Ab Initio Theory of Phonon Magnetic Moment Induced by Electron-Phonon Coupling in Magnetic Materials

Abstract

Circularly polarized phonons, characterized by nonzero angular momenta and magnetic moments, have attracted extensive attention. However, a long-standing critical issue in this field is the lack of an approach to accurately calculate phonon magnetic moments resulting from electron-phonon coupling (EPC) in realistic materials. Here, based on the linear response framework, we develop an ab initio theory for calculating EPC-induced magnetic properties of phonons, applicable to both insulating and metallic materials. Our method can precisely calculate phonon Zeeman splittings in magnetic metals with significant EPC, as demonstrated by the remarkable agreement with recent experimental observations of phonon Zeeman splitting in the ferromagnetic Weyl semimetal Co3Sn2S2. In addition, the long-sought magnetic phonon spectra across the entire Brillouin zone are obtained, facilitating the study of magnetic phonon transport and topology. Specifically, by constructing an inertially decoupled lattice model, we propose candidate materials exhibiting intrinsic phonon Chern states with robust unidirectional edge phonon currents. Our work paves the way for investigating novel phonon phenomena in magnetic quantum materials.

Figures (4)

• Figure 1: Schematic of the phonon Zeeman splitting driven by (a) the classical PMM and (b) the EPC-enhanced PMM. In (a), the phonon band crossing point consists of two degenerate phonon modes vibrating in two different directions (black arrows), which can be recombined into left-circularly (red) and right-circularly (blue) polarized phonons. The classical phonon Zeeman splitting deduced from the point-charge model ($Z_{\text{eff}}$ is the effective charge) is negligible as a result of the significant atomic mass, whereas the EPC-induced phonon Zeeman splitting (b) is orders of magnitude larger.
• Figure 2: Lattice structure and calculated magnetic phonon dispersions of $\mathrm{Co_3Sn_2S_2}$. (a) The conventional cell (black lines) and primitive cell (green lines) of $\mathrm{Co_3Sn_2S_2}$. (b) The magnetic phonon dispersion with phonon Zeeman splittings along $\Gamma$-T path. (c) The magnetic phonon dispersion of $E_g$ modes. The color gradient from red to blue represents the phonon angular momentum. Upper insets: The phonon magnetic moment for the two branches of $E_g$ mode along $\Gamma$-T path.
• Figure 3: Inertially decoupled model (IDM) and intrinsic phonon Chern states. (a) Schematic of circularly polarized phonon modes in the high frequency regime: When $m\sim M$, both atoms contribute to high-frequency modes (upper panel); in the IDM condition ($m \ll M$), high-frequency modes are primarily contributed by light atoms (lower panel). (b) Phonon dispersions of nonmagnetic honeycomb lattice and ferromagnetic triangular lattice. (c-e) TRS-breaking phonon dispersions of the 2D IDM by combining the honeycomb and triangular lattices (c,d) and the 3D IDM by stacking 2D IDM along the $z$-direction (e). (f) Edge states (cyan/magenta) of the 2D IDM. Parameters are detailed in the SM supp.
• Figure 4: Lattice structure and TRS-breaking phonon dispersions of ferromagnetic $\mathrm{EuSi_2}$. (a,b) The front view (a) and the side view (b) of the lattice structure of $\mathrm{EuSi_2}$. (c) The Brillouin zones of bulk $\mathrm{EuSi_2}$ and the projected side plane. (d,e) The phonon dispersions of bulk $\mathrm{EuSi_2}$ without (d) and with (e) the phonon Zeeman splittings, respectively. The calculated PMMs at $K(K')$ and $H(H')$ point are $5.7\times 10^{-3} \mu_B$ and $2.3\times 10^{-2} \mu_B$, respectively. (f) The phonon dispersion of bulk $\mathrm{EuSi_2}$ under open-boundary conditions. (g,h) The phonon dispersion of the sandwiched slab enclosed by the black rectangle in (b) without (g) and with (h) the phonon Zeeman splittings, respectively. The PMM at $K(K')$ point is $2.2\times 10^{-2} \mu_B$. (i) The phonon dispersion of the sandwiched slab under open-boundary conditions. The color gradient from red to blue represents the phonon angular momentum. Cyan and magenta colors in (f) and (i) label topological boundary modes residing on opposite boundaries.