Phononic-crystal cavity magnomechanics

Acoustic phonons are possibly means for controlling magnons on-chip because of their micro-/nanometer wavelengths (similar to those of magnons), low-loss property, and negligible cross-talk Li et al. (2021); Zhang et al. (2016); Kikkawa et al. (2016); Berk et al. (2019); An et al. (2020); Potts et al. (2021). The pioneering studies on magnomechanical technology used surface acoustic wave (SAW) devices Weiler et al. (2011); Dreher et al. (2012); Kobayashi et al. (2017); Sasaki et al. (2017); Hern$\acute{\rm a}$ndez-M$\acute{\rm i}$nguez et al. (2020); Xu et al. (2020); Kawada et al. (2021); Hatanaka et al. (2022). They succeeded in generating various magnetoelastic phenomena like acoustic spin pumping Kobayashi et al. (2017); Matsuo et al. (2013) and nonreciprocal transport Sasaki et al. (2017); Xu et al. (2020). However, their large cavity structures are unsuitable for system integration and have difficulty taking full advantage of phonons Hatanaka et al. (2022).
    A phononic crystal (PnC) is a promising platform that enables acoustic phonons to be guided and trapped in a tiny wavelength-scale acoustic cavity and waveguide Maldovan (2013); Benchabane et al. (2006); Mohammadi et al. (2009); Otsuka et al. (2013); Pourabolghasem et al. (2018); Baboly et al. (2018); Hatanaka and Yamaguchi (2020). Individual cavities are efficiently and finely driven and thereby can be used to control magnetic elements embedded in a PnC circuit. Moreover, the cavity sustains various spatial distributions of vibrational strains among multiple acoustic resonances, so it can be used to adjust the magnetoelastic effect Hatanaka and Yamaguchi (2021). We consider that PnCs will enable us to make full use of phonons in magnomechanical technology.
      Here, we demonstrate a PnC-based magnomechanical system sustained by cavity-waveguide coupled systems, as shown in Fig. 1(a). The nanomechanical vibrations confined in the cavity, excited through the waveguide, generate spin waves (magnons) in a nickel (Ni) film placed on its surface via magnetostriction. The acoustic spin pumping reversibly induces a frequency shift and damping modulation of the cavity resonances. Moreover, the magnon-phonon interaction can be tailored by selectively driving an appropriate cavity mode with specific strain distributions. PnC cavity magnomechanics is useful for on-chip control of magnons and phonons as well as their hybridized states (magnon polaron), and shows promise for extending the capabilities of classical and quantum information technologies.
    The PnC is fabricated in free-standing GaAs, as shown in Fig.1(b) (the details of the fabrication and structure are presented in the Methods.). It consists of a snowflake triangular lattice with full bandgaps between 0.5 and 0.8 GHz (Fig. 1(c) and 1(d)) Hatanaka and Yamaguchi (2020); Safavi-Naeini et al. (2014). Acoustic waves at frequencies within the bandgap propagate in a line-defect waveguide and drive a line-defect cavity. Measuring the cavity’s spectral response reveals three acoustic resonances (Fig. 1(e)). Their modal shapes exhibit complete confinement of the vibrations in the defect (top panels of Fig. 1(f)). Numerical calculations with the finite-element method (FEM) reproduce these modal shapes and verify the origin of the observed peaks. In this way, the PnC cavity can strongly confine vibrations that are remotely driven through the waveguide.
    The Ni film on the cavity surface has a magnetization whose precession is acoustically excited via magnetostriction. The magnetostrictive force that induces the precession is divided into two components $\mu_{0}h^{\rm am}_{1}$ and $\mu_{0}h^{\rm am}_{2}$ on the $1$- and $2$-axis, whose definitions are given in Fig. 2(a). The equilibrium magnetization axis ($m_{3}$) can be decomposed into in-plane $x$ and $y$ components and is at an angle ($\phi_{\rm m}$) from the waveguide direction $x$. The out-of-plane magnetostrictive force ($\mu_{0}h^{\rm am}_{1}$) is negligibly small because out-of-plane shear strains such as $\epsilon_{\rm xz}$ and $\epsilon_{\rm yz}$ vanish in the Ni at that location, whereas in-plane force ($\mu_{0}h^{\rm am}_{2}$) can be expressed as Dreher et al. (2012)

where $b_{\rm am}$ is the magnetostrictive coefficient and $\epsilon_{\rm ij}$ is the vibrational strain component. Thus, the magnetostriction is governed by two major factors, vibrational strain ($\epsilon_{\rm xx}$, $\epsilon_{\rm yy}$ and $\epsilon_{\rm xy}$) and the magnetization direction ($\phi_{\rm m}$), whose contributions to the system are theoretically investigated below.
    The magnetization angle ($\phi_{\rm m}$) can be predicted from the magnetic free-energy density normalized by the saturation magnetization ($M_{s}$), given by Dreher et al. (2012)

where $\mu_{0}\bm{H_{\rm ex}}$ and $\bm{m}$ are the external magnetic field and unit vector of magnetization, respectively. The thin-film structure of the Ni results in a perpendicular magnetic anisotropy ($B_{\rm d}$). The angle $\phi_{\rm m}$ is determined by estimating the minimum of $G$. In this calculation, the in-plane anisotropic field ($B_{\rm u}$) and its unit vector ($\bm{u}$) are introduced so as to reproduce the experimental magnetization curve (Fig. 2(b)). For instance, the response of $\phi_{\rm m}$ as a function of $\mu_{0}H_{\rm ex}$ at $\phi_{\rm h}$ = 0^∘ is shown in the top panel of Fig. 2(c). The magnetization is parallel to $\mu_{0}H_{\rm ex}$, i.e. $\phi_{\rm m}=0^{\circ}$, when the field strength stays in the high field region $|\mu_{0}H_{\rm ex}|$ $>$ 20 mT. However, it undergoes a rotation to $\phi_{\rm m}=90^{\circ}$ in the low field region $|\mu_{0}H_{\rm ex}|<$ 20 mT before reversing. In this way, the magnetization experiences a rotation and reversal while sweeping $\mu_{0}H_{\rm ex}$. This change in magnetization determines the magnon resonance frequency ($\omega_{\rm m}/(2\pi)$). The field response is shown in the bottom panel of Fig. 2(c). The frequency monotonically decreases with decreasing $\mu_{0}H_{\rm ex}$ and approaches the acoustic resonant frequency ($\omega_{\rm a}/(2\pi)$) at $|\mu_{0}H_{\rm ex}|$ = 7 mT, where the magnon-phonon frequency mismatch is minimized.
    Another aspect determining the magnetostriction is the spatial distribution and direction of vibration strains. The observed acoustic resonances can be decomposed into three strain components (Fig. 2(d)-2(f) shows the spatial profiles of the longitudinal ($\left|\epsilon_{\rm xx}-\epsilon_{\rm yy}\right|$) and shear strain components ($\left|2\epsilon_{\rm xy}\right|$) of the resonances). Shear (longitudinal) strain is dominant in the resonances at 0.581 GHz (0.593 GHz), whereas both strains are comparable at 0.552 GHz. The desired strain distributions can thus be generated by selectively actuating an appropriate resonance. Hereafter, the shear- and longitudinal-strain modes will be labeled S and L, while the mode with comparable strains will be labeled SL.
    The magnetization dynamics and spatial strain profiles allow us to estimate the magnetostrictive coupling mode volume ($V_{\rm c}$), which characterizes the interaction efficiency. determined by $\phi_{\rm m}$ and the magnon-phonon spatial mode overlap (expression and derivation in the Methods). Figures 2(g)-2(i) show the simulated field dependence of $V_{\rm c}$ of the L, SL, and S modes at $\phi_{\rm h}$ = 0^∘ and for various $\phi_{\rm h}$ between $0^{\circ}$ and $90^{\circ}$. For the L mode at $\phi_{\rm h}=0^{\circ}$, $V_{\rm c}$ mostly vanishes in the high field region because $\phi_{\rm h}=\phi_{\rm m}=0^{\circ}$ and $\left|2\epsilon_{\rm xy}\right|$ is negligibly small. In contrast, it increases dramatically as $\mu_{0}H_{\rm ex}$ decreases below 20 mT. This enhancement at $\mu_{0}H_{\rm ex}=6$ mT is caused by the magnetization rotation from $\phi_{\rm m}$ = 0^∘ to 45^∘; thereby, the term $(\epsilon_{\rm xx}-\epsilon_{\rm yy})\sin{2\phi_{\rm m}}$ becomes non-zero. $V_{\rm c}$ temporarily returns to almost zero as $\phi_{\rm m}$ reaches 90^∘ just before the magnetization reverses and then approaches the original value after the increase at $\phi_{\rm m}=225^{\circ}$ and $\mu_{0}H_{\rm ex}=-6$ mT. In contrast, the S mode exhibits the opposite field dependency, in which a finite $V_{\rm c}$ in the high field region is reduced in the low field region because the dominant magnetostrictive term is $2\epsilon_{\rm xy}\cos{2\phi_{\rm m}}$, not $(\epsilon_{\rm xx}-\epsilon_{\rm yy})\sin{2\phi_{\rm m}}$. Since the SL mode has almost equal contributions from both strain components, the variation with respect to $\mu_{0}H_{\rm ex}$ is moderate compared with the other two modes (Methods). As $\phi_{\rm h}$ increases, the susceptibility to $\mu_{0}H_{\rm ex}$ is cyclically modulated (right panels of Fig. 2(g)-2(i)). The change in direction of $V_{\rm c}$ while sweeping $\phi_{\rm h}$ is opposite between S and L modes, resulting from different dominant strain components. These results indicate that the cavity mode structures as well as the external field can be used to tune the magnetostrictive interaction.
    To experimentally show the magnetostrictive interaction, the field response of the cavity resonance was measured at $\phi_{\rm h}=0^{\circ}$. The observed resonant frequency shift ($\Delta\omega^{\prime}_{\rm a}/(2\pi)$) and quality factor ($Q_{\rm a}^{\prime}$) are plotted as a function of $\mu_{0}H_{\rm ex}$ in Fig. 3(a) and 3(b), where L and S modes are chosen for their opposite and distinct field susceptibilities. The response on the L mode exhibits dual dips at $|\mu_{0}H_{\rm ex}|$ = 5 mT with a reduction in $\Delta\omega^{\prime}_{\rm a}/(2\pi)$ and $Q_{\rm a}^{\prime}$ in the low field region. This behavior can be understood from the theoretical formula,

where $\rho$, $V_{\rm a}$, $f_{\rm d}$ and $\omega$ are mass density, acoustic mode volume, driving force density and angular frequency, and $\chi_{\rm m}$ is magnetic susceptibility (Methods). The theoretical predictions are in agreement with the experimental results, indicating that the acoustic modulation is due to the magnon-phonon interaction, as described by our model. This interaction enables acoustic excitation of spin-wave oscillations in Ni, which exerts back-action force on the cavity resonance and tunes $\omega_{\rm a}^{\prime}$ and $Q_{\rm a}^{\prime}$.
    Remarkably, ferromagnetic magnons were able to be driven by phonons in the tiny PnC cavity. The effective mode volume of the L mode is estimated to be $V_{\rm a}\approx 6.6$ $\mu$m³ = 0.54 $\lambda^{2}t$ with an acoustic wavelength $\lambda$ of 3.5 $\mu$m, which is $10^{5}-10^{6}$ times smaller than that of a SAW-based magnomechanical cavity system Hatanaka et al. (2022). The tiny-energy vibrations are confined by the high $Q$ resonance, so they hardly affect surrounding systems, unlike conventional electromagnetic-wave-based magnetic devices. We believe that the PnC cavity-waveguide system would be a building block for a magnomechanical system and could be used as a local magnon driver and phonon modulator.
    Strong back-action effects were also observed in the S mode, where shear strain dominate magnetostriction. A comparison with the effect in the L mode reveals the impact of the mode strain profiles on the interaction. Figure 3(b) shows that the dual dips in this mode at $\left|\mu_{0}H_{\rm ex}\right|$ = 10 mT are wider than those in the L mode because of the finite $V_{\rm c}$ in the high field region. In addition, a distinct center dip occurs around $\mu_{0}H_{\rm ex}=-1$ mT, an effect of the magnetization rotation. The field response is distinctly different from that in the L mode and can be simulated with our model. Thus, our cavity geometry is also used to selectively drive the magnetization dynamics utilizing the difference in strain distributions.
      To examine how the strain distribution affects the magnetoelastic modulation, the field responses of the acoustic resonant frequency ($\Delta\omega^{\prime}_{\rm a}/(2\pi)$) and damping rate ($\kappa_{\rm a}^{\prime}$) on the L and S modes were investigated at $\phi_{\rm m}$ ranging from $0^{\circ}$ to $90^{\circ}$ (Fig. 4(a)-4(d)). The magnetoelastic modulation regions in $\Delta\omega^{\prime}_{\rm a}$ and $\kappa_{\rm a}^{\prime}$ broaden as $\phi_{\rm h}$ increases from 0^∘ to 60^∘ in the L mode and shrink toward $\phi_{\rm h}$ = 90^∘. In contrast, the S mode shows the opposite dependency, in which the modulation regions become narrow around $\phi_{\rm h}$ = 45^∘. The theoretical calculations reproduce the experimental variations in both modes; here, the cyclic modulation while changing $\phi_{\rm h}$ is governed by $V_{\rm c}$, so the L and S modes show the opposite behaviors. Note that the experiment and theoretical model showed a moderate response to the magnetoelastic effect in the SL mode (Methods), supporting the validity of our model. We also found similar mode-sensitive magnetoelastic modulation in a PnC cavity with a different defect geometry (Methods). These results verify that the mode-tunable magnetostriction allows us to control magnomechanical states acoustically.
    In conclusion, the ease of designing the PnC cavity and the small spatial leakage of its vibrations are useful for constructing integrated magnomechanical systems, in which microwave signal operations such as sensing, memory and processing are performed using magnons and phonons. Moreover, it is possible to build a tiny magnomechanical system of magnon wavelength size to efficiently control and read-out the magnetic state of a micro-/nano-ferromagnetic system such as a magnetic tunnel junction Yuasa et al. (2004). We believe that PnC cavity magnomechanics will expand the use of magnons and phonons and related technologies.