  We show that higher-harmonics exchange scattering from a magnetic adatom on
the surface of a three dimensional topological insulator leads to a magnetic
anisotropy whose magnitude and sign may be tuned by adjusting the chemical
potential of the helical surface band. As chemical potential moves from the
Dirac point towards the surface band edge, the surface normal is found to
change from magnetic easy, to hard axis. Hexagonal warping is shown to diminish
the region with easy axis anisotropy, and to suppress the anisotropy
altogether. This indirect contribution can be comparable in magnitude to the
intrinsic term arising from crystal field splitting and atomic spin-orbit
coupling, and its tunability with chemical potential makes the two
contributions experimentally discernible, and endows this source of anisotropy
with potentially interesting magnetic functionality.
