Phase Diagram of the Disordered RKKY Model in Dilute Magnetic Semiconductors

We consider ferromagnetism in spatially randomly located magnetic moments, as in a diluted magnetic semiconductor, coupled via the carrier-mediated indirect exchange RKKY interaction. We obtain, via Monte Carlo calculations, the magnetic phase diagram as a function of the impurity moment density $n_i$ and the relative carrier concentration $n_c/n_i$. As evidenced by the diverging correlation length and magnetic susceptibility, the boundary between ferromagnetic and nonferromagnetic phases constitutes a line of zero temperature critical points which can be viewed as a magnetic percolation transition. In the dilute limit, we find that bulk ferromagnetism vanishes for $n_c/n_i>0.1$. We also incorporate the local antiferromagnetic direct superexchange interaction between nearest neighbor impurities and examine the impact of a damping factor in the RKKY range function.

Figure 1

$T=0$ magnetization and susceptibility, and $T_c$ plots vs $n_c/n_i$ for $x_i=0.1$; results are shown for the undamped RKKY with no AF superexchange coupling. Panels (a) and (b) display the zero temperature magnetization and susceptibility, respectively, for $⟨N⟩=1000$. Panel (c) depicts Curie temperatures vs $n_c/n_i$. The solid circles correspond to the undamped ($l=\infty$) case and the open circles to the damped ($l=a$) RKKY model. The inset of (c) shows MFT and MC $T_c$’s for $l=a$. Panel (d) is a graph of $T_c$ vs $x_i$ with $n_c/n_i=0.03$ for $l=\infty$ and $l=a$. For $l=\infty$, open circles are Monte Carlo results, while the solid line in (d) is a theoretical curve, given by $29.2x_i^{4/3}$; for $l=a$, closed circles are Monte Carlo $T_c$’s, and the solid line is a guide to the eye.

Figure 2

Phase diagrams for the RKKY model; the solid line corresponds to the undamped, full RKKY model, while the dashed line is for the damped case where $l=a$.

Figure 3

The graph in panel (a) displays $T_c$ vs $x$ for $n_c/n_i=0.05$, $l=a$, and $j^{\mathrm{AF}}=-10$. Panels (b) and (c) are phase diagrams for the RKKY model with a local AF coupling incorporated for $l=\infty$ (b) and $l=a$ (c); the open circles at the left denote locations in the diagrams where $T_c$ is optimized.