The Order of Longitudinal and Transverse Optical Phonon Modes in Monoclinic Oxides

In monoclinic crystals, as required by symmetry, the infrared-active phonon modes have their transition dipoles oriented parallel (A_u modes) and perpendicular (B_u modes) to the symmetry axis. In the absence of additional symmetry constrains, the B_u modes are distributed within the monoclinic plane, the pairs of the transverse optical (TO) and longitudinal optical (LO) modes (TO-LO pairs) do not need to be parallel to each other, and this leads to the apparent violation of the TO-LO rule. The analysis of the limiting frequencies within the Born-Huang approach reveals that TO-LO pairs either belong to inner or outer phonon modes. Inner modes are nested within the outer modes. The directional limiting frequencies are bound to within outer mode frequency regions not occupied by inner mode pairs. Hence, an unusual phonon mode order can occur where both lower-frequency as well as upper-frequency limits for the directional modes can be both transverse and/or longitudinal modes. This behavior will be demonstrated in the results of first principles calculations of phonon modes for three different monoclinic crystals: β-Ga₂O₃ (space group 12), CdWO₄ (space group 13), and Y₂SiO₅ (space group 15), and compared with experimental results based on infrared spectroscopic ellipsometry.