Energy spectrum and Landau levels in bilayer graphene with spin-orbit interaction

We present a theoretical study of the bandstructure and Landau levels in bilayer graphene at low energies in the presence of a transverse magnetic field and Rashba spin-orbit interaction in the regime of negligible trigonal distortion. Within an effective low energy approach (L\"owdin partitioning theory) we derive an effective Hamiltonian for bilayer graphene that incorporates the influence of the Zeeman effect, the Rashba spin-orbit interaction, and inclusively, the role of the intrinsic spin-orbit interaction on the same footing. Particular attention is spent to the energy spectrum and Landau levels. Our modeling unveil the strong influence of the Rashba coupling $\lambda_R $ in the spin-splitting of the electron and hole bands. Graphene bilayers with weak Rashba spin-orbit interaction show a spin-splitting linear in momentum and proportional to $\lambda_R $, but scales inversely proportional to the interlayer hopping energy $\gamma_1$. However, at robust spin-orbit coupling $\lambda_R $ the energy spectrum shows a strong warping behavior near the Dirac points. We find the bias-induced gap in bilayer graphene to be decreasing with increasing Rashba coupling, a behavior resembling a topological insulator transition. We further predict an unexpected assymetric spin-splitting and crossings of the Landau levels due to the interplay between the Rashba interaction and the external bias voltage. Our results are of relevance for interpreting magnetotransport and infrared cyclotron resonance measurements, including also situations of comparatively weak spin-orbit coupling.