14 anisotropic properties. Secondly, all first-principle cal- culations suffer from the well-known limitations of DFT when applied to semiconductors. In particular, the es- timates of J depend on the version of DFT used, e.g. LSDA vs. GGA. When the GGA+U approach is used to account for electronic correlation effects, the value of the parameter U , which has as strong influence on the results, can only be determined indirectly by comparing with experiment. In Ref. [35] the electronic structure of (Ga,Mn)As was calculated from first principles including disorder via the coherent-potential approximation. The magnetic force theorem and one-electron Greens functions then al- low mapping onto an effective Heisenberg Hamiltonian. Their results for the exchange interaction strength re- veal a strong dependence on the doping concentration x . In particular, the effective J of the nearest 〈 110 〉 pair is highly sensitive to doping and increases dramatically with decreasing Mn concentration, ranging from J = 19 meV for x = 0 . 08 to J = 55 meV for x = 0 . 001. The lowest concentration of 0.1% Mn, agrees very well with our estimate. Note that our calculations correspond to a slightly smaller effective Mn concentration x = 0 . 0006. In Ref. [9] the authors employed a self-consistent LSDA atomic-spheres approximation 36 and their result of the effective J = 55 meV for the 〈 110 〉$_{d}$$_{=0}$$_{.}$$_{71}$$_{a}$ pair agrees remarkably well with our values, despite the small super- cell corresponding to x = 0 . 009. With spin-orbit interac- tion this value drops to 48 meV. In Ref. [37] the authors employ the GGA+U method within the Projector Aug- mented Wave (PAW) ab initio approach, treating U as a parameter. For 〈 110 〉$_{d}$$_{=0}$$_{.}$$_{71}$ and x = 0 . 03 they observe a decreasing J as a function of U , with J = 34 meV for U = 0 and J = 22 meV for U = 6 eV. As the Mn concen- tration is reduced from x = 0 . 03 to x = 0 . 008 at U = 0, an increase in J of approximately 10 meV is observed. A larger U parameter causes the d levels to emerge deeper in the valence band, the acceptor wave function becomes more delocalized and p - d exchange decreases $^{38}$. Photoe- mission experiments 39 indicate that the t$_{2}$$_{g}$ -states should be approximately 4 eV below the valence band maximum, which corresponds to U = 3 − 4 eV. Ref. [37] shows that the calculated J depends on the extension of the hole wave function as per chosen U . We now turn to the 〈 100 〉$_{d}$$_{=}$$_{a}$ pair, which has a very low experimental acceptor splitting (see Fig. 13). In con- trast to our results (see Fig. 14), ab initio calculations generally predict 35,37,40,41 a dip in the curve of J vs Mn separation occurring for this pair, with a lower value than the two following points, 〈 211 〉$_{d}$$_{=1}$$_{.}$$_{21}$$_{a}$ and 〈 110 〉$_{d}$$_{=1}$$_{.}$$_{41}$$_{a}$ . Similar but smaller dips are also found for the other 〈 100 〉 pairs. 55 However, as mentioned above, all these calcula- tions show a significant dependence on impurity concen- tration. For example, Ref. [35] finds that the value of J for the 〈 100 〉$_{d}$$_{=}$$_{a}$ pair steadily increases from approx- imately J = 1 meV for x = 0 . 1, to J = 7 meV for x = 0 . 02. So do the next two points for 〈 211 〉 and 〈 110 〉 which end up at approximately 10 meV and 11 meV for x = 0 . 02. When U = 0 and x = 0 . 008, the GGA + U approach finds J = 6 meV, well below the values for the next two pairs 〈 211 〉$_{d}$$_{=1}$$_{.}$$_{21}$$_{a}$ and 〈 110 〉$_{d}$$_{=1}$$_{.}$$_{41}$$_{a}$ , J = 14 meV and J = 9 meV respectively, which are also com- parable to our results. Upon increasing U to the physi- cally reasonable value of 6 eV, for both 〈 211 〉$_{d}$$_{=1}$$_{.}$$_{21}$$_{a}$ and 〈 110 〉$_{d}$$_{=1}$$_{.}$$_{41}$$_{a}$ J = 7 meV, while J = 3 . 5 meV for the 〈 100 〉$_{d}$$_{=}$$_{a}$ pair. For even larger values of U the difference in J between the 〈 100 〉$_{d}$$_{=}$$_{a}$ and 〈 211 〉$_{d}$$_{=1}$$_{.}$$_{21}$$_{a}$ pairs decreases further; however, the 〈 100 〉$_{d}$$_{=}$$_{a}$ is still consistently lower. The GGA+U values of J for 〈 110 〉$_{d}$$_{=1}$$_{.}$$_{41}$$_{a}$ and 〈 211 〉$_{d}$$_{=1}$$_{.}$$_{21}$$_{a}$ , do not agree with the experimental acceptor splitting trend for sound values of U . A similar trend for GGA calcu- lations is reported in Refs. [42,43]. The LDA results 44 for 1% Mn including disorder in the coherent-potential approximation agree reasonably well with our results for 〈 100 〉$_{d}$$_{=}$$_{a}$ , 〈 211 〉$_{d}$$_{=1}$$_{.}$$_{21}$$_{a}$ and 〈 110 〉$_{d}$$_{=1}$$_{.}$$_{41}$$_{a}$ with J = 8 . 2, 9.1 and 7.9 meV. The 〈 100 〉$_{d}$$_{=}$$_{a}$ is still consistently lower for larger distances and the 〈 110 〉$_{d}$$_{=0}$$_{.}$$_{71}$$_{a}$ is relatively low at J = 18 meV, indicating a sensitivity of J on the chosen ab initio method. In summary, with the caveats mentioned above, the general trend of the ab initio J at reduced concentrations shows some qualitative agreement with STM experimen- tal acceptor splittings at shortest distances d = 0 . 71 a , d = a . For larger Mn separations the agreement is not so good. As corresponding theoretical estimates of the splittings are not reported in the literature, it is hard to conclude whether or not a relationship between the acceptor splittings and J is present in the ab initio cal- culations. Concerning the comparison between the ab initio values of J and our results, we can see that al- though the overall magnitudes agree reasonably well, the qualitative trend differs since at short Mn separations we find a monotonic decrease of J with distance. Given the strong dependence of J on Mn concentra- tion and the fact that the reported ab initio results are FIG. 18: The effect on ˜ J when increasing the Mn concen- tration from x = 0 . 0006 (shown in Fig. 14) to x = 0 . 0017 (filled circles). The empty circles show how ˜ J changes when increases x further for selected points. 110 110 100 100 100 211 211 211 111 1.0 1.5 2.0 2.5 3.0 3.5 0 10 110 50 R a x=0.0017 x=0.0028 x=0.0050 meV 2 6 4 8 x=0.0050 x=0.0028