Quasi-bound layer-breathing phonons inside perfect dislocations of lattice-relaxed twisted bilayers

Using multiscale modelling we investigate layer-breathing phonons in MX$_2$ bilayers (M=Mo,W; X=S,Se) containing dislocations specific for lattice-relaxed moir\'e superlattices. The dislocations, forming in the bilayers with parallel and antiparallel alignment of layers, bring about spatial modulation of the interlayer distance, generating effective potentials that promote the emergence of one-dimensional quasi-bound bands of layer-breathing modes inside perfect dislocations. For parallel MX$_2$ bilayers, perfect dislocations host multiple quasi-bound bands, with frequencies above the layer-breathing mode in rhombohedral-stacked domains. In contrast, antiparallel bilayers exhibit only a single quasi-bound band arising for orientations close to the edge-dislocation type, having frequencies above the layer-breathing mode in 2H-stacked domains.