Interplay between Structural, Electronic, and Topological Properties in Low-Dimensional Tellurium

We present a comprehensive first-principles investigation of the structural, electronic, vibrational, and topological properties of tellurium across its dimensional hierarchy, including bulk trigonal Te-I, two-dimensional tellurene polymorphs, and one-dimensional helical nanowires. Using density functional theory with full inclusion of spin-orbit coupling, we confirm that bulk Te-I is a narrow-gap semiconductor hosting Weyl nodes arising from broken inversion symmetry and degenerate phonon modes suggestive of chiral phonon behavior. In contrast, two-dimensional α- and β-tellurene are found to be topologically trivial ( $Z_2=0$ ), with no spin-orbit-driven band inversion in the occupied manifold. Beyond these established phases, we find that buckled kagome and buckled square tellurene lattices exhibit a nontrivial two-dimensional $Z_2=1$ topology of the occupied electronic bands, indicating incipient quantum spin Hall character in metallic systems. In contrast, one-sided hydrogen-passivated hexagonal tellurene realizes a fully gapped quantum spin Hall phase with a robust $Z_2=1$ invariant, preserved under applied strain and chemical functionalization. In the one-dimensional limit, helical tellurium nanowires preserve chirality and host edge-localized states accompanied by pronounced anisotropy in carrier effective masses. These results establish tellurium as a highly tunable platform for engineering topological phenomena across dimensionality, bridging three-dimensional Weyl physics, two-dimensional quantum spin Hall and incipient $Z_2$ phases, and one-dimensional helical systems.