Phase diagram and critical behavior of the random ferromagnet Ga${}_{1-x}$Mn${}_x$N

Molecular beam epitaxy has been employed to obtain Ga${}_{1-x}$Mn${}_x$N films with $x$ up to 10% and Curie temperatures $T_{\text{C}}$ up to 13 K. The magnitudes of $T_{\text{C}}$ and their dependence on $x$, $T_{\text{C}}\left(x\right)\propto x^m$, where $m=2.2\pm 0.2$, are quantitatively described by a tight-binding model of superexchange interactions and Monte Carlo simulations of $T_{\text{C}}$. The critical behavior of this dilute magnetic insulator shows strong deviations from the magnetically clean case ($x=1$), in particular, (i) an apparent breakdown of the Harris criterion, (ii) a nonmonotonic crossover in the values of the susceptibility critical exponent ${\gamma }_{\text{eff}}$ between the high temperature and critical regimes, and (iii) a smearing of the critical region, which can be explained either by the Griffiths effects or by macroscopic inhomogeneities in the spin distribution with a variance $\Delta x=(0.2\pm 0.1)$%.

Figure 1

Magnetic hysteresis at selected temperatures for two Ga${}_{1-x}$Mn${}_x$N samples with Mn content $x=5.5$% in panel (a) and 10% in panel (b).

Figure 2

Experimental Curie temperatures as a function of Mn content $x$ (circles), together with the experimental result of Refs. 10 and 7 (squares and triangle, respectively). The dotted line indicates the scaling dependence $T_{\text{C}}\propto x^m$ with $m=2.2$. Results of Monte Carlo simulations with exchange integrals from the tight-binding model[10] (stars) are also shown.

Figure 3

Temperature dependence of the magnetic response for Ga${}_{1-x}$Mn${}_x$N layers at $H=1$ kOe. The error bars (in matching colors for the samples with the highest and the lowest Mn concentrations) reflect the experimental uncertainty in the samples’ weight (i.e., the substrate contribution) and the shape correction factors.[28] The thick solid line traces the Curie law $\chi \left(T\right)\propto 1/T$. The dotted line indicates that at high temperatures $\chi \left(T\right)\propto T^{-\alpha }$, where $\alpha >1$, a dependence specific to random ferromagnets with short-range spin-spin interactions.

Figure 4

Critical behavior of $x=10$% Ga${}_{1-x}$Mn${}_x$N and Curie point determination from the inverse ($1/\chi$) and ac (${\chi }_{\text{ac}}$) magnetic susceptibility, coercive field ($H_{\text{c}}$), thermoremanent (TRM), and zero-field-cooled (0FC) magnetization $M$, as well as from the position of the inflection point of $M\left(T\right)$ (here, the maximum of numerically established $-dM/dT$), the latter pointing to a lower value of $T_{\text{C}}$. Both $1/\chi$ and $-dM/dT$ values are established from $M\left(T\right)$ measured at $H=1.3$ Oe.

Figure 5

Experimental (points) and theoretical (lines) temperature dependencies of the effective critical exponents for two Ga${}_{1-x}$Mn${}_x$N samples from different wafers. The values of $\beta$ and $\gamma$ as well two values of the variance $\Delta x$ in the Mn distribution employed in the calculations are displayed.