PH-NODE: A DFPT and finite displacement supercell based python code for searching nodes in topological phononic materials

Also, there are 21 energy values at which the node points are situated, corresponding to the full Brillouin zone. For LiCaAs, the previously reported literature claims that there is a node point along the W-X high symmetry direction between the highest longitudinal acoustic and the lowest transverse optical branch, while in our DFT calculations, a gap of 0.17 meV is found. Furthermore, ScZn hosts six nodal-ring TPs phonons at the boundary planes of the Brillouin zone in the vicinity of the M high-symmetric point.

In addition to this, straight-line TPs are also found along the Γ-X and Γ-R high symmetric directions. Moreover, for TiS, six Weyl node points (WP1, WP2, WP3, WP4, WP5 and WP6) are found along H-K high-symmetric direction. In CdTe, it is found that Weyl points are located along the X-W high-symmetry direction.

In the case of CsTe, a TP and a quadratic contact TP are found along the Γ-X direction and at the R high-symmetry point, respectively. The results obtained from the PH-NODE code are in good agreement with the experimentally and theoretically reported data for each material.