  Topological insulators represent a paradigm shift in surface physics. The
most extensively studied Bi$_2$Se$_3$-type topological insulators exhibit
layered structures, wherein neighboring layers are weakly bonded by van der
Waals interactions. Using first principles density-functional theory
calculations, we investigate the impact of the stacking sequence on the
energetics and band structure properties of three polymorphs of Bi$_2$Se$_3$,
Bi$_2$Te$_3$, and Sb$_2$Te$_3$. Considering their ultrathin films up to 6 nm as
a function of its layer thickness, the overall dispersion of the band structure
is found to be insensitive to the stacking sequence, while the band gap is
highly sensitive, which may also affect the critical thickness for the onset of
the topologically nontrivial phase. Our calculations are consistent with both
experimental and theoretical results, where available. We further investigate
tribological layer slippage, where we find a relatively low energy barrier
between two of the considered structures. Both the stacking-dependent band gap
and low slippage energy barriers, suggest that polymorphic stacking
modification may offer an alternative route for controlling the properties of
this new state of matter.
