  A number of transition-metal (TM) doped group-IV semiconductors,
R$_{x}$Y$_{1-x}$ (R=Cr, Mn and Fe; Y=Si, Ge), have been studied by the first
principles calculations. The obtained results show that antiferromagnetic (AFM)
order is energetically more favored than ferromagnetic (FM) order in Cr-doped
Ge and Si with $x$=0.03125 and 0.0625. In 6.25% Fe-doped Ge, FM interaction
dominates in all range of the R-R distances while for Fe-doped Ge at 3.125% and
Fe-doped Si at both concentrations of 3.125% and 6.25%, only in a short R-R
range can the FM states exist. In the Mn-doped case, the RKKY-like mechanism
seems to be suitable for the Ge host matrix, while for the Mn-doped Si, the
short-range AFM interaction competes with the long-range FM interaction. The
different origin of the magnetic orders in these diluted magnetic
semiconductors (DMSs) makes the microscopic mechanism of the ferromagnetism in
the DMSs more complex and attractive.
