Phononic metamaterials (PnM) are carefully man-engineered structures capable to control and tune both, sound (acoustic vibrations) [11, 35] and heat (thermal vibrations) [33, 36] propagation in ways not observed in nature. In that sense, one can referred to thermal or acoustic PnM depending on its internal structure by virtue of acoustic vibration are in the Hz-KHz frequency range while thermal vibration correspond to THz range [11, 33], such that, the PnM internal structure needs be sensible to these frequencies. Phononic Crystals (PnC) are referred as a PnM kind, an artificial periodic composite metamaterials consisting of sound or heat scatters, periodically disposed in a matrix with high impedance contrast of mass densities and/or elastic moduli with respect to that for scatters. PnC are the analogue photonic crystals but engineering for the phonons controls, photons in the second case. The contrast between the elastic features of the matrix and scatter can give rise to new and unusual phononic features and the existence of phononic band gaps (PnBG), coming from the periodic Bragg scattering as well as localized Mie scatterings from the individual scatters [36, 37]. Recent progress in nanofabrication technologies, however, has made it possible to engineering micro/nano-scale periodic structures allowing to the modulation of wavelength in the GHz frequency range, which are in the domain of thermal excitations [38]. Thus, the manipulation of thermal properties, such as thermal conduction and heat capacity, are possible [39, 40].
One of the main and first observed properties of phononic crystals is concerning with the fact that these composite media typically exhibit stop bands or phononic band gaps (PnBG) in its transmission spectra, i.e., frequency ranges where the propagation of sound and vibrations is strictly forbidden, like in the photonic band gaps in photonic crystals. The first reports on phonons propagating in 1D-periodic elastic composites were done by Dobrzynski and Djafari-Rouhani group in 80s, describing in a first paper an Al-W superlattice made from alternate layers of Al and W [41], founding the importance of the layer thickness on the vibrational properties of the composite. The same group reported the existence of phononic gaps and surface localized phonons in superlattice consisting of alternating slabs of two different crystals [42]. They develop the theory to easily study all the bulk and surface vibrational properties of a superlattice like the described. This last report is probably the first study on the existence of vibrational gaps.
Inspired in the existence of photonic crystals-like structures, starting 90s, Kushwaha and coworkers developed the first theoretical studies on the existence of phononic band gaps in 2D-periodic structures [43, 44].