  The magnetic behavior of insulating doped diluted magnetic semiconductors
(DMS) is characterized by the interaction of large collective spins known as
bound magnetic polarons. Experimental measurements of the susceptibility of
these materials have suggested that the polaron-polaron interaction is
ferromagnetic, in contrast to the antiferromagnetic carrier-carrier
interactions that are characteristic of nonmagnetic semiconductors. To explain
this behavior, a model has been developed in which polarons interact via both
the standard direct carrier-carrier exchange interaction (due to virtual
carrier hopping) and an indirect carrier-ion-carrier exchange interaction (due
to the interactions of polarons with magnetic ions in an interstitial region).
Using a variational procedure, the optimal values of the model parameters were
determined as a function of temperature. At temperatures of interest, the
parameters describing polaron-polaron interactions were found to be nearly
temperature-independent. For reasonable values of these constant parameters, we
find that indirect ferromagnetic interactions can dominate the direct
antiferromagnetic interactions and cause the polarons to align. This result
supports the experimental evidence for ferromagnetism in insulating doped DMS.
