  Simultaneous co-existence of room-temperature(T) ferromagnetism and
ferroelectricity in Fe doped BaTiO$_3$ (BTO) is intriguing, as such Fe doping
into tetragonal BTO, a room-T ferroelectric (FE), results in the stabilization
of its hexagonal polymorph which is FE only below $\sim$80K. Here, we
investigate its origin and show that Fe-doped BTO has a mixed-phase
room-temperature multiferroicity, where the ferromagnetism comes from the
majority hexagonal phase and a minority tetragonal phase gives rise to the
observed weak ferroelectricity. In order to achieve majority tetragonal phase
(responsible for room-T ferroelectricity) in Fe-doped BTO, we investigate the
role of different parameters which primarily control the PE hexagonal phase
stability over the FE tetragonal one and identify three major factors namely,
the effect of ionic size, Jahn-Teller (J-T) distortions and oxygen vacancies
(OVs), to be primarily responsible. The effect of ionic size which can be
qualitatively represented using the Goldschmidt's tolerance (GT) factor seems
to be the major dictating factor for the hexagonal phase stability. The
understanding of these factors not only enables us to control them but also,
achieve suitable co-doped BTO compound with enhanced room-T multiferroic
properties.
