Surface effect on the propagation of flexural waves in periodic nano-beam and the size-dependent topological properties

With the development of science and technology, phononic crystals can be fabricated at nanoscale. The advancement yields significant applications in thermo-electric, acousto-optical and electron-phonon devices. At nanoscale, the surface heterogeneity of structures is a result of the differences between the arrangement and environment of the surface atoms and those of their counterparts in the body. This characteristic plays an important role in determining the mechanical properties of the structures. The effects of surface heterogeneity on band structures of the phononic crystals at nanoscale are studied in this paper. We also investigate the topological properties of phononic nanobeams through theoretical and numerical approaches. The topological edge mode, which is affected by the surface heterogeneity, is observed. Particularly, a theoretical method based on the plane wave expansion method is proposed to conveniently calculate the Zak phase, which is an important constant describing the topological properties. The numerical results show that surface heterogeneity mainly influences the response frequency for the dispersion relations and the topological behaviors in the phononic nanobeam. The study may be beneficial for the understanding of the size-dependent behaviors of phononic crystals at nanoscale and for the design of nanodevices based on the dispersion and topological properties of phononic crystals.