Magnetic susceptibility of diluted magnetic semiconductors at low carrier densities

We calculate the static longitudinal and the transverse dynamic magnetic susceptibilities of (III,Mn)V diluted magnetic semiconductors, using the random phase approximation, for a simple impurity band model appropriate for the low charge carrier concentration regime. The magnetic susceptibilities are shown to depend sensitively on the amount of positional disorder of the Mn impurities. The results we obtain are consistent with previous studies of the spin wave spectrum and of the spatially inhomogeneous ferromagnetic state of these materials.

Figure 1

${\chi }_{\mathrm{Mn}}\left(T\right)$ (full) and ${\chi }_h\left(T\right)$ (dotted line), for an ordered Mn configuration with $N_d=512$, $x=0.00926$, and $p=10\%$. The inset shows the corresponding magnetizations.

Figure 2

Weak, medium and full disorder systems’ susceptibilities vs $T$, for $N_d=216$, $x=0.00926$, and $p=10\%$. Inset focuses on the low-temperature region.

Figure 3

${\chi }_{\mathrm{Mn}}\left(T\right)$ and ${\chi }_h\left(T\right)$ for single fully disordered impurity configuration. $N_d=512$, $x=0.00924$, $p=0.10$.

Figure 4

Left: total susceptibility of a disordered sample, near $T_c$. Right: $S_{\mathrm{Mn}}(i,T)$ for spins belonging to two different clusters. $N_d=125$, $x=0.00924$, $p=0.10$.

Figure 5

$\chi (\vec{q},\omega )$ for the ordered case. ${\vec{q}}_1=0$, ${\vec{q}}_2=(\pi ∕15a)(2,0,0)$ and ${\vec{q}}_3=2{\vec{q}}_2$. $N_d=125$, $x=0.00924$, $p=0.1$, $T=0$.

Figure 6

Spin-wave energies $\hslash {\omega }_{\vec{q}}\left(T\right)$ for ${\vec{q}}_1=(\pi ∕9a)(1,1,0)$, ${\vec{q}}_2=2{\vec{q}}_1$, and ${\vec{q}}_3=3{\vec{q}}_1$; $N_d=216$, $x=0.00924$, $p=0.1$.

Figure 7

$\chi (\vec{q},\omega )$ for an individual fully disordered configuration, for ${\vec{q}}_1=0$, ${\vec{q}}_2=(\pi ∕15a)(4,0,0)$, ${\vec{q}}_3=(\pi ∕15a)(8,0,0)$, and ${\vec{q}}_4=(\pi ∕15a)(8,8,8)$. $N_d=125$, $x=0.00924$ and $p=0.1$, $T=0$.

Figure 8

$\chi (\vec{q},\omega )$ for ordered (dashed) and weakly disordered (full) systems. $T=0$, $x=0.00924$, $p=0.10$, $N_d=125$, and ${\vec{q}}_1=(\pi ∕15a)(2,0,0)$, ${\vec{q}}_2=2{\vec{q}}_1$, ${\vec{q}}_3=(\pi ∕15a)(4,4,0)$, ${\vec{q}}_4=(\pi ∕15a)(4,4,4)$.

Figure 9

$\chi (\vec{q},\omega )$ for ordered (dashed) and weakly disordered (full) systems, for $\vec{q}=(\pi ∕15a)(2,0,0)$, $T=0$, $x=0.00924$, $p=0.10$, $N_d=125$. Inset: the high-energy tail for fully disordered configurations, with $\vec{q}=(\pi ∕15a)(4,0,0)$ $(N_d=125)$ and $\vec{q}=(\pi ∕18a)(4,0,0)$ $(N_d=125)$.