Theory of spin-orbit coupling in bilayer graphene is presented. The
electronic band structure of the AB bilayer in the presence of spin-orbit
coupling and a transverse electric field is calculated from first-principles
using the linearized augmented plane wave method implemented in the WIEN2k
code. The first-principles results around the K points are fitted to a
tight-binding model. The main conclusion is that the spin-orbit effects in
bilayer graphene derive essentially from the single-layer spin-orbit coupling
which comes almost solely from the d orbitals. The intrinsic spin-orbit
splitting (anticrossing) around the K points is about 24\mu eV for the
low-energy valence and conduction bands, which are closest to the Fermi level,
similarly as in the single layer graphene. An applied transverse electric field
breaks space inversion symmetry and leads to an extrinsic (also called
Bychkov-Rashba) spin-orbit splitting. This splitting is usually linearly
proportional to the electric field. The peculiarity of graphene bilayer is that
the low-energy bands remain split by 24\mu eV independently of the applied
external field. The electric field, instead, opens a semiconducting band gap
separating these low-energy bands. The remaining two high-energy bands are
spin-split in proportion to the electric field; the proportionality coefficient
is given by the second intrinsic spin-orbit coupling, whose value is 20\mu eV.
All the band-structure effects and their spin splittings can be explained by
our tight-binding model, in which the spin-orbit Hamiltonian is derived from
symmetry considerations. The magnitudes of intra- and interlayer
couplings---their values are similar to the single-layer graphene ones---are
determined by fitting to first-principles results.Comment: 16 pages, 13 figures, 5 tables, typos corrected, published versio
