8 Understanding the microscopic mechanisms of mag- netic anisotropy in DMS is crucial in order be able to manipulate the magnetization vector by magnetic and electric fields. 30 For a review of recent theoretical and ex- perimental works on magnetic anisotropy in (Ga, Mn)As see Ref. [31], where ferromagnetic samples with a rela- tively high Mn content of a few percent are examined. Comparison with our results, which instead pertain to the magnetic anisotropy of isolated Mn dimers, is not straightforward. Our results could on the other hand be directly compared with those of STM experiments using a spin-polarized magnetic tip 32 and an external magnetic field. As we show below, all the AFM pairs have a higher to- tal energy, which is expected on the basis of the heuristic toy model considerations. The AFM anisotropy land- scapes are qualitatively inverted with respect to the FM ones, i.e., the bistable minima along the Mn dimer axis in the FM variation are replaced by quasieasy planes per- pendicular to the dimer axis in the AFM variation. This inversion of hard and easy directions between FM and AFM can be understood by considering the FM configu- ration in the easy direction, where the spin-orbit splitting between the highest occupied and the lowest unoccupied levels is large, resulting in a lower total energy. In the corresponding AFM configuration, the two levels are of opposing spin character, such that the splitting is smaller and the total energy instead higher. In Fig. 9 (b), we see that the closer the Mn are, the smaller the anisotropy en- ergies tend to be. This figure gives us an idea of the range of strongly interacting Mn. Weaker interactions at larger distances in the AFM configuration tends to increase the anisotropy, as the effectively spin-polarized region in the lattice around each Mn increases. The anisotropy of the embedded Mn dimers is accu- rately reflected in the anisotropy of the two acceptor lev- els and their associated wave functions. In this section we therefore perform a detailed analysis of various prop- erties of the acceptor levels, such as their splitting and LDOS - quantities that are directly accessible by STM spectroscopy. From now on we use the convention to denote with ε$_{−}$$_{3}$ , ε$_{−}$$_{2}$ , ε$_{−}$$_{1}$ , ε$_{0}$ the energies of four highest occupied states, while ε$_{1}$ and ε$_{2}$ denote the energies of the two acceptor levels. When mixing with the conduction band is negligible, the anisotropy energy landscape can be ex- tracted from the the two acceptor levels using 2. Character of acceptors for Mn pairs in bulk GaAs where E$_{anis}$ ( ˆ Ω) is defined in Eq. (7) and the upper limit in the sum can be generalized to a greater number of ac- ceptor levels. Equation (8) follows from the fact that the additive 53 Coulomb correction on the common As neigh- bor, where the low and high barriers have increased to 9 and 14 meV. As we let the Mn move apart one step further along the 〈 110 〉 with distance d = 1 . 4 a, the high barrier attains the maximum value of all considered con- figurations of 21 meV, and a low barrier of 12 meV. At the longest considered separation along 〈 110 〉 , the low and high barriers are now 12 and 16 meV, respectively. The 〈 211 〉 Mn pairs follow a similar evolution of the high and low barrier, relative the connecting Mn line, now with a low barrier along the [110] direction that changes to the high barrier with increasing distance. Distances are longer than for the 〈 110 〉 pairs and therefore hy- bridization is weaker and the anisotropy energies drops by a few meV. At the larger separations along 〈 211 〉 the bistable minima are tilted away from the connecting line, an indication that the single Mn anisotropy is becom- ing comparable to the Mn-Mn interaction energy. Even with a supercell of 3200 atoms we cannot exclude pos- sible finite-size-effect contributions to the anisotropies. Nevertheless, we can get a good estimate of the barriers separating the generally bistable minima. The magnetic anisotropy in (Ga,Mn)As nanostructures, such as (Ga, Mn)As epilayers, is presently a topic of great interest. E$_{anis}$ ( ˆ Ω) = − 2 ∑ i =1 [ ε$_{i}$ ( ˆ Ω) − ε min i ] , (8) FIG. 9: High and Low magnetic anisotropy barriers. In these graphs, zero corresponds to the minimum energy on the unit sphere. The anisotropy increases when the Coulomb correc- tion is taken to be non-additive for 〈 110 〉 ∗ in the FM con- figuration (a). The 〈 110 〉$_{d}$$_{=1}$$_{.}$$_{41}$ shows enhanced anisotropy barriers, with strong interactions along this symmetry direc- tion. 〈 100 〉 has a single barrier for the closest distance, but exhibits quasi-easy-planes (low barrier is zero) perpendicular to the connecting Mn line for longer distances. In the AFM configurations (b), the anisotropies gradually increase with distance as cancellations between the two acceptor levels de- crease. 110 110 110 110 100 100 100 211 211 211 111 1.0 1.5 2.0 2.5 3.0 3.5 0 5 10 15 20 25 R a FM High Low Barrier meV (a) 110 110 110 110 100 100 100 211 211 211 111 1.0 1.5 2.0 2.5 3.0 3.5 0 5 10 15 R a AFM High Low Barrier meV (b)