Inter-atomic spin–orbit interaction in a p-orbital helical atomic chain

We derive the inter-atomic spin–orbit interaction (SOI) from a helical atomic chain composed of p-orbitals with intra-atomic SOI, which exhibits a helical state—a potential origin of the chiral-induced spin selectivity (CISS) effect. In this model, a strong uniaxial crystal field in the tangential direction of the helix leads to the formation of energetically separated σ- and π-bands. In the second-order process in terms of the nearest-neighbor hopping matrix element, a spin in the σ-orbital virtually hops to the π-orbital, flips its direction due to intra-atomic SOI, and then hops back to the σ-orbital in the neighboring atom due to the misalignment between the orientations of the σ- and π-orbital lobes, arising from the site-to-site rotation of the local uniaxial crystal-field axis. This process induces an inter-atomic SOI in the σ-band, which takes the form of a Rashba-type SOI generated by an electric field normal to the helical axis. The magnitude of the SOI is proportional to the curvature, the hopping energy, the intra-atomic SOI energy, and inversely proportional to the crystal-field strength. The second-order process also induces second-nearest-neighbor hoppings. We analytically derive the spin-split band structure in the zero-torsion limit.