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[Fig. 19 (a)-(c)] show qualitatively different landscapes - tilted quasi-easy planes with the hard direction approx- imately along [111]. Similarly for the AFM configura- tions, only 〈 110 〉$_{d}$$_{=0}$$_{.}$$_{71}$$_{a}$ and 〈 100 〉$_{d}$$_{=}$$_{a}$ [Fig. 19 (e)-(f)] dif- fer from the single Mn type landscape in Fig. 19 (d). In 〈 100 〉$_{d}$$_{=}$$_{a}$ the easy axis and the low barrier are inter- changed between the AFM and FM, but the hard direc- tion remains along the [111]. 

Fig. 20 shows that anisotropy energies are very small, typically one order of magnitude smaller than for the fully periodic systems. In both FM and AFM configu- rations the interactions between Mn for distances below two lattice constants, tend to reduce the anisotropy heav- ily. At larger distances, where all Mn pairs produce the [111] easy axis type landscape in Fig. 19 (d), the single barriers are around 2 meV. This limit of weakly interact- ing Mn is also reflected in the difference in barriers (see 20) between the FM and AFM, which becomes negligi- ble for weakly hybridized pairs above 1 . 5 a . As we will show, total energy differences are also small, causing a very small effective exchange. 

## 2. Character of acceptor levels for Mn pairs in a (110) surface 

Fig. 21 shows the group of six impurity levels for the Mn pairs in the surface. We see that for both FM and AFM configurations, two of the occupied levels are split around 0.5 eV below the others. For the closest 〈 110 〉 pair the acceptors are so high in energy that the upper 

acceptor has in fact crossed the first conduction band state. This is associated with a very large splitting of the acceptors. In 〈 110 〉 R the off-site Coulomb has been reduced even further (to 0.5 eV) in order to also make the upper acceptor appear in the gap. The overall spectrum for the AFM and FM configuration looks similar, indi- cating that interactions are very weak and the itinerant spin wave functions are very localized at the surface. In the STM experiment 4 Zn dopants can give rise to reso- nant tunneling between conduction states and the accep- tor level, thought to be responsible for the negative dip seen in the curve of the tunneling conductance vs. bias voltage. 46 The coupling to conduction-like states can lead to a more extended, bulk-like acceptor wave function seen at the (110) surface. As a first approximation our treat- ment of the surface should reveal some of the relevant properties. Our results indicate that the interactions in the surface are much weaker than in bulk, and magnetic anisotropies are one order of magnitude smaller. This is related to acceptor wave functions that are much more localized at the surface. 

So far there are relatively few papers that attempt to simulate STM images of Mn in the (110) surface. 47,48 The acceptor LDOS resulting from our tight-binding model for a few representative pairs are shown in Fig. 22 for the FM and Fig. 23 for the AFM configurations. The 

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