  We show theoretically that the magnetic ions, randomly distributed in a
two-dimensional (2D) semiconductor system, can generate a ferromagnetic
long-range order via the RKKY interaction. The main physical reason is the
discrete (rather than continuous) symmetry of the 2D Ising model of the
spin-spin interaction mediated by the spin-orbit coupling of 2D free carriers,
which precludes the validity of the Mermin-Wagner theorem. Further, the
analysis clearly illustrates the crucial role of the molecular field
fluctuations as opposed to the mean field. The developed theoretical model
describes the desired magnetization and phase-transition temperature $T_c$ in
terms of a single parameter; namely, the chemical potential $\mu$. Our results
highlight a path way to reach the highest possible $T_c$ in a given material as
well as an opportunity to control the magnetic properties externally (e.g., via
a gate bias). Numerical estimations show that magnetic impurities such as
Mn$^{2+}$ with spins $S=5/2$ can realize ferromagnetism with $T_c$ close to
room temperature.

Materials Science;Disordered Systems and Neural Networks