Theory of surface enhanced Raman scattering in colloidsRoberto Rojas V and F. Claro Citation: The Journal of Chemical Physics 98, 998 (1993); doi: 10.1063/1.464263 View online: http://dx.doi.org/10.1063/1.464263 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/98/2?ver=pdfcov Published by the AIP Publishing Articles you may be interested in The theory of surface-enhanced Raman scattering J. Chem. Phys. 136, 144704 (2012); 10.1063/1.3698292 Surface‐Enhanced Raman Scattering of Hydroxyproline in Gold Colloids AIP Conf. Proc. 1267, 922 (2010); 10.1063/1.3482891 A Unified Theory Of Surface Enhanced Raman Scattering AIP Conf. Proc. 1267, 62 (2010); 10.1063/1.3482717 On the chemical mechanism of surface enhanced Raman scattering: Experiment and theory J. Chem. Phys. 108, 5013 (1998); 10.1063/1.475909 Theory of surface enhanced Raman scattering J. Chem. Phys. 78, 2882 (1983); 10.1063/1.445247

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Theory of surface enhanced Raman scattering in colloids Roberto Rojas V Universidad Tecnica Federico Santa Maria, Departamento de Fisica, Casilla nO-v, Valparaiso, Chile

F. Claro Pontificia Universidad Cat6lica de Chile, Facultad de Fisica, Casilla 6177, Santiago, Chile

(Received 16 July 1992; accepted 23 September 1992)

We present a theory of surface enhanced Raman scattering (SERS) that gives a response enhancement of up to ten orders of magnitude for pyridine on silver. The signal obtained depends on the laser frequency and is consistent with available experimental data. Our theory emphasizes the effect of the molecule itself on the polarization of the environment as well as the effect of high multipoles, required by the nonuniformity of the local fields. The host could be metallic or insulating. We explain the laser frequency dependence of the relative size of Raman signals for different vibrational lines and predict the presence of a doublet at each electromagnetic resonance. The effect on the Raman signal of depositing a metallic or an oxide coating over a metallic host is discussed.

I. INTRODUCTION

Since the work by Fleischman et af. reporting a Raman signal from pyridine molecules adsorbed on a silver elec­trode, and later evidence of its enhancement by six orders of magnitude with respect to free molecules in a liquid environment, the field of surface enhanced Raman scatter­ing (SERS) has been very active and the interpretation of the effect controversial. l- 7 In spite of the enormous accu­mulated data, more experimental and theoretical work is at present necessary in order to fully understand its multiple aspects. Some excellent review articles discuss the various aspects that seem to play a role in the effect­electromagnetic enhancement (surface plasmon excita­tions), roughness scales (atomic, submicroscopic, and mi­croscopic), SERS-active surfaces (colloids, cold-deposited, and metal-island films), short-range effects, chemical en­hancement (charge transfer excitations), chemical speci­ficity, etc.3-7

Typical models of the electromagnetic contribution to SERS treat a molecule near a metallic sphere, or a mole­cule in between two metallic spheres forming a small clus­ter.8-18 Induced surface plasmon excitations at or near res­onance give rise to an electric field enhancement, its largest values being along the common direction of the external field and the axis joining the two or three particles.8 These treatments usually ignore the effect of the molecule in ac­counting for the response of the metal, or rely on the sim­ple dipole approximation. It is well known that the latter fails when the interacting particles (the molecule and the spheres) are very near each other,19-21 a regime certainly most important to SERS. High-order multipoles are ex­cited due to the nonuniformity of the field in the neighbor­hood of curved conducting surfaces; these muItipoles pro­duce their own short-range field and accurate calculations must take into account the self-consistent total field with adequate multipolar convergence. In this work, we develop a theory that treats the electromagnetic interaction in all detail, including the effect of the molecule itself on the self-consistent local fields, and using all multipoles needed

to obtain converged results, as required in SERS geome­tries where the enhancement is largest. The long wave­length (Rayleigh) limit is assumed.

Experimental data not well understood at present in­cludes the quenching of the enhancement by a thin layer of a second metal deposited over the Ag surface.22 By extend­ing our theory to coated metallic surfaces, we show that the quenching effect is a natural result in the electromag­netic approach, and it takes place whether the coating is metallic or insulating (an oxide), although with a peculiar frequency dependence for each case. Also, changes in the relative signal size for different vibrational modes of the molecule have been observed experimentally.23 As we show, our theory is capable of explaining this effect as well. Furthermore, we predict that the Raman signal size may be optimized if the physical environment of the molecule and the laser frequency are adjusted properly so that the latter and the plasmon resonance of the interacting system coincide. We also show that the latter is usually a doublet.

In Sec. II, a dipolar theory is developed, mostly to introduce the proper definition of the enhancement factor. We show that for very dilute colloids, the dipole approxi­mation may be used safely. We discuss in detail the range of validity of the usual approximation that makes the SERS signal proportional to the fourth power of the local field, and show that it often gives inaccurate results, most notably when clustering is involved. In Sec. III, we extend the theory to include arbitrary multipolar order, as needed for modeling clusters of spherical shapes. In Sec. IV, we apply our formula to obtain numerical results for pyridine on a Ag substrate free and coated with Pd. We also give results for tin, free and with an oxide coating. The conclu­sions are finally presented in Sec. V.

II. MOLECULE OVER A SPHERE: DIPOLAR APPROXIMATION

We consider a molecule near a metallic sphere of ra­dius b, both under the influence of an external electromag­netic wave of frequency co and wavelength A.. We are in-

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R. Rojas and F. Claro; SERS in colloids 999

teres ted in effects due to coupling between the particles, which are largest when the polarization of the wave is such that the electric field is parallel to the axis joining their centers,20,21 a geometry we shall adopt here. The particles are much smaller than the wavelength of the incident ra­diation, so that the usual long wavelength limit may be applied. In this approximation, the elastic scattering in the system is mediated by the dipole moments induced in both particles

(la)

(lb)

where Eo is the magnitude of the external field, a~«(i) is the elastic dipole polarizability of the molecule, as( (i) is the dipole polarizability of the sphere, and D is the distance between the centers of the sphere and the molecule. The square brackets are the local fields affecting each particle. By solving these coupled equations one obtains

Pm «(i) =a~«(i) )Eo ,

Ps«(i) =as«(i)Eo ·

(2a)

(2b)

We have expressed the dipole moments in terms of effective polarizabilities characterizing the response of molecule and sphere to the external field when in the presence of the other particle. They are defined by

1+{[2asC(i)]lD3} E

1-{[ 4a!«(i)asC(i)]I D6} am«(i), (2c)

Notice that when the center-to-center distance D is very large, these quantities acquire the value appropriate for the isolated particle. They thus include the effect of interac­tion. Using Eq. (2b) in Eq. (la), one obtains the actual local field affecting the molecule

EL«(i) =gl«(i)Eo , (3a)

where g 1 «(i) is the frequency-dependent factor

(3b)

The Raman scattering by a molecule is an inelastic process at frequency (i)R=(i)±(i)u' where (i)u is one of the characteristic vibration frequencies of the molecule. A di­pole moment is excited at (i) R and depends on the local field E L «(i) through

(4a)

Here a~ «(i) R,(i) is an inelastic polarlzability corresponding to th~ Raman emission. This new dipole itself creates a field EL«(i)R) at frequency (i)R that contributes to the over­all local field in its neighborhood. As a response to this

field, a dipole moment at the same frequency is in tum excited in the metallic sphere due entirely to the excitation in the molecule

PsC(i)R) =asC(i)R)EL«(i)R)

with

2Pm«(i)R) EL«(i)R) D3

(4b)

(5)

The total dipole moment p«(i),(i)u) producing the Raman signal is the sum of dipoles at the molecule and the sphere at the frequency (i) R' By using Eqs. (4) and (5), this dipole moment may be written as

p«(i),(i)u) =gl «(i) )g2 «(i) R)a~«(i) R,(i) )Eo

with

(6)

(7)

Note that in the absence of the sphere (D ..... 00 ), g 1 «(i) ) =g2«(i)R) = 1 and the total dipole moment at the Raman frequency is simply a~«(i)R,(i)Eo as expected.

Because the intensity of the radiation field is propor­tional to the square of the total dipole moment, there is a Raman signal enhancement G due to the presence of a sphere

(8)

This relation is the origin of a well-known approximation relating the Raman intensity to the fourth power of the local field. Because g 1 «(i) is the enhancement factor affect­ing the local field factor that excites the molecule, the ap­proximation g2«(i)R) =gl «(i) is required in order to assume a fourth power law in the Raman signal. This equality does not hold in general. In fact, the quantities gl and g2 could become very different, as shown below. Expanding the first in powers of D-3 and the second in powers of Y=(i)u /(i) and retaining only terms of lowest orders, one gets

gl «(i) = 1 +2asC(i) [1 +2am«(i)/ D3+ ... ]I D3,

x:; + .... ] / D3.

For the two quantities to be approximately equal, the sec­ond term in each set of parentheses must be small. Since am~R6< D3

, where Ro is the radius of the pyridine mol­ecule, the criterion will usually be satisfied in gl' In g2' however, near the plasmon resonance €+2€h::::::O, the cri­terion is not satisfied and g2=Fgl' Furthermore, as we shall see in Sec. IV, these quantities differ by orders of magni­tude in a broad range of frequencies when the molecule is placed in the narrow space between two metal particles and the approximation is in such case completely incorrect.

In this section, our treatment has only considered di­polar excitations. It is well known that this approximation may be applied safely to the case of two unequal particles

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1000 R. Rojas and F. Claro: SERS in colloids

when their edge-to-edge separation is of the order, or larger than the radius of the smaller particle.24 It is reasonable to believe that for a small molecule sitting on a large metallic sphere, such a criterion is satisfied, so we expect that for an isolated pair, such as will occur most frequently in dilute colloids, multipolar effects are small. This is confirmed by our numerical results as discussed in Sec. IV. A quite dif­ferent situation arises when a third particle is in the vicinity of the pair. Multipolar couplings may then become most important even when the above criterion is satisfied. This effect is discussed below.

III. MOLECULE IN A TRAP: MULTIPOLAR TREATMENT

In Raman scattering experiments, the excitation fre­quency w is normally far below the surface plasmon reso­nance of an isolated metallic sphere which occurs at wp /yj. It is at this resonance frequency that the field be­comes largest in its vicinity. A small shift towards lower frequencies is provided by the coupling of the metal with the active molecule, producing increased fields near the former and therefore a somewhat enhanced Raman signal from the molecule. Still larger fields and lower resonance frequencies occur in a system of two spheres very near each other.8,l9-21,25 A molecule trapped in the narrow space be­tween two curved metallic surfaces would thus give large Raman signals even at laser frequencies in the optical re­gion. This situation is thus a good candidate to explain large Raman signals in aggregated colloids and metal is­land films, where active molecules tend to be those in such traps.

It is well known that when the edge-to-edge separation between two spheres of the same size is smaller than their radius, the dipolar approximation fails and the excitation of higher-order multipoles must be taken into account. 19 In this section, we develop a complete mUltipolar theory for the Raman response of molecules when placed in between two spheres. In our treatment, we include for the first time the coupling of the spheres with the molecule itself. As we shall see, the red shift of the plasmon resonance by the multipolar terms is enough to place it in the optical region. Our results of this section can also be used to obtain a multipolar generalization of the expressions of the last sec­tion, which describe the case of a molecule near an isolated sphere.

Consider a molecule (index i=2) at z=O placed in between two spheres centered at z= - D and z= D (la­beled i=l and i=3, respectively). For simplicity, we as­sume the spheres to be identical. An external electric field Eo parallel to the z axis excites the system at frequency w. In terms of the multipolar moments excited in both spheres, the local electric field at the site of the molecule is given by [see Eq. (A7) in Appendix A]

'" ~. '[-Ql,l-t-(_)I+lQ1,3] EL(w)=Eo+ 1~1 V2l+1 (/+1) DI+2 ,

(9)

where the multipolar moments Ql,i obey Eqs. (Bl) (see Appendix B) appropriate for a system of three particles. In

analogy with the dipolar expression (3a), the local field may be written, using the results of Appendix B, as E dw) =g, (w ) Eo, where g, (w) is now given by

'" ~ (l+I)a1ali , ,. g,(w)=l+ ;::, i7:, D1+2ai' [(T- )i,'{

+ (-1 )1+l(T-')J,1 ]. (10)

Here T-' is the inverse of the matrix T defined in Eq. (B8), a',i is the dipole polarizability of particle i[a,,2 =a~(w)], and ai is its radius, and we have assumed a, =a3' a2=Ro. We are interested in the local field where the molecule sits, so Eq. (10) includes the effect of the latter only through the intermediacy of the spheres and this is accounted for by terms with i=2.

The electric field that excites the Raman response of the molecule is the local field at frequency w at the site the molecule is in, as discussed in relation to Eq. (4a). As before, we assume that the inelastic dipole polarizability is a;;'(wR,w), independent of the magnitude of the local field. Then its dipole moment is given by Eqs. (3a) and (4a)

Pm(WR) = a;;' (WR'W )g, (w )Eo , (11)

where g, (w) is now the three-particle fully mUltipolar field factor given in Eq. (10). This dipole moment in the mol­ecule excites the response of the spheres at the frequency of the Raman signal. The total dipole moment p(w,wv ) re­sponsible for the Raman signal is the sum of the dipole moments induced in the three particles at frequency w R,

and is proportional to the dipole moment Pm(WR) of the molecule. By using Eq. (11) and results of Appendix C, the total dipole moment may be written as in Eq. (6), with g, (w) given by Eq. (10) and

'" 3 3

g2(wR)=I+ l~' i~' i/~' :~ (S-')y:; iiJ,:', (12)

where the matrix S is defined in Eq. (C8). The overall enhancement factor of the Raman intensity due to the presence of the spheres is again expression (8), with g, (li)) and g2(WR) given by Eqs. (10) and (12), which include multipoles to all orders.

Results for the case of a molecule placed at the origin z=O and a single sphere centered at z= - D may be ob­tained simply by excluding from Eqs. (10) and (12) terms with particle indices i or i' equal to 3. This allows for a generalization to arbitrary multipole order of the' results of Sec. II.

IV. NUMERICAL RESULTS

As an illustration of the theory developed in the pre­vious sections, we shall present numerical results for the SERS enhancement factor for pyridine molecules in two different environments-near a single spherical metallic surface and placed in between two such surfaces. These are geometries found in colloid systems, the latter when the particles form clusters. They could also be used to model molecules placed near spherical bumps, as in metal island films. Our numerical results show enhancements of up to

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R. Rojas and F. Claro: SERS in colloids 1001

ten orders of magnitude if the metal is silver and even larger for tin. We also give results for a metal with a coat­ing.

The multipolar polarizabilities of a spherical surface of radius R and dielectric constant e(CU) has the form

(3[=/3[(e,eh)R2[+ I,

where

e(CU) + [(1+ 1)/I]eh·

(13a)

(13b)

We call this latter quantity the dimensionless polarizabil­ity. The system is assumed embedded in a homogeneous medium of dielectric constant eh. For the metallic surface, we use the Drude model

CU2

e(CU) =eh cu(cu:iy)' (14)

with eh a background dielectric constant associated with bound electrons in the metal (interband transition term), cup the plasma frequency, and y the electron collision fre­quency. The effective multipolar polarizabilities of the mol­ecule are estimated from the spherical particle model by writing /3[ in terms of 131,

31 {3[ (2/+ 1) + (I_1){3/1 .

(15)

For pyridine, we use al = 11.3 A? and Ro= 1.6 A, as given by Ford et al. 4

We shall also report results for coated spherical sur­faces, using as dimensionless polarizabilities,26

where e f( cu) is the dielectric function of the coating ma­terial, a is the radius of the metallic core, and b is the external radius of the coated particle.

Figure 1 shows the logarithm of the SERS enhance­ment factor G for a pyridine molecule placed over the sur­face of an isolated Ag sphere of radius 100 times bigger. We used the parameters appropriate for silver in the range 450-700 nm, eh=3.6, cup =75256 cm- I

, and y=468 cm -1.4 The center of the active portion of the molecule is assumed 2.4 A away from the metallic surface. In our examples, we have set €h= 1. The converged multipolar results (full line) as well as the dipole approximation (dashed line) have been included in the figure. The dipolar curve agrees with previous calculations done in this ap­proximation 18 and, as axpected from the discussion in Sec. II, is shown to depart little from the more accurate multi­polar results. The enhancement may thus be obtained in this case by simply assuming that only dipole moments are induced on the sphere and molecule, and multipoles need not be taken into account. Note that in Ag, the largest value is about seven orders of magnitude and is attained near cup! ~eh+2 'Z. 31800 cm- I in the ·ultraviolet. Th~se

0.2 0.5

5.0

(!)

S! .9'0.0

C) @

I .. 0 ,.1

-5.0 20 50

FIG. 1. The Raman intensity enhancement factor G for the 1005 cm- 1

line of a pyridine molecule over a silver sphere. The dipole approximation (dashed line) and the result including all relevant multipoles [/m§x=48 (solid line)] are shown. In the notation of the inset, b=160 A and D= 162.4 A. The heavy line in the upper horizontal axis of this and the next figures mark the visible region.

results are appropriate to a very dilute colloid, where clus­ter formation is improbable.

In Fig. 2, we show the case of a pyridine molecule placed in the narrow space between two 160 A radius Ag spheres very near each other. As in the previous case, the active center of the molecule is 2.4 A away from the me­tallic surfaces. In order to reach convergence in the numer­ical evaluation of Eqs. (10) and (12), it was necessary to include up to multipole order 1=48. For pure Ag (solid

FIG. 2. The same as Fig. 1 for pyridine between two uncoated silver spheres (solid line) and coated with b-a= 14.5 A thickness of Pd (dashed line). Only the multipolar results are shown. Dots mark wave­numbers used in Fig. 3.

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1002 R. Rojas and F. Claro: SERS in colloids

10.0 =--------------~~---,,---, --21,222 cm-1

9.0

<!)

!2 B.O Ol o

-.- 23,329 cm-1

--- 16,556cm-1

/ __ 0_°-7.0 / .-.-.-._._._._._._

! -------------------------j,.. ..... -

o 10 20 Paladium coverage (monolayers)

FIG. 3. Enhancement for the system and geometry of Fig. 2 as a function of Pd coating thickness. Three representative wavenumbers around the main peak are shown.

line), we find enhancements of up to ten orders of magni­tude in the optical region, where laser frequencies usually lie. We note that the main peaks that appear in the figure follow the plasma resonances associated with the excitation of multipolar modes in the system.25

As discussed in the previous sections, besides the local field, represented by the factor gl in Eq. (8), a second (generally smaller) factor g2 intervenes in the enhance­ment. The doublet appearing in each peak of Figs. 1 and 2 is due to the fact that each separate factor has a maximum at a different frequency, separated by the pyridine line be­ing considered (1005 cm- I in the figures). If, as is often done, the fourth power of the local field had been used to calculate the enhancement, a still larger maximum would have been obtained as high as 14 orders of magnitude in the case of Fig. 2. This example illustrates that use of this fourth power rule can yield grossly inaccurate results.

Figure 2 also shows the case of Ag coated with a 15 A layer ofPd, leaving a metallic core of 145 A (broken line). For Pd, we have used the Drude model (14) with param­eters Eb= 1, wp=60 495 cm-I, and -Jiy=6856 cm- I •

27 The largest enhancement found in the case of uncoated surfaces is lowered by about two orders of magnitude by the Pd coating and the peaked structure in the frequency depen­dence is essentially gone. Experiments in Ag island films coated with 2.5 A of Pd exhibit a reduction of SERS in­tensity by a factor of 50.22 Our results are consistent with this finding.

Figure 3 shows the SERS peak intensity reduction with coating thickness for the same geometry as in Fig. 2. We kept the external radius b of the spheres and the distance D between them fixed as the coating thickness increases. The continuous line was obtained at a laser frequency of2l 222 cm-I, the dashed-dotted line at 23 329 cm -1, and the dashed line at 16556 cm- I . They illustrate very different behavior that depends on frequency. The frequencies were chosen at the highest point of the leftmost resonance in

0.3 0.4

tin I tin oxide

-- uncoated

10.0 ---- coated

!2 Ol .9

5.0

15 25 Laser frequency ( 103 cm-1)

FIG. 4. The enhancement factor for the same geometry as in Fig. 2 for tin metal uncoated (full line) and coated with a 15 A layer of tin oxide (dashed line).

Fig. 2, at the bottom of the dip to the right of such a point, and at a point to its left, marked by dots in the figure. In all cases, there is a decrease in the enhancement at sufficiently thick coatings, although in the thin region, a local increase is possible. We remark that Fig. 2 is for a fixed geometry, whereas in a sample with disorder, molecules will be found in a range of gemetries that contribute differently to the enhancement. In our model, this variety may be repre­sented by a range of separations D, radii b, etc. For in­stance, an increase of D yields a frequency response as the one in Fig. 2, only that the resonances are shifted towards higher frequencies and somewhat less peaked. The distri­bution of responses by many molecules in different envi­ronments excited at the same frequency will roughly re­semble the response of a single molecule over a range of frequencies in the resonance region of Fig. 2. That being the case, it is clear from Fig. 2 that the coating (dashed curve) acts to suppress the largest contribution to the en­hancement in the resonance region, and thus the overall Raman signal. We also remark that since the surface res­onances of different metals occur roughly in the same fre­quency range, the detail effect of metallic coating will differ when different metals are combined, and a special analysis should be made using our theory for any given specific case.

At frequencies below the main resonance, a thin metal­over-metal coating has little effect over the enhancement, as shown in Figs. 2 and 3. The situation is different if the metallic surface has an oxide coating instead. Figure 4, obtained for tin covered with a 15 A layer of tin oxide, illustrates this point. The same geometry as in Fig. 2 has been assumed. For the metal, we used the parameter values [see Eq. (14)] Eb= 1, lUp=62108 seg- I

, and y=vFla, where VF= 1.24 X 106 mls is the Fermi velocity and a= 145 A (160 A) for the coated (uncoated) metal. For the oxide, we used the dielectric function E= 1.572+2.l5IF1 +0.05/

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R. Rojas and F. Claro: SERS in colloids 1003

W/Wp 0.2 0.3

10.0 fl'"'""-----"'P"'"-----... -------,

o

f

,.. I \

I \ I \

I \ / \ ' ..... _......... \

5.0 ""--______ --L.. _______ ----1~

15 20 Laser frequency ( 10

3 cm -1)

25

FIG. S. The enhancement factor for uncoated silver in the geometry of Fig. 2. Two pyridine lines are shown-IOOS (full line) and 30S5 cm- 1

(dashed line).

F2, with FI = 1-0.028/A? and F2= 1-0.094/..1.2 (A in mi­crometers), an expression appropriate in the range 350-10000 nm.2S The full (broken) line is for the uncoated (coated) system. Note that the low frequency enhance­ment is decreased by one to two orders of magnitude by the coating, while the peak structure is de-enhanced and blue shifted. These effects follow from the flat response of the insulated coating in the region shown. In this region, the coating actually acts mainly as a spacer, separating the molecule from the highly susceptible metal, thus decreas­ing the short-range multipolar couplings responsible for the red shift of the resonance structure and its multiplic­ity.20 This analysis suggests that as the insulating coating becomes thicker, for frequencies in the visible and above, the signal will decrease further since the spacing of mole­cule and metal increases as well. At the same time, an enhancement may appear in the infrared, where the oxide is likely to have its own plasmon resonance. 26

Figure 5 shows the SERS enhancement factor in the visible region for two different lines of pyridine, with the molecule placed between uncoated spheres in the same ge­ometry as in Fig. 2. The continuous line corresponds to a 1005 cm- I vibrational mode of the molecule, while the dashed line corresponds to a mode at 3055 cm -I. The fact that the curves intersect shows that different vibrational modes may invert the relative size of their Raman signal as the frequency is varied. In general, on the basis of our results, one expects the frequency dependence of the SERS enhancement factor to be different for different molecular vibrational modes, producing crossings as those exhibited for the case of pyridine on metal in Fig. 5. Because the enhancement factor of these modes also depends on the polarization of the excitation (laser line), we propose the above effect as an explanation of experimental results on the relative efficiency of sand p polarization on SERS.6,23

v. CONCLUSIONS

We have developed a new theory ofSERS based on the multipolar electromagnetic coupling of the molecule and the surface. We have treated a dilute colloidal system, modeled by a molecule over a sphere, and a system where clusters are present, modeled by a molecule placed in the space between two curved surfaces.

We emphasize that the models used in this work rep­resent SERS active centers only, i.e., geometries in which the molecule is giving a maximally enhanced signal. In this way, we have shown that in silver, enhancements as large as ten orders of magnitude may be obtained. The largest values occur when the molecule is fit snuggly in between two curved surfaces. Then the plasmon resonance is shifted to lower energies by the multipolar coupling of the metallic features, bringing it towards the optical region where the laser frequency is likely to lie.

When interpreting experimental data, one must take into account that in most situations, the environment of the adsorbed molecules will vary widely in detail geometry. For instance, in metallic colloids away from the dilute limit, the particles tend to form clusters of various sizes and shapes, where narrow spaces of all sorts between the aggregated particles are likely to be found. 29,30 Molecules that diffuse to such spaces will find themselves in environ­ments that cover a rather wide range of neighboring sur­face curvatures and separations, each yielding a particular enhancement of the Raman signal. Thus, a distribution of intensities will be obtained. For instance, in aggregated colloids and metal island films, only a fraction of the active molecules will be placed in the special geometries that op­timize the Raman enhancement for the given laser fre­quency. As our theory shows, the detail response is quite sensitive to geometry. Accurate results for a particular sample may thus require a statistical analysis that includes, with their appropriate weight, all geometries present. We propose, however, that even in the absence of such a de­tailed analysis, the insight and explicit expressions pro­vided by our theory gives a criterion for choosing the ex­perimental variables that optimize the signal size of a resonant mode in the adsorbed molecule. These variables include the laser frequency, the host particle size, the host dielectric function, etc.

In the case of very dilute colloids, where the geometry of a molecule over a spherical particle is well separated from the rest predominates, we predict that the Raman signal is enhanced maximally when the laser frequency matches the plasmon resonance given in the Drude model by w/ ~Eb+2Eh. In well dispersed Ag particles immersed in a medium of dielectric constant Eh~ 1, the required laser frequency should approach 31 450 cm -I, which is in the ultraviolet region. The resonance may be brought towards the visible by increasing the dielectric constant of the host dielectric function. For instance, setting Eh=7.1, the reso­nance takes place at a laser wavenumber of about 18 000 cm -I in the visible spectrum.

Free as well as coated metallic surfaces have been con­sidered, and we have shown that an oxide coating in gen-

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1004 R. Rojas and F. Claro: SERS in colloids

eral decreases the Raman intensity in a sample with suffi­cient randomness that a range of geometrical situations pertaining to the position of the molecule is encountered. The coating acts to suppress the multipolar modes charac­teristic of curved surfaces. If the coating is metallic and uniform, the most noticeable effect is a suppression of the peak structure in the resonance region, unless the coating is sufficiently thick that by itself it is capable of sustaining surface modes.26 If the coating is an oxide, it acts mainly as a spacer, thus decreasing the effective coupling of curved metallic surfaces among themselves and with the molecule. This means a signal decrease and a shift of the electromag­netic resonances to higher energies away from the optical region.

Our theory assumes that the curvature of the metallic surfaces is much smaller than the wavelength of light A, so that the long wavelength limit may be applied. At length scales of order A, the electric field in the laser beam is nonuniform and may excite by itself multipoles higher than the dipole in particles whose curvature is of the same order. This, together with the coupling to neighboring surfaces, will make the structure in the detailed frequency response of an adsorbed molecule in such situations more complex and difficult to characterize. For small molecules such as pyridine, however, this effect is important when the surface curvature is so large compared to the size of the molecule that the local gemetry may be well approximated by a planar surface, an unfavorable situation for the signal en­hancement. The long wavelength limit is thus sufficient for the analysis of most cases where large enhancements in SERS take place.

ACKNOWLEDGMENTS

This work has been partially supported by Fondo Na­cional de Investigaci6n Cientifica y Tecnol6gica (Chile) grant 0375-90 and by Universidad Tecnica Federico Santa Maria Grants 90110 1 and 91110 1. We would like to thank Dr. T. Lopez-Rios, Dr. A. Cabrera, and Dr. R. E. Clavijo for illuminating discussions, and Dr. Ronald Fuchs for providing us with the data for tin.

APPENDIX A: INDUCED FIELD

The static electric potential outside a particle of mul­tipolar moments ql,m located at the origin of the coordinate system is given by the relation

41T YI,m (e,¢> ) cI>(r,e,¢» = L L 21+ 1 ql,m 11+1 ,

I m (Al)

where YI,m( e,¢» are spherical harmonics and (r,e,¢» are spherical coordinates. When an external uniform electric field is applied in the z-axis direction, only multipolar mo­ments labeled with m=O are induced on a polarizable par-

ticle with spherical symmetry. The corresponding induced potential has azimuthal symmetry and its value on the z axis is

where

~ ql,o VI(z) = VU+1l z li+1

if z>a

if z< -a, (A2)

(A3)

and a is the radius of the particle. The z component of the electric field intensity is given by the relation

dV(z) Ez(z) = - dZ ' (A4)

which gives

E(z) = I~I-EI(Z) {( _ :;1+1,

where VI(z)

EI(z) = (/+ 1) TzI'

if z>a

if z< -a, (A5)

(A6)

The corresponding electric potential and electric field for a particle located on the z axis at coordinate Zi is obtained from the previous relations replacing z by Z-Zi' Many­particle systems require the superposition principle, and for a system of two particles located at zl and z2 (with z 1 < Z2), the induced electric field intensity in between them is

where zl +al <z <z2-a2' ql,i is the / polar moment, and ai is the radius of particle i. The previous relation also gives the electric field produced by a single particle. By replacing ql,2 =0, the electric field of an isolated particle placed any­where in the z axis is obtained.

APPENDIX B: MUL TIPOLE MOMENTS EXCITED AT THE LASER FREQUENCY

We consider a molecule (index i=2) centered at z=O and a couple of metal spheres centered at Z= - D and z=D and labeled i=1 and i=3, respectively. A uniform external field is set along the z axis of magnitude Eo and frequency liJ.

The multipolar moments ql,i resulting from the mutual interactions among the particles under the action of the external electric field are given by

(B1)

where i and i' label the particles, and I and /' are indices labeling the pole order and extending over all positive in­tegers. The coupling coefficients are21

I',;' _ k+I'+1 2/+1 -(/+I')! al,;Cl-oi,i') _. BI,i -(-) 2/'+1 l!1'! 1 1'+/'+1' - Zi-Zi'

(B2)

J. Chern. Phys., Vol. 98, No.2, 15 January 1993 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

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R. Rojas and F. Claro: SERS in colloids 1005

where

k= {O,

I,

if Zi<ZjI

if Zi>Zi' (B3)

with the 8 symbol representing a Kronecker delta and al,i

being the I-polar polarizability of particle i. The coefficients FI,i represent the external electric field and are the multi­polar moments in the absence of interactions

Fu= {343

au E0811' t V4V, , (B4)

Equation (Bl) forms a system oflinear equations that we solve up to polar order L, with L chosen as the smallest integer giving converged results for the electric field. It is convenient to use the normalized quantities26

(B5)

(B6)

I' -I' '1 21' + I ai' 1'" B ,1- ___ B ,I

I,i - 21 + I 7. I,i' I

(B7)

where ai is the radius of particle i. The normalized multi­polar moments are given in terms of the inverse of matrix T, defined by its elements,

1' " -1' " Ti,i' =81,II8i,i,-Bi,i'. (B8)

The normalized multipolar moments are given by

L

(if,i= 2, 2, (T-1)r/' FI"i" 1'=1 i'

APPENDIX C: MUL TIPOLE MOMENTS AT THE RAMAN FREQUENCY

(B9)

As in Appendix B, we consider a molecule in between two metallic spheres. Given the dipole moment qi,2 of the molecule

qi.z=G;Pm({i)R)' (CI)

we assume it couples to the spheres through (3L-1) re­lations similar to Eq. (B1), giving the system multipoles q;',i' at frequency (i)R' Terms like FI,i in Eqs. (B1) are omitted in this case because there is no external field at the Raman frequency, so that the induced multipole moments are given by [(l,i)*(1,2)],

, ~ /',i' I

qu= £oJ BI,i ql'i" , I',i' , (C2)

Equation (C2) constitutes an inhomogeneous system of (3L-1) linear equations because of the known term qi.z. By substitution of the (L - I) equations correspond­ing to the molecular multipoles qi,2 into the remaining equations, we obtain 2L equations for the multipolar mo­ments on the spheres (i=1,3),

where

and

J B I,2q' I,i= I,i 1,2'

L

ct',i' - 2, Bk,2 Bl' ,i' I,i - k=2 I,i k,2 •

(C3)

(C4)

(C5)

(C6)

As in Appendix B, we define normalized quantities in Eq. (C3). In defining q', J, and .0, we follow Eqs. (B5), (B6), and (B7) and get the normalized multipolar moments of the spheres

L " " - 1/"'-qli= £oJ £oJ (S- ) i,i' JI',i',

, 1'=1 i'=/=2

(C7)

where matrix S is defined by its elements

(C8)

with

(C9)

and

(ClO)

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