Theory of electron mediated Mn-Mn interactions in quantum dots.

We present a theory of interaction of magnetic Mn ions depending strongly on the number (Ne) of electrons in a quantum dot. For closed electronic shells, we derive the RKKY interaction and its dependence on magnetic ion positions, quantum dot energy quantization omega0, and the number of filled shells Ns. For partially filled shells, the many-electron magnetopolaron effect leads to effective carrier mediated ferromagnetic Mn-Mn interactions. The dependence of the magnetopolaron energy on magnetic ion positions, quantum dot energy quantization omega0, and the number of electrons Ne is predicted.