In Figs. 2 and 3, the dependences of the real, \(R_{j}^{\prime}\), and
imaginary, \(R_{j}^{\prime\prime}\), parts of the function \(R_{j}\) on the normalized
velocity \(V_{0}\) and frequency \(\Omega_{0}\), respectively, are depicted.
Figures 2,\(a\) and 3,\(a\) bring us to a conclusion that, for the selected fixed
parameters, the frequency of oscillations in the system is a little (in
comparison with \(\omega_{0}\)) shifted toward lower values in the frequency
\(x\)- and \(y\)-branches, and to both sides in the \(z\)-branch. Figure 2 also
demonstrates that \(R_{x}^{\prime\prime}\) changes its sign at \({V_{0}\approx 0.8}\), which corresponds to the drift velocity \({v_{0}\approx 3.4\times 10^{7}}\) \(\mathrm{cm/s}\). If condition (\ref{inst-cond-2}) is
also obeyed at that, an instability must emerge, and oscillations in the
system start to grow. The figure also illustrates that the \(x\)- and
\(y\)-branches merge together at low enough drift velocities, which is
associated with the equivalence of those directions at \({V_{0}\rightarrow 0}\).
Figure 3 testifies that the lower the frequency of dipole oscillations, the
larger is the instability increment. As to the \(y\)- and \(z\)-branches,
\({R_{y}^{\prime\prime}<0}\) and \({R_{z}^{\prime\prime}<0}\) in the considered
ranges of frequencies and drift velocities; therefore, the corresponding
oscillations in the system attenuate here.
As was shown in works \cite{Burghoon_1994,Kalkman_1996,Allen_2005}, the inverse lifetime of donor electrons at shallow
donors in GaAs can be of the order of \(10^{7}~{}\mathrm{s}^{-1}\). For
\({V_{0}\approx 0.94}\) and \({\Omega_{0}\approx 0.58}\), we obtain
\({\frac{\Lambda}{\Omega_{0}}R_{x}^{\prime\prime}\omega_{pl}\approx 5.28\times 10^{9}}~{}\mathrm{s}^{-1}\). In this case,
criterion (\ref{inst-cond-2}) is satisfied well, and the indicated
instability effects must be observed for the hybrid system
concerned. Note that the increase in the electron concentration or
the distance \(h\) leads to a reduction of the instability increment,
because \({\Lambda\propto n_{0}^{-1}}\) and \({\Lambda\propto h^{-2}}\).
We also verified that all effects discussed depend weakly on the
parameter \(\Gamma_{p}\).
Therefore, using an InAs quantum well and a shallow hydrogen-like donor in the
GaAs barrier as an example, we showed that an instability can take place in
the \(x\)-branch of collective oscillations in the system. Oscillations
corresponding to the \(y\) and \(z\) frequency branches attenuate.