Engineering of dispersion curves and band gaps in a phononic crystal made of pillars on a membrane or substrate
We present the recent progress in our theoretical work about the acoustic properties of phononic crystals constituted by a periodic array of pillars on a membrane or on a substrate. In the case of a thin membrane, we have shown, in addition to the Bragg gaps, the possibility of a much lower frequency gap related to the geometrical parameters of the structure. We discuss the origin of the latter and more generally the high bending and flatness of the lowest dispersion curves in this structure. When the thickness of the membrane is increased and progressively approaches the case of a semi infinite substrate, some of the bent or flat dispersion curves remain below the sound cone of the substrate in some part of the Brillouin zone and become surface acoustic waves. It becomes possible to find absolute band gaps between the surface wave dispersion curves, and then a waveguide or a cavity located at the surface can give rise to surface confined phonons. The above effects can be investigated through the calculation of the band structures and transmission coefficients. A second part of the work is dealing with the effect of the shape of the pillars on the engineering of the band gaps. For instance, we have shown that pillars of conic shape can significantly widen the band gaps and lower their frequencies. The maximum effect is obtained for some angle of the cone with respect to a cylindrical shape. We study also other geometries such as multilayer pillars or pillars deposited on both faces of a membrane. The last part of the work is dealing with the calculation of the effective properties of the phononic crystal when the wavelength is much larger than the period, for instance to investigate the possibility of negative mass density in the vicinity of local resonances. We use a numerical method developed by Zhu et al (Phys. Rev. B 86, 144307 (2012) where the dynamic mass tensor is calculated by applying an external displacement field perturbation on the boundaries of a unit cell and the induced forces exerted on these boundaries are evaluated at each frequency. A similar approach is used to obtain the effective elastic constants.