 7.

*= 0.062THz, and the radiative damping in the open-mode is estimated to be Γ*

_{D}*≈ 0.20THz, which leads to the important finding that the radiative damping is predominant in determining the Q-factor of meta-resonances.*

_{R}We compare our results with the study of an Al film planar LC-structured metamaterial reported in Ref. [3], where a 14% increase in Q-factor of the resonances was experimentally observed when cooled down to liquid nitrogen temperature, and a 40% increase in Q-factor has been estimated at cryogenic temperature by a numerical simulation. In the symmetric metamaterial superlattice, the increase in the Q-factor is limited by the defect scattering for the trapped mode and by the radiative damping for the open-mode.

## 4. Conclusion

Without the presence of asymmetry either in the meta-particle or in the array of meta-particles, it is shown that a trapped mode excitation takes place in a symmetric metamaterial superlattice, where the double-split-ring-resonators are oriented in a wavy arrangement. Finite-difference-time-domain simulation showed that a cancelation of counter-flowing current densities among the nearest-neighboring meta-particles is responsible for the trapped mode excitation. The temperature dependence of meta-resonance in the symmetric metamaterial superlattice is investigated by a THz time-domain-spectroscopy measurement at the cryogenic temperature. When compared with the room temperature measurement, a 16% increase in Q-factor is observed for the trapped mode, while a 7% increase for the open-mode. The finding here provides a new scheme of designing metamaterials with a high-Q factor meta-resonance and controlling the Q-factor value, when high-sensitivity sensor applications of metamaterials are envisioned.

## Acknowledgments

This work is supported by the Quantum Metamaterial Research Center program ( Ministry of Education, Science, and Technology, Republic of Korea). We thank a fruitful discussion with Dr. Jean-Yves Bigot (The Institute of Physics and Chemistry of Materials of Strasbourg, UMR 7504, the CNRS and the University of Strasbourg, France) under the CNRS-Ewha International Research Center Program.

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