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III. THEORETICAL MODELS OF SERS

III. THEORETICAL MODELS OF SERS

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Surface-Enhanced Raman Scattering (SERS)



323



Eq. (1) denotes an average over all possible orientations and positions of the species which scatter the light. For instance, we may

envisage molecules adsorbed on a heterogeneous surface, each

sensing a local field, characterized by a polarizability and associated

with a propagator, all of which may be position dependent.

The enhancement of the Raman scattering, R, is given in these

terms by the ratio

R = (ISD/NS)/(I°D/NQ)

(2)

where N is the total number of molecules and the index 0 denotes

a measurement in the bulk, characterized by a 0 , Go, n0, No, and

Ep = £„ when Et is the laser electric field. For a homogeneous field

bulk system, to which we generally make reference, one remains

only with the average over the orientations of a single molecule:

I°D = Av{|G0K, fts) • ao(eus, a>n H5, fl.) • Et(a>n n,)|2}

(3)

In many cases in the SERS studies one does not have precise

knowledge of the number of adsorbed molecules but only an estimate, Ns. Consequently, an apparent enhancement factor will be

calculated which is different from the actual enhancement by a

factor NR = NJ Ns. Obviously, if NR » 1, then it will seem that

there are surface enhancements. In the inverse case (special active

sites) the real enhancements are higher than the apparent ones. In

the following we assume that the surface coverage is known exactly.

Any one of the terms in Eqs. (1) and (2) can be different on

the surface than it is in the bulk, and thus result in R ^ 1. For

SERS, as considered here, R > 100 and, generally, R > 104.

The difference between the orientational averages in solution

and on the surface can be the reason (or one of them) for the

specific angular and polarization properties of SERS.24"26'309

However, it can affect the intensity only to an insignificant degree

on the scale of the total enhancement. At the most it can cause an

enhancement of 3 x for a molecule with only one large Raman

polarizability principal axis which is oriented perpendicular to the

surface of a planar reflecting metal. Thus, for the sake of simplicity,

we disregard the angular average and the vectorial properties of

all the quantities involved.

Thus, one must search the three quantities Gs, a5, and Ep, for

the source of SERS. Ep and Gs are essentially electromagnetic in

nature and can be obtained by solving the Maxwell equations with



324



S. Efrima



the proper input of the dielectric properties of the various components of the system. Rigorous solutions can be obtained in many

model cases. However, these solutions use (sharp) boundary conditions at the face of the solid, which may limit their validity very

close to the surface, where a gradual transition from phase to phase

is apparent.

The surface changes in the polarizability, a, can be due to

electromagnetic interactions, or to "chemical" effects. By electromagnetic interactions is meant interactions given by macroscopic

Maxwell equations, or their microscopic modification, when the

adsorbed molecules and the substrate retain their own chemical

identity and integrity. Chemical interactions involve some overlap

of the molecular orbitals and those of the substrate.

One may classify the various proposed models in several ways.

One way is to differentiate between models that focus on the role

of the electric field E and the emission G terms (these two are

related), on the one hand, and those that emphasize the role of

changes in the Raman polarizability tensor, on the other. The former

discuss the enhancement in terms of amplified fields, due to the

presence of the surface, which act on the scattering molecule and

its emission being further amplified by the surface. These are the

local field and emission enhancement models (LFE). The difference

between the various models which belong to this group is in the

identification of the specific excitation in the solid which is responsible for the amplification: plasmon polaritons, shape resonances,

electron holes, etc.

The second main group of theoretical models associates the

SERS effect with a large intrinsic polarizability through a resonance

mechanism (RE). The resonating species depend on the specific

model and can be the molecule modified by the presence of the

metal, the metal itself affected by the molecular vibrations, or some

mixed molecule-metal states.

Another classification of the theoretical models can be based

on the nature of the interaction: "electromagnetic" vs. "chemical"

(for which we will be satisfied with the simplistic intuitive definition

given above). Of course, one does not always have a clear

delineation between these two types of interactions.

One should note that most of the theoretical models suggested

are not mutually exclusive, at least not in principle. Thus, it is



Surface-Enhanced Raman Scattering (SERS)



325



possible that the real situation is a composite of several mechanisms.

The exact blend is, at present, a matter of dispute. There is also

the possibility that this blend changes from system to system.

2. Resonance Models (RE)



To this group belong several models of very different kinds such

as the "image" model (RE-IE), the charge transfer (RE-CT) model,

the electron-hole excitation model (RE-EH), and the Raman reflectivity model (RE-RF). These models have very little in common

except that they all lead to enhancements by virtue of a resonance

scattering mechanism. The validity of the last statement is not

always realized by people, but it will be shown below to hold true.

An example of the models in this category is that proposed by

Moskovits.310'311 He proposed that the SERS was a result of a

resonance or preresonance Raman scattering, where the resonating

moieties are the metal roughness features—the "bumps." The localized, plasmonlike states in these bumps are excited by the incident

field, and the vibrations of the molecules which are adsorbed on

their surface scatter as if they are in a resonating molecule. Moskovits showed that the characteristic frequencies of the excitations

in small coinage metal particles should be in the visible, especially

if interactions among them are considered. Thus the special role

of silver, gold, and copper was stressed. Moskovits did not specify

the mechanism that couples the molecular vibrations to the metallic

bump. This mechanism can be any of those described below as

resonance mechanisms. However, this same general outline can

lead to SERS through enhancements of the local fields, which we

classified as a LFE mechanism. This comes to emphasize the point

that all the enhancement mechanisms suggested involve some type

of resonative process. The difference between them is based on

their major effect: large fields (and amplified emission) vs. resonance of the scattering center itself.

(i) Renormalization ("Image'") Model (RE-IE)

One of the earliest models proposed to explain SERS was

suggested and developed by Efrima and Metiu24'26'312"315 and also

by King et al.316 It is based on the classical theory of Raman



326



S. Efrima



scattering and on classical electrodynamics. The main idea is that

an excited (or virtually excited) molecule can be described by a

classical oscillating electric dipole. It is induced by the incoming

laser field. This dipole exerts fields on the surroundings. One component decreases as the first power of the distance and is the

radiative, scattered field. The other components, the so-called near

field,317 decrease as the square and the cube of the reciprocal

distance. They are extremely small far from the molecule, but can

become exceptionally strong at close proximity to the molecule.

This field is responsible for energy transfer processes such as the

Forster mechanism. The oscillating dipole, near a surface, strongly

polarizes the metal which exerts a field back on the molecule, which

further polarizes it. The larger molecular dipole induces additional

polarization in the metal and so on. The net result is a much larger

induced molecular dipole than would be achieved in the absence

of the highly polarizable surface. Such a large oscillating dipole

scatters much more light which amounts to a measured enhancement. In molecular terms, as will be shown below, this selfpolarizing effect means a shift of the molecular levels to attain a

state separation which is, under certain circumstances, in resonance

with the incident light. Thus the molecule which may be a normal

Raman scatterer in solution turns into a resonance scatterer on the

surface.

The simplest presentation of this mechanism starts from an

expression which gives the induced dipole /JL in terms of the total

field ET operating on it:

\x = a0ET

(4)

where a0 is the molecular Rayleigh polarizability, in the absence

of the surface. The total field is composed of the incident laser field

(perhaps modified by the surface) Ep and a field exerted by the

induced polarization in the metal which is proportional to the dipole

moment and falls off as the distance to the surface, Z, cubed:

ET = Ep + yfJL/(2Z)3



(5)



Here y denotes a metal response function and the factor 2 was

written in order to obtain the form of an image interaction. From

Eqs. (4) and (5) one obtains:

M = OLSEP



(6)



Surface-Enhanced Raman Scattering (SERS)



327



with the surface polarizability as given by

<** = «o/[l " 7«o/(2Z) 3 ]



(7)



In the absence of the metal, y = 0 and as = a0. However, in its

presence, y may be large so that the whole denominator becomes

very small and the polarizability very large. This is what is generally

called the image mechanism. In this form it amounts to calculating

y and checking whether it can be large enough and in the appropriate direction, y generally has an imaginary part, which remains

even when the denominator of Eq. (7) vanishes. This will determine

the maximum value of the surface polarizability. For large enhancements, one needs small contributions of the imaginary part. This

is also a matter of concern in all calculations of y.

Before discussing calculations of % which is at the core of this

mechanism, we show the equivalence of this model to a resonance

enhancement mechanism. We take a simple form of the molecular

polarizability,

a0 = (fe2/m)/(co20 -co2-



2icoT)



(8)



where / is the oscillator strength, e the electronic charge, m the

electron mass, co0 a self-frequency of the molecule (an electronic

transition frequency in a molecule in a quantum-mechanical picture), F the transition width, and to the frequency of the exciting

light. Inserting this expression for the polarizability into Eq. (7)

one obtains

as = (fe2/m)/[co2 -co2-



2cobs - 2ico(T + Ts)]



(9)



Here we defined the surface shift A5 and the surface width Ts as

the real and imaginary parts of (fe2/m)y/(2Z)3,

respectively. In

the absence of the surface, the resonance condition is attained for

co = COQ. On the surface, the resonance condition is approximately

co = co0 - A 5



(10)



and the total width has increased by Ts. This establishes the connection between the "image" model and a surface-induced resonance

Raman process. Incidentally, such a picture was developed even

prior to the discovery of SERS by Philpott,318 and enhancements

were predicted assuming increased widths (not shifts).



328



S. Efrima



The Raman polarizability will be obtained from as by differentiating with respect to a vibrational coordinate Q, which gives the

leading term in the scattered intensity

I'D ~ klViD ~ yao/(2Z)3f\dao/dQ\2

24 26



(11)



316



Efrima and Metiu, " and King et al. suggested that y could

be calculated on the basis of reflection from a plane metal surface.

They find

y = 2(eM - es)/(eM + es)



(12)



Here eM and es are the frequency-dependent dielectric constants

of the metal and the solution, respectively. Using known pyridine

polarizability and silver dielectric data,319 large enhancements could

be obtained (up to 107). In terms of the molecular picture, a

several-eV decrease of level spacing was involved. This shift,

however, strongly depends on the frequency, through the dielectric

constant of the metal. This is a dynamic shift and the resonance is

really a joint metal-molecule-photon excitation. This is different

from a shift of levels under static fields. This point has often been

misunderstood.

The consequences of this model are:

1. For realistic molecular polarizabilities a close proximity

to the surface is needed. For a normal Raman scatterer,

generally only first-layer enhancement is expected (as seen

below, roughness can increase this range to 1-2 nm); for

a preresonance scatterer, the effect can be at longer distances, however, the enhancement will be smaller than for

a NR scatterer.

2. No further enhancement is expected of a molecule which

is already in resonance. This is a common feature of all

RE mechanisms.

3. Coinage metals will support SERS at appropriate frequencies, due to their dielectric properties. Silver is especially

good due to the relatively small imaginary part of the

dielectric constant. Other metals may be SERS active, Hg

for instance, especially for preresonance scatterers (i.e.,

when the polarizability is already relatively large).

4. The excitation spectrum should be strongly related to the

dielectric response of the metal-molecule system. For a



Surface-Enhanced Raman Scattering (SERS)



329



NR scatterer, the maximum of enhancement should appear

on the long-wavelength side of the metal substrate excitation.

5. No direct dependence on surface roughness! (However,

note below that roughness does increase the "image" interaction and changes the dielectric response.)

6. The dependence on electric potential stems from the

dependence of the dielectric properties on it, and it can

be a strong dependence as the enhancement is nonlinear

with the dielectric response.

The model has the disadvantage (and advantage) of being

extremely simplistic: classical electrodynamics, plane surface, point

dipole, etc., concepts are involved. From the point of view of theory,

using a classical electromagnetic treatment at a distance of 0.1 nm

from the surface can send shudders down the spine of any decent

theoretician. Using a point-dipole approximation does not help

either. Hilton and Oxtoby320 indeed criticized this model for the

use of the dipole approximation. In a series of papers,321323 Maniv

and Metiu used a quantum-mechanical RPA (random-phase

approximation), infinite-barrier method to investigate the validity

of macroscopic electrodynamics at the surface (see also Feibelman324"326). They find that the macroscopic results, the Fresnel

reflectivity, for instance, are exact outside of the metal electronic

cloud, are still reasonable about halfway between the infinite barrier

and the jellium edge, and for closer distances are simply inapplicable. For a dipole near the surface,327329 they find that the simple

image interaction is not a good approximation. Outside the electron

cloud, it is too small, because of "spillout" effects (the polarizable

electrons are closer to the molecule and the "image plane" is closer;

see also Efrima330). As the molecule penetrates the electronic cloud,

strong screening sets in and the interaction becomes much smaller

than the "image." Their conclusion is that a corrected image model

could still produce, somewhat reduced, but still substantial,

enhancements, but only at higher frequencies than those predicted

on the basis of the clasical image, probably in the blue (for silver).

This model still uses the point-dipole approximation and the surface

is taken as flat.

Weber and Ford331"333 introduce finite-size effects and also

close-range dispersion for a rough surface in a classical electrody-



330



S. Efrima



namic model. To simulate the size, they perform their calculation

for a polarizable sphere, and the dispersion is introduced via a

Lindhard dielectric function.334 They find that the "image" enhancement is much reduced, but still considerable,~1000.

Agarwal et al.335 showed, using a point-dipole, classical electrodynamic calculation, that the image can be considerable near a

metal sphere (simulating a roughness feature) of about 2-3 nm and

with similar molecule-sphere distances. At such distances the

approximations are tolerable, so that their results are dependable.

The reason for the increase in the interaction is a coupling to the

surface plasmon (or shape resonance), which in the sphere can

occur at lower frequencies than for a flat surface.

Incidentally, any mechanism which lowers the plasmon

frequency (by shape effects or by interaction between small particles

sustaining the local plasmon oscillations) will result in a stronger

"image" interaction, provided dissipation does not increase to any

large extent. Furthermore, roughness promotes efficient coupling

also by providing wavenumber components that help in conserving

momentum (i.e., conservation of "crystal" momentum).

Wood336 notes that the static image interaction with a sphere

is smaller than with a plane. Note, however, that the coupling at

optical frequencies is stronger due to the effects mentioned above.

Gersten337 showed that at the tip of a perfect metal ellipsoid of

aspect ratio of 10:1 to 50:1 the "image" is small at any reasonable

distance. Gersten and Nitzan338 show, on the other hand, that at

the tip of an ellipsoid conductor characterized by silver dielectric

properties, the image can be of importance at distances of 0.10.25 nm. Kerker et al339 remark that in the limit of the small

spherical particle (particle radius smaller than the wavelength) the

image enhancement is small, but they give no details.

Eesley and Smith340 have carried out a classical electrodynamic

calculation, which is essentially an "image" model. They also

consider the interaction between neighboring molecules, using a

dipole-dipole coupling including images. The most significant

result they present is a dependence of the scattered intensity on

the second power of the incident intensity! No other model has

this prediction, and it can easily be checked experimentally. Another

point of interest is the coverage dependence, at the submonolayer

level. Due to the depolarizing effect of the neighboring molecules,



Surface-Enhanced Raman Scattering (SERS)



331



the Raman cross section should decrease at coverages higher than

0.1-0.3. This effect can also be represented simply by an average

dielectric constant of the adsorbed layer, decreasing the field

intensity sensed by the individual molecules. However, an increase

in the dielectric constant of the surface layer affects the metal

excitations as well, a factor which was not considered in this work.

Arunkumar and Bradley341 carry out a similar calculation, but

include the polarizing and depolarizing effect of the other molecules

and the roughness features. They assume that atomic-scale surface

roughness can reflect the near field of the molecules that are neighbors to the site of the scattering molecule. If the polarizing effect

is larger than the depolarizing effect, then enhancements are expected. Large surface-roughness features are found to increase the

contribution of the polarizing reflections (probably due to allowing

favorable molecular orientations with respect to each other). The

expected enhancements are >105 for the atomic-scale roughened

surface and higher by several orders of magnitude, when microscopic features are present.

Li,342 in a classical treatment based on the Drude equation,

considers also nonlinear effects. The model is essentially an "image"

model.

There are several theoretical models which are essentially

image models, but apply quantum-mechanical treatments. Lee and

Birman343345 find "coupled-system eigenstates" created from the

electrodynamic interaction of the molecular-induced dipole and

the polarizable metal medium, which are analogous to the shifted

molecule-metal-photon states of the classical treatment. They

include dispersion, which is important at short distances, but still

retain a planar surface. Note that they have a sharp boundary, i.e.,

no allowance is given to the penetration of the polarizable electron

cloud outside the positive-charge background of the metal, an effect

which generally increases the interaction. They find a peaked dependence of the enhancement on the frequency, near the surfaceplasmon excitation or even on the short-wavelength side of it, and

a maximum enhancement of a 1000. If surface structure will push

the surface excitations into the visible range, then this calculation

predicts "image" enhancements of this order of magnitude.

The treatment of Eguiluz346 for the "image" interaction strength

also shows that dispersion tends to decrease it.



332



S. Efrima



Ferrell,347 using a simple quantum-mechanical approach to

investigate the case of a molecule adsorbed on a small sphere, finds

renormalized states, which enable a resonance process to arise at

lower energies than in the solution. He predicts an enhancement

of 104, but does not state at what molecule-surface distances it is

calculated, nor whether such distances which cause a resonance

are realizable.

Arya and Zeyher348 develop a general many-body theory for

SERS. In the limit of a planar surface and only plasmon contributions to the interactions, they find an enhancement of a factor of

100 for the "renormalization" model and it is relatively weakly

dependent on the excitation frequency. This is for silver of course.

For Cu or Au, they predict even smaller "image" effects, <10.

To summarize the "renormalization," "image" enhancement

mechanism, it is important to note that all estimates predict

maximum enhancements ranging from 100-108, located at frequencies close to the metal excitations (plasmon or shape resonances

or electron hole, for that matter). If such excitations are possible

in the visible range, then this mechanism should be an important

contributor to SERS. The "image" mechanism, also, contrary to

popular belief, is sensitive to surface roughness, and generally

favorably so. It therefore can be stronger at specific SERS-active

"sites" on the surface. It is also molecule sensitive through the

molecular (Rayleigh) polarizability.

(ii) Charge Transfer Models (RE-CT)

Basically, this set of models claims that SERS is a result of a

resonance Raman process, where the molecule avails itself of the

unoccupied states of the metal, or vice versa. Thus one can envisage

a virtual intermediate transition to occur not between pure

molecular states, but between a molecular and a metal state. This

model has the advantage of a clear chemical picture, but it does

not easily lend itself to rigorous calculations leading to quantitative

predictions.

Aussenegg and Lippitsch349 have explained Raman scattering

enhancements in charge transfer complexes as compared to the

separate molecules, on the basis of this charge transfer. They found

enhancement factors of about 2, in agreement with experiment.



Surface-Enhanced Raman Scattering (SERS)



333



They suggested that the same model is valid for SERS, where due

to the high polarizability of the metal electrons, large enhancements

are expected. They also predict that the molecule-metal vibrational

mode will be exceptionally intense, as it strongly affects the degree

of charge transfer.

King and Schatz350 investigated the effect of charge transfer

on the transition dipoles (oscillator strengths) determining the

Raman activity. This charge transfer, in their model, is promoted

by static electric fields at the surface. For the case of small oscillator

strengths, a large enhancement is predicted, due to the strong

coupling through the metal states.

Burstein et al.351 discussed several possible mechanism involving electron-hole pairs and charge transfer (CT). They expect a

threshold behavior at a frequency where the photon energy plus

the molecular ground-state energy surpass the Fermi level in the

solid, for a molecule-to-metal charge transfer. In the case of metalto-molecule charge transfer, an onset should appear at the frequency

that equals the difference between the Fermi level and the lowest

unoccupied molecular state (LUMO).

Gersten et al352 treat this model using an Anderson model.

They note a threshold behavior, as mentioned above. A dependence

on the electric potential is also predicted. When the virtual excited

state resides on the molecule (metal-molecule CT), then more

positive potentials will require higher excitation frequencies. For

molecule-to-metal CT, more positive potentials should be associated with lower frequencies for SERS. The CT model also rationalizes the continuum background as real transitions which occurred

and had suffered some decay in the metal.

Ueba et al353 35S have presented a comprehensive treatment

of the CT model. Based on a Fano-type formalism they predict an

enhancement of ~ 100 for pyridine on smooth silver. When roughness is added, in the form of ad-atoms, the enhancement increases

to 1000. The excitation profile exhibits a resonance shape, not an

onset. Ueba359'360 applied his model to the Raman scattering of

molecules adsorbed on semiconductors and interacting with them

via the excitonic states. Persson361 and Kirtley et al362 using similar

but not identical models predict the same order of magnitude of

CT enhancements, ~ 100. Adrian363 calculated enhancements of

10-1000, again, provided the threshold conditions discussed above



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