Edge-spin tilt and bound states in curved multiorbital graphene nanoribbons: Dominant role of d bands

Intrinsic spin-orbit coupling (SOC) in graphene is dominated by the <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"> <a:mi>d</a:mi> </a:math> orbitals, and hence, effective three orbital models with <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"> <b:msub> <b:mi>p</b:mi> <b:mi>z</b:mi> </b:msub> </b:math> and <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"> <c:mi>d</c:mi> </c:math> 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 <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"> <d:msub> <d:mi>p</d:mi> <d:mi>z</d:mi> </d:msub> </d:math> - and <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"> <e:mi>σ</e:mi> </e:math> -band orbitals, owing to the assumption that the rather empty <f:math xmlns:f="http://www.w3.org/1998/Math/MathML"> <f:mi>d</f:mi> </f:math> orbitals have a marginal effect. We show in this work that the role of <g:math xmlns:g="http://www.w3.org/1998/Math/MathML"> <g:mi>d</g:mi> </g:math> 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 <h:math xmlns:h="http://www.w3.org/1998/Math/MathML"> <h:mi>s</h:mi> </h:math> , <i:math xmlns:i="http://www.w3.org/1998/Math/MathML"> <i:mi>p</i:mi> </i:math> , and five <j:math xmlns:j="http://www.w3.org/1998/Math/MathML"> <j:mi>d</j:mi> </j:math> orbitals of bent graphene nanoribbons with zigzag edges. We scrutinize the effects of the <k:math xmlns:k="http://www.w3.org/1998/Math/MathML"> <k:mi>p</k:mi> </k:math> or <l:math xmlns:l="http://www.w3.org/1998/Math/MathML"> <l:mi>d</l:mi> </l:math> 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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>d</m:mi> </m:math> 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 <n:math xmlns:n="http://www.w3.org/1998/Math/MathML"> <n:mi>d</n:mi> </n:math> orbitals in the theoretical modeling of spintronic devices and experiments in graphene-based nanostructures.