# Chapter 14 Plasmonics (Research)

**Learning Objectives:**

In this chapter you will basically learn:

\(\bullet\) Maxwell’s Equations with Static Fields.

\(\bullet\) Maxwell’s Equations with Time-Varying Fields in Vacuum.

\(\bullet\) Optical properties of metals: Drude model, interaction of electromagnetic waves with metals.

\(\bullet\) Surface Plasmon Polaritons: surface plasmon polaritons, excitation of surface plasmon polaritons, surface plasmon polariton waveguides, bulk plasmons.

\(\bullet\) Localized surface plasmon resonance-I: resonances in small metal nanoparticles: Quasistatic approximation, fabrication and optical characterization of plasmonic structures.

\(\bullet\) Localized surface plasmon resonance-II: resonances in small metal nanoparticles: complex shapes and structures, light scattering on metal nanoparticles: beyond quasistatic approximation.

\(\bullet\) Applications of plasmonics: nanoparticle ensembles for light localization and guiding, plasmonics applications for light emission enhancement, plasmonics for sensing, nonlinear optics, and optomechanics applications.

## 14.1 Electromagnetic Field & Plasmonics

To introduce the subject of Plasmonics we need to understand the theory of electromagnetic field. As we discussed in Chapter-13 Electromagnetic Waves, that an Electromagnetic field is a vector field, and it is described by electric field \(\vec E\) and vector of magnetic induction \(\vec B\). In the general case these two vectors are functions of time t and coordinate r. In static fields there is no time dependence. On the other hand if fields are independent of coordinates, the fields are called homogeneous fields. These two vectors satisfy four Maxwell’s equations that are founding pillars of Plasmonics.

Maxwell’s equations were first formulated about one and a half century ago by James Clerk Maxwell. The first Maxwell’s equation states that the source of rotor electric field is a magnetic induction alternating in time. Actually, this is just a part of the law written in the differential form. The second Maxwell’s equation is just Ampere’s circuital law which states that the source of rotary magnetic induction is electrical current described by the density J and electric field alternating in time. The third Maxwell’s equation states that the source of electric field are electrical charges described by density Rho. This is a Gauss’s law. And the last Maxwell’s equation is a Gauss’s law for magnetic induction, and it states that there is no magnetic charges in nature. The main property of electromagnetic field is that if we put a particle into electromagnetic field, the force will act on it. This force is called Lorentz force, or general Lorentz force, and this force has two parts: a part from the electric field and a part from the magnetic induction, where V is the velocity of the particle. This is a continuity equation for density of electrical current. These are the simple consequences of two of these equations. Electromagnetic field could transfer energy, and the flux of the energy is given by the Poynting vector which is given by the cross product of electric field and magnetic induction. The physical meaning of the absolute value of the Poynting vector is the amount of energy crossing the unit area per unit time. Electromagnetic field could also accumulate energy, and the density of the cumulated energy is given by this expression. Here we can see that the Maxwell’s equations contain two constants: epsilon zero and mu zero. These are vacuum permittivity and vacuum permeability. Electric field is measured in volts per meter, and magnetic induction is measured in tesla. Let’s discuss a couple of interesting facts about electric field and magnetic induction, about their magnitudes. The first example is our planet. Actually, the earth is a charged ball, and the total charge of our planet is about 100,000 Coulomb. This huge charge results in the appearance of electric field, of the static electric field in the vicinity of our planet. And the strength of this electric field is about 120 volts per meter. It means that the voltage between the top of my head and my foot is about 220 volts. It’s comparable to the voltage in the wall socket. But why don’t I feel this voltage? This is very simple: because actually I’m a conductor as you are. So, my skin has a potential. Beside the electric field our planet produces static magnetic field. The strength of this magnetic field is enough to rotate the arrow of a compass. Another example is microwave oven, we use it for warming food. When we put food inside, electromagnetic waves penetrate inside the food and tilt water molecules, and it results in heating. And because of this electric field is alternating in time, it results in the appearance of the magnetic field. The same story is with a wi-fi router. It also radiates electromagnetic waves. This is the value of strength of electric field and magnetic induction. Another example is an air breakdown, electrical breakdown. If you apply some voltage between two claddings, if you achieve enough value of electric field, you can observe electrical breakdown accompanied by a flash. And the voltage necessary to break down one centimeter of air is about 30,000 of volts. I think, the most strong magnetic field which you could meet in your daily life is a MRI machine. Usually a MRI machine has the strength of magnetic induction of about 1.5 tesla. But in some countries a MRI machine with 3 tesla is also allowed. But the strongest electric and magnetic field could be observed inside atom.

## 14.2 Maxwell’s Equations with Static Fields

Before we get to the time-varying fields, let us review the electrostatic model in which the fields have no time dependence. The electric field \(\vec E\) and magnetic field \(\vec B\) are uncoupled. The Maxwell’s equations that apply to electrostatics are

\[\begin{equation} \vec\bigtriangledown\times \vec E = 0 \tag{14.1} \end{equation}\]

\[\begin{equation} \vec\bigtriangledown\times \vec H = \vec J \tag{14.2} \end{equation}\]

The constitutive relation for \(\vec B\) and \(\vec H\) in linear and isotropic media is

\[\begin{equation} \vec H = \frac{1}{\mu}\vec B \tag{14.3} \end{equation}\]

\[\begin{equation} \vec\bigtriangledown\cdot \vec D = \rho \tag{14.4} \end{equation}\]

where for linear and isotropic (may not be homogeneous media) \(\vec D = \epsilon\vec E\).

\[\begin{equation} \vec\bigtriangledown\cdot \vec B = 0 \tag{14.5} \end{equation}\]

We may observe that \(\vec E\) and \(\vec D\) in the electrostatics are not coupled with \(\vec B\) and \(\vec H\) in the magnetostatic. However, in the conducting media, static electric field and static magnetic field exist and form an electromagnetostatic field. A static electric field in a conducting media creates steady current to flow and that creates a static magnetic field.

## 14.3 Maxwell’s Equations with Time-Varying Fields in Vacuum

\[\begin{equation} \vec\bigtriangledown\times \vec E =-\frac{\partial \vec B}{\partial t} \tag{14.6} \end{equation}\]

\[\begin{equation} \vec\bigtriangledown\times \vec B = \mu_0\vec J+\mu_0\epsilon_0\frac{\partial \vec E}{\partial t} \tag{14.7} \end{equation}\]

\[\begin{equation} \vec\bigtriangledown\cdot \vec E = \frac{\rho}{\epsilon_0} \tag{14.8} \end{equation}\]

\[\begin{equation} \vec\bigtriangledown\cdot \vec B = 0 \tag{14.9} \end{equation}\]

where the electric field is \(\vec E(\vec r,t)\) in V/m and magnetic induction, magnetic flux density (magnetic field) is \(\vec B(\vec r,t)\) in T. The permittivity of free space is \(\epsilon_0=8.85\times 10^{-12}\space F/m\) and permeability of free space is \(\mu_0=4\pi\times 10^{-7}\space H/m.\)

## 14.4 Maxwell’s Equations in Medium

\[\begin{equation} \vec\bigtriangledown\times \vec E = -\frac{\partial \vec B}{\partial t} \tag{14.10} \end{equation}\]

\[\begin{equation} \vec\bigtriangledown\times \vec H = \vec J_{ext}+\frac{\partial \vec D}{\partial t} \tag{14.11} \end{equation}\]

\[\begin{equation} \vec\bigtriangledown\cdot \vec D = \rho_{ext} \tag{14.12} \end{equation}\]

\[\begin{equation} \vec\bigtriangledown\cdot \vec B = 0 \tag{14.13} \end{equation}\]

where \(\vec H\) is the magnetic field and \(\vec D\) is the electric displacement.

\[\begin{equation} \vec D = \epsilon_0\vec E + \vec P \tag{14.14} \end{equation}\]

\[\begin{equation} \vec H = \vec B/\mu_0 \tag{14.15} \end{equation}\]

where \(\vec P\) is the polarization (electric dipole moment per unit volume) and \(\vec M\) is the magnetization (magnetic dipole moment per unit volume).

**Example 14-1**
Consider a gas in a static homogeneous electric field. Find the absolute value of polarization \(|\vec P|\) (in SI unit) if the absolute value of the average induced dipole moment of each molecule is $|p|=7^{−35} C⋅m and concentration of the molecules in the gas is \(n=3\times 10^{25} m^{-3}\).

**Solution:**

\[|\vec P| = n|\vec p | = (3\times 10^{25}~ m^{-3})(7\times 10^{−35} ~C⋅m) = 21\times 10^{-10}~C/m^2~(Answer)\]

## 14.5 Harmonic Oscillator Model:

Charges in a material are treated as harmonic oscillators

\[m\vec a_{el}=\vec F_{E,Local}+\vec F_{damping}+\vec F_{spring}\] \[m\frac{d^2r}{dt^2}+m\gamma\frac{dr}{dt}+kr = -eE_0e^{-i\omega t}\] \[\vec J = -ne\vec v\] \[\begin{equation} \frac{d\vec J}{dt}+\gamma \vec J = \left(\frac{Ne^2}{m_e}\right)\vec E \tag{14.16} \end{equation}\]

Applied electric field and the conduction current density:

\[\vec E = \vec E_0e^{-\omega t}\] and \[\vec J = \vec J_0e^{-\omega t}\] Substituting in Eq. 16, we get

\[\begin{equation} (-i\omega +\gamma)\vec J = \left(\frac{ne^2}{m_e}\right)\vec E \tag{14.17} \end{equation}\]

For static fields (\(\omega\) = 0) hence \[\vec J = \left(\frac{ne^2}{m_e\gamma}\right)\vec E\] and \[\vec E = \sigma\vec E\], where \(\sigma = \left(\frac{ne^2}{m_e\gamma}\right)\) is the static conductivity.

In an oscillating applied field:

\[\begin{equation} \vec J = \left[ \frac{\sigma}{1-(\frac{i\omega}{\gamma}}\right]\vec E = \sigma(\omega)\vec E \tag{14.18} \end{equation}\]

where \(\sigma(\omega)\) is the dynamic conductivity.

For very low frequencies, \(\frac{\omega}{\gamma}\ll 1\), the dynamic conductivity is purely real and the electrons follow the electric field.

As the frequency of the applied field increases, the inertia of electrons introduces a phase lag in the electron response to the field, and the dynamic conductivity is complex. For very high frequencies \(\frac{\omega}{\gamma}\gg 1\), the dynamic conductivity is purely imaginary and the electrons oscillations are \(\pi/2\) out of phase with the applied field.

## 14.6 Optical properties of metals

### 14.6.1 Drude model

Considering free electrons in the field of a plane electromagnetic wave is called Drude Model.

\[\begin{equation} m_e\ddot r = -eE_0e^{-\omega t} \tag{14.19} \end{equation}\]

\[\begin{equation} r(t) = \frac{e}{m_e\omega^2}E_0e^{-i\omega t} \tag{14.20} \end{equation}\]

Polarization:

\[\begin{equation} \vec P = n\vec p = -n|e|\vec r(t) = -\frac{ne^2}{m_e\omega^2}\vec E_0e^{-\omega t} \tag{14.21} \end{equation}\]

\[\begin{equation} \vec P = \epsilon_0\chi_e\vec E_0e^{-\omega t} \tag{14.22} \end{equation}\]

Equating Eq. 14-21 & Eq. 14-22, the electric susceptibility is

\[\begin{equation} \chi_e = -\frac{n|e|^2}{\epsilon_0m_e\omega^2} \tag{14.22} \end{equation}\]

\[\begin{equation} \mathcal{E}(\omega) = 1 + \chi_e=1-\frac{n|e|^2}{\epsilon_0m_e\omega^2}=1-\frac{\omega_p^2}{\omega^2} \tag{14.23} \end{equation}\]

\[\begin{equation} \omega_p = \sqrt\frac{n|e|^2}{\epsilon_0m_e} \tag{14.24} \end{equation}\]

is called the plasma freuency. The plasma frequency represents a plasma oscillation resonance or plasmon.