Phononic hybrid-order topology in semihydrogenated graphene

In recent years, first -order and second -order topological phonons have been discovered in crystalline materials, which has aroused great interest. In the present work, through symmetry analysis and bulk polarization calculation, we demonstrate that semihydrogenated graphene (i.e., graphone) is an ideal platform intrinsically possessing hybrid -order (first -order and second -order) phonon topology. Based on the acoustic sum rule correction of the phonon tight -binding Hamiltonian, we show that the first -order optical and acoustic topological phonon edge states as well as second -order topological phonon corner states can coexist in graphone. Interestingly, the twofold degeneracy of topological acoustic phonon edge modes in graphene is eliminated due to spatial inversion breaking in graphone. In addition, there are three topological corner states within the wide gap of the phonon spectrum of the zigzag -edged nanodisk. Our results not only show hybrid -order topological phonon properties in graphone but also pave the way for a more complete understanding of topological phonons in other crystalline materials.