**3. Extinction cross section of silver and gold nanosphere: semianalytical approach**

The optical property of silver and gold nanosphere has been discussed in terms of extinction efficiency which is the ratio of extinction cross section to geometrical cross section. Wavelength-dependent extinction efficiency of silver and gold nanosphere for three different radii ranging from 5 to 7 nm has been plotted as shown in **Figure 2**. It can be observed from the spectrum that the choice of two different metals would cover two different parts of electromagnetic spectrum. For silver, SPR resonance was observed at wavelength 410 nm, while for gold, it was around 560 nm. The magnitude of extinction is a function of nanosphere radii, while its SPR peak positions are almost independent of the chosen radii. The nanoplasmonic coupling to the silica (N = 1.54) has been studied in terms of extinction efficiency and SPR resonances. The two different metals exhibit their SPR resonances in two different regimes of solar spectrum due to different optical constant and Frohlich conditions.

The optical properties are expressed in terms of optical cross sections such as scattering and absorption, and it can be calculated by deriving the Poynting vector from the reference [8]:

*Cext* = *Cscat* + *Cabs* (8)

If the cross sections are normalized by their geometrical cross section, then it is called by a new name known as Q-extinction. For spherical geometry, geometrical cross section is *πa* <sup>2</sup>

There are several parameters involved in the study of optical signature of plasmonic geometry. Out of these parameters, optical constant of metal is one of most important parameters. Therefore, we have given a special attention to the same. This optical constant has a dual character: one at the bulk level and the other at the nanolevel. The nanolevel character comes via the size of the geometry, which has been derived from Drude-Lorentz model, which can

> 2 \_\_\_\_\_\_\_\_ *<sup>ω</sup>*<sup>2</sup> <sup>+</sup> *<sup>j</sup> <sup>γ</sup>bulk <sup>ω</sup>* <sup>−</sup> *<sup>ω</sup><sup>p</sup>*

is the bulk plasmon frequency, *ω* is the frequency of incident light photon and *τbulk* <sup>=</sup> 1/*γbulk*

is the damping constant of bulk silver metal. Where *γ* is the effective relaxation time, *vf* <sup>=</sup> 1.39 <sup>×</sup>

The optical property of silver and gold nanosphere has been discussed in terms of extinction efficiency which is the ratio of extinction cross section to geometrical cross section. Wavelength-dependent extinction efficiency of silver and gold nanosphere for three

 m/s is the Fermi velocity of electron in silver, A is geometrical parameter and its value lies between 2 to 1 (in our case we have chosen A = 1) [10] and *a* is the radius nanoparticle. Using

2 \_\_\_\_\_\_ *ω*2 − *j*

<sup>|</sup> *<sup>ε</sup>* <sup>−</sup> *<sup>ε</sup>* \_\_\_\_\_*<sup>m</sup> ε* + 2 *εm*|

*<sup>ε</sup>* <sup>−</sup> *<sup>ε</sup>* \_\_\_\_\_*<sup>m</sup>*

2

*<sup>ε</sup>* <sup>+</sup> <sup>2</sup> *<sup>ε</sup>m*] (7)

*<sup>π</sup><sup>a</sup>* <sup>2</sup> (9)

(6)

;

(10)

<sup>6</sup>*<sup>π</sup>* <sup>|</sup>*α*<sup>|</sup> 2 = \_\_\_ <sup>8</sup>*<sup>π</sup>* <sup>3</sup> *k*<sup>4</sup> *a* <sup>6</sup>

The sum of these two cross sections will give rise to the extinction cross section:

〈*Cscat*〉 <sup>=</sup> *<sup>k</sup>*<sup>4</sup> \_\_\_

156 Emerging Solar Energy Materials

therefore, Q-extinction for sphere is

be expressed as [8, 9]

**analytical approach**

where *ω<sup>p</sup>*

10<sup>6</sup>

〈*Cabs*〉 <sup>=</sup> *<sup>k</sup>* Im{*α*} <sup>=</sup> <sup>4</sup>*πka*<sup>3</sup> Im[

*Qextn* <sup>=</sup> *<sup>C</sup>*\_\_\_*ext*

*<sup>ε</sup>*(*ω*) <sup>=</sup> *<sup>ε</sup>bulk*(*ω*) <sup>+</sup> *<sup>ω</sup><sup>p</sup>*

*<sup>γ</sup>* <sup>=</sup> *<sup>γ</sup>bulk* <sup>+</sup> *<sup>A</sup> vf* \_\_

the optical constant of metal at the nanolevel, extinction spectrum is studied.

**3. Extinction cross section of silver and gold nanosphere: semi-**

*a*

The simulated extinction spectra as shown in the above figures of silver and gold nanosphere clearly give the idea of extinction magnitude and SPR wavelength which can be used to compute the electric field distribution near the surface of metal nanosphere. **Figure 3**a shows the electric field profile of silver nanosphere embedded in silica environment at SPR wavelength 410 nm. The legend in the figure shows the normalized field (E/E<sup>0</sup> ) magnitude in y-z plane. The computation of electric field has been done by using COMSOL Multiphysics software with triangular fine grid. The red region shows the high-intensity zone which can be utilized for various applications [11–15].

Further, we have also done the analysis to visualize the electric field distribution of gold nanosphere of radius 50 nm embedded in silica medium as shown in **Figure 3**b. The near field has been computed at SPR wavelength 560 nm. From the field distribution, it was observed that the magnitude of field is different for silver and gold due to different SPR wavelengths. These different magnitudes of fields can be used to increase the electron hole or exciton generation rate inside the thin film of solar device.

The above semi-analytical model has certain restrictions that it is valid only for the smallersized metal nanoparticle. Therefore, for the analysis of optical properties of larger-sized metal nanoparticle, we required some numerical approach like discrete dipole approximation (DDA), finite-difference time-domain (FDTD), finite element method (FEM) and surface integral equation (SIE). In this chapter, we have used the FDTD technique to simulate the optical properties

**Figure 2.** Wavelength-dependent extinction spectra of (a) silver and (b) gold metal nanosphere embedded in medium having refractive index N = 1.54.

**Figure 3.** Electric field distribution of (a) silver and (b) gold nanosphere of radius 7 nm surrounded by silica matrix having N = 1.54.

of spherical-shaped metal nanoparticle [16]. We have used the Lumerical-based finite-difference time-domain technique to study the optical properties of noble metals. The metals are silver and gold whose optical constants are taken from the literature [8, 9]. These metals are surrounded by silica environment having constant dielectric constant. The objective of the work is to analyze the distribution of electric field near the metal surface in a resonance condition.
