Improper ferroelectricity in 134-type A A 3 ′ B 4 O 12 perovskites

The 134-type $A$-site ordered ${\\mathrm{LaMn}}_{3}{\\mathrm{Cr}}_{4}{\\mathrm{O}}_{12}$, that belongs to the family of $A{A}_{3}^{\\ensuremath{\u0027}}{B}_{4}{\\mathrm{O}}_{12}$ perovskites, with cubic structure and magnetically driven improper polarization is puzzling because the conventional mechanisms leading to the creation of polarization in type-II multiferroics do not apply. By symmetry analysis and first-principles simulations, we discover an original and previously overlooked energetic term that naturally explains the origin of improper polarization in ${\\mathrm{LaMn}}_{3}{\\mathrm{Cr}}_{4}{\\mathrm{O}}_{12}$. In addition, we demonstrate the existence of three other energetic invariants in $A{A}_{3}^{\\ensuremath{\u0027}}{B}_{4}{\\mathrm{O}}_{12}$ perovskites, for which other physical quantities (such as oxygen tiltings or shear strain) mediate the interaction between $G$-type antiferromagnetic vectors to create and even enhance improper polarization.