  Adding a high-frequency ac component to the bias field of a superlattice
induces a synchronous modulation of the velocity with which the electrons
traverse the Brillouin zone. In the presence of inelastic scattering, the
k-space velocity modulation causes a wave-like bunching of the electrons in
k-space, which in turn introduces a high-frequency component in the real-space
current, synchronous with the drive field, but phase-shifted relative to the
latter. For a drive frequency equal to the Bloch frequency, the phase shift is
less than 90 degrees (implying a positive ac conductivity), increasing to 90
degrees in the limit of a vanishing scattering (a purely reactive current). If
the drive frequency is lowered below the Bloch frequency, and the inelastic
scattering rate remains sufficiently low, the phase shift can increase beyond
90 degrees, implying a negative ac conductivity, which peaks at a drive
frequency not far below the Bloch frequency.
