Hierarchical compositional ordering in lead-based perovskite relaxors

Lead-based relaxor ferroelectric perovskites with the formula $\mathrm{Pb}({\mathrm{B}}_{1/3}{\mathrm{B}}_{2/3}^{\ensuremath{'}}){\mathrm{O}}_{3}$, such as $\mathrm{Pb}({\mathrm{Mg}}_{1/3}{\mathrm{Nb}}_{2/3}){\mathrm{O}}_{3}$, can have different degrees of compositional ordering even for the same stoichiometry. Despite intensive investigation, determining the nature of the B and ${\mathrm{B}}^{\ensuremath{'}}$ ordering remains a challenging problem. Here, we reveal a hierarchical ordering in $\mathrm{Pb}({\mathrm{B}}_{1/3}{\mathrm{B}}_{2/3}^{\ensuremath{'}}){\mathrm{O}}_{3}$ on top of the well-known ${\ensuremath{\beta}}^{\text{I}}$ and ${\ensuremath{\beta}}^{\text{II}}$ sublattices, which is the first level of the proposed hierarchy. Such additional ordering, which is found by considering additional nearest neighboring interactions in the model enriches our understanding of the complex nature of the chemical ordering in lead-based ferroelectric relaxors. In addition, we show that $\mathrm{Pb}({\mathrm{Cd}}_{1/3}{\mathrm{Nb}}_{2/3}){\mathrm{O}}_{3}$ has a higher degree of ordering than that of $\mathrm{Pb}({\mathrm{Mg}}_{1/3}{\mathrm{Nb}}_{2/3}){\mathrm{O}}_{3}$ or $\mathrm{Pb}({\mathrm{Zn}}_{1/3}{\mathrm{Nb}}_{2/3}){\mathrm{O}}_{3}$ in which antiphase domains exist, revealing the possible different degree of compositional ordering arising from intralattice ordering.