# Nonreciprocal spin-wave propagation in anisotropy-graded iron films prepared by nitrogen implantation

L. Christienne, J. Jiménez-Bustamante, P. Rovillain, M. Eddrief, Y. Zheng, F. Fortuna, Massimiliano Marangolo, M. Madami, R.A. Gallardo, P. Landeros, et al.

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## To cite this version:

L. Christienne, J. Jiménez-Bustamante, P. Rovillain, M. Eddrief, Y. Zheng, et al., Nonreciprocal spin-wave propagation in anisotropy-graded iron films prepared by nitrogen implantation. Physical Review Applied, 2025, 24 (6), pp.064040. 10.1103/1qrj-gtjb. hal-05419676

HAL Id: hal-05419676

https://cnrs.hal.science/hal-05419676v1

Submitted on 16 Dec 2025

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Nonreciprocal Spin-Wave Propagation in Anisotropy-Graded Iron Films Prepared by
Nitrogen Implantation

L. Christienne,¹ J. Jiménez-Bustamante,² P. Rovillain,¹,* M. Eddrief,¹ Y. Zheng,¹ F.

Fortuna,³ M. Marangolo,¹ M. Madami,⁴ R. A. Gallardo,² P. Landeros,²,† and S. Tacchi⁵,‡

¹Sorbonne Université, CNRS, Institut des Nanosciences de Paris, INSP, F-75005 Paris, France

²Departamento de Física, Universidad Técnica Federico Santa María, Avenida España 1680, Valparaíso, Chile

³Université Paris-Saclay, CNRS, Institut des Sciences Moléculaires d'Orsay, 91405 Orsay, France

⁴Dipartimento di Fisica e Geologia, Università di Perugia, I-06123 Perugia, Italy

⁵Istituto Officina dei Materiali del CNR (CNR-IOM), Sede Secondaria di Perugia,
c/o Dipartimento di Fisica e Geologia, Università di Perugia, I-06123 Perugia, Italy

(Dated: December 16, 2025)

Gradual modification of the magnetic properties in ferromagnetic films has recently been proposed as an effective method to channel and control spin waves for the development of new functionalities in magnonic devices. Here, we investigate graded Fe:N films prepared by low-dose nitrogen implantation of Fe epitaxial thin films. Combining Brillouin light scattering measurements and a spin-wave theoretical approach, we show that nitrogen implantation induces a graded profile of both the in-plane and the perpendicular anisotropies along the film thickness. This graduation leads to a significant modification of the spin-wave spatial localization across the thickness and generates a marked frequency asymmetry in the spin-wave dispersion. Moreover, we find that the anisotropy profile, and as a consequence the dispersion relation, can be tuned on changing the implantation dose, opening a way for the potential use of the graded Fe:N films in magnonic applications.

# I. INTRODUCTION

A significant portion of contemporary magnetism research is focused on manipulating the magnetic anisotropy of thin films and nanostructures. This meticulous control holds significant promise for advances in spintronics and magnonics applications, where tailoring both in-plane and out-of-plane magnetic anisotropy is crucial. One notable application is the enhancement of perpendicular magnetic anisotropy (PMA), essential for downsizing devices for higher-capacity storage [1]. Moreover, PMA has recently emerged as pivotal in stabilizing chiral textures for magnetic recording and data processing [2]. In the field of magnonics, the fine-tuning of magnetic anisotropy can enable the manipulation of magnetization dynamics and facilitate controlling spin-wave propagation [3, 4]. Several techniques have been proposed to control magnetic anisotropies, such as substrate-induced stress [5], ion doping during growth [6], voltage-controlled piezoelectric effect [7], interface effects [8], and ion implantation [9]. The modification of magnetic properties in low-damping materials can manifest in various ways, including uniform, graded, and lateral adjustments. A prime illustration is the uniform Bismuth doping of Y₃Fe₅O₁₂ (BiYIG) films—a material renowned for its low losses and high PMA [6, 10]. In this context, graded magnetic materials, where physical properties such as saturation magnetization [11–20], exchange constant [21–25], Dzyaloshinskii-Moriya strength [26–32],

or magnetic anisotropy [33–40] gradually change along a specific direction, also represent an exciting class of systems. Graded materials have indeed been proposed to control magnetism in 3D [41], offering a promising approach for guiding and manipulating spin-wave propagation in magnonic systems.

Graduating a magnetic property through a given direction, indeed, breaks the spatial symmetry and, as a consequence, affects both the allowed spin-wave frequencies and the spatial mode profiles. In particular, it has been theoretically predicted that a graded variation of the magnetic properties along the film thickness causes heterosymmetric mode profiles and a modification of the conventional quantization condition associated with the perpendicular standing spin waves, inducing a significant frequency nonreciprocity of counterpropagating waves [17, 42]. This latter behavior is currently of great interest in the research area of magnonics since nonreciprocal spin-wave propagation is crucial for the implementation of unidirectional devices, such as spin-wave circulators and diodes, which serve as basic building blocks for magnonic circuits [43, 44]. Nonreciprocal spin-wave propagation has been experimentally demonstrated for ferromagnetic bilayers consisting of different magnetic materials in direct contact or spaced by a thin nonmagnetic layer, which can be regarded as a class of graded films with a sharp interfacial profile in saturation magnetization [45–53]. Beyond bilayers, tailoring spin-wave transport in 3D magnonic architectures has also been explored through periodic vertical structuring and meander-type conduits [54, 55], as well as in multilayer synthetic antiferromagnets [56]. In these works, the spin-wave dispersion is engineered via geometric periodicity rather than magnetic graduation. For thick enough films, different spin waves hybridize and lead to inter-

<footnote>* pauline.rovillain@insp.upmc.fr</footnote>

<footnote>† pedro.landeros@usm.cl</footnote>

<footnote>‡ tacchi@iom.cnr.it</footnote>
esting phenomena such as mode anti-crossings and anti-Larmor precession of the magnetization in some regions [52, 57].

Recently, ion implantation has emerged as a powerful tool for realizing graded systems. This technique allows the introduction of different kinds of atoms with high precision in dosing, thereby enabling a controlled adjustment of the magnetic parameters such as the saturation magnetization or the magnetic anisotropy [58–60]. In addition, depth control can be finely regulated by modifying the energy of impinging ions. This precise tuning can be accomplished uniformly across large samples or at the nanoscale level through focused ion beam (FIB) implantation. Although graded materials have been reported by other techniques, such as oxidation implantation [27], laser writing [20], alloying method [61], and co-deposition during molecular beam epitaxy (MBE) growth [62], ion implantation offers the unique advantage of combining high spatial and depth resolution, allowing for localized and flexible grading.

In this work, we present a detailed study of the spin wave properties in nitrogen-implanted Fe:N thin films. In previous studies, it was demonstrated that N-implantation induces an increase of the out-of-plane anisotropy in Fe films while preserving the film epitaxy, local magnetization, and low damping, showing the potential of the methodology for magnonic applications [63–65]. Here, we focus on the influence of graded anisotropy on the spin-wave dispersion of Fe:N thin films by means of Brillouin light scattering measurements and theoretical calculations. We show that low-dose nitrogen implantation induces a gradual increase of the perpendicular anisotropy and a decrease of the in-plane cubic one in the surface region of a Fe film. These anisotropy variations affect the spatial localization of the spin-wave modes along the film thickness and, as a consequence, significantly modify their dispersion relation. In particular, we observed a marked frequency asymmetry between spin-wave modes propagating in opposite directions, which can be tuned by changing the irradiation dose.

## II. EXPERIMENTS

### A. Sample Preparation and Structural Characterization

Iron films having a thickness of 54 nm were grown using MBE on non-doped GaAs (001) wafers (see Fig. 1). Prior to Fe deposition, a thin layer (10 nm) of ZnSe was applied to prevent migration [66], which could create a magnetic “dead layer”. The growth of Fe layers occurred with the in-plane [100]-Fe axis perfectly aligned azimuthally with that of the underlying GaAs substrate, while the [001] axis was out-of-plane. A gold capping layer (8 nm) was deposited in a separate chamber in order to prevent oxidation of the iron layer. Samples, measuring about 10 × 5 mm, were cleaved from the wafers and resized

for various measurements. Two of these samples underwent two different implantation doses at the IN2P3 JAN-NuS platform using a Nier-Bernas 190 kV ion implanter. The acceleration was set at 26 keV, and the chosen doses were 1.0 and $1.5 \times 10^{16}$ N$_2$/cm$^2$. In this study, nitrogen implantation parameters were carefully chosen to avoid the formation of weak magnetic stripe domains, which are known to emerge at doses above $2.0 \times 10^{16}$ N$_2$/cm$^2$ under normal incidence and significantly alter the magnetic ground state [63, 64]. Transport of Ions in Matter (TRIM) simulations provide rough estimates of the implantation depth, as shown in Fig. 2. It is important to note that the N implantation within the Fe film is not homogeneous. N ions are mainly concentrated in the upper part of the Fe overlayer underlying the Au/Fe interface. To ensure this, we limited the dose and fixed the implantation temperature at room temperature to suppress nitrogen diffusion. This choice guarantees an asymmetric nitrogen depth profile, which is essential for the magnetic effects under investigation, while a deeper implantation energy that would reduce this asymmetry was avoided.

During the nitrogen implantation, the emergence of Fe:N phases with crystalline parameters differing from bcc iron is expected [64]. X-ray diffraction measurements are conducted on our samples to determine the present phases and to extract the associated lattice parameters. Measurements are performed using a Rigaku SmartLab 5-axis diffractometer. From a fit of the X-ray reflectivity oscillations, the mean thickness of the iron layer, *d*, of around 54 nm was obtained. Figure 3 shows the $\theta-2\theta$ measurements of the (001) plane of the three different samples as pristine Fe, first dose ($1.0 \times 10^{16}$ N$_2$/cm$^2$) and second dose ($1.5 \times 10^{16}$ N$_2$/cm$^2$) irradiated samples. Another sample with a higher dose ($2 \times 10^{16}$ N$_2$/cm$^2$) was also studied to put into evidence the tetragonal distortion by a reciprocal space mapping (see Appendix A). These measurements allow access to the out-of-plane lattice swelling, notably by tracking the peaks identified as $\alpha'$-Fe:N, a bcc-phase with lattice parameters very close to pristine Fe ones. An $\alpha'$-Fe:N phase is also observable, corresponding to a phase trending towards $Fe_8N_1$ stoichiometry [67–69]. We notice that for $\alpha'$-Fe$N_x$ and

<figure><img src="image_3.png" /><figcaption>FIG. 1. Sketch of the sample: Fe or Fe:N (54 nm thick) epitaxied on ZnSe(001)/GaAs(001). **B** is the in-plane applied magnetic field, and **M** is the magnetization.</figcaption></figure>
$\alpha'$-Fe$_8$N$_{1-x}$, only diffraction peaks corresponding to crystal planes parallel to the (001) surface of the GaAs substrate are observed. This indicates that the [001] directions of these two phases are parallel to the [001] direction of GaAs. The corresponding in-plane and out-of-plane lattice parameters are reported in Table I. Both N-rich phases exhibit an increase in the lattice parameter c, with a significantly greater expansion observed in the $\alpha'$-Fe$_8$N$_{1-x}$ phase, as shown in Table I. Regarding the in-plane parameter, the measured values for both phases are slightly higher than that of bulk Fe. Additionally, we observe that the lattice parameters of the $\alpha'$-FeN$_x$ phase are very close to those of Iron, making it effectively Fe-like.

<table><caption>TABLE I. Lattice parameters.</caption><thead><tr><th rowspan="2"></th><th colspan="2">α'-Fe:N (Fe-like)</th><th colspan="2">α'-Fe:N</th></tr><tr><th>c (Å)</th><th>a (Å)</th><th>c (Å)</th><th>a (Å)</th></tr></thead><tbody><tr><td>Pristine Fe</td><td>2.86</td><td>2.870</td><td>-</td><td>-</td></tr><tr><td>First dose</td><td>2.87</td><td>2.877</td><td>2.96</td><td>-</td></tr><tr><td>Second dose</td><td>2.88</td><td>2.872</td><td>3.07</td><td>2.862</td></tr></tbody></table>

To illustrate the tetragonal distortion and demonstrate the preservation of pristine Fe epitaxy conditions, the (112) diffraction spot of a sample at higher dose (2.0 × 10$^{16}$ N$_2$/cm$^2$) along the $Q_x$ (||[110]) and $Q_z$ (||[001]) directions is mapped. A reciprocal lattice map is presented and discussed in Appendix A. The results indicate that the epitaxial conditions of the pristine Fe/GaAs(001) remain intact after N$_2$ implantation. These findings are consistent with those previously reported in Ref. 64 for higher implantation dose.

<figure><img src="image_4.png" /><figcaption>FIG. 2. TRIM simulation of the implantation with N$_2$ ions accelerated at 26 keV and implanted in a Fe film epitaxied on ZnSe/GaAs(001) substrate. The dose is 1×10$^{16}$ N$_2$/cm$^2$.</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>FIG. 3. $\theta-2\theta$ measurements. Colors correspond to different N$_2$ doses implanted into Fe films epitaxially grown on ZnSe/GaAs(001); black for pristine Fe, orange for the first dose (1.0 × 10$^{16}$ N$_2$/cm$^2$) and green for the second dose (1.5 × 10$^{16}$ N$_2$/cm$^2$). Two distinct Fe:N phases are clearly observed, both exhibiting tetragonal distortion.</figcaption></figure>

## B. Magnetic Characterization

### 1. Vibrating Sample Magnetometry

Vibrating-sample magnetometry (VSM) was employed to measure the magnetization loops of the pristine Fe and implanted samples (see Fig. 4). A magnetic field up to 3 T was applied perpendicular to the film surface. For each sample, the saturation magnetization $M_s$ was first extracted from the measured saturation value. Then the out-of-plane anisotropy constant was determined using the Stoner-Wohlfarth method, where the energy density can be written $\epsilon = -\mu_0\mathbf{H} \cdot \mathbf{M} + (\mu_0 M_s^2/2 - K_u) \cos^2 \theta$, with $\mathbf{H}$ the applied magnetic field, $\mathbf{M}$ the sample magnetization, $\theta$ the angle between the magnetization direction and the (001) plane, and $K_u$ the out-of-plane uniaxial anisotropy constant (see Fig. 1).

### 2. Brillouin Light Scattering Experiments

Brillouin light scattering (BLS) measurements (Fig. 5 and Fig. 6) were performed focusing about 200 mW of monochromatic light with a wavelength $\lambda = 532$ nm onto the sample surface, using a camera objective of numerical aperture 0.24 and focal length 50 mm. The backscattered light was analyzed using a Sandercock type (3 + 3)-pass tandem Fabry-Pérot interferometer. First, spin wave frequency was measured as a function of the in-plane angle $\phi$ between the applied magnetic field and the [110] Fe direction. The intensity of the magnetic field was set to $\mu_0H = 200$ mT to ensure the in-plane saturation of the sample, while the incidence angle of light was normal to the sample plane in order to probe spin waves with zero
<figure><img src="image_6.png" /><figcaption>FIG. 4. (a) Out-of-plane VSM measurements (First dose: 1.0 × 10¹⁶ N₂/cm²) compared with the Stoner-Wohlfarth model. The inset is a zoom to show the quality of the fit. (b) Out-of-plane anisotropy obtained by fitting the VSM data.</figcaption></figure>

wave vector. Then, the SW dispersion relation was measured in the magnetostatic surface spin-wave (MSSW) configuration sweeping the in-plane wave vector along the perpendicular direction. Due to the conservation of the in-plane momentum in the scattering process, the wave vector magnitude $k$ is linked to the incidence angle of light $\theta$ by the simple relation $k = \frac{4\pi}{\lambda} \sin \theta$. BLS spectra were recorded at different incidence angles of light between 0° and 70°, corresponding to a variation of the in-plane transferred wave vector from 0 to $2.3 \times 10^7$ rad/m.

### III. THEORETICAL MODEL

The measured dispersion relation of the spin waves was evaluated with the Landau-Lifshitz equation translated as an eigenvalue problem $i\omega\mathbf{m}(\mathbf{r}) = \gamma\mu_0\tilde{\mathbf{A}}\mathbf{m}(\mathbf{r})$, where matrix $\tilde{\mathbf{A}}$ contains all the magnetic interactions in the system. The theory of spin-wave dynamics in magnetization-graded films was developed by Gallardo et al. [17], where the dynamic matrix method [70] was used to account for the variation of the saturation magnetization along the thickness $d$ for different profiles. In this work, the dynamic matrix theory is extended to the case of a thick film, where the anisotropies $K_u$ and $K_c$ are spatially modulated to study the irradiated Fe:N samples. The new matrix elements $A_{\eta\nu\eta_p}(\eta = y', z')$, including the anisotropy terms for the $\nu$-th sublayer, are given by (see

details in Ref. 17)

$$ 
\begin{aligned}
A_{y'_{\nu} y'_{\nu}} &= A_{z'_{\nu} z'_{\nu}} = 0, \\
A_{y'_{\nu} z'_{\nu}} &= -H - \frac{J}{\mu_0 M_s d_s} \sum_i (\delta_i^{\nu-1} + \delta_i^{\nu+1}) \\
&\quad - \frac{2A_{\text{ex}}}{\mu_0 M_s} k^2 - \frac{2M_s}{|k|d_s} \sinh\left(\frac{|k|d_s}{2}\right) e^{-|k|\frac{d_s}{2}} \\
&\quad + \frac{2K_{u\nu}}{\mu_0 M_s} - \frac{2K_{c\nu}}{\mu_0 M_s}, \\
A_{z'_{\nu} y'_{\nu}} &= H + \frac{J}{\mu_0 M_s d_s} \sum_i (\delta_i^{\nu-1} + \delta_i^{\nu+1}) + \frac{2K_{c\nu}}{\mu_0 M_s} \\
&\quad + \frac{2A_{\text{ex}}}{\mu_0 M_s} k^2 + M_s - \frac{2M_s}{|k|d_s} \sinh\left(\frac{|k|d_s}{2}\right) e^{-|k|\frac{d_s}{2}}.
\end{aligned}
$$

For different sublayers $\nu$ and $p$ ($\nu \neq p$), the matrix elements are

$$ 
\begin{aligned}
A_{y'_{\nu} y'_{p}} &= -A_{y_{\nu} y_{p}} = \\
&\quad -ik \frac{2M_s}{|k|d_s} \sinh^2\left(\frac{|k|d_s}{2}\right) e^{-|k||\nu-p|d_s} \operatorname{sgn}(\nu - p), \\
A_{y'_{\nu} z'_{p}} &= \frac{2M_s}{|k|d_s} \sinh^2\left(\frac{|k|d_s}{2}\right) e^{-|k||\nu-p|d_s} \\
&\quad + \frac{J}{\mu_0 M_s d_s} (\delta_p^{\nu-1} + \delta_p^{\nu+1}), \\
A_{z'_{\nu} y'_{p}} &= \frac{2M_s}{|k|d_s} \sinh^2\left(\frac{|k|d_s}{2}\right) e^{-|k||\nu-p|d_s} \\
&\quad - \frac{J}{\mu_0 M_s d_s} (\delta_p^{\nu-1} + \delta_p^{\nu+1}).
\end{aligned}
$$

Here, $K_{u\nu}$ and $K_{c\nu}$ are the uniaxial and cubic anisotropy constants of sublayer $\nu$, $\delta_i^j$ is the Kronecker delta function ($\delta_i^j = 0$ for $i \neq j$ and $\delta_i^j = 1$ for $i = j$), $\operatorname{sgn}(\xi) = \xi/|\xi|$ is the sign function and $J = 2A_{\text{ex}}/d_s$ [17, 71], with $A_{\text{ex}}$ being the exchange constant and $d_s$ the thickness of each sublayer. Note that the total thickness of the film is given by $d = N_s d_s$, where $N_s$ is the total number of sublayers. Then, the eigenvalue problem can be numerically solved with these matrix elements to obtain the frequencies and spatial profiles of the dynamic magnetization that support the experimental data. It is worth noting that the orthogonal coordinate system ($x', y', z'$) is chosen such that $z' = z$ is the normal axis, the equilibrium magnetization aligns with $x'$ that points along [100] (easy axis), and $y'$ lies in the film plane. For all the sample, having a thickness $d = 54$ nm, a number of sublayers $N_s = 30$ was used, where a convergence of the modes was obtained.

### IV. RESULTS AND DISCUSSION

In the first step, the saturation magnetization ($M_s$) of all samples was estimated using VSM measurements. Both pristine Fe and the implanted samples exhibited $M_s$ values closely matching the expected value of bcc
<figure><img src="image_7.png" /><figcaption>FIG. 5. Brillouin light scattering spectra of three investigated samples taken at $k = 6$ rad/µm, applying a magnetic field $\mu_0H = 200$ mT along the [100] direction.</figcaption></figure>

Iron [72], i.e., $M_s = 1713$ kA/m, within an error margin of ±5% mainly attributed to the evaluation of the thin film volume. This confirms previous studies, where it was demonstrated that iron epitaxial thin films grown on ZnSe/GaAs (001) exhibit the expected magnetic moment for bcc iron [66] and that no significant change in $M_s$ is detected in Fe:N thin films [64]. Consequently, the uniaxial anisotropy constant, $K_u$, was extracted using the Stoner-Wohlfarth model described above and assuming $M_s = 1713$ kA/m for all the samples. Figure 4(a) presents the fit performed for the first dose Fe:N sample, demonstrating the effectiveness of this approach. It was observed that $K_u$ increases with increasing irradiation dose. As shown in Fig. 4(b), which depicts the variation of $K_u$ as a function of the measured c-axis lattice parameter, the enhancement of PMA can be attributed to the tetragonal distortion induced by N implantation [63, 64]. The error bars are calculated by considering the lowest and highest values of $M_s$ compatible with VSM measurements, i.e., with $M_s \in [1627, 1799]$ kA/m for Fe:N, and $M_s \in [1627, 1713]$ kA/m for pristine Fe.

Then, the dynamic properties of the samples were investigated using BLS measurements. Fig. 5 reports typical BLS spectra taken for the three investigated samples at $k = 6$ rad/µm when an in-plane magnetic field $\mu_0H = 200$ mT is applied along the in-plane [110] direction. For the pristine Fe, three peaks are present on

<figure><img src="image_8.png" /><figcaption>FIG. 6. BLS measurements (points) performed at $k = 0$ as a function of the in-plane angle $\phi$ between the applied magnetic field and the [110] Fe direction. The lines are obtained from the theoretical approach.</figcaption></figure>

both the Stokes and anti-Stokes sides of the spectrum, corresponding to spin waves propagating in opposite directions. As discussed in more detail later, the peak on the anti-Stokes side, tagged by an asterisk, which corresponds to a much less intense peak on the Stokes side, can be identified as the Damon-Eshbach (DE) surface mode. The two modes having similar intensity in both sides of the spectrum, instead, correspond to the first and second perpendicular standing spin-wave (n-PSSW) mode, characterized by $n = 1$ and $n = 2$ nodes across the film thickness, respectively. The BLS spectra of the irradiated samples are characterized by a significative change on increasing the irradiation dose. In particular, the DE mode becomes less intense and exhibits a larger nonreciprocal BLS intensity, eventually being visible only on the anti-Stokes side for the second irradiation dose. In addition, the DE and 1-PSSW modes exhibit a frequency asymmetry between the Stokes and the anti-Stokes sides, while all the modes are observed to gradually shift towards lower frequency. Finally, it is interesting to note that in the BLS spectra of the irradiated samples, the signal-to-noise ratio and peak width are similar to those observed for pristine Fe, indicating that the irradiation preserves the film quality, in agreement with the X-rays results.

Figure 6 shows the dependence of the spin-wave frequency as a function of the in-plane angle $\phi$, between the applied magnetic field and the [110] Fe direction, measured by BLS for the pristine Fe film (open diamonds) and the two irradiated samples (dots and squares), applying a magnetic field $\mu_0H = 200$ mT. In order to avoid the influence of the frequency nonreciprocity between the Stokes and anti-Stokes peaks, observed in the irradiated samples, spin waves with zero wave vector were probed. As it can be seen, the pristine Fe exhibits an os-
<figure><img src="image_9.png" /><figcaption>FIG. 7. Analysis of the pristine Fe film. (a) Comparison between the BLS measured (diamonds) and the calculated (continuous lines) frequencies as a function of the wave vector $k$. The blue, orange and green lines denote the DE, 1-PSSW, and 2-PSSW mode, respectively. (b) Values of the magnetic anisotropy constants $K_u$ and $K_c$ (assuming a constant value along the film thickness for pristine Fe) obtained from the theoretical calculations. Panel (c), precessional amplitudes (in arbitrary units) of the three spin wave modes calculated at $k = -18, 0, \text{and } 18 \text{ rad/µm}$. The bottom row represents the Damon-Eshbach (DE) mode, while the first and second perpendicular standing SW modes are illustrated in the middle and upper rows, respectively.</figcaption></figure>

cillating behavior in the spin wave frequency of all three modes, indicating an in-plane cubic anisotropy typical of the bulk Fe, with the easy and hard axes along the [100] and [110] directions, respectively. In the irradiated samples, all spin wave modes show a shift toward lower frequencies and a reduction of the in-plane oscillation, indicating an increase in out-of-plane anisotropy and a reduction in the cubic one, respectively, which become more pronounced with increasing implantation dose. The observed decrease of the in-plane cubic anisotropy in the irradiated samples is consistent with the increase in mobility, as supported by XRD measurements (not shown). In addition the increase of the PMA with the irradiation dose can be ascribed to the tetragonal distortion of the unit cell caused by the N implantation, as previously discussed.

Figs. 7-9(a) show the spin-wave dispersion relation measured for the pristine Fe (Fig. 7) and the two irradiated samples (Figs. 8 and 9). The experiments were performed in MSSW geometry, with a magnetic field applied along the in-plane easy direction [100]. Positive (negative) values of $k$ correspond to the Stokes (anti-Stokes) side of the experimental BLS spectra. One can note that for the pristine Fe all the magnon modes exhibit a reciprocal dispersion relation, while for the irradiated samples the dispersion relation of the 1-PSSW and the DE modes is characterized by a noticeable nonreciprocal behavior, which becomes more marked when the implantation dose

increases.

In order to gain better insight into these findings, the dynamic-matrix approach previously described was exploited to calculate the spin-wave frequency (solid lines) and the spatial profiles through the film thickness of the dynamic magnetization components for each mode [Figs. 7-9(c)]. First, the experimental results of the pristine Fe were analyzed (Fig. 7), assuming uniform magnetic parameters through the film thickness. The saturation magnetization was set to the value $M_s = 1713$ kA/m, obtained from the VSM measurements. As it can be seen in Figs. 6 and 7(a), both the dispersion relation and measurements as a function of $\phi$ are well reproduced for constants values of the anisotropy $K_c = 49.6$ kJ/m$^3$, $K_u = 100$ kJ/m$^3$ [see Fig. 7(b)], and $A_{ex} = 19$ pJ/m (see Table II.). One can note that a dispersionless behavior characterizes both the 1-PSSW (orange) and 2-PSSW (green) modes, while the DE mode (blue) exhibits a large positive frequency dispersion and group velocity. In addition, a crossing of the DE mode with the 1-PSSW (at about 2 rad/µm) and the 2-PSSW one (at about 18 rad/µm) is clearly observed in both the experiment and the calculations.

Fig. 7(c) reports the spatial profiles of the dynamical magnetization calculated for selected values of the in-plane wave vector $k$, corresponding to relevant points of the calculated spin-wave dispersion. For $k = 0$, the lowest frequency mode (diamond) corresponds to DE mode
<figure><img src="image_10.png" /><figcaption>FIG. 8. Analysis of the Fe:N film irradiated with the first dose. (a) Comparison between the BLS measured (dots) and the calculated (continuous lines) frequencies as a function of the wave vector *k*. The blue, orange and green lines denote the DE, 1-PSSW, and 2-PSSW mode, respectively. (b) Evolution of the magnetic anisotropy constants $K_u$ and $K_c$ along the film thickness obtained from the theoretical analysis. Gray dashed lines indicate the value of the anisotropy constant of pristine Fe [73]. (c) Precessional amplitudes (in arbitrary units) of the three spin wave modes calculated at $k = -18, 0, \text{ and } 18 \text{ rad/µm}$.</figcaption></figure>

which assumes a uniform spatial profile at zero wave vector. In contrast, the second and third modes are characterized by one node (cylinder) and two nodes (trapeze) across the thickness, respectively, as expected for the 1-PSSW and 2-PSSW modes. As the wave vector *k* increases, the DE mode acquires a surface character. It localizes at the bottom ($z = 54$ nm) and top ($z = 0$) surfaces for positive and negative wave vectors, respectively, reflecting the well-known amplitude nonreciprocity of the DE mode [74]. In addition, one can observe that its profile is hybridized with that of the 1-PSSW and the 2-PSSW modes. The latter, indeed, maintain their oscillating behavior, but have a different amplitude at the two film surfaces.

The experimental results of the two irradiated samples were then analyzed (Figs. 8 and 9). Based on the measurements performed as a function of the in-plane direction of the applied magnetic field, indicating that both the in-plane and the out-of-plane anisotropies are affected by the N implantation, a graded profile that mimics the TRIM simulations (Fig. 2) was assumed for both $K_u$ and $K_c$. In particular, $K_u$ ($K_c$) was set to increase (decrease) in the irradiated part of the film ($z = 0$), with the maximum (minimum) value being reached at the depth where a maximum number of implanted N ions is expected, corresponding to 7.2 nm and 7.5 nm for the first and second doses, respectively. Also, for both implantation doses, $K_u$ and $K_c$ were assumed to recover the values of the pristine Fe at approximately a depth of 25 nm from the

surface plane, as can be seen in the graded-anisotropy profiles shown in Figs. 8(b) and 9(b). Saturation magnetization was assumed to be uniform through the thickness and set to the value $M_s = 1713$ kA/m. Besides, a slight decrease in the exchange constant $A_{ex}$ was required to improve the fitting of the experimental data. Table II summarizes the parameters used to fit the data.

As it can be seen in Fig. 6 and in the dispersion relations in Figs. 8(a) and 9(a), a good agreement between BLS data and the theoretical calculation was obtained for both samples. The spatial graduation profiles of $K_u(z)$ and $K_c(z)$ through the film thickness are illustrated in the panels (b) of Figs. 8 and 9. A rapid evolution of both the perpendicular and the in-plane anisotropy with the implantation dose can be observed. In particular, the perpendicular anisotropy constant is found to increase up to a maximum value of 318 kJ/m$^3$ and 596 kJ/m$^3$ for the first and second dose, respectively. Conversely, the cubic anisotropy constant decreases, reaching the minimum value of 17.8 kJ/m$^3$ for the first dose and 6.9 kJ/m$^3$ for the second one. Note that the out-of-plane anisotropy averaged over the whole film thickness assumes the values $\langle K_u \rangle = 162$ kJ/m$^3$ and $\langle K_c \rangle = 240$ kJ/m$^3$ for the first and second implantation doses, respectively, in reasonably good agreement with the values estimated in the VSM analysis. Concerning the exchange interaction between sublayers, in the dynamic-matrix formulation, the inter-sublayer exchange parameter $J = 2A_{ex}/d_s$ primarily controls the curvature of the Damon–Eshbach branch
<figure><img src="image_11.png" /><figcaption>FIG. 9. Analysis of the Fe:N film irradiated with the second dose. (a) Comparison between the BLS measured (squares) and the calculated (continuous lines) frequencies as a function of the wave vector *k*. The blue, orange and green lines denote the DE, 1-PSSW, and 2-PSSW mode, respectively. (b) Evolution of the magnetic anisotropy constants $K_u$ and $K_c$ along the film thickness obtained from the theoretical fit of the data. Gray dashed lines indicate the value of the anisotropy constant of pristine Fe [73]. (c) Distribution of precessional amplitudes (in arbitrary units) along the thickness of the three spin wave modes calculated at $k = -18, 0, \text{ and } 18 \text{ rad/µm}$.</figcaption></figure>

<table><caption>TABLE II. Magnetic parameters used in the theoretical calculations. For the irradiated samples the maximum and minimum values of the graded profiles of the uniaxial and cubic magnetic anisotropy are reported, respectively.</caption><thead><tr><th></th><th><i>M</i><sub>s</sub> (kA/m)</th><th><i>A</i><sub>ex</sub> (pJ/m)</th><th><i>K</i><sub>c</sub> (kJ/m<sup>3</sup>)</th><th><i>K</i><sub>u</sub> (kJ/m<sup>3</sup>)</th><th><i>d</i> (nm)</th></tr></thead><tbody><tr><td>Pristine Iron</td><td>1713</td><td>20</td><td>49.6</td><td>100</td><td>54</td></tr><tr><td>First Dose</td><td>1713</td><td>17</td><td>17.8</td><td>318</td><td>54</td></tr><tr><td>Second Dose</td><td>1713</td><td>14</td><td>6.9</td><td>596</td><td>54</td></tr></tbody></table>

and the frequency spacing among perpendicular standing spin-wave modes. Thus varying *J* shifts the DE-PSSW hybridization points, but does not affect the sign of the nonreciprocity, which is governed by the anisotropy gradient. For irradiated films, the best fit procedure indicates a moderate reduction of $A_{ex}$, from 20 pJ/m in pristine Fe, to 17 pJ/m and 14 pJ/m for the first and second doses, respectively (see Table II), consistent with the near-surface tetragonal distortion and Fe-N phase formation revealed by X-ray analysis.

A deeper insight into the influence of the N implantation on the spin-wave dispersion relation can be achieved looking at the panels (c) of Figs. 8 and 9, which show the spatial profiles of the dynamical magnetization, calculated for selected values of the in-plane wave vector *k* according to Figs. 8(a) and 9(a). One can observe that the local variation of the effective magnetization $\mu_0 M_{eff} = \mu_0 M_s - 2K_u/M_s$ along the film thickness, in-

duced by the graded magnetic anisotropy, strongly affects the spatial profiles of the DE and the 1-PSSW mode. In particular, at $k = 0$, the lowest frequency mode corresponds to the mode without nodes along the thickness as for the pristine Fe. However, it has a nonuniform spatial profile assuming a larger amplitude in the top region of the film ($z = 0$), where $\mu_0 M_{eff}$ has a lower value. Also, the spatial profile of the 1-PSSW evaluated at $k = 0$ is no longer symmetric through the film thickness and tends to localize in the bottom part of the film ($z = 54$ nm) due to the repulsion with the DE mode. On increasing *k*, the DE mode assumes a surface character. In particular, for positive *k*, it localizes at the bottom region of the film, where $\mu_0 M_{eff}$ has a higher value. In contrast, for negative *k*, it localizes in the top region where $\mu_0 M_{eff}$ assumes a lower value. This behavior explains the marked nonreciprocal dispersion of this mode, which assumes higher (lower) frequency values for positive (negative) wave vec-
<figure><img src="image_12.png" /><figcaption>FIG. 10. Frequency nonreciprocity $\Delta f = f(+\mathbf{k}) - f(-\mathbf{k})$ for DE (blue), 1-PSSW (orange), and 2-PSSW (green) modes in Fe-N films irradiated with (a) first and (b) second doses. The wave vectors values $k_c^+$ and $k_c^-$ denote the anticrossing points between DE and 2-PSSW modes for negative and positive propagation directions, respectively.</figcaption></figure>

tors. It can also be observed that at small wave vector values the profile of the DE mode hybridizes with that of the 1-PSSW mode. As a consequence, the latter tends to localize on the opposite surface with respect to the DE mode, and assumes a dispersive behavior characterized by an opposite nonreciprocity, where for $k < 0$ it oscillates at higher frequencies compared to $k > 0$. In contrast, the spatial profile of the 2-PSSW mode is only slightly affected by the N implantation. Consequently, this mode exhibits a dispersionless behavior similar to that of the pristine Fe.

It is interesting to note that the more pronounced nonreciprocity observed for the second dose can be explained considering that this sample is characterized by a more significant local variation of $\mu_0 M_{\text{eff}}$ through the thickness due to the larger change of the perpendicular anisotropy induced by the N implantation. The calculated frequency difference $\Delta f$ between positive and negative $k$ is summarized in Fig. 10, showing a similar behavior for the (a)

first and (b) second doses. As can be seen, the DE and 1-PSSW modes exhibit opposite nonreciprocity, which increases linearly with $k$ reaching an absolute value of about 2.2 GHz (4.3 GHz) at $k = 20 \text{ rad}/\mu\text{m}$ for the first (second) dose. Moreover, an abrupt discontinuity of $\Delta f$ can be observed for the DE mode at $k \approx k_c^{\pm}$ (see vertical dashed lines), due to the anticrossing with the 2-PSSW mode. The latter, instead, exhibits zero $\Delta f$ for all wave vectors except in the range between $k_c^{\pm}$, where it shows a sharp discontinuity, induced by the coupling with the DE mode, assuming an almost constant value of about 0.3 GHz (0.7 GHz) for the first (second) irradiation dose. As mentioned before, the opposite sign of the frequency shift for DE and 1-PSSW modes arises because the graded anisotropy makes the effective magnetization smaller in the implanted top surface. For $k > 0$ ($k < 0$) the DE mode localizes in the non-implanted bottom (top) region having higher (lower) $\mu_0 M_{\text{eff}}$, while hybridization forces 1-PSSW to localize on the opposite surface, leading to an opposite nonreciprocity for the two modes. The 2-PSSW mode, instead, is only weakly affected by the anisotropy gradient, and remains nearly reciprocal except within the DE-2-PSSW anticrossing window. Finally, one can observe that as the implantation dose increases, the decrease in frequency of the 2-PSSW mode and the increase in nonreciprocity of the DE lead the crossing points $k_c^{\pm}$ to shift toward lower frequencies and move apart, respectively. This result emphasizes the potential for tailoring spin-wave properties through controlled modifications of magnetic anisotropy, which is essential for the design of tunable magnonic devices.

V. CONCLUSIONS

In this work, spin wave dynamics in the nitrogen-implanted Fe films have been investigated by combining BLS measurements and theoretical calculations. We show that the low-dose nitrogen implantation, causes in the Fe:N film a gradual increase of the perpendicular anisotropy and a decrease of the in-plane cubic one. This local variation of the magnetic anisotropies along the film thickness, which becomes more marked on increasing the implantation dose, breaks the symmetry and induces a preferential localization of the spin-wave modes in certain regions of the sample, strongly affecting the magnon coupling and the dispersion relation. In particular, the mode at lower frequencies, having a dispersionless behavior in the pristine Fe, is found to assume a dispersive character with opposite group velocity for counterpropagating spin waves. Furthermore, we find that by modifying the implantation conditions, the anisotropy profile changes, which in turn allows for tuning the dispersion relation of the spin-wave modes.

The observed nonreciprocal spin-wave propagation offers clear opportunities for device applications in magnonics. By enabling efficient transmission in one direction while suppressing propagation in the opposite di-
rection, this effect can protect sensitive circuit elements
from back-reflected signals, thereby reducing interfer-
ence in spin-wave logic architectures [48]. This direc-
tional control also supports the concept of the magnonic
diode, which allows unidirectional spin-wave flow and
could serve as a fundamental component for signal rout-
ing and isolation in nanoscale magnonic networks [46, 49].
These results demonstrate that nitrogen implantation is
a viable method for engineering magnetic anisotropy gra-
dients in iron films while preserving the film epitaxy and
manipulating spin-wave propagation properties, repre-
senting a highly promising technique in the development
of magnonics and 3D nanomagnetism.

VI. ACKNOWLEDGMENTS

J. J. B. acknowledges the support of USM-PIIC Nº
054/2023. Support from FONDECYT, Grants No.
1250803 and 1241589, and Basal Program for Centers
of Excellence, Grant CIA250002 (CEDENNA), is highly
acknowledged. This work was funded by the Euro-
pean Union within the HORIZON-CL4- 2021-DIGITAL-
EMERGING-01 Grant No. 101070536 MandMEMS. P.
R. and M. M. acknowledge the French National Re-
search Agency, ANR-22-CE24-0015 SACOUMAD. The
authors also acknowledge Paola Atkinson for preparing
the GaAs substrate by MBE, Jean-Eudes Duvauchelle
and the staff of the MPBT (physical properties – low tem-
perature) platform of Sorbonne Université for their sup-
port. M.M. and S.T. acknowledge PRIN 2022 “Metrol-
ogy for spintronics: A machine learning approach for the
reliable determination of the Dzyaloshinskii-Moriya in-
teraction (MetroSpin)”, 2022SAYARY, and PRIN 2022
”ThreEE-dimensional Processing tecHnique of mAgNetic
crYstals for magnonics and nanomagnetism (TEEP-
HANY) ID 2022P4485M”. This work has been funded
by the European Union - NextGenerationEU, Mission
4, Component 2, under the Italian Ministry of Univer-
sity and Research (MUR) National Innovation Ecosys-
tem grant ECS00000041 - VITALITY - CUP Nos.
J97G22000170005 and B43C22000470005.

Appendix A: Reciprocal space mapping

A reciprocal space mapping of the 2.0 × 10¹⁶ N₂/cm²
dose was performed by scanning the Qₓ direction (par-
allel to [110]) and the Qᵧ direction (parallel to [001])
around the (112) Fe peak. Results are reported in Fig. 11.
It is observed that the epitaxial conditions of pristine
Fe/GaAs are preserved after N₂ implantation. Along
the Qₓ direction, the (112) reflections of α'-Fe₈N₁-ₓ and
α*-FeNₓ are located very close to that of α-iron within

the thin film plane. Along the *Q*<sub>*z*</sub> direction, while α*-FeN<sub>*x*</sub> is very close to the pristine cubic structure of α-Fe, the α' phase exhibits a clear tetragonal distortion. This indicates that nitrogen ion implantation is capable

<figure><img src="image_13.png" /><figcaption>FIG. 11. Reciprocal lattice maps showing the (112) diffraction spots of GaAs (or ZnSe) for an implanted sample Fe film (2 × 10¹⁶ N₂ atoms/cm²). The GaAs (or ZnSe) and (112) of α'-Fe₈N₁-ₓ and α*-FeNₓ are clearly observed.</figcaption></figure>

<figure><img src="image_14.png" /><figcaption>FIG. 12. θ-2θ measurements for the 2.0 × 10¹⁶ N₂/cm² Fe:N sample and comparison with pristine Fe.</figcaption></figure>

of forming the α'-Fe₈N₁-ₓ and α*-FeNₓ phases without
causing significant modifications to the crystalline struc-
ture of the target α-Fe, despite the fact that numerous
atomic displacements were likely occurring during colli-
sion cascades. Thus, the tetragonality of the α'-Fe₈N₁-ₓ
phase, discussed in Refs. 63, 64, and 67, is confirmed.
The in-plane and out-of-plane lattice parameters for all
doses are reported in Table I. The tetragonality of the
α'-Fe₈N₁-ₓ phase reaches values of 7% and 8.4%, re-
spectively, for second (1.5×10¹⁶ N₂/cm²) and the high-
est (2.0×10¹⁶ N₂/cm²) implantation doses. Figure 12
shows the θ-2θ measurements of the (001) plane for the
2 × 10¹⁶ N₂/cm² sample.
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