Spin–Orbit Gaps in Carbynes

Abstract

The effect of spin–orbit interaction on the band structures of the monatomic carbon chains, called the carbynes, is calculated in terms of a linear augmented cylindrical wave method. Because of the cylindrical symmetry of carbynes, the twofold orbitally degenerate π bands correspond to the semiclassical clockwise and anticlockwise rotational motion of electrons around the symmetry axis. In the absence of spin–orbit interaction with the two possible directions of spin, the π bands would be the fourfold degenerate ones. The spin and orbital motion of electrons are coupled, thereby splitting the fourfold degeneracy. Each π sub-band still has the twofold degeneracy, the spin polarization direction between degenerate two bands being opposite to each other. In a cumulenic carbyne with the double bonds (...═C═C═...), the splitting of π band at the Fermi energy region is equal to 2.4 meV, but the metallic character of band structure is not broken by spin–orbit interaction. In the semiconducting polyynic carbyne with alternating single and triple bonds (...–C≡C–C≡C–...), the spin–orbit gaps are different for the highest valence band (3.1 meV) and the lowest conduction band (2.1 meV). The spin–orbit gaps in carbyne are about 2 or 3 times smaller than the spin–orbit splitting (6 meV) in the carbon atom. In carbyne, the spin–orbit interaction is larger than that in carbon nanotubes because of the larger curvature of electron orbits encircling the carbyne chains; the larger spin–orbit coupling can be attractive for new experiments and applications.