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 Goldschmidts 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.
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