Indirect interaction of impurity spins on the surface of topological insulators

In this work, a mathematical model to study the indirect interaction in topological insulators was constructed. Analysis of the model was carried out numerically. We have calculated the indirect exchange interaction in the film of a topological insulator, for example Bi$_2$Te$_3$, within the s-d model. The calculations showed that the magnetic ordering of the impurity spins varies periodically with increasing distance between atoms, asymptotically decreasing to zero. $\lambda$ is a parameter associated with hexagonal distortion and is a component of the dispersion relation. The dependence of the constants of the effective exchange interaction upon the $\lambda$ parameter is shown; this parameter characterizes the crystal lattice geometry for a topological insulator.