  Surfaces of topological insulators host a new class of states with Dirac
dispersion and helical spin texture. Potential quantum computing and spintronic
applications using these states require manipulation of their electronic
properties at the Dirac energy of their band structure by inducing magnetism or
superconductivity through doping and proximity effect. Yet, the response of
these states near the Dirac energy in their band structure to various
perturbations has remained unexplored. Here we use spectroscopic mapping with
the scanning tunneling microscope to study their response to magnetic and
non-magnetic bulk dopants in Bi2Te3 and Bi2Se3. Far from the Dirac energy
helicity provides remarkable resilience to backscattering even in the presence
of ferromagnetism. However, approaching the Dirac point, where the surface
states' wavelength diverges bulk doping results in pronounced nanoscale spatial
fluctuations of energy, momentum, and helicity. Our results and their
connection with similar studies of Dirac electrons in graphene demonstrate that
while backscattering and localization are absent for Dirac topological surface
states, reducing charge defects is required for both tuning the chemical
potential to Dirac energy and achieving high electrical mobility for these
novel states.
