Carrier-induced ferromagnetism in two-dimensional magnetically doped semiconductor structures

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 pathway 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 ${\mathrm{Mn}}^{2+}$ with spins $S=5/2$ can realize ferromagnetism with $T_c$ close to room temperature.

Figure 1

Comparison of the range functions for the RKKY interaction potentials in the 2D (black, dashed line) and 3D (red, solid line) spatial dimensions. The inset provides a magnified view of the 3D RKKY range function.

Figure 2

FM phase-transition temperature $T_c$ vs the chemical potential $\mu$ (both in units of $T_0$) in a 2D DMS with $S=5/2$. The abscissa also corresponds to a ratio between the free-carrier and magnetic ion densities $n_e$ and $n_i$ (except a constant). The dashed line on the left end projects the condition at which the FM ordering ceases to exist in the nondegenerate regime. By contrast, the vertical dashed-dotted line at ${\mu }_S$ ($\approx 4.6T_0$) distinguishes the regions where the MFA and RKKY treatments are used to obtain the solutions (i.e., curves 1 and 2), respectively. The inset illustrates additional details of both solutions beyond the crossover point ${\mu }_S$. With $T_c$ staying at the asymptotic value $T_0$ with no/little dependence on $\mu$ (see $\mu >{\mu }_S$ for curve 1 and $\mu <{\mu }_S$ for curve 2), it is apparent that a single model cannot describe the entire range of $\mu$ adequately.