Complex nodal structure phonons formed by open and closed nodal lines in CoAsS and Na₂CuP solids

Topological phononic states with nodal lines not only have updated our knowledge of the phases of matter in a fundamental way, but also have become a major frontier research direction in condensed matter physics. From a mathematical perspective, nodal line phonons can be divided into open and closed types. The present attempt is a report on the coexistence of such open and closed nodal line phonons in two realistic solids, CoAsS and Na₂CuP, based on first-principles calculations. Furthermore, it is shown that the closed and the open nodal line states in CoAsS and Na₂CuP have touching points and can form a complex nodal structure phonon in a momentum space. Due to the topologically non-trivial behavior of the complex nodal structure in both phonons, evident phononic surface states occur in the projected surfaces of both materials. In this way, these states, arising from the projected crossing points, can benefit experimental detection in follow-up studies. It has been stated that the open and closed nodal line states are formed by the crossings of two phonon branches and, hence, these two types of nodal line phonons are coupled with each other. The results obtained here could be considered as a breakthrough in clearly demonstrating the coexistence of the open and closed nodal line states in phonons and, for this reason, may inspire researchers seeking materials with such topological states in other bosons, such as photons.