Cladded phononic nodal frame state in biatomic alkali-metal sulfides

With rapid progress, the current study of topological properties in condensed matter systems has been further extended from the electronic scope to the phononic perspective. Based on first-principles calculations, we present a systematic investigation on topological phononic states in a series of biatomic alkali-metal sulfides AB $(\mathrm{A}=\mathrm{Na},\mathrm{K}; \mathrm{B}=\mathrm{S},\mathrm{Se},\mathrm{Te})$. Taking NaSe as an example, clear topological features of the nodal line and nodal surface states are revealed, and these two states, originating from the same two bands, concatenate together into a cladded nodal frame structure. Detailed symmetry arguments and effective Hamiltonian analysis show that the occurrence of the cladded nodal frame structure is determined by the combination of both symmorphic and nonsymmorphic symmetries. The phonon surface spectrum is calculated on the (010) surface and clean surface states are observed, which not only confirms their nontrivial topological feature, but also benefits their experimental verification. This cladded nodal frame structure can provide an ideal platform to examine the exotic topological phononic states and, with the present material candidates, especially those that have been synthesized, the corresponding experimental studies can be immediately advanced.