Unveiling hidden ferrimagnetism and giant magnetoelectricity in polar magnet Fe$_{2}$Mo$_{3}$O$_{8}$

Polar magnets, belonging to the polar crystallographic symmetry groups and containing magnetic ions, can exhibit non-trivial magnetoelectric (ME) effects below magnetic ordering temperatures due to the broken time reversal and space inversion symmetries. Mono-domain polar single crystals can often be grown, and eliminate the need for any poling procedures to reveal the possible ME response. Here, we report a giant ME effect in a polar magnet Fe$_{2}$Mo$_{3}$O$_{8}$ at temperature as high as 60 K. Polarization jumps of 0.3 $\mu $C/cm$^{2}$~and repeated mutual control of ferroelectric and magnetic moments with differential ME coefficients on the order of 10$^{4\, }$ps/m are achieved. The sign of the ME coefficients can be switched by changing the direction of the applied ``bias'' magnetic field. Importantly, no electric or magnetic poling is needed, as necessary for applications. Using first principles calculations, we show that exchange striction is the leading mechanism responsible for the observed ME effect.