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In this work, we present a theoretical research on the lattice relaxations,phonon properties, and relaxed electronic structures in magic-angle twistedbilayer graphene (TBG). We construct a continuum elastic model in order tostudy the lattice dynamics of magic-angle TBG, where both in-plane andout-of-plane lattice displacements are take into account. The fully relaxedlattice structure calculated using such a model is in quantitative agreementwith experimental measurements. Furthermore, we investigate the phononproperties in magic-angle TBG using the continuum elastic model, where both thein-plane and out-of-plane phonon modes are included and treated on equalfooting. We identify different types of moir\'e phonons including in-planesliding modes, soft out-of-plane flexural modes, as well as out-of-planebreathing modes. The latter two types of phonon modes exhibit interestingmonopolar, dipolar, quadrupolar, and octupolar-type out-of-plane vibrationpatterns. Additionally, we explore the impact of the relaxed moir\'esuperlattice structure on the electronic band structures of magic-angle TBGusing an effective continuum model, which shows nearly exact agreement withthose calculated using a microscopic atomistic tight-binding approach. Our worklays foundation for further studies on the electron-phonon coupling effects andtheir interplay with $e$-$e$ interactions in magic-angle TBG.

Lattice distortions, moir\'e phonons, and relaxed electronic band structures in magic-angle twisted bilayer graphene