Lattice Dynamics Simulation of Ionic Crystal Surfaces: I.~Criteria for Vibrational Mode Localisation and Surface Modes

A simple function, called attenuation parameter, is presented that can be used to extract quantitative information on the degree of localisation of normal modes at locations of arbitrary dimensionality from eigenvector components calculated in lattice dynamics simulations. Normal modes are identified as localised if their attenuation parameter values exceed a certain threshold value. As examples for localisation at two-dimensional locations, modes localised at the KBr(001), MgO(001) and CaF2(111) surfaces modelled by periodic shell model slabs are determined using the attenuation parameter aplane for surface localisation. This allows to gain an overview of the frequencies of the crystalline vibrations localised at the surface which is important for the vibrational interaction with adsorbed molecules. Surface modes exist at frequencies unoccupied by the bulk spectrum (in lagoons of MgO and CaF2 and in the vibrational gap in KBr) but also overlapping with it (resonant surface modes). Non-resonant normal modes are generally localised more strongly than resonant local modes. Ambiguities exist due to the finite size of the supercell model but physically relevant criterion cutoffs can be extracted from histogrammatic considerations. Experimentally measured surface modes that were not identified by previous theoretical studies are seen to have intermediate surface aplane values.