Normal Mode Determination of Perovskite Crystal Structures with Octahedral Rotations: Theory and Applications

TL;DR

This paper systematically analyzes the normal vibrational modes of perovskite structures with octahedral rotations, providing detailed tables and schematics to aid in understanding their lattice dynamics and physical properties.

Contribution

It introduces a comprehensive set of normal mode tables for all Glazer tilt systems in perovskites, linking symmetry, atomic displacements, and physical phenomena.

Findings

Generated 705 atomic displacement patterns for all tilt systems.

Provided methods to identify octahedral rotations and distortions from vibrational modes.

Clarified the physical meaning of symmetry-unique vibrational modes.

Abstract

Nuclear site analysis methods are used to enumerate the normal modes of $ABX_{3}$ perovskite polymorphs with octahedral rotations. We provide the modes of the fourteen subgroups of the cubic aristotype describing the Glazer octahedral tilt patterns, which are obtained from rotations of the $BX_{6}$ octahedra with different sense and amplitude about high symmetry axes. We tabulate all normal modes of each tilt system and specify the contribution of each atomic species to the mode displacement pattern, elucidating the physical meaning of the symmetry unique modes. We have systematically generated 705 schematic atomic displacement patterns for the normal modes of all 15 (14 rotated + 1 unrotated) Glazer tilt systems. We show through some illustrative examples how to use these tables to identify the octahedral rotations, symmetric breathing, and first-order Jahn-Teller anti-symmetric breathing distortions of the $BX_{6}$ octahedra, and the associated Raman selection rules. We anticipate that these tables and schematics will be useful in understanding the lattice dynamics of bulk perovskites and would serve as reference point in elucidating the atomic origin of a wide range of physical properties in synthetic perovskite thin films and superlattices.