Coupled-mode theory for the interaction between acoustic waves and spin waves in magnonic-phononic crystals: Propagating magnetoelastic waves

We have investigated codirectional and contradirectional couplings between spin wave and acoustic wave in a one-dimensional periodic structure (the so-called magphonic crystal). The system consists of two ferromagnetic layers alternating in space. We have taken into consideration materials commonly used in magnonics: yttrium iron garnet, CoFeB, permalloy, and cobalt. The coupled mode theory (CMT) formalism has been successfully implemented to describe the magnetoelastic interaction as a periodic perturbation in the magphonic crystal. The results of CMT calculations have been verified by more rigorous simulations with the frequency-domain plane-wave method and the time-domain finite-element method. The presented resonant coupling in the magphonic crystal is an active in-space mechanism which spatially transfers energy between propagating spin and acoustic modes, thus creating a propagating magnetoelastic wave. We have shown that CMT analysis of the magnetoelastic coupling is an useful tool to optimize and design a spin wave–acoustic wave transducer based on magphonic crystals. The effect of spin-wave damping has been included to the model to discuss the efficiency of such a device. Our model shows that it is possible to obtain forward conversion of the acoustic wave to the spin wave in case of codirectional coupling and backward conversion in case of contradirectional coupling. That energy transfer may be realized for broadband coupling and for generation of spin waves which are of different wavelength (in particular, shorter) than exciting acoustic waves.

Figure 1

The geometry of considered system. The transverse acoustic wave described by displacement $u_3$ and the spin wave described by dynamic magnetization $\vec{m}=(m_1,m_2)$ propagate along $x_1$, which is the direction of periodicity of the one-dimensional magphonic structure. The external magnetic field and thus saturation magnetization $M_s$ is along $x_3$.

Figure 2

(a) Codirectional anticrossing of acoustic wave (A) and spin wave (S) in CoFeB calculated by CMT (red triangles) and PWM (black dots). Solid lines mark dispersion relations in the absence of magnetoelastic coupling. Arrows mark the definition of coupling coefficient ${\kappa }_{\text{max}}$. (b) Spatial distribution of acoustic-wave displacement $u$ and spin-wave dynamic magnetization $m_1$ in the codirectional coupling in CoFeB. Arrows indicate the propagation direction of waves.

Figure 3

(a) Contradirectional C2 anticrossing of acoustic wave (A) and spin wave (S) in Py1/Py2 calculated by CMT (red triangles) and PWM (black dots). Solid lines mark dispersion relations in the absence of magnetoelastic coupling. Arrows mark the definition of coupling coefficient ${\kappa }_{\text{max}}$. (b) Spatial distribution of amplitudes $U\left(x\right)$ and $M\left(x\right)$ in the contradirectional coupling in Py1/Py2. Arrows indicate the propagation direction of waves. Insets show acoustic-wave displacement $u$ and spin-wave dynamic magnetization $m_1$ in the fragment of the structure.

Figure 4

Dependence of coupling coefficient $\kappa$ on the filling factor $f$ in the Py1/Py2 magphonic crystal for the four different crossings. The lines are result of CMT calculations, while the points come from PWM simulations.

Figure 5

(a) Contradirectional C2 anticrossing of acoustic wave (A) and spin wave (S) in Py1/Co calculated by CMT (red triangles) and PWM (black dots). Dashed lines mark dispersion relations for the effective medium in the absence of magnetoelastic coupling. (b) Spatial distribution of acoustic-wave displacement $u$ and spin-wave dynamic magnetization $m_1$ in the contradirectional coupling in Py1/Co. Arrows indicate the propagation direction of waves.

Figure 6

Dependence of coupling coefficient $\kappa$ on the filling factor $f$ in the Py1/Co magphonic crystal for the four different crossings. The lines are result of CMT calculations, while the points come from PWM simulations.

Figure 7

(a) Dispersion relation for Py1/CoFeB magphonic crystal optimized for codirectional coupling obtained by PWM simulations. (b) Spatial distribution of acoustic displacement $u$ (top) and spin-wave dynamic magnetization $m_1$ (bottom) in the codirectional coupling in Py1/CoFeB. Blue and red are oscillations without damping; black dashed lines and solid lines are absolute amplitudes of the waves with damping from FEM simulations and CMT calculations, respectively. Black horizontal line at the $x$ axis indicates the area of the magphonic crystal; the acoustic wave is excited at point S; patterned areas are damping edges.

Figure 8

Amplitudes $U\left(x\right)$ (blue line) and $M\left(x\right)$ (red line) in the Py1/CoFeB system without damping (solid lines) and with damping (dashed lines) in (a) codirectional coupling and (b) contra-directional C2 crossing.