  We investigate the spin-orbit torque exerted on the magnetic moments of the
transition-metal impurities Cr, Mn, Fe and Co, embedded in the surface of the
topological insulator Bi$_{2}$Te$ _{3} $, in response to an electric field and
a consequent electrical current flow in the surface. The multiple scattering
problem of electrons off impurity atoms is solved by first-principles
calculations within the full-potential relativistic Korringa-Kohn-Rostoker
(KKR) Green function method, while the spin-orbit torque calculations are
carried out by combining the KKR method with the semiclassical Boltzmann
transport equation. We analyze the correlation of the spin-orbit torque to the
spin accumulation and spin flux in the defects. We compare the torque on
different magnetic impurities and unveil the effect of resonant scattering. In
addition, we calculate the resistivity and the Joule heat as a function of the
torque in these systems. We predict that the Mn/Bi$_{2}$Te$_{3}$ is optimal
among the studied systems.
