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.