| Summary: | The motion of a quantum particle in a periodic potential can generate rich dynamics in the presence of a driving field. Such systems include, but are not limited to, semiconductor superlattices which exhibit a very anisotropic band structure that results into pronounced nonlinearities and high carrier mobility. In this thesis, we investigate the semiclassical dynamics and electron transport in a spatially periodic potential driven by a propagating wave. Firstly, we examine the transport features of an electron in a single miniband superlattice driven by a high-frequency acoustic plane wave. In this system, the nonlinear electron dynamics crucially depends on the amplitude of the acoustic wave. The transport characteristics are studied by means of a non-linearised kinetic model. In particular, to provide a realistic description of the directed transport, we employ the exact path-integral solutions of the Boltzmann transport equation. The calculated electron drift velocity and the time-averaged velocity show a nonmonotonic dependence upon the amplitude of the acoustic wave with multiple pronounced extrema. We found out that the changes in the velocity-amplitude characteristics are directly associated with a series of global bifurcations due to topological rearrangements of the phase space of the system. These dramatic transformations are connected with superlattice intraminiband transitions, and accompanied by inelastic emission (absorption) of the quantum particle. The bifurcations also signify the transitions between different dynamical regimes, involving unconfined electron motion, wave-dragging and phonon-assisted Bloch oscillations. Each regime has a characteristic spectral fingerprint, which manifests itself in appearance of specific high-frequency components in the spectra of the corresponding averaging trajectory. Secondly, we consider to use the acoustically pumped superlattices for an amplification of THz electromagnetic waves, involving the mechanisms similar to the Bloch gain in electrically biased superlattices. In particular, we predict the tunable THz gain due to nonlinear oscillations which are associated with the localised motion of electrons confined by a propagating potential wave. Traditionally, one of the key issues which emerges from considering different schemes for achieving small signal gain in superlattices, is the control of electric stability. Here, it is shown that for our case of the fast miniband electrons driven by an acoustic wave, terahertz gain can occur without the electric instability. Additionally, we find that the characteristic changes in the averaged velocities are connected to the shape of gain profiles. Consequently, the analytic findings, which determine the transitions between different dynamical regimes at the bifurcations, hold up for the behaviour of amplification of high-frequency electromagnetic waves. The increase of the miniband width, results in an enhancement of the effect of phase space restructuring on the drift velocity and high-frequency gain. Finally, we analyse the case for a superlattice device utilising acoustic waves with a very slow propagation speed. Benefiting from a simple solution of the Boltzmann equation, here we clarify the role of spatial nonlinearity both in miniband electron dynamics and in amplification of an electromagnetic wave. We show that nonlinear Bloch oscillations occur at a single critical value of the wave amplitude, inducing high negative differential drift velocity. Within this model, we also explain how the amplification of a high-frequency signal can arise below the threshold for an excitation of Bloch oscillations.
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