Nonreciprocal magnetoacoustic waves in synthetic antiferromagnets with Dzyaloshinskii-Moriya interaction

The interaction between surface acoustic waves (SAWs) and spin waves (SWs) in a piezoelectric/magnetic thin film heterostructure yields potential for the realization of novel microwave devices and applications in magnonics. In the present work, we investigate the SAW-SW interaction in a Pt/Co($2\mathrm{nm}$)/Ru($0.85\mathrm{nm}$)/Co($4\mathrm{nm}$)/Pt synthetic antiferromagnet (SAF) composed of two ferromagnetic layers with different thicknesses separated by a thin nonmagnetic Ru spacer layer. Because of the combined presence of interfacial Dzyaloshinskii–Moriya interaction (iDMI) and interlayer dipolar coupling fields, the optical SW mode shows a large nondegenerate dispersion relation for oppositely propagating SWs. Due to SAW-SW interaction, we observe nonreciprocal SAW transmission in the piezoelectric/SAF hybrid device. The equilibrium magnetization directions of both Co layers are manipulated by an external magnetic field to set a ferromagnetic, canted, or antiferromagnetic configuration. This has a strong impact on the SW dispersion, its nonreciprocity, and SAW-SW interaction. The experimental results are in agreement with a phenomenological SAW-SW interaction model, which considers the interlayer exchange coupling, iDMI, and interlayer dipolar coupling fields of the SWs.

Figure 1

(a) Interlayer dipolar coupling and iDMI can give rise to a very high nonreciprocity $\mathrm{\Delta }{f}_{\pm }=f\left(k^{+}\right)-f\left(k^{-}\right)$ of the SW dispersion in a synthetic antiferromagnet (SAF) with iDMI. Thus, the SAW-SW resonance differs for counter-propagating SAWs. (b) Illustration of the investigated magnetoacoustic hybrid device and the coordinate system used. The nonreciprocity of the magnetoacoustically excited SWs in the Pt/Co(2)/Ru/Co(4)/Pt SAF is characterized by different transmission magnitudes of oppositely propagating waves ${\mathbf{k}}^{+}$ and ${\mathbf{k}}^{-}$. For moderate excitation frequencies ($f<7\mathrm{GHz}$) solely the optical SW mode can be excited in the antiferromagnetic (AFM) configuration. This is schematically shown by the trajectory of the precessing magnetic moments (black arrows) in the Damon-Eshbach geometry (${\varphi }_0^l=\pm 90{}^{\circ }$).

Figure 2

(a) The $M-H$ hysteresis loop of the Pt/Co(2)/Ru/Co(4)/Pt SAF with the magnetic field applied in the $xy$ easy magnetization plane along ${\varphi }_H=0{}^{\circ }$. The magnetization hysteresis loops of experiment and simulation agree well. Both show a FM, C, and AFM configuration, which are additionally schematically depicted by the blue (layer $A$) and green (layer $B$) magnetization vectors. The parameters of the simulation are given in Table 1 and ${\varphi }_H=0{}^{\circ }$ is assumed. (b) The magnetization dynamics and the effective magnetoacoustic driving fields are described in individual (${\mathbf{e}}_1^l,{\mathbf{e}}_2^l,{\mathbf{e}}_3^l$) coordinate systems of both layers. Here, the ${\mathbf{e}}_3^l$ axes are defined by the equilibrium orientations ${\varphi }_0^l$ of the magnetizations ${\mathbf{M}}^l$.

Figure 3

(a) The calculated resonance frequency of the optical SW mode as a function of the external magnetic field ${\mu }_{0}H$ shows nondegenerated frequencies for counter-propagating SWs with wave vectors $k^{+}$ and $k^{-}$. Calculated for ${\varphi }_H=45{}^{\circ }$ and $|k^{\pm }|=13.1\mu {\mathrm{m}}^{-1}$. (b) The large frequency nonreciprocity $\mathrm{\Delta }{f}_{\pm }=f\left(k^{+}\right)-f\left(k^{-}\right)$ can be understood as an additive interplay of both individual contributions of iDMI and IDC [23]. (c) The equilibrium orientations ${\varphi }_0^l$ of the magnetizations ${\mathbf{M}}^l$, which are schematically depicted by the arrows govern the nonreciprocal SW resonance.

Figure 4

Magnetoacoustic driving field and impact on the excited SW modes in the investigated SAF sample. [(a) and (b)] The dominating in-plane component $h_2^l$ of the effective magnetoacoustic driving field of a Rayleigh-type SAW in the individual magnetic layers of the SAF is calculated with Eq. (4) on basis of the equilibrium magnetization directions ${\varphi }_0^l$ and the parameters in Table 1. The symmetry of $|h_2^l|$ follows the well-known fourfold symmetry [48] in the FM and AFM configuration, but is more complicated in the C configuration. [(c)–(e)] Schematic drawings of the excited optical SW modes and driving fields $h_2^l$, which can potentially be excited in the FM, C, and AFM configuration for ${\varphi }_H=45{}^{\circ }$. Here, the acoustic SW modes cannot be magnetoacoustically excited since its resonance frequency is much higher ($f>20\mathrm{GHz}$) than the SAW excitation frequency ($6.77\mathrm{GHz}$). While the magnetoelastic excitation of SWs in the AFM configuration is efficient, it is not efficient in the FM configuration. The drawing of the C configuration depicts only one possible situation. The direction of $h_2^l$ is defined by the direction of the ${\mathbf{e}}_2^l$ axes (see Fig. 2).

Figure 5

[(a)–(h)] The frequency dependence of the magnetoacoustic transmission $\mathrm{\Delta }{S}_{21}({\mu }_{0}H,{\varphi }_H)$ was determined as a function of external magnetic field magnitude and orientation for all four harmonic resonances of the acoustic delay line. Experiment (Expt.) and simulation (Simu.) show good agreement. [(i)–(l)] The SW resonances $f$ (blue), SW linewidths indicated by $f\pm \mathrm{\Delta }{f}_{\text{HWHM}}$ (light blue), and SAW excitation frequencies (orange) at ${\varphi }_H=45{}^{\circ }$. Both, simulations [(e)–(h)] and SW resonances [(i)–(l)] were calculated with the parameters of Table 1 and the corresponding wave vector of the SAW ($k_{\text{SW}}=k_{\text{SAW}}$). The nonreciprocity of $\mathrm{\Delta }{S}_{21}({\mu }_{0}H,{\varphi }_H)$ and of the SW resonance $f\left({\mu }_{0}H\right)$ decrease with decreasing frequency.

Figure 6

Change of the SAW transmission $\mathrm{\Delta }{S}_{ij}({\mu }_{0}H,{\varphi }_H)$ for counter-propagating magnetoacoustic waves with the wave vectors $k^{+}$ and $k^{-}$ at $6.77\mathrm{GHz}$. The large frequency nonreciprocity $\mathrm{\Delta }{f}_{\pm }$ of about $2\mathrm{GHz}$ at ${\varphi }_H=\pm 45{}^{\circ }$ in the AFM configuration (see Fig. 3) results in a pronounced nonreciprocity of the SAW transmission. [(a) and (c)] Experiment (Expt.) and [(b) and (d)] corresponding simulations (Simu.) show good overall agreement.

Figure 7

The out-of-plane magnetoacoustic driving field ${\tilde{h}}_1^l\propto b_2^l{\tilde{a}}_{xz}$ is changed in simulations to probe the impact of the SAW-SW helicity mismatch effect on the SAW transmission $\mathrm{\Delta }{S}_{ij}({\mu }_{0}H,{\varphi }_H)$ at $6.77\mathrm{GHz}$. (a) The SAW-SW helicity mismatch effect vanishes for an in-plane linear polarized driving field. The results in (a) are almost identical to the results in (b), for which the same simulation parameters as in all previous simulations were used. Thus, the impact of the SAW-SW helicity mismatch effect is small in the simulations in Figs. 5 and 6. If the driving field ${\tilde{h}}_1^B\propto b_2^B{\tilde{a}}_{xz}$ of the top Co layer is artificially low (c) or high (d), the SAW-SW helicity mismatch effect modulates slightly the SAW transmission nonreciprocity.