7 allel or antiparallel. 1. Magnetic anisotropy energy 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 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- 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. FIG. 8: (Color online) Magnetic anisotropy in collinear vari- ation of parallel Mn spins on the unit sphere. Generally, an easy axis is found to be parallel to the connecting Mn line, with the exception of the 〈 100 〉$_{d}$$_{=2}$$_{a,}$$_{3}$$_{a}$ . The 〈 110 〉 ∗ shows the effect of non-additive Coulomb correction on the common As neighbor, yielding larger anisotropy barriers. In panel (a) the crystal axes have been marked and the same labels apply to all other panels. Also indicated is the (110) plane that con- tains all Mn pairs. The Mn are numbered in accordance with Fig. 7. The AFM configurations all have a higher total energy and the variations are inverted with respect to the FM ones. E$_{anis}$ ( ˆ Ω) ≡ E$_{tot}$ ( ˆ Ω) − E min tot , (7) (b) (c) (d) (f ) (g) (i) (e) (k) (a) (h) (j) [100] (110) [010] [110] [001] 0 1 0 1 0 2 0 3 0 1 0 2 3 0 0 1 0 2 0 1 0 3