# Reduction of magnetoelastic constant B2 in Fe thin films by nitrogen implantation, as determined by surface acoustic waves

A. Sharma, L. Christienne, F. Millo, M. Eddrief, E. Dandeu, J.-Y. Duquesne, C. Gourdon, L. Thevenard, P. Rovillain, Massimiliano Marangolo

<figure><img src="image_1.png" /></figure>

## To cite this version:

A. Sharma, L. Christienne, F. Millo, M. Eddrief, E. Dandeu, et al., Reduction of magnetoelastic constant B2 in Fe thin films by nitrogen implantation, as determined by surface acoustic waves. *Applied Physics Letters*, 2026, 128 (6), 10.1063/5.0289237. [hal-05507690](https://hal.science/hal-05507690v1)

HAL Id: hal-05507690

https://hal.science/hal-05507690v1

Submitted on 12 Feb 2026

HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

<figure><img src="image_2.png" /></figure>

Distributed under a Creative Commons CC BY 4.0 - Attribution - International License
# Reduction of Magnetoelastic Constant $B_2$ in Fe Thin Films by Nitrogen-Implantation, as determined by surface acoustic waves

A. Sharma,¹ L. Christienne,¹ F. Millo,¹ M. Eddrief,¹ E. Dandeu,¹ J.-Y. Duquesne,¹ C. Gourdon,¹ L. Thevenard,¹ P. Rovillain,¹, a) and M. Marangolo¹

Sorbonne Université, CNRS, Institut des Nanosciences de Paris, INSP, F-75005 Paris, France

(Dated: 12 February 2026)

This article presents an original approach to both modify and measure the magnetoelastic coupling constant $B_2$ in epitaxial nitrogen-implanted Fe thin films presenting the $\alpha' - Fe_8N_{1-x}$ phase. Once the magnetocrystalline anisotropy has been determined by standard techniques, we measure the field-dependence of Rayleigh surface acoustic wave (SAW) velocity variations. The magnetoelastic coupling constant is extracted by fitting the field-angle dependence of these variations. Using the magnetic and magneto-elastic parameters thus determined, the variations in SAW velocity as a function of the external field seen in surface acoustic wave induced ferromagnetic resonance (SAW-FMR) experiments can be accurately reproduced, both in pristine and nitrogen-implanted Fe thin films. This ensemble of results confirms a reduction by almost half of the magnetoelastic coupling under doping level $2 \times 10^{16} N_2/cm^2$.

In recent years, particular attention has been given to the magnetoelastic coupling between surface acoustic waves (SAWs) and magnetic media [1]. This renewed interest echoes early observations made nearly fifty years ago [2], which had already highlighted the potential of acoustic waves for magnetic sensing [3]. While initially underexplored, this line of research has experienced a revival, driven by new perspectives in spintronics and magnonics, as well as by promising applications in areas such as magnetic field sensing and acoustic-wave-based devices [4–6]. In this context, recent research has focused on the surface acoustic wave-assisted ferromagnetic resonance (SAW-FMR) effect, where SAWs efficiently couple to magnetization dynamics [7–13]. The SAW-FMR coupling becomes particularly significant when the external magnetic field $B_{ext}$ satisfies the resonance condition, i.e. $B_{ext} = B_{res}$, leading to an intersection between the dispersion relations of the SAWs and spin waves (SWs). This resonance is characterized by a sharp change in SAW velocity and increased acoustic absorption, resulting from enhanced SW excitation [14]. This well-understood phenomenon arises from the dynamic effective magnetic field induced by the Rayleigh wave via both magnetoelasticity (ME) and magnetorotation (MR) [12]. The magnetoelastic coupling is characterized by the magnetoelastic constants that reflect the coupling between magnetization and strain. Modifying these constants allows the ME coupling to be optimized for applications such as stress sensors, piezomagnetic actuators, or magnonics devices, for example [5, 6, 15, 16]. This coupling can be tuned during growth via alloys (NiFe, GaMnAs [17]), or atomic substitution (Fe-Ga) [18]. This ME coupling constant requires advanced approaches to be measured, as detecting the shift in the resonant field is difficult, particularly for thin films. Indeed, the most reliable and widely used technique remains the optical or the capacitive cantilever method, which detects the curvature of the substrate and the clamped thin film under an applied magnetic field [19–22].

However, this technique presents significant experimental constraints. In particular, for thin films, this requires the substrate to be sufficiently thin to allow bending elastically with a measurable curvature change. [19].

In this article, we present an alternative way to tune magnetoelasticity after material growth in a controlled manner i.e. by N-implantation. To obtain the $B_2$ magnetoelastic constant we develop a method relying on field-dependent SAW propagation in a magnetic layer. This approach allows us to accurately investigate the dependence of the SAW velocity on the magnetic field, especially in the SAW-FMR regime. We apply this method to a system that has been extensively studied in the literature: iron with interstitial nitrogen atoms (Fe:N). In particular, we investigate epitaxial layers grown on GaAs(001) substrates in the presence of the $\alpha' - Fe_8N_{1-x}$ ($x \sim 0$) phase. This system was chosen for two main reasons. Firstly, the insertion of nitrogen preserves the epitaxy of iron on the substrate [23, 24]. It thus offers an ideal playground, as no radical changes occur in the crystal structure apart from a tetragonal distortion. The subtle variations in magnetoelastic coupling can be systematically studied as a function of nitrogen insertion into the pristine bcc structure of iron. Secondly, FeN alloys are attracting considerable interest in the scientific community, due to the still debated increase in the magnetic moment of the ordered phase, i.e. the so-called $\alpha'' - Fe_8N$ — a property that could, one day, enable the replacement of rare-earth elements in specific applications [25]. In addition, this study allows us to track how nitrogen implantation subtly modifies the SAW-FMR condition, thereby opening possibilities for magnonic SAW-based devices in which resonances can be locally tuned via controlled nitrogen insertion as recently proposed in Ref. [26].

In Fig. 1, we show the sketch of the sample under study. A single-crystal Fe thin film of thickness $t = 54$ nm was deposited through Molecular Beam Epitaxy (MBE) on a ZnSe (001) layer of 10 nm, grown on a GaAs (001) substrate. The epitaxial conditions are then Fe(001)//ZnSe(001)//GaAs(001)

<footnote>a) Author to whom correspondence should be addressed: P. Rovillain pauline.rovillain@insp.upmc.fr</footnote>
<figure><img src="image_3.png" /><figcaption>FIG. 1. Sample under study. The SAWs are generated using Al Split-52 IDTs at $f_{SAW} = 1.8$ GHz by applying an input signal at IDT1. $\mathbf{B}_{ext}$ is the in-plane applied magnetic field, and $\mathbf{M}$ is the Fe magnetization. x // [100], y // [010], and z // [001] (red axes) correspond to the cubic frame, and x and y are the magnetic easy axes. X // [110], Y // [-110], and Z // [001] (black axes) correspond to the rotated frame with X and Y being the magnetic hard axes.</figcaption></figure>

and Fe[100]//ZnSe[100]//GaAs[100]. The ZnSe layer prevents the diffusion of Ga atoms into the Fe layer [27]. An Au capping layer of 12 nm was deposited to prevent oxidation of the Fe layer [28]. Two pristine Fe samples were implanted with different concentrations of $N_2^+$ at the IN2P3 JANNuS platform, using a 190 kV Nier-Bernas ion implanter. The acceleration energy was set to 26 keV, with selected doses of 1.0 and $2.0 \times 10^{16} N_2/cm^2$. Simulations using the Transport of Ions in Matter (TRIM) indicate that the nitrogen implantation is not uniform across the Fe film; N ions are predominantly concentrated in the upper region of the Fe layer, just beneath the Au/Fe interface [29, 30]. The pristine Fe sample and the implanted ones are subsequently etched into a mesa of a few mm$^2$ for interdigitated transducers (IDTs) lithography on the piezoelectric GaAs substrate [31]. More details on the sample can be found in [14] and [10]. Table I provides the magnetic parameters, i.e. the in-plane cubic anisotropy ($K_c$) and out-of-plane uniaxial anisotropy ($K_u$), of the three samples. These values are obtained by Vector Network Analyser-based Ferromagnetic Resonance (VNA-FMR) and out-of-plane Vibrating Sample Magnetometry (VSM) measurements, which are included in the supplementary. As discussed in previous articles [23, 24, 32], the increase in $K_u$ with nitrogen implantation is accompanied by a decrease in $K_c$. X-ray diffraction (XRD) measurements presented in the Supplementary Material reveal that the former is due to nitrogen insertion into the octahedral sites of Fe [33] leading to tetragonal distorted bcc phase ($\alpha^*-\text{Fe:N}$) and to the $\alpha' - \text{Fe}_8\text{N}_{1-x}\text{N}_x$ phase. In contrast, we attribute the reduction in $K_c$ to a slight increase in crystal mosaicity induced by nitrogen implantation. Notably, XRD measurements in the Supplementary Material indicate that the mosaicity remains low (of the order of $1^\circ$) and the film preserves its epitaxial conditions after implantation.

SAWs are generated and detected with split-52 IDTs [34] at an acoustic frequency of $f_{SAW} = 1.8$ GHz. Pulsed excitation of 200 ns duration every 200 µs is used, and the acoustic signal is acquired with a digital oscilloscope to make direct time domain measurements. In the following, we show the relative SAW velocity variation, $\frac{\Delta V}{V}$, obtained with SAWs

<table><caption>TABLE I. In-plane cubic ($K_c$) and out-of-plane uniaxial ($K_u$) anisotropy constants. $K_c$ is deduced from VNA-FMR experiments. $K_u$ is calculated by fitting the out-of-plane VSM measurements using the Stoner-Wolfarth model.</caption><thead><tr><th>N<sub>2</sub> dose level</th><th>K<sub>c</sub> (kJ/m<sup>3</sup>)</th><th>K<sub>u</sub> (kJ/m<sup>3</sup>)</th></tr></thead><tbody><tr><td>Pristine Fe</td><td>50 ± 2</td><td>50 ± 50</td></tr><tr><td>Fe:N (1.0 × 10<sup>16</sup> N<sub>2</sub>/cm<sup>2</sup>)</td><td>43 ± 3</td><td>200 ± 70</td></tr><tr><td>Fe:N (2.0 × 10<sup>16</sup> N<sub>2</sub>/cm<sup>2</sup>)</td><td>34 ± 7</td><td>355 ± 80</td></tr></tbody></table>

excited along the [110] direction ($\mathbf{k}_{SAW}$//[110]) as a function of an external magnetic field, $B_{ext}$, for various angles, $\psi$ between $\mathbf{B}_{ext}$ and $\mathbf{k}_{SAW}$. $\frac{\Delta V}{V}$ is defined as $\frac{V(B_{ext}) - V(B_{ref})}{V(B_{ref})} = \frac{\beta(B_{ext}) - \beta(B_{ref})}{2\pi f_{SAW} \tau_{FeN}}$ [35]. Here $\beta$ is the phase of the acoustic signal detected at IDT$_2$, and $\tau_{FeN}$ is the SAW travel time in the Fe:(N) layer. The reference field, $B_{ref}$, was set equal to 320 mT for all measurements, ensuring a complete magnetic saturation of magnetization along the applied magnetic field. Starting from $B_{ref}$, the magnetic field was ramped downward to prevent hysteretic SAW-FMR effects [36]. We study $\frac{\Delta V}{V}$ for different levels of N implantation doses and interpret the evolution of the SAW-SW interaction as a function of the N dose.

It has been established that SAW-SW interaction is governed by magnetoelasticity (ME) and magnetorotation (MR) [7, 12, 37]. Their energy density terms normalized to the saturation magnetization ($M_s$) can be written as follows,

$$ U_{\text{cub}}^{\text{ME}} = \frac{B_1}{M_s} (\varepsilon_{xx} m_x^2 + \varepsilon_{yy} m_y^2 + \varepsilon_{zz} m_z^2) + \frac{2B_2}{M_s} (\varepsilon_{xy} m_x m_y + \varepsilon_{yz} m_y m_z + \varepsilon_{xz} m_x m_z), \quad (1) $$

$$ U_{\text{Fe}}^{\text{MR}} = \zeta \mu_0 M_s m_z (\omega_{yz} m_y + \omega_{xz} m_x), \quad (2) $$

where $\varepsilon_{ij}$ refers to the strain in the i direction, $\varepsilon_{ij}$ is the shear strain associated with directions i and j, $m_i = M_i/M_s$ denotes the reduced magnetization components along the i direction where $i = x, y, z$ and $\omega_{ij}$ are the anti-symmetric elements of the strain tensor representing infinitesimal local rotations associated with directions i and j. Equation 2 should, in principle, include correction terms related to $K_c$ and $K_u$ [38]. However, since $\mu_0 M_s^2 \approx 3.6 \times 10^6$ J/m$^3$ in Fe is much larger than $K_u$ and $K_c$ terms, these corrections can be safely neglected in our case.

The effective magnetic field acting on the magnetization can be derived from the energy density $U$ from the following equation: $\mathbf{b} = -\nabla_m U$. In the rotated frame for SAWs propagating along the [110] direction, the effective magnetic field corresponding to the SAW-SW interaction ($b_{\text{ME-MR}}$) is given by,

$$ b_{\phi} = - \frac{B_2 \varepsilon_{XX}}{M_s} \sin(2\phi_0), \quad (3) $$

$$ b_{\theta} = - \left( \frac{2B_2 \varepsilon_{XZ}}{M_s} + \zeta \mu_0 M_s \omega_{XZ} \right) \cos(\phi_0), \quad (4) $$
where $b_\phi$ and $b_\theta$ are the in-plane and out-of-plane components of $b_{ME-MR}$ and X, Y, and Z axis are parallel to [110], [-110] and [001] directions, respectively. It is important to note that the magnetoelastic effective field is exclusively determined by the magnetoelastic coupling constant $B_2$, which quantifies the strength of the interaction under the deformation induced by SAWs propagating along the [110] direction. We discuss in the supplementary material how a similar experiment with the SAW propagating along [100] would allow for the determination of the longitudinal magnetoelastic coupling constant, $B_1$. The factor $\zeta$ in $b_\theta$ is equal to 0.45 [39] and has been taken from ref. [40]. Physically, $\zeta$ is a measure of the dipolar field generated over the two faces of the thin film, averaged over its thickness.

Since the relative SAW velocity variation is of the order of $10^{-5}$, the SAW dispersion is only slightly perturbed, and the coupling regime is weak. An important consequence of the weak coupling regime is that the acoustic power transmitted to the magnetic layer ($\Delta P$) and the relative SAW velocity variation are related directly as follows [14, 37],

$$ \frac{\Delta V}{V} = \frac{R_{\text{eff}}}{2k_{\text{SAW}} d_{\text{Fe}} P_{\text{ac}}^{\text{layer}}} \text{Re}(\Delta P) \quad (5) $$

where

$$ \Delta P = -\frac{\omega M_s}{2\mu_0} \int_{V_0} (b_{\theta}^*, b_{\phi}^*) \overleftrightarrow{\chi} \begin{pmatrix} b_{\theta} \\ b_{\phi} \end{pmatrix} dV. \quad (6) $$

In Eq. 5, $P_{\text{ac}}^{\text{layer}}$ is the total power carried by the SAW in the magnetic layer, and $d_{\text{Fe}}$ is the propagation distance of the SAW in the magnetic layer. In Eq. 6, $V_0$ and $\overleftrightarrow{\chi}$ denote the volume of the Fe layer and the Polder matrix, respectively. The dimensionless factor $R_{\text{eff}}$ in Eq. 5 is a scaling factor for the available magnetic power in the layer and is directly proportional to the SAW frequency. Its value is determined for pristine Fe [14] and then kept for the implanted samples. The complete phenomenological approach has been discussed in ref. [14].

Figures 2 (a) and (b) show the experimental relative variation in SAW velocities for the three samples at $f_{\text{SAW}} = 1.8$ GHz and their calculated SW frequencies for $k_{\text{SW}} = 3.9 \text{ rad}/\mu\text{m}$ as a function of the external field, $B_{\text{ext}}$, at $\psi = 0^\circ$. The data exhibits three distinct magnetic field regions:

(i) **High-field region:** The magnetization vector is aligned along $B_{\text{ext}} \parallel [110]$. The SW frequencies lie well above the SAW frequency, and a weak variation of SAW velocity is observed.

(ii) **SAW-FMR region:** The magnetization gradually rotates toward the in-plane easy axes (i.e., the [100] and/or [010] directions), leading to a softening of the SW frequency. This softening causes the SW frequency to intersect the SAW frequency at two distinct field values, resulting in a sharp $\Delta V/V$ drop characterized by two dips. As previously reported in ref. [14], although not shown in this article, the sharp transition and the appearance of the dips occur only when $B_{\text{ext}} \parallel k_{\text{SAW}} \parallel [110]$. A slight misalignment of just 0.2° leads to a broadening of the

<figure><img src="image_4.png" /><figcaption>FIG. 2. (a) Experimental (symbols) and calculated (curves) SAW relative velocity variation vs. in-plane external magnetic field, for pristine Fe (black square), intermediately (red circle) and highest (blue triangle) implanted doses, Fe:N, for $\psi = 0^\circ$ at $f_{\text{SAW}} = 1.8$ GHz. (b) Calculated spin wave frequency for $k_{\text{SW}} = 3.9 \text{ rad}/\mu\text{m}$ vs. in-plane external magnetic field. The green line indicates the SAW frequency at 1.8 GHz.</figcaption></figure>

transition and the disappearance of the two dips, as the SW frequencies increase and the resonance conditions are no longer fulfilled. In contrast to our previous work in ref. [14], these two dips are observed thanks to an increase in SAW frequency to 1.8 GHz.

(iii) **Low-field region:** The magnetization fully aligns along the easy axes, leading to an increase in SW frequencies and an overall increase of the SAW velocity as the resonance conditions are no longer satisfied.

The first key experimental result of this study is that nitrogen implantation reduces the in-plane magnetic anisotropy $K_c$ (see Table I) thereby shifting the SAW-SW resonance field toward lower values as reflected in Fig. 2(b). As a consequence, the SAW-SW resonance condition experiences a rigid shift: the sharp variation in relative velocity, marked by the two characteristic dips, moves toward lower magnetic field values with increasing doping, as shown in Fig. 2(a).

We then propose a method to evaluate the $B_2$ magnetoelastic constant by measuring the SAW velocity as a function of the angle $\psi$ between the SAW propagation direction and the applied magnetic field. Specifically, we record $\Delta V/V$ as a function of $B_{\text{ext}}$ for angles $\psi$ ranging from 0° to 360° in steps of 10°, and then plot the experimental $\Delta V/V$ as a function of $\psi$ at a fixed external field (see Fig. 3).
<figure><img src="image_5.png" /></figure>

FIG. 3. SAW relative velocity variation ($\frac{\Delta V}{V}$) vs. $\psi$ for pristine Fe (black, square), intermediately (red, circle) and highest (blue, triangle) doped Fe:N at $B_{ext} = 150$ mT. At this field, the magnetization is completely saturated along the direction of the field. The solid line curves denote the calculated $\frac{\Delta V}{V}$ vs. $\psi$ for different values of $B_2$ as function of the N dose and at a fixed $R_{eff} = 0.22$.

We select $B_{ext} = 150$ mT, corresponding to complete magnetic saturation, since a close examination of Eqs. 3, 4, 5, 6 shows that, under saturation, $\frac{\Delta V}{V}$ depends only on the magnetoelastic coefficient $B_2$. In Fig. 3, we simulate the experimental data of pristine Fe by fixing $B_2 = 7$ MJ/m$^3$ (value already known [41, 42]) and by adjusting the $R_{eff}$ parameter to get a good agreement between the experimental points and the theoretical curve [43]. It is worth noting that the term $\zeta$, introduced in the driving fields of Eq. 4 effectively captures the asymmetry observed in the $\frac{\Delta V}{V}$ vs. $\psi$ measurements as shown in the supplementary material. In particular, the asymmetry in velocity variation values on either side of the main peak at $\psi = 180^\circ$ is accurately reproduced.

For the two implanted samples, we fix $R_{eff} = 0.22$ determined by fitting the measurements on pristine Fe, and adjust $B_2$ to fit their $\frac{\Delta V}{V}$ vs. $\psi$ measurements. The experimental results and the corresponding calculations reveal that $B_2$ decreases with increasing doping concentration, as illustrated in Fig. 3. The $B_2$ values are 7, 6 and $4.2$ MJ/m$^3$ for implantation levels of 0 (pristine sample), $10^{16}$ and $2 \times 10^{16}$ N$_2$/cm$^2$ and represented in Fig. 4. The corresponding magnetostriction coefficient can easily be computed using $\lambda_{111} = -B_2/3C_{44}$, assuming the nitrogen implantation does not modify the elastic constant of Fe (C$_{44}$ = 115.9 GPa): $|\lambda_{111}| = 20.1, 17.2$ and 12 ppm for doping levels of 0, $10^{16}$ and $2 \times 10^{16}$ N$_2$/cm$^2$, respectively. Note that values of up to $\lambda_{100} \approx -120$ ppm have been recorded for the Fe$_4$N phase though [44, 45].

In the following, we corroborate the magnetic parameters extracted from experiments by comparing the measured curves with calculated ones (Fig. 2 (a)) using Eq. 5. The sim-

<figure><img src="image_6.png" /></figure>

FIG. 4. Evolution of the magnetoelastic constant $B_2$ with N-implantation dose in Fe. Inset: linear dependence of $\Delta V/V$ with $B_2^2$ at the saturation field of 150 mT, taken at $\psi = 200^\circ$ for the three samples. The main source of error in $B_2$ is due to the error in $M_s$, which stems from the uncertainty in the thickness of the magnetic material as explained in [29].

ulations are based on the magnetic anisotropy constants listed in Table I and the $B_2$ values reported in Fig. 4. These results demonstrate that the experimentally extracted parameters form a consistent and reliable set for reproducing the field dependence of $\frac{\Delta V}{V}$, as shown in Fig. 2 (a), in particular the position of the resonance. This supports the validity of our approach, which consists in measuring $B_2$ at high magnetic fields — i.e., outside the SAW-FMR resonance regime — in order to decouple the measurements from the influence of magnetic anisotropy constants.

However, in Fig. 2(a), a deviation is observed between the calculated and experimental $\frac{\Delta V}{V}$ vs. external field curves in the low field region, where the measured SAW velocity variation is significantly lower than predicted. This discrepancy stems from the presence of perpendicular magnetic anisotropy (PMA), which is not fully accounted for in our model. The effect is particularly pronounced in the highest dose sample, where PMA can lead to the formation of weak magnetic stripe domains [24]. Additionally, some discrepancies are observed near the resonance field, which may arise from factors not included in the model, such as the mosaicity of the epitaxied thin film (see supplementary material) and a possible slight misalignment of the applied magnetic field known to be critical for SAW-FMR on epitaxial Fe [10].

Moreover, the inset of Fig. 4 shows a linear dependence between $\frac{\Delta V}{V}$ and $B_2^2$ in agreement with the high field expression of $\frac{\Delta V}{V}$, the procedure of this plot is detailed in the supplementary material.

In the following we put forward some hypotheses to explain the observed reduction in the $B_2$ magnetoelastic coefficient with increasing nitrogen content. Let us recall that nitrogen implantation in epitaxial Fe stabilizes the so-called
$\alpha'$-Fe$_8$N$_{1-x}$ phase. Nitrogen atoms occupy octahedral interstitial sites, leading to an increase in Fe-Fe nearest-neighbor distances [46]. In our Fe:N thin films, epitaxy on GaAs(001) is preserved, with a slight increase in the in-plane lattice parameter (2.87 Å compared to 2.86 Å for pristine Fe) and a pronounced out-of-plane tetragonal distortion (c = 3.11 and 3.14 Å for intermediate and highest N-implantation doses, respectively), as previously reported [24, 29].

We then address the microscopic origin of the magnetoelastic coupling. To second order in the spin density field and first order in the displacement field associated with long-wavelength acoustic phonons, the magnetoelastic energy and coefficients $B_i$ can be written as the sum of two terms [47]: a first term that arises from relativistic effects, such as dipole-dipole interactions and spin-orbit coupling, and a second term that becomes nonzero only when the magnetization is non-uniform. This second term vanishes for a uniformly magnetized Fe lattice but can become non-zero when a nitrogen atom is introduced into the unit cell, perturbing the collinearity of the spins. In the case of the $\alpha'$-Fe$_8$N$_{1-x}$ phase, ab initio calculations have shown that the orbital magnetic moment of Fe is strongly influenced by the presence of N atoms, leading to the emergence of perpendicular magnetic anisotropy (PMA), as experimentally observed in our sample [33]. We thus hypothesize that the observed reduction in the $B_2$ coefficient originates from the same mechanism as PMA — namely, a variation in the orbital moment of Fe atoms and a modification of the crystal field and of relativistic terms caused by the presence of N atoms in the unit cell. This remains a tentative interpretation and should be considered with caution. Detailed calculations based on the tetragonally distorted $\alpha'$-Fe$_8$N$_{1-x}$ phase would be required to confirm this mechanism.

In conclusion, this work demonstrates the possibility of modifying the SAW-FMR conditions and the magnetoelastic coupling by implanting nitrogen after Fe growth. This was demonstrated by determining the magnetoelastic constant $B_2$ by an alternative method using SAW-FMR experiments. The magnetoelastic constant $B_2$ is extracted by investigating the angular dependency of SAW velocity variation $\Delta V/V$ at saturating magnetic field. Beyond this main result, we have also validated two theoretical predictions. First, the role of the dipolar fields generated on the two surfaces of the thin film in producing the asymmetry observed in SAW propagation; second, that $\Delta V/V$ varies linearly with $B_2^2$, the magnetoelastic constant. These insights contribute to a deeper understanding of the magnetoelastic coupling in nitrogen implanted Fe systems and provide a robust framework to study future SAW-based technological applications.

## SUPPLEMENTARY MATERIAL

See supplementary material for details on the characterization of samples and the extraction of anisotropy parameters, the $\Delta V/V$ dependence with $B_2^2$, the influence of the magneto-rotation on the SAW-SW driving fields, and the experimental

geometry used to determine $B_1$. Also included is a table of acoustic parameters.

## ACKNOWLEDGMENTS

P. R., F. M. and M. M. acknowledge the French National Research Agency, ANR-22-CE24-0015 SACOUMAD. The authors acknowledge the France2030 program PEPR SPIN, under Contract No. ANR-22-EXSP-0008 and ANR-22-EXSP-0004. The authors also acknowledge Paola Atkinson for preparing the GaAs substrate by MBE, Franck Fortuna for N-implantation at the IN2P3 JANNuS platform, Serge Vincent, Yunlin Zheng, Jean-Eudes Duvauchelle and the staff of the MPBT (physical properties – low temperature platform of Sorbonne Université) for their support, and Mathieu Desquirez for VNA-FMR data analysis program.

<footnote n="1">¹J. Puebla, Y. Hwang, S. Maekawa, and Y. Otani, "Perspectives on spintronics with surface acoustic waves," Applied Physics Letters **120** (2022).</footnote>

<footnote n="2">²A. Ganguly, K. Davis, D. Webb, C. Vittoria, and D. Forester, "Magnetically tuned surface-acoustic-wave phase shifter," Electronics Letters **11**, 610–611 (1975).</footnote>

<footnote n="3">³A. Ganguly, K. Davis, D. Webb, and C. Vittoria, "Magnetoelastic surface waves in a magnetic film–piezoelectric substrate configuration," Journal of Applied Physics **47**, 2696–2704 (1976).</footnote>

<footnote n="4">⁴P. Smole, W. Ruile, C. Korden, A. Ludwig, E. Quandt, S. Krassnitzer, and P. Pongratz, "Magnetically tunable saw-resonator," in *IEEE International Frequency Control Symposium and PDA Exhibition Jointly with the 17th European Frequency and Time Forum, 2003. Proceedings of the 2003 (IEEE, 2003) pp. 903–906.</footnote>

<footnote n="5">⁵A. Barman, G. Gubbiotti, S. Ladak, A. O. Adeyeye, M. Krawczyk, J. Gräfe, C. Adelmann, S. Cotofana, A. Naeemi, V. I. Vasyuchka, et al., "The 2021 magnonics roadmap," Journal of Physics: Condensed Matter **33**, 413001 (2021).</footnote>

<footnote n="6">⁶P. Delsing, A. N. Cleland, M. J. A. Schuetz, J. Knörzer, G. Giedke, J. I. Cirac, K. Srinivasan, M. Wu, K. C. Balram, C. Bäuerle, T. Meunier, C. J. B. Ford, P. V. Santos, E. Cerda-Méndez, H. Wang, H. J. Krenner, E. D. S. Nysten, M. Weiß, G. R. Nash, L. Thevenard, C. Gourdon, P. Rovil lain, M. Marangolo, J.-Y. Duquesne, G. Fischerauer, W. Ruile, A. Reiner, B. Paschke, D. Denysenko, D. Volkmer, A. Wixforth, H. Bruus, M. Wiklund, J. Reboud, J. M. Cooper, Y. Fu, M. S. Brugger, F. Rehfeldt, and C. Westerhausen, "The 2019 surface acoustic waves roadmap," Journal of Physics D: Applied Physics **52**, 353001 (2019).</footnote>

<footnote n="7">⁷M. Weiler, L. Dreher, C. Heeg, H. Huebl, R. Gross, M. S. Brandt, and S. T. B. Goennenwein, "Elastically driven ferromagnetic resonance in nickel thin films," Phys. Rev. Lett. **106**, 117601 (2011).</footnote>

<footnote n="8">⁸L. Thevenard, C. Gourdon, J.-Y. Prieur, H. J. Von Bardeleben, S. Vincent, L. Becerra, L. Largeau, and J.-Y. Duquesne, "Surface-acoustic-wave-driven ferromagnetic resonance in (ga, mn)(as, p) epilayers," Physical Review B **90**, 094401 (2014).</footnote>

<footnote n="9">⁹A. Hernández-Mínguez, F. Macià, J. Hernández, J. Herfort, and P. Santos, "Large nonreciprocal propagation of surface acoustic waves in epitaxial ferromagnetic/semiconductor hybrid structures," Physical Review Applied **13**, 044018 (2020).</footnote>

<footnote n="10">¹⁰J.-Y. Duquesne, P. Rovillain, C. Hepburn, M. Eddrief, P. Atkinson, A. Anane, R. Ranchal, and M. Marangolo, "Surface-acoustic-wave induced ferromagnetic resonance in Fe thin films and magnetic field sensing," Physical Review Applied **12**, 024042 (2019).</footnote>

<footnote n="11">¹¹P. G. Gowtham, T. Moriyama, D. C. Ralph, and R. A. Buhrman, "Traveling surface spin-wave resonance spectroscopy using surface acoustic waves," Journal of Applied Physics **118** (2015).</footnote>

<footnote n="12">¹²M. Xu, K. Yamamoto, J. Puebla, K. Baumgaertl, B. Rana, K. Miura, H. Takahashi, D. Grundler, S. Maekawa, and Y. Otani, "Nonreciprocal surface acoustic wave propagation via magneto-rotation coupling," Science advances **6**, eabb1724 (2020).</footnote>
¹³F. Millo, R. L. Seeger, C. Chappert, A. Solignac, and T. Devolder, "Symmetry of the dissipation of surface acoustic waves by ferromagnetic resonance," AIP Advances 15, 045230 (2025).

¹⁴P. Rovillain, J.-Y. Duquesne, L. Christienne, M. Eddrief, M. G. Pini, A. Rettori, S. Tacchi, and M. Marangolo, "Impact of spin-wave dispersion on surface-acoustic-wave velocity," Physical Review Applied 18, 064043 (2022).

¹⁵X. Liang, C. Dong, H. Chen, J. Wang, Y. Wei, M. Zaeimbashi, Y. He, A. Matyushov, C. Sun, and N. Sun, English "A review of thin-film magnetoelastic materials for magnetoelectric applications," Sensors 20, 1532 (2020).

¹⁶M. Matsuo, E. Saitoh, and S. Maekawa, "Spin-mechatronics," Journal of the Physical Society of Japan 86, 011011 (2017), https://doi.org/10.7566/JPSJ.86.011011.

¹⁷S. C. Masmanidis, H. X. Tang, E. B. Myers, M. Li, K. De Greve, G. Vermeulen, W. Van Roy, and M. L. Roukes, "Nanomechanical measurement of magnetostriction and magnetic anisotropy in (ga,mn)as," Phys. Rev. Lett. 95, 187206 (2005).

¹⁸L. T. . W.-F. M. Summers, E.M., "Magnetostriction of binary and ternary fe-ga alloys," J Mater Sci 42, 9582-9594 (2007).

¹⁹D. Sander, "The correlation between mechanical stress and magnetic anisotropy in ultrathin films," Reports on Progress in Physics 62, 809 (1999).

²⁰M. Coisson, W. Hüttenes, M. Cialone, G. Barrera, F. Celegato, P. Rizzi, Z. H. Barber, and P. Tiberto, "Measurement of thin film magnetostriction using field-dependent atomic force microscopy," Applied Surface Science 525, 146514 (2020).

²¹F. Zighem and D. Faurie, "A review on nanostructured thin films on flexible substrates: links between strains and magnetic properties," Journal of Physics: Condensed Matter 33, 233002 (2021).

²²J.-P. Jay, F. Le Berre, S. Pogossian, and M. Indenbom, "Direct and inverse measurement of thin films magnetostriction," Journal of magnetism and magnetic materials 322, 2203-2214 (2010).

²³T. Amarouche, L.-C. Garnier, M. Marangolo, M. Eddrief, V. H. Etgens, F. Fortuna, Y. Sadaoui, M. Tamine, J. Cantin, and H. von Bardeleben, "Influence of ion implantation parameters on the perpendicular magnetic anisotropy of fe-n thin films with stripe domains," Journal of Applied Physics 121 (2017).

²⁴L.-C. Garnier, M. Eddrief, S. Fin, D. Bisero, F. Fortuna, V. H. Etgens, and M. Marangolo, "Perpendicular magnetic anisotropy in fe-n thin films: Threshold field for irreversible magnetic stripe domain rotation," in Spin, Vol. 6 (World Scientific, 2016) p. 1640014.

²⁵J.-P. Wang, "Environment-friendly bulk fe16n2 permanent magnet: Review and prospective," Journal of Magnetism and Magnetic Materials 497, 165962 (2020).

²⁶I. Ngouagnia Yemeli, L. Christienne, P. Rovillain, J.-Y. Duquesne, A. Anane, M. Marangolo, and D. Stoeffler, "A micromagnetic study of surface-acoustic-wave-driven excitation of spin waves in an iron-based conductant," Journal of Applied Physics 137 (2025).

²⁷M. Marangolo, F. Gustavsson, M. Eddrief, P. Sainctavit, V. H. Etgens, V. Cros, F. Petroff, J. George, P. Bencok, and N. Brookes, "Magnetism of the fe/znse (001) interface," Physical review letters 88, 217202 (2002).

²⁸I. S. Camara, S. Tacchi, L. C. Garnier, M. Eddrief, F. Fortuna, G. Carlotti, and M. Marangolo, "Magnetization dynamics of weak stripe domains in fe-n thin films: a multi-technique complementary approach," Journal of Physics: Condensed Matter 29, 465803 (2017).

²⁹L. Christienne, J. Jiménez-Bustamante, P. Rovillain, M. Eddrief, Y. Zheng, F. Fortuna, M. Marangolo, M. Madami, R. Gallardo, P. Landeros, and S. Tacchi, "Nonreciprocal spin-wave propagation in anisotropy-graded iron films prepared by nitrogen implantation," Phys. Rev. Appl. 24, 064040 (2025).

³⁰L.-C. Garnier, Couches minces en Fe-N élaborées par implantation ionique: Propriétés structurales et magnétiques, Ph.D. thesis, Université Paris Saclay (COmUE) (2019).

³¹The size of the Fe mesa is 2 × 0.5 mm², and the delay line for this sample is 1000 µm. For the N-implanted samples, the mesa are 2 × 2 mm² each, with delay lines of 2500 µm. Split-52 IDTs were used in our experiment with following parameters: acoustic aperture (w) = 450 µm, number of electrode pairs (Np) = 25, fundamental frequency (fo) = 895 MHz, SAW velocity in GaAs (vSAW) = 2767 m/s.

³²P. Gupta, H. Fiedler, S. Rubanov, and J. Kennedy, "Magnetisation and magnetic anisotropy of ion beam synthesised iron nitride," Journal of Magnetism and Magnetic Materials 517, 167388 (2021).

³³H. Zhang, I. Dirba, T. Helbig, L. Alff, and O. Gutfleisch, "Engineering perpendicular magnetic anisotropy in Fe via interstitial nitrogenation: N choose k," APL Materials 4 (2016).

³⁴F. J. Schülein, E. Zallo, P. Atkinson, O. G. Schmidt, R. Trotta, A. Rastelli, A. Wixforth, and H. J. Krenner, "Fourier synthesis of radiofrequency nanomechanical pulses with different shapes," Nature nanotechnology 10, 512-516 (2015).

³⁵Since the phase β is a function of the external field, sweeping the field downward from Bref to 0 leads to a phase shift. At fixed frequency, this phase shift corresponds to a change in kSAW, and therefore, to a change in the acoustic velocity. By taking the relative velocity variation (ΔV/Δt), we subtract any contribution of the velocity variation arising from the substrate. Thus, the SAW travel time (τFeN) is taken only for the magnetic layer.

³⁶A. K. Vythelingum, T. Tremblais, D. B. Nguyen, D. Ourdani, E. Dandeu, V. Laude, C. Gourdon, and L. Thevenard, "Role of coercivity in surface acoustic wave driven ferromagnetic resonance," ACS Applied Electronic Materials 7, 4055-4064 (2025).

³⁷L. Dreher, M. Weiler, M. Pernpeintner, H. Huebl, R. Gross, M. S. Brandt, and S. T. Goennenwein, "Surface acoustic wave driven ferromagnetic resonance in nickel thin films: Theory and experiment," Physical Review B—Condensed Matter and Materials Physics 86, 134415 (2012).

³⁸E. d. Trémolet de Lacheisserie, "Magnetostriction: theory and applications of magnetoelasticity," (No Title) (1993).

³⁹ζ is a dimensionless factor equal to

$$ \frac{1 - e^{-|k_{SAW}|t}}{2|k_{SAW}|t}, $$

and it quantifies the dipolar field produced by the two film surfaces that survives after averaging through the thickness t, and this factor should be accounted for to obtain the correct weight of the asymmetry observed in ΔV/Δt of the SAW propagation.

⁴⁰K. Yamamoto and S. Maekawa, "Magnetostatic field induced by mechanical deformations," Annalen der Physik 536, 2300395 (2024).

⁴¹M. Barturen, D. Sander, J. Milano, J. Premper, C. Helman, M. Eddrief, J. Kirschner, and M. Marangolo, "Bulklike behavior of magnetoelasticity in epitaxial fe 1-x ga x thin films," Physical Review B 99, 134432 (2019).

⁴²G. Bayreuther, J. Premper, M. Sperl, and D. Sander, "Uniaxial magnetic anisotropy in fe/gaas (001): Role of magnetoelastic interactions," Physical Review B—Condensed Matter and Materials Physics 86, 054418 (2012).

⁴³We checked that $R_{eff}$ does not vary notably with $B_2$ by using the fully magnetoelastic approach described at length in [14]. We found that $R_{eff}$ varies relatively by less than $10^{-4}$ when $B_2$ varies from 7 to 1 MJ/m³.

⁴⁴K. Sin and S. Wang, "Stress and magnetic properties in high moment fen thin films," JOURNAL OF APPLIED PHYSICS 79, 5901-5903 (1996), 40th Annual Conference on Magnetism and Magnetic Materials, PHILADELPHIA, PA, NOV 06-09, 1995.

⁴⁵K. Ito, I. Kurniawan, Y. Shimada, Y. Miura, Y. Endo, and T. Seki, "Giant tunability of magnetoelasticity in fe4n system as a platform to unveil correlation between magnetostriction and magnetic damping," COMMUNICATIONS MATERIALS 6 (2025), 10.1038/s43246-025-00784-5.

⁴⁶J. Coey and P. Smith, "Magnetic nitrides," Journal of magnetism and magnetic materials 200, 405-424 (1999).

⁴⁷A. Rückriegel, P. Kopietz, D. A. Bozhko, A. A. Serga, and B. Hillebrands, "Magnetoelastic modes and lifetime of magnons in thin yttrium iron garnet films," Physical Review B 89, 184413 (2014).
# Supplemental material: Reduction of Magnetoelastic Constant B₂ in Fe Thin Films by Nitrogen-Implantation, as determined by surface acoustic waves

A. Sharma,¹ L. Christienne,¹ F. Millo,¹ M. Eddrief,¹ E. Dandeu,¹ J.-Y. Duquesne,¹ C. Gourdon,¹ L. Thevenard,¹ P. Rovillain,¹ and M. Marangolo¹

Sorbonne Université, CNRS, Institut des Nanosciences de Paris, INSP, F-75005 Paris, France

(*Electronic mail: pauline.rovillain@insp.upmc.fr)

(Dated: 12 February 2026)

## I. EXPERIMENTS TO DETERMINE THE MAGNETIC PARAMETERS OF Fe:N

This section presents the experimental methods for the determination of the magnetic parameters of Fe:N. The magnetic parameters determined are the in-plane cubic anisotropy ($K_c$) and out-of-plane uniaxial anisotropy ($K_u$) constants using broadband ferromagnetic resonance with a vector network analyzer (VNA-FMR) and vibrating sample magnetometry (VSM) experiments, respectively. In the rotated (X, Y, Z) reference frame of Fig.1 in the main article, the magnetic free-energy density normalized to the saturation magnetization $M_s$ is given by,

$$U = U_{\text{Zeeman}} + U_{\text{dip}} + U_{\text{cubic}} + U_{\text{uniaxial}} + U_{\text{exchange}}, \quad (S1)$$

$$U_{\text{Zeeman}} = -B_{\text{ext}} \left( \sin \theta \sin \theta_B \cos(\phi - \psi) + \cos \theta \cos \theta_B \right),$$

$$U_{\text{dip}} = -\frac{1}{2} \mathbf{B}_{\text{dip}} \cdot \mathbf{m},$$

$$U_{\text{cubic}} = \frac{1}{2} B_c \left( \sin^2 \theta \cos^2 \theta + \frac{1 + \cos(4\phi)}{8} \sin^4 \theta \right),$$

$$U_{\text{uniaxial}} = \frac{1}{2} B_u \sin^2 \theta,$$

$$U_{\text{exchange}} = B_{\text{ex}}(ka)^2,$$

where $B_{\text{ext}}$ is the external magnetic field, $\theta_B$ is the angle between $B_{\text{ext}}$ and the [001] direction (z-axis), $B_{\text{dip}}$ is the dipolar field, $B_c = 2K_c/M_s$ and $B_u = 2K_u/M_s$ are the in-plane cubic anisotropy field and out-of-plane uniaxial anisotropy field, respectively, and $B_{\text{ex}} = \frac{2A_{\text{ex}}}{M_s a^2}$ is the field corresponding to the exchange energy density, where $A_{\text{ex}}$ is the exchange coupling constant and a is the length of the unit cell of Fe.

The second derivatives of U with respect to $\theta$ and $\phi$ give the resonance frequency known otherwise as the Smit & Beljers resonance frequency,

$$\omega_0^2(k) = \frac{\gamma^2}{1 + \alpha^2} \left[ U_{\theta\theta}(k) U_{\phi\phi}(k) - U_{\theta\phi}(k) U_{\phi\theta}(k) \right] \Big|_{\theta=\theta_0, \phi=\phi_0}. \quad (S2)$$

## A. Broad-band ferromagnetic resonance detected with vector network analyzer (VNA-FMR)

VNA-FMR was performed to experimentally determine the cubic anisotropy $K_c$ of the Fe:N samples. In this work, a two-port VNA was used to generate an RF current that produced an RF field on top of the center conductor of a coplanar waveguide (CPW). The sample is placed in a flip-chip geometry immediately above the center conductor, minimizing the spacing that would otherwise attenuate the RF field [1]. The VNA swept the frequency to over a range of 8 GHz, at a constant in-plane external magnetic field along [110] Fe-direction, with the measurement then repeated for a range of magnetic fields and plotted after normalizing to a reference value, with $B_{\text{ref}} = 150$ mT. We record the transmission parameter $S_{12}$ as a function of the frequency and the external field, with the corresponding plots for the three samples shown in Fig. S1. $K_c$ can be obtained for each sample
from the VNA-FMR measurements by fitting $\omega_0^2(k)$ at $k=0$ as a function of $B_{ext}$ - using the perpendicular anisotropy constant $K_u$ (determined in section I B), to the experimental curves as shown in Fig. S1.

<figure><img src="image_7.png" /></figure>

FIG. S1. $S_{12}$ as a function of the frequency and external field for Fe: N ($1 \times 10^{16} \text{ N}_2/\text{cm}^2$) (in red) and Fe:N ($2 \times 10^{16} \text{ N}_2/\text{cm}^2$) (in blue). In each curve, the FMR frequency reaches its minimum when the external field equals the sample's in-plane cubic anisotropy field, $B_c = 2K_c/M_s$ with $M_s$ determined by VSM (see next section). Notably, $B_c$ shifts to lower fields as the N-content in Fe increases, demonstrating that N-implantation reduces the intrinsic cubic anisotropy of the Fe film.

## B. Magnetometry using vibrating sample magnetometry (VSM)

VSM measurements were conducted using an out-of-plane magnetic field to experimentally determine $K_u$ of the Fe:N samples (Fig. S2). A magnetic field of up to 3 Tesla was applied perpendicular to the film surface. $K_u$ was determined using the Stoner-Wolfarth model, with the energy density given by: $-B_{ext}M_s \cos\theta + (\frac{\mu_0 M_s^2}{2} - K_u) \cos^2\theta$, where $B_{ext}$ is the applied magnetic field along the [001] axis, and $\theta$ is the angle between the magnetization direction and the [001] axis.

<figure><img src="image_8.png" /></figure>

FIG. S2. Out-of-plane VSM measurements (red dashed curves) compared with the Stoner-Wolfarth model (green curves) for Fe:N ($1 \times 10^{16} \text{ N}_2/\text{cm}^2$) (a) and Fe:N ($2 \times 10^{16} \text{ N}_2/\text{cm}^2$) (b). Optimal agreement is reached with uniaxial anisotropy constants $K_u = 200$, and 355 kJ/m$^3$ for the two N-implantation levels. The in-plane VSM measurements with the field applied along [100] for the implanted samples are shown as magenta dashed lines for the first dose (a) and second dose (b).

From these measurements, we obtain the saturation magnetization $M_s = 1.7 \times 10^6$ A/m for the Fe:N samples within an error bar of ±5%, consistent with the literature [2, 3].

## C. X-ray diffraction measurements

Figure S3 shows the X-ray diffraction measurements conducted on our samples to determine the phases present and to extract the associated lattice parameters. Measurements were performed using a Rigaku SmartLab 5-axis diffractometer ($\lambda \approx 1.540593$ Å). The key features of these measurements are:
1. The sharp, intense peak at ~ 65.5° is the GaAs (004) substrate reflection.

2. The α'-Fe:N phase, corresponding to a phase trending towards Fe<sub>8</sub>N stoichiometry [4]. Its diffraction peak grows in width in the Δω direction (Fig. S3) as we go from the pristine Fe film (black) to the highest N-dosed film (blue) i.e. the α'-Fe<sub>8</sub>N<sub>1-x</sub> (x ≈ 0) phase, which corresponds to the tetragonally distorted phase [3]. Although there can be many possible reasons for this broadening, like non-uniform distribution of the lattice strain, interfacial roughness, etc., the most probable signature of this broadening can be related to the microscopic tilting of crystal planes induced by mosaic structures (~ 1°) in the α'-Fe:N-phase (see Fig. S3.) [5].

The lowering of the in-plane K<sub>c</sub> with N implantation in Fe has been attributed to increased mosaicity—evidenced by the XRD measurements—and to the out-of-plane tetragonal distortion of the bcc Fe lattice induced by N-implantation.

<figure><img src="image_9.png" /><figcaption>FIG. S3. θ-2θ scans of pristine Fe and Fe:N films on GaAs(001) for pristine Fe (in black), Fe:N (first dose, in red) and Fe:N (second dose, in blue), with pronounced peak broadening due to increased mosaicity for the α'-phase corresponding to the highest N-dosage, as seen in the Δω curves.</figcaption></figure>

## II. SAW-FMR: SUPPLEMENTARY MATERIAL

### A. Dependence of ΔV/V on B<sub>2</sub> at saturation

Eqs. [3, 4] of the main text give the SAW-SW coupling field for k<sub>SAW</sub> propagating along the Fe hard axis [110] [6]. Further simplification of Eq. 5 using Eqs. [3, 4] of the main text gives the following relation between ΔV/V and B<sub>2</sub> at magnetic saturation (φ<sub>0</sub> = ψ):

$$ \frac{\Delta V}{V} = a_0 + a_1 B_2 + a_2 B_2^2, \qquad (S3) $$
<figure><img src="image_10.png" /><figcaption>FIG. S4. Left: SAW relative velocity variation (ΔV/V) vs. ψ for pristine Fe (black, square), intermediately (red, circle) and highest (blue, triangle) doped Fe:N at B$_{ext}$ = 150 mT. Right: demonstration of the linear dependence of ΔV/V with B$_{2}$ at the saturation field of 150 mT, taken at ψ = 200° for the three samples. The field angle of 200° was chosen because the variation in ΔV/V is the strongest at this angle.</figcaption></figure>

where,

$$a_0 = - \frac{M_s^3 R_{\text{eff}} V_0 \zeta^2 \mu_0 \omega_{\text{SAW}}}{4 d_{\text{Fe:(N)}} k_{\text{SAW}} P_{\text{ac}}} \chi_{11} \tilde{\omega}_{\text{XZ}}^2 \cos^2 \psi,$$

$$a_1 = - \frac{M_s R_{\text{eff}} V_0 \omega_{\text{SAW}} \zeta}{4 d_{\text{Fe:(N)}} k_{\text{SAW}} P_{\text{ac}}} (4 \tilde{\epsilon}_{\text{XZ}} \chi_{11} \tilde{\omega}_{\text{XZ}} \cos^2 \psi - 2i \epsilon_{\text{XX}} \chi_{12} \tilde{\omega}_{\text{XZ}} \cos \psi \sin(2\psi)), \quad (S4)$$

$$a_2 = - \frac{R_{\text{eff}} V_0 \omega_{\text{SAW}}}{4 d_{\text{Fe:(N)}} k_{\text{SAW}} P_{\text{ac}} \mu_0 M_s} (4 \tilde{\epsilon}_{\text{XZ}}^2 \chi_{11} \cos^2 \psi - 4i \epsilon_{\text{XX}} \tilde{\epsilon}_{\text{XZ}} \chi_{12} \cos \psi \sin(2\psi) + \epsilon_{\text{XX}}^2 \chi_{22} \sin^2(2\psi)).$$

where V$_{0}$ is the volume of the magnetic film, χ$_{ij}$ are matrix elements of the Polder matrix, ω$_{SAW}$ is the angular frequency of the SAW, d$_{Fe:(N)}$ is the length of the Fe:(N) layer, ω̃$_{XZ}$ = -iω$_{XZ}$, and ε̃$_{XZ}$ = -iε$_{XZ}$. Fig.S4 (left) shows the experimental and calculated ΔV/V vs ψ at 150 mT (magnetic saturation) for the three samples. The dashed brown line at ψ = 200° intersects the curves at three distinct points. The corresponding ΔV/V values at the intersection points are then plotted against their B$_{2}$ values. In our case, since the weight of the third term (a$_{2}$B$_{2}$) of ΔV/V in Eq. S3 is maximum compared to the first two terms for all ψ ∈ [0°, 360°], we get an almost perfect linear dependence of ΔV/V with B$_{2}$ plotted at ψ = 200° for the three samples, as shown in Fig. S4 (right). Upon substitution of all the parameters in Eq. S4, the coefficient a$_{2}$ is found to be almost identical for the three samples. Thus, it is legitimate to fit linearly the three points corresponding to the three samples as shown in Fig. S4 (right). Another important point to note is that these coefficients, in general, depend on the equilibrium magnetization angle φ$_{0}$ . Under the specific case of complete magnetic saturation, φ$_{0}$ → ψ (the field orientation), the coefficients no longer inherit a dependence on the magnetic parameters through φ$_{0}$ , and ΔV/V depends on B$_{2}$ , thus isolating the magnetoelastic constant in our analysis.

## B. Influence of the magneto-rotation on the SAW-SW driving fields

We remind the reader that the SAW-SW driving field components are:

$$b_{\phi} = - \frac{B_2 \epsilon_{XX}}{M_s} \sin(2\phi_0), \quad (S5)$$

$$b_{\theta} = - \left( \frac{2 B_2 \epsilon_{XZ}}{M_s} + \zeta \mu_0 M_s \omega_{XZ} \right) \cos(\phi_0), \quad (S6)$$

In the absence of the shear strain ε$_{XZ}$ and lattice rotation ω$_{XZ}$ , the amplitude of the SAW-SW driving field should be identical on either side of ψ =180°. Instead, we see velocity variations to be slightly asymmetrical (Fig. S5). Because ω$_{XZ}$ , which determines the magnetorotation (MR) coupling is one order of magnitude larger than ε$_{XZ}$ (see Table. S1), getting its correct weight and sign is crucial to fit our measurements properly. We parameterize this weight by the factor ζ , adapted from [7]. Fig.
S5 shows that taking $\zeta=0$ (i.e ignoring magneto-rotation) poorly accounts and gives the incorrect sign for the asymmetry of the velocity variations either side of $\psi=180^{\circ}$ (e.g. $\psi=160$ and $200^{\circ}$). Instead, the value of $\zeta \sim 0.4 - 0.5$ reproduces correctly the data. We independently verified this range using the expression $\zeta = \frac{1-e^{-k_{SAW}t}}{2k_{SAW}t}$ from Ref. [7], which gives $\zeta = 0.45$. Using this same $\zeta$ for all three samples yields a good fit of the $\Delta V/V$ variations, which validates this value.

<figure><img src="image_11.png" /><figcaption>FIG. S5. Experimental and calculated $\Delta V/V$ vs $\psi$ for Fe:N (second dose) at $B_{ext} = 150$ mT. Both curves are plotted for $B_2 = 4.2$ MJ/m$^3$, but at different values of $\zeta$. The solid blue curve corresponds to $\zeta = 0.45$, while the solid dark yellow curve corresponds to $\zeta = 0$ (no magneto-rotation). We observe that $\zeta = 0.45$ provides the most accurate match to the data, in agreement with its mathematically derived form in the literature [7].</figcaption></figure>

### C. The acoustic parameters

Table. S1 gives the acoustic parameters used for the calculations in this paper. The components of the strain tensors are average values over the Fe film thickness.

<table><caption>TABLE S1. Acoustic parameters at $f_{SAW} = 1.8$ GHz</caption><thead><tr><td>Acoustic parameter</td><td>Value</td></tr></thead><tbody><tr><td>⟨ε<sub>XX</sub>⟩</td><td>3.39 × 10<sup>-3</sup></td></tr><tr><td>⟨ε<sub>XZ</sub>⟩</td><td>2.63 × 10<sup>-4</sup> i</td></tr><tr><td>⟨ω<sub>XZ</sub>⟩</td><td>-6.52 × 10<sup>-3</sup> i</td></tr><tr><td>P<sup>layer</sup><sub>ac</sub> (W)</td><td>0.112</td></tr></tbody></table>

### D. Determination of the $B_1$ constant

As detailed in the main text, the strength of our approach is that by saturating the magnetization in the plane (fixed $\theta = \pi/2$) and rotating the field (varying $\psi = \phi_0$), the magneto-acoustic fields are mainly determined by the $B_2$ constant (Eqs. 3, 4 of the main text). If instead we wish to evaluate $B_1$, minimizing the contribution of $B_2$, one option is to have the SAW propagate along a cubic [100] axis, while the magnetization is rotated from out of the plane $\theta = 0$ to in-plane $\theta = \pi/2$ at fixed in-plane projection $\phi_0=\psi=0$. In that case, the magneto-acoustic fields read:

$$b_{\phi} = 0, \qquad (S7)$$

$$b_{\theta} = - \left( \frac{2 B_2 \epsilon_{XZ}}{M_s} + \zeta \mu_0 M_s \omega_{XZ} \right) \cos(2\theta) + \frac{B_1 (\epsilon_{ZZ} - \epsilon_{XX})}{M_s} \sin(2\theta), \qquad (S8)$$
Because $\epsilon_{XX}$ and $\epsilon_{ZZ}$ exhibit opposite signs and are both at least an order of magnitude higher than $\epsilon_{XZ}$ [6], the variation with the field angle $\theta$ will this time be dominated by the $B_1$ constant.

<footnote n="1">¹C. Bilzer, *Microwave susceptibility of thin ferromagnetic films: metrology and insight into magnetization dynamics*, Ph.D. thesis, Université Paris Sud-Paris XI (2007).</footnote>

<footnote n="2">²L. Christienne, J. Jiménez-Bustamante, P. Rovillain, M. Eddrief, Y. Zheng, F. Fortuna, M. Marangolo, M. Madami, R. Gallardo, P. Landeros, and S. Tacchi, "Nonreciprocal spin-wave propagation in anisotropy-graded iron films prepared by nitrogen implantation," Phys. Rev. Appl. **24**, 064040 (2025).</footnote>

<footnote n="3">³L.-C. Garnier, M. Eddrief, S. Fin, D. Bisero, F. Fortuna, V. H. Etgens, and M. Marangolo, "Perpendicular magnetic anisotropy in fe–n thin films: Threshold field for irreversible magnetic stripe domain rotation," in *Spin*, Vol. 6 (World Scientific, 2016) p. 1640014.</footnote>

<footnote n="4">⁴L.-C. Garnier, *Couches minces en Fe-N élaborées par implantation ionique: Propriétés structurales et magnétiques*, Ph.D. thesis, Université Paris Saclay (COmUE) (2019).</footnote>

<footnote n="5">⁵K. Nakashima, "A new version of the method for analysing peak broadening caused by local tilt distribution in double-crystal x-ray diffraction measurements," *Applied Crystallography* **33**, 1376–1385 (2000).</footnote>

<footnote n="6">⁶P. Rovillain, J.-Y. Duquesne, L. Christienne, M. Eddrief, M. G. Pini, A. Rettori, S. Tacchi, and M. Marangolo, "Impact of spin-wave dispersion on surface-acoustic-wave velocity," *Physical Review Applied* **18**, 064043 (2022).</footnote>

<footnote n="7">⁷K. Yamamoto and S. Maekawa, "Magnetostatic field induced by mechanical deformations," *Annalen der Physik* **536**, 2300395 (2024).</footnote>