Critical temperatures of the two-band model for diluted magnetic semiconductors

Using dynamical mean field theory and Monte Carlo simulations, we study the ferromagnetic transition temperature $\left(T_c\right)$ of a two-band model for diluted magnetic semiconductors (DMS), varying coupling constants, hopping parameters, and carrier densities. We found that $T_c$ is optimized at all fillings $p$ when both impurity bands fully overlap in the same energy range, namely when the exchange couplings $J$ and bandwidths are identical. The optimal $T_c$ is found to be about twice larger than the maximum value obtained in the one-band model, showing the importance of multiband descriptions of DMS at intermediate $J$’s.

Figure 1

(Color online) $T_c$ vs the carrier concentration $p$, at various $J_2∕J_1$, obtained with the DMFT technique. Here, $x=0.05$, $W_1∕W_2=1$, and $J_1∕W_1=0.5$. The inset shows the corresponding DOS at $T=0$.

Figure 2

(Color online) Results obtained with the DMFT approximation: (a) $T_c$ vs $J_2∕J_1$, at $W_1∕W_2=1$, and $J_1∕W_1=0.5$, for the values of $p$ indicated in (b). (b) $T_c$ vs $J_1∕W_1$, at $W_1∕W_2=1$ and $J_1∕W_1=J_2∕W_2$, for the $p$’s indicated. At $p=0.02$ and $J_2∕J_1=0$, a finite $T_c∕W_1\sim 0.0037$ is caused by the $l=1$ band. At $p=0.05$, $T_c$ is not zero for $J_2∕J_1∊(0.6,1.35)$, and it increases significantly due to band overlap. The case $p=0.07$ $(>x)$ corresponds to the first IB completely filled. (c) $T_c$ vs $p$ at different ratios $W_2∕W_1$, fixing $J_1=J_2=2$. In all the frames $x=0.05$.

Figure 3

(Color online) $T_c$ vs $p$ at different ratios $J_1∕W_1$, obtained using DMFT. The parameters $x$, $W_1$, $W_2$, and $J_1$ are as in Fig. 1. The inset contains the DOS at $T=0$.

Figure 4

(Color online) (a) MC magnetization (in absolute value) vs temperature $\left(T\right)$, with $J_1=J_2=4$ and $t_1=t_2=1$, for the hole densities indicated. The dashed line is the exact asymptotic high-temperature value $M_{\infty }$, which tends to 0 only in the bulk limit. In this paper, the $T_c$ on the $5^3$ cluster was (arbitrarily) defined as the $T$ where $M$ reaches $\sim 5\%$ of the $1-M_{\infty }$ value [indicated by arrows in (a) and (b)]. Other criteria lead to very similar conclusions. (b) Magnetization vs $T$ for $p_h=1$, $t_1=t_2$, and several (equal) magnetic couplings $J$. (c) Curie temperature vs hole density for $J=4$ and different ratios of the band hoppings. (d) Curie temperature versus the (equal) magnetic interactions $J$ for several hole densities and equal band hoppings $t_1$ and $t_2$. Results for $6^3$ $\left(5^3\right)$ lattices are indicated by closed (open) symbols. Error bars due to the disorder (up to seven samples) are only shown for a few points for clarity.

Figure 5

(Color online) (a) $T_c$ vs $p_h$ obtained with MC on a $6^3$ $\left(5^3\right)$ lattice with $t_1=t_2=1$ for the values of $J_2∕J_1$ indicated by the closed (open) symbols. $J_1$ is fixed to 4, i.e., when the IB are about to separate from the valence band for band 1 [inset Fig. 5b]. For $J_2<J_1$ (e.g., $J_2∕J_1=0.4$ curve), $T_c$ is regulated by the IB in band 1, since the IB for band 2 is deep into the valence band. In this case, a single-band behavior is observed, with $T_c$ maximized for $p_h=0.5$. For $J_2>J_1$, both IB play a role. For $J_2⪢J_1$ (see $J_2∕J_1=2$), the two IB do not overlap, and for $0⩽p_h⩽1T_c$ is determined by the band-2 IB reaching a maximum at $p_h=0.5$ and vanishing at $p_h=1$. For larger $p_h$, $T_c$ is controlled now by the IB 1, and it raises again passing through a maximum at $p_h=1.5$ and vanishing at $p_h=2$. For $J_2∕J_1=1.25$, the two IB overlap and we observe residual local maxima at $p_h=0.5$ and 1.5, related to the single band physics, and a new local maximum at $p_h=1$ due to IB overlap for the corresponding value of the chemical potential. $T_c$ at $p_h=0.5$ is boosted by the partial IB overlap as well. (b) $T_c$ vs $p_h$ for $t_2∕t_1=1$ and several $J$’s. Inset: low-$T$ DOS.