The electronic band structure of graphene monolayer and bilayer in the presence of spin-orbit coupling and transverse electric field is analyzed emphasizing the roles of three complementary approaches: first-principles calculations, symmetry arguments and tight-binding approximation. In the case of graphene monolayer, the intrinsic spin-orbit coupling opens a gap of 24 µeV at the K(K´)-point. The dominant physical mechanism governing the intrinsic spin-orbit interaction originates from d and higher carbon orbitals. The transverse electric field induces an additional extrinsic (Bychkov-Rashba-type) splitting of typical value 10 µeV per V/nm. In the case of graphene bilayer; the intrinsic spin-orbit coupling splits the band structure near the K(K´)-point by 24 µeV. This splitting concerns the low-energy valence and conduction bands (two bands closest to the Fermi level). It is similar to graphene monolayer and is also attributed to d orbitals. An applied transverse electric field leaves the low-energy bands split by 24 µeV independently of the applied field, this is the interesting and peculiar feature of the bilayer graphene. The electric field, instead, opens a semiconducting band gap separating these low-energy bands. The remaining two high-energy bands are directly at K(K´)-point spin-split in proportion to the electric field; the proportionality coefficient is given by value 20 µeV. Effective tight-binding and spin-orbit hamiltonians describing graphene mono-and bi-layer near K point are derived from symmetry principles.
"Spin-orbit coupling in graphene structures", Proc. SPIE 8461, Spintronics V, 84610L (9 October 2012); doi: 10.1117/12.930947; https://doi.org/10.1117/12.930947