  We identify a possible mechanism of the plasmon instabilities in periodically
gated two-dimensional electron systems with a modulated electron density
(plasmonic crystals) under direct current. The instability occurs due to the
amplified reflection of the small density perturbations from the gated/ungated
boundaries under the proper phase matching conditions between the crystal unit
cells. Based on the transfer-matrix formalism, we derive the generic dispersion
equation for the travelling plasmons in these structures. Its solution in the
hydrodynamic limit shows that the threshold drift velocity for the instability
can be tuned below the plasmon phase and carrier saturation velocities, and the
plasmon increment can exceed the collisional damping rate typical to III-V
semiconductors at 77 K and graphene at room temperature.
