Interaction between a Shallow Donor and Drifting Two-Dimensional Electrons

In this section, we apply the results obtained above to a specific hybrid system consisting of a shallow impurity center and a heterostructure with a quantum well.
It is well known that one-particle Coulomb impurities in semiconductors are characterized by low binding energies, and the allowed photo-induced dipole transitions between impurity states correspond to the THz spectral range. Such impurities can be regarded as hydrogen-like atoms, the energy spectrum and the wave functions of which can be calculated in the effective mass approximation (see, e.g., work \cite{Mitin}). The energy difference between the basic, \(1S\), and excited, \(2P\), states is evaluated as \({E_{S-P}={3e^{2}}/{8\kappa a_{B}}}\), where \({a_{B}=\kappa\hbar^{2}/me^{2}}\) is the radius in the ground state. For example, for GaAs with \({m=0.067m_{0}}\) and \({\,\kappa=12.9}\) (\(m_{0}\) is the free electron mass), we obtain \({E_{S-P}\approx 4.12~{}}\)meV and \({a_{B}\approx 10}\) nm. This energy difference corresponds to the frequency \({\omega_{0}\approx 6.2\times 10^{12}}\) s\({}^{-1}\) (\(\approx 0.99\) THz). The phototransition \({S\leftrightarrow P}\) is an allowed electric dipole transition with the transition matrix element \({\langle 1|x|0\rangle}\approx{0.52}\,{a_{B}^{\ast}}\). The cited parameters allow the polarizability of a Coulomb impurity to be calculated using relation (\ref{beta0}).
To achieve conditions needed for the instability excitation, a substance for the heterostructure must be selected, which would be characterized by high electron velocities. Consider a quantum well on the basis of InAs with GaAs-barriers. It is known [24] that the effective electron mass in InAs is small, \({m\approx 0.023m_{0}}\), and the electron mobility is high even at room temperature, \({\mu\approx 8\times 10^{4}}~{}\mathrm{cm}^{\mathrm{2}}/(\mathrm{V\times s})\). It gives rise to very high drift velocities of electrons, up to \({v_{0}\approx 6\times 10^{7}}\) \(\mathrm{cm/s}\) \cite{Kuchar,Krotkus}. In quantum wells on the basis of InGaAs, drift velocities of the same order are observed. The difference between the dielectric constants of the quantum well and the barrier can be neglected \cite{Kukhtaruk_UJP}. For numerical calculations, let us choose such physical parameters that criteria (\ref{crit}) are satisfied. In particular, let us fix the electron concentration \({n_{0}=10^{11}}~{}\mathrm{cm}^{-2}\) and the distance from the 2DEG to the donor \({h=4\times 10^{-6}}\) \(\mathrm{cm}\). The corresponding characteristic parameters, which were introduced by relations (\ref{dimensionless}), are \({\omega_{pl}\approx 1.07\times 10^{13}}\) s\({}^{-1}\), \({\Lambda\approx 0.0013}\), and \({\Gamma_{p}\approx 0.03}\). The drift velocity of charge carriers is normalized by the quantity \({\omega_{pl}h\approx 4.28\times 10^{7}}\) \(\mathrm{cm/s}\).