1. Introduction

Since the pioneering works of J. Pendry et al. [1], metamaterials have been recognized as one of the most promising way to a full control of matter electromagnetic parameters with the realization of non-natural and counterintuitive material properties such as negative refraction [2,3], superlensing [4], hyperlensing [5] or electromagnetic cloaking [6,7]. In this rapidly evolving field, there is today a strong motivation for translating first demonstrations of negative index materials into the optical domain. Among the different approaches reported to date, the use of “microwave inspired” structures combining thin metallic wires and split ring resonators (SRR) [3] has received a particular attention. The negative permittivity in such structures is associated to the diluted plasma formed by the electrons of the thin conducting wires [8], while the negative permeability at microwave frequencies is due to the non- zero magnetic moment induced in the coil of each SRR by a magnetic field parallel to the SRR axis [1]. The achievement of an unambiguous negative index of refraction for an electric field parallel to the wires and a magnetic field parallel to the SRR axis actually requires that ε′µ″+ε″µ′<0 where ‘and “respectively denote the real and imaginary parts for each of the two effective parameters [9]. First studies to scale down these metamaterials to optical wavelengths were concentrated onto SRR arrays whose magnetic response was claimed in the THz domain [10] and then in the near-infrared and visible domains [11]. A somewhat different topology for achieving a magnetic resonance under normal-to-plane propagation was proposed, where the metamaterial structure consisted of periodically arranged paired metallic nanorods separated by a thin dielectric layer [12]. This approach was validated in microwave experiments where metallic cut wires of millimeter length were used instead of metallic nanowires [13–14]. Recently, a theoretical study showed that optical resonances in both SRR arrays and cut wire arrays could be interpreted in terms of plasmon resonances [15]. A gradual shift in the SRR resonance frequencies was predicted when reducing the length of SRR legs to the point where each SRR was transformed into a single wire piece. The magnetic and electric properties of these modified versions of SRRs were theoretically investigated in [16].

Following these theoretical studies, we present here an experimental and numerical analysis of the infrared response of metamaterials made of continuous nanowires and split ring resonators whose geometry is gradually altered. The metamaterial structure is fabricated on low-doped silicon. The impact of the geometric transformation of the SRRs on the spectra of the composite metamaterial is measured in the 20–400 THz frequency range (i.e., in the 1.5–15 µm wavelength range) for the two field polarizations under normal to plane propagation. Coupling effects between the SRRs and the continuous nanowires are analyzed for different spacings between them. The results of our study are expected to provide useful guidelines for the design and engineering of negative index metamaterials on silicon.

 

Fig. 1. Schematic representations (top) and scanning electron microscope images (middle and bottom) of the four metamaterial structures fabricated on silicon. The top and bottom pictures show the elementary unit cell of each structure. From left to right, the length of SRR legs is reduced to the point where the SRR is transformed into a single wire piece. The middle pictures show the regularity achieved in the fabrication of the periodic arrays of nanowires and split ring resonators

Download Full Size | PDF

2. Design, fabrication, characterization and modeling of metamaterial structures

Four structures consisting of a two-dimensional periodic array of gold nanowires and gold SRRs were fabricated on a 280 µm thick silicon substrate (Fig. 1). The fabrication steps included e-beam lithography, high vacuum electron beam evaporation of 5 nm thick titanium and 40 nm thick gold films, and a lift-off process. It is worthwhile noticing that all structures were fabricated in the same run, thereby allowing a meaningful comparison of their optical characteristics. As seen in Fig. 1, the four structures only differ in the shape of SRRs, which are gradually transformed into simple cut wires from structure 1 to structure 4. In the intermediate cases of structures 2 and 3, SRRs appear to be U-shaped with smaller legs than in the standard case of structure 1. Except for this resonator shape, all the other geometrical parameters of the four structures are identical. In each case, the lattice period is ~600 nm, the width of all wires (continuous and discontinuous) is ~ 50 nm, and the continuous wires are parallel to the SRR bases with a separation of 115±20 nm between each continuous wire and the closest SRR base. The SRR gap width in structure 1 is ~100 nm while the length of the two SRR legs is ~280 nm. This length is reduced to ~190 and 110 nm in structures 2 and 3, respectively. The scanning electron microscope (SEM) images reported in Fig. 1 (middle row) illustrate the excellent regularity of the four fabricated structures.

The transmission and reflection spectra of the fabricated structures were measured under normal-to-plane incidence with a FTIR (Fourier transformed Infrared spectrometer) BioRad FTS 60 equipped with a Cassegrain microscope. The FTIR beam was polarized using a KRS5 polarizer adapted to the wavelength region from ~1.5 to 15µm (i.e. to frequencies varying from 20 to 200 THz). A diaphragm was used in such a way as to produce a light spot smaller than 100×100 µm² onto the sample (i.e. smaller than the surface of each periodic structure). Measurements were performed for two field polarizations of the incident beam, the parallel polarization with the illuminating electric field parallel to both the continuous wires and the SRR gaps and the perpendicular polarization with the electric field perpendicular to the continuous wires and SRR gaps. The measured transmission spectra were normalized versus the transmission of an unprocessed part of the silicon substrate. The measured reflection spectra were normalized versus the reflection of a 40 nm thick gold film deposited on silicon.

Numerical simulations of the spectral responses of the four structures were performed with a finite element software (HFSS from Ansoft [17]). Periodic boundary conditions were applied to the lateral sides of the elementary lattice cell (Fig. 1). The silicon substrate was assumed to be lossless with a constant permittivity equal to 11.9. A Drude model was used to simulate the permittivity and loss tangent of the gold wires:

where ω_p and ω_c are the plasma and collision frequency of the gold film, respectively. The values of ω_p and ω_c chosen in the simulations were: ω_p=1.367.10¹⁶s^-1 (f_p=2176 THz) and ω_c=6.478 10¹³s^-1 (f_c=10.3 THz). Actually, the collision frequency can be considered to a certain extent as a fit parameter. An increase of the collision frequency results in higher absorption losses of the structures, while it does not change the spectral positions of resonances. The value reported above for ω_c is 2.6 times larger than in bulk gold. This increase is supposed to account for additional scattering experienced by electrons at the metal surfaces.

3. Results of measurements and simulations

The results of our measurements and simulations are shown in Fig. 2 for the four structures. Results for the parallel polarization are gathered in the series of figures from (a) to (d). Those for the perpendicular polarization are gathered in the series of figures from (e) to (h). In all these figures, resonances manifest themselves as reflection maxima correlated with transmission minima. A slow decrease (resp. increase) of the transmission level (resp. reflection level) is also observed at low frequencies for the parallel polarization. This latter evolution can be readily attributed to the plasmon-like band associated to the periodic array of continuous wires [8].

For the parallel polarization and the structures with the SRRs of larger sizes (structures 1 and 2), two resonances are observed within the spectral window of measurements [Figs. 2(a) and 2(b)]. Only the first resonance is experimentally observed for structures 3 and 4. The SRR resonances actually shift towards higher and higher frequencies as the whole SRR length is decreased. In the same time, their amplitude becomes smaller and smaller. For the first resonance, the maximum reflection R_max decreases from ~0.63 to ~0.5 while the minimum reflection R_min remains close to ~0.35 [Fig. 2(b)]. Accordingly, the minimum transmission T_min increases from ~0.12 to ~0.25 while the maximum transmission T_max remains close to ~0.6 [Fig. 2(a)]. It is worthwhile noticing that the values of R and T out of resonance correspond to those expected for a single face of silicon wafer partially covered with ~10–15% of highly reflecting metal. Whereas (R+T) approaches unity in this case (R_min+T_max~0.95), its smaller value at resonance (R_max+T_min~0.75) clearly indicates the presence of dissipative losses in metallic elements. The frequency positions of resonances, the values of R and T out of resonance, the values of (R+T) in general as well as the shapes of experimental curves in Figs. 2(a) and 2(b) are very well reproduced by numerical simulations [Figs. 2(c) and 2(d)]. The only discrepancy between experiments and simulations stems from the smaller amplitudes of resonances measured in experiments, especially those at high frequencies. The second resonance predicted at ~180 THz (λ~1.7 µm) for structure 3 is even not resolved in the experiments. Actually, minute deviations of the geometry from unit cell to unit cell and particularly residual surface roughness of the SRRs and continuous wires can explain the damping and inhomogeneous broadening of resonances as well as the increasing importance of these effects at high frequencies.

 

Fig. 2. Measured and simulated transmission/reflection spectra of the four structures depicted in Fig. 1. Curves in red, blue, green and black are for the 1^st, 2^nd, 3^rd and 4^th structures, respectively. - (a) and (b): transmission and reflection spectra measured for the parallel polarization (incident electric field parallel to the SRR gap). - (c) and (d): numerical simulations corresponding to (a) and (b), respectively. - (e) and (f): transmission and reflection spectra measured for the perpendicular polarization (incident electric field perpendicular to the SRR gap). - (g) and (h): numerical simulations corresponding to (e) and (f), respectively.

Download Full Size | PDF

For the perpendicular polarization and for structures 1, 2 and 3, a single resonance is observed within the spectral range of measurements [Figs. 2(e) and 2(f)]. In contrast, no resonance is detected for the structure with cut wires. The evolution of the SRR resonance with the SRR length is actually similar to that observed for the parallel polarization. Smaller SRR lengths simultaneously lead to higher resonance frequencies and smaller resonance amplitudes. A good agreement is found between experimental results and numerical simulations [Figs. 2(g) and 2(h)]. Previous remarks made for the parallel polarization apply to the perpendicular polarization. Minute deviations from unit cell to unit cell and surface roughness of metallic elements are likely to explain the broader resonances with smaller amplitudes observed in the experiments. The second resonances predicted for structures 1, 2 and 3 in the frequency region from 170 to 200 THz [Figs. 2(g) and 2(h) manifest themselves only as smooth maxima (resp. minima) in the measured reflection (resp. transmission) spectra of Fig. 2(f) (resp. Fig. 2(e)]. Supplementary measurements between 200 and 250 THz (not shown here) did not reveal any other resonance.

 

Fig. 3. Magnitude of the normal electric field component (|E_z|) calculated at the bottom surface of the metallic elements for each of the resonant plasmon modes observed in the different spectra of Fig. 2. The different colours, from blue to red correspond to increasing magnitudes of the field component. Modes are classified according to the resonance order and to the polarization of the incident electric field. As expected, both the energy and the number of field nodes increase with the resonance order.

Download Full Size | PDF

4. Discussion

4.1 E field distribution of the resonant modes

Figure 3 shows the distribution of the electric field calculated at the bottom surface of metallic elements for each of the resonant modes observed in Fig. 2. These results are in agreement with previous calculations reported by Rockstuhl et al. [15]. When the incident field is polarized parallel to the SRR gap, resonant modes possess an odd number of nodes along the entire SRR. This number is equal to one for the first resonance, while it is equal to three for the second resonance. The first resonance also identifies to the so-called LC resonance as defined in previous studies at microwave and far infrared frequencies [18, 19]. It simply identifies to the dipolar mode in the case of the cut wire. When the incident field is polarized perpendicular to the gap, resonant plasmon modes possess an even number of nodes. The first resonant mode in this polarization thus exhibits one additional node as compared to the first resonance in the parallel polarization. This in turn requires higher energies of the light field to excite this mode. The frequency of the first resonant mode in the perpendicular polarization is typically two times higher than that of the first resonant mode in the parallel polarization.

4.2 Frequency positions of resonances

At this stage, it is interesting to compare the frequency positions of resonances reported in Figs. 2 and 3 to those reported in previous works for similar structures with gold SRRs on glass substrate. For instance, the first resonance calculated in [15] for the parallel polarization and U-shaped SRRs with 400 nm long base and 190 nm long legs was found to be close to 3800 cm^-1, i.e. close to 115 THz instead of 65 THz measured in our experiments for U-shaped SRRs with similar sizes [Figs. 2(a), 2(b), and Fig. 3(b)]. The first resonance calculated in [15] for the parallel polarization and U-shaped SRRs with the same base but with 110 nm legs was found to be near 4800 cm^-1, i.e. near 144 THz instead of 80 THz measured in our experiments [Figs. 2(a), 2(b) and Fig. 3(c)]. Actually, all the mode frequencies calculated in [15] are 1.7–1.9 times higher than those reported in this work, whatever the resonance order and the field polarization are. Approximately the same ratio is obtained when comparing the first resonance measured in [10] for SRRs of standard shape (~100 THz) to that reported in Fig. 3(a) (~60 THz) for standard SRRs with the same total length (l_m~960 nm). This ratio is actually of the same order of magnitude than the ratio between the refractive index of glass and that of silicon. This suggests that in our case the electromagnetic field at resonance largely extends into the silicon substrate.

From a practical point of view, it is also of interest to compare the frequency measured for the first resonance in the standard case of Fig. 3(a) to theoretical estimates that can be made using the “electrical engineering” approach. Following this approach, the gap of the SRR is regarded as a capacitance C, and the “U” as a single winding of an inductance loop L. Coupling to the LC-resonance occurs when the electric field vector has a component normal to the plates of the capacitance, i.e. a component parallel to the gap. Analytical formulas have been proposed in [20, 21], where both L and C can be written as the sum of two components: L=L_m+L_eff and C=C_m+C_eff. The L_m and C_m components are related to the geometrical parameters of the SRR:

where w is the width of the metallic wire, t its thickness, g the length of the gap, and l the effective length of the axis of the SRR: l=πl_m/2 with l_m the physical length of the SRR. The expression for C_m presently accounts for the dielectric permittivity of the silicon substrate (ε_r=11.9). The L_eff component, which is related to the kinetic energy of the electrons in the SRR, expresses as a function of the plasma frequency [22]:

where S is the effective cross-section of the wire. S may somewhat differ from the geometrical cross-section (S₀=wt) due to the presence of skin effect in the metal and to the asymmetrical current distribution in the SRR. The capacitance C_eff originates from the displacement current in the SRRs, and is given by [20]:

The frequency of the first resonance thus simply writes:

The resistive term associated to the collision term of the Drude model can be neglected in first approximation [21]. Using w=50 nm, t=40 nm, l=1500 nm, g=130 nm, ω_p=1.367.10¹⁶s^-1, S=S₀ and applying formulas (2), (3), (4) and (5), we thus find: L_m=2.3.10^-12 H, C_m=1.62.10^-18 F, L_eff=4.5.10^-13 H, C_eff=1.18.10^-20 F and finally f ₁≈75 THz, which value is not far from that measured in Fig. 2. While C_eff is shown to have a weak influence on the result, the correcting term Leff cannot be neglected in comparison with L_m. Clearly, the precision of the result is limited by the uncertainties on the physical parameters w, t, l and g, but the relatively good agreement between the LC-model and experiments confirms the important role of the substrate permittivity on the frequency position of the first resonance. It is also verified that a decrease of the SRR legs and thus of L_eff and L_m shifts this resonance to higher frequencies in agreement with the experimental evolutions shown in Fig. 2. The value of the frequency calculated from formulas (2)–(5) is compared to the measured one for each of the four structures in Table 1. As expected the discrepancy between calculated and measured values increases from structure 1 to 4. The application of formulas (2)–(5) becomes even questionable in the case of structure 4 based on single wire piece. The frequency of the dipolar mode in this structure is actually better estimated from the effective dipole length L_d, which itself is related to the cut-wire length L_c-w. This gives: f~c/2nL_c-w~120 THz. Besides, it is evident that the LC-model cannot account for high order resonances in its present form.

Table 1:. comparison between the frequency calculated from formulas (2)–(5) and that measured at the first resonance for each of the four structures studied in this work.

View Table

4.3 Coupling effects between continuous wires and SRRs

In the preceding sections, it has been implicitly assumed that the presence of continuous wires had no influence on the resonant response of the structures except for a slow decrease of the transmission observed at low frequencies for the parallel polarization. However, a careful examination of the field distributions in Fig. 3 indicates that, at least for the second resonance, the electromagnetic field extends well in the region comprised between the SRR and the closest wire. We thus performed a numerical analysis to investigate in more detail the possible existence of coupling effects between the SRRs and continuous wires. For this purpose, the spectral responses of the four structures were calculated for different distances d between the SRR base and the closest wire. They were also compared to the spectral response of a periodic array of SRRs only.

Results of our calculations are shown in Figs. 4(a) and 4(b) for structure 1 with three values of d, for a periodic array of SRRs without continuous wires and for a periodic array of wires without SRRs. The dimensions of SRRs are the same for the first four structures. The lattice period is identical for all the structures. Calculations are performed within the same spectral range as in Fig. 2, and the incident electric field is polarized parallel to the continuous wires and/or to the SRR gaps. Figure 4(a) represents the transmission spectra calculated for the different structures. Figure 4(b) shows the electric field distributions calculated for the different resonances observed in Fig. 4(a).

As seen in Fig. 4(a), the position of the first SRR resonance is not modified by the presence of the continuous wires whatever the separation between SRRs and wires is. Only the shape of the resonance is modified, and it becomes asymmetric with the presence of the wires.

 

Fig. 4. (a). Transmission spectra calculated for a periodic array of SRRs (red curve), a periodic array of continuous wires (pink dashed curve) and periodic arrays of SRRs and wires with different separations between SRRs and wires: d=130 nm (blue curve), d=50 nm (green dashed curve), d=10 nm (black dashed curve). In each case, the incident electric field is polarized parallel to the gap, the SRR dimensions and lattice period as the same as for structure 1 in Fig. 1(b): Magnitude of the normal electric field component (|E_z|) calculated at the bottom surface of the metallic elements for each of the resonant modes observed in the transmission spectra of Fig. 4(a). The different colours, from blue to red correspond to increasing magnitudes of the field component. Modes are classified according to the resonance order and to the separation d between the SRR and the closest wire.

Download Full Size | PDF

This asymmetry mainly results from the fact that the optical response of the wires [pink dashed curve in Fig. 4(a)] adds to that of the SRRs. A weak coupling between SRRs and wires only occurs at the smallest separations between the two metallic elements as shown from the calculated distribution of the electric field at the first SRR resonance (Fig. 4(b), second column).

The evolution of the second SRR resonance is quite different. For a sufficiently large separation between the SRRs and wires (d≥100 nm), the frequency position of this second resonance is still rather independent of the presence of the wires. This justifies our previous interpretations concerning the results of Fig. 2, where the different spectra were obtained for d≈130 nm. However, for small separations between the SRRs and wires, strong coupling effects exist, which lead to a splitting of the second SRR resonance into two components [Fig. 4(a)]. These components are well separated for the smallest value of d (black dashed curve in Fig. 4(a), d=10 nm). As seen in Fig. 4(b) (second and third columns), the modal field of the low frequency component is found to be concentrated in the region between the SRR base and the closest wire. That of the high frequency component is rather concentrated in the SRR legs.

Calculations above were repeated for structures with U-shaped SRRs as structures 2 and 3 and for structures with cut wires as structure 4 (Fig. 1). The same behavior was found in the case of structures with U-shaped SRRs. The frequency position of the first resonance was not modified when the distance between SRRs and continuous wires was varied. The second resonance split into two components for small values of d. In contrast, the structures with cut wires did not exhibit the same “robustness” of the first resonance against coupling effects. The resonant mode split into two distinct components when the separation between cut wires and continuous wires was smaller than 50 nm. This different behavior can be simply explained by the fact that unlike structures with true SRRs, the field of the first resonant mode is obviously concentrated in the close neighborhood of the continuous wires, i.e. in the cut wires themselves.

Coupling effects were also investigated from numerical simulations for the perpendicular polarization. We only considered the first SRR resonance since it was the only one clearly observed in the experiments (Fig. 2). The frequency position of this resonance was found to be rather independent of the presence of continuous wires whatever the separation between the SRRs and wires was. However, a small splitting of the resonance was observed for very small separations (d=10 nm) in the case of U-shaped SRRs with small legs (structure 3).

4.4 Application to metamaterials with negative refraction on silicon

It is now well established that the use of an array of continuous metallic wires allows obtaining a negative permittivity over the whole plasmon-like band when the electric field is polarized parallel to the wires [8]. Figures 2 and 4(a) presently show that this band can extend well up to near-infrared frequencies for a sufficiently small period of the wire lattice. On the other hand, it has been demonstrated that an array of metallic SRRs can exhibit a magnetic response in the optical domain with a negative permeability at certain frequencies [11]. However, this situation only occurs at SRR resonances and for an oblique or grazing incidence, i.e. for an incident magnetic field with a non-zero component along the SRR axis. An additional condition is that the resonant plasmon mode must possess an odd number of field nodes along the SRR [16]. Concerning this latter aspect, our experimental results confirm that the first resonant mode, the so-called LC resonance, is by far the most exploitable due both to its strength and to its robustness against parasitic coupling effects. They also show that its frequency position can be finely tuned by adjusting the total length of SRRs and using for instance U-shaped SRRs.

One solution to achieve a magnetic response at normal incidence with respect to the sample plane consists in using a stack of SRR layers [22] or simpler, a stack of cut-wires as originally proposed in [12]. Coupling between adjacent SRRs or between adjacent cut-wires leads to the formation of hybridized plasmon modes of opposite symmetry. Antisymmetric plasmon modes can exhibit a magnetic response, and lead to a negative permeability in certain frequency regions. Our experimental results in Fig. 2 show that the resonance associated to the dipolar mode of cut-wires is well pronounced for the fabricated structures. Coupling between two such modes in a multilayer stack should thus allow obtaining a magnetic response at normal incidence. One advantage in using stacked cut-wires instead of stacked SRRs stems from the possibility of achieving more easily a magnetic response at (high) near-infrared frequencies. This is all the more true when metallic nanostructures are fabricated on a high permittivity substrate such as silicon. All the plasmon resonances are shifted to low frequencies, and the realization of very-small-size SRRs operating at telecommunication wavelengths on silicon would require pushing the lithographic techniques to their present limits.

Conclusion

We have experimentally investigated the impact of geometric transformations of split ring resonators (SRR) on the infrared spectra of composite metamaterials made of gold nanowires and SRRs deposited on silicon. To our knowledge, the periodic arrangements of continuous nanowires and SRRs of variable shapes considered in this work have not been studied in the mid- and near- infrared domains yet. The comparison between spectral measurements at normal incidence and numerical simulations has shown that the frequency evolutions of the first two SRR resonances with the SRR geometry were quite predictable for both the parallel and perpendicular polarizations, thus allowing a true engineering of these resonances on silicon. The numerical study has also revealed the robustness of the first resonance against parasitic interactions between SRRs and continuous wires while splitting effects were observed for the second resonance and for small distances between the SRRs and wires. The control of resonances is of great interest for applications such as the realization of negative refraction metamaterials on silicon or that of electromagnetic cloaks at optical frequencies where the effective parameters need to be spatially adjusted by varying the geometry of elementary motifs [6, 7]. We have shown that simple motifs such as cut-wires could certainly be used instead of SRRs for operating at optical frequencies near normal incidence, with the supplementary condition, however, that a sufficient number of cut-wire layers could be stacked. More generally, the use of a silicon substrate as proposed in this work is believed to open the route toward active devices based on metamaterials.