Quantitative completion of the scale-invariant theory of Raman enhancement in fractal metal clusters

The scale-invariant theory of enhanced Raman scattering by fractal clusters (Stockman, Shalaev, Moskovits, Botet, and George, Phys. Rev. B 46, 2821 (1992)) shows that the Raman enhancement factor per monomer, $G^{\mathrm{RS}}$, of a fractal aggregate of resonant metal nanoparticles scales in the small-Stokes-shift regime as $G^{\mathrm{RS}}\sim Q^{3}(R_{0}^{3}|X|)^{d_{o}+3}$, where $Q$ is the monomer resonance quality factor, $R_{0}$ the inter-monomer spacing, $X$ a spectral variable, and $d_{o}$ the optical spectral dimension. The theory, however, leaves four quantitative ingredients undetermined: the numeric prefactor of the closed-form estimate $G^{\mathrm{RS}}\approx\tfrac{1}{2}Q^{3}(R_{0}^{3}X)^{4}\operatorname{Im}\chi(X)$; the effect of short-range couplings absent from the dipole-dipole model (higher multipoles, exchange, and conduction between touching monomers); the crossover behavior when the derived validity inequalities are only marginally satisfied, including radiative far-field interference; and a first-principles computation of $d_{o}$ itself, which is currently taken from independent numerical modeling. The problem is to complete the theory so that it predicts the absolute magnitude of $G^{\mathrm{RS}}$, not only its scaling, for specified classes of fractal aggregates.

Background

Surface-enhanced Raman scattering (SERS) is the enormous intensification (factors of $10^{4}$–$10^{6}$ or more) of the Raman signal of molecules adsorbed on nanostructured noble-metal surfaces, produced mainly by the concentration of the local optical field near resonantly excited surface plasmons — the collective electron oscillations of the metal. Aggregated colloidal silver and gold form fractal clusters: random, statistically self-similar aggregates built of $N\sim10^{2}$–$10^{4}$ nearly identical spherical monomers, in which the number of particles within a distance $R$ grows as $R^{D}$ with Hausdorff (fractal) dimension $D\approx1.7$–$2$ for the usual cluster–cluster aggregation and random-walk geometries. Because a fractal is dilute on large scales, its optical eigenmodes are localized on subsets of monomers, which distinguishes its response from both an isolated particle and a compact aggregate.

The standard theoretical model is the coupled-dipole (discrete-dipole) approximation: each monomer carries an induced dipole $\mathbf{d}_{i}$ obeying $Z\,d_{i\alpha}=E_{\alpha}^{(0)}-\sum_{j}(i\alpha|W|j\beta)d_{j\beta}$, where $Z=\chi_{0}^{-1}$ is the inverse monomer polarizability, $W$ is the quasistatic dipole-dipole coupling (proportional to $r_{ij}^{-3}$), and $\mathbf{E}^{(0)}$ is the driving field. For a spherical monomer of radius $R_{m}$, $\chi_{0}=R_{m}^{3}(\epsilon-\epsilon_{0})/(\epsilon+2\epsilon_{0})$ in terms of the metal dielectric function $\epsilon$. It is convenient to use the spectral variable $X\equiv-\operatorname{Re}Z$ and the dissipation parameter $\delta\equiv-\operatorname{Im}Z>0$, so that $Q=(R_{0}^{3}\delta)^{-1}\gg1$ is the quality factor of the isolated monomer resonance. The theory of Markel, Muratov, Stockman, and George (Phys. Rev. B 43, 8183 (1991)) established that the cluster's linear response scales in a window $\delta,\;R_{0}^{-3}N^{-(3/D-1)/(1-d_{o})}\ll|X|\ll R_{0}^{-3}$ with a critical index $d_{o}$, the optical spectral dimension ($0<d_{o}<1$), the analogue, for the dipolar plasmon modes of a fractal, of the Alexander–Orbach fracton dimension (the exponent governing the density of vibrational excitations on a percolation fractal); $d_{o}$ was obtained numerically ($\approx0.3$ for cluster–cluster aggregates, $\approx0.4$ for random-walk clusters) and is the exponent controlling the density of localized dipolar eigenstates and hence the absorption per monomer $\operatorname{Im}\chi$.

In this framework Stockman, Shalaev, Moskovits, Botet, and George (Phys. Rev. B 46, 2821 (1992)) derived an exact relation for the Raman enhancement factor per monomer in the large-Stokes-shift regime (Stokes frequency well outside the cluster absorption band), $G^{\mathrm{RS}}=\delta(1+X^{2}/\delta^{2})\operatorname{Im}\chi\sim Q(R_{0}^{3}|X|)^{d_{o}+1}$, showing that there the enhancement is purely the enhancement of local-field intensity at the excitation frequency. For small Stokes shifts (Stokes frequency inside the band), the induced Stokes-frequency dipoles re-interact with the cluster, and scale invariance yields the stronger law

$$G^{\mathrm{RS}}\sim Q^{3}\,(R_{0}^{3}|X|)^{d_{o}+3},$$

together with the approximate closed form $G^{\mathrm{RS}}\approx\tfrac{1}{2}Q^{3}(R_{0}^{3}X)^{4}\operatorname{Im}\chi(X)$. Monte Carlo simulations (with decimation, i.e. random dilution of the clusters, and ensemble averaging) supported the exponent $d_{o}+3$ (fitted exponents 3.30 and 3.27 for cluster–cluster aggregates, 3.45 for random-walk fractals), the independence of $N$, and the $Q^{3}$ scaling; the predicted spectral shape reproduced measured wavelength-dependent SERS enhancement of silver colloids up to an overall normalization. The results also correct a widespread SERS heuristic: the small-Stokes enhancement is not the square of the large-Stokes one. The general product form $|E(\omega_{L})|^{2}|E(\omega_{S})|^{2}$ ('E$^{4}$') of the SERS electromagnetic enhancement factor was later rigorously justified within classical electrodynamics by Le Ru and Etchegoin (Chem. Phys. Lett. 423, 63 (2006)); its fractal-specific counterpart, which includes the self-consistent re-interaction of the Stokes-frequency dipoles, is precisely the $Q^{3}$ law with exponent $d_{o}+3$ above.

The theory's validity rests on derived inequalities — the scaling window above (Eq. (12) of the paper), the condition $R_{0}^{3}|X|\gg(R_{0}/\lambda)^{(3-D)/(1-d_{o})}$ ensuring the eigenmode localization (coherence) length $L_{X}\sim R_{0}(R_{0}^{3}|X|)^{-(1-d_{o})/(3-D)}$ is smaller than the wavelength $\lambda$ so the incident field is spatially uniform across a mode (Eq. (39)), and, for clusters of total size $R_{c}\gg\lambda$ (with $N\sim(R_{c}/R_{0})^{D}$), the conditions $|X|\gg(R_{0}/\lambda)^{3-D}$ for $D<2$ or $|X|\gg(R_{0}/\lambda)N^{1-2/D}$ for $D>2$ ensuring that radiative far-field ($r_{ij}\gg\lambda$) coupling between monomers can be neglected (Eq. (40)). Within the model, two exact sum rules constrain $\operatorname{Im}\chi$ and the local-field correlations, and the paper used them only to estimate — not derive — the prefactor $\tfrac{1}{2}$, noting that its true value depends on how the sum-rule weight is partitioned between the scaling region and the non-scaling parts of the spectrum, and that the simulations were actually better fit with $\tfrac{1}{4}$. The model itself omits all short-range physics of real aggregates: higher-multipole couplings, exchange interactions, and Ohmic (or tunneling) conduction between touching monomers — effects that electron micrographs show are present in real colloidal aggregates and that can renormalize the enhancement magnitude and possibly the accessible scaling window. Finally, $d_{o}$ is an input taken from independent numerical modeling of linear responses rather than a quantity computed from the microscopic aggregation physics.

Problem Statement

Complete the scale-invariant theory of enhanced Raman scattering by fractal clusters so that it predicts the absolute magnitude — not only the scaling — of the Raman enhancement factor per monomer $G^{\mathrm{RS}}$ for specified classes of fractal metal aggregates (e.g. cluster–cluster aggregates with $D\approx1.7$, $d_{o}\approx0.3$, and random-walk clusters with $D=2$), within the framework of Stockman, Shalaev, Moskovits, Botet, and George (Phys. Rev. B 46, 2821 (1992)). The four requested ingredients are jointly in service of this single objective — the prefactor fixes the magnitude, the short-range physics fixes how nonuniversal that magnitude is, the crossover analysis fixes where the prediction applies, and $d_{o}$ fixes the input exponent — and each one remains undetermined in the cited theory:

1. Determine the numeric prefactor in the small-Stokes approximate formula $G^{\mathrm{RS}}\approx\tfrac{1}{2}Q^{3}(R_{0}^{3}X)^{4}\operatorname{Im}\chi(X)$, in particular how it depends on the partitioning of the exact sum rules for $\operatorname{Im}\chi$ and the local-field correlations between the scaling region and the non-scaling parts of the spectrum (the source's own numerics suggested $\tfrac{1}{4}$ fits better than $\tfrac{1}{2}$ for cluster–cluster aggregates).
2. Incorporate quantitatively the short-range couplings omitted from the dipole-dipole model — higher-multipole couplings, exchange interactions, and Ohmic/tunneling conduction between touching monomers — and establish how they renormalize the enhancement magnitude, whether they change the prefactor, and whether they shrink or shift the accessible scaling window.
3. Characterize the transition behavior when the validity inequalities of the theory — the scaling window (Eq. (12)), the localization-length condition $R_{0}^{3}|X|\gg(R_{0}/\lambda)^{(3-D)/(1-d_{o})}$ (Eq. (39)), and the conditions for neglecting radiative far-field coupling in clusters larger than the wavelength (Eq. (40)) — are only marginally satisfied, including the role of radiative interference between monomers in the intermediate regimes.
4. Compute the optical spectral dimension $d_{o}$ from first principles — from the microscopic cluster-generation protocol (aggregation kinetics and the statistics of the resulting fractal geometry) together with the short-range physics — rather than taking it from independent numerical modeling of linear responses, and specify the range of aggregation models for which the resulting $d_{o}$ supports the scaling predictions.

Each of the four ingredients is individually required for completion — in particular, the short-range couplings must be incorporated quantitatively rather than qualitatively, and $d_{o}$ must be derived from the aggregation physics rather than taken from published numerical modeling — so that partial completion of any single ingredient does not resolve the task. Completing the theory therefore means fixing the absolute magnitude of $G^{\mathrm{RS}}$ for the aggregate classes treated, together with the parameter window within which the prediction holds and the origin of $d_{o}$, so that the comparison with measured SERS enhancement of fractal colloidal aggregates no longer requires an overall fitted normalization.

References

1. M. I. Stockman, V. M. Shalaev, M. Moskovits, R. Botet, and T. F. George, 'Enhanced Raman scattering by fractal clusters: Scale-invariant theory', Phys. Rev. B 46, 2821–2830 (1992). DOI: 10.1103/PhysRevB.46.2821, https://doi.org/10.1103/PhysRevB.46.2821 (source paper)
2. V. A. Markel, L. S. Muratov, M. I. Stockman, and T. F. George, 'Theory and numerical simulation of optical properties of fractal clusters', Phys. Rev. B 43, 8183–8195 (1991). DOI: 10.1103/PhysRevB.43.8183, https://doi.org/10.1103/PhysRevB.43.8183 (introduces the optical spectral dimension $d_o$)
3. M. I. Stockman, L. N. Pandey, L. S. Muratov, and T. F. George, 'Giant fluctuations of local optical fields in fractal clusters', Phys. Rev. Lett. 72, 2486–2489 (1994). DOI: 10.1103/PhysRevLett.72.2486, https://doi.org/10.1103/PhysRevLett.72.2486
4. M. I. Stockman, L. N. Pandey, L. S. Muratov, and T. F. George, 'Optical absorption and localization of eigenmodes in disordered clusters', Phys. Rev. B 51, 185–195 (1995). DOI: 10.1103/PhysRevB.51.185, https://doi.org/10.1103/PhysRevB.51.185
5. V. M. Shalaev, 'Electromagnetic properties of small-particle composites', Phys. Rep. 272, 61–137 (1996). DOI: 10.1016/0370-1573(95)00076-3, https://doi.org/10.1016/0370-1573(95)00076-3 (review continuing the program)
6. S. V. Karpov, V. S. Gerasimov, I. L. Isaev, and V. A. Markel, 'Local anisotropy and giant enhancement of local electromagnetic fields in fractal aggregates of metal nanoparticles', Phys. Rev. B 72, 205425 (2005). DOI: 10.1103/PhysRevB.72.205425, https://doi.org/10.1103/PhysRevB.72.205425
7. Z. Naeimi and M. Miri, 'Optical properties of fractal aggregates of nanoparticles: Effects of particle size polydispersity', Phys. Rev. B 80, 224202 (2009). DOI: 10.1103/PhysRevB.80.224202, https://doi.org/10.1103/PhysRevB.80.224202
8. J. A. Sánchez-Gil and J. V. García-Ramos, 'Calculations of the direct electromagnetic enhancement in surface enhanced Raman scattering on random self-affine fractal metal surfaces', J. Chem. Phys. 108, 317–325 (1998). DOI: 10.1063/1.475394, https://doi.org/10.1063/1.475394
9. K. Arya, 'Scattering T-matrix theory in wave-vector space for surface-enhanced Raman scattering in clusters of nanoscale spherical metal particles', Phys. Rev. B 74, 195438 (2006). DOI: 10.1103/PhysRevB.74.195438, https://doi.org/10.1103/PhysRevB.74.195438
10. E. C. Le Ru and P. G. Etchegoin, 'Rigorous justification of the |E|4 enhancement factor in Surface Enhanced Raman Spectroscopy', Chem. Phys. Lett. 423, 63–66 (2006). DOI: 10.1016/j.cplett.2006.03.042, https://doi.org/10.1016/j.cplett.2006.03.042
11. L. Wu, H. Duan, P. Bai, M. Bosman, J. K. W. Yang, and E. Li, 'Fowler–Nordheim Tunneling Induced Charge Transfer Plasmons between Nearly Touching Nanoparticles', ACS Nano 7, 707–716 (2013). DOI: 10.1021/nn304970v, https://doi.org/10.1021/nn304970v
12. M. Banik, P. Z. El-Khoury, A. Nag, A. Rodriguez-Perez, N. Guarrottxena, G. C. Bazan, and V. A. Apkarian, 'Surface-Enhanced Raman Trajectories on a Nano-Dumbbell: Transition from Field to Charge Transfer Plasmons as the Spheres Fuse', ACS Nano 6, 10343–10354 (2012). DOI: 10.1021/nn304277n, https://doi.org/10.1021/nn304277n
13. M. Moskovits, 'Persistent misconceptions regarding SERS', Phys. Chem. Chem. Phys. 15, 5301 (2013). DOI: 10.1039/c2cp44030j, https://doi.org/10.1039/c2cp44030j