Colloidal nanocrystal superlattices as phononic crystals: Plane wave expansion modeling of phonon band structure

Colloidal nanocrystals consist of an inorganic crystalline core with organic ligands bound to the surface and naturally self-assemble into periodic arrays known as superlattices. This periodic structure makes superlattices promising for phononic crystal applications. To explore this potential, we use plane wave expansion methods to model the phonon band structure. We find that the nanoscale periodicity of these superlattices yield phononic band gaps with very high center frequencies on the order of 10² GHz. We also find that the large acoustic contrast between the hard nanocrystal cores and the soft ligand matrix lead to very large phononic band gap widths on the order of 10¹ GHz. We systematically vary nanocrystal core diameter, d, nanocrystal core elastic modulus, E_{NC core}, interparticle distance (i.e. ligand length), L, and ligand elastic modulus, E_ligand, and report on the corresponding effects on the phonon band structure. Our modeling shows that the band gap center frequency increases as d and L are decreased, or as E_{NC core} and E_ligand are increased. The band gap width behaves non-monotonically with d, L, E_{NC core}, and E_ligand, and intercoupling of these variables can eliminate the band gap. Lastly, we observe multiple phononic band gaps in many superlattices and find a correlation between an increase in the number of band gaps and increases in d and E_{NC core}. We find that increases in the property mismatch between phononic crystal components (i.e. d/L and E_{NC core}/E_ligand) flattens the phonon branches and are a key driver in increasing the number of phononic band gaps. Our predicted phononic band gap center frequencies and widths far exceed those in current experimental demonstrations of 3-dimensional phononic crystals. This suggests that colloidal nanocrystal superlattices are promising candidates for use in high frequency phononic crystal applications.