15 obtained for larger concentrations, it is interesting to in- vestigate how our value of J changes when we increase the Mn concentration. In Fig. 18 the effect of decreas- ing the supercell size such that the Mn concentration changes from x = 0 . 0006 to x = 0 . 0017 is shown. The 〈 100 〉$_{d}$$_{=}$$_{a}$ J is now closer to the 〈 211 〉$_{d}$$_{=1}$$_{.}$$_{21}$$_{a}$ value. In- creasing the concentration further causes the two pairs to obtain equal J , but the 〈 100 〉$_{d}$$_{=}$$_{a}$ value never drops below the 〈 211 〉$_{d}$$_{=1}$$_{.}$$_{21}$$_{a}$ value, indicating that there is a fundamental difference with respect to ab initio . Generally, ab initio predicts a qualitative trend that agrees better with our effective J obtained by occupying the lower acceptor (see Fig. 15). The 〈 100 〉$_{d}$$_{=}$$_{a}$ highest occupied level and lower acceptor are quasi-degenerate, with a gap that varies between 14-46 meV. In ab initio methods, a common technique used to speed up evalu- ation of k-point sums, is to introduce a fractional occu- pation of the unoccupied levels controlled by an occupa- tion smearing parameter. At the end of the calculation, the limit of zero smearing is taken to calculate the to- tal ground state energy. It is still possible that the end result depends on the choice of smearing parameter. If this is the case, the effect can cause the J for the quasi- degenerate 〈 100 〉$_{d}$$_{=}$$_{a}$ to decrease. So far we have studied Mn pairs embedded in bulk GaAs. In this last Section we consider the experimentally more relevant situation where the two Mn atom substi- tute two Ga atoms on the (110) surface. We consider again a 3200-atom supercell, but this time we apply pe- riodic only in the two directions in the plane of the (110) surface. This corresponds to a 38 × 38 A 2 surface and a supercell cluster that has 20 atomic layers along the surface normal separating the two (110) surfaces, such that surface-surface interactions are negligible. The loss of coordination and hybridization with the surface states will cause the impurity levels to appear very deep in the gap. The depth is subject to experimental uncertainty due to a band-bending 45 effect when imaging a semi- conductor surface with a metal tip. In our model, the depth is very sensitive to the off-site Coulomb correc- tion, V$_{off}$ , and we use this parameter to reproduce the acceptor level of a single Mn on the (110) surface at the experimentally observed position 4 at 850 meV. The pa- rameter V$_{off}$ has therefore been reduced from 2.4 eV to 1.57 eV. We proceed again to study the magnetic anisotropy energy of the systems, the properties of the mid-gap ac- ceptor states, the effective exchange interaction and its connection with the acceptor splitting. Fig. 19 shows the qualitatively distinct anisotropy landscapes for Mn pairs on the surface and Fig. 20 the magnitude of the barriers. The surface geometry com- pletely dominates the anisotropy and for most separa- tions the anisotropy landscape is qualitatively the same as for a single Mn in the surface 10 [exemplified in Fig. 19 (d)]. This type of anisotropy landscape occurring for sep- arations larger than ≈ 1 . 5 lattice constants is an indica- tion that the acceptor state hybridization is so weak that the resulting anisotropies are essentially the sum of two independent Mn atom anisotropies. Only for the closest pairs do different anisotropies appear. In the FM con- figurations only 〈 110 〉$_{d}$$_{=0}$$_{.}$$_{71}$$_{a}$ , 〈 110 〉$_{d}$$_{=1}$$_{.}$$_{41}$$_{a}$ and 〈 100 〉$_{d}$$_{=}$$_{a}$ 1. Magnetic anisotropy energy FIG. 19: (Color online) The magnetic anisotropy landscapes for Mn pairs in the (110) surface. The transparent plane indicates the (110) surface. (a)-(d) show the distinct types of landscapes in the FM configurations. Only along the 〈 110 〉$_{d}$$_{=0}$$_{.}$$_{71}$$_{a,}$$_{1}$$_{.}$$_{41}$$_{a}$ in (a) and (b), and 〈 100 〉$_{d}$$_{=}$$_{a}$ in (c), do the landscapes differ from the result of weakly interacting pairs in (d), which is just the sum of two isolated Mn anisotropies. Similarly, in the AFM configurations, only 〈 110 〉$_{d}$$_{=0}$$_{.}$$_{71}$$_{a}$ (e) and 〈 100 〉$_{d}$$_{=}$$_{a}$ (f) deviate qualitatively from the landscape (d). B. Mn-Mn interactions in the (110) GaAs surface M AFM (a) (b) (c) (d) (e) (f ) [100] (110) [010] [110] [001] 0 1 0 2 0 1 0 2 0 1 0 1 M M M M