# Spectroscopy of the spin waves of a synthetic antiferromagnet grown on a piezoelectric substrate

G. Thiancourt, S. Ngom, N. Bardou, T. Devolder

► To cite this version:

G. Thiancourt, S. Ngom, N. Bardou, T. Devolder. Spectroscopy of the spin waves of a synthetic antiferromagnet grown on a piezoelectric substrate. AIP Advances, 2025, 15 (3), pp.035058. 10.1063/5.0250787 . hal-05128323

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# Spectroscopy of the spin waves of a synthetic antiferromagnet grown on a piezoelectric substrate

G. Y. Thiancourt (✉), a) S. M. Ngom, N. Bardou, and T. Devolder (✉, b)

Université Paris-Saclay, CNRS, Centre de Nanosciences et de Nanotechnologies, 91120, Palaiseau, France

(Dated: 24 June 2025)

Efficient coupling between phonons and magnons requires a precise knowledge of their dispersion relations on material platforms that contain magnetic multilayers with versatile high-frequency properties grown on piezoelectric substrates with large electromechanical coupling coefficients. One of these systems is the CoFeB/Ru/CoFeB Synthetic antiferromagnet grown on Lithium Niobate substrate. We investigate its microwave magnetic properties using a combination of ferromagnetic resonance and propagating spin wave spectroscopy, from which we extract the dispersion relation of the acoustic branch of spin waves. The frequency and the linewidth of this spin wave resonance, its field dependence and its dispersion relation indicate that the magnetic properties are as good as when grown on standard non-piezoelectric substrates, as well as being in line with theory. The knowledge of the exact dispersion relation of the spin waves in this new material platform opens opportunities to extend microwave acousto-magnonics beyond the use of single layer magnets.

## I. INTRODUCTION

Devices exploiting surface acoustic waves (SAWs) play a significant role in numerous modern technological applications, ranging from wireless communications¹ to medicine², sensors³, and RF signal processing in telecommunications. However, they have certain limitations, such as a reciprocal propagation and the near-absence of tunability. Circumventing these limitations may be possible by coupling SAWs with magnons⁴⁻¹² since the latter feature tunability and non-reciprocity. In particular, the spin waves (SWs) within magnetic bilayers have attracted interest¹³⁻¹⁶ because of their convenient high frequency SWs with long propagation lengths. Ensuring the high quality of both the magnetic thin film and the piezoelectric substrate is crucial to benefit from the SAW-SW coupling. In addition, getting a significant hybridization of SAWs and SWs requires a crossing of their dispersion relations which therefore needs to be characterized¹⁷,¹⁸.

In this study, we investigate SWs within a synthetic antiferromagnet (SAF) grown on a piezoelectric lithium niobate substrate LiNbO₃. The SAF consists of two magnetic layers of CoFeB, separated by a non-magnetic spacer layer mediating antiferromagnetic coupling. The SAF exhibits two eigenmodes: the SW optical mode and the SW acoustic mode, characterized by out-of-phase and in-phase precession of the magnetizations, respectively¹⁹⁻²⁴. It is known that the SW properties are very sensitive to the material structure, the roughness and the interdiffusion inside the stack²⁵. Here we measure the properties of SAF acoustic SW mode using inductive techniques. We detail how to rigorously measure the SW dispersion relation from propagating

spin wave spectroscopy data and demonstrate that it is possible to grow high-quality SAFs on high performing LiNbO₃ piezoelectric substrates.

## II. MATERIALS AND DEVICES

We use a SAF of composition LiNbO₃ (Y-cut substrate)/Ta /Co₄₀Fe₄₀B₂₀(tᵢₘₐg = 17 nm) /Ru(0.7nm) /Co₄₀Fe₄₀B₂₀(tᵢₘₐg) /Ru /Ta(cap) [Fig. 1.(a)]. The CoFeB has a magnetization μ₀Mₛ = 1.7 T and an interlayer exchange field μ₀Hⱼ ≈ 100 mT. We pattern²⁴ the SAF into stripes of width wᵢₘₐg = 20 µm and much longer length. We cover the sample with Si₃N₄ and fabricate two micrometer-sized single wire antennas of widths wᵢₐnt and variable center-to-center distances r from 3.4 to 6 µm [Fig. 1(c)]. This device will be used to perform propagating spin wave spectroscopy (PSWS) and compare the material quality to existing benchmarks. The analysis of PSWS data will require the knowledge of the resonant frequencies ω(k = 0) of uniform SWs, for which we fabricate an additional device to perform VNA-FMR²⁶. To ensure comparability, the VNA-FMR device is built on the same chip as the PSWS device. It consists of an array of circular 4 µm dots²⁷ inductively coupled to a much wider coplanar waveguide [Fig. 1(b)], which excites the uniform SW modes. The linewidth and the resonant frequencies of the uniform acoustic and optical spin wave modes shall provide the value of Hⱼ and of the damping parameter α using the standard formula for SAFs²⁵,²⁸.

## III. FERROMAGNETIC RESONANCE OF THE SAF ACOUSTIC SPIN WAVE MODE

Fig. 2 summarizes the results obtained by VNA-FMR. The resonant frequency of SAF acoustic SW mode (pink curve) of the dots evolves quasi-linearly with the applied

<footnote>a) Electronic mail: gael-yann.thiancourt@universite-paris-saclay.fr;</footnote>

<footnote>b) Electronic mail: thibaut.devolder@cnrs.fr</footnote>
<figure><img src="image_2.png" /><figcaption>FIG. 1. (a) Sketch of the setup of propagating spin wave spectroscopy (PSWS). (b) Scanning electron microscopy image of a PSWS device. Light gray: single-wire antennas. Medium gray: Synthetic antiferromagnet conduit. (c) of array of SAF dots covered by a coplanar waveguide meant to measure VNA-FMR.</figcaption></figure>

field. The resonance has the classical Lorentzian line shape (inset in Fig. 2), with a full width at half maximum that is almost independent of the field, being for instance 370 MHz for an applied field of $\mu_0H_x = 50$ mT leading to a resonant frequency of $f_{dots}(k=0) = 6.49$ GHz. The analysis of the linewidth using ref. 29 yields a Gilbert damping of $\alpha = \Delta\omega_0/(\gamma_0(M_s + H_j)) = 0.006 \pm 0.001$, where $\gamma_0$ is the gyromagnetic ratio.

Because of shape anisotropy, the uniform resonance of the dots differs from the uniform resonance of the stripe on which PSWS will be performed. Let us model this difference using the demagnetizing factors of our geometries $\bar{N} = \{N_x, N_y, N_z\}$. Using Eq. 11 of 30, the demagnetizing factor along z for a single magnetic dot is written as:

$$N_z = 1 + \frac{\tau_{a.r.}}{\pi} \left(1 + \ln\left(\frac{\tau_{a.r.}^2}{16}\right)\right) \quad (1)$$

where $\tau_{a.r.}$ = thickness/diameter. Therefore, we assume that the demagnetizing factor of each layer of the dot are $\bar{N}_{dot} = \{0.01, 0.01, 0.98\}$. Using 31, the demagnetizing tensor of each layer of the stripe is $\bar{N}_{stripe} = \{0.004, 0.001, 0.995\}$. Since $N_y \le N_x \ll N_z$, $\pm y$ is the easy axis and we can define a shape anisotropy field as $H_k = (N_x - N_y)M_s > 0$. We also define the saturation fields $H_{sat,x} = H_j + H_k$ and $H_{sat,y} = H_j - H_k$ for saturation by fields in either the x or the y direction.

For a field of magnitude $H_k < H_x < H_{sat}$ applied in the x direction, the SAF stripe is in a scissors state with the layers' magnetizations being:

$$\left\{ \begin{array}{l} m_{x1} = m_{x2} = \frac{H_x}{H_{sat,x}} \\ m_{y1} = -m_{y2} \\ m_{z1} = m_{z2} = 0 \end{array} \right. \quad (2)$$

<figure><img src="image_3.png" /><figcaption>FIG. 2. VNA-FMR measurements of the uniform acoustic spin wave mode resonance in the array of 4 µm dots. Left scale: Resonant frequencies as function of the applied field. The data correspond to the frequencies at which $||\tilde{S}_{21}||$ is maximal (inset). Right scale: Full widths at half maximum of these resonances.</figcaption></figure>

Linearization of the equations of motions can be used to derive the frequency of the SAF acoustic SW mode:

$$\omega_{acou} = \gamma_0 H_x \sqrt{H_j + (N_z - N_y)M_s} \times \sqrt{\frac{H_{sat,x}}{H_{sat,y}^2} - \frac{H_k}{H_x^2}} \quad (3)$$

We have deduced $f_{acou}^{stripe}(H_x)$ from $f_{acou}^{dots}(H_x)$, $\bar{N}_{stripe}$, $\bar{N}_{dot}$ and Eq. 3. The expected uniform resonance frequencies of the magnetic stripes for an applied field $\mu_0H_x = 50$ mT is for instance $f_{acou}^{stripe}(H_x) = 6.44$ GHz. The resonance frequencies and their linewidth are very similar to that of samples grown on non-piezoelectric substrates$^{32,33}$. This is a first indicating that high quality SAF can be grown also on piezoelectric substrates. Let us determine the spin wave dispersion relation and its field dependence to assess this statement in a more quantitative way.

## IV. SPIN WAVE SPECTROSCOPY OF THE ACOUSTIC SPIN WAVE MODE OF THE SYNTHETIC ANTIFERROMAGNETS

### A. Propagating spin wave measurements and methods

To get the dispersion relations, we performed propagating spin wave spectroscopy. We use the standard method and connect the two antennas A1 and A2 to the two ports of a calibrated VNA to measure the forward $\tilde{S}_{21}$ and backward $\tilde{S}_{12}$ transmission parameters for various applied field [Fig. 1(a)]. The SWs of SAF in this field configuration are known (and were checked) to be frequency-reciprocal$^{23}$. The data processing meant to maximize the signal-to-noise ratio is carried out following
<figure><img src="image_4.png" /><figcaption>FIG. 3. Propagating Spin Wave Spectroscopy of the acous- tic spin wave mode channeled in the SAF stripe. (a) Im- pulse response of the antenna-to-antenna forward transmis- sion coefficient. The black line is a guide to the eye that tracks the time at which the signal of the SAF acoustic SW wavepacket is maximum. (b) Field derivative of the forward transmission parameter ˜ S$_{21}$ ( f ) for an applied field µ$_{0}$H$_{x}$ = 50 mT after time-gating. (c) Comparison of the experimental phase at µ$_{0}$H$_{x}$ = 50 mT with Eq. 6. Starting from the experimental estimates of Eq. 3, the parameters were fitted to be L$_{att}$ = 3.6 µ m, r = 4.2 µ m, v$_{g}$ = 4.4 km.s − 1 and f ( k = 0) = 6.43 GHz.</figcaption></figure>

the time-gating and field-differentiation methods detailed in refs. 34 and 24.

Fig 3.(a) shows the impulse response s$_{12}$ ( t ) for a SW propagation distance r = 4.2 µ m. The wavepackets of the SAF acoustic SW reach the second antenna in a travel time of typically t$_{g}$ ≈ 1 ns. This group delay is essen- tially independent of the applied fields for fields inducing a scissor state 35 [see the dashed line in Fig 3(a)]. This group delay yields a first estimate of the group velocity of v$_{g}$ ≈ r/t$_{g}$ = 4.2 km.s − 1 and a first estimate of the atten- uation length of L$_{att}$ = 2 v$_{g}$ Δ ω$_{0}$ = 3.6 µ m of the propagating SAF acoustic SWs.

Fig 3(b) shows a representative processed transmission spectrum around the frequency of the SAF acoustic SW mode. The phase of the signal is reported in Fig 3(c). Transforming these phases into dispersion relations re- quires the modeling of the device’s response.

B. Model of ˜ S$_{21}$ for reciprocal spin waves

For reciprocal waves with a ∨ -shaped dispersion rela- tion, the contribution ˜ S yy 21 of the diagonal terms of the magnetic susceptibility to the forward transmission coef- ficient can be deduced from the sum of the experimental forward and reverse transmission coefficients (see the sec- tion IX.B of ref. 36):

$$
\tilde{S}_{21}^{yy}(\omega, r, \nabla) = \frac{1}{2} (S_{21}^{\exp}(\omega) + S_{12}^{\exp}(\omega)) \quad (4)
$$

We consider in addition the limit of an ultra-thin antenna placed close to the magnetic film. Using the formalism of ref. 36, we can reformulate an expression of ˜ S$_{21}$ ( ω ) for re- ciprocal SWs with a ∨ -shaped dispersion relation. Apart very near the uniform acoustic SW mode resonance, the scattering parameters can be expressed as 36:

$$
\tilde{S}_{21}^{yy}(k > \frac{\Delta\omega_0}{v_g}, r, \nabla) = \sqrt{\frac{\pi}{2}} \mathcal{L}(k) \left( P_2(r - w_{\text{ant}}, k) + P_2(r + w_{\text{ant}}, k) - 2P_2(r, k) \right) \quad (5)
$$

for k > Δ ω$_{0}$ v$_{g}$ , with L ( k ) ∝ v − 2 g ( k − i/L$_{att}$ ) − 2 a Lorentzian function centered at k = 0 and of width related to the inverse attenuation length $^{37}$. The propagation term is expressed using the function P$_{2}$ ( x, k ) = | L$_{att}$ | e $^{ikx}$e − | x | L$_{att}$ | that expresses the phase rotation and the decay of the wave upon its propagation.

Using Eq. 3 and 5 and taking into consideration that w$_{ant}$ < r ≪ L$_{att}$ , we can state that for k > 2 L$_{att}$ the experimental data can be linked to the SW wavevector of the SAF acoustic SW branch by:

$$
\arg \left( \frac{\partial \tilde{S}_{21}^{yy}}{\partial H_x}(\omega) \right) = kr - \tan^{-1} \left( \frac{kL_{att}}{4} \right) + 2n\pi, \quad (6)
$$

with n ∈ Z . The value of n can be found from the frequency of the uniform mode ( k = 0) determined in section II. The equation 6 expresses that the phase of the field derivative of ˜ S yy 21 ( ω ) parameter evolves linearly with the wavevector with a distance independent cor- rection term. This correction term can be rewritten as tan − 1 ( ω − ω$_{0}$ 2Δ ω$_{0}$ ) to evidence that it tends towards π 2 when working at frequencies ω > ω$_{0}$ + 2Δ ω$_{0}$ . Fig. 3(c) shows the comparison of the experimental phase of the field derivative of ˜ S$_{21}$ ( f ) with Eq. 6 for input values adjusted according to experimental measurements. The agree- ment is satisfactory and the dispersion relation can thus be deduced from a fit to Eq. 6.

C. Dispersion relation of the acoustic spin wave mode

Fig. 4(a) gathers the dispersion relations of the SAF acoustic SWs measured for applied field spanning from 30 mT up to 139 mT. This extends up to the maximum wavevector k$_{max}$ ≈ 3 rad/ µ m that can be investigated with our antenna width. The k$_{y}$ < 0 part of the disper- sion relations are constructed by symmetry.

For each applied field, the quasi-linear V-shape of the dispersion relation allows to define a characteristic group velocity. Fig. 4(b) reports the group velocity of SW de- duced from linear fits done in the 0.7 to 1.3 rad/ µ m interval of wavevector. For µ$_{0}$H$_{x}$ = 50 mT the group velocity is 4.422 ± 0.062 km.s − $^{1}$. It increases up to 5.220 ± 0.041 km.s − 1 for µ$_{0}$H$_{x}$ = 85 mT. The group ve- locities deduced from samples with different propagation
distances match together within a standard deviation of typically 200 m/s [inset of Fig. 4(b)]. This consistency among different devices is indicative of a good homogeneity of the magnetic properties despite their growth on non-standard substrates.

The group velocity is found to increase with the applied field until it saturates and decreases slightly for $\mu_0H_x > 100$ mT. This behavior is in semi-quantitative agreement with the predictions for a 2-macrospin SAF$^{23}$, for which when $\vec{H}_x \perp \vec{k}_y$, the group velocity is expected to be $\frac{1}{2}\gamma_0M_s t \frac{H_x}{H_j} \frac{M_s}{\sqrt{H_j(H_j+M_s)}} \operatorname{sgn}(k_y)$ for $H_x < H_j$ (corresponding to the scissors state), and then to be slightly decreasing following $\frac{1}{2}\gamma_0M_s t \frac{M_s}{\sqrt{H_x(H_x+M_s)}} \operatorname{sgn}(k_y)$ for larger fields (corresponding to the saturated parallel state). This is an additional indication of proper synthetic antiferromagnetic behavior.

## V. SUMMARY AND CONCLUDING REMARKS

In summary, we have measured several metrics representative of the dynamical properties of synthetic antiferromagnets grown on LiNbO$_3$ substrates and patterned into magnonic devices. The measurements include the uniform resonance of the SAF acoustic spin wave mode, the dispersion relation of these spin waves, and consequently their group velocities and attenuation lengths. These metrics are indicative that the SAFs grown on LiNbO$_3$ have very similar properties as the SAFs grown on standard non-piezoelectric substrates, for instance on silicon substrates$^{23,32}$. The knowledge of the dispersion relation in this new material platform shall enable to efficiently couple the magnetics degrees of freedom to the elastic ones, for instance to confer non-reciprocity to surface acoustic wave delay lines$^{38,39}$.

## ACKNOWLEDGMENTS

This work was supported by the French RENATECH network, by the French National Research Agency (ANR) as part of the "Investissements d'Avenir" and France 2030 programs. This includes the SPICY project of the Labex NanoSaclay: ANR-10-LABX-0035, the MAX-SAW project ANR-20-CE24-0025 and the PEPR SPIN projects ANR 22 EXSP 0008 and ANR 22 EXSP 0004. We acknowledge Aurélie Solignac for material growth and Loukas Kokkinos, Nessrine Benaziz, Joo-Von Kim and Maryam Massouras for helpful comments.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

<figure><img src="image_5.png" /><figcaption>FIG. 4. (a) Dispersion relation of the acoustic spin wave mode for $\vec{H}_x \perp \vec{k}$ obtained for a Propagating spin wave spectroscopy device with antenna width $w_{ant} = 2$ µm and antenna to antenna distance $r = 4.2$ µm. (b) Group velocities obtained for $k=1$ rad/µm and $r = 4.2$ µm (violet), $r = 6$ µm (black) and $r = 3.4$ µm (green). Inset: standard deviation among these three measurements.</figcaption></figure>

### Author Contributions

**G. Y. Thiancourt**: Data curation (lead); Formal analysis (lead); Writing—original draft (equal); Writing—review and editing (equal); Measurements (supporting); Investigation (equal). **S. M. Ngom**: Resources (equal); Investigation (equal); Measurements (equal). **N. Bar-dou**: Resources (equal). **T. Devolder**: Funding acquisition (lead); Investigation (equal); Measurements (equal); Writing—original draft (equal); Writing—review and editing (lead); Conceptualization (lead).
DATA AVAILABILITY

Data supporting the findings of this study are available
in the article or obtainable from the corresponding author
upon reasonable request.

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