Coexistence of phononic sixfold, fourfold, and threefold excitations in the ternary antimonide Zr 3 Ni 3 Sb 4

Three-, four-, and sixfold excitations have significantly extended the subjects of condensed-matter physics. There is an urgent need for a realistic material that can have coexisting three-, four-, and sixfold excitations. However, these materials are uncommon because these excitations in electronic systems are usually broken by spin-orbit coupling (SOC) and are normally far from the Fermi level. Unlike the case in electronic systems, phonon systems with a negligible SOC effect, not constrained by the Pauli exclusion principle, provide a feasible platform to realize these excitations in a wide frequency range. Hence, in this work, we demonstrate by first-principles calculations and symmetry analysis that perfect three-, four-, sixfold excitations appear in the phonon dispersion rather than the band structures of ${\mathrm{Zr}}_{3}{\mathrm{Ni}}_{3}{\mathrm{Sb}}_{4}$, which is a well-known indirect-gap semiconductor with an ${\mathrm{Y}}_{3}{\mathrm{Au}}_{3}{\mathrm{Sb}}_{4}$-type structure. This material features a threefold-degenerate quadratic contact triple-point phonon, fourfold-degenerate Dirac point phonon, and sixfold degenerate point phonon. Moreover, these nodal point phonons are very robust to uniform strain. Two obvious phonon surface arcs of the (001) plane are extended in the whole Brillouin zone, which will facilitate their detection in future experimental studies. The current work provides an ideal model to investigate rich excitations in a single material.