1. Introduction

Coupling to a metallic surface can be used for the enhancement of optical emission or absorption [1, 2], with applications in fields such as surface-enhanced Raman scattering (SERS) [3], bio-imaging [4, 5], light-emitting diodes (LED) [6], photovoltaics [7, 8] or single-photon sources [9]. In order to take advantage of surface-plasmon-polariton (SPP) modes, which are not coupled to far-field radiative modes in the case of a planar metallic surface, many studies have introduced a periodic corrugation to the metallic surface. The phase-matching condition is then fulfilled by the grating wave vector. For instance, LED or solar-cell devices have been proposed with a corrugated metal surface for SPP-assisted in- or out-coupling [6,10].

The realization of such plasmonic crystals with a sub-micrometric period often involves clean room technologies (e-beam lithography and nano-imprint [11]), although other techniques have been used, such as holography [12, 13]. For many applications, in particular for solar cells as reviewed in Ref. [7], it is necessary to obtain plasmonic samples of macroscopic size, with broadband coupling to visible light and little dependence on the sample orientation.

Self-assembly provides an interesting bottom-up alternative for the fabrication of ordered nanostructures. Opal samples (three-dimensionally ordered stacks of sub-micrometer balls) have been the most studied. Their fabrication, which uses soft chemistry reactions for the synthesis of silica balls and self-assembly for the ordering of the lattice, is now a mature technology [14, 15]. Three-dimensional metallic structures have also been fabricated either by direct self-assembly of metallic nanoparticles [16] or by metal deposition in a dielectric opal used as a template (inverse opal [17–20]).

Self-assembled silica balls have also been used for the fabrication of two-dimensional metallic gratings.

In some cases, a monolayer of silica balls served as a mask for the evaporation of triangular nanoparticles [21–23] or as a mould for the electrodeposition of spherical nanovoids. This latter type of structures has been the subject of comprehensive optical studies [24] which evidenced a dominant role of localized surface plasmon (LSP) modes, and applications for SERS [25] or solar cells [26] were demonstrated.

In other cases, a metallic layer was deposited on top of self-assembled colloidal spheres. For thin opal slabs covered by a thin metallic film, extraordinary optical transmission (EOT) was demonstrated and attributed to SPP mediation [27,28], and combined photonic-crystal and EOT effects were studied [29,30]. With thicker metallic layers, the opal is purely a template for fabricating an opaque two-dimensional corrugated metallic grating. This is the approach which is followed in the present study. Such structures have been used in SERS experiments : good-quality wafer-scale samples, reproducible spectra and very large enhancements were evidenced [31–33]. There has been however no extensive characterization of their optical properties. This is the purpose of the present article.

In this paper, we describe the fabrication of a plasmonic crystal of centimetric size, by evaporating a thick layer of gold on an artificial silica opal used as a periodic template. We combine structural characterization (atomic-force and electron microscopies) with optical angle- and polarization-resolved characterizations. We analyze the role of disorder inside our plasmonic crystal and show that its macroscopic properties are little sensitive to its in-plane orientation. A strong broadband absorption by SPP modes is measured, with absorptions as high as 95 %. The broadband nature of the plasmon coupling is related to the dispersive character of the SPP modes.

In the first section, we present the fabrication and the structural characterization of our samples, and explain how the corrugation depth can be tuned through the deposition parameters. We examine further the two-dimensional order of the samples in the second section by analyzing electron microscopy images, and show a transition from microscopic order to macroscopic isotropy. We present in the third section an optical reflectometric characterization of the samples and evidence both LSP and SPP modes, with a very strong absorption from the SPP mode. In the last section, we theoretically define and calculate a density of coupled SPP modes and show that broadband coupling to the SPP modes can be expected.

2. Sample fabrication

The principle of the sample fabrication (Fig. 1(a)), is to evaporate a layer of gold onto a self-assembled periodical array of silica spheres (artificial opal). The gold thickness is larger than the SPP skin depth, so the gold layer is optically infinite : the silica structure is used purely as a template to impose a periodic corrugation to the gold layer.

 

Fig. 1 (a) General structure of the fabricated samples : a gold layer evaporated on an artificial opal, with an optional silica smoothing layer. (b) AFM image of a sample with 500 nm gold layer (and no smoothing layer). (c) Profile of the sample along the purple line indicated in (b). A groove depth h = 90 nm is measured.

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In a first step, we characterized the opal template, fabricated following the sedimentation protocol described in [34]. The opal lattice is face-centered cubic, with its (111) planes parallel to the sample surface. Thus, the surface appears as a compact two-dimensional triangular lattice. The diameter of the spheres was estimated to a = 395 ± 5 nm by reflectometry as previously described in [35].

In a second step, the gold layer was evaporated on the silica balls template. Our typical sample size was 1–2 cm. An atomic force microscopy (AFM) image of a sample with 500-nm gold thickness is displayed in Fig. 1(b). It shows that the opal surface is completely covered by the gold layer, in agreement with Ref. [31] where a similar silver deposition was characterized. The template two-dimensional structure is preserved in the corrugated surface and well-defined grooves are observed (the sharpness of the junctions between the grooves was possibly underestimated due to the AFM tip size, which is of the order of 20 nm). We measure by AFM a spheres diameter of 390 ±10 nm, in agreement with the value obtained by reflectometry.

An AFM profile of the sample is shown in Fig. 1(c). We define the groove depth h as the amplitude of this curve. A value of h = 90 ±10 nm was obtained for this sample.

The groove depth is a key parameter for adjusting the SPP dispersion relation. As compared to the 500-nm-gold sample, h can be reduced by using thinner gold depositions, or increased by previous addition of a silica smoothing sublayer (Fig. 1(a)). By varying the two layer thicknesses (as summarized in Table 1), we were able to tune the groove depth h from 55 nm to 150 nm.

Table 1. Control of the groove depth h (measured by AFM) by the thickness of the gold and silica layers.

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3. Two-dimensional cristalline order

Let us analyze more closely the two-dimensional grating order, with the example of the same sample as in Fig. 1. A 20×20 μm² scanning electron microscopy (SEM) image is shown in Fig. 2(a). It reveals the good homogeneity of the gold deposition and the almost-perfect flatness of the opal template.

 

Fig. 2 (a) SEM image of a plasmonic sample with a gold layer of 500 nm. (b) FTs of three sample portions of sizes 60×60, 40×40 and 20×20 μm². (c) Autocorrelation curves A(ϕ) of the three FTs, as defined in the text.

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We plot in Fig. 2(b) the two-dimensional Fourier transform (FT) of the sample SEM image, for the overall 60×60 μm² image, and for two 40x40 and 20x20 μm² sub-portions (the latter being the one shown in Fig. 2(a)).

For the smallest portion, the six reciprocal nodes appear clearly and form a hexagon. We extract from it a spheres diameter of a = 390 ± 10 nm, which matches well with the values of 390 and 395 nm previously obtained respectively by AFM and reflectometry. This portion of the sample is thus practically a single crystalline domain. For the larger portions, the FT pattern is no longer hexagonal and tends to become a circle due to the presence of several ordered monodomains with different random orientations.

In order to clarify the growing complexity of the FT patterns, we plot their angular autocorrelation in Fig. 2(c). For this purpose, we consider the FT, expressed in polar coordinates F(k,ϕ), at a radius $k=2\pi /\sqrt{3}a$ corresponding to the distance of the lattice nodes, and we define the function $A\left(\varphi \right)=<F\left(2\pi /\sqrt{3}a,\alpha \right)F\left(2\pi /\sqrt{3}a,\alpha +\varphi \right)>$. This function shows a transition from the ordered to the isotropic situations. For the 20×20 μm² portion, on the one hand, the plot exhibits a single 60°-periodicity corresponding to the hexagonal order. For the 60×60 μm² portion, on the other hand, the quasi-isotropic nature of the sample is evidenced by the almost constant autocorrelation function. A weak 15°-periodicity is also observed, which indicates the presence of several monodomains tilted by multiples of 15° from each other.

The conclusion of this section is that the periodicity of our structure is maintained over typical distances of 20 μm, larger than the SPP propagation length (3 μm at 600 nm for a flat gold surface [36]) : this sample constitutes indeed a plasmonic crystal. On the other hand, due to the presence of multiple domains of random orientations, the behavior of this sample at millimetric scale is isotropic : there is no preferred azimuthal orientation (as opposed to one-dimensional metallic gratings for instance).

4. Optical characterization

In this section, we characterize the SPP modes of the plasmonic samples by optical polarization-resolved reflectometry.

A collimated white-light beam was passed through a 1-mm diaphragm and a polarizer and directed onto the sample with an incident angle θ. The reflected beam was passed through a 2 mm diaphragm (chosen larger than the former diaphragm because of the slight beam divergence) in order to separate reflection from low-angle scattering, collected by an optical fiber and analyzed by a spectrometer. All spectra were normalized by the spectrum of the illumination lamp (measured with the 2-mm diaphragm and fiber facing the incident beam).

Figure 3 shows the reflection spectra for incident angles θ from 20 to 80° for the same plasmonic crystal sample. We also plot in dashed lines the reflection spectra (at 20 and 70°) of a plane gold surface, deposited under the same conditions. The plane gold reflectivity decreases below 500 nm due to bulk gold absorption by interband transitions. The same effect is observed for the plasmonic crystal. At high wavelengths, the reflectivity of the plasmonic crystal is maximum, but lower than the reflectivity of the plane gold surface, which we attribute to the scattering by the surface roughness.

 

Fig. 3 Full lines : p- and s-polarized reflection spectra, at incidence angles θ ranging from 20 to 80°, with steps of 10°, of a plasmonic sample with groove depth h = 90 nm. Dashed lines : p- and s-polarized reflection spectra of a plane gold surface for θ = 20 (black) and 70° (grey) (divided by 2 for clarity because the reflectivity of the plane sample is larger).

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The p-polarized spectra show a dip moving to longer wavelengths as θ increases (from 560 nm at 20 to 650 nm at 80), while the s-polarized spectra show a fainter dip around 570 nm. Both dips indicate the absorption of the incoming light by plasmonic modes of the sample, which are then dissipated or re-radiated in different directions.

As SPP modes are transverse-magnetic, the s-polarized incident beam should not be coupled to these modes. Moreover, the position of the dip in s-polarization shows little dependence on the incidence angle θ. We thus attribute this dip to a LSP mode, introduced by the sharpness of the sample structure. For metallic nanovoids, such non-dispersive modes were also observed and attributed to LSP modes, which was confirmed by numerical modeling [24].

The p-polarized dip, on the other hand, corresponds to the SPP modes, as evidenced by its strong position dependence in the incidence angle. The reflectivity at the bottom of this dip is only a few percent, as compared to typically 40 % above 700 nm. We conclude that, at the SPP wavelength, about 95 % of the incoming beam is absorbed by the plasmonic sample. This demonstrates the excellent coupling of the SPP modes with light.

The same measurement was performed for samples with different groove depths h. The results are summarized in Fig. 4(a) which plots the SPP-dip wavelength λ as a function of θ (circles), along with the theoretical SPP-coupling wavelength (full line), calculated at the zero-th order in h as explained in the next section. As expected, the experimental curves converge towards the theoretical one as h is decreased, which confirms our attribution of the reflection dip to SPP modes and validates the model.

 

Fig. 4 (a) Dots : experimental wavelengths λ of the p-polarized reflection dip as a function of θ for six samples with different groove depths h. Full line : theoretical position of the dip, as calculated in the last section for at the zero-th order in h. (b) Relative difference (λ_exp. – λ_theo.)/λ_theo. between the measured and theoretical wavelengths, as a function of the ratio of the groove depth h over the SPP skin depth in air δ. The colors correspond to the same values of h as indicated in (a).

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The difference between the calculations and the experimental data is larger for lower θ. We suggest that this occurs because h is then not negligible as compared to the SPP skin depth in air δ. In order to verify this, we plot in Fig. 4(b) the relative difference between the experimental and theoretical λ, as a function of the ratio h/δ (δ being obtained from the experimental values of λ and θ, without any assumption on the SPP dispersion relation [37]). We find an excellent correlation between these two quantities, demonstrating that h/δ is indeed the parameter which determines to which extent the zero-th order model is valid. This model remains valid within a 5% margin as long as h/δ remains below 0.6. It is the case at θ > 40° for h = 90 nm, and for all data points for h = 55 nm.

5. Theoretical two-dimensional dispersion relation

We finally develop a model for the calculation of the SPP-coupling relation λ(θ). The results, which were discussed above for comparison with the experimental data of Fig. 4(a), will here uncover further characteristics of our samples.

At the zero-th order in h, the coupling conditions are given analytically by the SPP dispersion relation of a flat gold surface and by the phase-matching condition of the triangular lattice.

The dispersion relation for a flat gold surface is $k_{SPP}\left(\omega \right)=k_0\left(\omega \right)\sqrt{{\varepsilon }_m^{\prime }{\varepsilon }_d/\left({\varepsilon }_m^{\prime }+{\varepsilon }_d\right)}$, where, ɛ_m(ω) = ɛ′_m + iɛ″_m, ɛ_d and k₀(ω) = ω/c are respectively the gold and air dielectric constants and the wave vector of light in vacuum. For a given frequency ω, the norm of the SPP wave vector k⃗_SPP is always larger than the norm k_// = k₀ sinθ of the in-plane component of the incoming beam wave vector, forbidding any interaction with propagating fields. The phase matching is provided by the vectors of the reciprocal lattice. The six first of these vectors, of norm $4\pi /\sqrt{3}a$, are defined in the inset of Fig. 5 and labelled G⃗_i. For an azimuthal angle ϕ between the lattice x-axis and the incoming beam, the phase-matching condition writes [11] :

We plot in Fig. 5(a) the corresponding calculated λ(θ) for a lattice parameter a = 390 nm, the gold dispersion ɛ_m(ω) being measured by ellipsometry on a plane gold surface. We limit ourselves to values of ϕ from 0 to 30° because of the symmetries of the system, and to the G⃗₄ and G⃗₅ vectors because the other bands are below 500 nm, where gold absorption is dominant.

 

Fig. 5 (a) Calculated values of the SPP-coupled wavelength as a function of the incidence angle θ, for the phase-matching vectors G⃗₄ and G⃗₅, for a = 390 nm and for various grating orientations ϕ. The line ϕ = 30° is a symmetry axis : λ_G4(60 – ϕ) = λ_G5(ϕ). (b) Density of coupled SPP modes for three angles θ, calculated as explained in text.

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The incident beam diameter is about 1 mm, so that, following the discussion of section 2, it probes an isotropic distribution of monodomain orientations ϕ. By calculating the curves λ(θ) for N_ϕ =175 values of ϕ equally spaced from 0 to 30° and counting the number N_λ,θ of these curves which pass through a [λ, λ + dλ] interval, we can define a density of coupled modes ρ(λ,θ) = N_λ,θ/(dλ.N_ϕ). We then convolve it by a gaussian distribution of values of a centered on 390 nm and of width 10 nm, and we plot the obtained ρ(λ,θ) in Fig. 5(b) for three values of θ. We check that ∫ ρ(λ)dλ = 2 : two modes (coupled to G⃗₄ and G⃗₅) are above 500 nm.

These densities of coupled SPP modes exhibit two peaks at lower and higher wavelengths and a continuum of coupled modes at intermediate wavelengths.

The peak at lower wavelength is close to 500 nm and could not be observed experimentally. The higher-wavelength peak is attributed to the experimental reflectivity dip, with a good agreement in the peak position and dependence on θ, as was already plotted and discussed in Fig. 4(a). The width of the experimental dip (about 50–70 nm) is similar to the width of the calculated peak, and it also increases with θ. It is much broader, for instance, than the SPP absorption dip measured on one-dimensional gratings [38].

We note in the p-polarized experimental spectra of Fig. 3 that, in the spectral interval between the bulk absorption at 500 nm and the dip, the reflectivity is much lower than in the spectral region above the dip - for the s-polarization, the short-wavelength side of the dip is also lower, but only because of its proximity to the bulk absorption edge. This interval of lower p-polarized reflectivity may be attributed to the continuum of SPP-coupled modes in Fig. 5(b). This continuum at intermediate wavelengths constitutes a large portion (two thirds for θ = 50°) of the total coupled SPP modes, and extends down to 500 nm. Absorption by SPP-coupling should thus occur over a very broad range of the visible spectrum, opening new opportunities for applications where broadband light coupling is required, such as multiplexed bioimaging or solar cells [7].

6. Conclusion

In this article, we described the fabrication of self-assembled plasmonic crystals of centimetric scale. The depth h of the profile, measured by AFM, ranged between 55 and 150 nm and was tuned by changing the gold and silica thicknesses. SEM images showed that, due to the presence of monodomains of random orientation, the microscopic crystalline order was averaged at the macroscopic scale so that the sample properties are isotropic. Under p-polarized illumination, a strong broadband absorption was measured, which was attributed to the coupling of light to the SPP modes of the plasmonic crystal. We detailed calculations at the zero-th order in h and showed that they were sufficient as long as h was lower than the SPP skin depth. These calculations showed that, due to the random orientation of the crystalline monodomains, SPP-assisted absorption occurs on a very broad continuum of the visible spectrum. Future work will focus on taking advantage of this isotropic broadband absorption in hybrid structures including a dielectric waveguide or an active medium.