Line group approach for quantum chemical study of intrinsic helical twist of ultrathin tellurium nanorods

Abstract

The results of quantum chemical modeling of the structure and electronic properties of tellurium nanorods using two different approaches to describing the symmetry of these objects are presented. The first one is based on the consideration of tellurium nanorods within the framework of subgroups of the P3₁21 space group of a bulk tellurium crystal (crystallographic rod groups L3₁21 and L3₁), which specify the order of the helical axis strictly equal to 3. The second one considers nanorods within the framework of the line group theory, which allows for a continuous change in the order of the helical axis for 1D objects and by this way to discuss the effects of the nanorod torsion. It is shown that in the absence of symmetry restrictions, the order of the helical axis of tellurium nanorods is greater than the crystallographic value 3, but tends to it with increasing of nanorod thickness. Therefore, the corresponding rod groups should be considered only as groups to which the true line group symmetry of the nanorods asymptotically tends as their thickness increases. For nanorods obtained experimentally, the difference between the true and crystallographic helical axis orders is so small that it is practically invisible at the nanometer level, but manifests itself at the mesoscale, leading to a visually observable helical twist of tellurium nanorods (see A. Londoño-Calderon et al. Nanoscale, 2021, 13, 9606). The dependences of the electronic band gap of nanorods on torsion are presented. Nanorod of a certain thickness has its own features of the electronic structure, which can be explained by the evolution of the topology of electronic bands in the helical Brillouin zone.