Ferromagnetic properties of $p$-(Cd,Mn)Te quantum wells: Interpretation of magneto-optical measurements by Monte Carlo simulations

In order to single out dominant phenomena that account for carrier-controlled magnetism in $p{\text{-Cd}}_{1-x}{\text{Mn}}_x\text{Te}$ quantum wells we have carried out magneto-optical measurements and Monte Carlo simulations of time-dependent magnetization. The experimental results show that magnetization relaxation is faster than 20 ns in the paramagnetic state. Decreasing temperature below the Curie temperature $T_C$ results in an increase of the relaxation time but to less than $10\mu \text{s}$. This fast relaxation may explain why the spontaneous spin splitting of electronic states is not accompanied by the presence of nonzero macroscopic magnetization below $T_C$. Our Monte Carlo results reproduce the relative change of the relaxation time on decreasing temperature. At the same time, the numerical calculations demonstrate that antiferromagnetic spin-spin interactions, which compete with the hole-mediated long-range ferromagnetic coupling, play an important role in magnetization relaxation of the system. We find, in particular, that magnetization dynamics is largely accelerated by the presence of antiferromagnetic couplings to the Mn spins located outside the region, where the holes reside. This suggests that macroscopic spontaneous magnetization should be observable if the thickness of the layer containing localized spins will be smaller than the extension of the hole wave function. Furthermore, we study how a spin-independent part of the Mn potential affects $T_C$. Our findings show that the alloy disorder potential tends to reduce $T_C$, the effect being particularly strong for the attractive potential that leads to hole localization.

Figure 1

Experimental. Temporal decay of magnetization in the ${\text{Cd}}_{0.96}{\text{Mn}}_{0.04}\text{Te}$ quantum well at various temperatures determined as a difference in photoluminescence intensities collected for two circular polarizations after a pulse of the magnetic field. The Curie temperature is 2.5 K.

Figure 2

Experimental. Temporal decay of magnetization in the ${\text{Cd}}_{0.96}{\text{Mn}}_{0.04}\text{Te}$ quantum well determined as a difference in photoluminescence intensities for two circular polarizations collected at various wavelengths after a pulse of the magnetic field. The Curie temperature is 2.5 K.

Figure 3

Monte Carlo simulations. The $z$ component of the absolute spin projections $⟨\left|S_z\right|⟩$ and $⟨\left|s_z\right|⟩$ (open symbols—dotted lines; left scale) and the spin susceptibilities ${\chi }_S$ and ${\chi }_s$ (full symbols—solid lines; right scale) [(a) and (c)] for the Mn ions and [(b) and (d)] for the holes in the absence of alloy disorder (no AD, $\delta V=0$) and for the repulsive $\left(+\text{AD},\delta V>0\right)$ alloy disorder potential. A maximum in the temperature dependence of spin susceptibility is identified as the Curie temperature below which the holes induce a long-range ferromagnetic order. Here the number of holes [(a) and (b)] $N_h=13$ and [(c) and (d)] 29, i.e., $p=2.54\times {10}^{10}$ and $5.66\times {10}^{10}{\text{cm}}^{-2}$, respectively.

Figure 4

Monte Carlo simulations. the $z$ component of the absolute spin projections $⟨\left|S_z\right|⟩$ and $⟨\left|s_z\right|⟩$ (open symbols—dotted lines; left scale) and the spin susceptibilities ${\chi }_S$ and ${\chi }_s$ (full symbols—solid lines; right scale) for the [(a) and (c)] Mn ions and for the [(b) and (d)] holes in the absence of alloy disorder (no AD, $\delta V=0$) and for the attractive $\left(-\text{AD},\delta V<0\right)$ alloy disorder potential. A maximum in the temperature dependence of spin susceptibility is identified as the Curie temperature below which the holes induce a long-range ferromagnetic order. Here the number of holes [(a) and (b)] $N_h=13$ and [(c) and (d)] 45, i.e., $p=2.54\times {10}^{10}$ and $8.78\times {10}^{10}{\text{cm}}^{-2}$, respectively.

Figure 5

Curie temperature $T_C$ as a function of the hole area density $p$. Solid symbols show results of Monte Carlo simulations carried out for classical spins, QW width $L_W\approx 5.2\text{nm}$, and Mn content 4% neglecting carrier correlations in the absence of alloy disorder ($\delta V=0$, squares) as well as for repulsive and attractive alloy potentials ($\delta V>0$, triangles and $\delta V<0$, circles, respectively). Results obtained within the mean-field approximation (i.e., disregarding thermal and chemical fluctuations) are shown by the solid line. Dashed line presents the MFA results taking into account carrier correlations with the Landau parameter $A_F=2$.

Figure 6

Curie temperature $T_C$ as a function of the hole areal density $p$. Solid symbols show results of Monte Carlo simulations carried out for classical spins, QW width $L_W\approx 5.2\text{nm}$, and the Mn content 4% in the absence of alloy disorder ($\delta V=0$, squares) and for repulsive alloy potentials ($\delta V>0$, triangles) obtained with the Fermi-liquid Landau parameters $A_F=1.5$ and $A_F=3$. Experimental results (Ref. 6) for QW width $L_W\approx 8\text{nm}$ and Mn content 3%–5% for two samples (open circles and triangles).

Figure 7

Monte Carlo simulations. $z$ component of the absolute spin projection of Mn spins $⟨\left|S_z\right|⟩$ at 0.1 K in the absence of alloy disorder ($\delta V=0$, squares) and for repulsive alloy potential ($\delta V>0$, triangles). Dashed straight line represents a dependence expected within the MFA.

Figure 8

Monte Carlo simulations. Time evolution of $S_{\text{Mn}}^z\left(t\right)/S$ at different temperatures in a ${\text{Cd}}_{0.96}{\text{Mn}}_{0.04}\text{Te}$ quantum well. The hole concentration is $p=0.4\times {10}^{11}{\text{cm}}^{-2}$ $\left(N_h=21\right)$. Time is measured in the Monte Carlo steps per Mn site.

Figure 9

Monte Carlo simulations. The magnetization autocorrelation function $G\left(\tau \right)$ of Mn ions in a ${\text{Cd}}_{0.96}{\text{Mn}}_{0.04}\text{Te}$ quantum well calculated at different temperatures. The concentration of holes $p=0.4\times {10}^{11}{\text{cm}}^{-2}$ $\left(N_h=21\right)$. Time is measured in the Monte Carlo steps per one Mn site. Thin solid lines describe exponential decays fitted to the Monte Carlo results

Figure 10

Monte Carlo simulations. The magnetization autocorrelation function $G\left(\tau \right)$ for selected values of hole concentrations in a ${\text{Cd}}_{0.96}{\text{Mn}}_{0.04}\text{Te}$ quantum well at $T=0.7\text{K}$. Time is measured in the Monte Carlo steps per Mn site.

Figure 11

Monte Carlo simulations. The magnetization autocorrelation function $G\left(\tau \right)$ of Mn ions in a ${\text{Cd}}_{0.96}{\text{Mn}}_{0.04}\text{Te}$ quantum well at $T=0.7T_C$. The hole density is $p=0.4\times {10}^{11}{\text{cm}}^{-2}$ $\left(N_h=21\right)$. Open circles and full squares: antiferromagnetic interactions are neglected $\left(J_{ij}=0\right)$ and taken into account $\left(J_{ij}\ne 0\right)$, respectively. Time is measured in the Monte Carlo steps per Mn site.

Figure 12

Monte Carlo simulations. The magnetization autocorrelation function $G\left(\tau \right)$ of Mn ions in a ${\text{Cd}}_{0.96}{\text{Mn}}_{0.04}\text{Te}$ quantum well at $T=0.7T_C$. The hole density is $p=0.4\times {10}^{11}{\text{cm}}^{-2}$ $\left(N_h=21\right)$. Open circles and full squares: Mn located in the central layers of the QW only and evenly distributed within the whole QW, respectively. Time is measured in the Monte Carlo steps per Mn site.

Figure 13

Monte Carlo simulations. The two-point connected correlation function $G_c^{\left(2\right)}\left(R\right)$ of valence-band holes at various temperatures in a ${\text{Cd}}_{0.96}{\text{Mn}}_{0.04}\text{Te}$ quantum well. The hole concentration is $p=0.4\times {10}^{11}{\text{cm}}^{-2}$ $\left(N_h=21\right)$.

Figure 14

Monte Carlo simulations. The two-point connected correlation function $G_c^{\left(2\right)}\left(R\right)$ of Mn ions in a ${\text{Cd}}_{0.96}{\text{Mn}}_{0.04}\text{Te}$ quantum well at $T=0.2T_C$. The hole concentration is $p=0.4\times {10}^{11}{\text{cm}}^{-2}$ $\left(N_h=21\right)$.

Figure 15

Monte Carlo simulations. The long-distance parts of two-point connected correlation functions $G_c^{\left(2\right)}\left(R\right)$ of Mn ions at various temperatures in a ${\text{Cd}}_{0.96}{\text{Mn}}_{0.04}\text{Te}$ quantum well. The hole concentration is $p=0.4\times {10}^{11}{\text{cm}}^{-2}$ $\left(N_h=21\right)$.