Modeling the Effect of Small Gaps in Surface-Enhanced Raman

FEATURE ARTICLE pubs.acs.org/JPCC

Modeling the Effect of Small Gaps in Surface-Enhanced Raman Spectroscopy Jeffrey M. McMahon, Shuzhou Li, Logan K. Ausman, and George C. Schatz* Department of Chemistry, Northwestern University, Evanston, Illinois 60208-3113, United States ABSTRACT: The electromagnetic mechanism of surface-enhanced Raman spectroscopy (SERS) involves plasmon enhancement of the optical-frequency electric fields on the surfaces of silver or gold (or sometimes other) nanoparticles where molecules that exhibit Raman scattering are located. It has long been recognized that the largest electric fields are associated with small gaps (∼1 nm) between two or more nanoparticles (including fused particle structures) that are 20100 nm in size. Recent advances in electromagnetic theory calculations, which we overview in this article, provide a clear quantitative picture of the SERS enhancement associated with this effect. These advances include: (1) recognition of the nanoparticle structures that give the highest enhancement factors for a given gap size; (2) determination of the dependence of enhancement factor on gap size in the small-gap limit; (3) the use of finite-element methods, rather than cubic grid-based methods, to evaluate enhancement factors; (4) evaluation of the dipole reradiation effect on enhancement factors for small gaps; and (5) the use of nonlocal dielectric functions to describe a material’s electrodynamic response. These advances have demonstrated that the “hot spots” for 1 nm or smaller gaps often have a multipolar plasmon resonance character, reflecting the short propagating plasmon wavelength that exists in them. This leads to SERS excitation spectra that are not correlated with extinction or scattering spectra. In addition, these results show that the electromagnetic SERS enhancement factor has an approximate 1/gap2 dependence on gap (size) for optimally chosen particles.

I. INTRODUCTION Surface-enhanced Raman spectroscopy (SERS) is perhaps the best example of a plasmon-enhanced spectroscopic technique in which the intensity of an optical transition, Raman scattering from an adsorbed molecule in this case, is enhanced as a result of the interaction with the electromagnetic field of a plasmon excitation in a nearby metal (typically silver or gold). In the original experiments, which date to the middle 1970s,13 it was found that the enhancement factor associated with silver particles near 500 nm wavelength is about 106. This was large enough so that SERS could be used to study adsorbates at submonolayer coverages, and therefore to use Raman spectroscopy as an analytical tool. However, commercial applications did not immediately occur, as the substrates needed to optimize this effect have taken a long time to develop; however, in the past few years, there has been growing activity in the use of SERS as an analytical tool for biomolecules4 as well as for tagging and labeling applications.5 References 613 provide recent reviews of SERS applications. In passing, we note that there is also much interest in other plasmon-enhanced spectroscopic methods, including those related to fluorescence, IR, and second harmonic generation, among others. There have been many proposed theories of the SERS effect: refs 1416 review the older literature, while refs 1722 describe more recent work. A popular idea is that the overall enhancement factor involves the product of an electromagnetic enhancement factor EEM that is associated with plasmon-enhanced fields near the r 2011 American Chemical Society

surface of a metal and a chemical enhancement factor ECH that arises from a charge transfer between the surface and the molecule. It was realized in the late 1970s that EEM must reflect intensity enhancement of the field both at the incident frequency ω and at the scattered (Stokes shifted) frequency ω0 . A commonly used expression for this enhancement factor is EEM ¼ jEðωÞj2 jEðω0 Þj2

ð1Þ

where E is the enhancement of the electric field (the ratio of the local field to the incident field), which asymptotically is a plane wave, evaluated at the position of the molecule. This expression, which we will call the Plane Wave (PW) electromagnetic enhancement factor, is only an approximate expression, as Kerker noted long ago;23 improvements to it will be discussed later. For qualitative applications, it is sufficient to further assume that ω ≈ ω0 (valid as long as the plasmon resonance is broad enough such that the field enhancement is approximately the same at both frequencies), which means that EEM ≈ |E|4 (where the ω designation has been dropped). However, for quantitative work, the frequency difference between incident and scattered light leads to noticeable effects, usually decreasing the enhancement factor with increasing Stokes shift. In addition, there can be important differences in polarization effects for a molecule on a surface (where there is significant Received: August 10, 2011 Revised: November 11, 2011 Published: November 17, 2011 1627

dx.doi.org/10.1021/jp207661y | J. Phys. Chem. C 2012, 116, 1627–1637

The Journal of Physical Chemistry C depolarization) compared to the same molecule in solution. This reflects the fact that the local field at a surface is largely perpendicular to it, independent of incident polarization; thus, if the molecule is also oriented in a fixed direction, such as through chemisorption, then there can be important changes in its Raman spectrum (as has been discussed24). The chemical contribution to SERS has also been studied in a number of publications.17,2527 One aspect of this is that charge transfer between the adsorbed molecule and a nanoparticle in the ground electronic state of the moleculenanoparticle system changes the static Raman spectrum. Recently, Morton and Jensen26 have demonstrated that this effect can contribute a factor of ECH ∼ 10, with systematic variations with the energy of the lowest unoccupied molecular orbital of the adsorbate. There can also be contributions from charge-transfer excited states, which contribute to SERS via a resonance-Raman mechanism.27 Lombardi and Birke have discussed how the electromagnetic and chemical contributions to the overall enhancement factor can be calculated using a unified theory.25 However, in this article, we will only consider the electromagnetic contribution. Equation 1 depends on the location of the molecule on the surface, so for all except single-molecule SERS (SMSERS) studies it should be averaged over the locations of all of the molecules being observed. However, the spatially averaged enhancement factor is often dominated by “hot spots”, where the plasmonenhanced field is unusually large, typically because there is a gap or crevice between two (or more) particles.28 The importance of gaps, crevices, and related features was discussed theoretically in the early SERS literature.29,30 However, experimental confirmation of this was generally indirect, until quite recently. One recent contribution to this was the work by Fang et al.,31 where SERS measurements were performed before and after removing molecules selectively from hot spots, leading to measurements of the intensity distribution as a function of the enhancement factor. They demonstrated that 0.1% of the surface area of the particles being considered appears to be responsible for over 20% of the observed signal, a result that is also supported by calculations.28 There have also been studies of gap structures fabricated using the OWL (on-wire lithography) method, which have shown what happens at the junction of very large rods where roughness in the gap plays an important role.32 Other evidence for the importance of junctions comes from SMSERS measurements, where enhancement factors of 1015 or more have been observed.3335 While there has been uncertainty as to whether the enhancement factor in SMSERS involves factors beyond what is described by the standard electromagnetic mechanism, a recent series of papers3638 have argued that SMSERS can be understood quantitatively based on a combination of eq 1 and very large resonance Raman enhancement factors for the molecules being studied. Also, a recent study in the Van Duyne group39 has observed SERS spectra for nonresonant scatterers adsorbed onto dimers and other small clusters of ∼90 nm spherical gold nanoparticles. They demonstrated that SERS intensities are not correlated with the wavelengths of plasmon resonances or with the number of particles in a cluster. Instead, they inferred that the dominant factor in determining the SERS enhancement factor is either the gap between the particles or if the particles are fused then the radius of curvature at the point of intersection. Although correlated experimental studies of SERS enhancements and nanoparticle structures are starting to appear,13 the quantitative relationship between SERS enhancement and gap size has not yet been determined. However, finite-element method (FEM) calculations

FEATURE ARTICLE

for the same nanoparticle clusters that were studied in the above experiments also show a strong sensitivity to gap,39 providing some quantitative insight. The gap structures described above are examples of electromagnetic antennas, so it is useful to consider them as such and the predictions of antenna theory with regards to the dependence of EEM on the structural parameters of an antenna. Here we note that antenna theory is normally concerned with metallic structures that are perfect electrically conducting (PEC) metals (i.e., those with a dielectric constant of ∞), and the complete relationship between the optical response of gaps in PEC structures and plasmonic ones has not yet been established. Nonetheless, in the PEC case, a simple argument for a thin-wire antenna (i.e., two metal strips or rods with a gap) is that the magnitude of the field in the gap region should vary as 1/gap (where gap is its size) due to an oscillating voltage across it. If correct, then |E|4 ∼ 1/gap4 (where the EM subscript has been dropped) would constitute a singular SERS enhancement as gap f 0. However, the connection of this prediction with more rigorous electrodynamics has only recently been examined,40 and the result is more complex and closer to |E|4 ∼ 1/gap2. We will examine this point later. Another structural parameter of interest within the antenna analogy is its overall size, which in standard theory dictates should be chosen to be a multiple of approximately half of the wavelength λ. This issue has recently been studied by Mirkin and co-workers using rod-shaped nanoparticles fabricated by the OWL method, where gold rods were aligned end-to-end with a diameter of about 300 nm.41 They observed SERS signals using a Raman microscope with a fixed excitation wavelength of 630 nm, finding that the signal oscillates periodically with rod length, with the period being determined by those of propagating surface plasmon waves (∼600 nm). In addition, the results demonstrated that the shortest rod for which peak intensities occur is about 120 nm, which is closer to λ/4 than to λ/2.41 Computations yield |E|4 enhancements which are consistent with these results, showing that the periodic variation of intensity with rod length L follows the formula L = (n + 1/4)λ, where n is an integer.32,42 This shows that the λ/2 result from antenna theory is modified to λ/4 for plasmonic antennas, which is a result that can be understood in terms of a phase shift43 of the propagating plasmon wave that occurs when it reflects from the ends of the rods.32 The Mirkin results41 also demonstrated that the SERS enhancement at a fixed wavelength increases with decreasing gap size down to 30 nm, but then decreases below that. Electromagnetic calculations confirm these results,41 and show that this behavior arises from changes in the characteristics of the hot spot between particles as the gap is narrowed. Pushing the particles together tunes the dipole resonance to the red and leads to the excitation of higher-order multipoles to the blue of the dipole resonance. These multipoles give lower enhancement factors due to electromagnetic depolarization effects, so at a fixed excitation wavelength, the intensity maximizes for a gap that makes it match the dipole resonance and then decreases as the gap does. It should be noted that roughness at the gap can further modify this behavior in the small-gap limit, as mentioned above, as then the hot spots are associated with the coupling between asperities on each side.32 In this article, we describe a number of recent advances in the modeling of SERS spectra for structures with gaps, with emphasis on understanding electromagnetic enhancement effects 1628

dx.doi.org/10.1021/jp207661y |J. Phys. Chem. C 2012, 116, 1627–1637

The Journal of Physical Chemistry C

FEATURE ARTICLE

Figure 1. Contours of EEM obtained from DDA calculations for: (a) isolated particles: a circular disk (the diameter and thickness are 55 and 7 nm, respectively), triangular prisms for two orientations (the edge lengths and heights are 60 and 12 nm, respectively), and a cylindrical rod with no end-caps (the diameter and length are 19 and 52 nm, respectively); (b) a dimer of triangular prisms with a 2 nm gap; (c) dimers of rods with 2 nm gaps: (1) headto-head, (2) perpendicular, and (3) side-by-side. Numbers in (b) and (c) (e.g., 10 ) refer to the increase in when going from the monomer structure (which is repeated for clarity) to the dimer. The color z-scale for EEM is the same for all figures, as is the white scale bar representing 40 nm. Also, the field is asymptotically normalized to unity, and all fields inside the particles are artificially set to 0.01 for visual clarity. Note that the grid spacing used was 1 nm for all calculations.

for surface-averaged enhancement factors. We begin by examining a variety of isolated and dimer structures, as well as the gap dependence of the dimer results, using the discrete dipole approximation (DDA), FEM, the finite-difference time-domain

(FDTD) method, as well as other theoretical techniques. We then describe and use Mie theory to study the dipole reradiation correction to the standard |E|4 enhancement factor for dimers of spheres. Finally, we describe the inclusion of spatially nonlocal 1629



FEATURE ARTICLE

dielectric effects in electromagnetic calculations using FDTD, and with it examine |E|4 enhancement factors for structures with small gaps, showing how nonlocal effects can significantly reduce electromagnetic enhancement factors.

II. ELECTROMAGNETIC THEORY PREDICTIONS FOR ISOLATED PARTICLES AND GAP STRUCTURES To illustrate the importance of small gaps in determining the electromagnetic contribution to the SERS enhancement, we begin by presenting DDA calculations of EEM (for zero Stokes shift) for a variety of isolated silver particles (all in water, with an index of refraction of 1.331), in which particle dimensions for a chosen shape have been adjusted to give a plasmon resonance close to 700 nm. We then examine dimers consisting of the same nanoparticles. Although calculations of this type have been done many times,4446 the constraint to a particular wavelength has not been presented, and in some cases this influences the conclusions as to which particle is best. Note that 700 nm is close to two very popular excitation wavelengths (633 and 785 nm), so this choice is particularly relevant to what is commonly used in experiments. The DDA method is an approximate electrodynamics method,47,48 which is relatively easy to use and gives semiquantitative estimates of local fields for arbitrary particle shapes (including gap structures). The approximation that EEM = |E|4 rather than EEM = |E(ω)|2|E(ω0 )|2 leads to enhancement factors that are systematically higher by a factor of about three compared to what would be obtained for a Stokes shift of 1000 cm1, but the results should otherwise be qualitatively the same. At the same time, the DDA method is incapable of determining realistic electric field values at the surface of a structure. Thus, an approximation is often made that takes advantage of the fact that the nanoparticle is described using a cubic array of elements, so it is easy to calculate the field at the center of grid points that are immediately outside of the particle grid.49 This means that the grid points at which fields are evaluated are always half of a grid spacing away from the “true” surface. The accuracy of this approach has been studied,49 and it typically leads to enhancement factors that are lower by a factor of about three compared to what might be expected if the field was evaluated right at the surface. These two factors therefore effectively cancel each other, so the resulting EEM may be relatively accurate. However, these issues are secondary for describing SERS anyway, as it is often only order-of-magnitude estimates of enhancement factors that are important. Figure 1 presents DDA-calculated field enhancement factors associated with the wavelength where the extinction maximizes for several particles, including: (a) isolated particles: a circular disk, triangular prisms (two orientations), and a rod (b) a dimer of the triangular prisms (c) dimers of rods: (1) head-to-head, (2) perpendicular, and (3) side-by-side. In each panel of Figure 1, the wavelength of maximum extinction and the choice of polarization direction are indicated, and below each panel, the average and peak values of EEM, which we label and EEMp, respectively, are listed; note that in some cases, the change in is indicated rather than the actual value of . Note also that the wavelength of maximum extinction is usually shifted (to the red) relative to the wavelength of maximum enhancement. However, this effect is

Figure 2. Average EEM as a function of gap size d (in nm) for the rod structures in Figure 1(c).

minor,46 so herein it is ignored, and all results refer simply to the wavelength of maximum extinction. The results for the isolated particles indicate that the highest average enhancement factors are obtained for triangles and rods, with values of around 106. These results are consistent with measured enhancement factors for triangular nanoparticles,50 assuming ECH ∼ 10. The EEMp values are about 108 for the same particles, and are associated with the sharp tips or edges. Not shown are results for spherical and coreshell particles, but we note that these have average and peak enhancement factors that are orders of magnitude below what are obtained for triangular prisms or rods. Factors that lead to larger enhancement factors for anisotropic particles compared to spheres and coreshell structures have been discussed elsewhere.46,49 Figure 1 shows that dimers of triangles and rods have significantly higher and EEMp values than for isolated particles, with surface average enhancements close to 108 for the head-to-head rod dimer. These larger average enhancement factors occur in spite of the fact that the enhancement is only large near the gap, so most of the exterior surface of the nanoparticles contributes relatively little. This is a consequence of the |E|4 enhancement factor, where only a few percent of the surface area of the structure can completely dominate the result.28 The gap in Figure 1 was constrained to be 2 nm. In Figure 2, we show the dependence of on gap size for several rod dimer structures, and it can be seen that for the head-to-head and perpendicular dimers, the largest enhancement factor is associated with the smallest gap. This provides significant motivation to develop SERS substrates with gaps of a few nanometers and below, including fused structures where analogous behavior is found, and recently there has been progress in this direction.28,39 There is, however, an exception to the trend observed for the gap structures. The side-by-side rod dimer in Figure 1(c) shows no significant hot spot between the particles, and the plasmon resonance is blue-shifted relative to the isolated particle resonance. This results from destructive interference between symmetric dipoles induced in each particle, and demonstrates that the mere presence of a gap does not guarantee a high enhancement. In general, however, this behavior is not likely to be common (e.g., in experiments, where such special symmetry is unlikely to occur). An interesting result shown in Figure 2 is that for the head-tohead dimer, a loglog plot of |E|4 (technically, ) vs gap is nearly linear with a slope of 1.8. This means that |E|4 ∼ 1/gap1.8 or roughly |E|2 ∼ 1/gap. Note that while this result refers to the 1630



FEATURE ARTICLE

Figure 3. FEM calculated (a) extinction cross sections for gold dimers with separations ranging from 0.5 to 10 nm; (b) maximum |E|2/|E0|2 values; and (c) average |E|2/|E0|2 values. The gaps used for these calculations are, in the top plots, 0.50 nm (black), 0.25 nm (red), 0.25 nm (blue), 0.50 nm (green), and in the bottom plots, 1 nm (black), 2 nm (red), 5 nm (blue) and 10 nm (green).

Figure 4. (top) FEM calculated |E|2/|E0|2 contour profiles at 1.7 eV (714 nm) and 2.1 eV (592 nm) for gold dimers that are 50 nm in diameter with a 0.25 nm separation. (bottom) Contour profiles showing the junction regions of the top figures (the top and bottom enlargements correspond to the left and right images, respectively). The internal fields were set to zero in the lower plots for clarity.

variation of the enhancement with gap at fixed wavelength, there is only a small shift in the plasmon resonance wavelength with gap for the structures considered in the plot, and thus the wavelengthoptimized peak enhancement-factor shows a similar functional dependence. This |E|2 ∼ 1/gap behavior is quite different from the |E|2 ∼ 1/gap2 result predicted from antenna theory (see again the Introduction). Recently, McMahon et al.40 have studied the gap dependence of the field enhancement factor using both FEM calculations and an analytical theory developed by García-Vidal and co-workers51 for studying millimeter-size slits in a PEC film at much longer wavelengths. They demonstrated that in small gaps,

field enhancements fundamentally arise from the confinement of waveguide modes, and even for PEC structures a result of the form |E|2 ∼ 1/gapp with p ∼ 1 is obtained. Plasmonic structures lead to somewhat higher powers than PEC ones, but the power-law dependence is still found for gaps down to ∼1 nm. It was also demonstrated that the power-law flattens (i.e., p f 0) and E f 0 for gaps below 0.5 nm. In this regime, however, other factors (nonlocal effects,52 orbital overlap effects, surface effects, chemical effects due to ligands,53etc.) become important, so the conventional application of Maxwell’s equations is probably not appropriate anyway. We will return to this point below. To study structures with gaps smaller than approximately 2 nm, including those that are slightly fused, it is necessary to go beyond the accuracy of grid-based methods, such as DDA or FDTD, and perform electrodynamics calculations using FEM, where analytical expressions are used to determine the electromagnetic fields. Figure 3 shows FEM calculations for the case of two parallel (relative to the incident light) 50 nm diameter gold cylinders with gaps of 0.50 and 0.25 nm, as well as fused structures in which the overlap (separation, technically) is 0.25, 0.50, 1, 2, 5, and 10 nm. Note that these results used the same FEM method and are similar to calculations previously presented for 90-nm cylinders,28 but in this case our focus is on providing a more detailed analysis about the behavior of the fields in small gaps. The results in Figure 3(a) show that the extinction spectrum varies dramatically with gap size, with the dipole (lowest energy) mode shifting to the red as the gap is decreased from 0.50 to 0.25 nm, which shifts back to the blue as the overlap is then increased from 0.25 to 10 nm. Also, there are multipole resonances that appear to the blue of the dipole resonance, whenever the energy of the latter occurs much below 2 eV (a wavelength of approximately 621 nm). Figure 3(b) shows the peak value of |E|2 in the gap for the same spacings as in Figure 3(a), while Figure 3(c) shows the corresponding average value. The results in Figures 3(b) and 3(c) show similar 1631


The Journal of Physical Chemistry C dependence on energy for a given gap size, but with the peak fields being about two orders of magnitude higher. However, the actual shapes of the curves in Figures 3(b) and 3(c) are only weakly related to those in Figure 3(a). For example, for a gap of 0.25 nm, the extinction cross section shows peaks at 1.8 and 2.2 eV, while there is only a single peak in Figures 3(b) and 3(c) (near 1.8 eV). As another example, for the fused structure with an overlap of 1 nm, the extinction spectrum shows peaks at 1.5, 1.8, and 2.1 eV, while in Figures 3(b) and 3(c), there are only two peaks (near 1.6 and 2.1 eV). To help understand this behavior, in Figure 4 we show |E|2 contours for the structure with a gap of 0.25 nm close to the extinction maxima at energies of 1.7 eV (714 nm) and 2.1 eV (592 nm). A magnified view of the region around the junction is presented at the bottom, and it can be seen that the 1.7 eV peak has a single region of high intensity, as might be expected for a dipole mode in which one-half of a wavelength (i.e., one lobe of a propagating plasmon) is confined in the gap. However, the 2.1 eV contours in Figure 4 show three regions of high intensity. This is the expected behavior for a plasmon multipole in which three-halves of a wavelength of the propagating plasmon is confined. Therefore, considering half-wavelength resonances, we can assign the 1.7 eV peak to the n = 1 resonance and that at 2.1 eV to n = 3. This means that somewhere near 1.9 eV should be a quadrupole (n = 2) resonance. However, this would be an optically forbidden mode for the chosen polarization, as the induced field has quadrupolar character (and would thus not appear in the extinction spectrum). Also, the hot spot for n = 2 occurs off-axis, as noted long ago for the quadrupole mode in spherical particles.46 Figure 4(b) shows that for 0.25 nm separation, |E|2 has only a single peak at 1.8 eV, which indicates that the largest enhancements are associated with energies close to the dipole resonance. This is understandable, since there are no interferences when |E|2 is averaged over the surface of the nanoparticles; thus, unlike the extinction spectrum where the n = 2 resonance is forbidden, all multipoles contribute to |E|2, thereby (partially) washing out any structure. Although this discussion has focused on symmetry effects and the connection between extinction spectra and near-field structure, there are also issues related to the comparison of absorption versus scattering spectra. In particular, the relative contributions of the various excitations to these properties can be quite different. For example, the quadrupolar mode makes a much stronger contribution to absorption than scattering, whereas the dipolar mode can contribute strongly to both. This is because the absorption cross section arises from the incoherent contributions of absorbed light throughout the nanoparticle, while scattering arises from the coherent superposition of scattered and incident light. Thus, scattering is sensitive to interference effects, while absorption is not. The results for the fused structures are largely analogous to those discussed above and with roughly comparable peak enhancements. In these cases, the highest enhancements are again associated with the dipole mode, but additional peaks also occur. However, these peaks are not easily connected to the energies of specific multipole resonances, and a full analysis and discussion is beyond the scope of this work.

III. DIPOLE RERADIATION EFFECTS All of the enhancements calculated to this point have been based on the PW formula, eq 1. This is in fact an approximation

FEATURE ARTICLE

Figure 5. Coordinates used to study DR effects for a sphere dimer, showing both molecule and detector locations, as described in the text. The polarization vector is taken to be aligned with the interparticle axis, and the wavevector is perpendicular to the plot.

to a more accurate expression, which we call the dipole reradiation (DR) formula Gðθs , ϕs Þ ¼

jFR ðθs , ϕs Þj2 jFR, 0 ðθs , ϕs Þj2

ð2Þ

in which the dipole induced in a molecule by the incident field is imagined to emit radiation at the Raman frequency ω0 , which then scatters off of the nanoparticle to generate a far-field outgoing spherical wave of the form FR(θs,ϕs)eikrs/rs, the latter which depends on the scattering angles θs and ϕs, the wavevector magnitude k, the radial coordinate rs, and the scattering amplitude FR. The subscript 0 in eq 2 indicates that the enhancement is given by the ratio of the square of the amplitude of FR in the presence of the nanoparticle to that in the absence of it. Although eq 2 seems very different from eq 1, Kerker demonstrated23 that in the quasistatic limit (i.e., when the particle is much smaller than the wavelength of light, usually around 50 nm) eq 2 reduces to eq 1 as a result of the reciprocity theorem. Note that the theorem itself is exact, but Kerker’s treatment was quasistatic (and only applies in that limit). The theorem states (in this application) that the field of the dipole emitter, when evaluated at the center of the particle, is the same as the field of a dipole at the center of a particle that is evaluated at the position of the dipole emitter, and the latter field is just that obtained from the PW approximation. Ausman and Schatz54 have compared results obtained from eq 2 to those from eq 1 for a number of spherical nanoparticle structures and scattering geometries, by evaluating dipole scattering from the nanoparticle exactly using an extension of Mie theory. Here we present similar results for dimers of 50 nm diameter silver nanoparticles with gaps of 2, 5, and 10 nm. The coordinates for this calculation are defined in Figure 5, and it is imagined that there is only one molecule present at 0.25 nm from the surface of one of the spheres. The molecule is at a location that is denoted on-axis, which is along the axis between them (also the polarization direction). In the other set of calculations that is presented, denoted off-axis, the molecule is taken to be rotated by 30° away from the interparticle axis, as also indicated in Figure 5. Figure 6 presents PW and DR enhancement estimates from these calculations. The top panels show the PW |E|4 profiles on a log scale for both the on- and off-axis cases, while the middle and 1632



FEATURE ARTICLE

Figure 6. (a) and (b) show PW SERS excitation profiles (denoted |Eloc|4) for varying gap sizes between two 50 nm silver spheres. The molecule is located 0.25 nm from the surface of one of the spheres along this direction in (a) and rotated 30° about the sphere from the incident polarization in (b). (c), (d), (e), and (f) show a comparison between the PW (also denoted |Eloc|4) and DR enhancement factors for two different detector locations, ϕs = 90° (red dashed-line) and ϕs = 45° (green dashed-dotted line), as well as two different molecule locations; (c) and (e) correspond to the location in (a), and (d) and (f) correspond to the location in (b). The gap size for both (c) and (d) is 2 nm, and for (e) and (f) it is 10 nm. In all cases, θs = 90°.

bottom panels show PW and DR enhancements on a linear scale for 2 (middle) and 10 nm (bottom) gaps. The DR enhancements depend on the angles θs, ϕs, and we have chosen these to be either 90°, 90° or 90°, 45° to demonstrate the sensitivity of the results to the choice of angle. Note that 90°, 90° corresponds to a scattering direction that is perpendicular both to the wavevector and (on-axis) polarization direction, and thus the two particles scatter symmetrically. The top panels of Figure 6 illustrate typical behavior, in which there is a strong dipole resonance near 390 nm for a 10 nm gap, which red-shifts as the spacing is decreased, as well as a weaker quadrupole resonance at 360 nm that only slightly red-shifts but becomes much stronger as the gap decreases. The results also show that the dipole |E|4 increases rapidly with decreasing gap for the on-axis case, exceeding 1010 for a 2 nm gap. However, the off-axis enhancement decreases with decreasing gap, approaching 106 also for the 2 nm gap. These results reflect the fact that the region of high intensity shrinks as the gap distance is decreased. The key result in the middle and bottom panels of Figure 6 is that there is relatively little difference between the PW and DR results, especially when the enhancements are high. The most

significant differences appear in the middle-right panel, in which enhancements for the off-axis case and 2 nm gap are seen to be small. Furthermore, it can be seen that the DR results can be either higher or lower than the PW results, especially for the quadrupole resonance for 90°, 45°. This arises because the DR intensity in this case involves emission of a dipole field by the offaxis dipole, leading to the excitation of a different quadrupole mode than is excited by either the PW or on-axis fields. Differences between the PW and DR results have been described elsewhere for other particle structures and scattering angles.54 A general rule is that the differences are greater for larger particles, where high-order multipole excitations are more important. Also, any asymmetry in scattering by the two particles can lead to interference effects, such as those that arise in the present case for 90°, 45°, which can also give important differences. In the end though, these effects are not particularly important for the gap structures in Figure 6, as a gap between spheres only gives rise to a single hot spot, and the same resonances are excited no matter what the excitation source. Thus, it seems that dipole reradiation effects are of secondary 1633



FEATURE ARTICLE

Figure 7. FDTD calculated extinction cross sections for gold dimers parallel to the incident field (100 nm cylinders) with gaps of 5, 2, 1, and 0.5 nm, (a)(d), respectively, showing local (red broken-lines) and nonlocal (blue solid-lines) calculations.

importance for the consideration of enhancements for simple gap structures.

IV. NONLOCAL EFFECTS Up to this point, all electrodynamics calculations have been done using a dielectric function for the metal ε that has been taken to be a function only of the frequency ω but not of the wavevector k. This is often referred to as a “local” dielectric function, as it leads to an induced polarization at any point that depends on the field only at that point. This is an approximation, as the quantum mechanical description of any material leads to an induced polarization at any point that can arise from applying a field away from that point. In this section, we examine some of the effects that result from including for a nonlocal response. It is easiest to compare local and nonlocal theories for metals by comparing the dielectric functions based on the Drude model. In the local theory, this model assumes that the electrons move as free particles with forces that are determined by the applied electric field E and are damped due to electronelectron and other scattering events m

dv ¼ eE m γv dt

ð3Þ

where v is the velocity, γ is the inverse scattering time, e is the charge of an electron, and m is the effective electron mass. If the quantities are time harmonic with an oscillation eiωt, then the induced polarization of the electrons is proportional to E, with that proportionality being the Drude dielectric function εðωÞ ¼ 1

ω2p ωðω þ i γÞ

where ωp is the bulk plasmon frequency given by ωp = ((n0e2)/(ε0m))1/2, where n0 is the equilibrium electron density and ε0 is the permittivity of free space. When one takes into account that electrons in reality have a nonuniform density, then eq 3 needs to be supplemented with a pressure gradient term m

dv m β2 ¼ eE m γv ∇n1 dt n0

ð5Þ

where n1 is the linear perturbation of the electron density that arises from the applied field and β is a parameter proportional to the Fermi velocity of an electron gas. To go further with this, the continuity equation can be used to relate the electron density to its velocity ∂n1 þ n0 ∇ 3 v ¼ 0 ∂t

ð6Þ

Then, for a time-harmonic field, this expression can be used to write 3n1 as n0 ð7Þ ∇n1 ¼ ∇∇ 3 v iω Often, the gradient of the divergence of the velocity is approximated by the Laplacian, 33 3 v = 32 3 v.55 However, some issues associated with making this approximation have recently been discussed.56 If the Laplacian form of eq 7 is inserted into eq 5 and then solved in Fourier space to determine the induced polarization, the so-called hydrodynamic Drude dielectric function results εðω, kÞ ¼ 1

ð4Þ 1634

ω2p ωðω þ i γÞ β2 k2

ð8Þ



FEATURE ARTICLE

Figure 8. FDTD calculated |E|2/|E0|2 values for gold dimers parallel to the incident field (100 nm cylinders) with gaps of 5, 2, 1, and 0.5 nm, (a)(d), respectively, showing local (red broken-lines) and nonlocal (blue solid-lines) calculations.

Figure 9. FDTD calculated |E|2/|E0|2 contour profiles for a gold dimer (100 nm cylinders) with a 0.5 nm gap, showing (left) local and (right) nonlocal calculations.

in which there is an explicit dependence of the dielectric function on the wavevector magnitude k. Thus, in real space (the spatial Fourier-transform of this expression), the dielectric function is nonlocal, and the parameter β controls the extent of the nonlocality (local behavior results for β f 0). See ref 57 for a more complete discussion of the hydrodynamic Drude model, including its derivation and implementation into FDTD calculations (see also below). Note that neither eq 4 nor eq 8 includes for conservation of the electron number in the relaxation of plasmons to electron hole pair excitations, which is an effect that is only included in more sophisticated models.58 Recently, McMahon et al. 52 have incorporated the nonlocal dielectric function in eq 8 into the FDTD method through the use of auxiliary equations and have used this to study two coupled cylinders with a gap. In Figure 7, we show new results using the same method, the extinction spectra of 100 nm diameter gold cylindrical nanoparticle dimers with gaps from 0.5 to 5 nm, and in Figure 8 we show the corresponding |E| 2 enhancement factors. The figures show

local and nonlocal results, where in all cases the Drude model is used to represent the free-electron part of the dielectric function, but the overall local dielectric function (including interband terms, etc.) is derived from experiment. The results show that including nonlocal effects leads to blueshifted plasmons and reduced field enhancements. For 0.5 nm separation, the peak |E|2 value is reduced by a factor of ∼4, leading to an order of magnitude reduction in the SERS enhancement factor |E|4. This suggests that the very high peak field enhancements that we have noticed earlier for subnanometer gaps will be somewhat reduced by nonlocal effects. The results are otherwise qualitatively the same, as well as the field profiles, contours of which are presented in Figure 9. Of course, nonlocal effects are only one of many factors which will change local fields in the vicinity of a gap compared to conventional electrodynamics, as summarized in the next section.

V. DISCUSSION This article provided an overview of the properties of electromagnetic fields that are present in the gaps between two or more plasmonic nanoparticles in the presence of optical irradiation. Such fields are often enhanced by many orders of magnitude, and as such they can play a dominant effect in SERS and many other optical processes. However, the lack of experimental control over gaps between nanoparticles at the 1 nm level (and smaller) has made it hard to provide an experimental quantitative understanding of the variation of the local fields with nanoparticle and gap structure, and therefore the contribution of such features to spectroscopic observables. Thus, much of our current understanding comes from theoretical and numerical studies, such as those that were discussed herein. 1635


The Journal of Physical Chemistry C We presented results obtained from computational electrodynamics methods, as well as analytical theory, to study the variation in local fields with nanoparticle structure and gap size, the wavelength dependence of the enhancements, and their lack of correlation with optical spectra due to multipolar interference effects. We demonstrated that |E|2 varies inversely with gap raised to a power that is typically close to unity, except in the limit of extremely small gaps where more complex behavior occurs and the singularity disappears. Our studies of DR effects demonstrated that deviations from the PW approximation are often relatively minor for gap structures, which means its application to SERS enhancements can be used with confidence, as long as classical electrodynamics methods are valid. Our study of nonlocal effects demonstrated that they do not change the qualitative behavior of enhanced fields near gap structures, but can reduce the maximum SERS enhancement that is achievable by roughly an order of magnitude for very small (∼0.5 nm) gaps. There are additional factors that we did not describe but which can also influence the behavior of local fields in the gaps between plasmonic nanoparticles. The metal dielectric functions used in our calculations were taken to be independent of position, but it is known that they are different close to the surfaces of nanoparticles, in part because electron densities close to an interface have complex behavior that includes electron spill out, decoupling of interband and intraband transitions, and ultimately the smooth decrease of the electron density to zero as one exits the interface. Such factors will likely reduce the size of electromagnetic enhancements in gaps smaller than we have described. Nonetheless, in spite of these factors, large electromagnetic enhancements have already been observed in SERS and other experiments (with enhancement factors of greater than 10 8 now common), and these results agree with the present theoretical calculations to within an order of magnitude.

’ BIOGRAPHIES

Jeffrey M. McMahon received a B.S. in Chemistry from Western Washington University in 2005. In 2010, he received his Ph.D. from Northwestern University under the supervision of George C. Schatz. He is currently a postdoctoral research associate in the Department of Physics at the University of Illinois at Urbana—Champaign. His current interests involve quantum simulations of physically and chemically important systems and fundamental optical properties of small nanostructures.

FEATURE ARTICLE

Shuzhou Li received his B.S. from Nankai University in China, M.S. from Beijing University in China, and Ph.D from University of Wisconsin, Madison, USA. After postdoctoral study at Northwestern University, USA, he joined the school of Materials Science and Engineering in Nanyang Technological University, Singapore. His research interests include surface-enhanced Raman spectroscopy, self-assembly, and plasmonics.

Logan Ausman received his B.S. in chemistry from the University of Wisconsin—Eau Claire in 2004 and his Ph.D. in theoretical chemistry from Northwestern University in 2010. His Ph.D. research was conducted under the direction of Professor George Schatz. After completing his degree he became a Research Staff Member at IDA in Alexandria, Virginia. His research interests include the induction of localized surface plasmon resonances in nanoparticles by an adsorbed dipole emitter.

George C. Schatz has a B.S. from Clarkson University and Ph.D. from Caltech. He is currently Morrison Professor of 1636


The Journal of Physical Chemistry C Chemistry and Professor of Chemical and Biological Engineering at Northwestern University. His research broadly covers nanomaterials and biomaterials.

’ ACKNOWLEDGMENT We acknowledge DOE Basic Energy Sciences grant DESC0004752 for generous support. ’ REFERENCES (1) Jeanmaire, D. L.; Van Duyne, R. P. J. Electroanal. Chem. 1977, 84, 1–20. (2) Fleischmann, M.; Hendra, P. J.; McQuillian, A. J. Chem. Phys. Lett. 1974, 26, 163–166. (3) Albrecht, M. G.; Creighton, J. A. J. Am. Chem. Soc. 1977, 99, 5215–5217. (4) McCreery, R. L. Raman Spectroscopy for Chemical Analysis; Wiley-Interscience: New York, 2000; Vol. 157. (5) Golightly, R. S.; Doering, W. E.; Natan, M. J. ACS Nano 2009, 3, 2859–2869. (6) Jain, P. K.; Huang, X. H.; El-Sayed, I. H.; El-Sayed, M. A. Acc. Chem. Res. 2008, 41, 1578–1586. (7) Camden, J. P.; Dieringer, J. A.; Zhao, J.; Van Duyne, R. P. Acc. Chem. Res. 2008, 41, 1653–1661. (8) Bell, S. E. J.; Sirimuthu, N. M. S. Chem. Soc. Rev. 2008, 37, 1012–1024. (9) Porter, M. D.; Lipert, R. J.; Siperko, L. M.; Wang, G.; Narayanana, R. Chem. Soc. Rev. 2008, 37, 1001–1011. (10) Wu, D. Y.; Li, J. F.; Ren, B.; Tian, Z. Q. Chem. Soc. Rev. 2008, 37, 1025–1041. (11) Kneipp, J.; Kneipp, H.; Kneipp, K. Chem. Soc. Rev. 2008, 37, 1052–1060. (12) McNay, G.; Eustace, D.; Smith, W. E.; Faulds, K.; Graham, D. Appl. Spectrosc. 2011, 65, 825–837. (13) Henry, A. I.; Bingham, J. M.; Ringe, E.; Marks, L. D.; Schatz, G. C.; Van Duyne, R. P. J. Phys. Chem. C 2011, 115, 9291–9305. (14) Metiu, H.; Das, P. Annu. Rev. Phys. Chem. 1984, 35, 507–536. (15) Schatz, G. C. Acc. Chem. Res. 1984, 17, 370–376. (16) Moskovits, M. Rev. Mod. Phys. 1985, 57, 783–826. (17) Morton, S. M.; Silverstein, D. W.; Jensen, L. Chem. Rev. 2011, 111, 3962–3994. (18) Lombardi, J. R.; Birke, R. L. Acc. Chem. Res. 2009, 42, 734–742. (19) Zhao, J.; Pinchuk, A. O.; McMahon, J. M.; Li, S.; Ausman, L. K.; Atkinson, A. L.; Schatz, G. C. Acc. Chem. Res. 2008, 41, 1710–1720. (20) Jensen, L.; Aikens, C. M.; Schatz, G. C. Chem. Soc. Rev. 2008, 37, 1061–1073. (21) Tong, L. M.; Zhu, T.; Liu, Z. F. Chem. Soc. Rev. 2011, 40, 1296–1304. (22) Halas, N. J.; Lal, S.; Chang, W. S.; Link, S.; Nordlander, P. Chem. Rev. 2011, 111, 3913–3961. (23) Kerker, M.; Wang, D.-S.; Chew, H. Appl. Opt. 1980, 19, 4159–4174. (24) Yang, W. H.; Schatz, G. C. J. Chem. Phys. 1992, 97, 3831–3845. (25) Lombardi, J. R.; Birke, R. L. J. Phys. Chem. C 2008, 112, 5605–5617. (26) Morton, S. M.; Jensen, L. J. Am. Chem. Soc. 2009, 131, 4090–4098. (27) Arenas, J. F.; Tocon, I. L.; Otero, J. C.; Marcos, J. I. J. Phys. Chem. 1996, 100, 9254–9261. (28) McMahon, J. M.; Henry, A. I.; Wustholz, K. L.; Natan, M. J.; Freeman, R. G.; Van Duyne, R. P.; Schatz, G. C. Anal. Bioanal. Chem. 2009, 394, 1819–1825. (29) Aravind, P. K.; Nitzan, A.; Metiu, H. Surf. Sci. 1981, 110, 189–204. (30) Liver, N.; Nitzan, A.; Gersten, J. I. Chem. Phys. Lett. 1984, 111, 449–454.

FEATURE ARTICLE

(31) Fang, Y.; Seong, N.-H.; Dlott, D. D. Science 2008, 321, 388–392. (32) Li, S.; Pedano, M. L.; Chang, S. H.; Mirkin, C. A.; Schatz, G. C. Nano Lett. 2010, 10, 1722–1727. (33) Kneipp, K.; Wang, W.; Kneipp, H.; Perelman, L. T.; Itzkan, I.; Dasari, R. R.; Feld, M. S. Phys. Rev. Lett. 1997, 78, 1667–1670. (34) Nie, S. M.; Emory, S. R. Science 1997, 275, 1102–1106. (35) Michaels, A. M.; Nirmal, M.; Brus, L. E. J. Am. Chem. Soc. 1999, 121, 9932–9939. (36) Camden, J. P.; Dieringer, J. A.; Wang, Y. M.; Masiello, D. J.; Marks, L. D.; Schatz, G. C.; Van Duyne, R. P. J. Am. Chem. Soc. 2008, 130, 12616–12617. (37) Dieringer, J. A.; Wustholz, K. L.; Masiello, D. J.; Camden, J. P.; Kleinman, S. L.; Schatz, G. C.; Van Duyne, R. P. J. Am. Chem. Soc. 2009, 131, 849–854. (38) Kleinman, S. L.; Ringe, E.; Valley, N.; Wustholz, K. L.; Phillips, E.; Scheidt, K. A.; Schatz, G. C.; Van Duyne, R. P. J. Am. Chem. Soc. 2011, 133, 4115–4122. (39) Wustholz, K. L.; Henry, A.-I.; McMahon, J. M.; Freeman, R. G.; Valley, N.; Piotti, M. E.; Natan, M. J.; Schatz, G. C.; Van Duyne, R. P. J. Am. Chem. Soc. 2010, 132, 10903–10910. (40) McMahon, J. M.; Gray, S. K.; Schatz, G. C. Phys. Rev. B 2011, 83, 115428. (41) Qin, L. D.; Zou, S. L.; Xue, C.; Atkinson, A.; Schatz, G. C.; Mirkin, C. A. Proc. Natl. Acad. Sci. U.S.A. 2006, 103, 13300–13303. (42) Pedano, M. L.; Li, S.; Schatz, G. C.; Mirkin, C. A. Angew. Chem. 2010, 49, 78–82. (43) Barnard, E. S.; White, J. S.; Chandran, A.; Brongersma, M. L. Opt. Express 2008, 16, 16529–16537. (44) Zhao, L. L.; Zou, S. L.; Hao, E.; Schatz, G. C. Electrodynamics in Computational Chemistry. In Theory and applications of computational chemistry: the first 40 years; Dykstra, C. E., Frenking, G., Kim, K. S., Scuseria, G., Eds.; Elsevier: Amsterdam, 2005; pp 4766. (45) Zhao, J.; Pinchuk, A. O.; McMahon, J. M.; Li, S.; Ausman, L. K.; Atkinson, A. L.; Schatz, G. C. Acc. Chem. Res. 2008, 41, 1710–1720. (46) Kelly, K. L.; Coronado, E.; Zhao, L. L.; Schatz, G. C. J. Phys. Chem. B 2003, 107, 668–677. (47) Draine, B. T.; Flatau, P. J. J. Opt. Soc. A 1994, 11, 1491–1499. (48) Draine, B. T.; Flatau, P. J. User Guide for the Discrete Dipole Approximation Code DDSCAT 6.1; http://arxiv.org/abs/astro-ph/ 0409262v2, 2004. (49) Hao, E.; Schatz, G. C. J. Chem. Phys. 2004, 120, 357–366. (50) McFarland, A. D.; Young, M. A.; Dieringer, J. A.; Van Duyne, R. P. J. Phys. Chem. B 2005, 109, 11279–11285. (51) Bravo-Abad, J.; Martin-Moreno, L.; García-Vidal, F. J. Phys. Rev. E 2004, 69, 026601. (52) McMahon, J. M.; Gray, S. K.; Schatz, G. C. Phys. Rev. Lett. 2009, 103, 097403. (53) Peng, S.; McMahon, J. M.; Schatz, G. C.; Gray, S. K.; Sun, Y. G. Proc. Natl. Acad. Sci. U.S.A. 2010, 107, 14530–14534. (54) Ausman, L. K.; Schatz, G. C. J. Chem. Phys. 2009, 131, 084708. (55) Boardman, A. D. Hydrodynamic theory of plasmon-polaritons on plane surfaces. In Electromagnetic Surface Modes; Boardman, A. D., Ed.; Wiley: New York, 1982; p 176. (56) Raza, S.; Toscano, G.; Jauho, A. P.; Wubs, M.; Mortensen, N. A. Phys. Rev. B 2011, 84, 121412. (57) McMahon, J. M. Topics in Theoretical and Computational Nanoscience: From controlling light at the nanoscale to calculating quantum effects with classical electrodynamics; Springer: New York, 2011. (58) García de Abajo, F. J. Rev. Mod. Phys. 2010, 82, 209–275.

1637


Modeling the Effect of Small Gaps in Surface-Enhanced Raman

Recommend Documents