## Abstract

We investigate the electromagnetic mechanism in surface-enhanced Raman scattering (SERS) from randomly rough metal surfaces with Gaussian statistics and Gaussian correlation function. By means of rigorous numerical calculations, large average SERS enhancement factors (above 10^{4}) are encountered when the correlation length is of the order of (or lower than) a hundred nanometers, with excitation in the visible and near infrared. These Gaussian-correlated metal surfaces can be used as SERS substrates. Furthermore, local SERS enhancement factors are obtained of up to 10^{8} that make them appropriate for resonant SERS single molecule detection.

©2002 Optical Society of America

## 1 Introduction

The observation of surface-enhanced Raman scattering (SERS) has stimulated numerous investigations on the optical properties of rough metal surfaces and adsorbates [1, 2]. Active SERS supports, such as electrodes, colloids, and metal islands are employed in a variety of situations, depending on the characteristics of the molecular system being studied. For instance, the tunability of the electrode potential adds flexibility to the acquisition of SERS spectra with electrodes, and such substrates have been used to study a large number of molecules. However, the technique becomes impractical when the molecule is oxidized, reduced, or degraded. Metal colloids, on the other hand, possess short-range roughness that make them more active. Nonetheless, besides the difficulties in controlling their adequate preparation, SERS on colloids is hampered by the solubility of the adsorbate in water or other solvents. And in the case of solids, SERS spectra can be obtained with substrates containing metal island structures, which can be fabricated using Langmuir-Blodgett films or self-assembled monolayers of different molecules [3].

An important problem shared by most of these SERS supports is that the fabrication process is elaborate and the results cannot be easily controlled and reproduced. In addition, in the case of randomly rough substrates, the resulting geometries can be difficult to characterize; they are multiscale and possess ill-defined statistics. Thus, comparison with theoretical work is difficult, complicating studies for the optimization of their parameters. For these reasons, alternative SERS supports in the form of metallized rough substrates have been studied. Particularly interesting in this respect is the case of metallic random surfaces whose profiles constitute Gaussian random processes with a Gaussian correlation function. Samples with such characteristics can be fabricated on photoresist [4, 5, 6, 7, 8], and covered with a metal coating using evaporation techniques. Surface features down to the sub-100 nm scale could be photofabricated by using deep and extreme UV lithographic techniques [9, 10]. Interestingly, such metal surfaces with Gaussian statistics and correlation function are often used in theoretical (linear) scattering work [11, 12, 13, 14].

From the theoretical standpoint, dipolar approximations (either retarded and non-retarded) have been used to describe the electromagnetic (EM) response of SERS fractal metal substrates [15, 16, 17, 18]. In recent years, numerical methods that are capable of dealing with full EM problems involving complex structures have been developed. Typically, however, these have been implemented only for one dimensional profiles due to limitations in computer memory. Among the structures considered there are: deterministic surfaces, such as periodic gratings [19], dimers of nanoparticles [20, 21] or nanowires of complex shape [22, 23]; or randomly rough, self-affine fractals [24, 25, 26].

In this work we investigate the EM mechanism in SERS from randomly rough metal surfaces with Gaussian statistics and Gaussian correlation function, expressed as:

where *a* is the 1/*e* correlation length and *δ* the rms deviation of surface heights. In our numerical study, we make use of the exact formulation of the scattering integral equations based on the application of Green’s second integral theorem. We restrict the analysis to one-dimensional surfaces *ζ*(**r**) = *ζ*(*x*), which are constant along the *y* direction, illuminated by an EM wave whose propagation vector is perpendicular to the grooves. The two fundamental modes of polarization are the transverse (to the plane of incidence) electric and magnetic ones (*s* and *p* modes, respectively). In this manner, the three-dimensional EM scattering problem is reduced to a two-dimensional scalar one; no changes in the state of polarization take place in the interaction with the surface. Thus, the formulation becomes feasible from the numerical standpoint [25], while the physics underlying the SERS EM mechanism is reproduced, except for the fact that with one-dimensional surfaces only *p*-polarized radiation can couple into surface-plasmon polaritons, leading to large surface field enhancements. Furthermore, randomly rough substrates with controlled statistics fairly invariant along one direction can be also fabricated [7, 8]. Interestingly, such substrates could be useful to carry out polarization-assisted SERS spectroscopy.

## 2 Model

A monochromatic, *p*-polarized Gaussian EM beam of frequency *ω* and width *W* impinges with the angle *θ*
_{0} on a rough interface *z* = *ζ*(*x*) separating vacuum from a semi-infinite Ag volume occupying the lower half-space [*z* < *ζ*(*x*)] and characterized by an isotropic, homogeneous dielectric function *ϵ*(*ω*), obtained from Ref. [27]. The surface electric field **E**(*x*|*ω*) = **E**>(*x*, *ζ*(*x*)|*ω*) is numerically calculated as detailed in Ref. [25]. The enhancement of the surface electric field intensity is given by

only the center half of the illuminated area is considered when calculating *σ* in order to avoid irrelevant enhancements at the wings of the incoming beam. The SERS enhancement factor due to the EM mechanism is then defined as [1]

where *ω _{R}* is the frequency of the Raman-shifted signal and, due to the closeness of the frequencies

*ω*and

*ω*, the enhancement of the intensity of the Raman-shifted signal [

_{R}*σ*(

*ω*)] emitted by a molecule placed on the substrate is assumed to be identical to that of the surface electric field intensity at the pump frequency

_{R}*ω*.

Ensembles of *N _{rea}* realizations of random profiles are numerically generated with Gaussian statistics and Gaussian correlation function as mentioned above. The correlation length

*a*is varied in the range of a hundred nanometers and below, since it is expected that roughness-induced surface-plasmon polariton excitation, the mechanism responsible for the occurrence of large EM fields, be more efficient in the visible and near IR for such subwavelength correlation lengths. Single realizations are shown in Fig. 1 with

*δ*= 102.8 nm and

*a*= 514,102.8, 51.4, and 25.7 nm. Monte Carlo simulation calculations based on the above mentioned rigorous scattering formulation are thus done for the ensemble of realizations, thereby obtaining the average SERS enhancement factor from

*N*=

*N*/2 ≥ 10

_{rea}N_{x}^{5}data points (

*N*≥ 10

_{x}^{3}being the number of sampling points per surface realization and

*N*= 100).

_{rea}## 3 Results

The spectral dependence of 〈𝓖* _{SERS}*〉 for different roughness parameters

*a*and

*δ*are shown in Fig. 2. We have also included for the sake of comparison the results for self-affine fractals with fractal dimension

*D*= 1.9, lower scale cutoff

*ξ*= 25.7 nm, and

_{L}*δ*= 257 nm; metal aggregates, films, or surfaces with quasi-fractal properties are well known to be SERS active [1, 16, 17, 26]. First, note the strong dependence of 〈𝓖

*〉 on*

_{SERS}*a*. Whereas in practice there is no relevant enhancement for correlation lengths of the order of the visible wavelength even for large

*δ*(see the results for

*a*=

*δ*=514.5 nm), significant SERS enhancement builds up as the correlation length is decreased to

*a*=102.8 nm, 〈𝓖

*〉 becoming even larger for nanoscale*

_{SERS}*a*. This tendency is evidenced by the results for fixed

*δ*and

*a*= 102.8, 51.5, and 25.7 nm. With regard to the rms height dependence for given

*a*, the larger is

*δ*, the more intense is the SERS enhancement as expected also from the results for self-affine surfaces [24].

Therefore, it should be remarked in light of the results in Fig. 2 that Gaussian-correlated Ag surfaces with correlation lengths of the order of a hundred nanometers or lower exhibit very large SERS enhancement factors, comparable to or even larger than those widely assumed for well known SERS substrates [1]: 〈𝓖* _{SERS}*〉 ~ 10

^{4}. (Recall there is an additional enhancement factor due to the charge transfer mechanism, typically assumed to account for another ~ 10

^{2}enhancement factor, which can in turn be substantially larger if an electronic resonance of the molecule is excited.) Furthermore, the large values of the SERS enhancement factor are preserved for a wide spectral range in the visible and near IR (as shown in Fig. 2 up to λ = 2

*πc*/

*ω*= 2

*μ*m, being wider for larger

*δ*). Note also that our results show that the desired values of 〈𝓖

*〉 can be achieved with different combinations of roughness parameters*

_{SERS}*a*and

*δ*. Some of the combinations involved are within reach of photolithographic techniques but, undoubtedly, the fabrication of samples with the shorter correlation lengths would be more involved; still, we believe that it remains a realistic proposal.

Large SERS enhancement factors stem not only from the large mean electric field intensity 〈*σ*〉, but to a large extent from the enormous fluctuations of the surface electric field intensity Δ*σ* = (〈*σ*
^{2}〉/〈*σ*〉^{2} - 1)^{1/2}. The crucial role of fluctuations in SERS enhancements was predicted a few years ago for fractal samples [15] and has been discussed extensively in subsequent work [16, 25]. In this regard, it should be remarked that the large fluctuations (Δ*σ* can be up to 10^{2} while 〈*σ*〉 is only about 30) observed with the metal surfaces employed to obtain Fig. 2 occur for Gaussian-correlated surfaces which, unlike fractals, posses no scaling properties.

In addition to large average SERS enhancement, the occurrence of extremely large local enhancement factors at so called *hot spots* on the metal substrate can be extremely useful in SERS single molecule probing [3, 28, 29, 30], and has been reported in the case of fractal surfaces [16, 17, 26]. An example of such a hot spot is seen in the movie of Fig. 3: The calculated near-field distribution of electric field intensity, normalized to that of the normal incident beam, is shown as a function of the frequency in the vicinity of a small area of a surface realization belonging to a Gaussian-correlated Gaussian process with *a* = 51.4 nm and *δ* = 257 nm. In our two-dimensional geometry, an experimental near-field optical image of the surface would map only the intensity along a horizontal line (constant height mode) in the frames of Fig. 3. Although limited by the reduced dimensionality, the numerical calculations permit the observation of the whole near-field map, including points on the selvedge region, and even inside the metal.

It can be observed in the movie that the near-field intensity map goes through substantial changes in a relatively narrow frequency range, illustrating the sensitivity of the optical hot spots to the pump frequency. At the central frequency *ω*
_{0} = 2*πc*/λ =1.5 eV, corresponding to the wavelength λ = 826.6 nm, an optical mode localized at the center peak of the surface area is observed. Its deduced linewidth is Γ ~ 60 meV. The enhancement factor in the vicinity of the hot spot is *σ* ≈ 6 × 10^{3}, so that *𝓖 _{SERS}* ~ 10

^{7}. Incidentally, in some of the frames in the movie of Fig. 3, a slight mismatch in the surface-field intensity (typically at low-intensity areas) can be observed within a narrow region (a few nanometers wide) close to the surface profile: As has been described in Ref. [25], this is related to the numerical interpolation method between the calculated surface and near EM fields (actually the near field is expressed as an integral over the surface profile involving the surface EM field, which plays the role of source function self-consistently calculated), and has no further physical implications.

In this regard, we present in Fig. 4 the maxima of the local SERS enhancement factors for the same Gaussian-correlated Ag surfaces used in the calculations of Fig. 2. Large *𝓖 _{SERS}* are found for excitation in the visible and near IR (up to λ = 2

*μ*m), originated by optical modes occurring at different sites and frequencies. The qualitative behavior of

*𝓖*is very similar to that of 〈

_{SERS}*𝓖*〉. Such local enhancement factors of up to nine orders of magnitude, although lower than those claimed in first experimental accounts on SERS single molecule probing [29], can be considered sufficiently large and realistic as to provide active sites for resonant single molecule SERS detection [3]. Additional intensification of the SERS signal can be accomplished by resonant excitation of molecular electronic transitions.

_{SERS}On the other hand, it should be mentioned that the choice of a Gaussian correlation function is motivated by its theoretical simplicity and by the existing experimental procedure of fabrication that we have mentioned, but is not crucial to the results; other correlation functions, associated with smoothly varying single scale surfaces can have similar consequences. Finally, though not shown here, gold or copper substrates with the same characteristics as those studied also produce similar SERS enhancement factors (except for λ ≤ 600 nm).

## 4 Conclusions

In summary, our rigorous calculations for the EM mechanism of SERS enhancement provide conclusive results in favor of the use of rough metal surfaces with simple statistics (Gaussian random process with a Gaussian correlation function) as SERS substrates. Typically, average SERS enhancement factors of four orders of magnitude and even larger are found. For such enhancements, correlation lengths of the order of a hundred nanometers are necessary, with an rms height greater than 50 nm. Moreover, hot spots with local SERS enhancement factors of up to 10^{9} occur, and these are candidates for electromagnetically active sites for SERS single molecule detection. In addition to being attractive for SERS studies, these kind of single scale surfaces are known to enhance other nonlinear optical phenomena [31, 32].

## Acknowledgments

This work was supported by the Spanish Dirección General de Investigación (BFM2000-0806 and BFM2001-2265) and Comunidad de Madrid (07M/0111/2000).

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