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Electrodynamics of surface-enhanced Raman scattering

We examine SERS from two perspectives: as a phenomenon described by the Laplace Equation (the electrostatic or Rayleigh limit) and by the Helmholtz Equation (electrodynamic or Mie limit). We formulate the problem in terms of the scalar potential, which simplifies calculations without introducing approximations. Because scattering is not usually calculated this way, we provide the necessary theoretical justification showing that the scalar-potential description is complete. Additional simplifications result from treating the scatterer as a point charge q instead of a dipole. This allows us to determine the consequences of including the longitudinal (Coulomb) interaction between q and a passive resonator. This interaction suppresses the mathematical singularities that lead to the unphysical resonant infinities in first and second enhancements. It also modifies the effective restoring-force constant of a resonant denominator, which permits us to explore the possibility of dual resonance through a molecular pathway. We apply the formalism to spherical inclusions of radius a for q located at polar and equatorial positions. For small a, of the order of 1 nm or less, the low-l multipole terms are important. For the more relevant case of radii of the order of 10 nm and larger, the q-sphere interaction can be approximated by a model where q interacts with its image charge for a dielectric plane, and the singularity shifts in a discrete manner from \epsilon(\omega)=-2 to \epsilon(\omega)=-1. These results are supported by more accurate calculations taking retardation into account, although the use of only one spherical-harmonic term does not fully represent the difference between forward- and backscattering.


I. INTRODUCTION
Surface-enhanced Raman scattering (SERS), discovered over 30 years ago, [1,2] has experienced renewed popularity as a result of the present interest in nanostructures and reported observations of SERS from single molecules. [3,4] A surprising aspect is the apparently extreme amplification of Raman intensities, with factors of the order of 10 14 being claimed. [3,4] These factors are well beyond those that can be realized by enhancing the driving field by plasmonic resonances. [5][6][7][8] Additional mechanisms appear to be required to explain the reported enhancement factors. [9][10][11] The concept of "hot spots" has been introduced to account for possible concentrations of the field in specific regions between dimer/multimer particles, sufficiently curved surfaces, or selfsimilar fractal aggregates. [10,[12][13][14][15] Of concern is that the average enhancement in a sample should be measurable as a bulk dielectric-response effect if the "hot spots" comprise a significant part of the sample, [16,17] as modeled for example by effective-medium theory. [18,19] An alternative view is that the electromagnetic enhancement factors are in fact modest, and that other mechanisms give rise to the observed scattering. [17] In this case, the molecular resonance may be important in a separate mechanism, such as resonance Raman. In parallel work, we have used the time-correlator formalism for calculation of resonance Raman cross sections [20] to show that molecular resonances are the most likely origin of the so-called chemical mechanism of SERS. [21][22][23] Yet another view has been presented that treats the local field, molecular resonance and charge transfer interactions in a Herzberg-Teller or vibronic coupling approach. [24] Almost every treatment of this problem concedes that multiple mechanisms are required in order to achieve an enhancement factor of 10 14 . Although the electromagnetic mechanism is considered to be the dominant mechanism by many researchers, essentially every theory also includes the chemical mechanism. The nature of the chemical mechanism itself is also a subject of controversy, in part because the methods required to calculate resonance Raman cross sections have also been under development. We have advanced the view that explicit calculation of excited state displacements is essential for calculation of resonance Raman scattering. [21][22][23] The alternative to explicit calculation of the excited state displacements is the use of polarizability derivatives to calculate the Raman cross section. [25][26][27] However, the polarizability derivative approach is only valid far from resonance. Therefore, polarizability derivatives cannot accurately estimate the role played by the resonance Raman effect in the chemical mechanism. Explicit calculation of excited-state displacements has shown that pyridine resonance can be enhanced by optical excitation of Pyr-Ag clusters of up to 20 Ag atoms. [23] The role of excited-state displacements has been verified experimentally to be consistent with resonance Raman scattering from molecule-cluster supermolecules. [28] Regardless of the method used to calculate its magnitude, the chemical mechanism arises from a direct, i.e. bonding, interaction of the metal cluster with the adsorbate that leads to greater Raman enhancement. The electromagnetic theory can be interpreted as an enhancement of molecular polarizability that arises from the polarization of the inclusion by the incident radiation. [7,8] Since the molecule is close to a curved metal surface, a self-consistent solution of Maxwell's Equations leads to a local-field contribution that can be cast into an effective polarizability for the bound adsorbate. [29][30][31] The polarizability tensor couples the incident and scattered fields. Theories of electromagnetic enhancement have required enhancement in both of these fields to explain the reported giant enhancements. To simplify the physics, investigators have examined the incident and scattered field enhancements separately, and given the names first and second enhancement, respectively.
Electromagnetic, or more accurately, electrostatic, theory is invoked to explain observed large enhancements as the contribution of the outgoing Raman-scattered radiation by a second plasmonic resonance. [6,32] Since the intensity of the incident and scattered radiation are each proportional to the square of the field, dual plasmonic resonances would allow large overall gains to be achieved with electric-field enhancements of the order of 10 3 per stage. However, this argument fails when bandwidth is taken into account. To resonantly enhance two signals at somewhat different frequencies, it is necessary to trade Q for bandwidth. This places an upper limit to possible enhancements by purely plasmonic resonances. This aspect has been ignored in prior work with two exceptions. [33,34] Yet it is evident that large enhancements are possible only if the quality factor Q of the resonance is extremely high, which requires the resonance itself to be extremely narrow. [33] Almost every prior study has ignored damping so that the resonance was treated in the limit of an infinitely narrow and infinitely large enhancement.
However, infinities are nonphysical. In a proper formulation of resonance enhancement they should not occur, and their appearance indicates that the elementary theory is deficient in some way. We show that the deficiency results from neglecting the longitudinal (Coulomb) interaction between the charge q and the passive resonator. The inclusion modifies the response of q so that at a resonance its motion is effectively frozen. While our calculations show that this occurs under conditions too extreme to have any practical effect, they identify and eliminate an objection to the elementary approach. We also provide an explicit general formulation of the enhancement problem in the electrostatic limit, then use it for spherical resonators to demonstrate our conclusions regarding intrinsic limits to first and second enhancements. As a consequence only one of the interacting electromagnetic waves, incident or scattered, can be at maximum gain.
These calculations show that for molecules on a single isolated nanoparticle (inclusion), electromagnetic enhancement is limited to 10 6 on Ag and significantly less on Au.
Our approach takes advantage of a point-charge model, which in anisotropic-bond form provided new insights into the physics of second-harmonic generation and other nonlinearoptical processes. [35] We start with a configuration C and subdivide it into q and a remaining system S, which we take to be a passive resonator of dielectric function ε(ω) embedded in a dielectric matrix. We calculate the radiation emitted by C by evaluating the four basic steps that occur whenever a passive illuminated configuration radiates light: (1) determine the local field at q; (2) solve the force equation to obtain the acceleration of q; (3) calculate the resulting farfield radiation emitted by q, and (4) superpose the radiation from all q. In the present application, step (3) is expanded to include the radiation emitted by S as well as q, and step (4) can be ignored. This general approach was originally applied to linear optics by Ewald, [36] and Oseen, [37] and is formally related to the derivation of the Lorentz polarizability as an optical response.
It is not intended to replace a full quantum-mechanical treatment, but to elucidate the underlying physics.
As shown by numerous examples, [38][39][40][41][42] this approach is particularly effective when the emerging radiation is orders of magnitude weaker than the driving radiation and occurs at a different frequency, as is the case here. Under these conditions all four steps are independent, so the self-consistency difficulties that complicate linear-optics applications do not arise. However, the Raman-scattering problem is complicated by the presence of the passive resonator, which changes the resonant conditions of q even though it nominally amplifies incoming fields through plasmon resonances and serves as a secondary radiator at the Raman-shifted frequencies. While the ratio of field strengths can be described by the reciprocity theorem, as recently noted by Le Ru and Etchegoin, [43] this is irrelevant because it does not take into account the reduction in emitted radiation caused by the connection between q and the resonator. A self-consistent solution is required, although this solution occurs at a different level from that of Ewald and Oseen.
Our approach of casting the calculation in terms of a bond charge instead of the usual point dipole is a seemingly minor detail, but it significantly reduces mathematical complexity. By selectively postponing differentiation until the final step, we avoid the need to use two arbitrarily large Green functions of opposite sign with infinitesimally small separation between singularities.
This allows us to obtain for example the solution of a Raman-active molecule at an equatorial site on a spherical inclusion, which is relevant for aromatic molecules that are expected to lie flat on a surface. We find that the polarization of the resonator by q, essentially a self-consistent mass-renormalization interaction from the perspective of q, is present at all frequencies, driving as well as radiating, and can act to move native molecular excitations on or off resonance. Thus for single inclusions our calculations reveal four electromagnetic routes to enhancement: plasmon resonances in the passive resonator for both incoming and outgoing waves, and resonant enhancement of the associated electronic transitions at the incoming and outgoing frequencies in the Raman-active molecule. However, finite-bandwidth effects make it obvious that all cannot be active at once. [33] Taken together, our results show that the extreme enhancements reported, if real, cannot be achieved with single particles but must involve a combination of the mechanisms discussed above in configurations involving two or more interacting particles. [15,44] We shall discuss this case elsewhere. The net result is a detailed understanding of the electromagnetic enhancement mechanisms of SERS.

A. Preliminaries
In this section we justify our use of the scalar potential φ as a sufficient descriptor in the current development. We begin by noting that electromagnetic scattering is described by the can be identified as the vector part of the term proportional to in the expansion of the local scalar potential. In the radiation zone , so we ignore the term in the denominator and expand the exponential to obtain . (2.7) Thus the dipole obtained from the term in the expansion of the local potential is the appropriate dipole for in the radiation zone. (2.8) We still need to show that it is sufficient to work only with φ. From the same expansion . (2.9) With the r dependence appearing only in the prefactor, we apply the definition of the Lorentz gauge (2.10) and obtain . That is, φ is simply the longitudinal component of . Since ρ and are related by charge conservation, the other two components of are connected to φ as well.
Hence we can work with either φ or . The electric field in the far-field zone is then . (2.11) As the final step, standard treatments employ boundary conditions on and . However, in the present application is continuous everywhere, so the boundary conditions on φ are sufficient.

B. General electrostatic formulation.
As noted above, we consider a configuration C consisting of a Raman-active molecule and a passive resonator, where the resonator is a polarizable inclusion of dielectric function ε = ε(ω) that is embedded in a uniform background medium of dielectric function ε a = 1. Setting ε a = 1 simplifies the equations with no loss of generality, since ε is always normalized to ε a . We let the relevant part of the molecule be a point charge q, which may be an element of a continuum charge density. We define the system S to be everything but q, although in the present development the only other system element is the passive resonator. The response of C to an externally applied field then consists of the responses of S and q to , conditioned by their mutual interaction. Here, is the location of the observer in a coordinate system where the origin is at the center of mass of the inclusion.
If the inclusion is near or at resonance at ω o , the field in the vicinity of the inclusion is larger than (first enhancement). For simple configurations the calculation of the enhancement is routine. Because we are focused here on the implications of the q-S interaction on other enhancement mechanisms, we assume that first enhancement is already pesent, and that the field in the vicinity of the resonator is , where is the enhancement due to S. We describe by the potential . For simplicity we assume that all boundaries are fixed, and ignore retardation and spatial dispersion. In this case the driving field for q is Retardation is relevant for determining the upper limits to resonator sizes; we will discuss the special case of spherical resonators below.
When q is inserted at location , new potentials that involve q and the associated polarization of S arise. We write the new overall potential as , is the contribution of the polarization induced in S by q, and a time dependence of is assumed. By separating the overall potential into these three terms, we identify and hence can calculate the consequences of the interaction between q and S, and therefore the response of C to the external driving field. The respective fields associated with each potential are given as usual by their negative gradient with respect to .
An equilibrium position is established by balancing the forces exerted on q by the fields and an intrinsic restraint that we represent for now with a constant second-rank Hooke's-Law restoring-force tensor . Because the driving field has the time dependence , all potentials considered so far will also have this time dependence. Because the restraint is not perfectly rigid, q oscillates about with the same time dependence. Accordingly, to first order we write , where is a constant. The subscript o on denotes that this displacement occurs at frequency ω o . When Raman scattering is introduced, we will define additional displacements associated with the Raman frequencies , where ω ν is a phonon frequency.

C. Equation of motion and far-field radiation
To proceed further we determine . From Newton's Law of Motion . (2.13) The quantity in braces is the local field, completing step 1. If the restoring force is isotropic such that , where is the unit second-rank tensor, then is related to the classical polarizability α by the Lorentz model . (2.14) To solve Eq. (2.13) for , we linearize it by expanding all terms to first order in .
When this is done the first-order expansion of the second term has the time dependence .
Since we are only interested in the linear response, this term is discarded. Upon expansion, this yields the radiating dipole .
In principle arbitrarily large second-enhancements occur because in the absence of damping , and therefore and , can become arbitrarily large at resonance.
However, Eqs. (2.5a) and (2.7) show that a corresponding divergence occurs in the prefactor to , so an overall singularity is avoided. Because we reach this conclusion independent of any specific configuration, the result is general.

A. Spherical inclusions, q-S interaction.
To illustrate the above, we consider S to be a spherical inclusion of radius a and dielectric function . We take the incident beam to be polarized along z, and assume that all dimensions are small compared to the wavelengths λ o and λ ± associated with ω o and ω ± , respectively. The electrostatics problem for a small sphere in a uniform field is standard, and the resulting potential for includes only the driving field and the dipole term from the sphere: where the subscript o on means the ε is evaluated at the driving frequency ω = ω o . The plasmon enhancement due to the sphere and its singularity at are evident. A primary characteristic of the above is the dependence of the magnitude of first enhancement on the sphere volume, as encoded in a 3 . This is a consequence of being proportional to the integral of , which is constant within the sphere. In practice this is reduced to a linear dependence, because the molecule can never get closer to the sphere than the radius a.
We next consider the interaction between q and S. Calculation of the associated potential is again straightforward. With q at , the general solution for the potential outside S at a frequency ω is , (3.2) where in the first term and a < r, in the second. The first term on the right is the series expansion of the potential due to q, and the second from the polarization that q induces in the sphere. The sums over the index m concern only the spherical harmonics, so we apply the sum rule (3.3a) where is a Legendre polynomial and . (3.3b) We obtain (3.3c) The field due to the polarization of the sphere induced by q is therefore To calculate the back-reaction of S on q we set , in which case . We obtain (3.5) The field at q is entirely radial, consistent with a longitudinal electrostatic interaction with retardation effects neglected. Hence for the q-S interaction the choice of coordinate system is irrelevant. Because is proportional to rather than to , lateral forces perpendicular to z will be generated when q is driven off-axis, relevant in the equatorial location discussed below.
The infinite number of singularities in Eq. (3.5) at values (3.6) represent the different plasmonic resonances to which q couples.
Unless the sphere is very small, the ratio , where d is the distance of q from the center of the sphere, will be very close to 1. In this case the sum is dominated by high-l terms.
Assuming that these dominate completely, we ignore the 1 in the denominator and rewrite Eq. To obtain the closed-form solution we assume that , which is certainly valid for Ramanactive molecules adsorbed onto spherical inclusions of nominally optimal enhancement sizes of ca. 50 nm.
We recognize Eq. (3.7) as the field at q due to its image charge at ( ) at a planar boundary of a semi-infinite medium of dielectric function ε o . This is physically reasonable, because in the limit of large diameters there is little distinction between q being adjacent to the sphere or to a semi-infinite plane. The function of the higher-order terms in the original expansion is also clarified: they narrow the lateral extent of the field to a width of the order of . A main point is that there exists a countably infinite set of resonances from to corresponding to the different multipoles as the sphere diameter increases. In principle any of these singularities forces the displacement of q to zero.

B. Spherical inclusion, first enhancement.
Although a solution can be obtained for any location of the Raman-active molecule, for illustrative purposes we consider the polar and equatorial-plane positions, with the molecule on the z axis at z = d > a in the former case and on the x axis at x = d > a in the latter case. The local field of the sphere is twice as large at a pole, but the equatorial location is more appropriate for Raman-active molecules that are physi-or chemisorbed to the sphere since many Raman-active dye molecules adsorb to metal inclusions such that their transition moments are parallel, rather than perpendicular to the surface. The negative gradient with respect to of Eq. Because r can never be less than a, the maximum fields are obtained with the molecule as close as possible to the sphere. The amplification is purely plasmonic, and independent of the size of the sphere. Although in the equatorial case q moves off the equatorial plane, we can use the zero-order driving fields because small off-axis motions generate harmonics that are not relevant to the current problem. Also, we assume that the local field at any region of the molecule is the same as that at its center, which is acceptable since the Raman response is already first order in small quantities. Hence we can use Eqs. (3.8) as-written.

C. Spherical inclusion, Second enhancement.
The role of the second enhancement at the Raman shifted frequency is central to the electromagnetic mechanism of SERS. To investigate this enhancement, we consider the polar and equatorial locations as above. The response at each location is divided into the charge motion and Raman radiation by the adsorbate.  The increase as the third power of the distance from the center of the sphere is understood in physical terms because at resonance for large d the q-S interaction effectively changes from a qplane interaction at small spacing to a q-dipole interaction with the dipole at the center of the sphere.

C.1 Charge motion, polar location
Before proceeding into a similar discussion on the Raman-radiation part of the problem, we consider some numbers. We suppose a 1 watt pump beam at λ = 514.5 nm focused onto a 10 µm

C.2 Raman radiation, polar location
To calculate the Raman radiation we replace the drive term in Eq. (3.9a) with and repeat the entire calculation for , keeping in Eq. (3.9b) but using in Eqs.

C.3 Charge motion, equatorial location
We retain along z and locate q on the x axis at , θ = π/2. The driving field is given by Eqs.

C.4 Raman radiation, equatorial location
The same discussion applies to the calculation of the Raman component of the displacement, with ε(ω) being evaluated at . The S-q radiation term is similarly reduced. The expressions for the dipoles are now , (3.15) so the apparent singularity at remains. However, as noted above, the expression remains finite because as , and the product remains finite. The capped value is now . (3.16) which is larger than the value given in Eq. (3.13) because the S-q restoring-force interaction is weaker.

A. Finite-size effects
As First, the vector spherical harmonics are defined as [45] (4.1) where .

(4.2)
is defined most conveniently in terms of raising and lowering operators where (ck 1 /ω) 2 = ε and (ck 2 /ω) 2 = 1 Matching boundary conditions at r = a leads to (4.12a) . (4.12b) Thus the singularity near remains, but the first consequence of finite size is to shift the resonance rather than decrease the signal. In fact whether the signal increases or decreases depends critically on the value of ε. With the coefficient actually increases with increasing sphere size. From the above expansion, if we pick k 1 a ~ 0.3 as a measure of where finite-size effects might start to become important, we find ~17 nm for 514.5 nm light. This is a surprisingly small value, but lends credence to the observations that enhancement is optimized in the 50 to 100 nm particle-size range.

B. Radiation damping
The above equations do not take radiation damping into account, although it is easily included by adding a viscous-friction term in the force equation. In principle the presence of the passive resonator improves the coupling of the radiating molecule to the outside world, with the molecule transferring energy to the passive resonator. However, this energy must come from the molecule. In the absence of dissipation the energy is traded between resonator and molecule as (4.13) As dissipation increases, the phase angle between and increases. However, there probably exists a maximum power level that the molecule can deliver. Inclusion of radiation damping has already been shown to lead to a finite bandwidth that defines the quality factor Q.
The most significant aspect of radiation damping is to put further constraints on the magnitude of enhancements, which are only upper bounds in the model here.

C. Alternative configurations
Many different configurations have been proposed in attempts to explain huge enhancement factors. In principle all can be treated by the general theory presented here, which requires only q and a system with which it interacts. Hence it applies to inclusions with depolarization factors other than 1/3, or for that matter a random collection of small particles. In a random collection there will probably always be a few for which resonant conditions are achieved. Note that the theory doesn't even require inclusions -it also describes random networks or contacting particles or whatever, as long as q can induce polarization in its surroundings.

D. Relationship to previous studies of the magnitude of the SERS effect
All previous publications recognize plasmonic amplication at ω o as a primary factor in SERS.
Surprisingly, none of them recognize the need for self-consistency at ω o although the need for self-consistency at the sideband frequencies is discussed in several papers. Even ignoring Δκ and the ω o issue, none of the previous works considered all of the remaining paths.
Gersten et al. [7,8] and Metiu et al. [29,30]  However, for quality factors Q high enough to explain claimed enhancements, only one of each type of enhancement is possible. This is because the Raman shift is sufficiently large for any important vibrational mode that it is not possible for both the incident and scattered frequencies to be near the maximum of their respective enhancement functions. Moreover, unless there is a loss mechanism in the conductor there will be no energy at the Raman frequency because excitation at ω o cannot store energy in the conductor at ω ± . Taken together these factors restrict SERS to the primary enhancement, which is dependent on , which is limited to factors of the order of 10 6 to 10 7 in the most favorable cases.
These results present a dilemma for the experimental observation of large scattering cross sections. Even adjacent inclusions would be unlikely to produce additional field amplifications of 10 7 −10 8 which would be required in order to achieve the reports of enhancement factors of 10 14 . The bandwidth restriction applies to any geometry, so geometries with very high quality factors have very narrowly peaked maximum gain regions. Moreover, large electric field gradients due to large curvatures can produce a considerable shift in the energy of the plasmon, which must be included in the theory for self-consistency. Thus, hot spots should shift the resonant energy significantly to the red in a very narrow bandwidth region. There is at present no direct experimental evidence for this phenomenon, which is required by the physics of the problem. An alternative view of the mechanism for enhancement includes the resonance Raman of the adsorbate molecule and the inclusion as one supermolecule. Our work suggests that holistic treatment of the problem as a resonance Raman phenomenon has a great deal of merit and should be systematically pursued.
Finally, we can postulate a situation where a dipole resonance occurs at one frequency and a higher multipole resonance at a different frequency. Because the radiation emerging from the passive radiator is due to the dipole term, any such higher multipole resonance must be associated with first enhancement. However, energy separations between the dipole and higher resonances are too large for this to be a viable explanation.

VI. ACKNOWLEDGMENTS
DEA is pleased to acknowledge the hospitality of KyungHee University and associated support by the World Class University (WCU) program through the Korea Science and Engineering Foundation funded by the Ministry of Education, Science, and Technology under grant R33-2008-0000-10118-0. SF wishes to acknowledge the hospitality and associated support of Zhejiang University and the Guangbiao Professor program.