Coexistence of zero-, one- and two-dimensional degeneracy in tetragonal SnO$_2$ phonons

TL;DR

This paper demonstrates the coexistence of zero-, one-, and two-dimensional topological phonon degeneracies in SnO$_2$, providing a new platform for studying complex phononic topological phenomena with observable surface states.

Contribution

First to predict and confirm the coexistence of multiple dimensional topological phonons in a single material, SnO$_2$, through symmetry analysis and first-principles calculations.

Findings

Coexistence of zero-, one-, and two-dimensional phonon degeneracies in SnO$_2$.

Presence of observable phonon surface states.

Negligible spin-orbit coupling effects in these topological phonons.

Abstract

Based on the dimension of degeneracy, topological electronic systems can roughly be divided into three parts: nodal point, line and surface materials corresponding to zero-, one- and two-dimensional degeneracy, respectively. In parallel to electronic systems, the concept of topology was extended to phonons, promoting the birth of topological phonons. Till date, few nodal point, line and surface phonons candidates have been predicted in solid-state materials. In this study, based on symmetry analysis and first-principles calculation, for the first time, we prove that zero-, one- and two-dimensional degeneracy co-exist in the phonon dispersion of one single realistic solid-state material SnO$_2$ with \textit{P}4$_2$/\textit{mnm} structure. In contrast to the previously reported electronic systems, the topological phonons observed in SnO$_2$ are not restricted by the Pauli exclusion principle, and they experience negligible spin-orbit coupling effect. Hence, SnO$_2$ with multiple dimensions of degeneracy phonons is a good platform for studying the entanglement among nodal point, line and surface phonons. Moreover, obvious phonon surface states are visible, which is beneficial for experimental detection.