Recipe for Dirac Phonon States with Quantized Valley Berry Phase in Two Dimensional Hexagonal Lattices

The topological quantum states in two-dimensional (2D) materials are fascinating subjects of research, which usually highlight electron-related systems. In this work, we present a recipe that leads to Dirac phonon states with quantized valley Berry phase in 2D hexagonal lattices by first-principles calculations. We show that candidates possessing the three-fold rotational symmetry at the corners of the hexagonal Brillouin zone host valley Dirac phonons, which are guaranteed to remain intact with respect to perturbations. We identify that such special topological features populated by Dirac phonons can be realized in various 2D materials. In particular, the monolayer CrI$_3$, an attractive 2D magnetic semiconductor with exotic applications in spintronics, is an ideal platform to investigate nontrivial phonons in experiments. We further confirm that the phonon Berry phase is quantized to $\pm \pi$ at two inequivalent valleys. The phonon edge states terminated at the projection of phonon Dirac cones are clearly visible. This work demonstrates that 2D hexagonal lattices with attractive valley Dirac phonons will extend the knowledge of valley physics, providing wide applications of topological phonons.