## Abstract

The optical transmission spectra of several samples of gold nanoparticle layers were examined using a modified Drude model proposed with a novel elastic scattering parameter, γ′. Although the measured transmission spectra deviated from the simple calculation from Mie scattering, it was explained well by the modified model assuming elastic and inelastic scattering in the form of the collision frequency of free electrons within a metal particle due to the particle boundary. The particle-size and inter-particle-distance dependences of γ′ were extracted within the framework of the proposed model from the curve of best fit of the transmittance spectra.

©2010 Optical Society of America

## 1. Introduction

Rapid progress in nanotechnology and the requirements for Green technology have stimulated considerable interest in metal nanostructures in wide range of applications, such as nano-sized passive waveguide, highly sensitive biosensor, and high efficiency light-emitting diodes based on plasmonic resonance, owing to their unique optical properties [1–5]. In particular, research on the optical anomalies and size effect of nano-sized metallic particles has become one of the most active areas [6–8]. Although the optical response for bulk metals is well understood using the Drude model of free electrons at a long wavelength regime, there is some controversy regarding nano-sized metallic particles in the visible and shorter wavelength regime partly due to difficulties in the optical measurement in metallic samples. This is complicated by the absorption itself as well as scattering by the surface roughness [9,10].

An analysis of the transmittance spectra for layered samples of nano-sized Au particles was performed assuming that each nano-sized particle forms a simple effective electric dipole to form an effective dielectric medium. The effective dielectric function was then derived from the modified Drude model with an additional parameter γ′ to represent the bound behavior of free electrons inside the particle. The phenomenological values of γ′ were estimated from the curve of best fit of the transmittance spectra. The particle density and size dependence of γ′ for the validity of the modified Drude model are discussed.

When the dielectric function of a bulk metal is comprised of two contributions, ε = ε_{b} + ε_{f}, ε_{b} is from the interband transition of bound electrons and ε_{f} is from the intraband transition of free electrons. The contribution ε_{f} dominates the long wavelength optical properties of a bulk metal, which can be described well by the simple Drude model with a damping constant γ [11,12]. The derivation of the dielectric function from the equation of motion for electrons gives the following

*ε*is the contribution from the interband transition of bound electrons and

_{b}*σ*is the conductivity of the metal in the form of $$\sigma =\left(\frac{n{e}^{2}}{m}\right)\cdot \left(\frac{1}{\gamma -i\omega}\right)\text{\hspace{0.17em}},$$ respectively. The quantities of σ have the same meaning as the remaining part of this work. The term

*ε*is

_{b}_{g}= ћω

_{g}, E

_{F}, T, and γ

_{b}are the proportionality constant, energy bandgap, Fermi energy, temperature, and collision frequency for the band-to-band transition, respectively [13,14]. F(x, E

_{F},T) is the Fermi distribution, and ω is the angular frequency of the applied optical field.

Because classical Mie theory using the simple Drude model and thin film analysis has difficulties in explaining the peak shift and broadening in the transmission spectra of a layer of nano-sized metallic particles, the size dependence of each contribution to ε_{b} and ε_{f} has been discussed in a variety of ways [14–17]. The complex electrodynamics of large sized metal particles has been discussed, and modifications of the damping constant γ caused by the limitation of the electron mean free path have been attempted [18–20]. While the trials of increasing γ have provided a reasonable explanation of the peak broadening of absorption according to size shrinkage, it has failed to explain the peak shift [21]. Therefore, the phenomenological modification of γ is introduced as a modified Drude model with a complex damping parameter, γ + iγ′, considering that a free electron within a metal particle experiences both elastic and inelastic scattering on the surface boundary according to the Fuchs-Sondheimer model [22]. The optical behavior of a metallic thin film was also described well within the framework of the modified Drude model, recently [23].

In this report, the transmission spectra measured for three samples of layered Au particles were in fairly good agreement with the curve fits with reasonable values of the physical parameters, as will be described later. As the parameter γ′ is related to elastic scattering at the surface boundary under the influence of inter-particle interactions as well as the electron-electron interactions within the particle, it is expected to have a size dependence and inter-particle distance. In addition, the corresponding physical model for the functional form of γ′ agrees quite well with the values extracted from the curve fit.

## 2. Theory

The relationships between the parameters required in the analysis were derived from the modified Drude model. The derivation of the dielectric function and the corresponding transmission spectra begins from the equation of motion for free electrons within a metal particle as shown below:

*m*,

*e*, $\overrightarrow{v}$, and $\overrightarrow{E}$ are the mass, charge, velocity of electron, and electric field of light applied. γ and γ′ is the inelastic and elastic collision frequency, respectively. It is straightforward to obtain $\overrightarrow{v}$, and the corresponding conductivity σ

*n*is the number density of free electrons within a particle. The effective dielectric function of the metal particle layer can be expressed as follows:

_{p}*ε*,

_{mi}*V*,

_{mi}*ε*,

_{sm}*V*and

_{T}*N*are the dielectric function, volume of the

*i-th*metal particle, dielectric function of the surrounding medium, total volume of the layer, and number of metal particles within the layer, respectively [20]. The inset in Fig. 1 shows a schematic cross-sectional view of the layer. As the contribution from bound electrons is unchanged, the first term in Eq. (5) can be rewritten as follows:

*ε*,

_{m,b}*σ*, and ε

_{mi}_{o}are the contribution to the dielectric function from the interband transition of bound electrons, local conductivity inside the

*i-th*metal particle, and free space permittivity, respectively. Considering

*Q*, the fraction of available free electrons participating in plasmonic resonance and

_{s}*ρ*, the size distribution, the second term in Eq. (6) becomes

*n*is the density of free electrons in a bulk metal, ${V}_{M}={\displaystyle \sum _{i=1}^{N}{V}_{mi}}$, ${\rho}_{i}={n}_{Vmi}{V}_{mi}/{V}_{M}$, and

*n*is the particle number of volume

_{Vmi}*V*. By substituting Eq. (6) and (7) into Eq. (5), the effective dielectric function can be obtained as follows:

_{mi}*ff*( = V

_{M}/V

_{T}) is the filling factor of volume and ω

_{p}is the bulk plasma frequency, ${\omega}_{p}^{2}=n{e}^{2}/m{\epsilon}_{o}.$

## 3. Experimental results and discussion

Samples of Au particle layers with three different size distributions were prepared by successive procedures of sputtering and annealing under carefully controlled conditions. Figure 1 compares the histograms of normalized size distribution of the three samples as a function of the effective radius r, and the corresponding scanning electron microscopy (SEM) images are shown in the inset. Note that the average radius of each sample is near the center of its effective radius distribution. Samples Au_{1}, Au_{2} and Au_{3} consist of gold nanoparticles on ITO/glass. The size distributions were analyzed using a commercial image analysis program, assuming that all particles are oblate spheroid (long axes // film surface, short axes ⊥ film surface). The volume filling factor *ff* considering the initial metal film thickness and effective oblate ratio η were estimated to *ff* = 0.330, 0.152, 0.316, and η = 0.228, 0.258, 0.386 for Au_{1}, Au_{2}, Au_{3}, respectively.

The optical transmittance simulated with this model and Mie theory were compared with the optical spectra measured for normal incident light as a function of the wavelength, as shown in Fig. 2
. The dielectric function ε_{sm} was set to ε_{sm} = (ε_{ITO} + ε_{air})/2 as the metal particles were sandwiched between the two dielectric media, ITO(ε_{ITO} = 3.8) and air(ε_{air} = 1) [24]. The values of the plasma frequency ω_{p} and inelastic collision frequency γ used in this study are 1.37 × 10^{16} rad/s and 1.05 × 10^{14} rad/s, respectively [25,26]. The term ε_{m,b} for the contribution from the interband transition, assuming a dispersionless *d* band and a parabolic *sp* conduction band, was obtained from Eq. (2) with the values Q = 2.3 × 10^{24} rad/s, E_{g} = 2.1 eV, E_{F} = 2.5 eV,T = 300 K, and γ_{b} = 2.4 × 10^{14} rad/s, respectively [13,14]. The curve fit from the model agrees well with the measured one, while the fit from Mie theory shows discrepancies in the broadening of the resonance peak, background intensity level, and peak wavelength. Note that the discrepancies in the Mie estimation are become severe as the particle size decreases. On the other hand, in the modified Drude model, elastic scattering accommodates the curve with the peak shift and broadening required by spreading the values of γ′. The inverse engineering of the curve fit for the optical transmittance provides the characteristic information on the physical principles for the parameters γ′ and Q_{s}.

As the modified Drude model used in this study utilizes the size-dependent scattering parameter γ′(r) and Q_{s}(r), the values extracted from inverse curve fit are displayed as a function of the effective radius of each metal particle in Fig. 3
. Interestingly, the values of Q_{s}(r) lie on almost straight lines, as shown in the inset, and the values of γ′(r) lie on several characteristic curves classified according to the average inter-particle distance of each sample, as shown in the figure. The slopes of Q_{s}(r) estimated for Au_{1}, Au_{2}, and Au_{3} were 2.32 × 10^{−3}, 2.28 × 10^{−3}, and 2.12 × 10^{−3} (1/nm), respectively. The size-dependent behavior for γ′(r) is simply characterized by the functional form of $\gamma \text{'}=\left(\frac{B}{{\epsilon}_{sm}}\right)\cdot \frac{{r}_{o}^{2}}{{r}^{2}+{r}_{o}^{2}}\text{\hspace{0.17em}},$ where B and r_{o} are the characteristic parameters. As the scattering is affected by both the electron-electron interaction and the inter-particle interaction, the limiting behavior as r→0 is represented by adding r_{o}
^{2} as a simple function of average inter-particle distance, ℓ. Note that B/ε_{sm} is the maximum value of the elastic scattering frequency for confined free electrons, and r_{o} is the reference length for the effective range of inter-particle interactions. The values of B/ε_{sm} extracted for Au_{1}, Au_{2}, and Au_{3} were 3.86 × 10^{15}, 4.03 × 10^{15}, and 4.10 × 10^{15} rad/s, respectively, as observed at the intersections of the y-axis and each curve. Interestingly the limiting value of γ′(r), B/ε_{sm}, remains almost constant for several samples with different size distributions, whereas B/ε_{sm} decreases slightly with increasing average size. On the other hand, the values of r_{o} extracted from the curve fit were 145, 79 and 49 nm, respectively, whereas the average inter-particle distances ℓ estimated from the Au_{1}, Au_{2}, and Au_{3} samples prepared assuming a regular polygon lattice structure were ~105, ~92, and ~42 nm, respectively. This suggests that the effective slope of γ′(r) depends on the inter-particle distance ℓ.

## 4. Summary

In summary, several gold nanoparticle layers were prepared by sputtering and annealing under carefully controlled conditions, and the optical transmission spectra were obtained and analyzed using a modified Drude model with a novel elastic scattering parameter γ′. The transmission curves reproduced by the inverse curve fit according to the modified Drude model agreed quite well with the measured transmission spectra, and the model explained the abnormal behavior using a few reasonable variables, r_{o}, B/ε_{sm} and ℓ for γ′(r). The elastic scattering due to the particle boundary is proposed in the form of the elastic collision frequency γ′ of free electrons within the metal particle, and the size dependence of γ′(r) was also extracted from the inverse curve fit. The ℓ-dependence of the slope of γ′ was also inferred within the framework of the proposed model from a comparison of the fit parameter r_{o} and ℓ.

## Acknowledgement

This study was supported by the Korea Science and Engineering Foundation through a Grant for the Integrated Photonics Technology Research Center (R11-2003-022).

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