Stability and thermodynamic properties of bound magnetic polarons in ferromagnetic semiconductors: Beyond the Gaussian approximation
The Gaussian approximation for thermodynamic fluctuations of host magnetization fails for ferromagnetic semiconductors like GdN near their Curie temperature $T_c$, producing unphysical instabilities in the bound magnetic polaron (BMP) free energy. We extend the Dietl-Spalek theory beyond the Gaussian level by incorporating full cubic and quartic anharmonic terms of the Ginzburg-Landau-Wilson functional. We develop two complementary generalizations: (i) a perturbative non-local treatment for finite spin correlation length, and (ii) a non-perturbative resummation of local fluctuations into a closed exponential form. The latter is rigorously justified by a novel polaronic Ginzburg criterion. By including variational optimization of the donor orbital radius, we quantitatively describe magnetic self-trapping. Applied to GdN ($T_c \approx 55$ K), our theory eliminates Gaussian divergences and predicts thermodynamically stable, ferromagnetically ordered BMPs with spontaneous internal spin splitting persisting deep into the paramagnetic phase. For realistic exchange coupling ($J_c = 400$ meV), this robust local ordering survives up to $T^* \approx 155-160$ K. Pronounced orbital contraction signals magnetic self-trapping, with a characteristic kink in the optimal Bohr radius at $T_{char} \approx 79$ K. We reveal a critical exchange threshold ($J_c \approx 330$ meV) triggering cooperative collapse into a highly localized small polaron state, and identify an optimal donor-concentration window near the metal-insulator transition maximizing $T^*$. These results establish a microscopic mechanism for persistent BMP-mediated ferromagnetism well above bulk $T_c$ via polaron percolation, suggesting clear experimental signatures (optical spin splitting, anomalous magnetoresistance) testable in GdN and related compounds.