Phase Diagram of a Model for Diluted Magnetic Semiconductors Beyond Mean-Field Approximations

A lattice spin-fermion model for diluted magnetic semiconductors (DMS) is investigated numerically, improving on previously used mean-field approximations. Curie temperatures are obtained varying the Mn spin $x$ and hole $n$ densities, and the impurity-hole exchange $J$ in units of the hopping $t$. Optimal values are found in the subtle intermediate regime between itinerant and localized carriers. At intermediate and large $J/t$, a “clustered” state is observed above the Curie temperature and ferromagnetism is suppressed. Formal analogies between DMS and manganites are discussed.

Figure 1

(a) $S(q=0)$ vs temperature, $T$, for $4^3$ (circles), $6^3$ (squares), and $8^3$ (diamonds) clusters at $J/t=2.0$, $x=0.1$, and $p=0.4$, using the MC technique. The inset shows the magnetization $|M|$ for the same clusters, with a vertical scale referred to the asymptotic $8^3$ $T=\infty$ value. (b) $|M|$ (circles) and spin-spin correlation $C$ at maximum distance (squares) vs $T$, on a $4^3$ cluster, at $J/t=1.0$, $x=0.25$, and $p=0.4$. The 0.25 horizontal line indicates $|M|$ value at $T=\infty$. (c) $S(q=0)$ vs $T$, at $J/t=2.0$, $x=0.25$, and $p=0.4$ using $4^3$ (circles) and $6^3$ (squares) clusters. In all cases $T_C$ is indicated.

Figure 2

(a) Magnetization $|M|$ (circles) and spin-spin correlation at maximum distance $C$ (squares) vs $T$, at $J/t=1.0$, $x=0.25$, and $p=0.1$, using a $6^3$ cluster. (b) $C(d)$ vs $d$ at $p\sim 3.0$, $x=0.25$, $J/t=1.0$, and $T/t=0.005$, using an $8^2$ cluster. The oscillations indicate an AF state. (c) $|M|$ (circles), spin-spin correlation at minimum distance $C(d_{\min })$ (squares), and at maximum distance $C(d_{\max })$ (diamonds) vs $J/t$, for a $4^3$ cluster at $x=0.25$, $p=0.4$, and $T/t=0.005$. (d) $\Delta E=E(\theta =\pi )-E(\theta =0)$ vs $J/t$ calculated exactly on finite but large clusters at $T=0$ for a system of two Mn spins and one electron, $\theta$ being the relative angle between the Mn spins. Results in one, two, and three dimensions are indicated. The spin distance is such that the associated effective $x\sim 0.1$ is the same in all cases.

Figure 3

(a) Exact $T=0$ local carrier density, for one carrier and two parallel spins at sites 10 and 20, of a 30-site chain, varying $J/t$. (b) $T_C$ vs $x$ for ${\mathrm{G}\mathrm{a}}_{1-x}{\mathrm{M}\mathrm{n}}_x\mathrm{A}\mathrm{s}$ [20] (thick line), and for Eq. (1) using $t=0.3\mathrm{e}\mathrm{V}$ (squares) and $t=0.5\mathrm{e}\mathrm{V}$ (circles). In both cases, $p\sim 0.4$ and $J/t=2.0$. Typical error bars are shown. Stars are for $p\sim 0.5$, $J/t=1$, and $t=0.5\mathrm{e}\mathrm{V}$. (c) Magnetization $|M|$ vs $x$ for model Eq. (1) (circles) at $T/t=0.005$, $J/t=2$, $p=0.4$, compared with the experimental value [1] for ${\mathrm{G}\mathrm{a}}_{1-x}{\mathrm{M}\mathrm{n}}_x\mathrm{A}\mathrm{s}$ at $x\sim 0.07$ and $T=2\mathrm{K}$ (squares).

Figure 4

(a) MC phase diagram in 2D varying $J/t$, at fixed $x$ and $p$. At large $J/t$, a broad scale $T^{*}$ corresponds to the formation of uncorrelated clusters. $T_C$ is the “true” transition temperature, defined as the $T$ where FM correlations develop at the largest distance in the clusters used. At small $J/t$, those temperatures are similar. The optimal $J/t$ is intermediate between itinerant and localized regimes. (b) Schematic phase diagram believed to be valid both in 2D and 3D, with the clustered and FM states indicated. (c) Numerically obtained $T_C$ vs $x$ and $p$, at $J/t=2.0$. Filled circles are from model Eq. (1) with $t=0.5\mathrm{e}\mathrm{V}$. The open square corresponds to the experimental value for ${\mathrm{G}\mathrm{a}}_{1-x}{\mathrm{M}\mathrm{n}}_x\mathrm{A}\mathrm{s}$ at $x\sim 0.1$.