Coexistent topological and chiral phonons in chiral RhGe: An ab initio study

TL;DR

This study demonstrates a coexisting regime of topological and chiral phonons in the noncentrosymmetric chiral crystal RhGe. Using ab initio density functional theory and symmetry-topology analysis, it reveals six spin-1 triply degenerate nodal points at the Γ point and six charge-2 double Weyl points at the R point in the phonon spectrum, mirroring the electronic structure without spin-orbit coupling. The work further shows that RhGe hosts extensive chiral phonon modes with finite phonon angular momentum and phonon magnetic moments, particularly along the chiral axis and near topological nodes, with all topological phonons identified as chiral. Temperature enhances these effects, suggesting experimental detectability via spectroscopic methods and opening avenues for phonon-based control of thermal and magnetic properties in CoSi-family materials.

Abstract

The CoSi-family of materials (CoSi, CoGe, RhSi and RhGe) forms a cubic chiral structure and hosts unconventional multifold chiral fermions, such as spin-1 and spin-3/2 fermions, leading to intriguing phenomena like long Fermi arc surface states and exotic transport properties. Recent interest on the phonon behavior in chiral materials is growing due to their unique characteristics, including topological phonons, protected surface states and the chiral phonons with non-zero angular momentums. In this study, we explore the topological and chiral phonon behavior in RhGe, using first-principles density functional theory calculations as well as the symmetry and topological analysis. In particular, we uncover six spin-1 triply degenerate nodal points at the $Γ$ point and six charge-2 double Weyl points at the R point in the Brillouin zone (BZ). Interestingly, these topological features are identical to that in the electronic band structure without the electron spin-orbit coupling, of the same material. We expect that this finding not only applies to the CoSi family but also is universal. Secondly, we find that chiral crystal RhGe hosts chiral phonon modes with a phonon angular momentum (PAM) and an associated phonon magnetic moment (PMM), everywhere in the BZ except at high symmetry points such as $Γ$, R, X and M. The PAM and PMM are large along the chiral rotation axis and also in the vicinity of the topological nodes. Our study also reveals that all the topological phonon modes are chiral. However, the reverse is not always true. Among other things, our finding of the coexistence of topological and chiral phonon modes in chiral RhGe not only deepens our understanding of the phonon behavior in the CoSi-family but also opens new pathways for developing advanced materials and devices.

Figures (6)

• Figure 1: (a) Crystal structure of RhGe in the cubic primitive cell. (b) The bulk BZ and the projected (001) and (111) surface BZ. (c) Top view along the [111] direction [the red line in (a)] of the crystal structure (2×2×2 supercell). Here the transparency of the atoms denotes the depth of the atomic positions from top to bottom. The green and indigo coloured arrows indicate the right-handed and left-handed helicity (chirality) of the Ge and Rh atoms, respectively.
• Figure 2: (a) Phonon dispersion along the symmetry lines in the BZ and (b) total and atom-projected phonon density of states (DOS) of RhGe. The red and blue boxes in (a) denote the locations of the spin-1 nodal points at $\Gamma$ and charge-2 double Weyl points at R, respectively. (c) Enlarged plot of the spin-1 nodal point around 7.92 THz at $\Gamma$ and (d) enlarged plot of charge-2 double Weyl point near 7.92 THz at R. In (c) and (d), $C$ denotes the Chern number of the phonon bands.
• Figure 3: (a) Bulk phonon dispersion, (b) (111) surface BZ, and (c) (111) surface phonon dispersion of RhGe. In (c), $\omega_1$ and $\omega_2$ are two different frequencies near the double Weyl point. (d) and (e) Surface arcs at $\omega_1$ and $\omega_2$ frequencies, respectively.
• Figure 4: (a) and (b) (001) surface phonon bands of RhGe along symmetry lines in the (001) surface BZ [see Fig. 1(b)]. Here $\omega_1$ and $\omega_2$ are two different frequencies near the double Weyl point. (c) and (d) Surface arcs at $\omega_1$ and $\omega_2$ frequencies, respectively. (e) and (f) Berry curvatures around the $\Gamma$ (source) and R/M (sink) points, respectively.
• Figure 5: (a) Rh atomic, (b) Ge atomic and (c) total phonon angular momentum (PAM) ($J_{111}$) of phonon bands of RhGe along the symmetry lines in the BZ at $T = 0$ K. Red and blue colors on the color bar indicates positive and negative values of PAM, respectively. Note that at high symmetry points $\Gamma$, R, X and M, the PAM values should be zero, as dictated by the symmetry. When plotting the PAM values on the phonon bands, although the symbols with the minimum size were used, the impression of non-zero values at these symmetry points may be created.