Extension of Discrete-Dipole Approximation Model to Compute

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Extension of Discrete-Dipole Approximation Model to Compute Nonlinear Absorption in Gold Nanostructures Mohan S. Dhoni and Wei Ji* Department of Physics, National University of Singapore, 2 Science Drive 3, Singapore 117542, Singapore ABSTRACT: Discrete-dipole approximation (DDA) model has been widely used to provide quantitative predictions on the linear optical absorption of metallic nanostructures (MNS) irrespective of their geometry. Here, we demonstrate that it can be extended to the computations of MNS’ nonlinear light absorption. In the extended DDA (e-DDA) modeling, Drude’s dielectric function (or standard dielectric function) for given MNS in the dark is employed for the computation of linear absorption in an ensemble of the MNS. As excited by intense laser light, the dielectric function is altered in the presence of photoexcited electrons in the MNS. With the altered dielectric function and quasi-equilibrium approximation, the DDA model is reapplied to acquire the quantitative simulation on the nonlinear optical absorption. The results of e-DDA modeling are in good agreement with experimental data for gold nanorods and nanospheres.

1. INTRODUCTION Plasmonic metallic nanostructures (MNS) have been the subject of intense research in scientific and industrial communities due to their unique electronic, catalytic, and optical characteristics.1
3 Among these, optical properties have been of scientific interest since first being acknowledged by Faraday,4 who tried to interpret why metal precipitates lead to the beautiful color of stained glasses. An interesting phenomenon manifested in MNS is surface plasmon resonance (SPR). SPR is now known to be caused by the coherent oscillation of conduction-band electrons in MNS induced by the resonant coupling of light’s electric field to MNS.5 Because of it, MNS exhibit fascinating optical behavior which gives rise to immense enthusiasm for potential applications. These applications include chemical and biochemical sensors, medical diagnostics and therapeutics, biological imaging and multiphoton microscopy, optical limiting, saturation absorption, plasmonics, and nanophotonics.6
18 In MNS with the highest geometrical symmetry, i.e, nanopsheres, there is one SPR. In nonspherical-shaped MNS, however, more than one SPR exist. For example, in the gold nanorods, there are two SPR bands: namely, (i) transversal surface plasmon resonance (TSPR) and (ii) longitudinal surface plasmon resonance (LSPR). TSPR and LSPR originate from the coherent and collective oscillation of the electrons along the directions perpendicular and parallel to the long axis of the nanorod, respectively. The spectral position of SPR depends on MNS’ size, shape, type of metal, and dielectric environment.19 The optical properties of MNS including the nonlinear part have been investigated experimentally.6
18,20 Theoretical understanding has also been developed to the extent as described in the following. Maxwell’s equations for the linear optical response of MNS to an electromagnetic field of light were first solved analytically for a homogeneous isotropic sphere by Mie21 in 1908. This method has a fundamental limitation as the exact r 2011 American Chemical Society

solutions are restricted only to highly symmetric particles such as nanospheres and nanospheroids. The linear optical response of nanoellipsoids was explained by Gans’ theory.22 For asymmetrically geometric MNS, numerically approximate methods are generally required and a number of computational methods have been evolved in recent decades based on more advanced scattering theories.19 Most popular and efficient approaches used for arbitrary shapes25 of MNS are as follows: discrete dipole approximation (DDA),19,25
33 finite difference time domain (FDTD),34
39 and boundary element method (BEM).40
43 Here, we focus on the DDA method because it is easier to be extended to the computations of nonlinear optical properties, i.e., light-intensity-dependent part in the optical properties. The DDA method was originally proposed by Purcel and Pennypacker25 for the application to the scattering problem of nonmetallic intersteller dust particles. It has been considered as one of the most powerful and flexible electrodynamic methods for computing the optical scattering of particles with an arbitrary geometry since a significantly improved numerical implementation by Draine and Flatau.19,27 Schatz and his co-workers have proved that the DDA method is valid for the calculations of linear optical properties of MNS with different geometries and environments through the extensive studies on the absorption spectra and the local electric field distribution in MNS.28
31 In the work presented here, we have adopted the open source code DDSCAT implementations of the DDA method,27,44
46 which has been demonstrated by many research groups to be capable of producing satisfactory simulations for the linear absorption spectra of gold nanospheres and nanorods. 19,31
33 Received: March 4, 2011 Revised: September 17, 2011 Published: September 19, 2011 20359

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Despite the above success in the modeling of linear optical properties, it should be pointed out that, to date, there has been no theoretical model that can provide the quantitative simulations of the nonlinear optical properties, which are of vital importance to many applications. In our present work, we demonstrate that the DDA method can be extended, via incorporating the photoexcited electrons, to compute the nonlinear absorption of gold nanorods (AuNRs) and Au nanospheres. To quantitatively describe the linear absorption of an ensemble, a composite system where a transparent matrix is embedded with AuNRs or Au nanospheres, we first employ the Draine and Flatau code with Drude’s dielectric function in the dark or under the condition that light intensity is too low to induce any significant light-intensity-dependent (or nonlinear) effects, see section 2. When the composite system is excited by high-intensity laser light, photoexcited electrons alter the dielectric function and, hence, the absorption, thereby giving rise to the nonlinear optical absorption. Under the quasi-equilibrium approximation in section 3, the extended DDA (e-DDA) modeling leads to the intensity-dependent part in the absorption of the composite system. The e-DDA modeling is validated by comparison with experimental observations reported by Hendry et al.15 and Boni et al.,16 as demonstrated in sections 4 and 5, respectively. The limitation of the e-DDA model is discussed in section 6.

2. DDA MODELING FOR LINEAR ABSORPTION IN MNS The DDA method is adopted in the code of DDSCAT 7.0,46 which can be modified to calculate the absorption and scattering of light (or electromagnetic waves) by targets with arbitrary geometries. Targets may be isolated entities (e.g., dust particles), one- or two-dimensional periodic arrays of “target unit cells”. Here, the above DDSCAT is employed to calculate the light absorption spectra of a composite system consisting of a transparent matrix in which MNS are embedded in a random fashion with an average spacing. The basics of DDA theory implemented in the DDSCAT are outlined as follows. In the DDA model,19,25
33 a particular MNS of interest is represented as a number of cubic cells, and each cell represents one-point dipole. The center position of a given point dipole is denoted by ri with polarizability αi, which is related to the dielectric function.25,27 Purcell and Pennypacker25 used the Clausius
Mossotti polarizabilities to relate with the dielectric function. The relation is given by αi ¼

3d3 εi
1 4π εi þ 1

ð1Þ

where d is the interdipole spacing and εi is the complex dielectric function of the MNS at location ri. Analytical modifications27 in eq 1 have been reportedly implemented in DDSCAT. The complex dielectric function for the case of MNS embedded in a transparent matrix can be explained with Drude’s theory.23,24 Drude’s theory suggests that the variation in the complex dielectric function (or standard dielectric function), εD of a metal with light frequency, ω is mainly reflected by the absorption of light due to free electrons in the metal, i.e., conduction-band electrons.47 It is expressed in the second right-side term of eq 2 below εD ¼ 1


ω2p ½ω2 þ iΓ0 ω

ð2Þ

where ωp is the plasmon frequency given by ω2p ¼

Ne e2 m 0 ε0

ð3Þ

where Ne is the density of conduction-band electrons, e is the electron charge, m0 is the effective electronic mass, and ε0 is the permittivity of free space. Γ0 in eq 2 is the free-electron
motion damping rate due to scattering of electrons with phonons, electrons, lattice defects, or impurities.For the MNS, dielectric function has the form23 ε ¼ εD þ εB

ð4Þ

where εB is the contribution of bound electrons and εD is Drude’s dielectric function given by eq 2. The εB term of dielectric function plays an important role in determining plasmon resonance wavelengths; however, the size dependence of the dielectric function is usually assumed to be determined by free-electron contribution. This leads to the correction to the bulk dielectric function give by ε ¼ εb þ

ω2p ½ω2 þ iΓ0 ω




ω2p ½ω2 þ iðΓ0 þ Γs Þω

ð5Þ

where εb is the bulk dielectric function from the literature and Γs is the damping rate contributed by nonlocal effects or scattering at the boundaries of MNS, when the size is comparable or less than the mean free path of the electron, and is given by23,24 Γs ¼ A

υf Leff

ð6Þ

where A is dimensionless parameter, normally assumed to be close to unity, vf is Fermi velocity, and Leff is effective mean free path of the electron which comprises the scattering of electron at the boundaries of MNS. To calculate the absorption by the DDA model, one may take the following semiempirical approach. In addition to employing Drude’s theory for the dielectric function values of bulk metals, one may also implement the experimentally measured values. The experimental values for the dielectric function of a noble metal like gold and silver in bulk were well-documented by Johnson and Christy.48 Their dielectric function values have been widely adopted to explain the linear optical properties of a composite system consisting of noble MNS by applying the DDA method.19,26,28
33 Both the approaches lead to the same results since the experimentally determined dielectric function is in agreement with Drude’s model.48 The polarization induced in each dipole as a result of interaction with a local electric field Eloc is given by Pi ¼ αi 3 Eloc ðri Þ

ð7Þ

where Eloc, for isolated particles, is the sum of an incident field (Einc,i) and a contribution from all other dipoles to the same particle (Edip,i). Eloc ðri Þ ¼ Einc, i þ Edip, i ¼ E0 eiKri


∑ Aij 3 Pj

ð8Þ

j6¼ i

E0 and K are the amplitude and wave vector of the incident wave, respectively, and the interaction matrix, A, has the 20360

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following form:

" # ð1
ikrij Þ eikrij 2 2 Aij 3 Pj ¼ 3 k rij  ðrij  Pj Þ þ  frij Pj
3rij ðrij 3 Pj Þg j 6¼ i rij rij2

ð9Þ Here, k = ω/c, and c is the speed of light. Extinction crosssection, σext of the MNS taken from the optical theorem,45 is related with the extinction efficiency or extinction of the MNS, Qext through the effective radius, aeff of the MNS, and has the following form: σext ¼

4πk n ImðEinc, j 3 Pj Þ ¼ Qext πa2eff jE0 jz j ¼ 1

∑

ð10Þ

The linear absorption coefficient, α0, of the composite system can then be determined by the equation below if the MNS concentration (or the number of MNS per cc), NMNS, is known from the experimental measurement. α0 ¼ σ ext NMNS

ð11Þ

3. MODELING FOR NONLINEAR ABSORPTION IN MNS As the composite system is excited by intense laser light, photoexcited electrons are promoted into the conduction band and are calculated by dΔNe α0 I ΔNe
¼ pω dt τ

ð12Þ

where ΔNe is the density of photoexcited excess electrons in the conduction band, I is the light intensity, and τ is the excited electron lifetime. There are two scenarios in which analytical solutions are obtainable. First, if the laser pulse duration (τL) is much less than τ, one can have Z ∞ α0 I dt 0
∞ ð13Þ ΔNe ¼ pω

Figure 1. Upper panel shows a schematic illustration of typical TSPR and LSPR in the linear absorption spectra of AuNRs. Lower panel shows a schematic of diagram of energy bands and transitions explaining the processes leading to TSPR and LSPR absorption in AuNRs and electron initial thermalization.

For the Gaussian temporal profile, I can be expressed as follows:

and

I ¼ I0 e


ðt=τL Þ2

Substituting eq 14 into eq 13, one can have pffiffiffi α0 I0 τL π ΔNe ¼ pω α0 I0 τ pω

ð15Þ

ð16Þ

By referring to eq 3, the change in the square of the plasmon frequency due to the creation of photoexcited electrons is given by Δω2P ¼

ΔNe e2 m 0 ε0

ReΔε ¼


ð14Þ

Second, if τL is considerably greater than τ, one can obtain ΔNe ¼

from eq 5 as

ð17Þ

ΔNe in the above equation may be computed either from eq 15 or eq 16 depending upon whether τL , τ or τL . τ. As a result, there is a change in the standard dielectric function, and it is given by Δε = ε0
ε, where ε0 is the modified dielectric function due to the photoexcitation. The change can be derived

ImΔε ¼

2ΔNe e2 m 0 ε0 ω 2

ΔNe e2 ð2Γ0 þ Γs Þ m 0 ε0 ω 3

ð18Þ

ð19Þ

In deriving eqs 19 and 20, we have assumed ω2. Γ20 and ω2 . Γ2s for all of our calculations. With the modified dielectric function, by ε0 = ε + Δε, the modified extinction efficiency under the photoexcitation is computed through eqs 1
11. In general, in noble metals like gold and silver, electrons can be excited from the d-band or p-band depending upon the photon energy used. The Fermi level, Ef, is within the p-band in gold and silver. Photons with energy at least 2.4 eV in gold and 4.1 eV in silver are required to overcome interband transition threshold energies, which can be explained from the band structure of the gold and silver at L symmetric point of Brillouin zone boundary.49,50 In AuNRs, there are the transversal surface plasmon resonance (TSPR) and longitudinal surface plasmon resonance (LSPR). When excitation’s photon energy is close to the LSPR, electrons in the p-band are photoexcited to the higher excited states in the same band depending on the excitation wavelength or energy used (Figure 1). At TSPR in AuNRs (∼520 nm or ∼2.3 eV), in addition to the 20361

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After the electrons are photoexcited to the higher excited states, they relax to the lower states in the p-band via the interactions like electron
electron scattering (internal thermalization), see Figure 1. The internal thermalization process could be as fast as a few femtoseconds, depending mainly on the excitation energy used.49
53 Subsequently, the photoexcited electrons reach a state that can be approximated as a quasi-equilibrium state. This quasiequilibrium disappears after electron recombination (or relaxation to the bottom states of the p-band and d-band) through electron
phonon interaction or energy relaxation to vibrational modes. We have assumed the excited electron lifetime, τ, to be 1 ps in our simulations. This assumption will be discussed in section 6. In our computations, the photoexcited electron density is calculated by eq 15 or 16, respectively, to meet the experimental conditions reported by Hendry et al. and by Boni et al. With the above calculated photoexcited electron density, the modified dielectric function is used to recalculate the modified extinction efficiency, Q0 ext, which leads to the modified absorption coefficient, α, for the composite system under the photoexcitation through the DDA method. The Fermi velocity, vf, is 1.39  108 cm s
1. The value of Γ0 used is 7.586  1013 s
1.24 Γs is set zero in our nonlinear absorption calculations. The effective mean free path, Leff, is approximated to be the effective radius, aeff in our case. From the modified absorption coefficient, the nonlinear absorption coefficient, α2, of the composite system at excitation intensity, I0, is given by α2 ¼

α
α0 I0

ð20Þ

4. SIMULATIONS OF THE EXPERIMENTAL RESULTS BY HENDRY ET AL.

Figure 2. (a) Aspect ratio (AR) distribution of AuNRs.15 (b) Experimental linear (—) and nonlinear (9) absorption spectra.15 (c) Computed linear and nonlinear absorption spectra for ellipsoid (black, purple), and capped cylinder (blue, red).

photoexcitation of the p-band electrons, photoexcitation of the d-band electrons is also responsible for SPR’s absorption band, as illustrated in Figure 1. In AuNRs, the TSPR absorption is less significant as compared to the LSPR absorption. Therefore, we focus on the LSPR for our nonlinear absorption calculations in AuNRs. Absorption saturation at the LSPR of AuNRs is caused by the bleaching of the lower p-band electrons under photoexcitation by laser light, as shown in Figure 1. In Au nanospheres, the SPR peak position is typically in the range 520
540 nm (or 2.29
2.38 eV), depending on their sizes. This gives rise to the absorption saturation, which results from the bleaching of electrons from the p-band and d-band, see Figure 1. To quantify the absorption saturation, we first apply the DDA method to calculate the extinction efficiency, Qext, and the linear absorption coefficient of the composite system with the standard dielectric function.48 Then, we calculate the modified dielectric function by using eqs 18 and 19, by considering electrons excited from the bottom states of the p-band or from the d-band which make no contribution to the linear absorption.

A. Experimental Section. Hendry et al.15 used an ensemble of gold nanorods (AuNRs) dispersed in water as the sample. As shown in Figure 2a, the average aspect ratio is 3.85, and the average diameter is 20 nm. The experimentally measured linear absorption spectrum of the sample is shown in Figure 2b, indicating that the LSPR band is peaked at 800 nm (λLSPR) with a fwhm of 176 nm and the TSPR band is peaked at 520 nm. Hendry et al.15 performed femtosecond (pulse width τL = 220 fs) Z-scan measurements at the intensity I0 = 0.5 GW/cm2 and observed the saturation absorption of AuNRs. The linear transmittance of 70% corresponds to a linear absorption coefficient, α0 = 2.73 cm
1 at 800 nm. As shown in Figure 2b, the nonlinear absorption coefficient, α2 in the units of cm/GW, is negative, indicating the saturation absorption. The maximum value of α2,
1.5 cm/GW, can be observed at the LSPR peak (800 nm), with a fwhm of 88 nm, see Table 1. B. Simulation of the Linear Absorption. To simulate the linear absorption spectrum of the composite system consisting of AuNRs and water, we consider two possible MNS shapes, namely, ellipsoid and cylinder with hemispherically end-caps (denoted as capped cylinder), see Figure 3. In the simulation, the AR is 3.85 with a diameter D = 20 nm. The numbers of dipoles for ellipsoid and capped cylinder are 16 188 and 22 236, respectively. The interdipole spacing, d, is 1 nm. The effective radius, aeff, is 15.7 nm for ellipsoid and 17.4 nm for capped cylinder. The damping rate, Γs, is 8.85  1013 s
1 for ellipsoid and 7.98  1013 s
1 for capped cylinder. The concentrations, NMNS, for ellipsoid and capped cylinder are 3.6  1010/cc and 5.1  1010/cc, respectively. The dielectric constant of water is 1.768. In our 20362

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Table 1. Computed Linear and Nonlinear Absorption Results for AuNRs and Au Nanospheres Together with Reported Experimental Data λLSPR

α0at LSPR

Δλabs

size

(nm)

(cm‑1)

(fwhm) (nm/eV)

dispersion (%)

Hendry et al. AuNRs

800

2.73

176/0.35

aspect ratio distribution Figure 2a

NA

our model orientation-random

796

2.73

62/0.12

0

828

2.73

64/0.11

0

Boni et al. Au nanosphere diameter = 16 nm

527b

NA

98/0.44

20

NA

0.7 ( 0.07

61/0.27

NA

our model Au nanosphere diameter = 16 nm

524b

4.44

68/0.31

0

2.5  1012

0.7 ( 0.07

56/0.26

0

Boni et al. AuNRs our model orientation-random capped

620 622

NA 5.27

94/0.31 53/0.17