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5912
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
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Publication Date (Web): March 23, 2009 | doi: 10.1021/jp808107h
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.
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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
Franzen
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Publication Date (Web): March 23, 2009 | doi: 10.1021/jp808107h
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:
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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
Franzen
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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
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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
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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 (<10 nm). Returning to the ring deformation mode at
1600 cm-1 (e.g., pyridine) as an example and using ωp ) 80 000
cm-1 (e.g., Ag) as an example, then the Raman shift is 0.02ωp
in reduced units. As is evident in Figure 2, this shift is greater
than the fwhm for the case ωp/Γ ) 200, which has a fwhm )
0.0035ωp and even when ωp/Γ ) 40 where the bandwidth is
0.016ωp. For ωp/Γ > 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.
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