  Bismuth-chalchogenides are model examples of three-dimensional topological
insulators. Their ideal bulk-truncated surface hosts a single spin-helical
surface state, which is the simplest possible surface electronic structure
allowed by their non-trivial $\mathbb{Z}_2$ topology. They are therefore widely
regarded ideal templates to realize the predicted exotic phenomena and
applications of this topological surface state. However, real surfaces of such
compounds, even if kept in ultra-high vacuum, rapidly develop a much more
complex electronic structure whose origin and properties have proved
controversial. Here, we demonstrate that a conceptually simple model,
implementing a semiconductor-like band bending in a parameter-free
tight-binding supercell calculation, can quantitatively explain the entire
measured hierarchy of electronic states. In combination with circular dichroism
in angle-resolved photoemission (ARPES) experiments, we further uncover a rich
three-dimensional spin texture of this surface electronic system, resulting
from the non-trivial topology of the bulk band structure. Moreover, our study
reveals how the full surface-bulk connectivity in topological insulators is
modified by quantum confinement.
