  We present a first principles study of the series of multiferroic barium
fluorides with the composition BaMF4, where M is Mn, Fe, Co, or Ni. We discuss
trends in the structural, electronic, and magnetic properties, and we show that
the ferroelectricity in these systems results from the "freezing in" of a
single unstable polar phonon mode. In contrast to the case of the standard
perovskite ferroelectrics, this structural distortion is not accompanied by
charge transfer between cations and anions. Thus, the ferroelectric instability
in the multiferroic barium fluorides arises solely due to size effects and the
special geometrical constraints of the underlying crystal structure.
