Strong trilinear coupling of phonon instabilities drives the avalanche-like hybrid improper ferroelectric transition in SrBi2Nb2O9

Improper and hybrid-improper ferroelectrics (FEs) host spontaneous polarization ($P$) owing to the anharmonic coupling between polar and nonpolar modes. In general, the unstable zone-boundary nonpolar mode is a primary order parameter that couples to the stable zone-center polar mode to induce $P$. However, in certain FEs, such as the Aurivillius family, the polar mode is also unstable. In such cases, the strength of coupling between polar and nonpolar modes governs whether FE phase transition is a single or double step. Here we investigate a single-step avalanche-like transition in ${\mathrm{SrBi}}_{2}{\mathrm{Nb}}_{2}{\mathrm{O}}_{9}$, an Aurivillius family compound, using inelastic neutron scattering and Raman spectroscopy combined with group theory and first-principles simulations. We find that a strong trilinear coupling constant $\ensuremath{\lambda}$ between the two nonpolar and one polar instabilities enables their single-step condensation at the FE transition temperature ${T}_{FE}$ and overcomes the effect of the positive bi-quadratic coupling that works against it. The ratio of $\ensuremath{\lambda}$ to the leading-order term in the energy invariant polynomial is two to six times larger than in iso-structural ${\mathrm{SrBi}}_{2}{\mathrm{Ta}}_{2}{\mathrm{O}}_{9}$, which undergoes a double-step transition. We identify and track polar and nonpolar instabilities and find their temperature dependence consistent with the theoretical prediction based on the strong trilinear coupling mechanism. From the sign ($\ifmmode\pm\else\textpm\fi{}$) of bi-quadratic coupling constants in the energy invariant polynomial, we further rule out the triggering mechanism to drive the FE transition. Moreover, we observe little change in phonons across ${T}_{FE}$ despite a first-order transition. However, a significant phonon broadening occurs even at 300 K, which we rationalize based on a large anharmonic vibrational amplitude of O atoms in the ${\mathrm{NbO}}_{6}$ octahedra.