Tight-binding model for topological insulators: Analysis of helical surface modes over the whole Brillouin zone

A tight-binding model is constructed for Bi₂Se₃-type topological insulators with rhombohedral crystal structure. The model takes full account of the spin-orbit interaction, and realizes both strong (S) and weak (W) topological insulators (TIs) depending on the mass parameter that causes the band inversion. It is found that there are two separate STIs with either a single or three Dirac cones on the surface, while the WTI realizes either zero or four surface Dirac cones keeping the same Z₂ indices. Closing of the bulk direct gap gives rise to transition between either STI and WTI, or TI and an ordinary insulator. On the other hand, closing of the indirect gap keeps intact the surface Dirac cones in both STIs and WTIs. As a result, helical modes can remain even in semimetals. It is found that reentrant helical modes appear in finite-momentum regions in some cases in STIs, and even in ordinary insulators with strong particle-hole asymmetry. All results are obtained analytically.