## Abstract

The wavelength dispersions of third-order nonlinear optical response for Cu nanoparticle materials have been experimentally evaluated from transient spectra measured with the pump-probe method. The evaluated dispersions were analyzed on hot electron contribution using the Maxwell-Garnett approximation with the Drude model for intraband transition and first principles calculation for interband transition. The wavelength dispersion didn’t directly reflect the dispersion of a local electric field factor. The interband transition term in hot electron contribution strongly dominated the dispersion around the surface plasmon resonance by Fermi smearing. Intrinsic interband contribution to the nonlinearity was suggested from the analysis. Particle-size and host-medium dependence of the nonlinearity were also simulated.

©2008 Optical Society of America

## 1. Introduction

Nanostructured metal materials are noteworthy optical materials because the localized surface plasmon excited by visible light causes an enhanced local electromagnetic field around the nanostructure and show unique optical properties. These notable properties promise to be applied to biological plasmonic sensors and key components of future optoelectronics devices. Optical nonlinearities of nanostructured metals are one of characteristic plasmonic properties [1–3] and have attracted a great deal of attention for physical interests and the applications [4, 5]. The third-order nonlinear optical response of metal nanomaterials is enhanced by the local electric filed at the surface plasmon resonance and is sensitively influenced by the morphology and combination of materials with the shift and change on intensity of the resonance. For the design of plasmonic devices and the understanding of optical properties in enhanced strong fields and of the origin of the nonlinearities, it is significant to evaluate the wavelength dispersion of nonlinear optical constants. Nevertheless, to our knowledge, theoretical and experimental wavelength dispersion of optical nonlinearity had rarely been reported [3,6–8]. We have experimentally evaluated the wavelength dispersion nonlinear dielectric function of Au:SiO_{2} nanoparticle materials and discussed hot electron contribution to the nonlinearity using the Maxwell-Garnett approximation with the Drude model for intraband transition and a simple potential model for interband transition [9]. And we have reported interband transition influences the spectra especially at a photon energy region higher than the surface plasmon resonance. The surface plasmon resonance of Cu:SiO_{2} nanoparticle materials is located in the interband transitions and the nonlinearities will be seriously affected by interband transition. Here we discuss experimental evaluation of the wavelength dispersion of nonlinear part of dielectric function of Cu nanoparticle materials and analyze detailed influence from interband transition calculated with first principles calculation in hot electron contribution. We also estimate the other contributions and simulate the particle-size and host-medium dependence of the nonlinear dielectric function.

## 2. Experimental detail

Negative ion implantation was applied to fabricate Cu nanoparticle sample in silica glass (KU-1®: OH 820 ppm and other impurities 6.3 ppm). The flux and total fluence of implanting Cu ions with 60 keV were 10 µ A/cm^{2}, 1·10^{17} ions/cm^{2}, respectively. The negative ion technique prevents a high surface discharge on the substrate during ion implantation and brings us easily to control the total fluence. The detail has been described elsewhere [10]. The Cu-implanted sample was annealed at 800 °C for 1 h in an Ar gas flow after the implantation to reduce irradiation damages in the substrate and to promote the growth of nanoparticles. The Cu nanoparticles were located near the surface with a depth of 40 nm from Rutherford backscattering spectrometry [11]. The thickness of Cu nanoparticle layer was analyzed by an ellipsometric measurement and was 45 nm. The average particle size analyzed by small angle X-ray scattering (SAXS) [12,13] was more than 30 nm. Steady-state optical transmission and reflection were obtained by measurements using a dual beam spectrometer (JASCO, V-570) with a strict correction of incoherent multiple reflections from the backside of the sample [14]. The refractive index was determined with a spectroscopic ellipsometer (HORIBA, UVISEL). Transient optical transmission and reflection spectra were sequentially measured using the pump-probe method with a tunable femtosecond-laser system (Spectra Physics). The duration of pumping pulses was 0.2 ps (150 fs) at 1 kHz-repetition rate. The pulse duration and repetition rate can avoid an effect of thermal refraction on nonlinearity. The wavelength and power density of pumping pulse were 573 nm (2.16 eV) and 0.9 GW/mm^{2} on the sample, respectively. The detail setup has been described elsewhere [9].

## 3. Experimental evaluation of nonlinear dielectric function

Steady-state and transient optical spectra are shown in Fig. 1. The surface plasmon resonance of Cu:SiO_{2} is located at 2.16 eV, where is higher than optical absorption edge (2.08 eV) of the interband transition.

The laser pumping modulates the transmission and reflection spectra as dashed lines shown in Fig. 1. For a small photo-induced modulation of spectra, the difference reflection and transmission can be given by total differential [9, 15, 16];

$$\Delta \epsilon \prime =2\left(n\Delta n-k\Delta k\right),\phantom{\rule{.2em}{0ex}}\Delta \epsilon "=2\left(n\Delta k+k\Delta n\right),$$

where Δ*n* and Δ*k* represent the nonlinear part of refractive index, and Δ*ε*=Δ*ε*′+*i*Δ*ε*″ the nonlinear part of the dielectric function, corresponding to optical third-order susceptibility *χ*
^{(3)}
_{eff}, as below;

$${\chi}_{\mathrm{eff}}^{\left(3\right)}\left(-{\omega}_{\mathrm{probe}};-{\omega}_{\mathrm{pump}},{\omega}_{\mathrm{pump}},{\omega}_{\mathrm{probe}}\right)=p{f}_{l}^{2}{\mid {f}_{l}\mid}^{2}{\chi}_{m}^{\left(3\right)}\left(-{\omega}_{\mathrm{probe}};-{\omega}_{\mathrm{pump}},{\omega}_{\mathrm{pump}},{\omega}_{\mathrm{probe}}\right),$$

where *E* denotes a pumping electric field, *p* the volume fraction of metal nanoparticles, *χ*
^{(3)}
_{m} the optical third-order susceptibility of metal nanoparticles in SI units [17]. The local field factor, *f _{l}*, is a ratio between the internal field in a spherical nanoparticle and the applied external field and is given in Maxwell-Garnett approximation by

Here, *ε _{d}(ω)* and

*ε*represent the complex dielectric function of a host medium and metal nanoparticles, respectively [1,2]. The third-order nonlinearity is proportional to the fourth power of the local electric field factor and intrinsic nonlinear constant,

_{m}(ω)*χ*

^{(3)}

_{m}, of metal nanoparticles themselves. From Eq. (1), we can evaluate the wavelength dispersion of Δ

*ε*[9] and show the result in Fig. 2.

The imaginary part of Δ*ε* indicates a minimal value near the surface plasmon resonance and the real part represents a type of differential curve around the resonance. The evaluated maximum values attain Δ*ε*′=0.35 at 2.20 eV and Δ*ε*″=-0.34 at 2.14 eV for a pumping power of 0.9 GW/mm^{2} (*E*=6.5·10^{8} V/m) and, corresponding to *χ*
^{(3)}
_{eff}′=7.9·10^{-11} esu and *χ*
^{(3)}
_{eff}″=-7.9·10^{-11} esu, respectively, where the relation between the susceptibilities [17] is:

The numerical calculation of *f*
^{2}
_{l}|*f _{l}*|

^{2}is shown in Fig. 3. From Eq. (2), Δ

*ε*strongly depends on the factor,

*f*

^{2}

_{l}|

*f*|

_{l}^{2}, however, the experimental dispersion of Δ

*ε*for Cu:SiO

_{2}material doesn’t directly reflect the

*f*

^{2}

_{l}|

*f*|

_{l}^{2}spectra in contrast with that of Au:SiO

_{2}materials [9]. The result indicates the wavelength dispersion of nonlinearity of particles,

*χ*

^{(3)}

_{m}(or Δ

*ε*), is not negligible for Cu nanoparticle materials.

_{m}## 4. Analysis with the Maxwell-Garnett model

The effective complex dielectric function of nanoparticle composite materials is defined by the Maxwell-Garnett model as below,

For noble metals, the dielectric function can be decomposed into two terms, a free electron (intraband transition) term and a bound electron (interband transition) term, as below;

The free electron term can be written in the Drude model as

$$\u045b{\omega}_{\tau}=\u045b{\omega}_{{\tau}_{0}}+\frac{A\u045b{v}_{F}}{R}$$

where ω_{p} is the bulk plasma frequency, ω_{τ} the dumping frequency, ω_{τ0} 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 is derived from a first principles calculation. The band structure for fcc Cu is calculated using the DFT (Density-functional Theory) and GGA (Generalized Gradient Approximation) with the plane-wave pseudopotential method [18–21]. The lattice constant and temperature are 0.361 nm for fcc Cu and 0 K, respectively. The dielectric function is calculated using a 36·36·36 **k**-point mesh and a number of bands of 32. The imaginary parts of *ε _{bound}* (

*ω*) are numerically evaluated with the Fermi-Dirac distribution at each electronic temperature by

where *p ^{k}_{ij,α}*=〈

*φ*|

^{k}_{j}*p*|

_{α}*φ*〉 represents a transition moment between each wavefunction,

^{k}_{i}*φ*, of i-th band at

^{k}_{i}*k*.

*f*(

*E*) indicates the Fermi-Dirac distribution function with an energy,

^{k}_{i}*E*, of i-th band at

^{k}_{i}*k*. α and β indicate Cartesian indices (x, y, z), and

*V*and

*m*unit cell volume, mass of electron, respectively [22–25]. The real parts can be evaluated using the Kramers-Kronig relation;

_{e}Figure 4 shows calculated dielectric function of interband term at 300 K with experimental data extracted from Ref. [26], where the Drude term with the plasmon energy of 8.9 eV and the damping energy of 0.095 eV is adopted by fitting on the magnitude.

The calculated *ε _{bound}* (

*ω*) with an offset of 0.53 eV for photon energy is well consistent with the extracted literature data. The offset originates from the minor difference of energy level between the calculated model and actual band structure. Calculated nonlinear dielectric function of bound electron term without the offset is shown in Fig. 5, where the hot electron contribution is defined by

The nonlinearity of bound electron term increases with the electron temperature of pumped state. Especially a marked modulation happens around the absorption edge due to Fermi smearing [1, 27] and the contribution stretches to the lower photon energy beyond the absorption edge.

For nanoparticle composite materials, the effective nonlinear dielectric function is also defined by

Hereinafter the calculated data with the offset for bound electron term is applied to the analysis in the narrow photon energy region.

Figure 6 shows calculated nonlinear dielectric functions, Δε_{m} and Δε_{MG}, for the experimental result in Fig. 2, where we decided the parameters from fitting on the linear effective dielectric function and the volume fraction, *p*, the averaged particle radius are 0.1, 15 nm, respectively [9]. The dielectric function of Cu nanoparticles with radii more than 10 nm, ε_{m}, is almost independent of the size. The volume fraction is consistent with the RBS measurement. And we adopt 2.6·10^{14} Hz as the dumping constant for free electron term in Eq. (7) and 1500 K as the pumped electron temperature for bound electron term in Eqs. (8) to fit in the magnitudes of the experimentally evaluated dispersion curve. The top, middle and bottom figures in Figs. 6(a) and 6(b) indicate the interband (bound electron) term, the intraband (free electron) term (Drude term) and the total spectrum including both terms, respectively. The interband term extends to 1.8 eV beyond the absorption edge (2.08 eV). The maximum values of the interband term are estimated to be Δ*ε _{m}*′=-1.1 at 2.2 eV and Δ

*ε*″=1.1 at 2.0 eV corresponding to

_{m}*χ*

^{(3)}

_{m}′=-2.5·10

^{-10}esu and

*χ*

^{(3)}

_{m}″=2.5·10

^{-10}esu, respectively. Meanwhile, the nonlinear dielectric function of Drude term monotonously increases by pumping and indicates positive sign at allover range. The estimated values of Drude term,

*χ*

^{(3)}

_{m}′ and

*χ*

^{(3)}

_{m}″, at 2.2 eV are 2.0·10

^{-11}, 1.1·10

^{-10}esu, respectively.

The experimental dispersion is almost consistent to the calculated bottom curve in Fig. 6(b) and the estimated maximum values for hot electron contribution indicate Δ*ε _{m}*′=-1.0 at 2.2 eV and Δ

*ε*″=1.9 at 2.0 eV, corresponding to

_{m}*χ*

^{(3)}

_{m}′=-2.2·10

^{-10}esu and

*χ*

^{(3)}

_{m}″=4.2·10

^{-10}esu, respectively. The dispersion is not reproduced only by the Drude term and the interband term strongly contributes the dispersion around the absorption edge.

The calculated results for experimentally evaluated Δε of Cu:SrTiO_{3} [28] are also shown in Fig. 7. The surface plasmon resonance of Cu:SrTiO_{3} is located about 1.95 eV lower than the absorption edge in contradiction to that of Cu:SiO_{2} material. The interband term strongly remains influential in Δε of Cu:SrTiO_{3}. The evaluated maximum values attain Δ*ε*′=1.56 at 2.10 eV and Δ*ε*″=-1.06 at 1.94 eV for a pumping power of 0.9 GW/mm^{2} (*E*=5.6·10^{8} V/m), corresponding to *χ*
^{(3)}
_{eff}′=4.7·10^{-10} esu and *χ*
^{(3)}
_{eff}″=-3.1·10^{-10} esu, respectively. The calculated dispersion is also consistent with the experimental result especially at the lower photon energy. However, these two experimental dispersions generally indicate a large magnitude at the higher photon energy in comparison with the calculated dispersions. These results probably suggest that the discrepancy originates from intrinsic interband contribution to the nonlinearities at the higher photon energy beyond the absorption edge. This magnitude of interband contribution, *χ*
^{(3)}
_{eff} and *χ*
^{(3)}
_{m}, is roughly estimated to be ~1–2·10^{-11} and ~1–2·10^{-10} esu, respectively and the dispersion is not complicate. The analysis also shows intrinsic intraband contribution is negligible small (*χ*
^{(3)}
_{eff}: less than 10^{-12} esu, *χ*
^{(3)}
_{m}: less than 10^{-11} esu) because of good fitting by hot electron contribution at the lower photon energy. F. Hache, *et al.*, theoretically estimated three contributions, which are hot electron contribution, intrinsic intraband and interband contributions, to the third-order optical response of gold nanoparticles and mentioned that hot electron mainly contributes to the third-order nonlinearities, intrinsic interband contribution is one order smaller than hot electron contribution and intraband contribution is negligible [2]. Our experimental results generally agree with estimated magnitudes on F. Hache’s theoretical studies but evaluated interband contribution is larger than theoretical estimation.

## 5. Simulated wavelength dispersions of nonlinear dielectric function

Figure 8 shows the particle-size dependence of simulated Δε_{MG} of Cu:SiO_{2} materials, where the Drude term in Eq. (7) only incrudes the size effect. Size effects for interband transition are negligible at this size range [1,2]. The other conditions are same as the calculation shown in Fig. 6. The nonlinear intensity will increases with the particle size and almost saturates with a size more than 10 nm as well as linear absorption intensity.

Figure 9 reprisents the ε_{d} dependence of Δε_{MG} of Cu nanoparticle materials, where we asuume ε_{d} is constant over the spectral range (*cf.* 2.1 for SiO_{2}, 5.6 for SrTiO_{3} and 6.8 for TiO_{2} at 2.0 eV). With increasing ε_{d}, the resonace shifts to red and is markedly enhanced due to deviation from the overlap with the interband transitions. Cu is one of low environment load materials and Cu nanoparticles embedded in a matrix with a high refractive index will be also expected to indicate large nonlinearity.

## 6. Conclusions

Optical nonlinearity of Cu nanoparticle materials was experimentally evaluated from transient optical spectra measured using the pump-probe method. The wavelength dispersions of nonlinear dielectric function were analyzed on hot electron contribution using the Maxwell-Garnett model with the Drude model for intraband transition and the first principles calculation for interband transition. The evaluated dispersion doesn’t directly reflect the local field factor. The overall experimental dispersions are generally consistent to the calculated dispersion based on hot electron contribution. Interband transition term in hot electron contribution dominates the dispersion and attains the lower photon energy beyond the absorption edge. The fitting results also suggest that intrinsic interband contribution is not negligible small.

## 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 formerly worked at Cybernet Systems Co., Ltd. for support of numerical calculations with Maple®.

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