Starting from a microscopic tight-binding model and using second order perturbation theory, we derive explicit expressions for the intrinsic and Rashba spin-orbit interaction induced gaps in the Dirac-like low-energy band structure of an isolated graphene sheet. The Rashba interaction parameter is first order in the atomic carbon spin-orbit coupling strength ξ and first order in the external electric field E perpendicular to the graphene plane, whereas the intrinsic spin-orbit interaction which survives at E = 0 is second order in ξ. The spin-orbit terms in the low-energy effective Hamiltonian have the form proposed recently by Kane and Mele. Ab initio electronic structure calculations were performed as a partial check on the validity of the tight-binding model.

We study the gate-voltage induced gap that occurs in graphene bilayers using ab initio density functional theory. Our calculations confirm the qualitative picture suggested by phenomenological tight-binding and continuum models. We discuss enhanced screening of the external interlayer potential at small gate voltages, which is more pronounced in the ab initio calculations, and quantify the role of crystalline inhomogeneity using a tight-binding model self-consistent Hartree calculation.

We predict that neutral graphene bilayers are pseudospin magnets in which the charge densitycontribution from each valley and spin spontaneously shifts to one of the two layers. The band structure of this system is characterized by a momentum-space vortex which is responsible for unusual competition between band and kinetic energies leading to symmetry breaking in the vortex core. We discuss the possibility of realizing a pseudospin version of ferromagnetic metal spintronics in graphene bilayers based on hysteresis associated with this broken symmetry.

Because graphene is an atomically two-dimensional gapless semiconductor with nearly identical conduction and valence bands, graphene-based bilayers are attractive candidates for high-temperature electron-hole pair condensation. We present estimates which suggest that the Kosterlitz-Thouless temperatures of these twodimensional counterflow superfluids can approach room temperature.

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