Disorder, spin-orbit, and interaction effects in dilute ${\mathrm{Ga}}_{1-x}{\mathrm{Mn}}_x\mathrm{As}$

We derive an effective Hamiltonian for ${\mathrm{Ga}}_{1-x}{\mathrm{Mn}}_x\mathrm{As}$ in the dilute limit, where ${\mathrm{Ga}}_{1-x}{\mathrm{Mn}}_x\mathrm{As}$ can be described in terms of spin $F=3∕2$ holes hopping between the Mn sites and coupled to the local Mn spins. We determine the parameters of our model from microscopic calculations using both a variational method and an exact diagonalization within the so-called spherical approximation. Our approach treats the extremely large Coulomb interaction in a nonperturbative way and captures the effects of strong spin-orbit coupling and Mn positional disorder. We study the effective Hamiltonian in a mean-field and variational calculation, including the effects of interactions between the holes at both zero and finite temperature. We study the resulting magnetic properties, such as the magnetization and spin-disorder manifest in the generically noncollinear magnetic state. We find a well-formed impurity band fairly well separated from the valence band up to $x_{\text{active}}\lesssim 0.015$ for which finite-size scaling studies of the participation ratios indicate a localization transition, even in the presence of strong on-site interactions, where $x_{\text{active}}<x_{\mathrm{nom}}$ is the fraction of magnetically active Mn. We study the localization transition as a function of hole concentration, Mn positional disorder, and interaction strength between the holes.

Figure 1

Spin-orientation dependence of the ground-state energy of one hole on a Mn dimer parallel to the $z$ axis for a spatial separation of $z_0=14\mathrm{\AA }$. The Hamiltonian (3) is diagonalized exactly with the parameters shown in Fig. 4. As the spins are rotated by an angle $\theta$ away from the $z$ axis the energy increases and reaches a maximum at $\theta =90°$ before again decreasing. This indicates the magnetic anisotropy is easy axis.

Figure 2

(Color online) Radial wave functions obtained from a variational calculation for $\mu =0.767$, the relevant value for GaMnAs. For $r\gtrsim 15\mathrm{\AA }$, the typical $\mathrm{Mn}–\mathrm{Mn}$ distance at $x=0.01$, $g_0\left(r\right)\approx f_0\left(r\right)$. The radial wave functions obey the normalization condition ${\int }_0^{\infty }dr{r}^2[f_0{\left(r\right)}^2+g_0{\left(r\right)}^2]=1$. From Eq. (7) this means the $d$-wave component of the wave function is important for typical Mn concentrations at $x=0.01$. It is thus expected that the nonhydrogenic nature of the wave function will strongly affect the magnetic and transport properties of dilute GaMnAs, a result supported by our numerical calculations presented in Sec. 4.

Figure 3

Polarization of a bound hole in the state $\mid F=3∕2,F_z=3∕2⟩$ in ${\mathrm{Ga}}_{1-\mathrm{x}}{\mathrm{Mn}}_{\mathrm{x}}\mathrm{As}$ around a Mn ion (dark arrow pointing downward represents the Mn $\mathrm{S}=\frac{5}{2}$ spin). Only the direction of the polarization is indicated. The magnitude falls off on a scale $\sim 10\mathrm{\AA }$, as indicated by Fig. 2.

Figure 4

(Color online) Parameters of the two-impurity Hamiltonian Eq. (15) omryobtained from the variational study of two Mn ions. The arrow indicates the typical $\mathrm{Mn}\mathrm{Mn}$ distance, $d_{\mathrm{typ}}$, for $x=0.01$ Mn concentration.

Figure 5

(Color online) Magnetization as a function of temperature, Mn concentration $x$, and Monte Carlo time $t_{\mathrm{MC}}$ for different hole fractions $f$. Fully polarized states have a magnetization of 1. Here $L=10a_{\mathrm{lat}}$, $U=0$, and 100 samples are averaged over. Top: Hole fraction $f=0.1$. Bottom: Hole fraction $f=0.3$. In both cases, as the Monte Carlo time increases, for fixed $x$, the saturation magnetization at zero temperature increases. For both values of $f$ the curves remain linear over a fairly wide temperature range, much the same as for experimentally measured curves for unannealed GaMnAs. The saturation magnetization never reaches more than $\sim 60\%$ of the fully saturated value. This is consistent with the wide spin distribution shown in Fig. 7 and indicates that the ferromagnetism is noncollinear.

Figure 6

(Color online) Magnetization as a function of temperature, hole fraction $f$, and Monte Carlo time $t_{\mathrm{MC}}$ for different Mn concentrations $x$. Fully polarized states have a magnetization of 1. Here $L=10a_{\mathrm{lat}}$, $U=0$ and 100 samples are averaged over. Compare to Fig. 5. Top: Mn concentration $x=0.005$. Middle: Mn concentration $x=0.01$. Bottom: Mn concentration $x=0.015$. The general trend is the same as in Fig. 5: Longer Monte Carlo times lead to larger zero temperature magnetizations. The saturation magnetization is roughly independent of Mn concentration $x$.

Figure 7

(Color online) Top: The dependence of the spin distribution function, $P\left(\cos \left(\theta \right)\right)$, on the on-site interaction, $U=U_N$, and the Monte Carlo time for $L=10a_{\mathrm{lat}}$, $x=0.01$, and $f=0.30$. We averaged over 10 samples. Here $\theta$ is the angle an individual spin makes with the net magnetization direction, as described in the text. Even with interactions and at large Monte Carlo times (small disorder) the spin distribution function remains broad. This is consistent with the strong reduction of the saturation magnetization $(\sim 60\%)$ observed in our calculations, independent of Monte Carlo time. Bottom: Dependence of the spin distribution on the hole fraction $f$, $U=0$, obtained after averaging over 100 samples.

Figure 8

(Color online) Top: The dependence of the density of states on doping $x$ for $L=10a_{\mathrm{lat}}$, $f=0.50$, and $t_{\mathrm{MC}}=0$. Data is the average of 50 sample realizations. The DOS is normalized to the volume of a unit cell, so the total number of states is proportional to $x$. The half-width of the impurity band ranges from $1000–2500\mathrm{K}$ and is centered around $-1100\mathrm{K}$, the binding energy of a hole at an isolated Mn. The shape of the density of states changes little with the Mn concentration, $x$, over the range of values shown. The value of the Fermi energy is $\approx -5000\mathrm{K}$. For comparison, the valence band density of states is also shown. Bottom: The dependence of the participation ratio on doping $x$. Data is averaged over 50 samples. Larger samples have larger values of the PR for delocalized states, while for localized states the PR is independent of system size for fixed $x$, $t_{\mathrm{MC}}=0$. The energy value that separates $L$-dependent PRs from $L$-independent PRs is the mobility edge. This depends on $x$ and is larger for larger $x$. For the disordered samples here, the mobility edge is not too sharp and lies in the tail of the density of states.

Figure 9

(Color online) Top: The dependence of the density of states on Monte Carlo time for $U=0$, $x=0.01$, $f=0.30$, and $L=10a_{\mathrm{lat}}$. Data is the average of 50 sample realizations. MC time $t_{\mathrm{MC}}=0$ means that the Mn positions are completely random; for $t_{\mathrm{MC}}=10$ the Mn ions form a nearly BCC lattice with a few point defects. The main effect of disorder is thus to broaden the impurity band. The width of the impurity band is proportional to the value of the dominant hopping parameter, $t_{3∕2}$, at typical Mn separations as can be seen from Fig. 4. Bottom: The dependence of the PR on the MC time for $x=0.01$ and $f=0.30$. Data is averaged over 50 samples. The mobility edge also moves up to higher energy values for more ordered Mn configurations and nearly all states become delocalized.

Figure 10

(Color online) Top: The dependence of the density of states on the on-site interactions, $U=2600\mathrm{K}$, for $x=0.01$, $f=0.30$, and $t_{\mathrm{MC}}$. The effect of interactions on the DOS is minimal: The overall shape remains largely unchanged by the interactions. Bottom: The dependence of the PR on the on-site interactions, $U$, for $x=0.01$, $f=0.30$, $L=10a_{\mathrm{lat}}$, and $t_{\mathrm{MC}}=0$. Data is averaged over 10 samples. The behavior of the participation ratio follows roughly that of the density of states shown in top; there is little shape change compared to the noninteracting case.

Figure 11

(Color online) Wave function overlaps for the two states on site 1 and site 2. The overlaps are computed from Eq. (B5).

Figure 12

(Color online) Interaction overlaps for the interactions defined in the text, Eqs. (B6, B7).

Figure 13

(Color online) The eight lowest-lying states of the $\mathrm{Mn}–\mathrm{Mn}$ pair as a function of Mn separation. Each state is twofold degenerate. At large distances the energies converge to the binding energy of a single hole on a single Mn ion, $-1120\mathrm{K}$.