Theory of polarization-dependent phonon pumping in ferromagnetic/non-magnetic bilayers

TL;DR

This work addresses polarization-dependent magnon-phonon coupling in ferromagnetic/non-magnetic bilayers when the magnetic and elastic symmetry axes are misaligned. It develops a minimal theoretical model that couples the uniform Kittel mode to elastic waves, incorporating rotation of the elastic tensor and boundary-condition–driven phonon pumping. The analysis reveals phononic birefringence with velocity-split transverse modes and magnon-phonon hybridization that imparts magnetic-field dependence to phonons, including tunable polarization transfer between phonons and the Kittel mode. The results quantitatively reproduce recent microwave-transmission experiments and provide a framework for engineering polarization- and mode-selective magnon-phonon interactions in layered heterostructures with potential applications in spintronics and quantum acoustics.

Abstract

We develop a theoretical model for polarization-selective phonon pumping induced by magnon-phonon coupling in a ferromagnetic/non-magnetic acoustic bilayer structure, focusing on the effects arising from a misalignment between the magnetic and crystallographic symmetry axes. Our model considers the coupled equations of motion describing uniform magnetization dynamics (the Kittel mode) and elastic waves in both layers, incorporating phonon pumping and boundary conditions at the interface. We show that even small misalignments lift the degeneracy of transverse shear elastic modes, resulting in phononic birefringence characterized by distinct propagation velocities for linearly polarized modes. Furthermore, our analysis reveals that magnon-phonon hybridization gives magnetic-field-dependent properties to otherwise non-magnetic phonons. We show that the polarization transfer between linearly polarized phonons and the circularly polarized Kittel mode can be tuned with an external magnetic field. Our theoretical results quantitatively reproduce recent experimental findings [1].

Figures (5)

• Figure 1: (a) The orientation of the coordinate systems and schematic representation of the two-layered structure composed of a ferromagnetic film with thickness $d$ and an elastic insulator with thickness $L\gg d$. The boundaries and interfaces are labeled by letters A, B and C. (b) Propagation direction of the elastic waves along $z$-axes with respect to the crystallgraphic axes of the hexagonal layer, $\theta=-\tilde{\theta}$. (c) Rotation of strain tensor.
• Figure 2: (a) Dispersion relations of elastic waves in an hexagonal crystal for $x$- (blue line) and $y$- (orange line) polarized elastic waves at a relatively large angle of deflection of the strain tensor $\theta=\pi/6$ for visualization purposes. (b) Difference in group velocities of the waves. The calculation parameters correspond to Al$_2$O$_3$ films, and are as follows: $\rho=3970\;{\rm kg/m^{3}}$, $c_{1,1}=5.00073\times10^{11}\;{\rm Pa}$, $c_{3,3}=5.02385\times10^{11}\;{\rm Pa}$, $c_{4,4}=1.51017\times10^{11}\;{\rm Pa}$, $c_{1,2}=1.61672\times10^{11}\;{\rm Pa}$, $c_{1,3}=1.11368\times10^{11}\;{\rm Pa}$, $c_{1,4}=-2.32604\times10^{10}\;{\rm Pa}$, and $\eta=0$. These stiffness are extracted from Ref. Tefft1966
• Figure 3: (a) Normalized theoretical power absorption $A_{\rm n}^{\text{theor}}$. (b) Normalized microwave transmission magnitude $A_{\rm n}^{\text{exper}}$ related to the absorption $\left|S_{21}\right|$ from Ref. Muller2024Chiral. (c) and (d) Comparison between the experimental data (blue) and theoretical calculation (red) of the relative absorption at $\mu_{0}H_{0}=3\;{\rm T}$ (c) and $\mu_{0}H_{0}=3.01\;{\rm T}$ (d). The experimental and theoretical data are normalized. The calculation parameters correspond to the Al$_2$O$_3$ and Co$_{25}$Fe$_{75}$ layers and they are $\gamma/\left(2\pi\right)=29.80\;{\rm GHz/T}$, $K_{1}=0$, $M_{{\rm s}}=1.91083\times10^{6}\;{\rm A/m},$$L=510\;\mu{\rm m}$, $d=30\;{\rm nm}$, $\rho=3970\;{\rm kg/m^{3}}$, $\tilde{\rho}=8110\;{\rm kg/m^{3}}$, $c_{1,1}=5.00073\times10^{11}\;{\rm Pa}$, $c_{3,3}=5.02385\times10^{11}\;{\rm Pa}$, $c_{4,4}=1.51017\times10^{11}\;{\rm Pa}$, $c_{1,2}=1.61672\times10^{11}\;{\rm Pa}$, $c_{1,3}=1.11368\times10^{11}\;{\rm Pa}$, $c_{1,4}=-2.32604\times10^{10}\;{\rm Pa}$, $\tilde{c}_{4,4}=8.15\times10^{10}\;{\rm Pa}$, $c_{tx}=6167.87\;{\rm m/s},$$c_{ty}=6167.37\;{\rm m/s},$$\tilde{c}_{tx}=\tilde{c}_{ty}=3170\;{\rm m/s}$$B_{\perp}=15.7\times10^{6}\;{\rm J/m^{3}}$, $\alpha=0.004$, $\eta=2\pi\times2.03\;{\rm GN/m^{4}}$, $\phi_e = 0.22798\;{\rm rad}$, $N_{xx}=N_{yy}=0$, $N_{zz}=1$ and $\theta=0.03^{\circ}$.
• Figure 4: Stokes parameter $V$ of magnetization at different polarization of exciting magnetic field: (a) $\emph{x}$-polarized field (b) $\emph{y}$-polarized field. The calculation parameters are the same as for Fig.$~$\ref{['fig:absorbtion']}.
• Figure 5: Susceptibility of hexagonal (a) and cubic (b) bilayers. The parameters for hexagonal structure consisting of Al$_2$O$_3$ and Co$_{25}$Fe$_{75}$ layers are the same as in Fig. \ref{['fig:absorbtion']}. The parameters for cubic structure comprised of Si and Co$_{25}$Fe$_{75}$ layers were taken as $\gamma/\left(2\pi\right)=29.80\;{\rm GHz/T}$, $K_{1}=0$, $M_{{\rm s}}=1.91083\times10^{6}\;{\rm A/m},$$L=675\;\mu{\rm m}$, $d=30\;{\rm nm}$, $\rho=2330\;{\rm kg/m^{3}}$, $\tilde{\rho}=8110\;{\rm kg/m^{3}}$, $c_{1,1}=c_{3,3}=161\;{\rm GPa}$, $c_{4,4}=76.1\;{\rm GPa}$, $c_{1,2}=c_{1,3}=64\;{\rm GPa}$, $\tilde{c}_{4,4}=8.15\times10^{10}\;{\rm Pa}$, $B_{\perp}=19.1\times10^{6}\;{\rm J/m^{3}}$, $\alpha=0.004$, $\eta=2.8\times10^{5}\;{\rm N/m^{4}}$, $N_{xx}=N_{yy}=0$, $N_{zz}=1$ and $\theta=0.03^{\circ}$.