  After more than a decade of intensive research in the field of diluted
magnetic semiconductors (DMS), the nature and origin of ferromagnetism,
especially in III-V compounds is still controversial. Many questions and open
issues are under intensive debates. Why after so many years of investigations
Mn doped GaAs remains the candidate with the highest Curie temperature among
the broad family of III-V materials doped with transition metal (TM) impurities
? How can one understand that these temperatures are almost two orders of
magnitude larger than that of hole doped (Zn,Mn)Te or (Cd,Mn)Se? Is there any
intrinsic limitation or is there any hope to reach in the dilute regime room
temperature ferromagnetism? How can one explain the proximity of (Ga,Mn)As to
the metal-insulator transition and the change from
Ruderman-Kittel-Kasuya-Yosida (RKKY) couplings in II-VI compounds to double
exchange type in (Ga,Mn)N? In spite of the great success of density functional
theory based studies to provide accurately the critical temperatures in various
compounds, till very lately a theory that provides a coherent picture and
understanding of the underlying physics was still missing. Recently, within a
minimal model it has been possible to show that among the physical parameters,
the key one is the position of the TM acceptor level. By tuning the value of
that parameter, one is able to explain quantitatively both magnetic and
transport properties in a broad family of DMS. We will see that this minimal
model explains in particular the RKKY nature of the exchange in
(Zn,Mn)Te/(Cd,Mn)Te and the double exchange type in (Ga,Mn)N and simultaneously
the reason why (Ga,Mn)As exhibits the highest critical temperature among both
II-VI and III-V DMS.
