7812
J. Phys. Chem. C 2010, 114, 7812–7815
Excitation Profiles and the Continuum in SERS: Identification of Fano Line Shapes John R. Lombardi* and Ronald L. Birke Department of Chemistry, The City College of New York, New York, New York 10031 ReceiVed: January 20, 2010; ReVised Manuscript ReceiVed: February 25, 2010
We extend a previous theoretical expression of SERS to include the possibility of coupling of the molecular levels with the continuum of levels in the conduction band in the metal. We modify the expression for excitation profiles to account for interference between nearby conduction band levels, producing an asymmetric Fano line shape. Both charge-transfer and molecular transitions are considered since both can contribute to the SERS enhancement. We test the theory by comparison with the observed profiles in pyridine, p-aminothiopehnol (PATP), and an azo dye (DABS). The experimental profiles are clearly asymmetric, dipping into the continuum on one side and approaching the continuum asymptotically on the other. In each case, an excellent fit of the Fano profile is obtained. Introduction Since the earliest days of surface-enhanced Raman spectroscopy (SERS), there have been periodic reports of a continuum background,1,2 which appears to accompany the observed Raman enhancement. For example, Otto found a strong background in the SERS spectrum of CN-, which was shown not due to Rayliegh scattering.3 We showed that the continuum displayed an electrochemical potential dependence that was identical to that of the SERS spectrum of pyridine4 and showed an asymmetric potential profile. This indicated strongly that the continuum was intimately connected to the SERS effect. We also showed that the anti-Stokes side of the continuum could be adequately fit by a temperature-dependent Fermi factor {1 + exp((ωL - ω)/kT)}-1. At the same time, Burstein et al.5 suggested a mechanism involving the photoexcitation of electron-hole pairs in the metal. However, this mechanism did not provide an adequate explanation of the observed potential dependence of the background. In a subsequent work,6 we suggested a mechanism involving photoexcited electron transfer from the molecule to the metal, followed by rapid recombination and emission (scattering). This explained both the existence of the continuum as well as the potential dependence. Still later, we expanded this theory by including a Herzberg-Teller coupling mechanism7 modeled after the theory of Raman scattering due to Albrecht.8 This was not only able to explain the potential dependence of SERS but also provided selection rules, which could be shown to predict the relative intensities of non-totally symmetric modes observed in SERS.9 By examining the ratio of the anti-Stokes to the Stokes contribution, Brolo, Sanderson, and Smith10 concluded that charge-transfer and plasmon resonances were required to explain their observations. Micheals, Jiang, and Brus11 demonstrated a correlation of peak intensities of SERS and the continuum with time. Later, further confirmation of the singular connection between the two signals was obtained by observing the coincidence of blinking behavior of SERS and the continuum.12 Both SERS and the continuum were observed in a single-molecule study by PeyserCapadona et al.,13 despite the undersized Ag clusters involved. Similarly, Bizzari and Cannistraro14 found, by correlating * To whom correspondence should be addressed.
statistical fluctuations in single-molecule SERS, that the continuum and SERS signal are linked. Still more recently, interest in the continuum has been rekindled in an elegant experiment described by Mahajan et al.,15 in which a substrate composed of sphere segment void (SSV) plamonic gold nanoparticles is used to carefully monitor the size dependence and the potential dependence of the background and SERS signal using both electron-withdrawing and electron-donating adsorbates. They showed once again the intimate connection between both SERS and the accompanying continuum. They confirmed that the continuum, which showed a broad maximum at around 600-700 cm-1, was not caused by nonlinear processes and furthermore identified and characterized a smaller feature (at around 2500 cm-1) which shifted with increasing thickness of the SSV substrate. This was associated with specific plasmon modes. Furthermore, they showed a strong dependence of the continuum intensity on the electronwithdrawing (more intense) or -donating (less intense) power of the adsorbed molecule. They also investigated the effect of an applied potential on the continuum and found results similar to those previously obtained.4 Their explanation of these observations extends the model of Burstein et al.5 to include specific reference to an image dipole. This is invoked to excite electron-hole pairs in the metal. It appears that in order to provide a convincing rational for the observations made, it is necessary to recognize that the molecule-metal system involves molecular orbital levels of the molecule, which can couple to the metal continuum. Such a coupling was originally examined in detail by Fano,16 in which he showed that an interaction of a discrete state with a continuum results in both phase shifts and interference effects giving rise to a characteristically asymmetric line shape with net reduction of intensity in some regions of the continuum. These ideas have recently been applied to J-aggregates and surface-plasmon-enhanced resonances in the Raman spectrum by Kelley.17,18 In this article, we will apply the coupling of the molecular levels to those of the continuum in the context of a unified theory of SERS,9 which includes contributions from the surface plasmon resonance, charge-transfer resonance, and molecular resonance in a single expression. This expression comes naturally out of considerations of the Herzberg-Teller vibronic coupling mechanism when applied to a molecule-
10.1021/jp100568b 2010 American Chemical Society Published on Web 04/14/2010
Identification of Fano Line Shapes
J. Phys. Chem. C, Vol. 114, No. 17, 2010 7813
metal system. We will show that application of the ideas of Fano to such a SERS system leads to excellent fits of the SERS excitation profile for both the electrochemical potential and excitation wavelength. Extension of the SERS Expression to Account for Coupling with the Continuum It is our intention to extend our previous unified description of SERS to include the possibility of coupling between excited states of the molecule-metal system and the continuum of states in the metal conduction band. By deriving an expression for the polarizability of a molecule-metal system which accounts for Herzberg-Teller coupling, we have shown that in SERS, there are three possible sources of resonantly enhanced surface Raman spectra.9 In the expression, the µ represent transition dipoles between various states of the molecule-metal system. I is the ground state of the system, F is an excited molecule-metal charge-transfer state, and K is an excited molecular state. The term hIF is the Herzberg-Teller vibronic coupling constant, and 〈i|Qk|f〉 is the vibrational matrix element for the normal mode represented by Qk. For a harmonic oscillator in the ground state, i ) 0 and f ) 1. The terms ε1 and ε2 are the real and imaginary dielectric constants of the metal, while ε0 is that of the surrounding medium. The laser frequency is given by ω, while ωFK and ωIK are optical transition frequencies. Corresponding damping factors are given by γ. The result is a typical sum over states, which is valid when far from any of the resonances. Near a resonance, however, only a single term (or at most a few) in the expression dominates. The combined expression when the exciting laser is near one or more of the resonances is given by RIFK(ω) ) ((ε1(ω) + 2ε0) + 2
µKIµFKhIF〈i|Qk |f〉 2 2 2 2 ε2)(ωFK - ω2 + γFK )(ωIK
2 - ω2 + γIK )
(1)
The SERS enhancement factor is proportional to |RIFK(ω)|2. Let us first examine the denominator, which involves the product of three terms, each of which presents a resonance contribution to SERS. The first (ε1(ω) + 2ε0)2 + ε22 is due to the plasmon resonance at ε1(ω) ) -2ε0. We choose this expression for a single spherical particle for illustrative purposes, recognizing that for particle aggregates with hot spots, a more complex expression containing a similar dielectric resonance expression will replace this.19 The second resonance, which may be potential (Fermi energy)-dependent and represents charge2 2 transfer resonance (ωFK - ω2) + γFK occurs at ω ) ωFK, and 2 2 2 the third (ωIK - ω ) + γIK represents the molecular resonance at ω ) ωIK. For electrochemical SERS, the expression for the second resonance predicts a positive slope for VMAX (the applied voltage at resonance) against ω for metal-to-molecule transfer or the opposite (negative) slope for molecule-to-metal transfer (pωFI,K ) EF(0) ( eVMAX).7,20 Note, in the third case, that if the resonance condition is fulfilled (ω ) ωIK), we have SERRS (surface-enhanced resonance Raman spectroscopy). This is the case for many single-molecule experiments, such as crystal violet,21 for which the molecular transition is also in resonance with the laser. This expression (eq 1) can be seen to depend crucially on the relative values of the resonance conditions (not to mention the damping factors ε2, γFK, and γIK) with respect to the laser wavelength. As the excitation wavelength is varied, the relative intensities of the Raman bands will also be expected
Figure 1. Levels of the molecule-metal system.
to vary. This is the explanation as to why the appearance of the SERS spectrum varies so much with excitation wavelength.22 Note that since all three resonances contribute in a multiplicative fashion, they cannot be considered as separate, despite the fact that at different wavelengths each may contribute a different amount to the total enhancement. In fact, by examination of the numerator, it can be seen that the three resonances are linked by the requirement that all four matrix elements must be simultaneously nonzero in order to observe a spectral band. This leads to strict selection rules9 which must be obeyed and determine the consequent appearance (relative intensities) of the SERS spectrum. The numerator involves two transition dipole moments, one for the charge-transfer state (F,K) and the other for the molecular state (I,K). In addition there is the Herzberg-Teller coupling constant (hIF), which determines the symmetry of the normal mode Qk. We are now in a position to consider the interaction of one of the unfilled excited levels of the molecule with one of the unfilled levels of the continuum of the conduction band of the metal; see Figure 1. Note that for many of the molecules observed in SERS, the Fermi level of the metal lies between the filled and unfilled levels of the molecule. Thus, when the molecule-metal system is in an excited state, either by a chargetransfer transition (F) or a molecular transition (K), the system can autoionize by coupling of the excited level with the metal continuum. (A similar case arises for molecule-to-metal charge transfer, which can be obtained with simple interchanges of the notation of I with K. We will not include that here for brevity.) This is the same as that considered by Fano. In a recent article, Christ et al.23 have examined the Fano interferences in a plasmonic lattice; therefore, we will confine ourselves instead to the interferences due to the molecular or charge-transfer states. Note that a charge-transfer resonance can be scanned by changing either the excitation wavelength or the applied potential, while for a molecular resonance, only the excitation wavelength can be used. We will not repeat our treatment for both, but since the charge-transfer resonance is more general, we will consider only that one. Following Fano, we assume now that |F〉 is an eigenfunction of the perturbed system, that is, the interaction of a discrete charge-transfer state with the continuum. Extracting only the portion of the eq 1 that is relevant to the charge-transfer resonance and expanding in partial fractions,24 we obtain
ωFK
µFK µFK ) - ω + iγFK γFK(i + ε)
(2)
7814
J. Phys. Chem. C, Vol. 114, No. 17, 2010
Lombardi and Birke
where ε ) (ωFK - ω)/γFK is the same as the definition of ε by Fano (γFK ) 1/2Γ). Further connection to Fano’s notation provides
µFK ) (ΨE |T|i) ) πaVE(q + ε)
(3)
In Fano’s notation, ΨE is the eigenfunction of the perturbed excited state of the system, while i represents the initial state involved in the transition. The parameter a is a normalization constant, VE is the coupling constant between the unperturbed excited molecular level and the continuum, and q is a measure of the ratio of transition probabilities from the initial state |i) to the modified discrete state (Φ) to that of unperturbed continuum states ψE, such that
1 2 2|(Φ|T|i)| 2 πq ) 2 |(ψE |T|i)| 2γFK
(4)
The parameter q may be regarded as an asymmetry parameter, which varies from 0 (where a simple Lorentzian function is obtained) to larger values for which increasing asymmetry is introduced into the profile, and is a measure to which extent the continuum states are perturbed by the discrete state causing interference. We then obtain the SERS enhancement profile by squaring
|RIFK | 2 ∝
|
µFK γFK(i + ε)
| ( 2
)c
(q + ε)2 1 + ε2
)
Figure 2. Best fit of the Fano profile to the voltage-dependent profile of the 1008 cm-1 line of pyridine. Experimental data (black squares) are from ref 4.
(5)
which is the familiar Fano profile. Note that c and q are regarded as a constant over the range of the profile, and for negative q, the sense of ε is reversed. These are plotted as a function of ε for various values of q in Fano’s Figure 116 and will not be reproduced here. As a test of this theory, in the next section, we fit several experimental SERS profiles to the expression in eq 5 using q and γ as fitting parameters. Comparison with Experiments It is actually difficult to find adequate experimental data with which to test this theory. This is especially true for electrochemical-potential-dependent sweeps since other effects, such as desorption, electrochemical reactions (oxidation or reduction), reorientation effects, or hydrogen evolution can distort the observed line shapes. Therefore, in order to ensure that such effects are not invasive, care must be exercised in looking for valid comparisons. We begin with the results for the 1008 cm-1 line of pyridine, which have been previously reported on a Ag electrode, in comparison with the potential dependence of the contunuum.4 The effective range of potentials was from -1.48 to 0.0 V (versus SCE), with an intensity maximum at about -0.6 V, which represents the charge-transfer resonance between the metal and molecular π* level, obtained with an excitation wavelength of 488 nm. There is no problem with the electrochemistry in these results, which were cut off at 0.0 V due to formation of AgCl on the electrode surface. It can be seen that there is a sufficient voltage range in which the full asymmetry of the profile is apparent (see the black squares in Figure 2). The optimum fit of the Fano profile occurred with a width of 0.25 eV and a value of q ) 5. Note that for sufficiently negative
Figure 3. Best fit of the Fano profile to the voltage-dependent excitation profile of the 1440 cm-1 line of p-aminothiophenol (PATP). Experimental data (black squares) are from ref 22.
ε, the profile dips below the continuum, while at higher positive values, it begins to approach the unperturbed continuum value. Figure 3 depicts the potential-dependent profile of the 1440 cm-1 line of the molecule p-aminothiophenol (PATP).22 The profile peaks at -0.50 V and once again displays the characteristic dip of intensity into the continuum on one side and asymmetrically flattens to the continuum level on the opposite side. The range of applied voltage is from -0.80 to +0.05 V. The fit parameters in this case are γ ) 0.18 eV and q ) 3.5. In Figure 4, we show the excitation wavelength profile reported by Siiman et al.25 for the azo dye 4-(dimethylamino)azobenzene-4′-sulfonyl aspartate (DABS) on a roughened silver electrode. Since this is a dye molecule, it has an absorption spectrum in the visible region. The surface-enhanced Raman profile was taken to be SERRS by the authors and distant from any charge-transfer transitions. Thus, this is a good test of whether the above theory also applies to molecular absorption (SERRS), as well as charge-transfer. This means that we should
Identification of Fano Line Shapes
J. Phys. Chem. C, Vol. 114, No. 17, 2010 7815 discrete molecular levels and the conduction band continuum and are responsible for the asymmetric excitation profiles observed in SERS. Acknowledgment. We are indebted to the National Institute of Justice (Department of Justice Award #2006-DN-BX-K034) and the City University Collaborative Incentive program (#80209). This work was further supported by the National Science Foundation under Cooperative Agreement No. RII9353488, Grants CHE-0091362 and CHE-0345987, Grant ECS0217646, and by the City University of New York PSCBHE Faculty Research Award Program. References and Notes
Figure 4. Best fit of the Fano profile to the excitation profile of the 1412 cm-1 line of DABS. Experimental data (black squares) are from ref 24.
replace ωFK and γFK with ωIK and γIK in the above expressions (eqs 2-5). For comparison, we used only the band at 1412 cm-1 (those at 1394, 1141, and 1106 cm-1 were also reported with a similar profile). Ten laser lines from 476.5 to 676.4 nm were utilized, with a maximum intensity found at about 520 nm. Once again a clear Fano profile is observed, and can be fit with a width of 0.086 eV (700 cm-1) and q ) 5. Conclusions We have extended a previous theoretical treatment of SERS to include the possibility of coupling of the molecular levels with the continuum of levels in the conduction band in the metal. Following the work of Fano, we modified the expression for excitation profiles to account for interference between the nearby molecule and the continuum of metal levels in the conduction band, producing an asymmetric line shape. Both charge-transfer and molecular transitions were considered since both can contribute to the SERS enhancement. From the observed profiles in Figures 2-4, we can see that in each case, an excellent fit is obtained. The experimental profiles are clearly asymmetric, dipping into the continuum on one side and approaching the continuum asymptotically on the other. This asymmetry has often been remarked upon (see, for example, ref 22), but for reasons discussed in the previous section, it is often difficult to find an experimental profile which is reliable over a sufficient range to be a good test of the theory. We feel, however, that the fits are of sufficient quality to show that the Fano theory may be applied to interaction between the
(1) Heritage, J. P.; Bergman, J. G.; Pinczuk, A.; Worlock, J. M. Chem. Phys. Lett. 1979, 67, 229. (2) Chen, C. Y.; Burstein, E.; Lundquist, S. Solid State Commun. 1979, 32, 63. (3) Otto, A. Surf. Sci. 1978, 75, L392. (4) Birke, R. L.; Lombardi, J. R.; Gersten, J. I. Phys. ReV. Lett. 1979, 43, 71. (5) Burstein, E.; Chen, Y. J.; Chen, C. Y.; Lundquist, S.; Tosatti, E. Solid State Commun. 1979, 29, 567. (6) Gersten, J. I.; Birke, R. L.; Lombardi, J. R. Phys. ReV. Lett. 1979, 43, 147. (7) Lombardi, J. R.; Birke, R. L.; Lu, T.; Xu, J. J. Chem. Phys. 1986, 84, 4174. (8) Albrecht, A. C. J. Chem. Phys. 1961, 34, 1476. (9) Lombardi, J. R.; Birke, R. L. J. Phys. Chem. C 2008, 112, 5605. (10) Brolo, A. G.; Sanderson, A. C.; Smith, A. P. Phys. ReV. B 2004, 69, 045424. (11) Michaels, A. M.; Jiang, J.; Brus, L. J. Phys. Chem. B 2000, 104, 11965. (12) Andersen, P. C.; Jacobson, M. L.; Rowlen, K. L. J. Phys. Chem. B 2004, 108, 2148. (13) Peyser-Capadona, L.; Sheng, J.; Gonsalez, J. I.; Lee, T.-H.; Patel, S. A.; Dickson, R. M. Phys. ReV. Lett. 2005, 94, 058301. (14) Bizzari, A. R.; Cannistraro, S. Phys. Chem. Chem. Phys. 2007, 9, 5315. (15) Mahajan, S.; Cole, R. M.; Speed, J. D.; Pelfrey, S. H.; Russell, A. E.; Bartlett, P. N.; Baumberg, J. J. J Phys Chem C 2010; DOI: 10.121/ jp907197b. (16) Fano, U. Phys. ReV. 1961, 124, 1866. (17) Kelley, A. M. Nano Lett. 2007, 10, 3235. (18) Kelley, A. M. J. Chem. Phys. 2008, 128, 224702. (19) Xu, H.; Aizpuruna, J.; KSˇll, M.; Apell, P. Phys. ReV. E 2000, 62, 4318. (20) Lombardi, J. R.; Birke, R. L.; Sanchez, L. A.; Bernard, I.; Sun, S. C. Chem. Phys. Lett. 1984, 104, 240. (21) Canamares, M. V.; Chenal, C.; Birke, R. L.; Lombardi, J. R. J. Phys. Chem. C 2008, 112, 20295. (22) Osawa, M.; Matsuda, M.; Yoshii, K.; Uchida, I. J. Phys. Chem. 1994, 98, 12702. (23) Christ, A.; Ekinci, Y.; Solak, H. H.; Gippius, N. A.; Tikhodeev, S. G.; Martin, O. J. F. Phys. ReV. B 2007, 76, 201405. (24) In order to make a connection with the Fano theory, taking a step back in the derivation, we must expand the denominator (ignoring γFK for 2 the moment). Then, (ωFK -ω2)-1 ) (2ωFK)-1 [(ωFK + ω)-1 + (ωFK-ω)-1], and choosing the term at resonance (i.e. (ωFK-ω)-1), the resulting term (including the γFK) is (2ωFK)-1 [(ωFK-ω) + iγFK]-1. (25) Siiman, O.; Smith, R.; Blatchford, C.; Kerker, M. Langmuir 1985, 1, 90.
JP100568B
