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J. Phys. Chem. C 2009, 113, 5912–5919
Intrinsic Limitations on the |E|4 Dependence of the Enhancement Factor for Surface-Enhanced Raman Scattering Stefan Franzen Department of Chemistry, North Carolina State UniVersity, Raleigh, North Carolina 27695 ReceiVed: September 11, 2008; ReVised Manuscript ReceiVed: December 21, 2008
Analysis of the bandwidth of the Clausius-Mosotti local field factor challenges the standard assumption that both incident and scattered fields are equally enhanced in surface-enhanced Raman scattering (SERS). The most common geometry for observation of SERS is on a nanoparticle or nanostructure where the localized surface plasmon resonance (LSPR) field enhancement arises from the electromagnetic environment produced by scattering off the conductor surface. Consequently, the electric field enhancement experienced by an adsorbate on the metal surface is a function of the magnitude of the transition dipole moment of the nanoparticle or nanostructure. Even in the treatment that considers the conducting nanostructure and molecule interactions as contributions to a collective scattering process, analytical expressions based on the Drude free-electron model reveal the importance of the bandwidth of the local field factor as a limitation on the SERS enhancement. These model calculations based on the Drude model are confirmed herein by explicit calculation using the dielectric functions for Au and Ag. The enhancement bandwidth depends on the ratio of the plasma frequency, ωp and the damping, Γ, such that the greater the enhancement ratio, ωp/Γ, the narrower the enhancement bandwidth. The relationship of the Raman shift to the enhancement bandwidth places severe constraints on the theoretical enhancement possible by the electromagnetic mechanism. Introduction The electromagnetic mechanism of surface-enhanced Raman spectroscopy (SERS) is a phenomenon that depends on the local field near the surface of a conductor.1,2 For SERS enhancement to occur, the excitation frequency must be near the plasma frequency, ωp. Surface-enhanced spectroscopy can occur when a molecule is bound to the surface of a substrate that supports a surface plasmon polariton (SPP) or near a surface that has a screened bulk plasmon polariton (SBPP).3 SPPs are observed directly on flat surfaces for an appropriate coupling geometry of the exciting light. However, electromagnetic calculations show that the resonance enhancement of molecules on flat surfaces is small.4 Large enhancement may also arise from optical excitation of a LSPR observed in suspensions of nanoparticles (colloids) in an insulating medium. The LSPR in nanoparticles is the analogue of the SBPP, which is observed as a decrease in reflectance at the plasma frequency in a conducting thin film, whose thickness is less than the skin depth.5,6 The interpretation of the optical properties of nanoparticle suspensions in heterogeneous media requires application of Maxwell-Garnett theory used to determine the average dielectric function.7 Rough surfaces with protrusions such as hemispheres, ellipsoids, or other shapes produce significantly larger local fields than that of flat surfaces.8-10 There has been great interest in identifying structures that may give rise to large local fields and thereby promote large SERS enhancements. Most treatments of the SERS effect favor an electromagnetic as opposed to a chemical enhancement mechanism.1,11 The electromagnetic mechanism arises from the dielectric response of the conductor that results in field amplification near the surface. The scattering intensity Iadsorbate ) σREi2, where σR is the Raman scattering cross-section of the adsorbate molecule. The time-dependent incident and scattered fields are Ei and Es, respectively. For compactness of presentation, the frequency
and spatial dependence according to Ei ) g(ωi)Ei0 exp(i[ωit kix]) will be assumed but not written explicitly in the following. The enhancement due to the local field effect is Ei ) g(ωi)Ei0, where g(ωi) is the local field factor obtained from the Clausius-Mosotti relation. According to the standard assumption, the electromagnetic effect in Raman spectroscopy depends on the fourth power of the electric field since both Ei and Es are enhanced. These are sometimes referred to as a first and second enhancement, respectively. While there is recognition that the enhancements require explanation there is no unified treatment of these two enhancements. The localized intensification of the incident plane wave field arises from scattering of the conducting particle or surface with intensity Iparticle ) σscaEi02. An appropriate geometry leads to a significant enhancement in the field, Ei, relative to the incident field, Ei0, in the vicinity of molecule on the surface. This first enhancement that leads to a Ei02 dependence of the scattering intensity is generally accepted. The maximum enhancement of incident radiation occurs on resonance with the screened plasma frequency of the conducting particle or surface. At the frequency of the scattered field, ωs, the stored energy is significantly less than the usual estimate based on the assumption that Es ) g(ωi)Ei0. Because g(ω) is a sharply peaked function with an intrinsic bandwidth, the local field in the conductor at Es ) g(ωs)Ei0 is significantly reduced relative to the value obtained at incident frequency, g(ωs) , g(ωi). On the other hand, the fourth power dependence of the SERS enhancement in almost every treatment assumes that g(ωs) ∼ g(ωi), with no consideration of the effect of the bandwidth. This approximation is valid only if the bandwidth of the local field factor, g(ω), is greater than the Raman shift ∆ω ) ωi ωs.12 This assumption will be discussed in the following with a theoretical treatment for a Drude free electron model, as well as explicit calculation for the two most important metals in the SERS field, Ag and Au.
10.1021/jp808107h CCC: $40.75 2009 American Chemical Society Published on Web 03/23/2009
Surface-Enhanced Raman Scattering Since the first experimental observation of SERS,8,13 the reported enhancement factors have continued to increase as new nanoparticle geometries were investigated.14-17 Since single molecule Raman scattering on a single particle14,18 is difficult to explain theoretically,19-22 there is a growing consensus that large SERS enhancement requires more than one particle in close proximity.19-21,23-27 Moreover, large Raman enhancements in a dimer or oligomer nanoparticle geometry can be explained by combining surface enhancement and resonance Raman spectroscopies.20,21,28,29 While laser excitation wavelengths are not always at the peak of the absorption spectrum of the adsorbate, they are usually within a range that is known as preresonant for most dyes used in SERS experiments. 22 Despite the recognition that molecular resonance may play a role, the requirement for large enhancement is seen mainly as a problem in geometry or plasmonic structure. Given the complex dynamic of surface topography and molecular motions, a SERS uncertainty principle that places limits on the simultaneous determination of spatial resolution and enhancement factor has been proposed.30 Although these important aspects that are requisite for reaching the extremely high enhancement factors needed for single molecule SERS have been considered, the relationship between the peak enhancement and the spectral bandwidth of the SERS effect has received little attention. It is generally accepted that SERS is largest near rough surfaces or nanoparticles when d , λ (d is the nanoparticle radius, and λ is the wavelength of the exciting light). The requirement for roughness arises from the requirement for spatial coupling of radiation into a surface in an appropriate geometry to drive the plasmon. Coupling refers to the requirement that the wave vector, ki, of the exciting plane wave be matched both in solution and in the conductor. On a planar surface, wave vector matching to create a SPP requires total internal reflection using the appropriate geometry. On the other hand, small spheres that have localized plasmons are often treated as analogous to transition dipoles in the optical absorption by molecules. For the condition d , λ, wave vector matching is less restrictive than on a surface, and electrostatic treatments are also valid. Analogous enhancement effects are observed for adsorbates on microspheres (d > λ) due to the evanescant field.31 The perpendicularly polarized SBPP in conducting metal oxides is the thin film analogue of a LSPR,32 in which the dipole is created by charge separation across the thin film.33 Both the SBPP (thin film) and LSPR (nanoparticle structure) can be approximated as dipolar plasmons that are distinct from surface plasmon polaritons (SPPs). One distinguishing characteristic is that they both contain loss due to absorption. The comparison of ITO with Au and Ag gives insight into the role of loss mechanisms on the dielectric response relevant to SPR and SERS.34 The wave vector consists of a real part, k1(ω), which is the in-phase or dispersion term that gives rise to SPPs and an outof-phase or imaginary contribution of the wave vector, k2(ω), which is the absorption coefficient. By analogy with the perpendicularly polarized SBPP of a conducting metal oxide thin film, the LSPR of a collection of nanoparticles is an absorption band. The pink color of suspensions of Au nanoparticles and yellowish luster of Au metal are both manifestations of plasma absorption, which is attributable to k2(ω). The plasma absorption is an intense broad absorption band that arises from a collective oscillation of the conduction electrons. Au nanoparticle absorptions are very broad because of the very short lifetime of the excited state. Ag nanoparticle absorption bands are narrower than Au35 but are qualitatively similar in that both noble metals have admixtures of band-to-band transition with
J. Phys. Chem. C, Vol. 113, No. 15, 2009 5913 the collective oscillations of conduction electrons. While the absorption bands arise due to the imaginary part of the dielectric response the SERS effect depends on an in-phase oscillation for local field amplification. In fact, the imaginary and real contributions to the wavevector give opposite effects. The absorption by nanoparticles and nanostructures decreases scattering by an absorbate. It is the in-phase or dispersive response that gives rise to a contribution to the molecular polarizability of the nanoparticle-adsorbate system and hence a contribution to Raman scattering.36 The resonant scattering of light from a collection of nanoparticles gives rise to enhancement of scattering of molecular adsorbates. The leading term in scattering from a conducting sphere is the dipolar term, and therefore this is often used to model the plasmonic absorption and scattering.16,37 Although the relationship between the bandwidth of resonance enhancement and the plasma absorption has recently been addressed experimentally using Au arrays,38 most of the computational effort has been expended calculating the spatial field dependence of the SERS effect, rather than the spectral bandwidth of the enhancement profile.39 The specific nature of field enhancement due to photons scattered from the adsorbate, which clearly do not impinge on the conductor as plane waves, is a more complex problem in both the spatial or frequency components. The spatial aspect of molecular interactions with the conductor is a problem in electrodynamics that is beyond the scope of this paper. Herein, I address the frequency dependence of the electric field enhancement in SERS. Theory. The theory of the local field relies on a model for the dielectric function of the conductor. The theory will first be explained using an analytical expression derived from the Drude free electron model and then using experimental data obtained for Au and Ag. The free electron or Drude model explains plasmonic phenomena in conducting metal oxides.33 Comparison with Au and Ag using experimental data provides insight into the role played by band-to-band transitions in the noble metal plasmon absorption bands.34 It has been shown that the free electron model is a reasonable approximation for Ag, but significant deviations are observed for Au.34 The deviations from the free electron model that arise from interfering band-to-band transitions are not mechanisms that provide additional enhancement but rather lead to greater damping and losses. As a result, the energy stored in the plasma absorption band of Au nanoparticles is rapidly lost as heat in a few picoseconds40-43 The large loss in Au and somewhat smaller loss in Ag nanoparticles both arise from the imaginary part of the dielectric function. The relationship between the Drude model and such absorptions is described in the Supporting Information. When only free carriers are present, one can express the dielectric function using the Drude model for conduction given in eq 1.
εc(ω) ) ε∞ -
ω2p ω2 + iωΓ
(1)
where ωp is the plasma frequency, Γ is the damping, and ε∞ is the static dielectric constant. Although any conducting material inherently has contributions from both the real (in-phase) and imaginary (out-of-phase) parts of the dielectric function, the analysis of SERS has focused on the real part. The enhancement is usually taken to arise from the local field factor g(ω), which is given by the generalized Clausius-Mosotti relation,44,45
5914 J. Phys. Chem. C, Vol. 113, No. 15, 2009
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εc(ω) - εs NR ) g(ω) ) ε0 ζ(εc(ω) + (1/ζ - 1)εs)
(2)
where ζ is the depolarization factor, which is a pure number (0 e ζ e 1) that depends only on the shape of the feature that gives rise to the polarization, εc(ω) is the frequency-dependent dielectric function of the conductor,6 and εs is the dielectric constant of the substrate or solvent. The function g(ω) is a local field factor that can apply for both dielectrics and conductors. The depolarization, ζ, is 1/3 for a sphere and smaller at the vertex of an ellipse, 1/2 for a cylinder, and 1/3 for a sphere. The basis of SERS enhancement is the large increase in local field at the frequency ω at the resonance condition, Re{εc(ω)} ) -(1/ζ - 1)εs. Both the enhancement, g(ω), and the plasma absorption, k(ω), occur near the spectral region of the screened surface plasma frequency in the free electron model, which is obtained by finding the maximum of eq 2. While SPPs can be driven at any frequency below the plasma frequency, enhancement and the SBPP are dependent on the free carrier density, which in turn determines the frequency at which the denominator of g(ω) is minimized (eq 2). Band-to-band transitions obscure the SBPP in Au and Ag. However, the theory can be readily verified by experimental observation of the SBPP and SPPs in ITO.33,34 Many studies have addressed the magnitude of the enhancement due to surface plasmons using analytical11,16,37 and numerical19,20,23,46-48 models. The expression in eq 2 is finite at the resonance condition because the dielectric function is complex and only the real part cancels out in the denominator. In the following, we consider the consequences of including the complex dielectric response in the Clausius-Mosotti relation relative to the frequency dependence of the local field correction applied to enhancement phenomena. Using the Drude model (eq 1) for conductors the complex dielectric response is ε1(ω) + iε2(ω), where the terms are given below.
ε1(ω) ) ε∞ -
ω2p
, ω2 + Γ2
ε2(ω) )
ω(ω2 + Γ2)
(3)
(
(1/ζ - 2)ε1(ω)εs)2 + ε22(ω)ε2s /ζ2 ζ2(ε21(ω) + 2(1/ζ - 1)εsε1(ω)+ (1/ζ - 1)2ε2s + ε22(ω))2
3λ4
|g*(ω)g(ω)|
(6)
The total intensity that impinges on a molecular adsorbate near the surface of the conducting sphere will be both the incident intensity I ) εsε0Ei02 and the much larger intensity due to particle scattering given by I ) σscaEi02. The interaction between the nanoparticle and adsorbate involves both image effects and local field effects.36 The role played by the image effect has been considered elsewhere. While there may also be a bandwidth to the induced fields by the image effect, the treatment of this case is beyond the scope of the present study. Considering the local field effect, eq 6 describes the origin of the first enhancement. The second enhancement is treated as part of the overall Raman scattering cross-section.36 Since the nanoparticle-adsorbate system has but one local field function given by eq 5, it is not possible for both the incident and the scattered intensities, Ei2 and Es2, to be at the peak of the modulus squared local field function. In addition to scattering by the conducting particle, absorption of a photon can occur as a competing process. The absorption cross-section of a conducting sphere is described by:
8π2d3√εs ) Im{g(ω)} λ
σabs
(7)
where
ζ
ε2(ω)εs 2 ε2(ω) + (1/ζ
+ - 1)2ε2s + 2(1/ζ - 1)εsε1(ω))
(ε21(ω)
2
(8)
(4)
The absorption, fluorescence, and Raman scattering of a molecule near the surface of a conductor is affected by the local field. The molecular absorption and scattering also involve two field interactions, one in-phase and a second out-of-phase. The Raman scattering cross-section relates the squared field of both the incident Ei2 and scattered Es2 waves. The local intensity enhancement factor for each of these fields will enter as modulus squared of g(ω) at frequency ω as given in eq 5.
|g*(ω)g(ω)| )
128π5d6ε2s
The above expressions can be combined to determine the ratio σsca/σabs, which provides an estimate of the minimum particle radius d that can lead to field enhancement by scattering.
ε1(ω) + iε2(ω) - εs ζ(ε1(ω) + iε2(ω) + (1/ζ - 1)εs)
(ε21(ω) - (1/ζ - 1)ε2s + ε22(ω)+
σsca )
Im{g(ω)} )
Γω2p
Using these definitions, the enhancement factor is:
g(ω) )
Equation 5 only applies when an incident field can excite a polariton inside the conductor. The enhancement factor for Raman spectroscopy is a result of scattering by a the nanoparticle-adsorbate system driven by an incident field.36 For the special case of ζ ) 1/3, the scattering cross-section of a conducting sphere is:
)
* σsca 16π3d3ε3/2 s |g (ω)g(ω)| ) ) 3 σabs lm{g(ω)} 3λ 2 2 16π3d3ε3/2 s (ε1(ω) - (1/ζ - 1)εs +
ε22(ω) + (1/ζ - 2)ε1(ω)εs + ε22(ω)ε2s /ζ2 3λ3(ε21(ω) + ε22(ω)+
(9)
(1/ζ - 1)2ε2s + 2(1/ζ - 1)εsε1(ω))ε2(ω)εs For a spherical particle the function becomes:
(ε21(ω) - 2ε2s + ε22(ω)+
(5)
σsca 16π3d3ε3/2 ε1(ω)εs)2 + 9ε22(ω)ε2s s ) σabs 3λ3 (ε21(ω) + ε22(ω) + 4ε2s + 4εsε1(ω))ε2(ω)εs
(10)
Surface-Enhanced Raman Scattering
J. Phys. Chem. C, Vol. 113, No. 15, 2009 5915
In early treatments the plasma absorption was not considered and the optical particle size was determined to be less than 10 nm.6 However, the large extinctions of Au and Ag nanoparticles compete with scattering until the particle radius exceeds a critical value.16,37 As the particle size increases, radiation damping causes a decrease in the intensity enhancement leading to an optimal particle size, which has been estimated to be in the range from 20-60 nm.16,37,45 To understand the effect of conductor geometry (depolarization factor) and dielectric, we can express both the position of the surface plasmon and the magnitude of the intensity enhancement in analytic formulas given in eqs 11 and 12. If the dielectric screening of the metal is accounted for the screened surface plasma frequency is:
ωsp )
ω2p - Γ2 ε∞ + (1/ζ - 1)εs
(11)
The maximum enhancement is obtained at ωsp, at which frequency the intensity enhancement factor is approximately:
|g*(ωsp)g(ωsp)| ≈
ω2p
(
ε2s
Γ ζ (ε∞ + (1/ζ - 1)εs) 2
4
3
)
(12)
Equation 12 indicates that the maximum enhancement is proportional to |ωp/Γ|2. However, it is modulated by the dielectric constant of the surroundings, εs, and the conductor geometry. The above derivation is valid provided εs > ε2(ω). The value of ζ is 1/3 for a sphere and decreases for molecules at vertex of an ellipse. As ζ f 0, the eccentricity of the ellipse increases, and the enhancement factor for a molecule at the vertex increases. These considerations have led to the concept of the “lightning rod effect” that produces enhancements significantly larger than those possible on a sphere. A small value of the depolarization, ζ, causes red shifts of ωsp, which must be taken into account in determining the resonance condition for SERS. While the analysis presented here does not explicitly consider dimers or aggregates that can have hot spots, similar considerations apply to those geometries. Equation 12 also indicates that dielectric screening plays a role. At small values of the medium dielectric function, εs, the response is dominated by the high frequency dielectric constant of the conductor, ε∞. The foregoing considerations indicate that the bandwidth for Raman enhancement, |g*(ω)g(ω)|, which we will write |g(ω)|2 in the following, and the plasma absorption coefficient, k2(ω), are not the same. This fact needs further investigation since the plasma absorption band is observed in many experiments and can easily be assumed to be the line shape relevant for the resonance enhancement. Recent experiments confirm that the SERS enhancement has a narrower excitation profile than the LSPR.38 In addition to the constraints discussed above, there are also distance and angular considerations. The field due to the LSPR decreases with distance as d3/r3 on a spherical surface, where d is the particle radius and r is the distance from a molecule to the center of particle. In the following, I treat the case where the molecule is on the surface of the nanoparticle. Hence, d ) r and the distance factor (d3/r3) is unity. The probability of absorption, and therefore also Raman scattering, is a function of the cos2 (θ), where θ is the angle between the dipole on the nanoparticle and transition dipole on
Figure 1. The calculated plasma absorption (solid) and dispersion curves (dotted) are shown for give relative values of the damping, Γ, and the plasma frequency, ωp. ωp/Γ ) 200, 100, 40, 20, and 10. In this calculation εs ) 1.0, ε∞ ) 4.0, ζ ) 1/2 in eq 5, and the screened bulk plasma frequency, ωsbp ) ωp/ε∞, is at 0.5ωp and the surface plasma frequency, ωsp ) ωp/(ε∞ + εs), is found at 0.44ωp. The plasmon band gap is indicated between ωsp and ωsbp.
the particle. On a single spherical particle, orientation averaging leads to a decrease in the intensity enhancement by a factor of 1/3. Furthermore, one must average over the solid angle for scattering of the Raman photon at ωs. In the simplest model this leads to a decrease by a factor 2 since there is a probability of 1/2 that Raman photons will be scattered in the hemisphere that coincides with the nanoparticle surface (see Supporting Information). Thus, orientational factors will reduce the overall enhancement by a factor of 18. The magnitude of this reduction is largely offset by the increase in enhancement due to the dielectric constant and increases in curvature of ellipsoidal surfaces (see Supporting Information). These factors are mentioned for completeness, but they are not included in the calculations of the enhancement factor given below. Results Figure 1 shows plots of k1(ω) and k2(ω) in reduced units of frequency for five values of the ratio ωp/Γ. The real part of the wave vector, k1(ω), represents dispersion and the imaginary part, k2(ω), represents absorption. On a flat conducting surface, the wave vector represents the requirement for spatial matching of incident radiation at the boundary between the medium and the conductor. However, for nanoparticles and rough surfaces, the angle dependence is lost and wave vectors can also be related to the real and imaginary components of the dielectric function of the conductor. As the ratio ωp/Γ decreases, the bandwidth of the absorption, k2(ω) increases. The dispersion curve, k1(ω), also becomes less sharp and loses the defined plasmon band gap shown in Figure 1. It is shown below using experimental data for Ag and Au that a realistic estimate for the ratio ωp/Γ is less than 40. The ratios of ωp/Γ used in Figure 1 span a range from 20 to 200 to be sure to capture the maximum possible enhancement effect. Recognizing that values of ωp/Γ > 40 are not realistic for metals such as gold and silver, the point of the comparison in Figure 1 is to demonstrate the nature of the tradeoff of peak enhancement and bandwidth for extremely large theoretical enhancements. Moreover, the large values of ωp/Γ are presented to accommodate large local field enhancements that are often assumed in experimental studies to account for large SERS effects.
5916 J. Phys. Chem. C, Vol. 113, No. 15, 2009
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2
Figure 2. The calculated intensity enhancement |g(ωi)| for a sphere (eq 5 with ζ ) 1/3, εs ) 1.0, ε∞ ) 4.0, d , λ) is shown. The calculated enhancement is shown for five relative values of ωp/Γ ) 200, 100, 40, and 20 scaled to have the same amplitude. The SERS enhancement is the product of the local field for the incident, |g(ωi)|2, and scattered, |g(ωs)|2, fields. An example is shown for which ωs ) ωi - 0.02ωp.
Figure 2 shows the Raman enhancement for spherical nanoparticles (ζ ) 1/3) decomposed into two potential enhancements at ωi and ωs, respectively, using the same parameters as used for the absorption curves shown in Figure 1. The maximum of the function |g(ωi)|2 is 15 000 for ωp/Γ ) 200 for the condition ωi ) ωsp. The enhancement at the scattered frequency, |g(ωs)|2, is 290 for this calculation, i.e., a factor of ∼50 less than the peak value. Moreover, the local field factor can only contribute if there is particle scattering at this frequency, which is shifted from the incident laser frequency, ωi. The example in Figure 2 is presented for a scattered photon, which has a Raman shift of 0.02ωp in reduced units. This would correspond to a typical high frequency mode, e.g., a ring breathing mode of pyridine at ∼1000 cm-1 for ωp ) 50 000 cm-1, or one of the intense in-plane ring deformations at ∼1600 cm-1 for ωp ) 80 000 cm-1. These ωp values are ∼6 and ∼10 eV, respectively, and correspond roughly to ωp in Au and Ag, respectively. If a loss process is assumed to result in particle scattering at frequency ωs ) ωi - 0.02ωp, we obtain overall enhancement factors 4.4 × 106, 1.1 × 106, 1.3 × 105, and 1.9 × 104 for values of ωp/Γ ) 200, 100, 40, and 20, respectively. The above calculations (Figures 1 and 2) correspond to a particle in vacuum (εs ) 1). Many experimental observations have been made on samples in water or embedded media such as silicon dioxide. For an index of refraction of n ) 1.5 (εs ) 2.25) the surface plasma frequency shifts to correspondingly lower values since ωsp ) ωp/(ε∞ + εs) ) 0.4ωp for a planar surface, and thus the plasmon band gap is increased since the screened bulk plasma frequency remains at ωsbp ) 0.5ωp. Likewise, the enhancement factor is increased by the factor given in eq 12. For example, for εs ) 2.25 relative to εs ) 1, the increase is a factor of 3.17. Inspection of these equations reveals that both the wavenumber shift of the surface plasmon and the enhancement depend on ζ and εs as well as ωp/Γ. However, the bandwidth does not change for εs and only gets narrower as ζ decreases, i.e., for greater curvature. Thus, any additional enhancement by virtue of the geometry only enhances the incident field intensity and not the scattered intensity. The modest dependence of the enhancement on ζ and εs is shown in the Supporting Information over the entire useful range of 1/6 < ζ < 1/3 and 1.0 < εs < 3.0. However, the effect of curvature on the position of the peak enhancement is a relatively large shift to lower energy. This effect is plotted in Figure 3 in reduced units of ωsp/ωp.
Figure 3. The dependence of the surface plasmon frequency on ζ is shown in reduced units. As the curvature increases, and the overall enhancement increases, the peak enhancement shifts to lower energy.
Figure 4. Calculation of the absorption coefficient k2(ω) and intensity enhancement |g(ω)|2 for (A) Au and (B) Ag. The calculated absorption coefficients for (A) Au and B(Ag) are scaled by a factor of 500 and 50, respectively, for comparison.
The analysis was applied to Ag and Au using the experimentally determined values of the optical constants n(ω) and κ(ω). The dielectric function was obtained from ε1(ω) ) n(ω)2 - κ(ω)2 and ε2(ω) ) 2n(ω)κ(ω) as described in a previous study.34 The dispersion properties of thin metal films and the plasma absorption spectra are predicted accurately using these dielectric functions. Figure 4 shows values calculated for |g(ω)|2 compared to the absorption coefficient Im{g(ω)} obtained using the methods used in Figures 1 and 2 for the Drude free electron model. The intensity enhancement is proportional to the particle scattering and therefore |g(ω)|2 (eq 6). The absorption of the nanoparticle is proportional to Im{g(ω)} (eq 7). Specifically, the imaginary wave vector component is k2(ω) ) 1000σabsNAC, where NA is Avagradro’s number and C is the nanoparticle concentration. Ag is much closer to a Drude free electron model
Surface-Enhanced Raman Scattering
Figure 5. Calculation of the cross-section for scattering relative to absorption using the optical constants for Au and Ag. The optical constants were used in eq 10 for a ratio d/λ ) 0.1.
than Au. The origin of the difference is the large contribution from bound electrons in Au, which can be seen in the large value of ε2(ω) for Au throughout the visible region. 34 Ag has a relatively small contribution from ε2(ω) seen at frequencies above ωsp. Calculations presented in the Supporting Information show that a Lorentzian absorption band, which contributes to ε2(ω), can be used to model the absorption in Au. The enhancement factor is significantly smaller for the model that includes an absorptive transition. Figure 5 shows the results of the calculation of σsca/σabs for Ag and Au. The greater contribution of ε2(ω) in Au and Ag than for a free electron conductor has the consequence that larger values of d/λ are required to cross the threshold required for intensity enhancement by the electromagnetic mechanism. Au and Ag cross the breakeven point (σsca/σabs ∼ 1) at values of d/λ ) 0.1 and 0.2, respectively. Discussion The magnitude of the SERS effect is determined by six factors: (1) tuning of the incident laser frequency, ωi ∼ ωsp, to the maximum value of the local field factor, |g(ωsp)|2, (2) the magnitude of the ωp/Γ ratio, which is a material property, (3) the magnitude of the bandwidth |g(ω)|2 relative to the Raman shift, (4) the role of the depolarization factor, ζ, (5) the dielectric constant of the medium, εs, and (6) orientation and distance effects. Factors 1 and 2 constitute the condition for resonance with the plasma absorption, which is of obvious importance for SERS. While factors 4, 5, and 6 are well-known, the enhancement bandwidth relative to the Raman scattered frequency (factor 3) has been hardly mentioned in the vast literature on the subject of SERS. Experimentally, SERS is often observed in the presence of a relatively large scattering background.1,49 The origin of the background is not known at present, but it is a reasonable assumption that the scattering background results from loss processes on the metal. These processes then can lead to Pnp(ωs) and enhancement at the scattered frequency, ωs, according to |g(ωs)|2. The image effect involving interaction of the molecular adsorbate and nanoparticle also plays a role. 36 In the following we will explore the consequences of a loss process for the SERS enhancement, taking into account the bandwidth of local field factor. The bandwidth limitation of SERS, which was explored systematically using the Drude model, is born out by the comparison of Au and Ag. Ag has a maximum intensity enhancement of 240 for a molecule on a single spherical particle
J. Phys. Chem. C, Vol. 113, No. 15, 2009 5917 (d/λ ∼ 0.1) if the incident laser frequency is resonant with ωsp, ωi ) ωsp. The fwhm of the intensity enhancement function, |g(ω)|2, is ∼2100 cm-1. The SERS enhancement is calculated to be 36 000 for a 1000 cm-1 vibrational mode ignoring orientation and distance effects. This calculated value and the bandwidth of the Raman enhancement in the small particle limit agree well with the calculations of Kerker et al.11 with the assumption that the enhancement can be assumed to involve fields at both frequencies ωi and ωs. It is difficult to assign a fwhm to the Au plasma absorption band because of its non-Lorentzian shape. However, using an estimate of ∼6100 cm-1 40 for the fwhm of Au nanoparticles, the maximum intensity enhancement for Au is ∼26 (Figure 4). If a loss process is assumed that permits enhancement at ωs, the maximum SERS enhancement could be as large as ∼520 for a single particle in the limit d , λ. The factor of the enhancement reduction of Au relative to Ag is ∼69, which is approximately equal to the ratio of the local field bandwidths, (fwhmAg/fwhmAu)4 ∼ 72. The increased bandwidth is consistent with the 3-fold decrease in the ratio ωp/Γ in Au relative to Ag. The greater damping in Au is likely a manifestation of the greater density of bound electrons, which is manifest in the significantly larger value of ε2(ω) of Au relative to Ag.34 The results obtained here are in agreement with relevant aspects of modeling using finite difference time domain methods.19 For example, using the parameters of Futamata et al. to model Ag nanoparticles we find that the widths of the field enhancement for a number of geometries are ca. 40 nm (3000 cm-1) with a peak for the Ag localized surface plasmon near 380 nm (26 300 cm-1), which gives an enhancement factor of ∼50 000 and fwhm of to around 1500 cm-1 using the standard assumption of a fourth power dependence on field. This calculation agrees reasonably well with our calculation for Ag with an enhancement factor of 36 000 and fwhm of 1800 cm-1. The Drude model calculation shown in Figure 4 is in this range if ωp/Γ ) 30, which is a reasonable estimate for metallic Ag. Models that use a parametrization of ωp/Γ ) 100 for Ag50 adequately represent the real part of the dielectric function, but do not properly capture the imaginary part of the dielectric function. Many of the studies using the FDTD method account for the enhancement in dimers and other more complex geometries.19,20,23,46-48 While these clearly have additional enhancement not discussed here, the basic feature that the enhancement and bandwidth vary inversely is also valid in those more complex systems. There has been a great deal of interest in the spatial distribution of the local field. One of the important considerations in deriving extremely large resonant Raman enhancements is the existence of the “lightning rod effect” or “hot spots”. Increased aspect ratio is one structural feature that can produce large local fields. The tip of an ellipse has a substantially higher local field than other locations along the length of the ellipse. The spaces between two nanoparticles and in fractal aggregates of nanoparticles have a large local field. Using the depolarization as a model of these effects, Figure 3 shows that while an increase curvature corresponding to a decrease in the depolarization factor from ζ ) 1/3 to 1/6 can increase the enhancement by a factor of 10, there is an accompanying shift in ωsp from 0.35 ωp to 0.26 ωp. Plots that include changes in the dielectric function as well are presented in the Supporting Information. Since the bandwidth of the local field factor remains narrow, the requirement for excitation is that the indicent laser wavelength must be tuned significantly to the red. For a typical laser excitation
5918 J. Phys. Chem. C, Vol. 113, No. 15, 2009 wavelength near 500 nm (e.g., a Ar ion laser), this would correspond to a shift to 674 nm for the change from ζ ) 1/3 to 1/6. The intensity enhancement scales as |ωp/Γ|2, and the fwhm scales roughly with Γ/ωp. In other words, to achieve a 10-fold gain in bandwidth, one loses a factor of 100 in intensity enhancement. Thus, for maximal effect both the molecular absorption and the laser frequency must be on resonance with the peak of g(ω). For example, in Figure 2 if the laser were tuned ∼10 nm from the optimum value, the enhancement factor drops by 1 order of magnitude. Similar considerations apply to elliptical geometries and dimers where the bandwidth is also inversely correlated with the intensity enhancement. Treatments that obtain a large enhancement of >109 for a single sphere use the assumption that the enhancement is proportional to |g(ω)|4.1,5,12,15,17-19,21,24,44,51,52 The results obtained here suggest that this model needs revision. In a more realistic model, the largest intensity enhancements of ∼15 000 on spherical particles (i.e., for ωp/Γ ) 200) can only be achieved under conditions where the bandwidth for SERS excitation is narrow ( 40, peak enhancement, |g(ωsp)|2, is high, but the enhancement at the Raman scattered frequency, |g(ωs)|2, is significantly reduced. These considerations place severe constraints on the “hottest” particles. Their relatively high peak enhancement is offset by their narrow bandwidth and shifts in ωsp that must be matched precisely by the incident laser frequency. Smaller values of ωp/Γ < 40 that are less restrictive because of a greater bandwidth cannot achieve the enhancements required to explain results reported for single nanoparticles with an electromagnetic mechanism. The computational results lead one to consider the implications for single molecule SERS. Starting with the first report of single molecule SERS,15,18 by far the most studied system has been the adsorption of Rhodamine 6G on Ag nanoparticles and nanostructures.20,28,51-61 These studies include evidence for the existence of single molecule Raman scattering based on a statistical analysis of scattering from isotopically labeled Rhodamine 6G.21 The lasers used in these experiments are usually in resonance with Rhodamine 6G itself so that resonance Raman scattering must be included in the description. One hypothesis is that Rhodamine 6G interacts strongly with Ag and Au nanoparticles as demonstrated by alteration of the adsorbate molecular spectrum.28,62,63 This strong interaction means that the adsorbate forms a supramolecular adduct with the nanoparticle in the same way that a ligand bound to a metal ion becomes part of a molecule. Similar comments apply to crystal violet and other planar aromatic molecules that have been used in SERS studies.22 If the molecule becomes part of the metal such that there is transfer of charge from the metal to molecule (ligand) during plasma oscillations, there is an additional resonant enhancement mechanism whose Raman excitation profile will approximately track the plasma absorption spectrum.64 This will be true for Franck-Condon active transitions and in the limit of small displacements of the vibrational modes in the excited state. The Raman scattering of the molecule can then be enhanced by resonant absorption of the nanoparticle. This hypothesis is a resonance Raman explanation for the chemical mechanism of SERS. The
Franzen Rhodamine 6G/Ag system has also been used to study the relationship of the first and second enhancements using a comparison of Stokes and anti-Stokes intensities for low and high frequency modes.65 While the data are seen as confirmatory of a second enhancement, they are also consistent with a SERS bandwidth as proposed here. In other words, low frequency modes are preferentially enhanced since they are closer to the excitation laser frequency, and the local field function has an intrinsic width. The approach taken here is consistent with such data, but the comparison would be significantly improved if experiments were conducted on nonresonant systems so that the role of normal resonance enhancement could be separated from local field enhancement in SERS. A recent detailed consideration of the experimental determinants of the enhancement factor in SERS concluded that measurements are consistent with enhancements of 1010 as an upper bound with typical values around 107 even for single molecule SERS.22 The theoretical debate over SERS enhancement is clearly not resolved, and the current estimates differ by over 7 orders of magnitude.66,67 In large measure this debate has been spurred by the experimental observation of Raman scattering from a single molecule. The results presented here are not consistent with reports of single molecule SERS on single spherical particles.18 Even for the optimum geometry and molecular orientation, the maximum theoretical enhancement is orders of magnitude too small for the electromagnetic mechanism to apply to a single conducting sphere. While the “lightning rod” effect, dimers, and aggregates may provide additional enhancement mechanisms, these would need to provide at least 5 orders of magnitude to account for the difference between the upper bound of 105 found here for a realistic single particle SERS enhancement and 1010, which is the lowest enhancement factor claimed for single molecule SERS. Dimers and complex structures must also meet the bandwidth requirement in order to account for the extremely large electromagnetic enhancements suggested to account for single molecule SERS. Conclusion Surface-enhanced spectroscopy continues to interest scientists and engineers both because of the interest in the fundamental physics of the spectroscopic effects and also the potential applications in sensor design. This paper applies a correction to the |E|4 enhancement dependence that is usually assumed to occur at both the incident and scattered frequencies. According to the Clausius-Mosotti local field, the maximum intensity enhancement for the incident photon at ωi ∼ ωsp is proportional to |ωp/Γ|2. The enhancement at ωs is determined by the relationship between the Raman shift, ∆ω ) ωi - ωs, and the bandwidth of |g(ω)|2. The bandwidth of the intensity enhancement function, |g(ωi)|2|g(ωs)|2, places severe constraints on the electromagnetic mechanism for the SERS effect in both Drude conductors as well as the noble metals. Contributions from the solvent dielectric constant εs and depolarization factor ζ can give rise to an additional enhancement by 1 order of magnitude but do not change this conclusion. When orientational factors are included, the upper limit for an electric field enhancement by a sphere is found to be approximately 105 by the Clausius-Mosotti approach using the standard approximation that both the incident and scattered fields are enhanced (assuming ωp/Γ ) 200, which is considered the upper limit for this ratio). More realistic values for Ag and Au are 2000 and 30, respectively, using experimental data for the dielectric function of these metals and including orientational averaging. These
Surface-Enhanced Raman Scattering values are consistent with much of the early work on SERS. Increases in peak enhancement due to geometry observed in more recent work will result in a shift of the frequency of the surface plasmon. Unless the incident laser frequency is appropriately matched to this shifted frequency within the narrow bandwidth where maximal is possible, the predicted enhancements for specific geometries cannot be realized. In conclusion, the bandwidth limitation of the local field needs to be considered in greater detail in the interpretation of experimental SERS data. Acknowledgment. Prof. David Aspnes is thanked for insightful comments on this manuscript. Supporting Information Available: Experimental details. This material is available free of charge via the Internet at http:// pubs.acs.org. References and Notes (1) Moskovits, M. ReV. Mod. Phys. 1985, 57, 783. (2) Zeman, E. J.; Schatz, G. C. J. Phys. Chem. 1987, 91, 634. (3) Schatz, G. C.; Vanduyne, R. P. Surf. Sci. 1980, 101, 425. (4) Knoll, W. Annu. ReV. Phys. Chem. 1998, 49, 569. (5) Wang, D. S.; Kerker, M. Phys. ReV. B 1981, 24, 1777. (6) Kerker, M. Acc. Chem. Res. 1984, 17, 271. (7) Aspnes, D. E. Phys. ReV. Lett. 1982, 48, 1629. (8) Jeanmaire, D. L.; Vanduyne, R. P. J. Electroanal. Chem. 1977, 84, 1. (9) Gersten, J.; Nitzan, A. J. Chem. Phys. 1980, 73, 3023. (10) Bakr, O. M.; Wunsch, B. H.; Stellacci, F. Chem. Mater. 2006, 18, 3297. (11) Kerker, M.; Wang, D. S.; Chew, H. Appl. Opt. 1980, 19, 4159. (12) Le Ru, E. C.; Etchegoin, P. Chem. Phys. Lett. 2006, 423, 63. (13) Moskovits, M. J. Chem. Phys. 1978, 69, 4159. (14) Krug, J. T.; Wang, G. D.; Emory, S. R.; Nie, S. M. J. Am. Chem. Soc. 1999, 121, 9208. (15) Kneipp, K.; Wang, Y.; Dasari, R. R.; Feld, M. S. Appl. Spectrosc. 1995, 49, 780. (16) Jiang, J.; Bosnick, K.; Maillard, M.; Brus, L. J. Phys. Chem. B 2003, 107, 9964. (17) Ward, D. R.; Grady, N. K.; Levin, C. S.; Halas, N. J.; Yanpeng, W.; Nordlander, P.; Natelson, D. Nano Lett. 2007, 7, 1396. (18) Nie, S. M.; Emery, S. R. Science 1997, 275, 1102. (19) Futamata, M.; Maruyama, Y.; Ishikawa, M. J. Phys. Chem. B 2003, 107, 7607. (20) Kall, M.; Xu, H. X.; Johansson, P. J. Raman Spectrosc. 2005, 36, 510. (21) Dieringer, J. A.; Lettan, R. B.; Scheidt, K. A.; Van Duyne, R. P. J. Am. Chem. Soc. 2007, 129, 16249. (22) Ru, E. C. L.; Blackie, E.; Meyer, M.; Etchegoin, P. G. J. Phys. Chem. C 2007, 111, 13794. (23) Sanchez-Gil, J. A.; Garcia-Ramos, J. V. Chem. Phys. Lett. 2003, 367, 361. (24) Jain, P. K.; El-Sayed, M. A. J. Phys. Chem. C 2008, 112, 4954. (25) Brown, R. J. C.; Wang, J.; Milton, A. J. T. J. Nanomater. 2007, Art. No. 12086. (26) Etchegoin, P. G.; Maher, R. C.; Cohen, L. F. New J. Phys. 2004, 6, Art. No. 142. (27) Markel, V. A.; Shalaev, V. M.; Zhang, P.; Huynh, W.; Tay, L.; Haslett, T. L.; Moskovits, M. Phys. ReV. B 1999, 59, 10903. (28) Zhao, J.; Jensen, L.; Sung, J. H.; Zou, S. L.; Schatz, G. C.; Van Duyne, R. P. J. Am. Chem. Soc. 2007, 129, 7647.
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JP808107H
