  Using a microscopic finite-cluster tight-binding model, we investigate the
trend of the magnetic anisotropy energy as a function of the cluster size for
an individual Mn impurity positioned in the vicinity of the (110) GaAs
surface,We present results of calculations for large cluster sizes, containing
approximately 104 atoms, which have not been investigated so far. Our
calculations demonstrate that the anisotropy energy of a Mn dopant in bulk GaAs
found to be non-zero in previous tight-binding calculations, is purely a finite
size effect, and it vanishes as the inverse cluster size. In contrast to this,
we find that the splitting of the three in-gap Mn acceptor energy levels
converges to a finite value in the limit of infinite cluster size. For a Mn in
bulk GaAs this feature is related to the nature of the mean-field treatment of
the coupling between the impurity and its nearest neighbors atoms. Moreover, we
calculate the trend of the anisotropy energy in the sublayers, as the Mn dopant
is moved away from the surface towards the center of the cluster. Here the use
of large cluster sizes allows us to position the impurity in deeper sublayers
below the surface, compared to previous calculations. In particular, we show
that the anisotropy energy increases up to the fifth sublayer and then
decreases as the impurity is moved further away from the surface, approaching
its bulk value. The present study provides important insight for experimental
control and manipulation of the electronic and magnetic properties of
individual Mn dopants at the semiconductor surface by means of advanced
scanning tunneling microscopy techniques.
