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
\({\omega_{pl}h\approx 4.28\times 10^{7}}\) \(\mathrm{cm/s}\).