The origin of stability of helical structure of tellurium

We present first-principles investigations of the origin of the stability of the helical structure of Te through determination of the structure and energetics of its clusters, the largest one comprising 12 Te atoms, and helical forms. Among all the clusters we have studied, Te8 with a closed-ringlike helical structure is the most stable one; it is even more stable than an infinitely long helix. The stability of the Te structures depends on the Te-Te-Te bond angle and coordination number of Te atoms. Wannier function-based analysis of the chemical bonding in Te helix reveals sp hybridization giving rise to a s- and a three-center-π-like bonding between neighboring Te atoms. While the former stabilizes a two-fold coordination in the helix, the latter stabilizes a specific Te-Te-Te angle. As a result, the helical structure is stabilized, and closed loops of helices are more stable than the open helical chains. A comparative study of the vibrational modes of bulk Te and that of a single helix shows that the rotational mode of bulk Te is significantly softened in the case of a single helix due to the absence of interhelical interactions in the latter. We estimate the interhelical interaction strength in bulk Te to be about 3.0 meV/helix. Relative energies of an infinite helix of Te with respect to a linear chain show that the coupling of transverse chiral acoustic wave with strain plays a crucial role in the formation of the helix. We find that bulk Te is more stable to slip deformation than stretch deformation