Spin- and valley-dependent commensurability oscillations and electric-field-induced quantum Hall plateaux in periodically modulated silicene

We study the commensurability oscillations in silicene subject to a perpendicular electric field Ez, a weak magnetic field B, and a weak periodic potential V=V0cos(Cy),C=2π/a0 with a0 its period. The field Ez and/or the modulation lift the spin degeneracy of the Landau levels and lead to spin and valley resolved Weiss oscillations. The spin resolution is maximal when the field Ez is replaced by a periodic one Ez=E0cos(Dy),D=2π/b0, while the valley one is maximal for b0 = a0. In certain ranges of B values, the current is fully spin or valley polarized. Additional quantum Hall conductivity plateaux arise due to spin and valley intra-Landau-level transitions.