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allel or antiparallel. 

We begin by looking at the magnetic anisotropy of the system, which is defined as the dependence of the to- tal ground-state energy E$_{tot}$ ( ˆ Ω) on the direction of the Mn-pair magnetic moment, ˆ Ω. For the AFM configura- tion, ˆ Ω is the direction of the staggered moment. Graphi- cal representations of the magnetic anisotropy landscapes E$_{tot}$ ( ˆ Ω) on the unit sphere of the all possible directions for ˆ Ω, are presented in Fig. 8 for the FM configurations. Each panel (a)-(k) refers to a Mn pair with a particular separation and orientation in the crystal, according to the notation defined in Fig. 7.In these figures, what we actually plot is 

as a function of the magnetic moment direction ˆ Ω. Each point of the “anisotropy surface” is obtained by mea- suring the ”distance” $^{(}$E$_{tot}$ ( ˆ Ω) − E min tot ) from the center of the reference parallelepiped defined by the cubic axes [100], [010], and [001]. The center of the parallelepiped is also taken to be the center of the unit sphere of direc- tions. Here E min tot is the minimum value of E$_{tot}$ ( ˆ Ω) upon varying ˆ Ω that occurs along one of the easy directions. The actual values of the high and the low energy bar- riers of these systems are shown in Fig. 9. In the FM configurations, the bistable easy directions are generally found to be parallel to Mn pair axis. The two excep- tions are the quasieasy planes perpendicular to the con- necting line, formed when the Mn are spaced at two and three lattice constants along 〈 100 〉 [see Fig. 8 (f) and (g)]. For these two configurations, the magnetic hard direction has 12-15 meV higher energy. The qualitatively different landscapes signal that the interactions along the 〈 100 〉 direction differ from the other crystalline directions. For the shortest separation along 〈 100 〉 , d = a = 5 . 65 A, a bistable easy axis parallel to the Mn-connecting line with a blocking barrier of around 11 meV is found. This means that a level crossing at the Fermi level occurs when the distance is increased from one to two lattice constants. Associated with this crossing is a change in the orbital character of the acceptors that results in a qualitatively different anisotropy. 

Focusing now on the 〈 110 〉 [with the Mn pair along [1 ¯ 10], see Fig. 8 (b)-(d)], for which closely spaced As and Ga provide more direct hopping paths between the Mn, we see that there is a low barrier along the [001] axis and a high barrier along [110]. At the closest spacing, d = 0 . 7 a , the anisotropy energy is very small with bar- riers of 1-3 meV [see Fig. 8 (b)]. This configuration is special, because the off-site Coulomb correction on the common nearest neighbor As between the Mn is additive, giving it a large on-site energy that reduces anisotropy. The 〈 110 〉 ∗ pair in Fig. 8 (a) shows the effect of a non- 

## 1. Magnetic anisotropy energy 

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