Anisotropic Gyromagnetic Ratio and Orthogonal Einstein-de Haas Effect

Abstract

We theoretically demonstrate an orthogonal Einstein-de Haas effect, where the rotation of ferromagnetic materials is caused by the change of magnetization in the direction orthogonal to the rotation axis. This amounts to an anisotropic gyromagnetic ratio. To reveal its microscopic origin, we treat the spin-orbit coupling as a perturbation, integrate out the electronic degree of freedom, and show that in collinear ferromagnets the phonon angular momentum admits a dipolar structure in the spin-order space due to the constraint of the spin group symmetry. The spin-flipping and spin-conserving parts of the spin-orbit coupling contribute differently to such a dipolar structure. All these features are exemplified in a lattice electron-phonon model with ferromagnetic order and $C_{1h}$ point group symmetry. Our work lays the ground for revealing the connection between phonon angular momentum and general spin-order configurations.

Figures (2)

• Figure 1: Schemes of (a) Conventional (parallel) Einstein-de Haas effect, and (b) Orthogonal Einstein-de Haas effect.
• Figure 2: The phonon angular momentum of the lattice model. (a) The unit cell of the lattice model. Bonds with different hopping amplitudes are showing with different colors. (b) The decomposition of phonon angular momentum into parallel and perpendicular components. (c) and (d) The $L_{ph,\parallel}$ and $L_{ph,\perp}$ with $\hat{\bm m}$ rotating in the $zy$ plane. (e) The $L_{ph,y}$ with $\hat{\bm m}$ rotating in the $xz$ plane. (f) The dipolar structure of the $L_{ph,\perp}$. (g) and (h) The $L_{ph,\perp}$ as a function of the strength of the spin-orbit coupling with $\hat{\bm m}\parallel \hat{z}$ and $\bm \delta$ in the spin-orbit coupling along $\hat{z}$ (in (g)) and $\hat{y}$ direction (in (h)).