In this study, we report giant nonreciprocal transmission of shear-horizontal surface acoustic waves (SH-SAWs) in a ferromagnetic bilayer structure with negative–positive magnetostriction configuration. Although the directions of magnetization in the neighboring layers are parallel, SH-SAWs can excite optical-mode spin waves (SWs) via magnetoelastic coupling at relatively low frequencies, which is much stronger than acoustic-mode SWs at high frequencies. The measured magnitude nonreciprocity or isolation of SH-SAWs exceeds 40 dB (or 80 dB/mm) at 2.333 GHz. In addition, maximum nonreciprocal phase accumulation reaches 188° (376°/mm). Our theoretical model and calculations provide an insight into the observed phenomena and demonstrate a pathway for further improving nonreciprocal acoustic devices toward highly compact microwave isolators and circulators.

## I. INTRODUCTION

Owing to the four orders of magnitude lower velocities, surface acoustic wave (SAW) devices can be greatly miniaturized compared to their electromagnetic wave counterparts. In addition, SAWs can be excited with a very high efficiency and travels in piezoelectric crystals with very low propagation losses.^{1} Therefore, they are very successful in RF filters and delay lines. Nonreciprocal microwave devices, such as isolators and circulators, have an important role in the front-end of most RF systems. These devices allow RF signal propagation in one direction and block propagation in the opposite direction. However, the typical frequency spectrum of SAW is reciprocal due to the fundamental time-reversal symmetry in the laws of mechanics.

Recent research has demonstrated that nonreciprocal transmission of SAWs can be realized by integrating magnetic layers on SAW delay lines and introducing strong magnetoelastic coupling (MEC).^{2} When both the resonance frequency and the wavevector of SAWs match those of spin waves (SWs), intense phonon–magnon coupling occurs with significant energy absorption due to magnetization precession; meanwhile, the propagation loss of SAWs remains low when SAWs and SWs are uncoupled. There are several ways to design nonreciprocal SAW isolators. For example, one can utilize the helicity mismatch between Rayleigh-type SAW and the chirality of the magnetization precession in ferromagnetic thin films,^{3–6} or the interfacial Dzyaloshinskii–Moriya interaction (DMI) between ferromagnetic thin films and heavy metals.^{7,8} However, the nonreciprocity caused by a helicity mismatch is relatively weak. DMI is generally accompanied by large magnetic damping due to a spin pumping effect, resulting in severe insertion loss.

Large nonreciprocity of SAW transmission has also been reported in magnetic bilayer structures. In these structures, the dispersion relation of SWs is nonreciprocal due to the break of the space-time inversion symmetry via interlayer dipolar coupling (IDC).^{9,10} Recently, researchers have studied nonreciprocal SAW transmission in NiFe/Au/CoFeB,^{11} FeGaB/Al_{2}O_{3}/FeGaB,^{12,13} and NiFeCu/FeCoSiB^{14} bilayers, and synthetic antiferromagnetic (SAFM) structures of Co/Ru/Co,^{15,16} CoFeB/Ru/CoFeB,^{17,18} and Ni/Ir/Ni.^{19} Of particular interest is that a giant nonreciprocal SAW transmission of >250 dB/mm and a very low magnetoacoustic insertion loss have been achieved in SAFM CoFeB/Ru/CoFeB with optimized structural parameters.^{17} However, SAFM requires precise control of the thickness $( < 1 nm)$ of the space layer over a large area. In addition, the requirements of IDC and SAFM on the thickness of magnetic layers are different.^{17,18} While high IDC prefers a large layer thickness or a wavenumber to boost SW nonreciprocity, the antiferromagnetic coupling becomes weaker with increasing layer thickness. Therefore, most SAW devices achieved large nonreciprocity at large wavenumbers or high frequencies at 5–8 GHz, much higher than commercial SAW devices (<3 GHz).

In this work, we investigated the SAW transmission in an anti-magnetostrictive bilayer structure of Ni/Ti/FeCoSiB with a negative–positive magnetostriction configuration. Distinct from SAFMs, SAWs can excite strong optical-mode SWs in Ni/Ti/FeCoSiB via MEC at frequencies lower than that of acoustic-mode SWs, although the static magnetization of two layers is parallel with each other. The negative–positive magnetostriction configuration can greatly enhance the nonreciprocal transmission of SAWs.

## II. THEORY

Figure 1(a) schematically illustrates a ferromagnetic bilayer in-between the interdigital transducers (IDTs) of a SAW delay line and the (x, y, z) coordinate system with the *x* axis parallel to the SAW propagation direction and the *z* axis perpendicular to the plane of the magnetic films. Here, $ \phi M$ and $ \phi H$ are the angles between the magnetic moment and the applied field with respect to the *x* axis, which is parallel to the SAW propagation direction.

_{A}(

*d*)/NM(

^{A}*s*)/FM

_{B}(

*d*), where the letters in the parentheses denote the layer thickness. The magnetization of the FM layers is represented as $ M X= M s X m X= M s X( m 0 X + \delta m X)$, where $X=A,B$. Here, $ M s X$, $ m 0 X$, and $\delta m X$ denote the saturation magnetization, the unit magnetization vector, and the small-amplitude magnetization precession of layer

^{B}*X*, respectively. When the frequency and wavenumber of SAWs match those of SWs, they can strongly couple with each other, giving rise to a magnetoacoustic wave. The magnetoacoustic wave vector is defined as $k= k \xb1x$, where $ k +$ and $ k \u2212$ represent the waves propagating in the positive and negative

*x*-directions, respectively. The magnetic energy density $ E X$ in layer X can be described by

^{20}

^{21}

*X*are described by the following spherical coordinate Landau–Lifshitz–Gilbert (LLG) equation:

*γ*is the gyromagnetic ratio and $\alpha $ is the damping factor. We assumed that the magnetization precession along the thickness direction is uniform. Subsequently, combining LLG equations of layers A and B, the magnetization response can be expressed as

Figure 2(a) plots the calculated SW dispersion curves for Ni, FeCoSiB, and Ni (16 nm)/Ti (8 nm)/FeCoSiB (16 nm). For a single-layered magnetic film, such as Ni or FeCoSiB, symmetrical dispersion curves are observed. The SWR frequency of Ni is much lower than that of FeCoSiB due to its lower saturation magnetization $ M s$. However, the SW dispersion curves of the Ni/Ti/FeCoSiB bilayer exhibit both in-phase (namely, acoustic mode, AM) and out-of-phase (optical mode, OM) branches. Additionally, the dispersion curves of both modes are nonreciprocal, which can be attributed to the interlayer dipolar interaction mentioned earlier. Of particular interest is that the OM frequency is lower than that of the AM, even lower than those of either Ni or FeCoSiB layers. This is due to the partial cancellation of the intralayer and interlayer dipole fields,^{11} and similar results have been reported on the Py (3 nm)/NM (5 nm)/Co (2 nm)/Pt multilayer.^{22} For $|k|\u22646\mu m \u2212 1$, the calculated OM frequencies are below 2.5 GHz, which benefits the coupling between SAWs and SWs. Figure 2(b) depicts the OM dispersion curves for different values of $ \phi M$. We define the frequency offset of OM at ±*k* as $\Delta f OM= f OM( + k)\u2212 f OM( \u2212 k)$, which gradually decreases when $ \phi M$ changes from 90° to 0° and vanishes at $ \phi M= 0 \xb0$. Therefore, the largest nonreciprocity of a spin wave appears at $ \phi M= 90 \xb0$.

*X*$( X = A or B)$. Clearly, the effective driving field of SH-SAW exhibits a $ cos( 2 \phi M)$ angle dependency. In other words, the strongest MEC is also observed at $ \phi M= 90 \xb0$. Therefore, SH-SAW is more suitable to generate large nonreciprocity compared to the well-studied Rayleigh wave since the effective driving field of the latter has a $ sin( 2 \phi M)$ angle dependency.

^{23–25}

*W*refers to the aperture of IDTs and

*L*denotes the length of the ferromagnetic layer. Using Eq. (12), we calculated the $ P abs$ of three bilayers, i.e., Ni (16 nm)/Ti (8 nm)/FeCoSiB (16 nm), Ni

_{81}Fe

_{19}(16 nm)/Ti (8 nm)/FeCoSiB (16 nm), and Ni

_{45}Fe

_{55}(16 nm)/Ti (8 nm)/FeCoSiB (16 nm) at the zero field. Notice that the $ B 2 X$ values in the upper layers change from +7 (Ref. 26) to 0 MPa and −5.7 MPa.

^{27}For all three bilayers, two power absorption peaks are seen in Fig. 3(a) with the low-frequency and high-frequency peaks corresponding to the OM and AM, respectively. In the negative–positive configuration of Ni/Ti/FeCoSiB, the OM resonance peak is much stronger than the AM resonance peak, while Ni

_{81}Fe

_{19}/Ti/FeCoSiB and Ni

_{45}Fe

_{55}/Ti/FeCoSiB exhibit an opposite behavior. As illustrated in Fig. 3(b), the enhanced OM in the negative–positive configuration can be attributed to the effective driving fields in the bottom and top layers excited by SH-SAWs. Although their magnetizations precess out-of-phase, $ h A \u2217\u22c5\delta m A$ always keep in-phase with $ h B \u2217\u22c5\delta m B$ due to the opposite effective driving fields.

## III. EXPERIMENTS

A 42°-rotated Y-cut X-propagation LiTaO_{3} substrate was selected as the piezoelectric substrate to excite shear-horizontal type SAWs (SH-SAWs). Interdigital transducers (IDTs) of Ti (5 nm)/Al (50 nm) were deposited on LiTaO_{3} via sputtering, and the spacing between two IDT pairs is 600 *μ*m. A split-finger design with a pitch width and an interval of 2.4 and 1.6 *μ*m was employed to suppress the reflection of SAWs and obtain high-order harmonics. Each IDT has five pairs of fingers. A 0.6 × 0.5 mm^{2} rectangular structure of Ni (16 nm)/Ti (8 nm)/FeCoSiB (16 nm)/Ti (10 nm) is then deposited and patterned between the two IDTs, as schematically illustrated in Fig. 4(a). The bottom Ti layer serves as a seed layer to facilitate the growth of subsequent layers. The central Ti layer separates the Ni and FeCoSiB layers, thereby inhibiting interlayer exchange coupling. During the sputtering process of both FeCoSiB and Ni layers, an *in situ* magnetic field about 150 Oe was applied along the *y* axis to induce uniaxial anisotropy; hence, $ \phi M$ is $ 90 \xb0 $under zero magnetic field. Figure 4(b) shows the photograph of our fabricated device.

Two-port transmission parameters of the delay lines were then measured by a vector network analyzer (VNA, Agilent N5230A). After performing time-domain gating, we can obtain the forward $( S 21)$ and backward $( S 12)$ transmission characteristics of the SAWs passing through the films. Although the fundamental resonant frequency of SH-SAWs is 257 MHz, high odd-order harmonic modes up to the ninth order (2333 MHz) are present due to the split-finger design of IDTs. The SAW wave vector $ k SAW$ can be obtained using the formula $| k SAW|=2\pi f SAW/ v SAW$, where $ v SAW$ is the SH-SAW velocity (∼4112 m/s). Figure 4(c) shows a wideband measurement of $ S 21$ and $ S 12$ from 500 to 2700 MHz under an applied field of 300 Oe. It can be seen that $ S 21$ almost overlaps with $ S 12$, indicating the absence of nonreciprocity under this field.

Since SAWs can only be excited at specific harmonic frequencies for a given wavelength, the resonance frequency of SWs needs to be tuned to match that of SAWs by varying the external magnetic fields. To verify the calculation above, we selected the SH_{9} mode $( k SAW = 3.56 \mu m \u2212 1)$ and measured the field-dependent transmission parameters. Although the acoustic insertion loss is large at this frequency, it is mainly caused by the low efficiency of higher-order harmonic modes and can be improved by reducing the pitch width of IDTs to excite the fundamental SH mode. The relative change of the background-corrected SAW transmission magnitude is defined as $\Delta S i j(H)= S i j(H)\u2212 S i j( 300 Oe)$ $( i j = 12 or 21)$. $ S i j(H)$ represents the transmission parameters at SH_{9} under different external magnetic fields, and $ S i j( 300 Oe)$ is the transmission parameters at a fixed field of 300 Oe, which is sufficient to saturate the bilayer.

Figures 5(a) and 5(b) show the polar plots of measured $\Delta S i j$ as a function of applied magnetic field for an SH_{9} mode. Strong power absorption is observed for $ \phi H$ around 0°, 90°, 180°, and 270°, confirming the expected $ cos( 2 \phi M)$ angle dependence of MEC. Additionally, comparing Figs. 5(a) and 5(b), one can find large variations in the corresponding SWR fields near 90° and 270°. Generally, the nonreciprocity of SAW transmission is defined as $ \Delta S 21\u2212 \Delta S 12$. As shown in Fig. 5(c), the strongest nonreciprocity (isolation) exceeds 40 dB near $ \phi H= 90 \xb0 and 270 \xb0$.

As shown in Fig. 5(d), the OM frequency is lower than that of the SH_{9} mode under zero field; therefore, there is no coupling between SWs and SAWs. Upon increasing $ H ref 1$ to 26 Oe, $ f OM( \u2212 k)$ can intersect with $ f SAW$ along the backward propagation direction, resulting in significant power absorption. Therefore, $ \Delta S 12$ reaches its minimum point of −33 dB in Fig. 5(e). Since there is a large SWR frequency offset $\Delta f OM$ according to Fig. 2(b), $ f OM( + k)$ is still lower than $ f SAW$ along the forward direction, thereby causing a large magnitude nonreciprocity $( \Delta S \xb1 = | \Delta S 21 \u2212 \Delta S 12 |)$ of 30 dB (or 60 dB/mm), as shown in the bottom panel of Fig. 5(e). A second SWR appears at $ H ref 2=57 Oe$ and is accompanied by an even more pronounced SAW power absorption. Compared to the previous resonance, a larger applied field causes $ f OM( + k)$ and $ f SAW$ to intersect with each other along the forward propagation direction $( \Delta S 21 = \u2212 43 dB)$, while $ f OM( \u2212 k)$ is now higher than $ f SAW$, thus exhibiting weak power absorption along the backward direction $( \Delta S 12 = \u2212 2 dB)$. When the direction of magnetic field changes, one can find that $ \Delta S i j( \u2212 H)= \Delta S j i( + H)$ since the reversal of the magnetic field is equivalent to the change of the SAW propagation direction. As shown in Fig. 5(e), the magnitude nonreciprocity of the second SWR reaches 41 dB (or 82 dB/mm). These values are comparable with those reported in synthetic antiferromagnetic systems^{15,16} at such a low frequency.

Our results open a way to develop efficient microwave SAW isolators. However, circulators and nonreciprocal phase shifters are also in great demand. A standard circulator scheme relies on the phase nonreciprocity; i.e., the phase accumulation for waves propagating in an opposite direction is not the same. Previously, Verba *et al.*^{28} proposed a SAW circulator, in which three nonreciprocal phase shifters based on SAFM heterostructures are combined in a SAW ring resonator. Next, we will show that the anti-magnetostrictive bilayer can also support nonreciprocal phase accumulation exceeding $ 180 \xb0$.

Clearly, $ \phi i j$ exhibits a sharp increase and decrease at the SWR fields corresponding to the peaks of $ \Delta S i j$ [Fig. 5(e)]. Strong phase nonreciprocity exceeding 180° is observed in Fig. 6(b) in the field range of ±(31–42) Oe. In particular, $| \phi 21 \u2212 \phi 12|$ reaches 188° (376°/mm), which is desired for SAW circulators. Moreover, the field range of the π-phase nonreciprocity does not overlap with those SWR fields but is lower than $ H ref 2$ and higher than $ H ref 1$. Therefore, a relatively low propagation loss $\Delta S i j=9.1 dB$ is obtained near 36 Oe.

However, a practical SAW circulator requires both π-phase nonreciprocity and low propagation loss below 3 dB. According to Verba *et al.*,^{12,28} the necessary conditions to fulfill these criteria are sufficiently large nonreciprocal splitting of the SW dispersion, $\Delta f OM\u226b\Delta f me\u226b \Gamma SAW \Gamma SW/2\pi $, and strong MEC larger than the damping rate of the SWs, $\Delta f me> \Gamma SW/2\pi $. Here, $\Delta f me$ represents the magnetoelastic coupling strength, and $ \Gamma SAW$ and $ \Gamma SW$ are the damping rate of SAWs and SWs, respectively. In our device, for an applied field of 57 Oe, we obtain $\Delta f OM=482 MHz$, $\Delta f me=52 MHz$. $ \Gamma SAW$ can be estimated using delay lines coated with different lengths of magnetic bilayers, and we obtained $ \Gamma SAW=560 kHz$ at 2.333 GHz. The damping rate of SWs was calculated using $ \Gamma SW=2\pi f\alpha \epsilon =2\pi \xd7182 MHz$ with an ellipticity-related factor $\epsilon =\delta \phi /\delta \theta \u22486.5$, following Ref. 28. Although the first condition of $\Delta f OM\u226b\Delta f me\u226b \Gamma SAW \Gamma SW/2\pi $ is satisfied, the second condition is not fulfilled due to the limitation of relatively large $ \Gamma SW$.

^{29,30}and using Eqs. (17) and (18), we can calculate the phase of $\Delta S i j$ as a function of

*H*. As shown in Fig. 6(a), the calculated results (dashed lines) are in good accordance with the measured ones (solid lines).

Figure 7(a) shows the calculated $| \Delta S i j|$ upon changing the thickness of the Ni and FeCoSiB layers from 16 to 20 nm. As can be seen, $\Delta H ref(= H ref 2\u2212 H ref 1)$ increases from 31 to 40 Oe, which helps that $\Delta S \xb1$ increases 8 dB and the insertion loss decreases 0.6 dB. Furthermore, reducing the Gilbert damping in the magnetic bilayer can also lower the insertion loss. When we change the damping factor of the Ni layer from 0.012 to 0.008, $\Delta S \xb1$ is further improved to 57 dB; meanwhile, the insertion loss is reduced to −1.3 dB, as shown in Fig. 7(b). Notice that FeCoSiB has a measured low damping factor of 0.008.^{31} Therefore, the key is to develop a negatively magnetostrictive film with a low Gilbert damping factor. For example, doping non-magnetic elements in nickel to form an amorphous phase may reduce the Gilbert damping, although there is currently limited research on this. Finally, utilizing TF-SAW technology, such as bonding LiTaO_{3} on a high-velocity substrate,^{32} is beneficial to better confine the acoustic wave to the substrate surface, thereby increasing $\Delta f me$.

## IV. CONCLUSION

In summary, we proposed a magneto-acoustic hybrid device integrated with an anti-magnetostrictive bilayer. Thanks to the opposite effective driving fields, SH-SAW excitation can greatly enhance optical-mode SWR but suppress the acoustic-mode one, although the directions of magnetization are parallel with each other in the neighboring layers. The lower frequencies of an optical mode migrate the requirement on nanolithography; meanwhile, the anti-magnetostrictive bilayer provides more freedom on structural design and a higher power handling capacity than synthetic antiferromagnets. A large magnitude and phase nonreciprocity of SAWs up to 82 dB/mm and 376°/mm have been demonstrated simultaneously. Our theoretical calculations based on the dynamic magnetoelastic coupling model are in good agreement with experimental results. Although the magnetoacoustic insertion loss is an obvious concern, it can be improved by optimizing the structure design of the anti-magnetostrictive bilayer to further increase the SWR frequency offset and employing a negative magnetostrictive film with a lower Gilbert damping factor.

## ACKNOWLEDGMENTS

This work was supported by the Natural Science Foundation of Sichuan Province under Grant No. 2022NSFSC0040 and the National Natural Science Foundation of China (Grant Nos. 61871081 and 61271031).

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**Zihan Zhou:** Investigation (equal); Methodology (equal); Validation (equal). **Wenbin Hu:** Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Validation (equal); Writing – original draft (equal); Writing – review & editing (equal). **Hao Wu:** Investigation (equal); Methodology (equal); Validation (equal). **Mingxian Huang:** Methodology (equal); Validation (equal). **Yutong Wu:** Formal analysis (supporting); Resources (supporting). **Yana Jia:** Formal analysis (supporting); Resources (supporting). **Wen Wang:** Formal analysis (supporting); Resources (supporting). **Feiming Bai:** Conceptualization (lead); Funding acquisition (lead); Investigation (equal); Methodology (equal); Project administration (lead); Resources (lead); Supervision (lead); Writing – review & editing (lead).

## DATA AVAILABILITY

The data that support the findings of this study are available within the article.