Shear and breathing modes of all layered materials

The low-energy part of the vibrational spectrum of van-der-Waals layered materials is characterised by two fundamental sets of normal-mode vibrations, where the layers oscillate as rigid units, either parallel (shear or C modes) or perpendicular (layer-breathing or LB modes) to each other. Their frequencies depend on the number of layers and can be used to characterise the layered materials by Raman or infrared spectroscopy. We present here a general approach to predict the fan diagram of optically-active C and LB modes of any layered material with any number of layers. Based on symmetry arguments, we describe the evolution of the point group as a function of the number of layers and of the spacegroup of the corresponding bulk system. We then combine group theory with a tensorial one-dimensional mechanical model to compute vibrational normal modes and identify which ones are Raman and/or infrared active. This procedure allows us to seamlessly provide the fan diagram of optically-active modes for any multilayer stack of any layered material. We implement this method and algorithms in an open tool that we make available online on the Materials Cloud portal, to assist any researcher in the prediction and interpretation of such diagrams.