18 FIG. 25: The effective exchange constant for Mn pairs in the (110) surface. The effective ˜ J falls off much faster at the surface, due to the highly localized acceptor wave functions. The 〈 211 〉$_{d}$$_{=1}$$_{.}$$_{22}$$_{a}$ is showing antiferromagnetic behavior. 110 110 100 100 100 211 211 211 111 1.0 1.5 2.0 2.5 3.0 3.5 0 R a 2 -2 56 110 meV -1 1 57 with the calculated acceptor splittings. The FM split- tings of the acceptor levels is shown in Fig. 24 (a). The very large splitting of 600 meV for the closest Mn pair seen in experiment (see Fig. 13) is now comparable to the calculated one of around 500 meV, when both acceptors are in the gap. The calculated splitting is small for the nearest 〈 100 〉 . It then increases to around 100 meV for the 〈 211 〉 , but after that splittings are very small and less than 50 meV, with the exception of the farthest 〈 110 〉 . In the AFM configurations, splittings are much larger at the surface compared to bulk. In particular, the closest 〈 211 〉 has a large splitting of around 70 meV. This is ex- pected, since a finite splitting in the AFM configuration is possible when inversion symmetry is broken, an effect that is clearly enhanced by the presence of the surface. The FM surface splittings do not match the long range experimental trend as well as the bulk calculations. In our calculations, we have accounted for the buckling of the GaAs (110) surface, but not the detailed relaxation of Mn environment. It is also not certain that the effective p - d exchange model holds so well for the surface. Not only the long range and short range Coulomb may differ at the surface, but the exchange splitting is also subject to change. This is because the p - d hybridization scheme in Fig. 1 will be altered by the surface states. The surface ˜ J are shown in Fig. 25. For the nearest 〈 110 〉 pair, ˜ J is strongly ferromagnetic and has a value similar to the corresponding pair in bulk. The 〈 100 〉 pair has a very low ˜ J of around half an meV and could easily become antiferromagnetic by a change in Coulomb pa- rameters. The SO-induced anisotropy of J on the unit sphere of magnetic moment directions is very weak, just fractions of an meV, due to the low anisotropy energies at the surface. Interestingly the 〈 211 〉 pair, which has a relatively large acceptor splitting of several tens meV in the AFM configuration (see Fig. 24), is antiferromag- netic. As mentioned above, this is an effect entirely due to the surface, where a significant acceptor splitting can exist even for antiferromagnetically aligned Mn spins and generate an energy gain that can compete with the FM configuration. The rest of the Mn pairs shown in Fig. 25 have a very small ˜ J , indicating that the Mn local mo- ments interact very weakly due to the high degree of lo- calization of the acceptor wave functions at the surface. The oscillatory behavior is indicative of Friedel oscilla- tions with a much shorter period than in bulk, which is connected to the strong localization of the acceptor wave functions at the surface. Finally, it is interesting to note that the magnitude of ˜ J (disregarding the sign), follows the trend of the experimentally observed split- tings, despite the fact that the calculated ones do not clearly do so. Disregarding the single point 〈 110 〉$_{d}$$_{=1}$$_{.}$$_{41}$$_{a}$ in Fig. 24 (a), the experimental trend but with smaller magnitudes is reproduced. It is also interesting to note the large AFM splitting for 〈 211 〉$_{d}$$_{=1}$$_{.}$$_{22}$$_{a}$ in Fig. 24 (a), which makes it difficult to distinguish from the FM con- figuration. There are however noticeable differences be- tween the 〈 211 〉$_{d}$$_{=1}$$_{.}$$_{22}$$_{a}$ FM and AFM LDOS (see Figs. 22 and 23). The FM acceptors each have spectral weight on both Mn sites, but the AFM acceptors are each associ- ated with one site. Given the large splitting between the AFM acceptors, this observation could perhaps be tested experimentally. Overall, it would appear that the exper- imental results for the (110) surface are intermediary to our surface and bulk calculations. In this paper we have investigated substitutional Mn pairs in GaAs, using a microscopic tight-binding model solved for large GaAs clusters. The model accounts realistically for spin-orbit interactions and includes the kinetic-exchange p - d coupling between the Mn local mag- netic moment and the spin of valence band electrons. One goal of this work was to assess whether or not this model can reproduce the main features of coupled accep- tor states observed in recent STM experiments, 4,5 and in this way further elucidate the nature of the effective exchange interaction between the two Mn magnetic mo- ments. When the two Mn atoms are located in the bulk, we find that the ground state of the system is generally one in which the two Mn magnetic moments are coupled ferromagnetically and aligned along the direction con- necting the Mn atoms. In the ferromagnetic configura- tion, the two topmost acceptor states are split, and their energy separation depends strongly on the Mn pair ori- entation with respect to the GaAs crystal structure and decreases with Mn separation. In particular, the largest splitting ( ∼ 300 meV) is found for the Mn pair oriented along the 〈 110 〉 direction. For separations larger than three lattice constants, the splitting is below 20 meV. The splitting is the result of hybridization of degener- ate acceptor states with identical spin character, giving rise to bonding and antibonding states. Our calculated acceptor splittings are typically a factor of two smaller than the experimental values. Coulomb correlation ef- IV. CONCLUSIONS AND OUTLOOK