**2. Theoretical model: scattering by small size metal nanosphere**

Theoretical model discuss the plasmonic properties of metal nanoparticle within electrostatic approximation wherein particle size is smaller than the wavelength of light. Under such assumption Laplace equation has been solved with suitable boundary condition to find out the optical parameters [13, 32]. The Laplace equation is expressed as

$$
\nabla^2 V = 0\tag{1}
$$

As we know when the incident electromagnetic field interacts with metallic nanostructure, some fraction of light gets absorbed and some of them gets scattered. The magnitude of the these absorbed and scattered light can be expressed in terms of scattering and absorption

<sup>3</sup> *k*<sup>4</sup> *a* <sup>6</sup>

where symbol *k* is the incident light wave vector, *ε*, *<sup>ε</sup>m* are the dielectric constant of metal and

A spherical shape metal nanoparticle is taken into account to observe the optical properties like scattering cross section and surface plasmon resonances. **Figure 2** represent the wavelength dependent normalized cross section of three different radii of spherical shape silver metal nanoparticle. We observed that as the size of nanoparticle increases corresponding magnitude of scattering cross section increases. But the position of SPR peak position is same

**Figure 2.** Wavelength dependent scattering cross section of silver metal nanoparticle of three different radii.

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

2

A Perspective on Plasmonics within and beyond the Electrostatic Approximation

*<sup>π</sup><sup>a</sup>* <sup>2</sup> (5)

http://dx.doi.org/10.5772/intechopen.81038

(4)

13

cross section as

surrounding medium.

for all radii which is at 364 nm.

〈*Cscat*〉 <sup>=</sup> \_\_\_ <sup>8</sup>*<sup>π</sup>*

*Qscat* <sup>=</sup> 〈*Cscat*〉 \_\_\_\_\_

and its solution in spherical polar coordinate is

$$\mathbf{V(r)} = \sum\_{l,\mathbf{m}} \mathbf{A}\_{l,\mathbf{m}} r^{-l-1} Y\_l^n(\theta, \varphi) + \sum\_{l,\mathbf{m}} B\_{l,\mathbf{m}} r^l Y\_l^n(\theta, \varphi) \tag{2}$$

Truncating the potential series only for dipolar term which correspond to *l* = 1 one can find the value of potential inside and outside the spherical surface as

$$V\_{\rm in} = -\frac{3}{\varepsilon + 2} \frac{\varepsilon\_{\rm m}}{\varepsilon\_{\rm m}} E\_0 r \cos \theta\_{\prime} \tag{2a}$$

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

Once we have potential profile, the calculation of field and polarizability can be easily obtained by *E* = −∇*V* expression.

The applied field polarizes the metal nanostructure and its polarization is in the same direction of applied electric field. If the size of nanoparticle is small enough then polarization is in the direction of applied field while for larger size nanoparticle the oscillation of electron is no longer symmetric and the polarization mechanism is split into component form such as transverse and longitudinal. Here the study reveals the optical properties of smaller size nanoparticle therefore, polarization of particle is in the same direction of applied field. The polarization is simply the dipole moment per unit volume which is expressed as

$$p = \varepsilon\_n \, aE\_0 \tag{2c}$$

where E<sup>0</sup> is the applied electric field and *α* is the polarizability of nanosphere expressed as

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

The symbol *a* is the radius of metal nanosphere, *ε*, *<sup>ε</sup>m* are the dielectric constant of metal sphere and surrounding environment. The expression of polarizability is an important parameter in the discussion of resonance physics. The concept of resonance brought into the picture from the denominator part of polarizability expression. The polarizability gets resonantly enhanced when |*ε* + 2 *εm*| = 0 which is known as Fröhlich condition.

As we know when the incident electromagnetic field interacts with metallic nanostructure, some fraction of light gets absorbed and some of them gets scattered. The magnitude of the these absorbed and scattered light can be expressed in terms of scattering and absorption cross section as

**2. Theoretical model: scattering by small size metal nanosphere**

the optical parameters [13, 32]. The Laplace equation is expressed as

*l*,m

the value of potential inside and outside the spherical surface as

A*<sup>l</sup>*,m *r* <sup>−</sup>*l*−<sup>1</sup> *Yl*

and its solution in spherical polar coordinate is

*Vin* <sup>=</sup> <sup>−</sup> <sup>3</sup> *<sup>ε</sup>* \_\_\_\_\_*<sup>m</sup>*

*Vout* <sup>=</sup> <sup>−</sup>*Eo <sup>r</sup>* cos*<sup>θ</sup>* <sup>+</sup> \_\_\_\_\_\_\_\_\_\_\_ *<sup>r</sup>* cos*<sup>θ</sup>*

*α* = 4*πa* <sup>3</sup>

V(r) = ∑

12 Plasmonics

by *E* = −∇*V* expression.

where E<sup>0</sup>

Theoretical model discuss the plasmonic properties of metal nanoparticle within electrostatic approximation wherein particle size is smaller than the wavelength of light. Under such assumption Laplace equation has been solved with suitable boundary condition to find out

∇<sup>2</sup> *V* = 0 (1)

*<sup>m</sup>*(*θ*, *ϕ*) + ∑ *l*,m

*<sup>r</sup>* <sup>3</sup> *<sup>a</sup>* <sup>3</sup> *<sup>E</sup>*0(

Truncating the potential series only for dipolar term which correspond to *l* = 1 one can find

*ε* + 2 *ε<sup>m</sup>*

Once we have potential profile, the calculation of field and polarizability can be easily obtained

The applied field polarizes the metal nanostructure and its polarization is in the same direction of applied electric field. If the size of nanoparticle is small enough then polarization is in the direction of applied field while for larger size nanoparticle the oscillation of electron is no longer symmetric and the polarization mechanism is split into component form such as transverse and longitudinal. Here the study reveals the optical properties of smaller size nanoparticle therefore, polarization of particle is in the same direction of applied field. The

*p* = *ε<sup>m</sup> αE*<sup>0</sup> (2c)

is the applied electric field and *α* is the polarizability of nanosphere expressed as

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

The symbol *a* is the radius of metal nanosphere, *ε*, *<sup>ε</sup>m* are the dielectric constant of metal sphere and surrounding environment. The expression of polarizability is an important parameter in the discussion of resonance physics. The concept of resonance brought into the picture from the denominator part of polarizability expression. The polarizability gets resonantly

polarization is simply the dipole moment per unit volume which is expressed as

enhanced when |*ε* + 2 *εm*| = 0 which is known as Fröhlich condition.

*Bl*,m *rl Yl*

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

*<sup>m</sup>*(*θ*, *ϕ*) (2)

*E*<sup>0</sup> *r* cos*θ*, (2a)

*<sup>ε</sup>* <sup>+</sup> <sup>2</sup> *<sup>ε</sup>m*) (3)

*<sup>ε</sup>* <sup>+</sup> <sup>2</sup> *<sup>ε</sup>m*) (2b)

$$
\left< \mathbf{C}\_{sat} \right> = \frac{8\pi}{3} \left| k^4 \, a^6 \left| \frac{\varepsilon - \varepsilon\_m}{\varepsilon + 2 \, \varepsilon\_m} \right| \right|^2 \tag{4}
$$

$$Q\_{\kappa at} = \frac{\langle \mathcal{C}\_{\kappa at} \rangle}{\pi a^2} \tag{5}$$

where symbol *k* is the incident light wave vector, *ε*, *<sup>ε</sup>m* are the dielectric constant of metal and surrounding medium.

A spherical shape metal nanoparticle is taken into account to observe the optical properties like scattering cross section and surface plasmon resonances. **Figure 2** represent the wavelength dependent normalized cross section of three different radii of spherical shape silver metal nanoparticle. We observed that as the size of nanoparticle increases corresponding magnitude of scattering cross section increases. But the position of SPR peak position is same for all radii which is at 364 nm.

**Figure 2.** Wavelength dependent scattering cross section of silver metal nanoparticle of three different radii.

managed by the lattice parameter. The chosen target in this technique is expressed in terms

where k is wave vector, lattice parameter and m is the complex refractive index of chosen target material. The main input parameter in DDA technique is the dielectric constant of target and surrounding media in which target is embedded. The complex dielectric function of

Here, the target is assumed as a point dipole situated at the each corner of cube. Further,

→ *i*

→ *<sup>i</sup>* = *α<sup>i</sup>* .*E* →

because of the radiation of all the others (N – 1) dipoles that set up the NPs. The field *E*

*<sup>i</sup>*,*app* = *E*<sup>0</sup> *ei*(*<sup>k</sup>*

*<sup>i</sup>*,*ind* = −∑ *j*=1 *N A* →

> *<sup>j</sup>*) <sup>+</sup> (<sup>1</sup> <sup>−</sup> *<sup>i</sup> <sup>k</sup> rij*) \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ *rij* <sup>2</sup> [*rij* 2 *p* → *<sup>j</sup>* − 3 *r* → *ij*(*r* → *ij*.*p* →

Dipole moment symbolizes the optical behavior of the target geometry. Therefore, the extinction and absorption cross section can be achieved after calculating the set of dipole moments

→ .*r*

→ *<sup>i</sup>*,*loc* = *E* → *<sup>i</sup>*,*app* + *E* →

→

→

→ *<sup>i</sup>* = *α<sup>i</sup>* . (Ε → *<sup>i</sup>*,*app* − ∑ *j*=1 *N A* → *ij*.*pj*

\_\_\_\_\_ *<sup>t</sup>*arg*et εmedium*

*<sup>i</sup>*,*ind* is the applied and induced field respectively acting on the ith individual

of the point dipoles need to be flexible for DDA

*<sup>i</sup>*,*loc* (7)

*<sup>i</sup>*,*ind* (8)

<sup>→</sup>−*t*) (9)

*ij*.*pj* (10)

) (12)

ing and the size of target is also expressed in term of effective radius as *aeff* = (3*V*/4*π*)

technique is applicable only when the following criteria is satisfied

composite system is provided by input file which can be expressed as

and polarizability *α*

calculation. Each entity is represented by a dipole moment as


*<sup>ε</sup>relative* <sup>=</sup> *<sup>ε</sup>*

*i*

*p*

*E*

*<sup>i</sup>*,*ind* is given by the following relation

*E*

*E*

→ *ij*.*p* → *<sup>j</sup>* <sup>=</sup> *ei <sup>k</sup> rij* \_\_\_\_\_\_\_\_\_ *rij* <sup>3</sup> {*k*<sup>2</sup> *<sup>r</sup>* → *ij* (*r* → *ij* × *p* →

*p*

where k = ∣*k*∣ represents the incident wave vector.

, where N is the number of discretized dipole and d is the lattice spac-

A Perspective on Plasmonics within and beyond the Electrostatic Approximation

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1⁄3 . The 15

(6)

→ *<sup>i</sup>*,*app* and

*<sup>j</sup>*)]} (11)

of volume as *V* = *Nd*<sup>3</sup>

placement position <sup>→</sup>*<sup>r</sup>*

*<sup>i</sup>*,*app* and *<sup>E</sup>* →

*A*

as given by

where *E* →

*E* →

**Figure 3.** Wavelength dependent scattering cross section of gold metal nanoparticle of three different radii.

In **Figure 3**, we have discussed the wavelength dependent scattering cross section of gold metal nanoparticle for three different radii. Here we observed that, as the radii of gold nanoparticle increases corresponding cross section magnitude increases with red shifted SPR resonance. In case of silver the SPR peak position is fixed at one wavelength while for gold the peak positions are red shifted with the radii. The SPR peak position for gold nanosphere surrounded by air (N = 1) is at 554 nm for radius 5 nm. The additional advantages of gold nanoparticle over the silver are analyzed in terms of SPR peak positions and corresponding spectral width known as full width at half maxima (FWHM).

These two different types of metal nanoparticle are described within the electrostatic approximation which is only valid for smaller size. Therefore, for the description of lager size nanoparticle we have used the numerical method discrete dipole approximation (DDA) which is valid within and beyond the electrostatic approximation.

### **3. Numerical approach: scattering by large size metal nanosphere**

The numerical technique that we have used to simulate the optical properties of large size nanoparticle is discrete dipole approximation method. In this method, target is discretized into large number of polarizable dipoles [33]. Each dipole situated at the corner of a cubic lattice and the relation between them i.e., separation or distance of one dipole to other are managed by the lattice parameter. The chosen target in this technique is expressed in terms of volume as *V* = *Nd*<sup>3</sup> , where N is the number of discretized dipole and d is the lattice spacing and the size of target is also expressed in term of effective radius as *aeff* = (3*V*/4*π*) 1⁄3 . The technique is applicable only when the following criteria is satisfied

$$|m|kd < 1$$

where k is wave vector, lattice parameter and m is the complex refractive index of chosen target material. The main input parameter in DDA technique is the dielectric constant of target and surrounding media in which target is embedded. The complex dielectric function of composite system is provided by input file which can be expressed as

$$\mathcal{E}\_{relative} = \frac{\mathcal{E}\_{target}}{\mathcal{E}\_{radius}} \tag{6}$$

Here, the target is assumed as a point dipole situated at the each corner of cube. Further, placement position <sup>→</sup>*<sup>r</sup> i* and polarizability *α* → *i* of the point dipoles need to be flexible for DDA calculation. Each entity is represented by a dipole moment as

$$
\vec{p}\_l = \alpha\_r \vec{E}\_{l\&c} \tag{7}
$$

$$
\vec{E}\_{\iota,loc} = \vec{E}\_{\iota,app} + \vec{E}\_{\iota,ind} \tag{8}
$$

where *E* → *<sup>i</sup>*,*app* and *<sup>E</sup>* → *<sup>i</sup>*,*ind* is the applied and induced field respectively acting on the ith individual because of the radiation of all the others (N – 1) dipoles that set up the NPs. The field *E* → *<sup>i</sup>*,*app* and *E* → *<sup>i</sup>*,*ind* is given by the following relation

$$
\vec{E}\_{l,app} = E\_0 e^{\ell \vec{k} \cdot \vec{s} - ut\ell} \tag{9}
$$

$$
\vec{E}\_{\cup, \text{ind}} = -\sum\_{\neq 1}^{N} \vec{A}\_{\neq} p\_{\neq} \tag{10}
$$

$$\vec{A}\_{\neq}\vec{p}\_{\neq} = \frac{e^{ikr\_{\neq}}}{r\_{\neq}^3} \left\{ k^2 \vec{r}\_{\neq} \left( \vec{r}\_{\neq} \times \vec{p}\_{\neq} \right) + \frac{(1 - ik \, r\_{\neq})}{r\_{\neq}^2} \left[ r\_{\neq}^2 \vec{p}\_{\neq} - 3 \, \vec{r}\_{\neq} (\vec{r}\_{\neq} \vec{p}\_{\neq}) \right] \right\} \tag{11}$$

$$
\vec{p}\_{\cdot} = \alpha\_{r} \left( \vec{\mathbf{E}}\_{\iota, \rho \mu p} - \sum\_{j=1}^{N} \vec{A}\_{\wedge} \vec{p}\_{\cdot} \right) \tag{12}
$$

where k = ∣*k*∣ represents the incident wave vector.

**Figure 3.** Wavelength dependent scattering cross section of gold metal nanoparticle of three different radii.

spectral width known as full width at half maxima (FWHM).

14 Plasmonics

which is valid within and beyond the electrostatic approximation.

In **Figure 3**, we have discussed the wavelength dependent scattering cross section of gold metal nanoparticle for three different radii. Here we observed that, as the radii of gold nanoparticle increases corresponding cross section magnitude increases with red shifted SPR resonance. In case of silver the SPR peak position is fixed at one wavelength while for gold the peak positions are red shifted with the radii. The SPR peak position for gold nanosphere surrounded by air (N = 1) is at 554 nm for radius 5 nm. The additional advantages of gold nanoparticle over the silver are analyzed in terms of SPR peak positions and corresponding

These two different types of metal nanoparticle are described within the electrostatic approximation which is only valid for smaller size. Therefore, for the description of lager size nanoparticle we have used the numerical method discrete dipole approximation (DDA)

The numerical technique that we have used to simulate the optical properties of large size nanoparticle is discrete dipole approximation method. In this method, target is discretized into large number of polarizable dipoles [33]. Each dipole situated at the corner of a cubic lattice and the relation between them i.e., separation or distance of one dipole to other are

**3. Numerical approach: scattering by large size metal nanosphere**

Dipole moment symbolizes the optical behavior of the target geometry. Therefore, the extinction and absorption cross section can be achieved after calculating the set of dipole moments as given by

$$\mathbf{C}\_{\text{ext}} = \frac{4\pi\mathbf{k}}{\left|\vec{E}\_{\text{o}}\right|^{2}} \sum\_{l=1}^{N} \text{Im}\left\{\vec{E}\_{\text{,loc}}^{\ast}\vec{p}\_{l}\right\} \tag{13}$$

**4. Conclusion**

**Acknowledgements**

**Conflict of interest**

**Author details**

**References**

of Technology, Delhi, India

2010;**9**:205-213

110096, India.

sensing, photovoltaic and Raman spectroscopy.

The authors do not have any conflict of interest.

\*Address all correspondence to: nileshpiitd@gmail.com

2 Maharaja Agrasen College, University of Delhi, Delhi, India

Nilesh Kumar Pathak1,2\*, Parthasarathi1

The work described the optical properties of plasmonic nanogeometries in terms of optical cross section and surface plasmon resonance. Two different types of metals like silver and gold are taken into account to see the optical properties. The surface plasmon resonance corresponding to these metals lies in visible range of electromagnetic spectrum wherein most of the applications exist. Therefore, the work guides to plamonic community to simulate various types of metal nanostructure which exhibit SPR in different part of electromagnetic spectrum. These tunable nature of surface plasmon resonances can be used in many purposes such as

A Perspective on Plasmonics within and beyond the Electrostatic Approximation

http://dx.doi.org/10.5772/intechopen.81038

17

The authors would like to thanks to Maharaja Agrasen College, University of Delhi, Delhi

, Gyanendra Krishna Pandey1

1 Plasma and Plasmonic Simulation Laboratory, Centre for Energy Studies, Indian Institute

[1] Catchpole KR, Polman A. plasmonic solar cell. Optics Express. 2008;**6**:21793-21800

[2] Pathak NK, Alok J, Sharma RP. Tunable properties of surface plasmon resonances: The influence of core–shell thickness and dielectric environment. Plasmonics. 2014;**9**:651-657

[3] Atwater HA, Polman A. Plasmonics for improved photovoltaic devices. Nature Mater.

and R.P. Sharma<sup>1</sup>

$$\mathbf{C}\_{\rm abs} = \frac{4\pi k}{\left|\vec{E}\_0\right|^2} \sum\_{l=1}^{N} \left\{ \text{Im}\left[\vec{p}\_{\rm r} \{\boldsymbol{\alpha}\_l^{-1}\}^\* \vec{p}\_{\rm r}^\*\right] - \frac{2}{3} k^3 \left|\vec{p}\_{\rm r}\right|^2 \right\} \tag{14}$$

$$\begin{aligned} \mathbf{C}\_{\text{sca}} &= \mathbf{C}\_{\text{ext}} - \mathbf{C}\_{\text{abs}}\\ \mathbf{Q}\_{i} &= \frac{\mathbf{C}\_{i}}{\pi a\_{\text{eff}}^{2}} \end{aligned} \tag{15}$$

where *Ci* signifies the optical cross section, *i* represents the running index includes extinction, absorption and scattering, *Qi* represents the normalized optical cross section and *aeff* effective radius of target.

The simulation effects could be visualized in terms of scattering cross section as shown in **Figure 4**. A 50 nm radius gold metal sphere is discretized into 4224 number of dipole to see the plasmonic properties like scattering cross section and surface plasmon resonance. The SRP wavelength of 50 nm gold nanosphere was observed at wavelength 560 nm.

**Figure 4.** Wavelength dependent scattering cross section of 50 nm gold nanosphere surrounded by air.
