Size-effect on band structures of nanoscale phononic crystals

The scaling law does not hold when the sizes of the phononic crystals reach the nanoscale dimension [Ramprasad et al., Applied Physics Letters 87 (2005) [9]]111101. This paper discusses the size-effect on the band structures of nanoscale phononic crystals. The transfer matrix method based on the nonlocal elastic continuum theory is developed to compute the band structures of a nanoscale layered phononic crystal. Detailed calculations are performed for a nanoscale HfO2–ZrO2 multilayer stack. It is shown that the nonlocal elastic continuum solution deviates from the classical elastic continuum one and finally approaches the first-principle result as the thickness of each individual layer decreases. When the thickness of each layer is much larger than several nanometers, the correspondence between the nonlocal and classical elastic methods is shown, and the size effects can be neglected. The developed nonlocal elastic continuum method is expected to overcome the limits of the classical continuum description for wave propagation in phononic crystals when dimensions are in nanometer-length scales.