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
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 = εb + εf, εb is from the interband transition of bound electrons and εf is from the intraband transition of free electrons. The contribution εf 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
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 εb and εf 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 . 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 . The optical behavior of a metallic thin film was also described well within the framework of the modified Drude model, recently .
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.
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: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:Eq. (6) becomesEq. (6) and (7) into Eq. (5), the effective dielectric function can be obtained as follows:
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 Au1, Au2 and Au3 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 Au1, Au2, Au3, 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 εsm was set to εsm = (εITO + εair)/2 as the metal particles were sandwiched between the two dielectric media, ITO(εITO = 3.8) and air(εair = 1) . The values of the plasma frequency ωp and inelastic collision frequency γ used in this study are 1.37 × 1016 rad/s and 1.05 × 1014 rad/s, respectively [25,26]. The term εm,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 × 1024 rad/s, Eg = 2.1 eV, EF = 2.5 eV,T = 300 K, and γb = 2.4 × 1014 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 Qs.
As the modified Drude model used in this study utilizes the size-dependent scattering parameter γ′(r) and Qs(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 Qs(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 Qs(r) estimated for Au1, Au2, and Au3 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 where B and ro 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 ro 2 as a simple function of average inter-particle distance, ℓ. Note that B/εsm is the maximum value of the elastic scattering frequency for confined free electrons, and ro is the reference length for the effective range of inter-particle interactions. The values of B/εsm extracted for Au1, Au2, and Au3 were 3.86 × 1015, 4.03 × 1015, and 4.10 × 1015 rad/s, respectively, as observed at the intersections of the y-axis and each curve. Interestingly the limiting value of γ′(r), B/εsm, remains almost constant for several samples with different size distributions, whereas B/εsm decreases slightly with increasing average size. On the other hand, the values of ro extracted from the curve fit were 145, 79 and 49 nm, respectively, whereas the average inter-particle distances ℓ estimated from the Au1, Au2, and Au3 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.
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, ro, B/εsm 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 ro and ℓ.
This study was supported by the Korea Science and Engineering Foundation through a Grant for the Integrated Photonics Technology Research Center (R11-2003-022).
References and links
2. M. L. Brongersma, J. W. Hartman, and H. A. Atwater, “Electromagnetic energy transfer and switching in nanopartice chain arrays below the diffraction limit,” Phys. Rev. B 62(24), R16356–R16359 (2000). [CrossRef]
3. G. Raschke, S. Kowarik, T. Franzl,, C. Sönnichsen, T. A. Klar, J. Feldmann, A. Nichtl, and K. Kürzinger “Biomolecular Recognition Based on Single Gold Nanoparticle Light Scattering,” Nano Lett. 3(7), 935–938 (2003). [CrossRef]
4. J. Vučković, M. Lončar, and A. Scherer, “Surface Plasmon Enhanced Light-Emitting Diode,” IEEE J. Quantum Electron. 36(10), 1131–1144 (2000). [CrossRef]
5. J. H. Sung, B. S. Kim, C. H. Choi, M. W. Lee, S. G. Lee, S. G. Park, E. H. Lee, and B. H. O, “Enhanced luminescence of GaN-based light-emitting diode with a localized surface plasmon resonance,” Microelectron. Eng. 86(4-6), 1120–1123 (2009). [CrossRef]
7. B. Khlebtsov, A. Melnikov, V. Zharov, and N. Khlebtsov, “Absorption and scattering of light by a dimer of metal nanospheres: comparison of dipole and multipole approaches,” Nanotechnology 17(5), 1437–1445 (2006). [CrossRef]
8. S. Berciaud, L. Cognet, P. Tamarat, and B. Lounis, “Observation of intrinsic size effects in the optical response of individual gold nanoparticles,” Nano Lett. 5(3), 515–518 (2005). [CrossRef] [PubMed]
9. M. Théye, “Investigation of the Optical Properties of Au by Means of Thin Semitransparent Films,” Phys. Rev. B 2(8), 3060–3078 (1970). [CrossRef]
10. D. E. Aspnes, E. Kinsbron, and D. D. Bacon, “Optical properties of Au: Sample effects,” Phys. Rev. B 21(8), 3290–3299 (1980). [CrossRef]
11. P. B. Johnson and R. W. Christy, “Optical Constants of the Noble Metals,” Phys. Rev. B 6(12), 4370–4379 (1972). [CrossRef]
12. H. Ehrenreich and H. R. Philipp, “Optical properties of Ag and Cu,” Phys. Rev. 128(4), 1622–1629 (1962). [CrossRef]
13. H. Inouye, K. Tanaka, I. Tanahashi, and K. Hirao, “Ultrafast dynamics of nonequilibrium electrons in a gold nanoparticle system,” Phys. Rev. B 57(18), 11334–11340 (1998). [CrossRef]
14. L. B. Scaffardi and J. O. Tocho, “Size dependence of refractive index of gold nanoparticles,” Nanotechnology 17(5), 1309–1315 (2006). [CrossRef]
15. U. Kreibig, “Electronic properties of small silver particles: the optical constants and their temperature dependence,” J. Phys. F Met. Phys. 4(7), 999–1014 (1974). [CrossRef]
16. D. Dalacu and L. Martinu, “Optical properties of discontinuous gold films: finite size effects,” J. Opt. Soc. Am. B 18(1), 85–92 (2001). [CrossRef]
17. S. Link and M. A. El-Sayed, “Size and Temperature Dependence of the Plasmon Absorption of Colloidal Nanoparticles,” J. Phys. Chem. B 103(21), 4212–4217 (1999). [CrossRef]
18. E. R. Encina and E. A. Coronado, “Resonance conditions for Multipole Plasmon Excitations in Noble Metal Nanorods,” J. Phys. Chem. C 111(45), 16796–16801 (2007). [CrossRef]
19. R. E. Hetrick and J. Lambe, “Optical properties of small In particles in thin-film form,” Phys. Rev. B 11(4), 1273–1278 (1975). [CrossRef]
20. C. G. Granqvist and O. Hunderi, “Optical properties of ultrafine gold particles,” Phys. Rev. B 16(8), 3513–3534 (1977). [CrossRef]
21. M. Xu and J. Dignam, “A new approach to the surface plasmon resonance of small metal particles,” J. Chem. Phys. 96(5), 3370–3378 (1992). [CrossRef]
22. K. Fuchs and N. F. Mott, “The conductivity of Thin Metallic Films according to the Electron Theory of Metals,” Proc. Camb. Philos. Soc. 34(01), 100 (1938); E. H. Sondheimer, “The Mean Free Path of Electrons in Metals,” Adv. Phys. 1(1), 1–42 (1952). [CrossRef]
23. J. S. Yang, S. G. Lee, S. G. Park, E. H. Lee, and B. H. O, “Drude Model for the Optical Properties of a Nano-Scale Thin Metal film Revisited,” J. Korean Phys. Soc. 55(6), 2552–2555 (2009). [CrossRef]
24. G. Xu, M. Tazawa, P. Jin, S. Nakao, and K. Yoshimura, “Wavelength tuning of surface plasmon resonance using dielectric layers on silver island films,” Appl. Phys. Lett. 82(22), 3811–3813 (2003). [CrossRef]
25. C. Kittel, Introduction to Solid State Physics, (John Wiley & Sons Inc., 2005).
26. N. K. Grady, N. J. Halas, and P. Nordlander, “Influence of dielectric function properties on the optical response of plasmon resonant metallic nanoparticles,” Chem. Phys. Lett. 399(1-3), 167–171 (2004). [CrossRef]