  We study energetic, magnetic, and electronic properties of diluted
substitutional Mn-pairs on the reconstructed $(001)$ GaAs surfaces. The studies
are based on first-principles calculations in the framework of the density
functional theory. We demonstrate that the stability of the systems strongly
depends on the position, orientation, and the distance between the Mn-atoms
constituting the pair. Independently of the considered surface reconstruction
pattern, the Mn-pairs with Mn-atoms being the nearest neighbors (NN) on
cationic sublattice turn out to be energetically more favorable than the pairs
with the larger distance between the Mn-atoms. However, the preferential
build-up orientation of the Mn-NN-pair depends on the surface reconstruction
and is parallel either to $[110]$ or $[1\bar{1}0]$ crystallographic direction.
We reveal also the mechanisms of the magnetic ordering of Mn-NN-pairs. The
Mn-NN-pairs along the $[110]$ crystallographic direction exhibit always
ferromagnetic alignment of Mn spins, whereas the spins in the Mn-NN-pairs along
$[1\bar{1}0]$ direction are mostly anti-ferromagnetically aligned. In the
electronic structure of the systems containing Mn-pairs with ferromagnetically
aligned spins, we observe the valence band hole states in the neighborhood of
Fermi energy. This indicates that the surface ferromagnetism in this prototype
of dilute magnetic semiconductors can be explained in terms of the $p$-$d$
Zener model.
