  We investigate the effect of potential barriers in the form of step edges on
the scattering properties of Bi$_2$Se$_3$(111) topological surface states by
means of large-scale ab-initio transport simulations. Our results demonstrate
the suppression of perfect backscattering, while all other scattering
processes, which do not entail a complete spin and momentum reversal, are
allowed. Furthermore, we find that the spin of the surface state develops an
out of plane component as it traverses the barrier. Our calculations reveal the
existence of quasi-bound states in the vicinity of the surface barriers, which
appear in the form of an enhanced density of states in the energy window
corresponding to the topological state. For double barriers we demonstrate the
formation of quantum well states. To complement our first-principles results we
construct a two-dimensional low-energy effective model and show that band
bending plays a significant role in the scattering process. Our findings are
discussed in the context of a number of recent experimental works.
