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

Takemitsu Kato, Yasuhiro Utsumi, Ora Entin-Wohlman, Amnon Aharony

TL;DR

This work investigates the origin of chirality-induced spin selectivity (CISS) by deriving an inter-atomic spin-orbit interaction (SOI) in a $p$-orbital helical chain with intra-atomic SOI. By a Schrieffer-Wolff transformation, the authors show that curvature-driven orbital mixing and a second-order $oldsymbol π$-$oldsymbol σ$ process generate a Rashba-like inter-atomic SOI in the $σ$-band, together with long-range second-nearest-neighbor hoppings; the strength scales as $ rac{2 ext{(curvature)} imes J imes ext{Δ}_{ m so}}{K_{m t}}$. In the zero-torsion limit, the Bloch Hamiltonian for the $σ$-band is analytically diagonalized, revealing a spin splitting proportional to $ rac{4 J_- ext{Δ}_{ m so}}{K_{m t}} rac{k}{N}$, while the $π$-band retains helical states. Finite torsion, as in DNA-like geometries, modifies the splitting and can reduce the effect, highlighting geometry as a tunable factor in CISS. Overall, the model provides a compact, analytically tractable framework linking curvature, crystal-field strength, and SOI to spin-selective transport in chiral systems.

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 $σ$- and $π$-bands. In the second-order process, 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 of $p$-orbitals along the helix. This process induces an interatomic 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 long-range second-nearest-neighbor hoppings. We analytically derive the spin-split band structure in the zero-torsion limit.

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

TL;DR

This work investigates the origin of chirality-induced spin selectivity (CISS) by deriving an inter-atomic spin-orbit interaction (SOI) in a -orbital helical chain with intra-atomic SOI. By a Schrieffer-Wolff transformation, the authors show that curvature-driven orbital mixing and a second-order - process generate a Rashba-like inter-atomic SOI in the -band, together with long-range second-nearest-neighbor hoppings; the strength scales as . In the zero-torsion limit, the Bloch Hamiltonian for the -band is analytically diagonalized, revealing a spin splitting proportional to , while the -band retains helical states. Finite torsion, as in DNA-like geometries, modifies the splitting and can reduce the effect, highlighting geometry as a tunable factor in CISS. Overall, the model provides a compact, analytically tractable framework linking curvature, crystal-field strength, and SOI to spin-selective transport in chiral systems.

Abstract

We derive the interatomic spin-orbit interaction (SOI) from a helical atomic chain composed of -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 - and -bands. In the second-order process, 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 of -orbitals along the helix. This process induces an interatomic 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 long-range second-nearest-neighbor hoppings. We analytically derive the spin-split band structure in the zero-torsion limit.

Paper Structure

This paper contains 7 sections, 31 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Model Hamiltonian: $p$-orbital helical atomic chain
  3. Inter-atomic SOI
  4. Results
  5. Conclusion
  6. Spin rotation
  7. Derivations of (\ref{['eqn:Heff_y']})

Figures (2)

  • Figure 1: (a) Schematic picture of a helical atomic chain. The model is a right-handed system in which the $z$-axis coincides with the helical axis. (b) Schematic picture of the local orthogonal coordinate system at site $n$. The basis set is chosen as $\{ -{\bm n}(\phi_n),{\bm t}(\phi_n),{\bm b}(\phi_n)\}$.
  • Figure 2: Energy dispersions for (a) $\alpha=1$, $\varphi=\Delta \phi$, $\tau \to 0$, $K_{\bm t}=5 J$ and $N=4$, (b) $\alpha=\sqrt{2}$, $\varphi=\pi/4$, $\tau \to 0$, $K_{\bm t}=5 J$ and $N=4$, and (c) $\alpha=\sqrt{2}$, $\varphi=\pi/4$, $\tau=0.48$, $K_{\bm t}=6 J$ and $N=10$. The chirality is $p=1$ and the spin-orbit interaction energy is $\Delta_{\rm so}=0.4 J$. The dotted and dashed lines indicate the eigenenergies of the $\sigma$-band effective Hamiltonian (\ref{['eqn:En_sig']}). The color scheme indicates the $z$ component of the average spin (red for $\uparrow$ spin and blue for $\downarrow$ spin).