# Spin dynamics of $3d$ and $4d$ impurities embedded in prototypical topological insulators

## TL;DR

This study investigates the spin excitations of 3d and 4d impurities in topological insulators, revealing high spin lifetimes, complex dynamics, and the influence of surface states, with implications for quantum applications.

## Contribution

It provides a systematic analysis of impurity-induced spin dynamics in topological insulators, including mapping to a Landau-Lifshitz-Gilbert model and comparison of magnetic anisotropy calculations.

## Key findings

- Impurity states significantly affect spin excitation spectra.
- High spin lifetimes up to microseconds observed.
- Surface states can alter impurity spin dynamics.

## Abstract

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.