The \({\epsilon}\)-magnitude varies from zero to 1. The ellipse transforms
into a circle at \({\epsilon\rightarrow 0}\) and into a line at \({\epsilon\rightarrow 1}\). The eccentricity of the ellipse, along which the dipole
circulates, evidently depends on the electron drift velocity and the frequency
of dipole oscillations, because the principal axes depend on \(K_{x}\).
Note that the time dependence of the dipole is very simple in the coordinate
system coupled with the principal axes, namely,
\begin{equation}
\begin{cases}\displaystyle{D_{x_{0}}=A\cos\tau_{x},}\cr\displaystyle{D_{z_{0}}=B\sin\tau_{x}.}\end{cases}\\
\end{equation}
Similar formulas can also be derived for solutions associated with
the dispersion \(z\)-branch. In this case, the dipole also circulates
along an ellipse, but now “clockwise”. The principal axes of this ellipse
depend on \(K_{z}(\omega_{z})\). To obtain the canonical ellipse
equation for the \(z\)-branch, the substitutions
\({a\leftrightarrow c}\), \({b\rightarrow-b}\), and \({K_{x}(\omega_{x})\rightarrow-K_{z}(\omega_{z})}\) should be made in the equations written
above. Now, the eccentricity is described by expression (\ref{eccentricity}),
but the principal axes swap (\({A\leftrightarrow B}\)), and the angle
\({\alpha_{x}}\) changes to \({\alpha_{z}}\). The eccentricity is the same for
both dispersion branches, because \({K_{x}(\Omega_{x})\simeq-K_{z}(\Omega_{z})}\).
Concerning the \(y\)-orientation of the dipole, the eccentricity of the corresponding
ellipse is equal to 1 for all \(V_{0}\), which is associated with the dipole
orientation strictly along the \(OY\) axis for any \(V_{0}\).
The dependences of eccentricity \(\epsilon\) on the electron drift velocity for
three different dipole oscillation frequencies are depicted in Fig. 9,\(a\). The
solid curve exposes the eccentricity that corresponds to the same frequency as
in all previous figures, where this frequency was fixed, i.e. \({\Omega_{0}\approx 0.58}\). Two other curves correspond to half as high and twice as
high dipole oscillation frequencies. The figure testifies that the
eccentricity is equal to 1, if the drift velocity equals zero, i.e. the
dipole oscillates with the frequency \(\Omega_{x}\) along the \(OX\) axis and with
the frequency \(\Omega_{z}\) along the \(OZ\) axis. Figure 9,\(b\) shows that the
lines, along which the dipole oscillates, are perpendicular to each other at
\({V_{0}=0}\). If \(V_{0}\) increases, the eccentricity decreases, and the line
transforms into an ellipse. The ellipse, which corresponds to the \(x\)-branch,
rotates “clockwise”; therefore, the absolute
value of the angle, by which its axes are rotated with respect to the \(OX\)
axis, grows. The other ellipse rotates “counter-clockwise”; therefore, the absolute value of its
orientation angle decreases. At a certain drift velocity, those ellipses coincide.
A similar situation takes place at other dipole oscillation frequencies as
well. The curves in Fig. 9 also demonstrate that the eccentricities have a
similar structure at different frequencies, and the smallest \(\epsilon\)-value
is realized for a curve that corresponds to \({\Omega_{0}\approx 0.58}\). At this
frequency, the function \(R_{j}\), the plot of which is exhibited in Fig. 2,
changes drastically in a vicinity of the drift velocity, which corresponds to
the eccentricity minimum (\({V_{0}\approx 0.43}\)). The plots of the functions
\(\alpha_{x}\) and \(\alpha_{z}\) intersect each other also in a vicinity of
\({V_{0}\approx 0.43}\). At other frequencies, such eccentricity minima take
place at different drift velocities, and the behavior of the function \(R_{j}\)
in vicinities of those minima (to be exact, in the region, where the
eccentricity diminishes sharply) is also drastic.
Let us elucidate the physical reason for the nontrivial behavior
demonstrated by the induced nanoparticle dipole. For this purpose,
consider the electric field of plasmons at a certain point
\(\left(x=y=0,z=h\right)\) remote from the 2DEG. The dispersion law
(\ref{plasma}) has two branches. However, for simplicity, let us
consider any of them, omitting the subscript at the frequency and
neglecting the plasmon damping. Let a plasmon with the frequency
\(\omega(k)\) and the two-dimensional wave vector \(\mathbf{k}=(k,0)\)
be excited. Then, at the distance \(h\) from the coordinate origin, it
creates a field
\begin{eqnarray}
\begin{cases}\displaystyle{E_{x}=-ik|A_{0}|e^{-kh}e^{-i\omega t}},\cr\displaystyle{E_{z}=k|A_{0}|e^{-kh}e^{-i\omega t}}\,,\end{cases}\nonumber \\
\end{eqnarray}
where \(\left|A_{0}\right|\) is an arbitrary amplitude. Only
the real part of the found field has a physical meaning. Its
components are
\begin{eqnarray}
\label{Replasmafield}\begin{cases}\displaystyle{E^{\prime}_{x}=-k|A_{0}|e^{-kh}\sin{(\omega t)}},\cr\displaystyle{E^{\prime}_{z}=k|A_{0}|e^{-kh}\cos{(\omega t)}}.\end{cases}\nonumber \\
\end{eqnarray}
We see that, at a certain distance from the coordinate origin, the
field created by plasmons circulates along a circle. It is evident
that the dipole moment of a nanoparticle, which is given by formula
(\ref{dipole}), when responding to an external force, will also
circulate under the influence of the plasmon field. When collective
oscillations take place in the hybrid system, not a single but
many plasmons with different \(\mathbf{k}\) are excited. The
total field has a more complicated time dependence, which results in
the dipole circulation along an ellipse. Hence, the dynamics of
the plasmon field is a key for understanding the behavior of
the nanoparticle polarizability.
It is important that the behavior of the induced dipole in time can be
experimentally observed owing to features of the emission
by the hybrid system. Really, having solved the problem of
an electrostatically coupled nanoparticle and the two-dimensional electron
gas, we determined the motion of charges in the hybrid system. The
motions of polarization charges in the nanoparticle and electrons
in the quantum well correspond to an electric current depending
on the coordinates and the time. The alternating current generates
radiation. Let the current density be designated as
\(\mathbf{j}(x,y,z,t)\). Then, the vector potential of the radiation
field can be determined by the
formula \cite{Landau_2}
\begin{equation}
\label{A}{\mathbf{A}(x,y,z,t)=\frac{1}{c_{0}}\int{\!dx_{1}dy_{1}dz_{1}\frac{\mathbf{j}(x_{1},y_{1},z_{1},t-\frac{R}{c_{0}})}{R}}},\\
\end{equation}
where \(c\) is the speed of light, and \(R=[(x_{1}-x)^{2}+(y_{1}-y)^{2}+(z_{1}-z)^{2}]^{1/2}\). Using formula (\ref{A}) for the far radiation zone,
we obtain the Fourier components of the vector potential in the form
\begin{equation}
\label{A_d}{\mathbf{A}(x,y,z,\omega_{j})=-\frac{ik_{0}\mathbf{d}(\omega_{j})e^{-i\omega_{j}t+ik_{0}R_{0}}}{R_{0}}}\,,\\
\end{equation}
where \(R_{0}=[x^{2}+y^{2}+(z-h)^{2}]^{1/2}\) is the distance to
the observation point, and \({k_{0}=\omega_{j}/c_{0}}\). The frequency
\(\omega_{j}\) belongs to one of three frequency branches. The dipole
moment, which determines \(\mathbf{A}\), is formed by the
self-consistent system composed of the electrons and the
nanoparticle. Hence, the radiation characteristics substantially
depend on such parameters of the system as the distance between the
dipole and the quantum well, the concentration of electrons, the
frequency of dipole oscillations, and the electron drift velocity.
At finite drift velocities of electrons, the induced dipole
circulates along an ellipse, and this fact must be reflected in the
radiation polarization.