“Hexagonal”\nPerovskites: From Stacking\nSequence to Space Group Symmetry and New Opportunities

The term “hexagonal perovskite” has been\nwidely used\nin literature to discuss the structure of perovskite-type compounds\nwith composition <i>ABX</i><sub>3–<i>y</i></sub> with partial <i>h</i>-type stacking of <i>AX</i><sub>3</sub> layers. Though the local surrounding of these <i>AX</i><sub>3</sub> layers resembles a hexagonal close packing,\nthe space group symmetries reported for some of these materials are\nnot belonging to the hexagonal crystal system but are trigonal or\northorhombic instead. Though reports and books with lists of hexagonal\nperovskites exist which assign stacking sequences and list a corresponding\nspace group, a concise guideline on how the maximum symmetry of the\nperovskite type material is related to the stacking sequence has not\nbeen reported. Clearly, such a systematization of space group symmetry\nand stacking notations would be desirable; for distortion or ordering\nvariants of the cubic perovskite, the whole research community benefits\nfrom concepts of group-subgroup relations or classification of tilting\nvia the Glazer notation. In this article, we derive a fully consistent\nguideline on how the aristotype space group symmetry can be determined\nfor any stacking sequence (e.g., <i>cchhchcch</i>, ...)\nin the Jagodzinski notation and provide a computer program which can\ndo this analysis. By this, one can narrow down the possible aristotype\nspace group symmetries for any perovskite stacking sequence with <i>h</i>-type layers to seven space groups in total (<i>R</i>3̅<i>m</i>, <i>P</i>3̅<i>m</i>, <i>P</i>6<sub>3</sub>/<i>mmc</i>, <i>P</i>6̅<i>m</i>2, <i>R</i>3<i>m</i>, <i>P</i>3<i>m</i>, and <i>P</i>6<sub>3</sub><i>mc</i>). The space groups can be directly derived\nfrom the Jagodzinski sequence. Remarkably, the possibility to obtain\npolar perovskites with <i>h</i>-type layers by stacking\nhas not been recognized in literature so far and provides the opportunity\nto develop new multiferroic materials by design of stacking.