Soft tilt and rotational modes in the hybrid improper ferroelectric Ca${}_3$Mn${}_2$O${}_7$

Raman spectroscopy is employed to probe directly the soft rotation and tilting modes, which are two primary order parameters predicted in the hybrid improper ferroelectric material Ca${}_3$Mn${}_2$O${}_7$. We observe a giant softening of the 107-cm${}^{-1}$ octahedron tilting mode by 26~cm${}^{-1}$, on heating through the structural transition from a ferroelectric to paraelectric orthorhombic phase. This is contrasted by a small softening of the 150-cm${}^{-1}$ rotational mode by 6~cm${}^{-1}$. In the intermediate phase, the competing soft modes with different symmetries coexist, bringing about many-faceted anomalies in spin excitations and lattice vibrations. Our work demonstrates that the soft rotation and tilt patterns, relying on a phase-transition path, are a key factor in determining ferroelectric, magnetic, and lattice properties of Ca${}_3$Mn${}_2$O${}_7$.