  Moir\'e-superlattices are ubiquitous in 2D heterostructures, strongly
influencing their electronic properties. They give rise to new Dirac cones and
are also at the origin of the superconductivity observed in magic-angle bilayer
graphene. The modulation amplitude (corrugation) is an important yet largely
unexplored parameter in defining the properties of 2D superlattices. The
generally accepted view is that the corrugation monotonically decreases with
increasing twist angle, while its effects on the electronic structure diminish
as the layers become progressively decoupled. Here we found by lattice
relaxation of around 8000 different Moir\'e-superstructures using high scale
Classical Molecular Simulations combined with analytical calculations, that
even a small amount of strain can substantially change this picture, giving
rise to more complex behavior of superlattice corrugation as a function of
twist angle. One of the most surprising findings is the emergence of an
ultra-flat phase that can be present for arbitrary small twist angle having a
much lower corrugation level than the decoupled phase at large angles. A
possible experimental realization of the ultra-flat state is revealed by
Scanning Tunneling Microscopy (STM) investigations of the graphene/graphite
system.
