Evaluation of Surface-Enhanced Raman Spectroscopy Substrates

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Evaluation of SERS Substrates from Single-Molecule Statistics Evan J. Kiefl, Robert F Kiefl, Diego P dos Santos, and Alexandre G. Brolo J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.7b08691 • Publication Date (Web): 20 Oct 2017 Downloaded from http://pubs.acs.org on October 24, 2017

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The Journal of Physical Chemistry

Evaluation of SERS Substrates from SingleMolecule Statistics Evan J. Kiefl1, Robert F. Kiefl2, Diego P. dos Santos3, Alexandre G. Brolo1,4* 1

Department of Chemistry, University of Victoria, Victoria, V8P 5C2, Canada

2

Department of Physics and Astronomy, University of British Columbia, V6T 1Z4,

Canada 3

Department of Physical Chemistry, Institute of Chemistry, State University of Campinas,

Campinas, São Paulo, CEP 13083-970, Brazil 4

Center for Advanced Materials and Related Technologies (CAMTEC), University of

Victoria, Victoria, BC, V8W 2Y2, Canada

Corresponding author: [email protected] KEYWORDS. SERS, single-molecule, Truncated Pareto distribution, plasmonics, enhancement factor, hotspot properties.

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Abstract Accurate quantification of substrate characteristics is a central pursuit within the field of surface-enhanced Raman spectroscopy (SERS). A theory based on single-molecule SERS (SMSERS) statistics was developed for comprehensive substrate evaluation. This approach is applicable to general substrates possessing many hotspots and is capable of quantifying hotspot strength variation using a minimal set of fitting parameters. The model was validated for simulated substrates and then applied to the SM-SERS statistics of a roughened silver electrode, for which the degree of hotspot uniformity was quantified. The fitted model parameters provide important information concerning the structure-activity relationship of hotspots and can be used to directly compare SERS substrates. Overall, our results present an experimentally determinable parameter set that potentially improves upon the widely used “average enhancement factor” metric currently used for SERS substrate evaluation.


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Introduction Surface-enhanced Raman scattering (SERS) originates from molecules experiencing enhanced optical fields in close proximity to nanostructured metal surfaces.1-5 The resulting Raman spectrum is amplified by local enhancement factors (𝐹) as large as 1010 .2, 6 The extreme sensitivity and specificity intrinsic to SERS has led to decades of research, resulting in an increasing number of applications in chemical sensing.7-12 Perhaps most important to all current and future applications are the SERS substrates (nanostructured metal surfaces that support the effect), as their properties are the largest determinant of SERS activity13. The efficiency of a SERS substrate depends critically on structure-activity relationships14 that must be finely controlled to optimally suit specific applications.7 Evaluation of the efficiency of SERS substrates is typically reported in terms of “average SERS enhancement factor” (⟨𝐹⟩). The experimental evaluation of ⟨𝐹⟩ requires assumptions for the number of probe molecules, both in Normal Raman and in SERS conditions,15 that can lead to large errors. An internal normalization procedure, using the Rayleigh scattering, has recently been suggested to minimize this problem.16 The thermodynamics of adsorption of the probe molecule should also have an effect on the SERS behavior and, consequently, on the substrate perceived efficiency.17 For many chemical applications, the ideal substrate possesses a large density of identical hot-spots (HSs), defined as regions on the nanostructure surface that exhibit enhanced local electric fields.15,

18, 19

Such a substrate would provide a high degree of sensitivity and

reproducibility for a given chemical analysis. Fabricating and synthesizing substrates that have minimal variation between HSs is a central pursuit in SERS research.7, 20-24 ⁠⁠ Researchers in the field face formidable challenges in unraveling the sensitive structure-activity relationships of



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HSs. Indeed, it has been shown theoretically and experimentally that key characteristics of gap HSs (formed by a metallic nanoparticle dimer, for instance), such as the maximum enhancement factor, FM (hereby called the “HS strength”), can vary by orders of magnitude in response to subnanometer alterations of the inter-particle gap distance.25, 26 The difficulty in developing uniform substrates is further compounded by challenges in quantifying the uniformity of HS characteristics. Since substrates have potentially a large number of HSs, quantifying uniformity is a statistical process by nature. However, no quantitative methods relating SERS measurements to HS uniformity currently exist. In this work, we developed a model capable of estimating the distribution of HS strengths of a substrate from single-molecule SERS (SM-SERS) intensity measurements. The distribution directly measures the a) average HS strength, b) spread of HS strengths within the substrate, and c) HS enhancement factor localization, i.e., parameters that can be used to quantitatively compare substrate performances. We propose that this approach significantly improves upon the “average enhancement factor” metric currently used for SERS substrate evaluation. This model is a natural extension of the work done by Le Ru et al.27 In a seminal paper in 200627, they developed a model to explain the SERS statistics of the simplest SERS substrate: two nearly touching metallic nanospheres that form a gap HS between them. Assuming a singlemolecule (SM) regime in which a SERS-active molecule adsorbs onto a random position of the substrate, the distribution of measured enhancement factor was accurately modeled by a truncated power-law distribution (otherwise known as a truncated Pareto distribution, i.e. TPD). TPD behavior has since been demonstrated for other substrate geometries2 as well as in experimental efforts.28,

29

While TPD behavior may not be a universal quality of SERS

substrates,30 most HS-bearing substrates have at least qualitatively similar “long-tail”


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distributions, making the TPD relevant both for its simplicity and general applicability. The model's potential use in quantitatively interpreting SERS measurements has always been recognized, but never appropriately applied due to the model being developed only for a single HS. However accurate the TPD is for a single HS, real substrates are composed of perhaps thousands of HSs that are active within the illumination volume of the exciting laser, and are; therefore, significantly more complex. Since extremely sensitive structural changes define the SERS activity of each HS, they all in practice have different characteristics and cannot be treated identically. In order to tackle this issue, we developed a multiple HS probability model that incorporates the fact that each individual HS is governed by its own TPD. Extending the model to this more realistic scenario, we demonstrate the ability to resolve important characteristics about the HS ensemble of a given substrate, which can be especially relevant for substrate performance comparisons.

Computational and experimental methods Extensive electrodynamics simulations to describe the enhancement factor distribution around different HSs were performed. In the case of spherical geometries, Mie theory simulations were performed using GMM-FIELD.31-34 In the case of nanorod structures, the simulations were performed using the discrete dipole approximation (DDA) method by DDSCAT 7.1 program.35 In all simulations the optical properties of silver and gold were taken from the tabulated data of Palik36 and Johnson and Christy37, respectively. More information on the procedure for the determination of the pdf for each HS is presented in the supplemental file. The experimental SM-SERS data were obtained for aqueous solutions of rhodamine 6G (R6G) at different concentrations on a roughed silver electrode. The roughening procedure was done by electrochemical oxidation and reduction cycles (ORC).38-40 Briefly, 3 cycles of oxidation 5 ACS Paragon Plus Environment


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and reduction were performed on a mechanically polished polycrystalline Ag electrode in the potential range -0.2 V to +0.2 V Vs Ag|AgCl|KCl(sat) at scan rate 50 mV.s-1. SERS measurements were performed at a 632.8 nm excitation and a laser power density of 0.25mW/m2 in a timeseries procedure that maintained a fixed illumination area on the electrode surface. In each timeseries measurement 3000 spectra were acquired. Since the signal-to-noise ratio can be small, especially at low concentrations, principal component analysis (PCA) were performed to minimize the noise. The first two PCA components, responsible for approximately 98% of the total variance, were selected to describe the data.39 The statistical treatment and data analysis were performed using the R program and its included packages for data visualization.41, 42

Results and discussion In this work, the TPD-HS model27 is generalized to include multiple HSs. In the supplemental information (“Multiple hot-spot theory”), the current state of the TPD-HS model, which is for a single HS, is summarized along with the mathematical formalism for extrapolating to multiple HSs. The main result is a parameterized probability distribution function (𝑔(𝐹; 𝐿M , 𝑘, 𝜎𝐿M )) that describes the probability of measuring an enhancement factor 𝐹 for a substrate containing multiple HSs which may be written: ∞

𝑔(𝐹; 𝐿M , 𝑘, 𝜎𝐿M ) = ∫0 𝑤𝐹M (𝐹M ; 𝐿M , 𝜎𝐿M )𝑝(𝐹; 𝑘, 𝐹M ) d𝐹M ,

(1)

where 𝑝(𝐹; 𝑘, 𝐹M ) is the probability distribution function (pdf) for a single HS and 𝑤𝐹M (𝐹M ; 𝐿M , 𝜎𝐿M ) is the adsorption probability for a molecule onto a given HS. Equation (1) has three parameters that characterize the properties of a SERS substrate: 𝐿M is the average logarithmic HS strength (the average logarithm of the maximum enhancement factor (FM)); 𝜎𝐿M


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is the standard deviation of the logarithmic HS strength; and 𝑘 describes the average field decay, a quantity that is related to HS enhancement localization, as will be later demonstrated. Equation (1) is a highly skewed distribution that in log-log form exhibits a pseudo-linear region at low enhancements and a gradual bending at high enhancements (Figure S3), as was predicted in ref.27. We refer to equation (1) as a “truncated Pareto mixture distribution” (TPMD), which is a continuum approximation to the average TPD of all HSs in a substrate. Therefore, as HSs become more uniform, the TPMD becomes increasingly TPD-like. Equation (1) was validated using a simulated substrate containing a collection of 3500 independent and non-identical HSs (silver nanosphere dimers) excited at their average dipolar resonance (see SI for details on the simulated substrate parameters). 𝑘, 𝐿M , and 𝜎𝐿M were estimated under the assumption of our model by numerically calculating the right-hand side of equation (1) for 1000 different (𝑘, 𝐿M , 𝜎𝐿M ) combinations and finding the combination that minimized the sum of squared residuals. Since the geometric characteristics of each HS was known, a pdf for each HS was calculated from which the true values of 𝑘, 𝐿M , and 𝜎𝐿M were determined and compared to the estimates obtained from equation (1). The results in Table 1 verify that the model estimates agree with the values known a priori. The least accurate determination was for 𝜎𝐿M , which is attributed to the fact that the slightly asymmetric distribution of 𝐿M was modeled by a Gaussian (Figure S4A). It should be noted that these are logarithmic quantities and therefore small differences between the fit parameters and the true parameters correspond to large differences on a linear scale. For example, even though the fit yielded 𝐿M = 9.67 (compared to the correct value of 𝐿M = 9.39), this corresponds to a 2-fold overestimation of the correct 𝐹M . These calculations are then order of magnitude estimates,



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which is in accordance with typical precisions seen for experimental estimates of average enhancement factors. Regardless, the model provides significant new information for substrate evaluation. For example, if one were to quantify the substrate via calculation of the logarithmic mean and standard deviation of the measured enhancements, one would obtain 7.47 and 1.06, respectively, from which little can be learned about the characteristics of the distribution of HSs. With Eq. (1), it is possible to distinguish HS strengths (i.e. the maximum enhancement) from all other measured enhancements, which allows extraction of the mean and standard deviation of HS strengths in a substrate.

TABLE 1. Enhancement Properties of a Simulated SERS substrate. 𝑘

Method

𝐿M

𝜎𝐿M

Exact calculation from the pdf of individual (independent) HSs

0.135 9.39 0.95

Least-squares fit from Equation (1)

0.142 9.67 0.58

The TPMD (equation (1)) has been developed under the assumption that exactly one molecule is adsorbed onto the substrate at a time (assuming that the whole substrate is illuminated by the laser); however, only rarely does experimental SM-SERS refer to this strict definition. More typically, SM-SERS refers to situations where potentially hundreds of molecules are adsorbed onto the illuminated surface, but the vast majority of the signal is contributed by just one molecule (or a few). Therefore, before modelling an experimental dataset, we evaluated the influence that multiple adsorbed molecules have on the multiple HS pdf (𝑔(𝐹)). Bi-analyte experiments are accepted as an experimental tool capable of determining the concentration regime at which SM-SERS is observed.43-45 It is then of interest to characterize 8 ACS Paragon Plus Environment

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how the pdf resulting from multiple adsorbed molecules deviates from equation (1) (which is valid for only one adsorbed molecule) when molecule numbers are commensurate with SMSERS concentrations determined in a bi-analyte experiment. To do so we considered the pdf for 𝑚 molecules distributed on the surface of 1000 HSs described by 𝑘, 𝐿M , and 𝜎𝐿M and conducted a bi-analyte-like simulation with hypothetical isotopologue molecules (𝑚𝑜𝑙1 and 𝑚𝑜𝑙2) at equal concentrations that were randomly placed onto the model substrate. Each molecule was considered to cover a surface area of 1 nm2 upon adsorption. The pdf was calculated via random sampling. Each sample in a simulation step (out of 106 steps) generated a total F-value that was calculated as the sum of all 𝑚F (m molecules adsorbed in different hotspots with their own Fvalues). This procedure was repeated for various 𝑚 values to reveal the effect of molecular surface concentration on the SM-SERS statistics. For each surface concentration, a distribution of F values was recorded, from which it was possible to extract the probability distribution. In SM-SERS experiments, typical histograms are presented in terms of normalized intensity (𝐼 ⁄⟨𝐼⟩). The experimental intensity is considered to be proportional to 𝐹 for the construction of 𝑔(𝐼 ⁄⟨𝐼⟩) distribution functions. 𝐹𝑚 = 1𝑥108 was chosen as the minimum enhancement factor and the “measured” 𝐹 values were normalized by an average ⟨𝐹⟩ to reflect the dynamic range observed in SM-SERS data, which typically spans ~2-3 orders of magnitude in 𝐼 ⁄⟨𝐼⟩. By applying such theoretical minimum and normalizing the resulting distribution by its average (𝐼 ⁄⟨𝐼⟩), it is possible to construct the 𝑔(𝐼 ⁄⟨𝐼⟩) from the normalized histogram (integrated total area equals 1). Figure 1A shows the results for the simulated 𝑔(𝐼 ⁄⟨𝐼⟩) for different densities of molecules per HS.



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Figure 1: A) Probability distribution functions for normalized SERS intensities (g(I/⟨𝑰⟩)) of molecules randomly adsorbed on the surface of the model substrate. The g(I/⟨𝑰⟩) curves were calculated for a different number of molecules per HS in the model substrate (1000 HSs in total). B) Characteristic histogram of the bi-analyte experiment for the conditions in (A) in terms of percentage of signal assigned to 𝒎𝒐𝒍𝟏, considering that the simulation is performed with the isotopologue molecules 𝒎𝒐𝒍𝟏 and 𝒎𝒐𝒍𝟐.

Figure 1A shows that for low densities of molecules, the TPMD (equation 1) is readily

observed. As the surface concentration of molecules in the system increases, the pdf clearly shifts from the characteristic TPMD shape. The behavior in Figure 1A is in accordance to observed experimental results that show a continuous change from an exponential-like to lognormal and finally to a normal intensity distribution as surface concentration increases.28,42 It is interesting to note in Figure 1A that departure from pure TPMD behavior is observed even for a density as small as 1 molecule/HS, most notably for small 𝐼 ⁄⟨𝐼⟩ values. For values of 𝐼 ⁄⟨𝐼⟩ above the average, the TPMD is recovered without prejudice. Strong deviations from TPMD behavior are apparent for densities larger than 5 molecules/HS. These results show that although the density of molecules may be low enough such that on average only 1 molecule is located at each HS, the high density of HSs in this simulated substrate leads to the simultaneous observation of signals from molecules located at different HSs, which causes the deviations from TPMD behavior.


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The above interpretation is corroborated by Figure 1B, which shows the probability distribution of the percentage of total intensity associated to 𝑚𝑜𝑙1 (w(pmol1)) in the bi-analytelike simulation. Such distributions were obtained from the histograms of the fraction of the total signal F (generated by the 𝑚 molecules) that can be solely attributed to 𝑚𝑜𝑙1. Therefore, in the extremes of the distribution in Figure 1B, only one kind of molecule; 𝑚𝑜𝑙1 (𝑝𝑚𝑜𝑙1 = 1) or 𝑚𝑜𝑙2 (𝑝𝑚𝑜𝑙1 = 0), contributed to the total signal. As can also be seen in Figure 1B, even for densities as low as 1 molecule/HS, there is a nonzero probability of observing events unassignable to true SM signals (𝑝𝑚𝑜𝑙1 = 0 or 𝑝𝑚𝑜𝑙1 = 1). This conclusion is corroborated in Figure 2, which presents the 𝑔(𝐼 ⁄⟨𝐼⟩) distributions for a density of 5 molecules/HS and varying number of identical HSs (all with equal 𝑘, and 𝐿M ).

Figure 2: Probability density functions for normalized SERS intensities (g(I/⟨𝑰⟩)) of molecules randomly adsorbed on the surface of model substrates with different densities of HSs. In all simulations the density of molecules were the same and equal to 5 molecules/HS.

For a large number of HSs the distribution strongly deviates from the TPMD model. However, as HS density decreases, the distribution gradually recovers the linear relationship indicative of a TPMD when viewed in a log-log plot. The observed fluctuation in 𝑔(𝐼 ⁄⟨𝐼⟩) for small HS numbers is probably due to a non-continuous description of 𝑤(𝐹𝑀 ) – see SI for details. 11 ACS Paragon Plus Environment


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Overall, these results suggest a simple way to compare different substrates in terms of HS density. If we consider two substrates formed by different numbers of HSs with similar strengths, the one that presents a larger density of HSs will show a larger deviation from TPMD statistics, requiring a decrease in analyte concentration if one is to recover the shape of a TPMD. In this sense, substrate performance comparisons could be done in terms of the analyte concentrations required to observe TPMD statistics. Such a comparison involves only the SM-SERS intensity fluctuations at different analyte concentrations and bypasses the necessity for explicitly calculating enhancement factors.6, 23, 43 In order to further illustrate the utility of the TPMD approach, we analyzed an experimental SM-SERS dataset involving rhodamine 6G (R6G) molecules adsorbed onto a rough Ag electrode (roughened by oxidation-reduction cycles (ORCs)).44 Normalized histograms of SERS intensities for the 1500 cm-1 band of R6G at different concentrations (Figure 3) were constructed by binning the data into 50 equally spaced bins, with zero SERS intensity events being discarded. The error bars in the histograms of Figure 3 are the standard deviation of a binomial distribution for the number of counts in each bin.


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Figure 3: Normalized histograms for SM-SERS intensity distributions of R6G in the ORC electrode at different concentrations ([R6G]). As can be seen in the figure, only at 10 nM was a TPMD-like shape of the distribution observed and, therefore, only for this concentration were the results fitted to Eq. (1), as shown in the figure. The first data point has been excluded from the fit for reasons discussed in the text. The parameters that minimized the 𝜒 2 were 𝑘 =0.11, 𝜎𝐿M =0.17, and 𝐿M =0.54, where for the case of the SM-SERS experiment 𝐿M = 𝑙𝑜𝑔10 (𝐼M ⁄⟨𝐼⟩). Error bars in the histograms are the standard deviation of a binomial distribution for the number of counts in each bin.

As it can be seen from Figure 3, the effect of the increase in [R6G] from 10nM to 5 µM is identical to that observed in the simulated substrate (Figure 1). TPMD-like behavior is observed for small [R6G] and is completely lost at 5 µM. Interestingly, [R6G] = 10 nM exhibits small deviation from the TPMD behavior at small intensities, which could be a result of a small fraction of events in which multiple molecules from different HSs contribute to the detected signal. This effect is indeed observed for this substrate in bi-analyte experiments.38,

40

Nevertheless, a least squared error minimization for [R6G] = 10nM yielded substrate parameter 13 ACS Paragon Plus Environment


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estimates of 𝑘 = 0.11, 𝜎𝐿M = 0.17 and 𝐿M = 0.54 and the resulting fit is presented by a red line in Figure 3. The mean scaling parameter of 𝑘=0.11 is in agreement with simulated 𝑘 values of nanodimer systems.27 The parameter 𝑘 describes how fast the pdf decreases with 𝐹, i.e., how likely it is to observe large 𝐹-values within a HS. Although this is an interesting description, a more physically relevant parameter for multiple HS characterization would be a measurement of the localization of the HS enhancement distribution. Such a parameter would depend not only on 𝑘 but also on 𝐹𝑀 . Hence, a description of HS enhancement localization should take into account both quantities. For a single HS, the average 𝐹 (⟨𝐹⟩) can be given as: 𝐹𝑀

⟨𝐹⟩ = ∫

𝐹𝑝(𝐹)𝑑𝐹 =

𝐹𝑚

𝐹𝑀 𝐷

(2)

with 𝐷=

1 − 𝑘 𝐹𝑀 𝑘 [( ) − 1] 𝑘 𝐹𝑚

(3)

𝐷 is defined as the degree of HS enhancement localization for a single HS. For the same 𝐹𝑀 a larger 𝐷 describes a smaller ⟨𝐹⟩ which means that most of 𝐹 on the HS surface is much smaller than 𝐹𝑀 , i.e., large field enhancements are confined to very small areas. The problem with this definition is that 𝐹𝑚 (the minimum enhanced factor) is not an experimentally determinable. The minimum 𝐹 experimentally accessible in SM-SERS measurements is the one provided by the instrument cutoff (𝐹𝑐𝑢𝑡𝑜𝑓𝑓 ). Therefore, we define the practical degree of HS enhancement localization for a single HS, 𝐷′ , as:


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𝑘

1−𝑘 𝐹𝑀 𝐷 = [( ) − 1] 𝑘 𝐹𝑐𝑢𝑡𝑜𝑓𝑓 ′

(4)

With such a definition, the experimental determination of 𝐷′ can be evaluated from the SM-SERS intensity distribution as: 𝐷′ =

𝐼𝑀 ⟨𝐼⟩

(5)

where 𝐼𝑀 is the maximum detected intensity in the SM-SERS fluctuations. The properties of a complex SERS substrate can now be entirely described in terms of the parameters 𝐿M (or 𝐹M ), 𝜎𝐿M and 𝐷′ . Whereas 𝐿M is related to the average HS strength, 𝜎𝐿M and 𝐷′ are related to the local field distribution and average enhancement localization, respectively. We hypothesize that a very efficient SERS substrate is one with high 𝐿M (or 𝐹M ) and low 𝜎𝐿M and 𝐷 ′ . Such a substrate would exhibit large field amplifications with small spatial variation that

spread over the SERS-active surface. Taken together, this would correspond to a large average enhancement factor, the quantity commonly used in the field to compare substrates. We have further validated the use of these parameters for substrate comparison by performing a large number of simulations from different types of isolated nanostructures. A comprehensive map of substrate efficiency was produced and it is presented in the SI (Figure S6). Since the SM-SERS intensities in Figure 3 are represented with arbitrary units, it is not possible to provide absolute metrics for the substrate without first converting from intensities to enhancement factors. Although not done here, note that a protocol for this type of conversion has been laid out by the Etchegoin group.6,

45

⁠ Nonetheless, comparisons between the roughened

electrode and the simulated substrate are still possible and provide a glimpse of how the model developed above can be used for substrate evaluation. Most notably is the relative standard



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deviation of HS strengths 𝜎𝐿M ⁄𝐿M , which is 6% for the simulated substrate and 31% for the roughened electrode. The small degree of HS strength variation observed for the simulated substrate is to be expected, given that the variation results from small perturbations from the mean dimer geometry. On the other hand, the large spread for the electrode reflects the high degree of nanostructural randomness manifesting from the stochastic electrochemical roughening process. Further simulations describing how substrate roughness affects the TPDM parameters can be found in the SI. The relatively large value of 𝐷′ obtained from the fit for a roughened electrode (𝐷′ = 8.97) indicated that highly enhancing adsorption sites are very spatially localized. This corroborates a large body of experimental evidence indicating that the roughened electrode is a mediocre substrate (average enhancement factors around 106).2,22,42 Beyond the substrate characteristics given by 𝐿M , 𝜎𝐿M and 𝐷′ , it is important to reiterate that information about the HS density (Figure 2) can also be obtained from SM-SERS measurements. The procedure presented above constitutes a powerful tool for SERS substrate performance comparisons that does not rely on simplifying assumptions, such as the number of illuminated molecules generally invoked in typical SERS experiments for calculating enhancement factors.46 However one should keep in mind that 𝐷′ depends on the instrumental parameter 𝐹𝑐𝑢𝑡𝑜𝑓𝑓 , which could limit its applicability across different instrument platforms.

Conclusion We have herein developed, tested, and applied a statistical model capable of extracting information on HS strength, density and uniformity from SM-SERS data. The results from this study open the possibility of direct correlation of the statistics of SM-SERS measurements to key substrate parameters, which may have important applications in substrate characterization and 16 ACS Paragon Plus Environment

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other analytical applications that depend on in-depth knowledge of the HSs properties. The results presented here also showed that fundamental characterizations of HS average local field properties can be extracted from SM-SERS intensity data and that this characterization correlates well with electrodynamics simulations.

Supporting Information Details of the theory for single hotspot and for multiple hotspots, evaluation of SERS efficiency from model single hotspot systems, effect of the roughness in the substrate efficiency and additional simulations are presented as supplemental information. This information is available free of charge via the Internet at http://pubs.acs.org

Acknowledgements This work was supported by the NSERC Discovery Grant. We also thank CFI and BCKDF for equipment grants. E. J. K also thanks NSERC for a summer undergraduate scholarship. D. P. S. gratefully acknowledges CNPq (process: 408985/2016-0) for financial support and CENAPAD-Unicamp (proj611) for computational resources.

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Evaluation of Surface-Enhanced Raman Spectroscopy Substrates

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