  We study theoretically the properties of buckled graphene-like materials,
such as silicene and germanene, in a strong perpendicular magnetic field and a
periodic potential. We analyze how the spin-orbit interaction and the
perpendicular electric field influences the energy spectra of these systems.
When the magnetic flux through a unit cell of the periodic potential measured
in magnetic flux quantum is a rational number, ? = p/q, then in each Landau
level the energy spectra have a band structure, which is characterized by the
corresponding gaps. We study the dependence of those gaps on the parameters of
the buckled graphene-like materials. Although some gaps have weak dependence on
the magnitude of the spin-orbit coupling and the external electric field, there
are gaps that show strong nonomonotic dependence on these parameters. For ? =
1/2, the spin-orbit interaction also opens up a gap at one of the Landau
levels. The magnitude of the gap increases with spin-orbit coupling and
decreases with the applied electric field.
