  We show that GHz acoustic waves in semiconductor superlattices can induce THz
electron dynamics that depend critically on the wave amplitude. Below a
threshold amplitude, the acoustic wave drags electrons through the superlattice
with a peak drift velocity overshooting that produced by a static electric
field. In this regime, single electrons perform drifting orbits with THz
frequency components. When the wave amplitude exceeds the critical threshold,
an abrupt onset of Bloch-like oscillations causes negative differential
velocity. The acoustic wave also affects the collective behavior of the
electrons by causing the formation of localised electron accumulation and
depletion regions, which propagate through the superlattice, thereby producing
self-sustained current oscillations even for very small wave amplitudes. We
show that the underlying single-electron dynamics, in particular the transition
between the acoustic wave dragging and Bloch oscillation regimes, strongly
influence the spatial distribution of the electrons and the form of the current
oscillations. In particular, the amplitude of the current oscillations depends
non-monotonically on the strength of the acoustic wave, reflecting the
variation of the single-electron drift velocity.
