  The study of spin-related phenomena in materials requires knowledge on the
precise form of effective spin-orbit coupling of conducting carriers in the
solid-states systems. We demonstrate theoretically that curvature induced by
corrugations or periodic ripples in single-layer graphenes generates two types
of effective spin-orbit coupling. In addition to the spin-orbit coupling
reported previously that couples with sublattice pseudospin and corresponds to
the Rashba-type spin-orbit coupling in a corrugated single-layer graphene,
there is an additional spin-orbit coupling that does not couple with the
pseudospin, which can not be obtained from the extension of the
curvature-induced spin-orbit coupling of carbon nanotubes. Via numerical
calculation we show that both types of the curvature-induced spin-orbit
coupling make the same order of contribution to spin relaxation in chemically
clean single-layer graphene with nanoscale corrugation. The spin relaxation
dependence on the corrugation roughness is also studied.
