Abstract
We present a theoretical study of the band structure 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 the (Löwdin 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 paid to the energy spectrum and Landau levels. Our modeling unveils the strong influence of the Rashba coupling λR 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 λR, but scaling inversely proportional to the interlayer hopping energy γ1. However, at robust spin–orbit coupling λR, the energy spectrum shows a strong warping behavior near the Dirac points. We find that the bias-induced gap in bilayer graphene decreases with increasing Rashba coupling, a behavior resembling a topological insulator transition. We further predict an unexpected asymmetric 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 situations of comparatively weak spin–orbit coupling.
Highlights
0, in addition to two non-degenerate modes at E1μ = μ 1 + 2 ̃ 2
We show within low-energy effective theory that for biased bilayer graphene (BLG) in which the Rashba effect is the dominant
Hamiltonian for BLG that includes both types of spin–orbit interaction (SOI) and the Zeeman effect is derived within the Lowdin partitioning theory
Summary
Using the Lowdin partitioning theory [43, 44], the full 8 × 8 Hamiltonian HK can be projected through a canonical transformation [45] into a 4 × 4 low-energy effective Hamiltonian H in an appropriate basis (see appendix A). Rashba-SOI (λR = 0), equation (12) decouples to the usual effective BLG Hamiltonian obtained within low-energy theory in the absence of trigonal warping effects [13]. Such a term gives rise to the well-known parabolic spectrum of the massive Dirac quasiparticles in BLG. The second term in H(1) is linear in momentum and can be viewed as a renormalization of the Rashba coefficient due to the presence of the higher bands. Note that it scales as the inverse of the interlayer hopping energy γ1.
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