Rayleigh Bound States in the Continuum in Shallow Surface Relief Phononic Crystals

We present an investigation into the band structure of acoustic waves in surface phononic crystals (SPnC), which comprise a square lattice of shallow cylinders on a mechanically isotropic semi-infinite substrate, utilizing the finite element method (FEM). The introduction of crystal periodicity to the surface modifies Rayleigh modes from non-dispersive to dispersive, thereby enabling the transformation of these modes into radiative or leaky forms. This spatial dispersion may facilitate the emergence of bound states in the continuum (BIC) by providing conditions appropriate for closing the radiative channels. A symmetry-protected BIC appears at the Γ point only when the periodicity of the crystal extends in the two dimensions of the surface plane. The decoupling from the radiative channels is due to symmetry incompatibility. An accidental BIC emerges in both one- and two-dimensional SPnCs at finite wave vectors. The partial-wave model applied to the empty lattice approximation shows that the underlying mechanism giving rise to the emergence of the accidental BIC is related to the simultaneous fulfillment of the nullification condition of the transverse radiative channel amplitude and the dispersion equation. Furthermore, the presence of the accidental BIC is not compromised by structural alterations that preserve the crystal symmetry, with only its frequency being influenced.