Local structure and its implications for the relaxor ferroelectric Cd$_2$Nb$_2$O$_7$

The relaxor ferroelectric transition in Cd$_2$Nb$_2$O$_7$ is thought to be described by the unusual condensation of two $Γ$-centered phonon modes, $Γ_4^-$ and $Γ_5^-$. However, their respective roles have proven to be ambiguous, with disagreement between $\textit{ab initio}$ studies, which favor $Γ_4^-$ as the primary mode, and global crystal refinements, which point to $Γ_5^-$ instead. Here, we resolve this issue by demonstrating from x-ray pair distribution function measurements that locally, $Γ_4^-$ dominates, but globally, $Γ_5^-$ dominates. This behavior is consistent with the near degeneracy of the energy surfaces associated with these two distortion modes found in our own $\textit{ab initio}$ simulations. Our first-principles calculations also show that these energy surfaces are almost isotropic, providing an explanation for the numerous structural transitions found in Cd$_2$Nb$_2$O$_7$, as well as its relaxor behavior. Our results point to several candidate descriptions of the local structure, some of which demonstrate two-in/two-out behavior for Nb displacements within a given Nb tetrahedron. Although this suggests the possibility of a charge analog of spin ice in Cd$_2$Nb$_2$O$_7$, our results are more consistent with a Heisenberg-like description for dipolar fluctuations rather than an Ising one. We hope this encourages future experimental investigations of the Nb and Cd dipolar fluctuations, along with their associated mode dynamics.

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