Magnetic fluctuations in topological insulators with ordered magnetic adatoms: Cr on Bi${}_2$Se${}_3$ from first principles

We consider a system consisting of a topological insulator with an array of magnetic adatoms interacting with the surface states of electrons. We find that the indirect coupling of the magnetic impurities results in a ferromagnetic ordering of the magnetic moments and is also responsible for the unusual linear dispersion of the surface magnons. We also show that the interaction of magnons with surface electrons essentially renormalizes the electron energy spectrum. The renormalized spectrum is nonlinear and can be characterized by a negative effective mass of electrons and holes for any $k\ne 0$. The electron velocity near the Dirac point depends on the electron-magnon coupling.

Figure 1

Surface band structure (left panel) of the topological insulator Bi${}_2$Se${}_3$ (0001), the crystal structure of which is depicted on the right panel.

Figure 2

Spin-resolved densities of states of Cr atoms on the Bi${}_2$Se${}_3$ (0001) surface.

Figure 3

Calculated MAE of Bi${}_2$Se${}_3$ (0001) surfaces doped with Cr atoms, which substitute for Bi atoms (red line) or are located in the surface interstitial position (blue line). $x=0\%$ corresponds to the case of a single impurity.

Figure 4

Calculated exchange interaction between Cr atoms on the Bi${}_2$Se${}_3$ (0001) surface, which substitute for Bi atoms (red line) or are located in the surface interstitial position (blue line). Calculations were obtained for the case $x=5\%$.

Figure 5

The energy spectrum of electrons renormalized due to the coupling to magnons calculated for various $\xi$: $\xi =0.00$ (red solid line), $\xi =0.02$ (blue dotted line), and $\xi =0.04$ (green dashed line). The gap $\Delta ={\lambda }_t\langle S_z\rangle \approx$4.5 meV is due to magnetization.

Figure 6

Spectral function of electrons for $\xi =0.001$. The broadening is proportional to $\mathrm{Im}{\Sigma }_{\mathbf{k}\varepsilon }$.