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2 Page 1 of 1 PCCP Dynamic Article Links 1 Cite this: DOI:.9/cxxx Image molecular dioles in Surface Enhanced Raman Scattering Cristian Mihail Teodorescu* a Received (in XXX, XXX Xth XXXXXXXXX XX, Acceted Xth XXXXXXXXX XX DOI:.9/bx The Surface Enhanced Raman Scattering (SERS effect is exlained using the interaction of a olarized molecule with its instantaneous image diole in a metal surface. This model exlains why SERS is obtained mostly on noble metals (Au, Ag, since these metals have usually lower inherent contamination as comared with other, more reactive, metals; thus, molecules may be found closer to the metal surface. It is shown how stronger SERS amlifications may be obtained using nanostructured surfaces, once the excited molecules are localized in concave sites. The deendence on the fourth ower of the incoming radiation electric field is obtained by taking into account the dynamics of adsortion/desortion rocesses of molecules. The SERS effect is maximal when the excitation frequency is red-shifted with resect to the bulk lasmon resonance. Also, the SERS amlification factor may be dictated by the olarizability of the investigated molecule α in a much more critical way than just a ower law α or even α. By comaring the diole induced charge density with the amlitudes of lasma waves, the domain of validity of the resent theory is derived to be in the low searation regime, where the distance between molecules and metal substrates are below a few nanometres. Some data from the literature are analyzed in the framework of this model, namely the distance, frequency and temerature deendence of the SERS signal, all confirming the validity of the model. Introduction FULL PAPER Although discovered almost four decades ago 1, the Surface Enhanced Raman Scattering (SERS did not get so far a fully satisfactory exlanation. The initial roosed exlanations were the 'chemical enhancement' (CE and the 'electromagnetic enhancement' (EME. As years assed, the latter was much more successful. At the same time, SERS data of increasing quality were available, achieving nowadays the ability to detect single molecules by this technique,6. Since the basic electromagnetic mechanism roosed to exlain the SERS effect stems in the use of 'surface lasmons' i.e. charge density waves in the metal oscillating in hase with the incoming radiation field, a whole domain named 'lasmonics' emerged during the last decade 7. The key henomenon in the EME theory may be summarized in three stes: (1 the incoming electromagnetic field generates a lasmon in the material, most often a surface lasmon (SP; ( the field generated by this lasmon olarizes the molecule, in addition to the incident field (if one neglects retardation effects, i.e. if the involved molecule-substrate distances are well below the wavelength of the light; ( the molecule emits Ramanshifted radiation. Therefore, in order to achieve high SERS amlifications, firstly a matching is sought between the frequency of the incident light and the SP frequency. Also, it was evidenced that surface nanostructuring yields to even enhanced SERS amlification factors 1,8-11. This yielded to the fabrication of secially adated SERS substrates 11,1, aiming to match the metal surface nanostructuring with either the light wavelength or with the SP wavelength. However, a disordered nanostructured goldcaed orous silicon substrate yielded a similar (even better SERS amlification factor as the commercial SERS nanofabricated substrates 11. Such observations yielded recently to the roosal of 'hotsots', mainly located in the concavities of metal surfaces or at the interface between metal nanoarticles 1,1. Other actual researches based on the revious three ste model of the SERS effect are the fabrication of nanoantennas 1,1 and studies of the light scattered by these nanostructures. The full theory of lasma excitation is rather comlicated 7 and difficult to be directly used in order to quantify the SERS effect. The normal way to roceed in simulation is based on discrete methods, either finite difference time domain (FDTD 1,1, two dimensional boundary element methods (DBEM 1 or discrete diole aroximations (DDA 1,16. The latter method already contains the ingredients of the method we will discuss in the following: but in the following we will roose that a single diole is induced in the metal, which may be regarded as the image of the diole induced in the investigated molecule, laced in the neighborhood of the metal surface. For arrays of ordered metal nanoarticles, a clear SERS enhancement is observed when the olarization of the exciting light is arallel to the rows of nanoarticles 17,18. It was comuted that the incident intensity (roortional to the square of the electric field E may be enhanced by a factor of several thousands for sheres of about nm diameters searated by a ga of 1 nm, right in the middle of the ga 17. Therefore, since the SERS enhancement factor may be arametrized as deending on E (see below, enhancement factors of ~ 6-7 are to be exected to be roduced by such hotsots. This journal is The Royal Society of Chemistry [year] [journal], [year], [vol], Acceted Manuscrit

3 Page of 1 1 The key quantity which will be comuted in the actual aer is the molecular olarizability α, which is the roortionality constant between the alied electric field and the diole moment induced in a molecule P α E. (Throughout this work we will neglect any eventual tensor character of the olarizability. Indeed, the Kramers-Heisenberg formula 19 shows that the intensity of the Raman scattering is roortional to α E, the roortionality with E originating from the intensity of the exciting field. A brief review of orders of magnitude imlied in SERS follow. The incoming electric field of a light source with flux of ~ W/m, roviding ~ x hotons/(m s with individual energy of ev, is of about x V/m. The molecular olarizability may be written as α π V, where V is the 'olarization volume of the molecule', on the order of 1- Å (Ref. [], and is the ermittivity of the vacuum. Therefore, the molecular olarizability is on the order of Fm, and in the above electric field a molecule will develo an instantaneous diole moment of about D (1 D. x - C m. e Å. The main features of SERS are: (i the enhancement factor is from to or even higher 1,,6 ; (ii the redominance of SERS when using noble metal surfaces (Au, Ag; (iii the further enhancement of SERS due to nanostructured substrates; (iv the deendence of SERS on the fourth ower of the electric field E ; (v the strong deendence on the distance d between the molecule and the substrate reorted so far ~ (d + const. -, according to Refs. [9,]; (vi the SERS enhancement when the exciting frequency aroaches the lasmon frequency of the metal 1,9,. In this aer we will resent a simler model, based on the interaction of an instantaneously olarized molecule with its image diole formed by the metal surface. This henomenon is suosed to occur instantaneously, as oosed to the three-ste model described a few aragrahs above (see the 'AOA' mechanism roosed in Ref. [1] for the surface-enhanced light scattering. Without discarding the resence of lasmons, it will be shown that the actual model, based on simle considerations on electrostatics, mechanics, and thermodynamics, accounts for the origin of the SERS effect and for all features (i-vi described above. From the very beginning, we must emhasize that the basic asects of the theory resented in the following are valid esecially when the searation distances from molecules to substrates are in (or below the nanometer range. These are the conditions for the field roduced by the image diole to be robust at the osition of the molecule. Then, the effects of the surface nanostructuring will be investigated, and the main result will be that nanocavities with tyical shaes in the nanometer range must be resent by the substrate, in order to give a further enhancement. The next Subsection resents the effects of dynamic adsortion/desortion statistics and here also one needs to consider quite low distances in order to obtain an effective agglomeration of the molecules towards the surface, yielding a roortionality with the fourth ower of the roduct between the exciting electric field and the olarizability. The next Subsection treats the dynamic case and ends with a comarison between the amlitudes of the charge modulations induced in the metal by the oscillating diole, comared with the lasma oscillations. Here it will be demonstrated that in the nanometer regime of moleculesubstrate distances the former contribution exceeds the later one. The final Subsection analyzes two ractical cases, showing (i how one can aly the model and derive the distance from analyzed molecules to the substrate when one knows the sectral deendence of the amlification factor and (ii hw one can analyze the temerature deendence of the SERS amlification factor and how one can extract from these data the enhancement factor. The last Section, Discussions and Conclusion, summarizes the findings from this basic considerations and resents a survey of exeriments to be achieved in the future to get more insight on the validity of the resent work. Phys. Chem. Chem. Phys., [year], [vol], This journal is The Royal Society of Chemistry [year] Results Image dioles Classical electrostatics rooses that any charge near a metal surface whose otential is ket constant will induce a olarization of the free electrons in the metal which may be assessed outside the metal by introducing a virtual 'image charge' 1. Therefore, dioles must induce image dioles. Such a situation is reresented in Fig. 1. It is easy to estimate that the induced electric field at the original molecule by its image is P/[π (d ]. Let α be the olarizability of the molecule in absence of the metallic wall. Therefore, the total diole moment given by the incident E and the induced E ind. electric fields is given by: α P P α ( E + Eind. α E + (1 π (d From here, the effective olarizability near the metal surface is 8 comuted simly as P αe, with: α α α 1 π (d π V V 1 (d Fig. 1. Image diole created into a metal surface. E is the external field, E ind. is the induced field by the image diole at the location of the original 9 diole. 9 These considerations are quite similar to the well known olarization catastrohe in dielectrics, yielding eventually to ferroelectric ordering, with occurrence of ermanent diolar moments in absence of an alied field. An imortant class of ferroelectrics (erovskites ABO are formed by combining low ionic radius A cations (e.g. Ti + with large ionic radius (and heavier cations in lower charge states (e.g. Pb +, with a clear covalent character of the B-O bonds. The BO lanes could be viewed as monolayer sheets of metallic character, which may ( Acceted Manuscrit

4 Page of 1 1 image diole moments roduced by rumlings of the AO atomic layers, yielding ferroelectricity in a quite similar mechanism as that described by eq. (. When the denominator of eq. ( is small, i.e. when d V 1/, the olarizability may increase dramatically. By taking into account the order of magnitude of the olarization volumes in absence of the metal surface (- Å, this imlies distances from the molecule to the substrate ranging from 1 to Å. Therefore, the roximity of the metal surface is essential for an increase in the molecular olarizability through this mechanism. That also imlied that any non-conductive layer (such as the inherent contamination of most metals, which may be of several nm thickness will significantly reduce the SERS amlification factor by imosing a larger searation distance d between the molecule and the metal able to roduce image dioles. This might be the reason why SERS is mostly observed on noble metal surfaces, since the reactivity of these surfaces with the ambient atmoshere is minimal. Indeed, any highly surface sensitive technique, such as X-ray hotoelectron sectroscoy, reveals that the contamination layers exceed about three times the hotoelectron escae deth (1-1. nm for most metals, excet for Au, Ag and Pt, where the inherent contamination layer is less than ~ 1.- Å thick (in the range of one single atomic layer. This is the case of metals in air and at atmosheric ressure; it is also true that the nature and the thickness of the contamination layer into a liquid (a solution, an electrolyte may change dramatically. When the excited molecule is located at equal distance between two metal walls, the induced field is almost twice the value from eq. (1, i.e. E ind. 1.8 P/[π (d ]. The factor of increase is not exactly two, owing to multile imaging yielding a series Σ k 1 (-1 k-1 /k.91 in the overall field induced. However, this effect may exlain the huge success of the ti enhanced Raman scattering (TERS method 6. A final remark on the above basic consideration is that the SERS amlification factor, as described by the square of the molecular effective; olarizability, eq. (, deends on the molecule investigated and not only on the distance to the metal substrate. Actually, several factors may be foreseen to influence the SERS signal: (i the nature of the molecule and its olarizability, via V ; note that the deendence is not just a ower law one (e.g. ~ α n ~ V n, but rather involves aroaching a divergence (ii the contamination of the metal substrate or any other rocess yielding layers which searate the molecules from the substrate, via d; (iii in absence of such contamination layer, the adsortion geometry of the molecule itself on the metal surface, suosing hat the molecule does not dissociate. In this last case, the chemical enhancement mechanism has also to be taken into account. In the following sections we will treat the other factors intervening in the SERS effect: the surface nanostructuring, statistical effects and the deendence on the excitation frequency. Surface nanostructuring Another feature of the SERS effect is the strong enhancement induced by surface nanostructuring 1,8,-1. The basic exlanation of this effect is that lasmons are couled to the geometry of the surface, therefore the lasma oscillation might be enhanced by a regular nanostructuring. In the following we will demonstrate that the actual model accounts for this effect in a quite natural way. Suose that a art of a nanostructured surface is formed locally by sherical cas, such as reresented in Fig.. One may comute the force exerted between a diole P q x δ and its image formed in such a sherical ca as: This journal is The Royal Society of Chemistry [year] Phys. Chem. Chem. Phys., [year], [vol], F a q ( d, 8π d 1 { 1+ [ δ /(d] } 1 / P π (d for δ << d ( A molecule of mass Daltons carrying a diole momentum of - D located Å away from the surface will suffer an attraction acceleration of about 18 g towards the metal surface. Therefore, a mechanism such as that reresented in Fig. may be roosed for migration of molecules towards the concavities. Molecules subject to the interaction with two image dioles will tend to migrate towards an equilibrium osition in a concavity. For a distance of nm to the surface the acceleration is lower (~ 1.8 x -6 m/s, but still imortant at the nanoscale level; the comlete migration of a molecule over nm takes lace in less than one second, so within the time scale of the measurement. Fig.. Mechanisms of localization of target molecules inside concavities Fig.. (a Image diole roduced by a concave metal surface. (b Enhancement factor f due to the resence of concavities (note the log scale. The next ste is to evaluate the induced field by an image diole created by a concave metal surface (Fig. a. The induced field will be enhanced by a factor which will be called nanostructuring factor f, defined as the ratio between the induced field in the case of a concave metal surface and the induced field by a lane metal surface: Acceted Manuscrit

5 Page of 1 1 E E i ( i d R d 1 R 1 d R d d R R δ + R δ R d δ R f, R R R d for δ << d, R ( The values obtained for this factor could go until several orders of magnitude when d R and δ R/. In ractice, when R is in the nanometre range, and by using tyical distances between the molecule and the surface of some Å, several units are derived for the shae enhancement factor f. The comlete function in eq. ( becomes infinite when d R, i.e. when the molecule is laced in a centre of a hemisherical cavity or of a sherical surface exceeding one half shere. This could corresond to the situation where the metal (at least half shere is filled with a fluid containing the analyzed molecules; thus, the strongest image diole induced field acts on molecules from the centre of this shere. A more realistic real situation imlies that the analyzed molecules may be found near the centre of the hemishere within some uncertainty d: d R ± d /, where d << R. Introducing this in eq. ( one finds, with a good aroximation that the enhancement factor f 8 [1 + (δ / d ]. If the molecules could be fixed near the centre of the hemishere within a better recision than the charge searation in the diole, the geometric enhancement factor may become considerable. For examle, if δ / d, then f 8 x. This stabilization should occur in a time scale on the order of the inverse of the light frequency (ν 1, in the range of -1 s for otical hotons. The osition fluctuation during one cycle of light may be written as d <v>/ν, where <v> is the average thermal seed of these molecules: (k B T/m 1/ ~ several hundreds of m/s in gases, where m is the molecular mass and k B T the Boltzmann factor. In liquids or solids, the average thermal seed may be lower. It follows that d is on the order of.1-.1 Å. Next, the evaluation of δ is rather difficult (ab initio comutations of the molecular electronic structure in an alied film are needed; from the otical roerties (diole moment all we know is its roduct with the charge q. However, one could make the ansatz that δ is at least on the order of a bond length, a few Ångströms. Thus, the estimated geometric enhancement factor in the centre of a hemishere might be even larger than the value estimated above. It may haen that when a fluid is accumulated inside a nanometre sized hemisherical cavity, most of the signal is rovided by molecules laced in the centre of this cavity. Thus, one may foresee a quite simle exlanation for hotsots. The formula for the olarizability modified by the nanostructured surface must be modified by including the nanostructuring factor f, as follows: α π V α α f ( d / R, δ / R V f ( d / R, δ / R 1 1 π (d (d Therefore, easier conditions are obtained for lowering the denominator of eq. ( if this factor is resent. An abbreviation such as CERS ( Cavity Enhanced Raman Scattering may be introduced to account for this henomenon [17]. Also, the ( resence of a convex surface (such as a ti alone, without an associated metal counter-electrode will yield to a decreasing factor f. Note also that, in any case, no regular array of concavities is needed to exlain the enhancement. The real surface nanostructuring with concavities as small as ossible (though not smaller than the tyical dimensions of the molecule is enough to rovide a robust enhancement, as evidenced in Ref. [11]. Gold substrates with ~ 1 nm nanogas rovided recently SERS enhancement factors in the range of 8-9, see e.g. Ref. [7]. When the denominator of eq. ( becomes negative, the molecule, which initially was assumed to not carry a ermanent diole moment, will become olarized or even dissociated near the metal surface. The mechanism is the following: first of all, even in absence of exciting radiation, molecules are undergoing 6 instantaneous olarizations owing to their interaction with thermal radiation or to vacuum fluctuations of the electromagnetic field. As soon as a molecule aroaches a metal surface, its eventual instantaneous olarization will be amlified by the effect described reviously. When the amlification factor {1 - V f/(d } -1 is strong enough, there are two ossibilities: (i the molecule is broken; (ii the molecule stabilizes in a olarized state. For understanding these two henomena, one should consider the nonlinearity of the olarization deendence P (E for the isolated molecule. Considering the next term in a series 7 develoment of the olarization vs field P (E αe + βe, when the arameter β >, any increase of the alied field (including the field due to the image diole yields a more dramatic increase of the diole moment, u to the dissociation of the molecule. When β <, one can imagine that the olarization saturates as 8 function on the alied field. Therefore, the molecule is stabilized in the olarized state, which does not increase anymore. Other interesting redictions are the natural induction of ferroelectricity in an ensemble of adsorbed molecules on a metallic substrate; these findings will be detailed in a subsequent work, which will 8 include also the analysis of olar molecules (molecules carrying a diole moment in absence of any interaction. Anyway, some sectroscoic evidences could be sought about the fact that molecules subject to SERS measurements are initially olarized. Actually, there are quite often sectral differences between SERS 9 and conventional Raman sectra of the same molecule; this is usually attributed to charge transfers or other chemical effects 8. As mentioned above, in he hysics of ferroelectrics, the olarization catastrohe is derived from a similar formalism and it yields to the stabilization of a sontaneous electric olarization 9 inside the material,9. Dynamic adsortion/desortion: E enhancement and deendence on distance to the metal substrate Once the adhesion force is comuted by eq. (, it is easy to estimate the adhesion energy, i.e. the work necessary to detach the molecule which is found initially at a distance z from the substrate: P z Fa ( z' dz' ε ( z (6 8π (z a ( z Phys. Chem. Chem. Phys., [year], [vol], This journal is The Royal Society of Chemistry [year] U Acceted Manuscrit

6 Page of 1 By using convenient units ε (mev 9 P (D/z (Å. As a consequence, even at sub-nanometre distances from the surface, molecules will have adhesion energies on the order or lower than the thermal energy (about mev at room temerature. Now suose that the target molecules are in a gas or in a liquid, obeying Botzmann statistics. The density increase near the surface (z may be exressed as: ε ( z P ( z n( z n ex 1 n kbt 8π (z kbt n α ( z E 8π (z k T B n being the initial density, α(z the z-deendent olarizability and E the exciting electric field. The last function from eq. (7 is a raidly decreasing function on z. We consider that the minimal distance from the molecule to the substrate d min. is close to (but larger than the singularity of α(z, i.e. d min. (V f 1/ + b, b being a small quantity. The overall SERS enhancement rovided 1 by molecules at a distance z in a layer dz may be written as: di SERS E α ( z E α ( z n( z dz S dz S (z T α E T (z [( z V f ] 9 dz S S being the area investigated. To comute the overall enhancement, one must integrate the above z deendence from z d min. to infinity. The integral can be comuted analytically and it follows that the deendence on the satial distance to the metal substrate is on the form: I SERS 1ξ + ξ ξ ( ξ 1 ( ξ 1 tan ξ ξ + π ( ξ 1 1log ξ + ξ + 1 (7 (8 (9 Fig.. Fits of the deendence of the SERS signal on the Al O sacer to Ag films grown on nanosheres (AgFON. The data are extracted from Refs. [9,]. This journal is The Royal Society of Chemistry [year] Phys. Chem. Chem. Phys., [year], [vol], 6 7 where ξ d min. /(V f 1/. This imlies a quite strong deendence on the searation distance between the molecules and the substrate. This is not exactly the (d + a - law roosed in Refs. [9,], but its behavior is quite close, i.e. it is a function strongly decreasing with d. The data from Refs. [9,] may be fitted quite well with formula (9, as shown in Fig.. An additional fitting arameter regarded as a dead layer (d was introduced additionally to imrove the quality of the fit, such that in this case t (d + d /(V f 1/. This dead layer of about 1. nm thickness may be seen as an inherent contamination of the Ag film or some other effects reventing yridine molecules to agglomerate exactly on the Al O sacer. The molecular olarizability was fixed, V 9. Å for yridine. The fit with the (d + a - law is also reresented. The fit by using eq. (9 is slightly better (and the number of fitting arameters is the same, the χ of the fit is lower by about %. To draw a definitive conclusion, we suggest that more such exerimental data on SERS signals deendence on insulating sacers need to be investigated. Note also the 1/T deendence. This could be a sign that imroved SERS signals may be obtained at low temeratures. The roortionality of the SERS intensity with the fourth ower of the olarizability and of the exciting electric field is a significant effect. A SERS enhancement factor of four orders of magnitude may occur if the olarizability increases by one order of magnitude only. In fact, for diole moments of ~ 1 D the adhesion energies may exceed the thermal energy, therefore in this case the develoment of the exonential in eq. (7 might not be aroriate. This should haen with molecules carrying ermanent diole moments (P. The overall deendence of SERS will be in this case as E α {ex[ct.(p +αe /T-1], more abrut than (αe. For diole moments of ~ D, i.e. with no ermanent diole moments and low olarizability, the adhesion energy is quite low comared with the thermal energy and it might haen that no significant increase of the density near the surface occurs; in this case, the SERS intensity remains roortional to (αe. Consequently, the 'E enhancement' might not be an universal law, according to the resent considerations. Some examles concerning the temerature deendence of the SERS signal will be discussed in a later aragrah. Charge density analysis in the dynamic case: image dioles vs. surface lasmons We will firstly consider the charge density in the metal for the static case. One assumes that the charge distribution on the metal surface generates an electric field which comensates the in-lane electric field generated by the external charge. By alying the in-lane divergence on this induced electric field, one may comute the associated induced charge density on the surface. For a diole P qδ consisting in a charge (+ q located at (x - δ /, y, z - d and a charge (- q located at (x δ /, y, z - d, the induced charge distribution on the surface (z is given by: Acceted Manuscrit

7 Page 6 of 1 1 q ρids ( x, y π δ δ x y d x y d / δ δ x + + y + d x + y + d Px π ( d x y 7 / ( x + y + d for δ << x, y / ( were the index ids stands for 'induced-diole-static'. These deendencies are reresented in Fig., as functions on x/d and y/d, for δ /d., which roughly corresond to an external diole or 1 D 1 e x. Å and to a searation distance d 1 nm. It may be seen that the aroximation is quite reasonable, consequently this aroximation will be used in the following. For lower values of the charge searation inside the diole δ, the aroximation is even better. Fig.. Charge density distribution due to a diole with δ /d.: (a exact formula, second term from eq. (; (b 'aroximate' formula in eq. (. The units of charge density are q/(πd x -. The next ste is to treat the dynamic case, i.e. to assume a eriodic electric field excitation of frequency : E cos (t E Re ex(it. All calculation will be erformed by using comlex numbers and the convention is to take the real art at the end. Within this formalism, the high frequency behaviour of the metal is described by the Drude-Zener conductivity : σ σ ( 1 + iτ 6 Phys. Chem. Chem. Phys., [year], [vol], This journal is The Royal Society of Chemistry [year] 1 (11 where τ 1 is the relaxation time, τ 1 <τ (ε F >, on the order of -1 s for most metals. Aart for τ 1, another time interval will occur in the formulas below, it is given by the ratio between the vacuum ermittivity and the static conductivity τ /σ ; this is on the order of s for good conductors. It is easy to introduce the lasma frequency as the inverse of the geometric mean of these two times; (τ τ 1-1/ (ne /m* 1/ ~ 16 s -1, where n is the electron density in the metal, and m* their effective mass. Now, for real exeriments, the excitation is close (though not exceeding and therefore τ 1 ~ τ 1 ~ >> 1 and τ ~ τ ~ - << 1. The following ste is to use at the same time the continuity equation and the first Maxwell equation, as follows: div j σ ( div ρids ( r σ ( e it ( E ( r, t + E ( r, t ext ind ρid ( r, t ρid + σ ( t (1 The first term is the external field roduced by the diole; its satial deendence is given by eq. (, where one simly relaces the static diole moment P by the amlitude of the oscillating diole P. Assuming now that the charge in metal oscillates with the same excitation angular frequency, relacing the Drude-Zener conductivity σ(, the charge induced in the metal is exressed as ρ id (r, t ρ (r ex(it, where ρ is comlex, to account for eventual hase shifts. Relacing ρ id in eq. (1, one obtains ρ (r ρ ids (r {1 + iτ (1 + iτ 1 } -1 and by taking the real art of ρ id (r, t, this yields the time-deendent charge induced at the metal's surface: 1 cost + τ sint ρid ( r, t ρids ( r 1 + τ 1 cost ρids ( r 1 + τ ρids ( rcost ρids ( r g( cost 1 (1 The molecular diole oscillates as cos t. Therefore, to a good aroximation, the charge in the metal oscillates in hase with the diole. Here and in the following we will neglect retardation effects, since these effects will occur when the distance d between the diole and the surface aroaches the wavelength, which in actual exeriments is of several hundreds of nm, whereas we consider d in the range of 1 nm. The intensity of the oscillations increases when the excitation frequency aroaches Acceted Manuscrit

8 Page 7 of 1 the lasmon frequency; however, it does not diverge when, as the last aroximation of eq. (1 suggests. The function g( exhibits extrema when (1 ± τ 1/ ± 1/(τ 1. In terms of wavenumbers, the extrema are shifted by aroximately 1/(8πcτ 1 with resect to the lasmon frequency. The term with (sin t becomes dominant when the excitation frequency aroaches the lasma frequency. In this case, the charge in the metal oscillates with a hase shift of π/ with resect to the excitation wave. However, this field, when transmitted back to the excited molecule which vibrates as (cos t, will roduce no net effect from classical theory in terms of absorbed ower from the radiation field, since the average of the roduct (sin t cos t is zero. In the Electronic Sulementary Information (ESI, the case of 1 corrections due to enetrating field into the metal is analyzed. The charge density in the (z lane oscillating in hase with the diole (~ P cost may be written in this case such as: ~ ρ( x, y, z, t 1 π ( + + x y d P xcost 7 / 1 + τ [ ( d x y κ d( x + y + d ] τ κ d ( x + y + d (1 where κ [/(τ ] 1/ /c is the inverse of the skin deth (see the ESI. It is easy to estimate that the correction term is imortant only when d becomes comarable with κ -1 ~ - nm. In Fig. 6 the satial deendence of the leading term of eq. (1 is reresented for several values of the roduct (κ d. This figure may be viewed as a 'thought exeriment' of what haens when the distance between the molecule and the metal surface is increased. At the beginning, the amlitude of the surface charge density is quite similar to that reresented described by eq. ( and lotted in Fig.. Starting with κ d 1, the charge density starts to exhibit riles which can also be described as surface lasmons. Interestingly, by about κ d the shae of the surface density is reversed, in that the charge density starts to have the same sign as the ion sitting on the area in question (electrons accumulate on the side of the negative ion. Physically, this inversion haens at distances of about nm. This could also corresond to a dynamic reulsion between the molecule and the surface. In the following, we will derive the critical distance where the surface charge density given by the direct excitation of lasmons with the incoming electromagnetic wave revails with resect to the charge density generated by the oscillation of the molecular diole. It will be seen that in this distance range (above nm the lasmon charge density dominates. The evaluation of the charge induced by the electromagnetic wave into the metal (the lasmons and the comarison of this charge modulation with the charge density induced by the diole is based on the following main ideas: (i The electromagnetic wave hits the surface at an angle θ ; the incidence lane is the (y lane. (ii The electromagnetic wave may resent two olarizations: s and, where stands for olarization in the incidence lane and s for olarization normal (senkrecht to this lane. It may be easily demonstrated that only the wave is able to excite surface lasmons. (iii The electromagnetic wave is totally reflected by the metal and the reflected wave acquires a hase shift of π with resect to the incident wave. This journal is The Royal Society of Chemistry [year] Phys. Chem. Chem. Phys., [year], [vol], 7 Fig. 6. Evolution of the amlitude of the surface charge density given by the leading term of eq. (1 with the roduct (κ d. We consider the coordinate z as the normal to the surface and the incidence lane is (y, while the x axis reresents the intersection of the incidence lane with the surface. By neglecting again any enetration inside the metal, the charge density of the surface lasmons is given by: Acceted Manuscrit

9 Page 8 of 1 1 ρ ( r, t E k sin(θ 1 sin 1 + τ sin Ek sin(θ 1 ( t kx sinθ τ cos( t kx sinθ ( t kx sinθ (1 where E is the amlitude of the wave and k π/λ the wavevector of the electromagnetic field, λ being its wavelength. It follows also that the most effective excitation of the surface lasmons occurs when θ. One has to comare the amlitudes of the diole induce charge modulations and of the lasmon amlitude: ρ ρ (max id (max.7p 1 πd 1 with 1 sin( E k θ 1 (16 The factor.7 is just three times the value of the maximum of the function exressed in eq. (: u( ( / u h( u u (17 By introducing again the olarizability and the olarization volume such that P α E π V, and by considering θ, one obtains a simle condition for more intense amlitudes of lasmons when comared to that induced by the diole: ρ (max (max 1/ id < ρ d > (.V λ (18 Several estimates may be done, but here let us just oint that for V Å, the diole induced charge amlitude variation is larger if the molecule is closer than about 1. nm to the substrate (λ nm. Note also that in the above considerations we introduced the 'standard' olarizability and olarization volume, in absence of the SERS effect, whereas one should take into account the SERS amlification factor. For instance, a olarization volume of 1 nm (i.e. the olarizability of a simle molecule increased by two orders of magnitude, corresonding to a SERS amlification factor of - 8 would induce revalence of the diole-induced charge variations for d <.8 nm. Finally, one has to introduce the frequency deendent enhancement function g(: 1 g( 1 + τ 1 1 (when τ << 1 (19 from the term with cos(t, eq. (1, into the derivation of the olarizability in resence of image dioles. Thus, the equation for olarizability ( has to be re-written as: Fig. 7. The frequency deendence of the overall SERS enhancement factor, for two values of V f/(d (red and blue series of curves and for different values of the roduct τ. Please note the log scale. α α α f ( d / R, δ / R V 1 1 π (d π V f ( d / R, δ / R g( (d 8 Phys. Chem. Chem. Phys., [year], [vol], This journal is The Royal Society of Chemistry [year] 6 ( Thus, the denominator contains, in front of the ratio between the olarization volume and (d the geometric enhancement factor discussed in the aragrah Surface nanostructuring together with a frequency enhancement factor g(. It is clear that the aroximate formula for g( exlains immediately the strong enhancement factor when the light frequency aroaches the lasma frequency in the metal. As stated above, from eq. (19 one can derive that the maximum of g( occurs when + (1 ± τ 1/ + 1/(τ 1, where we remind that τ 1 is the relaxation time in the metal. The maximum value of the frequency deendent enhancement function can be derived as being g max. 1/( τ - τ 1/( τ (1/ x (τ 1 /τ 1/ /(. Eq. ( can be used to model the SERS enhancement factor, starting with basic knowledge of the vacuum olarizability of the molecule, geometric characteristics of the substrate, nature of the metal used (manifested by its lasmon frequency and conductivity. In fact, knowing the geometric enhancement factor f, the olarization volume of the molecule V and the distance of the molecule to the substrate d, one may determine the frequency where the maximum enhancement factor occurs, m, by solving g( m (d /(V f. Some frequency deendencies of the overall SERS enhancement factor α are reresented in Fig. 7. This icture analyzes unfavorable cases, where τ (τ /τ 1 1/ is rather large and the factor V f/(d is away from unity. The frequency deendence is much sharer for low values of the roduct τ (τ /τ 1 1/. Therefore, good conductors are needed as SERS substrates and this suorts also the revalence of the effect on gold and silver substrates. Acceted Manuscrit

10 Page 9 of 1 1 Also, in cases of larger distances from the molecule to the surface, the density amlitude from eq. (1 may be emloyed. But, as stated above, in such cases also lasmons become imortant and one has to introduce both charge densities from eqs. (1 and (1, then to retrieve numerically the electric field at the origin of the molecule. In this case, the simle aroximation of interaction with the instantaneous image diole formed in the metal will not be that accurate. Examles of analyses Frequency deendence of SERS signals The distance of the molecules analyzed to the substrate may be estimated starting with eq. (. Knowing the sectral deendence of the SERS amlification factor, one can infer the maximum of the frequency amlification function g max. /(. Thus, by introducing the geometric enhancement factor f {R/(R - d}, according to eq. ( and the amlification factor of the olarizability A α/α, the average distance of the molecules to the substrate may be derived as: 1/ 1 V g max. A 1 d R 1 1 R 1 1 g R A 1 R ( V 1/ max. (1 where the aroximation holds for large amlifications. Additionally, for large values of the tyical radii R of the surface structures, the distance from the molecules to the substrate is given simly by: 1 Vgmax. A d A 1 1 / ( V g 1 / max. ( as it might be derived also directly from eq. (, introducing in addition the frequency deendent enhancement function. An alication of the above formula starting with the values from Ref. [9] is resented in Table 1. The analyzed molecule is benzenethiol with vacuum olarizability of 11.9 Å (Ref. [1]. Let us first remark that all data resented a blue shift of the maximum SERS amlification frequency with resect to the maximum of extinction, whereas from g(, eq. (19, when > the function g( becomes negative and the denominator of eq. ( increases, therefore no enhancement factor should be observed (α < α. The key to solve this uzzle is to remark that the maximum of extinction corresonds to the surface lasmon resonance (SPR, s, of lower frequency than the bulk lasma frequency (in the ideal case of a flat interface between a erfect metal with air or vacuum s / 1/. Indeed, for silver the lasmon frequency, as theoretically comuted, is 9. ev, corresonding to 1.96 PHz, whereas the measured value is.9 ev, corresonding to.9 PHz. From the shifts between the frequency of maximum amlification and the bulk or surface lasmon frequency (maximum of extinction one derives g max. as: This journal is The Royal Society of Chemistry [year] Phys. Chem. Chem. Phys., [year], [vol], g max. m s s m ( Then, eq. ( is alied. The result for distances from molecules to the metal (Table 1 is in the range Å, which imlies that the Ag layer is quite few oxidized. For instance, standard hotoelectron sectra of Ag foils as introduced yield the ration between the oxidized and the bulk metal around.1, therefore the oxide layer is at most % of the inelastic mean free ath (around 1 Å. XPS data obtained in our laboratory confirmed this ratio between the comonents corresonding to native Ag oxide layer and that of bulk Ag metal. Therefore, in average, the oxide layer has a thickness in the range of Å, corresonding to less than one layer of Ag atoms undergoing oxidation at the surface. Thus, the data from Ref. [9] may be analyzed by considering that the molecules are ractically adsorbed on the AgFON SERS substrates and that quite few oxide layer is resent on the surface, Consequently, the maximum SERS enhancement was obtained at rather low frequencies, as comared with the bulk lasma frequency. Incidentally, this frequency was close to the surface lasma frequency, but shifted towards higher frequencies. Now, if surface lasmons were the main resonsible for the SERS effect in this case, it can be simly demonstrated that their frequency cannot exceed s / 1/. Couling to the surface nanostructures would eventually stabilize surface lasmons with lower values of the wavevector, therefore with even lower energies. Another effect to be taken into account would be a dielectric constant of the outer medium ( r different from 1, which also lowers the surface lasmon frequency, in this case s /(1 + r 1/. Table 1. Analysis of SERS data of benzenethiol for different substrates 9, exhibiting different lasmon frequencies: λ, are lasmon wavelength and frequency, resectively; λ m, m are the wavelength and the frequency of maximum SERS enhancement; EF is the enhancement factor, A is the olarizability amlification factor; g max. the maximum of the frequency deendent amlification function; d is the estimated distance of molecules to the substrate. λ (nm λ m(nm EF AEF 1/ s(phz (PHz m(phz g max. d(å 89 8.x x x x Note that the analyzed data from Fig., taken from the same References, imlied the formation of a dead layer of about 1 nm. It may haen that when atomic layer deosition of Al O on AgFON is started, a further oxidation of the silver occurs rior to the deosition of alumina, owing to the resence of reactants and to a larger substrate temerature. Indeed, atomic oxygen reacts with silver yielding a limiting thickness of the oxide layer of about 1 Å, which is in line with the thickness of the dead layer derived in Fig.. Temerature deendence Figure 8 resents data extracted from Ref. [], reresenting SERS signals of ethylene deosited on atomically clean, steed Acceted Manuscrit

11 Page of 1 1 Cu(977, as function on temerature. The reason for analyzing these data is to investigate the validity of Eq. (8, that there should be a strong increase of the SERS signal at low temerature. The sectrum obtained at K was not considered in this lot, since it has a much higher intensity that sectra obtained at lower temeratures, 1 K and K. It may haen that, when the temerature is increased, more vibrational modes are excited and this may yield to an increase of the signal. However, the vibrational level investigated (18 cm -1 corresond to an excitation energy of about.16 ev, considerably larger than the thermal energy corresonding to K (1 mev. Another effect not taken into account is the eventual deendence of the lasmon frequency on the temerature, which may yield non-monotonous temerature deendence of the SERS signal 6. Such roblems need to be investigated in more details, but in the following just the data exhibiting monotonous deendence of the SERS signal with temerature will be analyzed. Fig. 8(a resents the raw data, Fig. 8(b reresents the SERS signal as function on the inverse of the temerature. It may be easily seen that one cannot conclude from these data that I SERS ~ 1/T. Fig. 8(c resents a log (I SERS vs. log (T lot, and from this reresentation it follows that I SERS ~ 1/T with a good aroximation. Fig. 8(d resents a lot of log(i SERS vs. 1/T and one can see that these data can also be fitted by a straight line. This imlies the validity of a model similar to that described by eq. (7, in the aroximation of strong binding of molecules to the metal substrate, and with no equilibrium concentration of the molecules in absence of the attraction to the metal substrate, i.e. n n ex 8π P (d kbt ( Fig. 8. Analysis of temerature deendencies of SERS signals of ethylene adsorbed on steed Cu(977. (a shows the extracted SERS signal as function on temerature, (b reresents a lot of the SERS signal as function on 1/T, (c reresents a log-log lot, together with a fit function, (d reresents the logarithm of the SERS signal as function on 1/T. Just to estimate the orders of magnitude, we take into account the vacuum (α molecular olarizability of ethylene, given by V. Å, therefore the minimum distance from molecules to the substrate yields d.8 Å. From the sloe of the curve from Fig. 8(d and introducing d (since molecules are deosited on atomically clean Cu, one obtains a diole moment of about 1. x - C m. D. The roduct between the amlitude of the exciting field and the enhancement factor yields as E x EF ~ P/(π V x 9 V/m, which is reasonable, for a tyical electric field of x V/m, as estimated at the beginning of this aer, and an enhancement factor of 6. This examle shows also how one can determine enhancement factors without having available Raman exeriments erformed on the same system, without the metal substrate. Discussions and conclusions Basic theoretical considerations discussed in this aer suggest that the SERS effect is favoured by any combination of the following situations: (i Small distances need to be achieved between the investigated molecules and the metal substrate, such that d aroaches V 1/. There is a strong deendence of the SERS signal on the distance d. Note that in Ref. [] the Raman signal saturates for 1 L of ethylene adsorbed on Cu(1 at K in ultrahigh vacuum (UHV. (ii Molecules should be laced into nanocavities (CERS, such that the geometric function f(d/r, δ /R may increase by several units, going u to one order of magnitude increase. There is no need for these cavities to be disosed in regular arrays. There is no need for exensive nanofabrication techniques. Therefore, the SERS substrates need to be nanostructured, without well-defined ordered atterning, but with tyical nanostructuring dimensions (e.g. radii of concavities as close as ossible to the tyical dimensions of the molecules aimed to be investigated (on the order of V 1/ 11. Therefore, basic nanostructuring methods, such as the use of orous silicon 11 or the nanoshere litograhy 9,7 may be sufficient to rovide low cost SERS substrates. (iii When the nanostructured surface resents concavities mimicking hemisheres, molecules laced in the centre of these hemisheres will be subject to a huge SERS enhancement factor. (iv The light frequency needs to aroach the lasmon frequency of the metal; however, the SERS effect will be the most vigorous when the excitation frequency is a few tens or hundreds of cm -1 higher than the lasmon frequency, corresonding to a correction of 1/(τ 1 in. This effect is well demonstrated in Refs. [9,], for examle. (v Good conductors are needed as SERS substrates, because the frequency deendent function sharen for low values of τ, i.e. for larger static conductivities. (vi Owing to the adsortion/desortion dynamics in a nonsolid medium (liquid or gas, the SERS effect is roortional to α E, therefore an increase of the olarizability by a factor or - suffices to exlain the observed SERS enhancement factors. At high temeratures or low olarizabilities, when no significant agglomeration of molecules is found near the substrates owing to their interaction with image dioles, the SERS effect remains roortional to E. Recent work roved exerimentally the high deendence of the SERS on the way molecules and colloids are diluted and mixed together, yielding the fact that inhomogeneities in the distribution of colloids carrying molecules may be crucial for the signal obtained 8. Phys. Chem. Chem. Phys., [year], [vol], This journal is The Royal Society of Chemistry [year] Acceted Manuscrit

12 Page 11 of 1 1 (vii The substrates must be chemically inert for both keeing the metallic surface close to the adsorbed molecules and for avoiding too strong interactions with the analyzed molecules, resulting in conformational changes or charge transfer. (viii There should be a noticeable temerature effect, if one takes into account eq. (8. Actually, the temerature intervenes in the dynamics of adsortion-desortion on the metal surface. Note also that most of the data resented in Ref. [] were obtained at low temeratures. Also, in Ref. [9] a clear enhancement of SERS signals at low temeratures was demonstrated from room temerature to 8. K, though not varying as 1/T. After eight years of develoment, a setu named Cryo In-SEM offered a stronger increase in the SERS features when the samles are cooled. It seems that this cryogenic area of Raman sectroscoy will have a considerable develoment in the near future. More theoretical work is needed to include other temerature effects, such as the oulation of the vibrational levels or the deendence on temerature of the lasmon frequency. (ix The SERS amlification factor is secific to the kind of molecules analyzed, since the olarization volumes enter in a critical way at the denominator of any formula describing the effect - static, eq. (, or dynamic, eq. (. If the surface lasmons were the main resonsible of the SERS effect, similar amlification factors would have been reorted for quite different molecules. The literature is still elusive regarding this asect. The actual considerations suggest a rather affirmative answer to the title of Cha. of Ref. [8]: Is SERS molecule deendent? It was shown also that both field enetration effects into the metal and lasmons affect the very simle model roosed in this work, for distance ranges starting with nm for simle molecules. Nevertheless, using image dioles in case of small searation distance could simlify considerably the evaluations of SERS data and may lead to more quantitative evaluations, once e.g. the distance to the substrate, the nanostructuring and the molecular olarizabilities are accurately determined by other methods. The actual results resent also a ossible drawback of SERS exeriments: working on too clean substrates (molecules too close to the metal, small d, or on too nanostructured ones (large f, with too large olarizabilities (V, or too close to the lasma frequency (large g may yield a too large factor V fg/(d, exceeding unity, therefore the molecule either will dissociate, or it will exhibit a ermanent diole moment (the case of a saturating function of the diole moment on the alied field, eventually with low olarizability, since in that case P/ E is exected to decrease. It seems that choosing the right conditions for a good SERS exeriment is rather delicate. The best is to have the ability to vary the excitation frequency, since the other factors are dictated by the molecules analyzed and by the substrate. Exerimental verifications of the assumtions from the resent work may be undertaken by combining standard surface science UHV techniques with Raman sectroscoy, as for examle in Ref. []. Such SERS exeriments must be combined with other techniques able to robe the geometry of the molecules and their adsortion site in a non-invasive way, e.g. by hotoelectron diffraction 1 or surface extended X-ray absortion fine structure, in order to assess recisely the molecule-substrate distance and the adsortion geometry. The SERS substrate need to be well characterized by scanning robe microscoic (SPM techniques before starting the exeriments. Last but not least, detailed SERS investigations as function on the temerature are needed, by further adatation of the setus resented in Refs. [9,] with other surface science analysis methods, ossibly also using synchrotron radiation. I am confident that such data will be available soon, in order to check the very basic assumtions from this elemental theory and to render the SERS method more quantitative. Finally, the actual considerations with image dioles seems to be alicable also to surface enhanced infrared absortion (SEIRA. In this case, the excitation frequency is far away from the (surface of bulk lasmon frequency of the metal, therefore g 1 and rather a static model outlined in the first sections of this aer aly. It is clear that in absence of the frequency deendent amlification factor, low values in the denominator of eq. ( will be more difficult to be achieved. For instance, in the case from Ref. [9] analyzed in Table 1, relacing g max. 1 in order to analyze the hyothetical case of low frequencies (SEIRA yields fv /(d.68, therefore the olarizability increases by a factor or.1, and the enhancement factor would increase by at most a factor of 9. SERS remains therefore in some cases a more sensitive method than SEIRA, esecially for small molecules, when the frequency deendent enhancement factor is needed to lower the denominator of eq. (. This journal is The Royal Society of Chemistry [year] Phys. Chem. Chem. Phys., [year], [vol], Nomenclature: 8 A amlification of the olarizability α/α (dimensionless α molecular olarizability (F x m α molecular olarizability in vacuum (F x m b small value to be added to (V f 1/ in eq. (8 to yield d min. (dimensionless 9 β second coefficient in the P(E series develoment ( P/ E (F x m /V dz differential of z (m d distance from a molecule to the metal substrate (m d min. minimum distance of molecules to the substrate (m 9 d dead layer, additional oxidized layer resent of the Ag metal rior to the growth of Al O (m δ searation distance between charges in a diole (m d uncertainity in the osition of a molecule with resect to the metal surface (m n(z deviation of the density of molecules from equilibrium owing to their attraction by the metal (m - S analyzed area in the exeriment (m deviation of the frequency for maximum enhancement from the bulk lasmon frequency - m (s -1 e elementary charge (1. x -19 C E electric field, in general (V/m E amlitude of the oscillating electric field (V/m E i electric field of the image diole inside a metal cavity (V/m E ( i electric field of the image diole near a metal lane (V/m 1 E ext. external electric field (V/m E ind. induced electric field by the image diole (V/m EF SERS enhancement factor (dimensionless ermittivity of vacuum (8.8 x -1 F/m r relative ermittivity of a material (dielectric constant 11 (dimensionless ε (z binding energy to a metal lane - U a (z (J Acceted Manuscrit