Anomalous phonon magnetic moments

Abstract

Circularly polarized phonons conventionally carry an angular momentum and a magnetic moment arising from circular motions of the atoms. Here, we present three anomalous cases that lead to phonon magnetic moments, which cannot be described in the conventional framework: rotationless axial phonons, which exhibit magnetic responses despite only carrying pseudo angular momentum, divergent gyromagnetic ratios of phonons, in which a magnetic moment is produced despite vanishing angular momentum, and anisotropic gyromagnetic ratios of phonons, which make the phonon angular momentum and magnetic moment noncollinear. Our results shed light on the origin and nature of phonon magnetism and suggest the existence of phononomagnetic hidden order.

Figures (7)

• Figure 1: Anomalous phonon magnetic moments. (a) Rotationless axial phonons can carry effective magnetic moments while containing only linear atomic motion. (b) Divergent gyromagnetic ratios produce finite magnetic moments despite vanishing angular momentum. (c) Anisotropic gyromagnetic ratios produce noncollinear phonon angular momentum and magnetic moments.
• Figure 2: Out-of-plane atomic displacements in monolayer h-BN. (a) Atomic displacement associated with the transverse acoustic modes, which correspond to the out-of-plane motion from phonons at $K/K'$ valleys with frequencies of 9.2 THz (top) and 17.9 THz (bottom). While all eigenvectors point out of the plane, their relative motion is phase delayed as shown in (b), resembling the motion of an Euler disk. (b) $K$ and $K'$ valley phonons have an opposite phase difference, resulting in an orbital PAM of $\pm 1$. Boron atoms are shown in green and nitrogen atoms are shown in yellow.
• Figure 3: Spin and orbital PAM in a kagome lattice. Phonons with out-of-plane motion in a kagome lattice can carry PAM due to intracell and intercell phase differences. A three-fold rotation of the phonon mode around a $C_3$ symmetric point in the lattice results in a phase of $\pm2\pi/3$. (a) Zone-center phonon, producing a $2\pi/3$ phase difference between the three atoms on a given lattice site, leading to a spin PAM of $1$ with the same sign for all sublattices under a three-fold rotation. (b) Valley phonons, with a $2\pi/3$ phase difference between the three lattice sites from neighboring unit cells, resulting in an orbital PAM under a three-fold rotation.
• Figure 4: Axial $E_{2u}$ mode and its magnetic response. (a) Displacement associated with the two orthogonal components of the $E_{2u}$ mode. (b) A circular superposition of the two components results in a relative phase difference between displacements of different atoms within the same unit cell, leading to a phonon spin pseudo angular momentum. (c) Phonon Zeeman splitting, $\Delta\omega_{ph}$ of the axial $E_{2u}$ mode in the presence of an external magnetic field. (d) Temperature dependence of the effective phonon magnetic moment, $m_\mathrm{eff}^{ph}$.
• Figure 5: (a) Phonon angular momentum, $l^{ph}$, and (b) phonon magnetic moment, $m^{ph}$, (in units of the nuclear magneton, $\mu_\mathrm{n}$) of the longitudinal acoustic (LA) and fast transverse acoustic (TA) bands of monolayer h-BN, shown as a function of wavevector along the $\mathrm{M} \ (\frac{1}{2} \ 0 \ 0)-\mathrm{K} \ (\frac{1}{3} \ \frac{1}{3} \ 0) - \mathrm{\Gamma} \ (0 \ 0 \ 0)$ trajectory in reciprocal space. (c) Atomic displacements in the TA band at the $\mathrm{K}$ point, where B and N atoms revolve in opposite directions around their equilibrium positions (transparent spheres). This leads to a near-vanishing net angular momentum, counter-aligned for the two sublattices, but a substantial net magnetic moment (blue arrows), co-aligned for the two sublattices due to their opposite effective charges.