Role of Disorder in Mn:GaAs, Cr:GaAs, and Cr:GaN

We present calculations of magnetic exchange interactions and critical temperature $T_c$ in ${\mathrm{G}\mathrm{a}}_{1-x}{\mathrm{M}\mathrm{n}}_x\mathrm{A}\mathrm{s}$, ${\mathrm{G}\mathrm{a}}_{1-x}{\mathrm{C}\mathrm{r}}_x\mathrm{A}\mathrm{s}$, and ${\mathrm{G}\mathrm{a}}_{1-x}{\mathrm{C}\mathrm{r}}_x\mathrm{N}$. The local spin-density approximation is combined with a linear-response technique to map the magnetic energy onto a Heisenberg Hamiltonion, but no significant further approximations are made. We show the following: (i) configurational disorder results in large dispersions in the pairwise exchange interactions; (ii) the disorder strongly reduces $T_c$; (iii) clustering in the magnetic atoms, whose tendency is predicted from total-energy considerations, further reduces $T_c$, while ordering the dopants on a lattice increases it. With all the factors taken into account, $T_c$ is reasonably predicted by the local spin-density approximation in Mn:GaAs without the need to invoke compensation by donor impurities.

Figure 1

Scaled pair exchange interactions $R_{ij}^{3}J(R_{ij})$, in $a^3\mathrm{m}\mathrm{R}\mathrm{y}$, for Mn:GaAs and Cr:GaN at $x=8.3\%$. Neighbor distance $R_{ij}$ is in units of the lattice constant $a$.

Figure 2

Dependence of $T_c(\mathrm{K})$ on $x$ in ${\mathrm{G}\mathrm{a}}_{1-x}{\mathrm{M}\mathrm{n}}_x\mathrm{A}\mathrm{s}$ and ${\mathrm{G}\mathrm{a}}_{1-x}{\mathrm{C}\mathrm{r}}_x\mathrm{A}\mathrm{s}$. Solid lines: $T_c$ computed from the MF ${\overline{T}}_c^{\mathrm{M}\mathrm{F}\mathrm{A}}$. Dotted line: $T_c$ extracted from spin-dynamics simulations of Eq. (1). Diamonds: $T_c$ computed from the Heisenberg cluster variation method.

Figure 3

Dependence of $T_c(\mathrm{K})$ on the pair correlation function $P_1$ in ${\mathrm{G}\mathrm{a}}_{0.92}{\mathrm{M}\mathrm{n}}_{0.08}\mathrm{A}\mathrm{s}$. The random (SQS) configuration corresponds to $P_1=0.7056$. (Two SQS structures were calculated.) Diamonds show $T_c$ computed with CVM; circles show ${\overline{T}}_c^{\mathrm{M}\mathrm{F}\mathrm{A}}$, and triangles show $T_c^{\mathrm{M}\mathrm{F}\mathrm{A}}$. The point at $P_1=2/3$ corresponds to the ordered compound.