Solutions of the Equations and Their General Properties

The dipole potential, being the solution of the second Poisson equation in system (\ref{2xPoisson}), has the well-known form
\begin{equation} \label{phi_d}{\varphi_{d}=\frac{1}{\kappa}\frac{d_{x}(\omega)x+d_{y}(\omega)y+d_{z}(\omega)(z-h)}{(x^{2}+y^{2}+(z-h)^{2})^{\frac{3}{2}}}},\\ \end{equation}
where \(d_{x}\), \(d_{y}\), and \(d_{z}\) are the components of the dipole moment \(\mathbf{d}(\omega)\). Below, for all quantities depending on coordinates, the two-dimensional Fourier transformation is used:
\begin{equation} \label{phi_e}{\varphi_{e}(\mathbf{r},z)=\!\int\!d^{2}\!k\,\varphi_{k}(z)\,e^{i\mathbf{k}\mathbf{r}}},\\ \end{equation} \begin{equation} \label{n_e}{n(\mathbf{r})=\!\int\!d^{2}\!k\,n_{k}\,e^{i\mathbf{k}\mathbf{r}}\,,\,\,\,\,\,\psi(\mathbf{r})=\!\int\!d^{2}\!k\,\psi_{k}\,e^{i\mathbf{k}\mathbf{r}},}\\ \end{equation}
where \({\mathbf{r}=(x,y)}\), \({\mathbf{k}=(k_{x},k_{y})}\), and \({\psi(\mathbf{r})=\left.\varphi_{d}(\mathbf{r},z)\right|_{z=0}}\). The quantities \(\varphi_{k}(z)\), \(n_{k}\), and \(\psi_{k}\) are the Fourier transforms of the electron potential, surface concentration, and potential induced by the dipole in the quantum-well plane (\(z=0\)), respectively. Then, the Poisson equation for \(\varphi_{k}(z)\) and the required boundary conditions are
\begin{equation} \begin{cases}\displaystyle{\frac{d^{2}\varphi_{k}^{\pm}}{dz^{2}}-k^{2}\varphi_{k}^{\pm}=0},\\ {\left.\varphi_{k}^{\pm}\right|_{z\rightarrow\pm\infty}\rightarrow 0,}\\ \displaystyle{\left.\varphi_{k}^{+}\right|_{z=0+\varepsilon}=\left.\varphi_{k}^{-}\right|_{z=0-\varepsilon},}\\ \displaystyle{\left.\frac{d\varphi_{k}^{+}}{dz}\right|_{z=0+\varepsilon}-\left.\frac{d\varphi_{k}^{-}}{dz}\right|_{z=0-\varepsilon}=\frac{4\pi en_{k}}{\kappa},}\end{cases}\\ \end{equation}
where \(\varphi_{k}^{+}(z)=\varphi_{k}(z)\) at \(z\geq 0\) and \(\varphi_{k}^{-}(z)=\varphi_{k}(z)\) at \(z\leq 0\), \(k=\sqrt{k_{x}^{2}+k_{y}^{2}}\), and \(\varepsilon\rightarrow+0\).
The solution of system (\ref{kpoisson}) and the functions \(\psi_{k}\) and \(n_{k}\) are
\begin{equation} \begin{cases}\displaystyle{\varphi_{k}=-\frac{2\pi en_{k}}{\kappa k}e^{-k|z|}}\\ {\psi_{k}=-\frac{1}{2\pi\kappa}\left(i\frac{d_{x}k_{x}+d_{y}k_{y}}{k}+d_{z}\right)e^{-kh}}\\ \displaystyle{n_{k}=-\frac{en_{0}k^{2}}{m}\frac{\left.\varphi_{k}\right|_{z=0}+\psi_{k}}{(\omega-\mathbf{v_{0}k})(\omega-\mathbf{v_{0}k}+i/\tau_{p})}}.\end{cases}\\ \end{equation}
Note that, in the limiting case where the dipole is infinitely far from the 2DEG (i.e. \({h\rightarrow\infty}\)), solutions (\ref{3eq}) easily bring about the dispersion law for collective excitations in the drifting 2DEG, i.e. drifting plasmons,
\begin{equation} \label{plasma}{\omega_{\pm}=\mathbf{v_{0}k}\pm\sqrt{\frac{2\pi e^{2}n_{0}}{\kappa m}k-\frac{\gamma_{p}^{2}}{4}}-\frac{i\gamma_{p}}{2},}\\ \end{equation}
where the notation \({\gamma_{p}=1/\tau_{p}}\) is used. On the other hand, the zero of the denominator of \(\beta_{0}(\omega)\) (Eq. (\ref{beta0})) corresponds to the frequency and the damping of a dipole, \(\omega=\omega_{0}-i\gamma_{0}\).
If the distance \(h\) is finite, system (\ref{3eq}) and relations (\ref{dipole}) and (\ref{beta0}) can be used to derive an integral equation, e.g., for the Fourier transform of the concentration of electrons interacting with the dipole,
\begin{equation} \par {I_{x}\!=\!\!\!\int\!\!d^{2}q\frac{q_{x}}{q}n_{q}e^{-qh}},\,\displaystyle{I_{y}\!=\!\!\!\int\!\!d^{2}q\frac{q_{y}}{q}n_{q}e^{-qh}},\displaystyle{I_{z}\!=\!\!\!\int\!\!d^{2}qn_{q}e^{-qh}}.\par \nonumber \\ \end{equation} \begin{equation} \label{integal equation}{n_{k}=-\frac{e^{2}n_{0}}{\kappa^{2}m}\frac{\beta_{0}(\omega)ke^{-kh}(k_{x}I_{x}+k_{y}I_{y}+kI_{z})}{(\omega-\mathbf{v_{0}}\mathbf{k})(\omega-\mathbf{v_{0}}\mathbf{k}+i\gamma_{p})-\frac{2\pi e^{2}n_{0}k}{\kappa m}}},\\ \end{equation}
where \({I_{x}}\),\(\ {I_{y}}\), and \({I_{z}}\) are the functionals of \(n_{k}\),
\begin{equation} \par {B_{0}=-\frac{e^{2}n_{0}}{\kappa^{2}m}\beta_{0}(\omega)},\par \nonumber \\ \end{equation}
With the notations
\begin{equation} \par {\Delta_{e}(\omega,k)=(\omega-\mathbf{v}_{0}\mathbf{k})(\omega-\mathbf{v}_{0}\mathbf{k}+i\gamma_{p})-\frac{2\pi e^{2}n_{0}k}{\kappa m}},\par \nonumber \\ \end{equation}
the integral equation (7) yields the following system of algebraic equations:
\begin{equation} \par {S_{x}=\int\frac{d^{2}kk_{x}^{2}e^{-2kh}}{\Delta_{e}(\omega,k)}}\,,\displaystyle{S_{y}=\int\frac{d^{2}kk_{y}^{2}e^{-2kh}}{\Delta_{e}(\omega,k)}},\par \nonumber \\ \end{equation} \begin{equation} \begin{cases}\displaystyle{I_{x}=B_{0}(S_{x}I_{x}+S_{0}I_{z}),}\cr{I_{y}=B_{0}S_{y}I_{y},}\cr\displaystyle{I_{z}=B_{0}(S_{0}I_{x}+S_{z}I_{z}),}\end{cases}\\ \end{equation}
where the notations
\begin{equation} \label{Sj}{S_{z}=\int\frac{d^{2}kk^{2}e^{-2kh}}{\Delta_{e}(\omega,k)}},\,\displaystyle{S_{0}=\int\frac{d^{2}kk_{x}ke^{-2kh}}{\Delta_{e}(\omega,k)}}\,.\\ \end{equation}
are used for the calculable integrals. Note that the parameter \({S_{0}=0}\), if the electron drift is absent (\(\mathbf{v}_{0}=0\)).
The zero value of the determinant of the system of equations (8) is a condition for the nontrivial solutions of the integral equation (7) to exist. This condition, which determines the frequency \(\omega\), will be referred to as a dispersion equation. If the electron velocity is directed along the \(OX\) axis, the dispersion equation is
\begin{equation} \label{main dispersion equation}\left[(1-B_{0}S_{x})(1-B_{0}S_{z})-B_{0}^{2}S_{0}^{2}\right]\left[1-B_{0}S_{y}\right]=0.\\ \end{equation}
The dispersion equation (10) describes all possible collective oscillations of electrons and the dipole. If electrons are in the equilibrium state (\(\mathbf{v}_{0}=0\) and \(S_{0}=0\)), the dispersion equation can be decomposed into three equations: \((1-B_{0}S_{x})=0\), \((1-B_{0}S_{y})=0\), and \((1-B_{0}S_{z})=0\). The solutions of each of them correspond to different orientations of the induced dipole. It is evident that the solutions of equations (the frequencies of collective dipole and electron oscillations), which correspond to the \(x\)- and \(y\)-orientations of the dipole, are identical, because these two directions are physically equivalent in the absence of charge carrier drift. The frequency of collective oscillations for the \(z\)-orientation of the dipole is, generally speaking, different from those corresponding to the \(x\)- and \(y\)-orientations.
\begin{equation} \par {(1-B_{0}S_{x})(1-B_{0}S_{z})=B_{0}^{2}S_{0}^{2}}\,.\par \nonumber \\ \end{equation}
In the general case, Eq. (10) is factorized into two separate equations. One of them is the equation
\begin{equation} \par {B_{0}S_{y}=1}\par \nonumber \\ \end{equation}
One can easily verify that, for the frequencies, which are the solutions of this equation, the electric field of electrons has a symmetry, at which the induced dipole of a nanoparticle lies in the \(x-z\) plane, i.e. it corresponds to a mixed \(x-z\) orientation. The roots of this equation are two frequency branches. The branch, which is associated at \({v_{0}=0}\) with the \(x\)-orientation of the dipole (i.e. which is determined by the equation \({B_{0}S_{x}=1}\)), will be called the \(x\)-branch. Accordingly, the other branch will be called the \(z\)-branch. As it was in the equilibrium case (\({v_{0}=0}\)), the solutions of the equation
correspond to the orientation of the dipole in parallel to the \(OY\) axis for any \(v_{0}\)-value; therefore, the corresponding frequency branch will be referred to as the \(y\)-branch.
The solutions of the dispersion equation (10) can be presented in the form
\begin{equation} \label{dimensionless}\par {\Omega_{j}=\frac{\omega_{j}}{\omega_{pl}}},\quad\displaystyle{\omega_{pl}=\sqrt{\frac{2\pi e^{2}n_{0}}{\kappa mh}}},\quad V_{0}=\frac{v_{0}}{\omega_{pl}h}\,,\par \nonumber \\ \end{equation} \begin{equation} \label{frequency}\Omega_{j}=\Omega_{0}-i\Gamma_{0}+\frac{\Lambda}{\Omega_{0}}R_{j}(V_{0},\Omega_{j},\Gamma_{p}),\\ \end{equation}
where the subscript \(j=x,y,z\) denotes the \(x\)-, \(y\)-, and \(z\)-branches, respectively. We also used the notations
\begin{equation} \par \Omega_{0}=\frac{\omega_{0}}{\omega_{pl}},\quad\displaystyle{\Gamma_{0}=\frac{\gamma_{0}}{\omega_{pl}}},\quad\displaystyle{\Gamma_{p}=\frac{1}{2\gamma_{p}\omega_{pl}}},\par \nonumber \\ \end{equation} \begin{equation} \par {\Lambda=\frac{e^{2}e^{*^{2}}{n}_{0}|\langle 1|x|0\rangle|^{2}}{\kappa^{2}mh^{4}\hbar\omega_{pl}^{4}}}\,.\par \nonumber \\ \end{equation}
The parameter \(\Lambda\) is responsible for the coupling between the 2DEG and the dipole; and \({R_{j}(V_{0},\Omega_{j},\Gamma_{p})}\) is a certain complex-valued function depending on the electron drift velocity, the frequency of dipole oscillations, and the plasmon damping. For the \(x\)- and \(z\)-branches, the expressions for the functions \(R_{x,z}\) are
\begin{equation} \par {R_{y}=S_{y}}\,.\par \nonumber \\ \end{equation} \begin{equation} \label{dimensionless}{R_{x,z}=\!\frac{1}{2}\!\left[S_{x}\!+\!S_{z}\!\pm(S_{x}\!-\!S_{z})\sqrt{1+\!\left(\frac{2S_{0}}{S_{x}-S_{z}}\right)^{2}}\right],\!}\\ \end{equation}
where the plus sign corresponds to the \(x\)-branch, and the minus sign to the \(z\)-branch. It is easy to obtain a simpler expression for the \(y\)-branch,
The real part of this function, \({R_{j}^{\prime}}\), describes a shift of the oscillation frequency of the system with respect to the eigenfrequency of the dipole owing to the interaction between the dipole and electrons. The corresponding imaginary part, \({R_{j}^{\prime\prime}}\), is responsible for the additional damping (or growing) of oscillations in the hybrid system.
\begin{equation} \par {R_{x,y}}=\pi\mathcal{P}\!\!\int{dk\frac{k^{3}e^{-2k}}{\Omega_{x,y}^{2}-k}-i\pi^{2}\Omega_{x,y}^{6}e^{-2\Omega_{x,y}^{2}},}\par \nonumber \\ \end{equation}
From the analysis of integrals (\ref{Sj}), it follows that, in the extremely high-frequency case (\ref{crit}), the functions \({R_{j}}\) practically do not depend on the parameter \(\Gamma_{p}\). (In the hydrodynamic approximation, which is used in this work, the plasmon damping is exclusively governed by the one-particle relaxation of electrons. The Landau damping for plasmons, which arises, if the electron kinetics is considered \cite{Landau}, is assumed small.) In the absence of electron drift, it can be shown analytically that, if \({\Gamma_{p}\rightarrow 0}\),
\begin{equation} {R_{z}=2R_{x,y}}.\\ \end{equation}
The nonzero value of \({R_{j}^{\prime\prime}}\) and, respectively, the oscillation damping (analogous to the Landau damping \cite{Landau}) in the hybrid system originate from the collective interaction. Really, since the spectrum of plasmons (Eq. (\ref{plasma})) is continuous, for any given frequency \(\omega_{0},\) there can always be found “resonance” plasmons, the charge waves of which are proportional to \(\exp[i(\mathbf{k_{r}r}-\omega t)]\), where the wave vector \(\mathbf{k_{r}}\) satisfies the condition \(\omega^{\pm}(\mathbf{k_{r}})=\omega_{0}\). At the same time, the electric field of the dipole is a sum of Fourier components \(\mathbf{E}_{d,\mathbf{k}}\exp[i\mathbf{k}\mathbf{r}-i\omega_{0}t]\), with \({\mathbf{k}=\mathbf{k}_{r}}\) inclusive. The dipole field waves with the wave vectors \(\mathbf{k}_{r}\) and the “resonance” plasmons are cophased, and they propagate with the same velocity. This means that the corresponding dipole field acts permanently (without variation in time) on “resonance” plasmon charges. As a result, the work of the dipole field over the charges is nonzero, and the dipole loses its energy. Under nonequilibrium conditions, the dipole can acquire the energy from the electron subsystem. A necessary condition for that is
\begin{equation} \label{inst-cond-1}{R_{j}^{\prime\prime}(V_{0},\Omega_{j},\Gamma_{p})>0\,.}\\ \end{equation}
The sufficient condition for the hybrid system to be unstable in whole is more rigorous:
\begin{equation} \label{inst-cond-2}{\frac{\Lambda}{\Omega_{0}}R_{j}^{\prime\prime}(V_{0},\Omega_{j},\Gamma_{p})>\Gamma_{0}\,,}\\ \end{equation}
If the inverse inequality is obeyed, either the system is stable or the oscillations in the system attenuate.