  Using a three-dimensional microscopic lattice model of a strong topological
insulator (TI) we study potential impurities and vacancies in surface and
subsurface positions. For all impurity locations we find impurity-induced
resonance states with energy proportional to the inverse of the impurity
strength, although the impurity strength needed for a low-energy resonance
state increases with the depth of the impurity. For strong impurities and
vacancies as deep as 15 layers into the material, resonance peaks will appear
at and around the Dirac point in the surface energy spectrum, splitting the
original Dirac point into two nodes located off-center. Furthermore, we study
vacancy clusters buried deep inside the bulk and find zero-energy resonance
states for both single and multiple-site vacancies. Only fully symmetric
multiple-site vacancy clusters show resonance states expelled from the bulk
gap.
