1. Permittivities of metals
Since surface plasmon polaritons can propagate along metal-dielectric interfaces the permittivity of metals play a huge role in the distribution of the fields. The frequently used mathematical models for the permittivities of materials are Lorentz model, Drude model and Extended Drude model. In this section we will first focus on the Lorentz model and then the Drude model.
1.1. Lorentz Model
In Lorentz model the electron is modeled as a damped harmonic oscillator. The damping term accounts for the fact that the electrons can loose energy by collisional processes. Here the motion of the nucleus is disregarded since it is massive compared to electrons. So we can start by writing the equation of motion for the electron.
\begin{equation}\label{eq:ElectronEOM}
m_0\frac{d^2x}{dt^2} + m_0\gamma\frac{dx}{dt} + m_0 \omega_0^2 x = -e \mathrm{E}
\end{equation}
Here $m_0$ is the mass of the electron, $\gamma$ is the damping rate, $e$ is the magnitude of the charge of the electron and $\mathrm{E}$ is the electric field of the incident electromagnetic wave. Mathematically this is a second-order, inhomogeneous ordinary differential equation. We also assume that the incident light is time harmonic, that is the electric field on the right hand side can be expressed as:
\begin{equation}\label{eq:Eharmonic}
\mathrm{E} = \mathrm{E_0}\cos(\omega t + \Phi) = \mathrm{E_0} \Re(\exp(-i\omega t – \Phi))
\end{equation}
Here $\mathrm{E_0}$ is the amplitude and $\Phi$ is the phase of the incident light which is the driving term in \eqref{eq:ElectronEOM}.
Here the assumption that the driving electric field will impose an oscillation of is own frequency is made, so the solutions sought are of the following form:
\begin{equation}\label{eq:EdispAnsatz}
x(t) = \Re(X_0 \exp(-i\omega t – \phi))
\end{equation}
Here $X_0$ is the complex amplitude of the oscillation of electrons. By plugging \eqref{eq:EdispAnsatz} into \eqref{eq:ElectronEOM} we get the following equation[1]
- The phase information is embedded in the amplitudes making them complex.}: \begin{equation}\label{eq:AlgebraicEquation} -m_0\omega^2 X_0 e^{-i\omega t} – i m_0 \gamma \omega X_0 e^{-i \omega t} + m_0 \omega_0^2 X_0 e^{-i\omega t} = -e \mathrm{E_0} e^{-i\omega t} \end{equation} Solving for $X_0$ yields: \begin{equation}\label{eq:ElectronDisplacement} X_0 = \frac{-e\mathrm{E_0}/m_0}{\omega_0^2-\omega^2-i\gamma\omega} \end{equation} This motion of electrons induces a microscopic, time dependent dipole moment p(t) which in turn yields a macroscopic polarization (dipole moment per unit volume) that can be incorporated in constitutive relations. The macroscopic polarization is related to the dipole moment via: \begin{gather} \begin{split} P_{induced} &= Np \\ &= -Nex \\ &= \frac{Ne^2}{m_0}\frac{1}{(\omega_0^2-\omega^2 – i \gamma \omega)} \mathrm{E} \end{split}\label{eq:Polarization} \end{gather} According to the above equation the magnitude of P is small unless the frequency is close to the resonance frequency $\omega_0$. In order to relate the polarization to the relative permittivity we make use of the relation between the electric displacement field $\mathbf{D}$\footnote{Bold face letters are used to denote vectors with three components in euclidean space.}, the electric field $\mathbf{E}$ and the polarization vector $\mathbf{P}$. With the assumption that we are dealing with an isotropic material this relation is as follows: \begin{gather} \begin{split} \mathbf{D} = \epsilon_0\mathbf{E} + \mathbf{P} &= \epsilon_0\mathbf{E} + \mathbf{P_{background}} + \mathbf{P_{induced}} \\ &= \epsilon_0 \epsilon_r \mathbf{E} = \epsilon_0 \mathbf{E} + \epsilon_0 \chi \mathbf{E} + \mathbf{P_{induced}} \end{split}\label{eq:dispol} \end{gather} Substituting \eqref{eq:Polarization} in \eqref{eq:dispol} yields the relative permittivity: \begin{equation}\label{eq:MetalPermittivity} \epsilon_r(\omega) = 1 + \chi + \frac{Ne^2}{\epsilon_0 m_0} \frac{1}{(\omega_0^2 – \omega^2 – i \gamma \omega)} \end{equation} 1.2. Drude Model The Drude model treats the metal as having immovable positive ions with a non-interacting free electron gas. Essentially this model is a special case of the Lorentz model with no restoring force hence the lattice potential and interaction of electrons among themselves are not taken into account [N/A]. The electrons are assumed to oscillate in response to the driving field. Collisions damped the motion of electrons and the collision frequency is denoted with $\gamma = 1/\tau$ from here on. $\tau$ is the relaxation time of the free electron gas, and it is on the order of $10^{-14}$ s at room temperature. So the equation of motion in the Lorentz model \eqref{eq:ElectronEOM} is simplified to: \begin{equation}\label{eq:DrudeEOM} m_0\frac{d^2x}{dt^2} + m_0\gamma\frac{dx}{dt} x = -e \mathrm{E} \end{equation} Solving for x(t), the displacement, results in: \begin{equation}\label{eq:DisplacementDrude} x(t) = \frac{e}{m_0(\omega^2+i\gamma\omega)}\mathrm{E(t)} \end{equation} $\mathbf{P_{background}}$ Using the same reasoning of deriving the electric permittivity from electric displacement field as before we have the following electric displacement vector \footnote{This time the inherent polarization is neglected.}: \begin{equation}\label{eq:DisplacementPolarizationDRUDE} \mathbf{D} = \epsilon_0\mathbf{E} + \mathbf{P} = \epsilon_0\mathbf{E} + \mathbf{P}_\text{free\ electrons} \end{equation} The permittivity follows from the above equation, \begin{equation} \begin{split} \epsilon(\omega) &= 1 – \frac{\omega^2_p}{\omega^2+i\gamma\omega} = 1 – \frac{\omega_p^2 \tau^2}{\omega^2\tau^2+i\omega\tau} \\ \epsilon_\text{real}(\omega) &= 1 – \frac{\omega_p^2 \tau^2}{1+\omega^2 \tau^2} \\ \epsilon_\text{imag}(\omega) &= \frac{\omega_p^2 \tau}{\omega(1+\omega^2\tau^2)} \end{split}\label{eq:DrudeMetal} \end{equation} Here for the notational convention $\omega_p = Ne^2/(\epsilon_0 m_0)$ is used. The behavior of \eqref{eq:DrudeMetal} can be examined under different frequency ranges. We start by restricting the discussion to frequencies below $\omega_p$, yet close enough to $\omega_p$ such that the product $\omega \gg \gamma$. In this frequency range the complex part of \eqref{eq:DrudeMetal} can be neglected. \begin{equation}\label{eq:DrudeWp} \epsilon(\omega) = 1 – \frac{\omega_p^2}{\omega^2} \end{equation} When the frequency is very low compared to the electron collision rate $\gamma$, that is $\omega \ll \gamma$. This is the frequency range in which metals are absorbing since $\epsilon_{imag} \gg \epsilon_{real}$ in \eqref{eq:DrudeMetal}. In this frequency range the real and complex refractive index calculated taking the square root of $\epsilon$ and they are similar in magnitude. \begin{equation}\label{eq:LowFreqMetal} n \approx \kappa = \sqrt{\frac{\epsilon_{imag}}{2}} = \sqrt{\frac{\tau \omega_p^2}{2\omega}} \end{equation} The power absorption coefficient is given as \begin{equation}\label{eq:PowerAbsorpCoef} \alpha = \kappa k_0 = \kappa \frac{2\pi}{\lambda} = \sqrt{\frac{2\omega_p^2 \tau \omega}{c^2 }} \end{equation} By using DC conductivity $\sigma_0 = Ne^2\tau / m_0 = \omega_p^2 \tau \epsilon_0$ this equation can be written as follows: \begin{equation}\label{eq:PowerAbsorpCoefDCcond} \alpha = \sqrt{2\sigma_0 \omega \mu_0} \end{equation} It is also useful to define a skin depth after which the fields decay by 1/e. \begin{equation}\label{eq:SkinDepth} \sigma = \frac{2}{\alpha} = \frac{c}{\kappa \omega} = \sqrt{\frac{2}{\sigma_0 \omega \mu_0}} \end{equation
References
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