Interatomic spin-orbit interaction in a $p$-orbital helical atomic chain

TL;DR

This paper derives the interatomic spin-orbit interaction in a helical p-orbital chain, revealing a Rashba-type SOI induced by the helical structure, which could explain the chiral-induced spin selectivity effect.

Contribution

It provides an analytical derivation of interatomic SOI in a helical p-orbital chain, highlighting the role of second-order processes and the resulting Rashba-type SOI.

Findings

Interatomic SOI is proportional to curvature, hopping energy, and intra-atomic SOI.

The process induces long-range second-nearest-neighbor hoppings.

Analytical spin-split band structure derived in the zero-torsion limit.

Abstract

We derive the interatomic 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 crystal field in the tangential direction of the helix leads to the formation of energetically separated $\sigma$- and $\pi$-bands. In the second-order process, a spin in the $\sigma$-orbital virtually hops to the $\pi$-orbital, flips its direction due to intra-atomic SOI, and then hops back to the $\sigma$-orbital in the neighboring atom due to the misalignment of $p$-orbitals along the helix. This process induces an interatomic SOI in the $\sigma$-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 long-range second-nearest-neighbor hoppings. We analytically derive the spin-split band structure in the zero-torsion limit.