  Improvement in magnetic and electrical properties of multiferroic BiFeO$_3$
in conjunction with their dependence on particle size is crucial due to its
potential applications in multifunctional miniaturized devices. In this
investigation, we report a study on particle size dependent structural,
magnetic and electrical properties of sol-gel derived
Bi$_{0.9}$Ba$_{0.1}$FeO$_3$ nanoparticles of different sizes ranging from
$\sim$ 12 to 49 nm. The substitution of Bi by Ba significantly suppresses
oxygen vacancies, reduces leakage current density and Fe$^{2+}$ state. An
improvement in both magnetic and electrical properties is observed for 10 %
Ba-doped BiFeO$_3$ nanoparticles compared to its undoped counterpart. The
saturation magnetization of Bi$_{0.9}$Ba$_{0.1}$FeO$_3$ nanoparticles increase
with reducing particle size in contrast with a decreasing trend of
ferroelectric polarization. Moreover, a first order metamagnetic transition is
noticed for $\sim$ 49 nm Bi$_{0.9}$Ba$_{0.1}$FeO$_3$ nanoparticles which
disappeared with decreasing particle size. The observed strong size dependent
multiferroic properties are attributed to the complex interaction between
vacancy induced crystallographic defects, multiple valence states of Fe,
uncompensated surface spins, crystallographic distortion and suppression of
spiral spin cycloid of BiFeO$_3$.
