Stable forms of two-dimensional crystals and graphene

We show that the "two-dimensional" graphene is stable due to transverse short-range displacements of carbon atoms, which may be described in a framework of Ising model with competing interactions. When temperature decreases, two transitions, high temperature disorder into order and order into low-temperature glass, arise. The graphene looks like a microscopic "washboard" with the wavelength of about 2–4 Å. Due to up–down asymmetry of the lattice distortions in graphene on substrate, a mini-bandgap arises. This leads to many new phenomena: a rectification of AC current induced by microwave or infrared radiation, the existence of self-trapping and a new type of fermionic mini-exciton-polaritons.