  Topological insulators are insulating bulk materials hosting conducting
surface states. Their magnetic doping breaks time-reversal symmetry and
generates numerous interesting effects such as dissipationless transport.
Nonetheless, their dynamical properties are still poorly understood. Here, we
perform a systematic investigation of transverse spin excitations of $3d$ and
$4d$ single impurities embedded in two prototypical topological insulators
(Bi$_2$Te$_3$ and Bi$_2$Se$_3$). The impurity-induced states within the bulk
gap of the topological insulators are found to have a drastic impact on the
spin excitation spectra, resulting in very high lifetimes reaching up to
${microseconds}$. An intuitive picture of the spin dynamics is obtained by
mapping onto a generalized Landau-Lifshitz-Gilbert phenomenological model. The
first quantity extracted from this mapping procedure is the magnetic anisotropy
energy, which is then compared to the one provided by the magnetic force
theorem. This uncovers some difficulties encountered with the latter, which can
provide erroneous results for impurities with a high density of states at the
Fermi energy. Moreover, the Gilbert damping and nutation tensors are obtained.
The nutation effects can lead to a non-negligible shift in the spin excitation
resonance in the high-frequency regime. Finally, we study the impact of the
surface state on the spin dynamics, which may be severely altered due to the
repositioning of the impurity-induced state in comparison to the bulk case. Our
systematic investigation of this series of magnetic impurities sheds light on
their spin dynamics within topological insulators, with implications for
available and future experimental studies as, for instance, on the viability of
using such impurities for solid-state qubits.
