Model and Basic Equations

Consider a hybrid system represented schematically in Fig. 1. It consists of a spherical nanoparticle, which can be polarized in an external field, and the heterostructure with a quantum well, in which two-dimensional electrons are localized. The distance between the nanoparticle and the quantum well is \(h\). The electrons and the nanoparticle interact by means of the electrostatic field \(\mathbf{E}=-\mathbf{\nabla}\phi\), where \(\phi(x,y,z,t)\) is a self-consistent electrostatic potential. The latter is described by the Poisson equation
\begin{equation} \label{Poisson}\frac{\kappa}{4\pi}\Delta\phi=e({n}-n_{0})\delta(z)+\left({\ \mathbf{D}}(t)\mathbf{\nabla}\right)\delta(x)\delta(y)\delta(z-h),\\ \end{equation}
where \(\kappa\) is the dielectric permittivity, \(e\) the electron charge, \({n}(x,y,t)\) the surface concentration of electrons, \(n_{0}\) the equilibrium surface concentration of electrons, \(\mathbf{D}(t)\) the electric dipole moment of a nanoparticle, and \(\delta(x)\) the Dirac delta-function.
It is convenient to express the scalar potential as a sum of two terms associated with the electrons and the NP, \({\phi(x,y,z,t)=\phi_{e}(x,y,z,t)+\phi_{d}(x,y,z,t)}\), where \(\phi_{e}\) and \(\phi_{d}\) are determined from the system of equations
\begin{equation} \label{2xPoisson}\begin{cases}\displaystyle{\kappa\Delta\phi_{e}=4\pi e({n}-n_{0})\delta(z),}\cr\displaystyle{\kappa\Delta\phi_{d}=4\pi\left({\mathbf{D}}(t)\mathbf{\nabla}\right)\delta(x)\delta(y)\delta(z-h).}\end{cases}\\ \end{equation}
To describe electrons in the quantum well (the plane \(z=0\)), we use Euler’s equation of motion and the continuity equation, which couple the surface concentration \({n}(x,y,t)\) and the electron velocity \(\mathbf{\mathfrak{v}}(x,y,t)\) with the electrostatic field. In the general case, we assume that electrons move with an average drift velocity \(\mathbf{v_{0}}\). Then, the equations are \cite{Dyakonov}
\begin{equation} \label{hydrodynamics}\begin{cases}\displaystyle{\frac{\partial\mathbf{\mathfrak{v}}}{\partial t}+\left({\mathbf{\mathfrak{v}}}{\mathbf{\nabla}}_{\parallel}\right){\mathbf{\mathfrak{v}}}=\left.\frac{e}{m}\mathbf{\nabla}\phi\right|_{z=0}-\frac{\mathbf{\mathfrak{v}-v_{0}}}{\tau_{p}}}\,,\cr\displaystyle{\frac{\partial{n}}{\partial t}+{\mathbf{\nabla}}_{\parallel}\left(\mathbf{\mathfrak{v}}{n}\right)=0}\,,\end{cases}\\ \end{equation}
where \(m^{\ast}\) is the effective mass of an electron, the term \({\frac{\mathbf{\mathfrak{v}-v_{0}}}{\tau_{p}}}\) makes allowance for the electron scattering by crystal defects, and \(\tau_{p}\) is the time of electron momentum relaxation. The subscript \(\parallel\) is used for quantities and differential operators dealing with the plane \(z=0\) only. For instance, the Laplace operator in the plane \(z=0\) is \({\Delta_{\parallel}=\frac{\partial^{2}}{\partial x^{2}}+\frac{\partial^{2}}{\partial y^{2}}}\). At the same time, the Laplace operator in the volume is \({\Delta=\Delta_{\parallel}+\frac{\partial^{2}}{\partial z^{2}}}\). We will also mark the real and imaginary parts of complex numbers as primed quantities, e.g., \({Z=Z^{\prime}+iZ^{\prime\prime}}\).
In the two-dimensional electron gas, there emerge collective oscillations of the charge density, which are called plasmons. The hydrodynamic model with Eqs. (\ref{hydrodynamics}) is valid, when the frequency of these oscillations, \(\omega\), and the absolute value of the wave vector, \(k\), satisfy the conditions
\begin{equation} \label{crit}\omega\tau_{p}\gg 1,\,\,ql_{p}\sim l_{p}/h\gg 1\,,\\ \end{equation}
where \(l_{p}\) is the mean free path of electrons. The formulation of the problem includes only one characteristic length scale; it is the distance \(h\) between the plane of the 2DEG and the NP. Therefore, one may expect that the wave vectors of plasmons, which make the major contribution to the interaction between the electrons and the NP, should be of the order of \(1/h\). Note that inequalities (\ref{crit}) correspond to the ballistic character of the electron motion in the spatial region, which is actual for the nanoparticle–plasmon interaction.
Suppose that the velocity of electrons is \({\mathbf{\mathfrak{v}}(x,y,t)=\mathbf{v_{0}}+\mathbf{v}(x,y)e^{-i\omega t}},\) and their concentration \({{n}(x,y,t)=n_{0}+n(x,y)e^{-i\omega t}}\), with \({v\ll v_{0}}\) and \({n\ll n_{0}}\). Let the time dependences of potentials also vary harmonically, i.e. \({\phi_{e}(x,y,z,t)=\varphi_{e}(x,y,z)e^{-i\omega t}}\) and \({\phi_{d}(x,y,z,t)=\varphi_{d}(x,y,z)e^{-i\omega t}}\). The frequency \(\omega\) in these formulas should be determined by solving the equations; generally speaking, it is a complex-valued quantity, which corresponds to the oscillations (damped or growing in time) in the hybrid system. (If \({\omega^{\prime\prime}>0}\), the system is unstable, and \({\omega^{\prime\prime}}\) is called the instability increment.) The quantities \(\mathbf{v}(x,y)\) and \(n(x,y)\) describe the spatial dependences of electron velocity and concentration perturbations, respectively.
\begin{equation} \par {\mathbf{D}(t)=\mathbf{d}(\omega)e^{-i\omega t},}\par \nonumber \\ \end{equation}
According to Fig. 1, the nanoparticle is located at the point \(\left(x=0,y=0,z=h\right)\). In an external electric field, it becomes polarized and can be characterized by an electric dipole moment, which is determined by the relation
\begin{equation} \label{dipole}{\mathbf{d}(\omega)=-\left.\beta_{0}(\omega)\mathbf{\nabla}\varphi_{e}\right|_{x=y=0,z=h},}\\ \end{equation}
where \(\beta_{0}(\omega)\) is the nanoparticle polarizability. Note that formula (\ref{dipole}) includes only the electron potential and, hence, the dipole self-action is excluded from consideration. If the nanoparticle is isotropic, its polarizability can be expressed in the standard form \cite{Davydov}
\begin{equation} \label{beta0main}{\beta_{0}(\omega)=-\sum\limits_{f}\frac{2\omega_{f}e^{\ast^{2}}}{\hbar}\,\frac{|\langle f|x|0\rangle|^{2}}{\omega^{2}-\omega_{f}^{2}+i\omega/\tau_{f}}}\,,\\ \end{equation}
where \(\hbar\) is the reduced Planck constant, \(\hbar\omega_{f}\) is the energy of the \(f\)-th nanoparticle level, \(\tau_{f}\) is the lifetime of this level, \(\langle f|x|0\rangle\) is the matrix element of the dipole transition between the ground and \(f\)-th excited states.
The largest contribution to polarizability (\ref{beta0main} ) is given by the transition between the ground and the first excited state. In this case, the polarization is
\begin{equation} \label{beta0}{\beta_{0}(\omega)=-\frac{e^{\ast^{2}}}{\hbar}\,\frac{|\langle 1|x|0\rangle|^{2}}{\omega-\omega_{0}+i\gamma_{0}}}\,,\\ \end{equation}
where the characteristic frequency of dipole oscillations, \({\omega_{0}}\), and the damping of dipole oscillations corresponding to the transition between the ground and the first excited state, \({\gamma_{0}=1/(2\tau_{1})}\), are introduced. While deriving formula (\ref{beta0}), an assumption was made that \(\omega\) is close to \(\omega_{0}\), because the main effects are expected to take place at frequencies close to the characteristic frequency of dipole oscillations.
Equations (\ref{Poisson})–(\ref{hydrodynamics}), (\ref{dipole}), and (\ref{beta0}) compose a basic system for the problem under consideration.