Behavior of the Dipole Induced in a Nanoparticle

Consider the behavior of the nanoparticle polarization during collective electric oscillations in the hybrid system. This behavior is described by the dynamics of the induced dipole moment (\ref{dipole}). To determine the dipole moment, we have to calculate the electric field created by electrons at the dipole localization point making use of definition (\ref{phi_e}), the first formula in system (\ref{3eq}), and the Fourier components of the concentration \(n_{k}\) (the latter were analyzed for various dispersion branches in the previous section).
First, let us consider the solutions corresponding to the dispersion \(x\)-branch, provided that the components of the dipole moment are
\begin{equation} \begin{cases}\displaystyle{d_{x}(\omega_{x})=-iC\frac{2\pi e\beta_{0}(\omega_{x})}{\kappa}\equiv\widetilde{C}_{x}},\cr\displaystyle{d_{y}(\omega_{x})=0,}\cr{d_{z}(\omega_{x})=i\widetilde{C}_{x}K_{x}},\end{cases}\\ \end{equation}
where \(\widetilde{C}_{x}\) is a new arbitrary constant. Whence, we obtain the law for the time evolution of dipole moment components,
\begin{equation} \begin{cases}\displaystyle{D_{x}=e^{\frac{\omega_{x}^{\prime\prime}}{\omega_{x}^{\prime}}\tau_{x}}(\widetilde{C}_{x}^{\prime}\cos\tau_{x}+\widetilde{C}_{x}^{\prime\prime}\sin\tau_{x}),}\cr\displaystyle{D_{z}=-e^{\frac{\omega_{x}^{\prime\prime}}{\omega_{x}^{\prime}}\tau_{x}}\left[(\widetilde{C}_{x}^{\prime}K_{x}^{\prime\prime}+\widetilde{C}_{x}^{\prime\prime}K_{x}^{\prime})\cos\tau_{x}+\right.}\cr\displaystyle{+\left.(\widetilde{C}_{x}^{\prime\prime}K_{x}^{\prime\prime}-\widetilde{C}_{x}^{\prime}K_{x}^{\prime})\sin\tau_{x}\right]}.\end{cases}\\ \end{equation}
System (\ref{dipole_x}) is a parametric equation of an ellipse: the end of the dipole moment vector moves along an elliptic trajectory in the plane \(y=0\). It is easy to verify that \({K_{x}^{\prime}>0}\) and \({K_{x}^{\prime\prime}>0}\) for the \(x\)-branch of the frequency dispersion; hence, this circulation is counter-clockwise. Since the dipole is in a self-consistent field with drifting electrons, the parameters of this ellipse depend on the drift velocity and the frequency of dipole oscillations.
Let us put the arbitrary constant \({\widetilde{C}_{x}=1}\) and, for simplicity, let \(\exp\left(\frac{\omega_{x}^{\prime\prime}}{\omega_{x}^{\prime}}\tau_{x}\right)\approx{1}\), i.e. we consider the behavior of the system within several periods. Then,
\begin{equation} \begin{cases}\displaystyle{D_{x}=\cos\tau_{x},}\cr\displaystyle{D_{z}=K_{x}^{\prime}\sin\tau_{x}-K_{x}^{\prime\prime}\cos\tau_{x}}.\end{cases}\\ \end{equation}
After excluding the parameter \(\tau_{x}\) from system (\ref{dipole_x-z}), we obtain the equation for this ellipse in the form
\begin{equation} \label{nonkanon}{aD_{x}^{2}+2bD_{x}D_{z}}+cD_{z}^{2}=1,\\ \end{equation}
where \({a=1+\left({K_{x}^{\prime\prime}}/{K_{x}^{\prime}}\right)^{2}}\), \({b={K_{x}^{\prime\prime}}/{K_{x}^{\prime^{2}}}}\), and \({c={1}/{K_{x}^{\prime^{2}}}}\). The rotation of the coordinate system by an angle \(\alpha_{x}\) determined by the equation
\begin{equation} \label{tg}{\tan\alpha_{x}=\frac{1-K_{x}^{\prime^{2}}-K_{x}^{\prime\prime^{2}}}{2K_{x}^{\prime\prime}}\pm\sqrt{\left(\frac{1-K_{x}^{\prime^{2}}-K_{x}^{\prime\prime^{2}}}{2K_{x}^{\prime\prime}}\right)^{2}+1}}\,,\\ \end{equation}
brings about the canonical ellipse equation,
\begin{equation} \par {A^{2}=\frac{1+\tg^{2}\alpha_{x}}{a+2b\tg\alpha_{x}+c\,\tg^{2}\alpha_{x}}},\par \nonumber \\ \end{equation} \begin{equation} \label{kanon}{\frac{D_{x_{0}}^{2}}{A^{2}}+\frac{D_{z_{0}}^{2}}{B^{2}}=1,}\\ \end{equation}
where \(D_{x_{0}}\) and \(D_{z_{0}}\) are the dipole components in the new coordinate system, and \(A\) and \(B\) are the principal ellipse axes,
\begin{equation} {B^{2}=\frac{1+\tg^{2}\alpha_{x}}{a\tg^{2}\alpha_{x}-2b\tg\alpha_{x}+c}}\,.\\ \end{equation}
The parameters \(A\) and \(B\) are used to introduce the ellipse eccentricity,
\begin{equation} {\epsilon=}\begin{cases}\displaystyle{\sqrt{1-\frac{B^{2}}{A^{2}}},\;\;A>B,}\\ \displaystyle{\sqrt{1-\frac{A^{2}}{B^{2}}},\;\;A<B}.\end{cases}\\ \end{equation}