  Dilute magnetic semiconductors (DMS) are nonmagnetic semiconductors doped
with magnetic transition metals. The recently discovered DMS material
(Ba$_{1-x}$K$_{x}$)(Zn$_{1-y}$Mn$_{y}$)$_{2}$As$_{2}$ offers a unique and
versatile control of the Curie temperature, $T_{\mathrm{C}}$, by decoupling the
spin (Mn$^{2+}$, $S=5/2$) and charge (K$^{+}$) doping in different
crystallographic layers. In an attempt to describe from first-principles
calculations the role of hole doping in stabilizing ferromagnetic order, it was
recently suggested that the antiferromagnetic exchange coupling $J$ between the
nearest-neighbor Mn ions would experience a nearly twofold suppression upon
doping 20\% of holes by potassium substitution. At the same time,
further-neighbor interactions become increasingly ferromagnetic upon doping,
leading to a rapid increase of $T_{\mathrm{C}}$. Using inelastic neutron
scattering, we have observed a localized magnetic excitation at about 13 meV,
associated with the destruction of the nearest-neighbor Mn-Mn singlet ground
state. Hole doping results in a notable broadening of this peak, evidencing
significant particle-hole damping, but with only a minor change in the peak
position. We argue that this unexpected result can be explained by a combined
effect of superexchange and double-exchange interactions.
