Coexistence of symmetry-enforced phononic Dirac nodal-line net and three-nodal surfaces phonons in solid-state materials: Theory and materials realization

In this study, using symmetry analysis, we reveal that symmetry-enforced three-nodal surfaces (NSs) and Dirac nodal-line (DNL) net phonons can coexist in two candidate space groups (SGs) - with SG Nos. 61 and 205 - out of 230 SGs. The phononic DNL net in SGs 61 and 205 contains three phononic DNLs, which share phononic nodal point R in the momentum space and lie at the hinge between two of the three ki(i=x,y,z) planes. More interestingly, using first-principle calculations, we find that the realistic materials ZnSb (with SG No. 61) and RuS2, P2Pt, and OsS2 (with SG No. 205) exhibit three NSs and DNL net phonons. In addition, we report that obvious phononic surface states can be observed in these mentioned realistic materials and will benefit their experimental detection. We hope that the proposed realistic materials can be viewed as ideal platforms to realize the DNL net and three-NSs phonons in solid-state materials.