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We have investigated wavelength dispersion of photo-induced nonlinear dielectric function of Au nanoparticle materials. Transient transmission and reflection spectra were sequentially measured by the pump-probe method with a femtosecond laser system. The dispersion of real and imaginary parts of the nonlinear dielectric function of Au:SiO2 nanoparticle material in the vicinity of the surface plasmon resonance was evaluated from these transient spectra with total differential. A local electromagnetic field factor and interband transition in Au nanoparticles directly dominate the dispersion.
©2007 Optical Society of America
1. Introduction
Nanostructured metal materials are noteworthy materials in the visible region because localized surface plasmon excited by light causes an enhanced local electromagnetic field around the nanostructure and a magnetic modulation with a resonator structure [1–5]. The notable plasmonic properties promise to be applied to key components of biological sensing applications and future optoelectronics devices. These nanostructured materials also have large optical nonlinearities enhanced in the vicinity of the surface plasmon resonance and the nonlinearities are applied to optical switching of a single nanoparticle by optical bistability [6,7] and so on. For the design of such optical devices and the understanding of optical nonlinearities, it is significant to evaluate the wavelength dispersion of nonlinear parts of the dielectric function of actual nanoparticle materials. Discussions, which are referred to the amplitude and the sign of nonlinear constants only measured at a single wavelength, are unessential to understand the nonlinear optical properties. It is also difficult to interpret transient transmission, reflection or absorption because these spectra are complicated functions of the real and imaginary parts of nonlinear dielectric function. Nevertheless, to our knowledge, theoretical and experimental dispersion of optical nonlinearity has rarely been reported [8–10]. Furthermore that only experimental dispersion for Ag nanoparticle materials was measured at several wavelengths and was not complicated because of isolation from interband transition [8]. The surface plasmon resonance of Au nanoparticle materials is located near the interband transitions and the overlapping with the interband transition may affect the plasmon resonance mode. Here we present the wavelength dispersion of nonlinear dielectric function evaluated from transient transmission and reflection spectra of Au nanoparticles in silica glass, fabricated by negative ion implantation. To evaluate accurate nonlinear constants, measurement of not only transient transmission but also transient reflection is necessary. These two spectra measured in the limited photon range bring us to derive the wavelength dispersion. We discuss the contribution of interband and intraband transitions as well as the effect of a local electromagnetic field to the dispersion.
2. Experimental detail
The Au nanoparticle sample used in this work was fabricated by negative ion implantation. Ion implantation is one of the most powerful techniques to fabricate metal nanoparticles in insulating substrates. Negative ion implantation has the advantage of controllability with less surface discharge and of applicability to various ion species and insulators with non-equilibrium process [11,12]. The selection of substrates and metals by ion implantation brings us to control the plasmonic properties [13–15]. Details of the technique can be found elsewhere [16]. Negative gold ions accelerated at 60 keV were implanted into amorphous SiO2 substrate (KU-1®: OH- 820 ppm and other impurities 6.3 ppm). The flux and total fluence of implanting ions were 17 μA/cm2, 1 × 1017 ions/cm2, respectively. Spherical Au nanocrystals precipitate and grow due to an ion-induced enhanced diffusion during implantation. A cross-sectional TEM image of Au nanoparticle composite in silica glass is shown in Fig. 1 as a reference, where the implantation condition is a flux of 1 μA/cm2. Under these conditions Au nanoparticles with a diameter of about 5-10 nm are located beneath the surface with 2-dimensional distribution of 27 nm in width [17].
Fig. 1. A cross sectional TEM image of Au nanoparticle composite and the layer structure for optical analysis.
Steady-state optical measurements were conducted in a photon energy range from 1.8 to 3.0 eV using a dual beam spectrometer (JASCO, V-570). Optical transmission and reflection spectra of the Au nanoparticle layer were obtained after strict correction of incoherent multiple reflections from the backside of the sample [18]. The refractive index (i.e. linear dielectric function) of the Au nanoparticle layer and the substrate was analyzed with a spectroscopic ellipsometer (HORIBA, UVISEL) and was determined. Transient optical reflection and transmission spectra were sequentially measured using the pump-probe method with a tunable femtosecond-laser system (Spectra Physics). The wavelength and pulse duration of the pumping pulse were 554 nm (2.23 eV) and 0.2 ps at 1 kHz-repetition rate, respectively. The beam waist of the pump and probe beams at the focal point were measured using a beam profiler and were both 150 μm. The pumping power was 4.7 mW corresponding to the peak power density focused on a sample of 0.7 GW/mm2. The transient spectra were obtained by dividing a laser pumped spectrum by a spectrum without a pumping. The chirping effect of the probe beam, which is due to the group velocity dispersion, was estimated and corrected with a cross-correlation method and nonlinear measurement of ZnO with fast response.
3. Evaluation of nonlinear dielectric function
Figure 2 shows (a) the steady-state transmission, T, the steady-state reflection, R, and (b) the effective refractive index, n+ik, of the Au nanoparticle layer. Spectral feature at 2.2 eV originates from the surface plasmon resonance. The effective refractive index of the Au nanoparticle layer is analyzed with ellipsometry, where the nanoparticle layer is assumed to be a homogeneous layer. The evaluated effective thickness of Au nanoparticle layer is 25 nm and is almost consistent with the cross-sectional TEM image as shown in Fig. 1. Laser irradiation stimulates electrons in a nanoparticle and transiently rises in electron temperature. The temperature rise modulates the optical properties as a nonlinear effect.
Fig. 2. (a) Steady-state reflection and transmission spectra and (b) the effective refractive index of Au implanted layer in silica glass.
Figure 3(a) shows the transient transmission, ΔT, and reflection, ΔR, of Au nanoparticle material right after laser pumping. The transmission increases around the surface plasmon resonance and the peak shifts to the higher photon energy from the resonance. The skirts of transient transmission indicate a negative sign. The transient reflection is not negligible because the amplitude of ΔR at the same level as that of ΔT. The transient reflection shows a minimum around 2.2 eV and turns to a positive sign across zero about 2.4 eV.
Fig. 3. (a) Transient reflection and transmission spectra and (b) the partial differential terms of Au implanted layer in silica glass.
For a small modification, the differential reflection and transmission can be written by total differential [10,19–21];
where Δn and Δk denote the nonlinear part of refractive index, and Δε’ and Δε’’ the real and imaginary part of the nonlinear dielectric function. From these equations, we can evaluate the nonlinear dielectric function from measurement in the limited photon energy range. Reflection and transmission of a thin layer with a coherent thickness on a substrate are defined by
where λ denotes wavelength of light, d the thickness of layer, N the complex refractive index and subindex, j, indicates 1: air, 2: Au nanoparticle layer and 3: silica glass as shown in Fig. 1. This definition almost catches on the measured optical feature shown in Fig. 2(a). The partial differential terms in Eq. (1) are calculated from Eq. (2) with Maple® software and the results are shown in Fig. 3(b). Finally the dispersion of nonlinear dielectric function evaluated from Eq. (1) is shown in Fig. 4. The evaluated dispersion doesn’t shows a simple Lorentz resonance curve and have asymmetric feature around the surface plasmon resonance. The real part of nonlinear dielectric function exhibits a minimum value of -0.3 at 2.15 eV and across zero at 2.3 eV. The imaginary part reflects transient transmission spectra and indicates a bottom at 2.25 eV higher than the surface plasmon resonance with a value of -0.32 for the pumping power of 0.7 GW/mm2.
4. Contribution of free electrons and bound electrons
For noble metals, the dielectric function can be decomposed into two terms, a free electron term and a bound electron term, as below;
The free electron term can be written in the Drude model as
where ωp is the bulk plasma frequency, ωτ the dumping frequency, ωτ0 the bulk dumping frequency, A the scattering factor, v F the electron velocity at the Fermi surface and R is the radius of the nanoparticle.
The bound electron term in this visible region is contributed by the interband transitions from d-band to the conduction sp-band near the L point in the Brillouin zone of gold [22]. Assuming that the curvature of d-band is ignored, the term can be simplified [23,24] as
where ħωg is the gap energy for gold, γee the damping constant in the interband transition and Q a proportional factor, and the simplified model has a single gap energy of 2.3 eV. The effective complex dielectric function of nanoparticle layer is defined by the Maxwell-Garnett model as below;
where p is the volume fraction of nanoparticles in a matrix and εd and εm represent the complex dielectric function of a matrix substrate and metal nanoparticles, respectively. The measured steady-state resonance shifts to red for the calculated position [17], because damages with a lack of oxygen generated by ion irradiation surely increase the refractive index of implanted area (i.e., SiOx) [25–27]. We adopt that the refractive index of the matrix is 20 % higher than that of original amorphous-SiO2 from the evaluated peak shift, corresponding to SiOx, x ≈ 1.5 [28]. Figure 5 shows (a) calculated and (b) measured linear dielectric function of Au nanoparticle layer. For simple fitting on the measured spectra we assume that Au nanoparticles with a constant radius of 3 nm and with a volume fraction of 0.1 are contained in a matrix with an effective dielectric function estimated from Fig. 1 and adopt the other parameters from Ref. [23]. The discrepancy between measured and computed spectra in the higher photon energy also denotes the same tendency of Ref. [23].
Fig. 5. (a) Calculated dispersion of and (b) the measured dispersion of the dielectric function of Au nanoparticle materials.
Figure 6 shows a calculated nonlinear dielectric function based on the result in Fig. 5(a), where we adopt 1.8 × 1014 Hz and 1200 K as the dumping constant for free electrons and the electron temperature at a laser-pumping state, respectively, to fit on the amplitude of the evaluated dispersion curve. The change on the dumping constant in the interband transition is assumed to be negligible [24]. The top, middle and bottom figures in Fig. 6 indicate the intraband (free electron) contribution, the interband (bound electron) contribution and the total spectrum including both contributions, respectively. The intraband term reflects the overall evaluated dispersion and mainly contributes at the photon energy lower than 2.4 eV. The interband transition contributes modulation in Δε around 2.5 eV. The calculated total dispersion relatively catches the evaluated dispersion at the lower photon energy, while the calculated dispersion doesn’t catch at the photon energy higher than 2.4 eV. We consider that the dispersion at the higher photon energy is due to contribution of interband excitation from the deeper level beyond the simplified model in Eq. (5).
5. Effect of Local field factor
The nonlinear dielectric function of nanoparticle composites is also defined by optical third-order susceptibility and is expressed by
where I represents pumping laser power, χm (3) denotes the optical third-order susceptibility of metal nanoparticles [8,29,30]. The local field factor, fl, dominates the optical nonlinearities. The local field factor is the ratio between an internal electromagnetic field in a nanoparticle and an applied external electromagnetic field and is given in Maxwell-Garnett approximation by
where L denotes depolarization factor and 1/3 for spherical material. The nonlinearity is proportional to the fourth power of the local electric field factor. The definition separates contributions of local field effect at the resonance and Drude term from intraband contribution as shown in the top figure of Fig. 6.
Figure 7 shows numerical calculation of fl 2∣fl∣2 of Au nanoparticle in silica glass (L = 1/3). The calculated dispersion and the sign roughly reflect the evaluated nonlinear dielectric function at the photon energy lower than the surface plasmon resonance, while at the higher photon energy the local field factor seems not to directly reflect the evaluated dispersion of Δε. Calculated Δεm, the nonlinear part of εm, corresponding to χm (3), spectra are shown in Fig. 8, where separated spectra by Drude term and interband contributions are also shown. The product of fl 2∣fl∣2 and Δεm almost reproduces the calculated total dispersion shown in Fig. 6. The Drude term indicates positive signs and flat structures in the whole region and hardly contributes the dispersion, while the interband contribution has large dispersed relation above 2.3 eV. Therefore the local field factor and interband transition mainly dominate over the nonlinear dielectric function. The local field directly reflects the nonlinear constant in the region without overlapping with interband transition, while the overlapping seriously affects the dispersion. The local field factor generally depends on not only dielectric function but depolarization factor, i.e. the shape of nano-structured metal. For example, oriented metal nanorods (L < 1/3) will show a lot of promise for a marked increase in the optical nonlinearity. It is also significant to be designed with interband contribution in mind for optical switching and other plasmonic applications with the nonlinearity.
6. Conclusions
We evaluated the dispersion of nonlinear dielectric function of actual Au nanoparticle materials from transient reflection and transmission spectra measured with the pump-probe method. The experimental evaluation clarifies the contribution of intraband and interband transitions to the optical nonlinearity. The local field factor and the interband transition directly contribute the dispersion. A design for the depolarization factor of metal nanomaterials will has efficacy for enhancement of the optical nonlinearity.
Acknowledgments
Part of this study was financially supported by the Budget for Nuclear Research of the MEXT, based on the screening and counseling by the Atomic Energy Commission. The authors would like to sincerely thank Ms. A. Terai of HORIBA Ltd. for analysis of ellipsometry and Mr. M. Nakasaka of Cybernet Systems Co., Ltd. for support of numerical calculations with Maple®.
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