Positional disorder, spin-orbit coupling, and frustration in ${\mathrm{Ga}}_{1-x}{\mathrm{Mn}}_x\mathrm{As}$

We study the magnetic properties of metallic ${\mathrm{Ga}}_{1-x}{\mathrm{Mn}}_x\mathrm{As}$. We calculate the effective Ruderman-Kittel-Kasuya-Yoshida interaction between Mn spins using several realistic models for the valence band structure of GaAs. We also study the effect of positional disorder of the Mn on the magnetic properties. We find that the interaction between two Mn spins is anisotropic due to spin-orbit coupling both within the so-called spherical approximation and in the more realistic six band model. The spherical approximation strongly overestimates this anisotropy, especially for short distances between Mn ions. Using the obtained effective Hamiltonian we carry out Monte Carlo simulations of finite and zero temperature magnetization and find that, due to orientational frustration of the spins, noncollinear states appear in both valence band approximations for disordered, uncorrelated Mn impurities in the small concentration regime. Introducing correlations among the substitutional Mn positions or increasing the Mn concentration leads to an increase in the remanent magnetization at zero temperature and an almost fully polarized ferromagnetic state.

Figure 1

(Color online.) Effective RKKY kernel in the four-band spherical approximation. The contribution from the heavy hole sector is also shown. At typical $\mathrm{Mn}-\mathrm{Mn}$ separations the structure of the kernel is very sensitive to the value of the hole fraction $f$. For $f=0.10$ one has $y^{\mathrm{typ}}=d^{\mathrm{typ}}{k}_{F,h}=1.6$, while for $f=0.25$ we have $y^{\mathrm{typ}}=d^{\mathrm{typ}}{k}_{F,h}=2.2$. In both cases the perpendicular part of the kernel is ferromagnetic, but for $f=0.25$ the parallel component is antiferromagnetic and is comparable in absolute value to the perpendicular component.

Figure 2

(Color online.) Effective exchange $J$ and anisotropy $\delta J$ for two Mn ions. We used the cutoff scheme (20) with $a_0=4\mathrm{\AA }$, and a cutoff energy $E_{\text{cutoff}}=3273\mathrm{K}$. The main parts show the obtained interaction for $L=23a$ and $29a$ at concentrations corresponding to “closed shells,” while the insets show the $L=29a$ results for all concentrations.

Figure 3

(Color online.) Spin orientation dependence of the ground state energy for a fixed number of holes corresponding to a hole concentration of $p=0.2{\mathrm{nm}}^{-3}$: (a) One of the spins is aligned along the $z$ axis and the other is rotated in the $xz$ plane. (b) The spins are parallel and rotated in the $xz$.

Figure 4

(Color online.) Comparison of the hole dispersion within the full six band model, the four band model, and the spherical approximation. The latter two are reasonable approximations for $E_F<0.15–0.2\mathrm{eV}$, which corresponds to a hole density of $p<0.25–0.4{\mathrm{nm}}^{-3}$.

Figure 5

(Color online.) Average anisotropy for $0.2{\mathrm{nm}}^{-3}<p<0.3{\mathrm{nm}}^{-3}$. We used the same parameters as in Fig. 2. The anisotropy increases with distance and becomes of the size $\sim 20\%$ for $R=2a$, which is around the typical $\mathrm{Mn}\mathrm{Mn}$ distance for $x=5\%$, indicated by the arrow.

Figure 6

(Color online.) Position- and hole concentration-dependence of the exchange and anisotropy of the RKKY interaction within the six band model. Distances are given in angstroms, Å.

Figure 7

(Color online.) Average nearest neighbor Mn distance vs Monte Carlo time for $L=12a$ and $17a$.

Figure 8

(Color online.) Magnetization vs temperature for hole fractions $f=0.10$ (top) and $f=0.25$ (bottom) as a function of Monte Carlo time. Note that for $f=0.1$ the isotropic kernel results in nearly full polarization even for the completely disordered sample demonstrating that the reduction of the magnetization is dominantly due to anisotropy effects.

Figure 9

(Color online.) Ground state spin distribution for $f=0.10$. The angle $\theta$ is measured with respect to the ground state magnetization.

Figure 10

(Color online.) Left panel: the $L=\infty$ extrapolated Curie temperature as a function of Monte Carlo time. The dashed curves are a guide to the eye. Inset to left panel: The scaling of the peak in susceptibility with system size $L$, for $t_{\mathit{MC}}=2.5$. Right panel: $M(T=0)$ for $f=0.10$ and $L=17a$.

Figure 11

(Color online.) $T=0$ temperature hysteresis loops for $x=0.05$ and $f=0.1$ and annealing time $t_{\mathit{MC}}=2.5$ as a function of system size $L=8a$, $L=14a$, and $L=17a$. Here $h$ is given in units of $E_J$ defined in Eq. (13). The coercive field is extremely small and the loop shrinks continuously, suggesting the absence of hysteresis in an infinite system.

Figure 12

(Color online.) Temperature dependence of the magnetization for a sample of linear size $L=9a$ with active Mn concentration $x=0.05$, hole fractions $f=0.25$ and 0.4 as a function of Monte Carlo relaxation time. The insets show the low temperature saturation of the magnetization.

Figure 13

(Color online.) Temperature dependence of the magnetization for a sample of linear size $L=10a$ with active Mn concentration $x=0.03$, hole fractions $f=0.25$ and 0.4 as a function of Monte Carlo relaxation time. The insets show the low temperature saturation of the magnetization.

Figure 14

The contours used in the evaluation of Eq. (B4). The dots indicate possible singularities of the integrand.