Mechanisms of optical orientation of an individual Mn$^{2+}$ ion spin in a II-VI quantum dot

We provide a theoretical description of the optical orientation of a single Mn$^{2+}$ ion spin under quasi-resonant excitation demonstrated experimentally by Goryca et al. [Phys. Rev. Lett. 103, 087401 (2009)]. We build and analyze a hierarchy of models by starting with the simplest assumptions (transfer of perfectly spin-polarized excitons from Mn-free dot to the other dot containing a single Mn$^{2+}$ spin, followed by radiative recombination) and subsequently adding more features, such as spin relaxation of electrons and holes. Particular attention is paid to the role of the influx of the dark excitons and the process of biexciton formation, which are shown to contribute significantly to the orientation process in the quasi-resonant excitation case. Analyzed scenarios show how multiple features of the excitonic complexes in magnetically-doped quantum dots, such as the values of exchange integrals, spin relaxation times, etc., lead to a plethora of optical orientation processes, characterized by distinct dependencies on light polarization and laser intensity, and occurring on distinct timescales. Comparison with experimental data shows that the correct description of the optical orientation mechanism requires taking into account Mn$^{2+}$ spin-flip processes occurring not only when the exciton is already in the orbital ground state of the light-emitting dot, but also those that happen during the exciton transfer from high-energy states to the ground state. Inspired by the experimental results on energy relaxation of electrons and holes in nonmagnetic dots, we focus on the process of biexciton creation allowed by mutual spin-flip of an electron and the Mn$^{2+}$ spin, and we show that by including it in the model, we obtain good qualitative and quantitative agreement with the experimental data on quasi-resonantly driven Mn$^{2+}$ spin orientation.