Using multiscale modeling we investigate layer-breathing phonons in ${MX}_2$ bilayers ($M=\text{Mo,}\text{W}$; $X=\text{S,}\text{Se}$) containing dislocations specific for lattice-relaxed moiré 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 quasibound bands of layer-breathing modes inside perfect dislocations. For parallel ${MX}_2$ bilayers, perfect dislocations host multiple quasibound bands, with frequencies above the layer-breathing mode in rhombohedral-stacked domains. In contrast, antiparallel bilayers exhibit only a single quasibound band arising for orientations close to the edge-dislocation type, having frequencies above the layer-breathing mode in 2H-stacked domains.

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• Phonons
• Twist deformation
• Twistronics
• Transition metal dichalcogenides