Kineo-Elasticity and Nonreciprocal Phonons by Rashba-induced Interfacial Spin-Lattice Coupling

Abstract

We identify a previously unrecognized spin-lattice coupling that is allowed in the presence of broken inversion symmetry that can be considered as a lattice analogue to the electronic Rashba spin-orbit coupling. In the low-frequency regime with magnons integrated out, the interfacial spin-lattice coupling is shown to engender a kineo-elastic term in the phonon Lagrangian that couples the strain on the lattice to its velocity and thereby gives rise to a nonreciprocity in transverse phonon velocity. We further analyze the full magnon-phonon spectrum and uncover directional hybridization and absorption, leading to asymmetric phonon propagation lengths for opposite directions. Our results indicate that such interfacial spin-lattice coupling can serve as an efficient route to achieve nonreciprocal phonon propagation properties in magnetic heterostructures with strong Rashba spin-orbit coupling.

Figures (4)

• Figure 1: Schematic of a 2D ferromagnet on a substrate used for the model. The system consists of a two-dimensional ferromagnetic film in the $xy$-plane bonded to a substrate that breaks inversion symmetry along the $\hat{z}$ direction. The equilibrium magnetization ${{\bm m}}_0$ lies in the $xz$-plane with an angle $\theta$ from the $x$-axis, given by ${{\bm m}}_0 = (\cos \theta, 0, \sin \theta)$. The wavy arrows represent the propagation of acoustic waves (phonons) along the $\hat{y}$ direction with wavevectors $\pm k$.
• Figure 2: Nonreciprocity of the phonon velocity. The relative velocity asymmetry $|\Delta v / c_T|$ is shown as a function of (a) the effective magnetic field $B_{\text{eff}}$ and (b) the magnetization orientation $\theta$.
• Figure 3: Directional magnon–phonon hybridization. Dispersion relations of the magnon-phonon spectrum as a function of the wave vector $k_y$, taken along the propagation direction ${\bm k} = k_y \hat{{\bm y}}$, for different magnetization orientations: (a) $\theta =0$, (b) $\theta = \pi/6$, (c) $\theta = \pi/3$, and (d) $\theta = \pi/2$. The color scale indicates the phonon (red) and magnon (blue) character of each hybridized mode. An effective magnetic field $B_{\text{eff}} = 0.2$ T is used.
• Figure 4: Nonreciprocal phonon absorption and propagation length. The absorption rate of phonon energy by magnons, $\Gamma_{\text{m-ph}}$, is shown as a function of the phonon wave vector $k_y$ for different phonon frequencies for (a) $\theta=0$ and (c) $\theta=\pi/2$. Panels (b) and (d) display the corresponding phonon propagation length for (b) $\theta=0$ and (d) $\theta=\pi/2$. An effective magnetic field $B_{\text{eff}} = 0.2$ T, Gilbert damping $\alpha = 0.05$, and $Q = 1000$ are used. The data are obtained for $k_x = 0$ with equal amplitudes of the in-plane phonon displacements, $u_x = u_y$.