https://doi.org/10.1140/epjs/s11734-025-02065-1
Regular Article
Inter-atomic spin–orbit interaction in a p-orbital helical atomic chain
1
Department of Physics Engineering, Faculty of Engineering, Mie University, 1577, Kurimamachiya-cho, 514-8507, Tsu, Mie, Japan
2
Department of Electrical and Electronic Engineering, Faculty of Engineering, Mie University, 1577, Kurimamachiya-cho, 514-8507, Tsu, Mie, Japan
3
School of Physics and Astronomy, Tel Aviv University, 6997801, Tel Aviv, Israel
Received:
31
March
2025
Accepted:
12
November
2025
Published online:
26
November
2025
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 
- and 
-bands. In the second-order process in terms of the nearest-neighbor hopping matrix element, a spin in the -orbital virtually hops to the -orbital, flips its direction due to intra-atomic SOI, and then hops back to the -orbital in the neighboring atom due to the misalignment between the orientations of the - and -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 -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.
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Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
