  A recently discovered magnetic semiconductor
Ba$_{1-x}$K$_{x}$(Zn$_{1-y}$Mn$_{y}$)$_2$As$_{2}$, with its decoupled spin and
charge doping, provides a unique opportunity to elucidate the microscopic
origin of the magnetic interaction and ordering in dilute magnetic
semiconductors (DMS). We show that (i) the conventional density functional
theory accurately describes this material, and (ii) the magnetic interaction
emerges from the competition of the short-range superexchange and the
longer-range spin-spin interaction mediated by the itinerant As holes. The
latter can be viewed as a high-doping extrapolation of with the
Schrieffer-Wolff $p-d$ interaction representing an effective Hund's rule
coupling, $J_{H}^{\text{eff}}$. The key difference between the classical double
exchange and the actual interaction in DMS is that an effective
$J_{H}^{\text{eff}}$, as opposed to the standard Hund's coupling $J_{H}$,
depends on the Mn $d-$band position with respect to the Fermi level, and thus
allows tuning of the magnetic interactions. The physical picture revealed for
this transparent system may also be applicable to more complicated DMS systems.
