Electric field tuning spin splitting in topological insulator quantum dots doped with a single magnetic ion

We investigate theoretically the electron spin states in disk-shaped HgTe topological insulator quantum dots (TIQDs) containing a single magnetic $Mn^{2+}$ ion. We show that the energy spectrum and the electron density distribution of the topological edge states in HgTe TIQD can be modulated significantly by the position of the magnetic $Mn^{2+}$ ion. The numerical results further demonstrate that the electric fields not only tune the spin splittings of edge states via the $\emph{sp-d}$ exchange interaction between the electron (hole) and the magnetic $Mn^{2+}$ ion but also give rise to the bright-to-dark transitions and anti-crossing behaviors in the photoluminescence (PL) spectra. Such spin properties of HgTe TIQDs with single $Mn^{2+}$ ion as illustrated in this work could offer a new platform for topological electro-optical devices.