Valley coupling in finite-length metallic single-wall carbon nanotubes

TL;DR

This paper investigates how boundary conditions, nanotube type, and spin-orbit interaction influence the energy level degeneracy and valley coupling in finite-length metallic single-wall carbon nanotubes, revealing distinct spectral features.

Contribution

It provides a detailed analysis of valley coupling effects and boundary influences on energy spectra in metallic nanotubes, introducing an effective 1D model for understanding these phenomena.

Findings

Metal-1 nanotubes show nearly fourfold degeneracy with small spin-orbit splitting.

Metal-2 nanotubes exhibit vernier-scale-like spectra due to strong valley coupling.

Boundary conditions significantly affect valley degeneracy and spectral features.

Abstract

Degeneracy of discrete energy levels of finite-length, metallic single-wall carbon nanotubes depends on type of nanotubes, boundary condition, length of nanotubes and spin-orbit interaction. Metal-1 nanotubes, in which two non-equivalent valleys in the Brillouin zone have different orbital angular momenta with respect to the tube axis, exhibits nearly fourfold degeneracy and small lift of the degeneracy by the spin-orbit interaction reflecting the decoupling of two valleys in the eigenfunctions. In metal-2 nanotubes, in which the two valleys have the same orbital angular momentum, vernier-scale-like spectra appear for boundaries of orthogonal-shaped edge or cap-termination reflecting the strong valley coupling and the asymmetric velocities of the Dirac states. Lift of the fourfold degeneracy by parity splitting overcomes the spin-orbit interaction in shorter nanotubes with a so-called minimal boundary. Slowly decaying evanescent modes appear in the energy gap induced by the curvature of nanotube surface. Effective one-dimensional model reveals the role of boundary on the valley coupling in the eigenfunctions.