1. Introduction

Unusual optical properties, such as negative index of refraction [1,2], magneto-inductive coupling [3–5], magnetoelectric coupling [6], extraordinary nonlinear effects [7–9], Fano resonance [10], and magnetism from nonmagnetic materials [11,12] not appeared in nature, are the phenomena while electromagnetic waves pass through artificial metamaterials. A classical metamaterial with negative indices is a composite material in which the metal straight wires and split-ring resonators (SRRs) possess both negative permittivity and negative permeability [1,2]. Beyond the negative indices, the glamorous characteristic of metamaterials is to obtain the desirable properties by tailoring the meta-atoms. Among the variously designed meta-atoms, SRRs have been investigated extensively since the appearance of strong artificial magnetism while resonating with external electromagnetic waves.

In spite of the fact that fascinating optical responses manifesting in three-dimensional (3D) SRRs were expected, former studies mainly focused on planar SRRs because of difficulties in fabricating stereo-structures. Few strategies, in particular of meta-atoms beyond micrometer scale, were reported as electroplating [13], two-photon reduction [14], stacked electron beam lithography [15–17], MEMS technique [18] and metal stress-driven method [19,20]. Due to the 3D geometry, magnetic dipoles can be easily induced by coupling to the incident magnetic field [21], several peculiar phenomena were demonstrated such as plasmon induced transparency [15], uniaxial anisotropic [20] and so on. These results suggest a new prospect comparing with the conventional planar SRRs.

The key factors to determine the performance of the metamaterials are the shape, size, and lattice arrangement of the meta-atoms. There have already been plenty of studies concerning interactions under different lattice arrangements both theoretically [22] and experimentally [23,24]. More specifically for planar SRR-based metamaterials, the coupling effects between SRRs have been well discussed as well by varying lattice arrangements of the meta-atoms [5,24–27]. However, rare reports disclosed the interactions between 3D SRRs [13,28] because of the challenging fabrication techniques. Here, we fabricated 3D SRRs with several lattice arrangements by utilizing metal stress-driven self-assembled method [19,20] to systematically study the mutual electric/magnetic couplings between the meta-atoms. For the rectangular-arrayed arrangements, the transmission spectra reveal that the resonant frequencies and strengths increase with decreasing inter-distances between 3D SRRs. For the interlacing-rectangular-arrayed arrangements, the transmission spectra show stronger resonances comparing to the rectangular ones. The numerical calculations reveal that both decreasing current densities and magnetic field strengths contribute to the blue-shift resonant frequencies for smaller inter-distances between 3D SRRs. We conclude that the electric coupling due to the accumulated charge interactions plays a major role rather than the magnetic couplings in the arrayed 3D SRRs.

2. Experiments

Figure 1(a) represents the rectangular-arrayed arrangement of the 2D template and the corresponding geometric parameters. The metal-stress-driven self-assembled method we adopted has the advantage of constructing 3D structures directly from two-dimensional patterns. The planar elements were fabricated by electron beam lithography in which Ni/Au ($10/60nm$) were deposited by thermal evaporation on a Si substrate. Self-assembly process is triggered by the process of ${\text{CF}}_4$ reactive ion etching to remove Si under both arms of each planar element from side-etching. They were thus released from the substrate and self-folded into 3D SRRs due to accumulated internal stress resulting from thermal depositions [20]. Also, the central connection pad is designed to be wider than its arms to prevent separation of a final 3D SRR from the substrate. By varying $P_y$, rectangular arrays with different filling factors were fabricated to discuss the mutual interactions between 3D SRRs. On the other hand, two interlacing-rectangular-arrayed arrangements of SRRs were also designed as Fig. 1(b) in which the neighboring rows were shifted by half $P_x$ to consider more complicate interactions. The 3D geometric parameters of our meta-atom, as shown in Fig. 1(c), were estimated from the SEM images of oblique-viewed 3D-SRR arrays.

 

Fig. 1 Schematics of designed planar SRR-arrays and a unit 3D-SRR with fixed geometric parameters of $P_x=5.8\mu m$, $l_{arm}=2.5\mu m$, $w_{arm}=125nm$, $l_{pad}=300nm$, $w_{pad}=250nm$: (a) rectangular arrays with $P_y$ of $0.8,1.3,2.3,3.3\mu m$. (b) interlacing-rectangular arrays by shifting the neighboring rows by half $P_x$ of rectangular arrays with $P_y=0.8,1.3\mu m$. (c) 3D geometric parameters with $G=1.3\mu m$, $D=2\mu m$, and $H=1.9\mu m$.

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The simulation results were retrieved by COMSOL Multiphysics with a finite element method to solve the Maxwell equations. The rectangular arrays of SRRs are simulated with ordinary rectangular periodic boundary conditions as Fig. 2(a). To simulate the interlacing-rectangular array as arranged in Fig. 2(b), the boundaries in a single unit cell are split into half and set with cross-referred periodic boundary conditions, as in Fig. 2(c). The refractive index of Si substrate was referred to Ref. 29. The metal film was set as Au with a thickness of $60nm$ to simplify the calculations. We adopted a Lorentz-Drude model with a $1.5$-time magnitude of ${\Gamma }_0$ to include the effect of roughness from the deposition and dry etching process [30]. Besides, the normally incident electric field from the top is x-polarized and parallel to the SRR gap to excite the resonance of a SRR.

 

Fig. 2 (a) The red dot and dashed boxes represent respectively the unit cells and boundaries of a rectangular array for numerical calculations. (b) The blue dot and boxes represent respectively the unit cells and boundaries of an interlacing-rectangular array for numerical calculations. (c) Three periodic boundary conditions for an interlacing-rectangular array with A-A, B-B, and C-C.

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3. Results and discussions

Figures 3(a–f) are the SEM images with tilting angle of 50 degrees of the 3D-SRR arrays, corresponding to the designs in Figs. 1(a) and 1(b). The transmission spectra of the 3D SRRs were carried out by a Fourier transform infrared spectrophotometer (FT/IR-6300FV, Jasco) with normally incident electric field parallel to the 3D-SRR gap as shown in Fig. 4(a). Note that the spectra were normalized by a bare Si substrate. For the rectangular arrays, the resonant frequencies and magnitudes of the fundamental mode increase with decreasing $P_y$, resulting in a blue-shift and lower transmission at their resonant frequencies. Similar trend can be observed in the interlacing-rectangular arrays as well. The stronger resonances leading to lower transmissions in the small-$P_y$ arrays can be directly explained by different filling factors of these arrays. To further clarify the physics of these phenomena, the numerical calculations were introduced as shown in Fig. 4(b). The simulated spectra, which were also normalized by a bare Si substrate, are in good agreement with the experimental results. The slight differences in resonant frequencies and transmittances could be attributed to the deviations of the geometric parameters observed by SEM.

 

Fig. 3 (a–f) Oblique-viewed SEM images of 3D SRRs. The rectangular arrays with $P_y$ of (a) 0.8, (b) 1.3, (c) 2.3, and (d) $3.3\mu m$. The interlacing-rectangular arrays by shifting the neighboring rows by half $P_x$ with $P_y$ of (e) 0.8, (f) $1.3\mu m$.

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Fig. 4 Transmission spectra of rectangular arrays with $P_y=0.8,1.3,2.3,3.3\mu m$ and interlacing–rectangular arrays with $P_y=0.8,1.3\mu m$. (a) Experimental characterizations by FTIR. (b) Numerical simulations by COMSOL Multiphysics.

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A SRR can be regarded as a LC circuit [31], its resonant frequency can then be simplified as

The numerical simulations thus give the information for the current densities (the capacitive component) and magnetic field strengths (the inductive component) to explain the resonant behaviors. The maximum current densities, $J_x$, as shown in Figs. 5(a) and 5(c), inside 3D SRRs decrease with decreasing $P_y$ both for the rectangular and interlacing-rectangular arrays. Besides, the maximum magnetic field strengths around the 3D SRRs decrease with decreasing $P_y$ as well, as shown in Figs. 5(b) and 5(c). The decreases of both capacitive component and inductive component lead to the blue-shift of resonant frequencies with decreasing $P_y$. The electric couplings can be explained by the repulsive force of charges between 3D SRRs in the rectangular arrays as shown in Fig. 6(a). As their inter-distances become smaller, the mutual repulsive electric force hinders the accumulation of charges inside SRRs. These reductions of charge accumulations then induce both diminishing current densities and thus diminishing capacitances. Moreover, the diminishing current densities inside SRRs further induce weaker magnetic field strengths in the SRR-plane. The electric couplings finally contribute to the blue-shift of resonant frequencies. On the other hand, the magnetic coupling can be explained by the mutual inductance as shown in Fig. 6(a). Decreasing inter-distances between SRRs give rise to strong magnetic field couplings and lead to the rising inductance of SRRs. Thus, a strong magnetic coupling should give red-shift resonant frequencies for small $P_y$. Based on the above interpretation, we believe that the mutual electric coupling plays a dominant role over magnetic couplings for the resonance of arrayed SRRs. Furthermore, although a weaker resonance reveals weaker current density and magnetic field in a single SRR under smaller $P_y$, significant resonant dips still appear in the rectangular arrays for small $P_y$. This is due to the different filling factors of 3D SRRs. The increasing number of SRRs per area compensate for the weaker resonant strength of each SRR, resulting in a final stronger resonance for a smaller $P_y$.

 

Fig. 5 (a) The current density, $J_x$, of 3D-SRR arrays. The inset shows the $J_x$ distribution inside a 3D SRR under the resonance. (b) The magnetic field norm around 3D-SRR arrays. The inset shows the H distribution around a 3D SRR under the resonance. (c) The maximum current densities/magnetic field strengths versus inter-distances between SRRs.

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Fig. 6 Schematic diagram of the mutual electric and magnetic couplings for (a) a rectangular array. (b) an interlacing-rectangular array.

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We then turn to the comparison between rectangular and interlacing-rectangular arrays to further discuss the mutual electric and magnetic couplings between SRRs. Since the interlacing-rectangular array with $P_y$ of $0.8/1.3\mu m$ has an approximate inter-distance in the y-direction and a doubled filling factor as a rectangular array with $P_y$ of $1.3/2.3\mu m$, as shown in Figs. 3(e)/(f) and 3(b)/(c), the experimental resonant frequency of the interlacing-rectangular array with $P_y$ of $0.8/1.3\mu m$ is close to the rectangular array with $P_y$ of $1.3/2.3\mu m$, as shown in Fig. 4. Significant transmission dips are due to the increased filling factors. Figure 6(b) represents the schematic diagram for the mutual electric and magnetic couplings in the interlacing-rectangular array. Different from the ones in rectangular arrays, in addition to the repulsive electric forces, the interlacing-rectangular arrays contain attractive electric forces. However, in Figs. 5(a) and 5(c), the maximum current density of the interlacing-rectangular array with $P_y$ of $0.8/1.3\mu m$ is just slightly different from the rectangular array with $P_y$ of $1.3/2.3\mu m$. This result indicates that the attractive electric force is relatively weak compared to the repulsive electric force. It’s also worth noting that the resonant strength is stronger in an interlacing-rectangular array with $P_y$ of $0.8/1.3\mu m$ than in a rectangular array with $P_y$ of $0.8/1.3\mu m$, which can be an important factor toward designing SRR metamaterials for the strongest resonance.

On the other hand, a relatively small value in the magnitude of maximum magnetic field strength of the interlacing-rectangular array with $P_y$ of $0.8/1.3\mu m$ compared with the rectangular array with $P_y$ of $1.3/2.3\mu m$ is observed both in Figs. 5(b) and 5(c). The reduction of magnetic field strength originates in the mutual magnetic couplings from the inserted SRRs in the x-direction, as shown in Fig. 6(b). Since the inserted SRRs stay outside the original rectangular-arrayed SRRs, the magnetic field lines of the inserted SRRs are on the opposite direction as those of the original ones, diminishing the magnetic fields within SRRs. The opposing effect of the mutual inductance from this reduction of magnetic field strength should lead to a decreasing effective inductance and thus an increasing resonant frequency; nevertheless, the resonant frequencies remain nearly the same between the interlacing-rectangular array with $P_y$ of $0.8/1.3\mu m$ and the rectangular array with $P_y$ of $1.3/2.3\mu m$. We then conclude that the electric response plays a dominant role for the resonance due to almost equal current densities between these two kinds of arrays. Here, we also point out that the results of mutual magnetic couplings in the interlacing-rectangular and rectangular arrays can give one the ideas to fabricate SRR metamaterials with a desired magnetic resonance.

Instead of the dominance of electric couplings, for multilayer double-ring SRRs, the magnetic response from mutual inductance dominates the resonant behavior [25]. Their effective inductance increases with decreasing inter-distances between different layers and then leads to a decreasing resonant frequency. Besides double-ring SRRs, magnetic inductive couplings also play a major role in traditional planar ones [24]. However, for structures of stereometamaterials [27] and vertical meander metamaterials [28], they exhibit the dominance of electric couplings, which is consistent with our results. To further clarify our claims, the scattering power of both electric and magnetic dipoles from 3D SRRs in both rectangular and interlacing-rectangular arrays with $P_y=0.8\mu m$ is calculated [32] and shown in Fig. 7. The result demonstrates that the scattering power of magnetic dipoles is about an order smaller than the one of electric dipoles and thus the scattering power of electric dipoles accounts for most of the total scattering power of 3D SRRs. As a result, a 3D SRR acts more as an electric dipole rather than a magnetic one despite its U-shaped structure, which confirms its dominant electric response in both arrangements.

 

Fig. 7 Scattering power of multipole moments in both rectangular and interlacing-rectangular arrays with $P_y=0.8\mu m$ where P stands for electric dipole and M for magnetic dipole.

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4. Conclusions

In conclusion, we experimentally investigated the electromagnetic coupling effects in infrared metamaterials consisting of densely-packed three-dimensional (3D) SRRs. Two different arrangements with different $P_y$, inclusive of rectangular and interlacing-rectangular arrays, were fabricated and characterized. The mutual electric and magnetic couplings are studied and have a conclusion that the electric coupling plays a dominant role of the resonant behavior for arrayed 3D-SRRs. We also indicate that the interlacing-rectangular arrays reveal the stronger resonances compared with rectangular ones with a similar or a half $P_y$. Moreover, the magnetic resonances of SRRs determined by the mutual inductive coupling can be varied by decreasing the inter-distances of rectangular/interlacing-rectangular arrays or adopting the interlacing-rectangular arrays. These results might give more degrees of freedom toward the designs of metamaterials which are based on 3D SRRs.