  Band inversion, one of the key signatures of time-reversal invariant
topological insulators (TIs), arises mostly due to the spin-orbit (SO)
coupling. Here, based on ab initio density-functional calculations, we report a
theoretical investigation of the SO-driven band inversion in isostructural
bismuth and antimony chalcogenide TIs from the viewpoint of its interplay with
the crystal-field effect. We calculate the SO-induced energy shift of states in
the top valence and bottom conduction manifolds and reproduce this behavior
using a simple one-atom model adjusted to incorporate the crystal-field effect.
The crystal-field splitting is shown to compete with the SO coupling, that is,
stronger crystal-field splitting leads to weaker SO band shift. We further show
how both these effects can be controlled by changing the chemical composition,
whereas the crystal-field splitting can be tuned by means of uniaxial strain.
These results provide a practical guidance to the rational design of novel TIs
as well as to controlling the properties of existing materials.
