Moiré superlattices*

We investigate phonon spectra in MoS₂ twisted bilayers and find substantial renormalization: over a range of small twist angles, the phonon spectra evolve rapidly due to ultra-strong coupling between different phonon modes and the atomic reconstruction of the moiré pattern. These phonon spectra also reveal three regimes of atomic reconstructions. We develop a new low-energy continuum model for phonons that overcomes the outstanding challenge of calculating phonon properties of large moiré supercells and successfully captures essential experimental observations. We show how simple optical spectroscopy experiments can provide information on strain and lattice distortions in moiré crystals for nanometer-sized supercells. Our newly developed theory provides a comprehensive and unified understanding of structural and optical properties of moiré superlattices.