Lattice dynamics of Bi$_2M_2$O$_7$ ($M$=Sn, Ti, and Hf) from first principles

Insulating bismuth pyrochlores with mixed cations randomly distributed on the B site, Bi$_2MM'$O$_7$, have been of interest primarily for their dielectric properties. As a way of helping to elucidate the effects of cation disorder from that of the highly polarizable Bi$^{3+}$ lone pair cation, systems like Bi$_2M_2$O$_7$ ($M$=Sn, Ti, and Hf) have beed studied. Far from being simple model systems, these single B-site cation materials have been show to display surprisingly complex and local structural distortions. While Bi$_2$Sn$_2$O$_7$ and Bi$_2$Hf$_2$O$_7$ undergo three and four different phases (where the ground state structure has 352 atoms), Bi$_2$Ti$_2$O$_7$ does not show any coherent structural distortions but rather the Bi$_2$O' simply becomes disordered. In this talk we will present a comparative first-principles study of the lattice instabilities throughout the BZ of the cubic prototype structures of Bi$_2M_2$O$_7$ ($M$=Sn, Ti, and Hf) . We then use the eigenvectors of the identified unstable force constants to perform a systematic search over all possible subgroup structure, performing full structural relaxations thereby constructing a picture of the energy landscape. Finally we studied the effect of biaxial strain along [100], [110], and [111].