1. Introduction

Surface enhanced Raman spectroscopy (SERS) is a highly sensitive method that can allow for single molecule detection [1]. A key to SERS is the large amplifications of the incident electromagnetic field in hot spots generated by surface plasmons [2] on rough metal surfaces or at the junctions of nanoparticles in metal colloids. However, the hotspots are often not easy to control and reproduce, reducing the efficacy of SERS. It is thus desirable to find a way of creating and controlling the electromagnetic amplifications in a more consistent, reproducible manner. Given advances in nanofabrication and lithography, one approach is to engineer nanostructured substrates with highly regular hot spots [3].

In this paper, we propose and model potentially robust SERS substrates (Fig. 1) composed of periodic arrays of chains of gold nanoparticles atop silver film structures. In each substrate, the periodic perturbations act as diffraction gratings for exciting surface plasmon polaritons (SPPs) tunable to the same frequency as local plasmon resonances (LSPs) of the metallic nanoparticles, taking advantage of multiplicative enhancements of light intensities for higher sensitivity and fidelity. Previous studies that consider the coupling between LSPs and SPPs have demonstrated hybrid LSP/SPP modes [4] and enhanced interparticle coupling [5]. In this paper, we use rigorous electrodynamics simulations to quantify the efficiency with which one can couple SPPs into LSPs and the extent of additional enhancements.

2. Computational details and preliminary considerations

We model the electromagnetic response of the structures in Fig. 1 with 3-D finite-difference time-domain (FDTD) [6] calculations. The electric, E(x,y,z,t), and magnetic, H(x,y,z,t), fields are represented on discrete, staggered grids and propagated in time using a leap-frog algorithm. Each grid point is assigned a relative dielectric constant, ε_R = ε_R(x, y, z). For water, glass, and poly(methyl methacrylate) (PMMA), we take ε_R = 1.77 , 2.25 and 2.23, respectively. Metallic regions involve a complex, frequency-dependent dielectric constant, ε_R = ε_R(ω). We use a Drude and two term Lorentzian model (Eq. (1) of Ref. [7]). The model parameters are fit to empirical data [8] in the 250 nm – 1000 nm range. The parameters for silver are given in Ref. [7]. Those for gold are, in the notation of Ref. [7], ε _∞ = 5.3983, ω_D = 9.2007 eV, γ_D = 0.06802 eV, δ= 2.5417, g ₁ = 0.2679, ω ₁ = 2.8131 eV, γ ₁ = 0.28654 eV, g ₂ = 0.7321, ω ₂ = 3.4394 eV, and γ ₂ = 0.4349 eV. These models are implemented with an auxiliary differential equations method [6, 7].

Referring to the coordinate systems in Fig. 1, the incident source is a finite duration x – polarized plane wave normal to the film surface injected using the total-field / scattered field approach [6]. Perfectly matched layers (PMLs) are used to absorb field components at the grid edges in the ± z-directions [6]. Relatively fine grid spacings of 1 nm are used, and simulation times of 150 fs allow for well converged results.

Structure I consists of 3-layers (glass ∣ silver (100 nm) ∣ water) with 50 nm wide slits within the silver layer at a periodicity P. Hexagonally packed pairs of 40 nm diameter gold nanospheres within the water layer were centered and located 4 nm above the silver film (Fig. 1(a)). Structure II consists of 4-layers (glass ∣ silver (100 nm) ∣ PMMA (d + 44 nm) ∣ water) where similar particle arrays were situated within 92 nm wide, 44 nm deep grooves within the PMMA layer (Fig. 1(b)). In both structures, the gap size between adjacent particles is 4 nm, and the slab thickness is chosen to restrict interference between SPPs on opposite sides of the film. Periodic boundary conditions in x and y are imposed on unit cells consistent with each structure to simulate the gratings and hexagonal packing of spheres. For structure I, incident light is injected normal from within the glass layer; for structure II, light is incident normal from within the water layer. The reason for exciting structure I from beneath was to remove LSP excitations due to incident and reflected light and consider only excitations via SPPs.

If ∣E∣ = ∣E(x,y,z)∣ denotes the electric field throughout the computational box for a given structure, then the electric field enhancement, g, is defined by ∣E∣ = g∣E ₀∣, with ∣E ₀∣ being the incident amplitude. While g = g(x,y,z), we focus on the largest values of g that typically occur in the gaps between particles. Reflection, R, is calculated as the ratio of the integrated normal energy flux to the integrated incident energy flux across an arbitrary plane in the −z direction.

 

Fig. 1. Proposed SERS substrates (a) Structure I (b) Structure II. See text for details.

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In addition to the reproducibility due to the use of well-ordered particle arrangements, it may be possible to increase the intensity of the hot spots between nanoparticles in structures I and II. These structures allow the use of SPPs, which themselves carry an enhancement of g _SPP over incident light, to excite LSP resonances, which also carry an enhancement g _LSP with respect to incident light. The overall enhancement in the electric field is therefore expected to be g_tot = α g_SPPg_LSP, where α, the coupling efficiency, can be ≤ 1 with the equality holding for perfect coupling.

In order to excite SPPs, one must overcome the momentum mismatch between incident light propagating within a dielectric medium and SPPs propagating at the interface of a metal and dielectric [3]. Diffraction gratings with periodicity, P, can provide wavevector components in the plane of the surface with magnitude 2πm / P, where m = 1, 2, …. For a fixed P, a properly chosen frequency or wavelength of incident light, λ ₀, can excite SPPs with k_SPP = 2πm /P. Using the well-known expression for k_SPP [3], one finds the implicit equation [8, 9] for λ₀,

The SPPs consistent with Eq. (1) are called Bloch-wave SPPs (BW-SPPs). Physically, the diffracted light excites SPPs with both ± k _SPP whose interference gives rise to standing waves. With P = 450 nm, the water/silver interface in structure I will have BW-SPPs excited with incident wavelengths near λ₀ = 633 nm (m = 1) and 403 nm (m = 2). For P = 394 nm, the PMMA/silver interface in structure II will have BW-SPPs excited with incident wavelengths near λ₀ = 633 nm (m = 1) and 416 nm (m = 2).

In addition to BW-SPPs, Rayleigh anomalies (RAs), sometimes called Wood or Rayleigh-Wood anomalies, can also be present [9, 10]. The condition for a RA is simply λ ₀ = nP/m′ where n is the refractive index of the medium interfacing the metal and m′ is an integer. An RA is not a plasmon but rather a form of diffracted light that is traveling along the metal surface. Intensities associated with RAs are generally weaker than those associated with BW-SPPs, and so they are less relevant to SERS applications.

3. Results and discussion

Before the FDTD analysis of structures I and II, we first calculated the reflection peak positions of the local plasmon resonances for a single ‘ribbon’ (i.e., PMLs were used in the x-direction) of hexagonally packed pairs of 40 nm diameter gold nanospheres in water 4 nm above a silver film. Reflection minima, not shown, corresponding to local plasmon excitations are visible at λ₀ = 533 nm and λ₀ = 618 nm, near the desired 633 nm.

3.1. Structure I

Figure 2(a) shows the calculated reflection spectrum for the periodic structure I without particles (red curve). The reflection minimum at λ₀ = 407 nm belongs to the m = 2 BW-SPP at the glass / silver interface (from Eq. (1), λ ₀ ≈ 438 nm). Examination of calculated fields reveal that the broad reflection minimum at λ₀ = 530 nm corresponds to local plasmon excitation associated with the grating slits, as does the sharp resonance at λ₀ = 736 nm which overlaps the m = 1 BW-SPP at the glass / silver interface near λ₀ = 722 nm. Resonances at λ₀ = 604 nm and at λ₀ = 686 nm are in good agreement with exciting a first order RA on the water side with λ₀ = nP = 1.33 × 450 nm = 599 nm, and a first order RA on the glass side with λ₀ = 1.5 × 450 nm = 675 nm. Now, the reflection minima near λ₀ = 633 nm are also in very good agreement with exciting the m = 1 BW-SPP at the water / silver interface. Consistent with incident light at wavelength λ₀ = 633 nm, Fig. 2(b) is a plot of the steady state electric field intensity for structure I with no particles present. A maximum SPP intensity of ∣E∣² _SPP ≈ 13∣E₀∣² (g _SPP = 3.6) is present at the water / silver interface. (Intensities can be > 20 ∣E₀∣² at the corners corresponding to local excitations.) This is consistent with the fact that propagating surface plasmons are typically much less intense than local plasmons [2].

Figure 2(a) also shows the reflection spectrum for structure I when particles are present (blue curve). The most striking and significant difference that arises from the presence of the gold nanoparticles is the increase in the reflection of the water / silver BW-SPP region near λ₀ = 633 nm. In Fig. 2(c) we plot the total steady state electric field intensity, ∣E₀∣², at λ = 633 nm for structure I with gold nanoparticles present, with a maximum total intensity of ∣E∣² _tot ≈ 1800 ∣E₀∣² (g = 42) within the particle gaps (note enhancements > 20 are again shown as yellow to allow visualization of the BW-SPPs). Plots of ∣E∣² for λ₀ = 618 nm and λ₀ = 656 nm (not shown) also predict similar enhancements.

As a final consideration for structure I, we applied PMLs [6] to the edges of the x grid, which negate the effect of periodicity. This allows us to calculate the intensities of LSPs within the particle gaps via light scattered from the corners. Figure 2(d) shows the steady state electric field intensity enhancements for incident light at λ₀ = 633 nm; we see a maximum intensity of ∣E∣² _LSP ≈ 400 ∣E₀∣² (g _LSP = 20) within particle gaps. Therefore, the coupling efficiency is given as α = g_tot / (g_SPP g_LSP) ≈ 0.6. Alternatively, α measures the extent to which the presence of the gold spheres disrupts the coupling into the BW-SPPs in the first place. This is seen by the increase in reflection amplitude of the peak at λ₀ = 633 nm in Fig. 2(a) when particles are present. Calculations which placed nanospheres at positions corresponding to the maxima in BW-SPP intensities actually show a more drastic decrease in the absorption in the λ₀ = 633 nm region and much lower enhancements within the particle gaps. The fact that the spheres in structure I are placed at a node minimizes this disruption and allows for a sizable coupling efficiency.

3.2. Structure II

With structure I, we showed that one can couple BW-SPPs on a silver film into the local resonances of gold nanospheres with an efficiency α ≈ 0.6. In considering structure II, we aim to verify this effect for a related but different structure and, furthermore, to increase the overall enhancements within the particle gaps. Structure II (see Fig. 1(b)) is essentially a 4-layer system (glass ∣ silver (76 nm) ∣ PMMA (d + 44 nm) ∣ water), with 92 nm wide / 44 nm deep grooves in the PMMA layer. The purpose of the grooves and spacer layer of height d is 3-fold: 1) to provide guides and capillary action for nanoparticle assembly, 2) to provide a grating for exciting BW-SPPs, 3) and to allow SPP propagation beneath the spheres. We consider values of d ranging from d = 0 nm to d = 40 nm in 10 nm increments. We note that BW-SPPs will now propagate primarily at the PMMA / silver interface, and for P = 394 nm, Eq. (1) predicts BW-SPPs at λ₀ = 633 nm (m = 1), 416 nm (m = 2). Absorption spectra, not shown, for structure II without particles present for each value of d indeed show two peaks corresponding to the m = 1 and m = 2 BW-SPPs. Figure 3(b) shows the steady state electric field intensity, ∣E∣² _SPP, for the m = 1 BW-SPP for d = 40 nm, with ∣E∣² _SPP ≈ 16∣E₀∣² (g_SPP ≈ 4.0) (similar values were observed for all d > 0). For d = 0 nm, not shown, coupling into the m = 1 BW-SPP is hindered by the discontinuous profile in the x-direction of the metal-dielectric interface, resulting in a smaller maximum intensity of ∣E∣² _SPP ≈ 9∣E₀∣² (g_SPP ≈ 3).

 

Fig. 2. (a) Reflection spectra for structure I without (red linepoints) and with (blue) particles present calculated via FDTD. Steady state electric field intensities at λ₀= 633 nm for the unit cell corresponding to Fig. 1(a) (b) without particles, (c) with particles, and (d) with particles and PMLs. See text for discussion.

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In Fig. 3(a), we present the reflection spectra for structure II with and without particles present for d = 0 (red) and d = 40 nm (blue). We also consider the effect of PMLs (red and blue linespoints). For d = 0 nm and PMLs (red linespoints), two resonances are visible, with the state at λ₀ = 667 nm corresponding to a mixed particle / film resonance. Calculated values for steady state electric field intensities (profiles not shown here) at λ₀ = 667 nm are ∣E∣² _LSP ≈ 1800 ∣E₀∣² (g_LSP = 42) within the particle gap and ∣E₀∣² _LSP ≈ 7000 ∣E₀∣² (g_LSP = 84) between the particles and film. The reflection spectrum corresponding to the periodic structure (red curve) shows a splitting of this resonance to give a small absorption peak at λ₀ = 587 nm and a larger peak at λ₀ = 687 nm. Figure 3(d) corresponds to a plot of ∣E∣² _tot for the resonance at λ₀ = 687 nm, showing ∣E∣² _tot ≈ 5000 ∣E₀∣² (g_tot = 84) within the particle gap and to ∣E∣² _tot ≈ 20000 ∣E₀∣² (g_tot = 84) between the particles and film. The coupling efficiency for this case, again, is given by α= 0.6.

For d = 40 (blue curve), one resonance at λ₀ = 623 nm is clearly visible and corresponds to the interaction of the tail end of the broad particle absorption (inset, blue) with the BW-SPP. A similar observation was reported in Ref. [4]. Calculated values for steady state electric field intensities within the particle gap at λ₀ = 623 nm for the isolated and periodic structure, respectively, are ∣E∣² _LSP ≈ 2200 ∣E₀∣² (g_LSP = 46.9) and ∣E∣² ≈ 6600 ∣E₀∣² (g_tot = 81.2) (see Fig. 3c), leading to a smaller coupling efficiency α = 0.4. This is probably due to the evanescent decay of the BW-SPPs in the z-direction giving rise to smaller fields 40 nm above the interface.

 

Fig. 3. (a) Reflection spectra for structure II with d = 0 nm (red linespoints) and d = 40 nm (blue). Inset corresponds to structure II with PMLs. Steady state electric field intensities (b) for the m = 1 BW-SPP with no particles present, (c) d = 40 nm with particles present, and (d) d = 0 with particles present. (All intensities > 30 ∣E₀∣² are saturated in yellow)

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In summary, we described calculations on potential SERS systems displaying multiplicative enhancements via interactions between LSP resonances of gold nanoparticles and BW-SPPs on silver film structures. While previous studies have shown coupling between SPPs and LSPs [4, 5], we considered somewhat different systems in which periodic gratings, quite independent of the presence of particles, give rise to BW-SPPs. We showed that the BW-SPP/LSP coupling efficiency can be α ≈ 0.6, with 1.0 representing perfect coupling. Overall SERS enhancements of g ⁴ = O(10⁸) were possible. (An average SERS enhancement factor determined by averaging such local enhancements over the sphere surfaces could be smaller by 2 orders of magnitude.¹¹) Further improvements to α and g are possible by considering different particle sizes, shapes, gratings, and gap sizes less than 4 nm.