**2. Optical properties of metal nanoparticles**

The optical properties of metal nanosphere embedded in semiconductor environment have been discussed under the quasi-static approximation, where we have assumed that the size of metal nanosphere is much smaller than the wavelength of incident light. In this approximation, we have taken metal nanosphere which is embedded in a semiconductor medium having constant electric field. The Laplace equation ∇<sup>2</sup> Φ = 0 with appropriate boundary condition was solved for finding out the potential, electric field, scattering and absorption and extinction cross section [8]. The schematics of modeled structure is shown in **Figure 1**, wherein the incident field interacts with the nanosphere of radius *a* and dielectric *ε*(*ω*) constant embedded in surrounding environment having dielectric constant *εm*.

The potential inside and outside the sphere that are obtained after solving the Laplace equation under appropriate boundary conditions are expressed as

$$\Phi\_{\rm in} = -\frac{3\,\varepsilon\_{\rm m}}{\varepsilon + 2\,\varepsilon\_{\rm m}} E\_0 \, r \cos\theta \,\text{and}\tag{1}$$

$$\Phi\_{out} = -E\_o r \cos \theta + \frac{r \cos \theta}{r^3} a^3 E\_0 \left(\frac{\varepsilon - \varepsilon\_n}{\varepsilon + 2\varepsilon\_n}\right) \tag{2}$$

After solving the potential profile, one can easily obtain the electric field expression as

$$E = -\nabla \Phi \tag{3}$$

The polarization of metal sphere under the influence of constant electric field is expressed as

$$P = \text{4 } \pi \varepsilon\_o \varepsilon\_m a^3 E\_0 \left( \frac{\varepsilon - \varepsilon\_m}{\varepsilon + 2 \varepsilon\_m} \right) \tag{4}$$

and polarizability of nanosphere is

in various fields. The optical properties of metal nanoparticles are highly dependent on its morphology and the surrounding medium [1–4]. These optical properties include scattering, absorption and extinction cross section which are obtained by solving the Maxwell's equation after the interaction of light with the nanogeometries. The optical cross sections are generally greater than the geometrical cross section. When the light interacts with the metals, two fundamental excitations are observed. These excitations are either propagating known as surface plasmon polaritons (SPP) or non-propagating known as surface plasmon resonance (SPR). These fundamental excitations have a wide range of applications. The candidates used as plasmonic elements are metals and few semiconductors, and they have their own optical features in different regimes of electromagnetic spectrum. These metals exhibit surface plasmon resonance (SPR) properties on interaction with the incident field. The SPR depends on several parameters such as size and shape of metal and choice of the surrounding media. The influence of surrounding media on the SPR peak positions has a great interest in material research society. One can tune these SPR peak positions by choosing different surrounding media. The tunable behaviors of SPR with size shape and surrounding environments would cover a wide range of applications such as biosensor, Raman spectroscopic,

The occurrence of the SPR resonances is due to the interaction of incident light frequency and its match with the frequency of collective oscillation of electrons inside the metal. Under such matching condition, a giant electric field is observed near the metal nanosurface, which could be used for various applications. The physics of SPR could be utilized in different fields of science and technology wherein the light metal interaction is taken into account. It could be utilized in biosensor field, thin-film plasmonic photovoltaic devices, surface-enhanced Raman

The work furnishes the plasmonic properties isolated in metal nanosphere and its interaction to the silica environment. This work explores the optical properties of isolated sphericalshaped medium and small-sized metal nanoparticle. The analysis of small-sized nanoparticle has been done using dipolar model, but the restriction with this model is that it can be applied for larger-sized nanoparticle. Therefore, we have to use numerical technique in full-wave analysis that needs to be taken into account. The purpose of selecting the silica environment is due to its frequent use in the photovoltaic devices. Silica is used as a spacer layer in thinfilm device fabrication, and it is also used as a core or coating material whose thickness plays an important role to tune the plasmonic resonance. Therefore, the aim of this work is to present systematic review of medium- and small-sized metal nanoparticle under the influence of

The optical properties of metal nanosphere embedded in semiconductor environment have been discussed under the quasi-static approximation, where we have assumed that the size of metal nanosphere is much smaller than the wavelength of incident light. In this approximation, we have taken metal nanosphere which is embedded in a semiconductor medium having

waveguide and thin-film photovoltaic device [5–7].

scattering field and communication field also [6, 7].

**2. Optical properties of metal nanoparticles**

silica environment.

154 Emerging Solar Energy Materials

$$a = 4\pi a^3 \left(\frac{\varepsilon - \varepsilon\_m}{\varepsilon + 2}\right) \tag{5}$$

where *ε*, *<sup>ε</sup>m* are the dielectric constant of sphere and medium, *a* is the radius of sphere and *α*, is the dipolar polarizability.

**Figure 1.** Optical properties of metal nanosphere placed in a uniform static electric field *E*<sup>0</sup> .

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]:

$$
\left< \mathbf{C}\_{\kappa at} \right> = \frac{k^4}{6\pi} \left| \alpha \right|^\ast = \frac{8\pi}{3} k^4 a^6 \left| \frac{\varepsilon - \varepsilon\_n}{\varepsilon + 2\varepsilon\_n} \right|^2 \tag{6}
$$

$$
\langle \mathcal{C}\_{ab} \rangle = k \operatorname{Im} \langle a \rangle = 4 \pi k a^3 \operatorname{Im} \left[ \frac{\varepsilon - \varepsilon\_n}{\varepsilon + 2 \varepsilon\_n} \right] \tag{7}
$$

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

$$\mathbf{C}\_{\rm cat} = \mathbf{C}\_{\rm sat} + \mathbf{C}\_{\rm abs} \tag{8}$$

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

Plasmonic Resonances and Their Application to Thin-Film Solar Cell

http://dx.doi.org/10.5772/intechopen.75015

157

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

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

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

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

) magnitude in y-z plane.

410 nm. The legend in the figure shows the normalized field (E/E<sup>0</sup>

Frohlich conditions.

for various applications [11–15].

rate inside the thin film of solar device.

having refractive index N = 1.54.

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> ; therefore, Q-extinction for sphere is

$$Q\_{extn} = \frac{C\_{nt}}{\pi a^2} \tag{9}$$

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 be expressed as [8, 9]

be expressed as [8, 9]

$$
\varepsilon(\omega) = \varepsilon\_{\text{bulk}}(\omega) + \frac{\omega\_{\text{p}}^{2}}{\omega^{2} + \dot{f}\gamma\_{\text{bulk}}\omega} - \frac{\omega\_{\text{p}}^{2}}{\omega^{2} - \dot{f}\gamma\omega} \tag{10}
$$

 $\gamma = \gamma\_{\text{bulk}} + A\frac{v\_{\text{f}}}{\overline{a}}$ 

where *ω<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> 10<sup>6</sup> 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 the optical constant of metal at the nanolevel, extinction spectrum is studied.
