Figure 4. Exchange parameter and Monte Carlo Simulations of a disordered 3D Ising model with exponential decaying RKKY interactions.

(a) Plot of the oscillating part of the exchange parameter [see equation (3)] for several distances between nanoparticles. The constant C dividing J^osc is the leading factor in equation (3). We plot both the case where no electron’s spin-flips are allowed (dashed curves) and the case where these are allowed (full curves). (b) Magnetization in terms of the temperature for different strengths of the indirect exchange coupling. (c) Comparison between the experimental and theoretical blocking temperature in terms of the strength of the spin-imbalance (the theoretical fitting parameters used were J₀ = 2.2 × 10⁶ and ξ = 5 nm). The Monte Carlo results of panels (b) and (c) were obtained for simulations (with 79507 Ising moments) starting from a highly ordered state (see supplementary information) that use Metropolis algorithm. They explore the phase space region in the vicinity of the global energy minimum, and indicate that the system can show long range order if the clusters are initially aligned by an external magnetic field.