Phononic (or acoustic) band structure calculations have been performed for a nanoscale HfO2ZrO2 multilayer stack using first-principles methods at the atomistic level and by solving the acoustic wave equation at the continuum level, as a first step toward determining the length scales when conventional continuum acoustic band-gap treatments become inadequate. Transverse acoustic waves are the focus of this study. The material parameters that continuum acoustic band gap methods require, such as the mass density and transverse wave velocity of the components of the acoustic crystal (i.e., for HfO2 and ZrO2), were determined using separate phonon calculations of the corresponding bulk materials. Comparison of the phononic band structure for a nanoscale HfO2ZrO2 multilayer stack calculated using first-principles and continuum methods indicates the need for careful treatments of wave propagation properties at these length scales.

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