Intrinsic spin-orbit coupling (SOC) in graphene is dominated by the $d$ orbitals, and hence, effective three orbital models with $p_z$ and $d$ orbitals faithfully reproduce the nontrivial band structure of the low-energy bands. However, bending and deformation are commonly described in the literature in terms of the $p_z$- and $\sigma$-band orbitals, owing to the assumption that the rather empty $d$ orbitals have a marginal effect. We show in this work that the role of $d$ orbitals is critical not only to determine spin-orbit gaps but also to account for curvature-induced spin splitting. Here we employ a multiorbital tight-binding model including the $s$, $p$, and five $d$ orbitals of bent graphene nanoribbons with zigzag edges. We scrutinize the effects of the $p$ or $d$ orbitals on the intrinsic SOC splitting and on the curvature-induced Rashba-type SOC due to the onset of orbital hybridization. Whereas the former causes the edge spin to align in the direction perpendicular to the graphene plane, the latter imposes an in-plane quantization. We find that the critical curvature to observe this in-plane quantization is increased by more than 100% when the effect of the $d$ orbitals is included, highlighting their importance. Moreover, we encounter localized bound states under sinusoidal corrugation, owing to an effective bending-induced gauge potential, which leads to a pseudomagnetic field and subsequent Landau quantization. We stress that our findings highlight the importance of $d$ orbitals in the theoretical modeling of spintronic devices and experiments in graphene-based nanostructures.

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• Edge states
• Electronic structure
• Electronic structure of atoms & molecules
• Landau levels
• Mechanical properties of membranes
• Spin-orbit coupling
• Symmetry protected topological states
• Topological insulators
• Topological materials
• First-principles calculations