Lone pairs in insulating pyrochlores: Ice rules and high-$k$ behavior

Pyrochlore dielectric materials such as (Bi$_{1.5}$Zn$_{0.5}$)(Nb$_{1.5}$Zn$_{0.5}$)O$_7$ (BZN) have generated interest because they combine high dielectric constants with small dielectric loss tangents and yet are cubic at all temperatures. The recent low-temperature preparation and structural characterization of Bi$_2$Ti$_2$O$_7$, which remains cubic down to 2 K, has provided a good model system for understanding the properties of Bi-based pyrochlores. In this contribution, the electronic structure of cubic Bi$_2$Ti$_2$O$_7$ is visualized and compared with the electronic structure of the Aurivillius phase ferroelectric SrBi$_2$Ta$_2$O$_9$ (SBT), which displays a ferroelectric distortion below 608 K associated with the tendency of lone pair active Bi$^{3+}$ to move off-center. Such coherent off-centering distortions are frustrated on the pyrochlore lattice, and this prevents a ferroelectric-paraelectric phase transition in Bi$_2$Ti$_2$O$_7$. Instead, Bi$^{3+}$ ions in Bi$_2$Ti$_2$O$_7$ are obliged to off-center in an \textit{incoherent} manner, that is compatible with the cubic structure being retained. Frustrated lone pair behavior in the defect pyrochlore Pb$_2$Sn$_2$O$_6$ is also described. Parallels between the well-studied frustration of certain types of \textit{magnetism} in pyrochlore compounds (spin-ice) and the striking paucity of ferroelectric pyrochlores, arising from the corner-connected tetrahedral topology of the pyrochlore lattice are pointed out.