Surface-Enhanced Raman Spectroscopy (SERS)
Electromagnetic SERS enhancement factor of a silver nanosphere: local field, maximum and surface-averaged |E|⁴ enhancement, resonance and distance decay.
SERS EF, maximum on the sphere
1.15e+4
At the pole (along the field)
SERS EF, surface average
2.79e+3
Molecules spread evenly over the sphere
Local intensity |E/E₀|² at λ_L
159 (pole)
23.3 at the equator
Raman (Stokes) wavelength λ_R
422.09 nm
Silver ε at λ_L / λ_R
-4.61 + 0.25i
-5.44 + 0.26i at λ_R (Yang et al. 2015)
Dipole resonance (Re ε = −2ε_m)
383.5 nm
Quasistatic, in n_m = 1.33
The enhancement factor of an isolated sphere is set by silver's permittivity and the medium, not by the radius in this quasistatic model: the radius only sets how fast the field decays away from the surface, as (a/(a + d))³ in the field. Gaps between particles (hot spots) give far larger but very local enhancements, which this model does not cover.
Enhancement factor vs laser wavelength (Raman shift fixed)
- EF, maximum over the surface
- EF averaged over the surface
- Current
Enhancement factor vs distance from the surface at 405 nm
- EF, maximum over the surface
- EF averaged over the surface
g = (ε − ε_m)/(ε + 2ε_m), s = (a/(a + d))³
|E/E₀|² = |1 + 2gs|² cos²θ + |1 − gs|² sin²θ
EF ≈ |E(λ_L)/E₀|² · |E(λ_R)/E₀|² (the |E|⁴ approximation)
⟨EF⟩ = A_L A_R/5 + 8 B_L B_R/15 + 2(A_L B_R + B_L A_R)/15, A = |1 + 2gs|², B = |1 − gs|²
A metal sphere much smaller than the wavelength acts as a dipole. Near the resonance, where Re ε = −2ε_m, the field at its poles grows by about 6ε_m/Im ε, and the Raman signal of a molecule there grows by about the square of the intensity enhancement, once at the laser and once at the Raman wavelength. Chemical (charge-transfer) enhancement adds to this and is not modelled. Silver data: Yang et al. (2015), template-stripped films; colloids and thin films are lossier, and spheres below about 10 nm add surface damping.
Model and references
Textbook approximation. A standard approximation from the literature (for example paraxial, plane-wave or Gaussian-beam), valid within the range the page states. About the tiers
Quasistatic (dipole) silver sphere with measured silver permittivity (Yang et al. 2015) and the |E|⁴ approximation, maximised over the polar angle and averaged over the surface. Not modelled: retardation and radiation damping (spheres above about λ/10), surface damping in very small spheres, particle gaps (hot spots) and chemical enhancement.
- Bohren C. F., Huffman D. R. (1983). Absorption and Scattering of Light by Small Particles, ch. 5. Wiley. Field of a sphere small compared with the wavelength. doi:10.1002/9783527618156
- Le Ru E. C., Etchegoin P. G. (2006). Rigorous justification of the |E|⁴ enhancement factor in Surface Enhanced Raman Spectroscopy. Chem. Phys. Lett. 423, 63–66. doi:10.1016/j.cplett.2006.03.042
- Le Ru E. C., Blackie E., Meyer M., Etchegoin P. G. (2007). Surface Enhanced Raman Scattering Enhancement Factors: A Comprehensive Study. J. Phys. Chem. C 111, 13794–13803. Maximum and surface-averaged enhancement factors. doi:10.1021/jp0687908
- Yang H. U. et al. (2015). Optical dielectric function of silver. Phys. Rev. B 91, 235137. Silver permittivity. doi:10.1103/PhysRevB.91.235137