Size Dependent ø(3) for Conduction Electrons in Ag Nanoparticles

Our theoretical study of the third-order susceptibility (ø(3)) for Ag dielectric composite reveals a critical role of saturation of optical transitions between discrete states of conduction electrons in metal quantum dots. The calculated size dependence of the ø(3) for Ag nanoparticles reproduces the published experimental results. Saturation effects lead to a decrease of the local field enhancement factor that is of particular importance for surface-enhanced phenomena, such as Raman scattering and nonlinear optical responses. Plasmonic nanomaterials have attracted much recent research interest because of their unique optical properties, such as nonlinear optical activity,1 the chirality of plasmon modes,2 and the quantum-size effect in two-photon excited luminescence.3 Current state-of-the-art nanofabrication techniques allow the development of novel applications based on such properties. Of particular importance for applications are the large local-field enhancements for metal particle aggregates that lead to surface-enhanced Raman scattering (SERS) and a number of nonlinear optical phenomena,4 including the polarization nonlinearities.5 The optical response of a nanosized metal particle is a core of all aforementioned phenomena. The confinement of electrons in a metal quantum dot leads to energy quantization of conduction band and appearance of collective plasmon modes. It is well-known that the energy quantization affects most of the physical properties of metal nanoparticles,6-8 and in particular its nonlinear optical response.9,10 The optical properties of a nanosized metal particle can be described in terms of electron transitions between the discrete energy states in a quantum well subjected to the enhanced local field. Large enhancements of the local field inside a particle can be realized at the plasmon resonance frequency. The local field inside a spherical particle, Ei, is related to the applied field, E0, by the local field (enhancement) factor f(ω) as follows:11 where m ) ′m + i ′′ m is the complex dielectric response of the metal, and h is the dielectric function of a host medium. Note that the zero in the denominator in eq 1 is the surface plasmon resonance condition for a spherical particle embedded in a host. In a composite with a small volume fraction of metal particles, the third-order nonlinear susceptibility can be computed by9 where p is a volume fraction of the metal particles and øm (3) is the nonlinear susceptibility term of the metal particle itself. It should be noted that both intraband (within conduction band) and interband (between dand s-p conduction bands) transitions contribute to øm . Utilizing the degenerate fourwave mixing technique, the ø(3) values and its size dependence were extracted from recent detailed experimental studies for nanosized Ag, Au, and Cu particles.9,12,13 Some of these results were taken to compare the findings with existing theoretical models in order to resolve the origin of the optical nonlinearity. Doing so, it was concluded that the conduction electron intraband transitions play a relatively minor role. This conclusion was based on a theoretical size dependence derived by Hache, Ricard, and Flytzanis (HRF),9 with the Hamiltonian that uses a description in terms of a vector potential and electron momentum. In this letter, we will demonstrate that the opposite conclusion can be made if one adopts the quantum well theory with the Hamiltonian of electron-field interaction taking the form where d is the dipole moment and E is the electrical field. * Corresponding author. Phone: 1-765-494-0628; Fax: 1-765-494-6951. e-mail: [email protected]. † Purdue University. ‡ New Mexico State University. Ei ) 3 h m + 2 h E0 ) f(ω)E0 (1) ø ) pf(ω) |f(ω)| øm (2) H ) -dE (3) NANO LETTERS 2004 Vol. 4, No. 8 1535-1539 10.1021/nl049438d CCC: $27.50 © 2004 American Chemical Society Published on Web 07/24/2004 Recently, Rautian10 showed that, for nanosized spherical particles, the use of the Hamitonian given in eq 3 is preferred, and that this Hamiltonian is no longer equivalent to the standard Hamiltonian in terms of a vector potential. Here, we compare the approaches based on the Rautian and HRF models and calculate the size dependencies of both øm (3) and f(ω) for nanosized Ag particles. Our results reaffirm the Rautian model, and we find good agreement of the sizedependent øm (3) with the experiment, a result that is not achieved with HRF’s approach. The characteristic separation between the levels near the Fermi energy, EF, can be estimated as δF ) 2xEFE0, where E0 ) p2/2ma2 is the energy separation found at the bottom of the conduction band of particles with radius a. Under the condition pω . δF, which is the case for the particle radii ranging from about 2 to about 100 nm and visible frequencies ω, one can distinguish two kinds of transitions between discrete states, resonant (ωij∼ω) and nonresonant (ωij,ω). The potential saturation of optical transitions between the discrete levels in metal nanoparticle is a second crucial factor. Saturation effects result in a decrease of the local field enhancement factor, and a subsequent decrease in the enhancements for SERS as well as for nonlinear effects. Using the degenerate electron gas model in an infinite spherical well in the limit (υF/2πc) λ , a , λ (where υF is the electron speed near the Fermi surface), Rautian was able to derive the linear and nonlinear dipole moments for a spherical particle induced by field component Ei: In eq 4, a denotes the particle radius, ω is the frequency of the field, m is the electron mass, e is the electron charge, N is the number of electrons in the particle, I0 ) ∑|Ei|2, Γ1 and Γ2 represent the relaxation rates for the population and coherence, respectively. We focus on a linearly polarized field, where Ai ) 2/5. For our case, the parameters F1, g1, F3, and g3 are only weakly size dependent (if at all): F1 is approximately unity, g1 ) 0.6 at pω/EF ≈ 0.5, F3 ranges from 0.30 to 0.33 for particles varying between 2 and 15 nm, and g3 ) 0.64. A detailed discussion of how to calculate these parameters can be found in ref 10. Basically, the parameters g1 and g3 result from the integration over the resonant states, whereas F1 and F3 result from the summation of the nonresonant terms close to EF. A component xxxx of the nonlinear tensor susceptibility, øm,xxxx (3) can then be written as the sum of the nonresonant, ømn , and resonant, ømr , contributions, i.e. Nonresonant contributions can be calculated by integrating over transitions close to the Fermi energy, and resonant contributions are derived by integration from zero to infinity when the energy difference is close to the photon energy. An analysis of eq 4 shows that for linear polarization the two terms in eq 5 are given by