# Dzyaloshinskii–Moriya Interaction in Fe 5 GeTe 2 Epitaxial Thin Films

João Sampaio, Antoine Pascaud, Edgar Quero, André Thiaville, Vincent Polewczyk, Alain Marty, Frédéric Bonell, Alexandra Mougin

► To cite this version:

João Sampaio, Antoine Pascaud, Edgar Quero, André Thiaville, Vincent Polewczyk, et al. Dzyaloshinskii–Moriya Interaction in Fe 5 GeTe 2 Epitaxial Thin Films. Nano Letters, 2025, 25 (39), pp.14341-14347. 10.1021/acs.nanolett.5c03547. hal-05321458

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Distributed under a Creative Commons Attribution 4.0 International License
Dzyaloshinskii-Moriya interaction in Fe<sub>5</sub>GeTe<sub>2</sub>
epitaxial thin films

João Sampaio,*,† Antoine Pascaud,† Edgar Quero,† André Thiaville,† Vincent Polewczyk,‡ Alain Marty,‡ Frédéric Bonell,‡ and Alexandra Mougin†

† Université Paris-Saclay, CNRS, Laboratoire de Physique des Solides, 91405 Orsay, France

‡University Grenoble Alpes, CNRS, CEA, IRIG-Spintec, F-38000 Grenoble, France

E-mail: joao.sampaio@cnrs.fr

**Abstract**

Van der Waals ferromagnets, such as Fe<sub>5</sub>GeTe<sub>2</sub>, offer a promising platform for spintronic
devices based on chiral magnetic textures, provided a significant Dzyaloshinskii-Moriya in-
teraction (DMI) can be induced to stabilise the textures. Here, we directly measure DMI in
epitaxial Fe<sub>5</sub>GeTe<sub>2</sub> thin films using Brillouin light scattering spectroscopy and observe a con-
sistent DMI (D = 0.04 mJ/m<sup>2</sup>) across various thicknesses. Its weak thickness dependence,
combined with the nominally symmetric film interfaces, suggests a bulk origin. Although we
do not determine the microscopic mechanism, our findings are compatible with *ab initio* calcu-
lations linking DMI to partial ordering of Fe split sites. Additionally, we find a low magnetic
dissipation (α < 0.02). The observed DMI, which could be further enhanced by optimising
the Fe site ordering, combined with low dissipation, makes Fe<sub>5</sub>GeTe<sub>2</sub> a strong candidate for
exploring the dynamics of chiral magnetic textures in two-dimensional materials.

**Keywords**

van der Waals magnets, Dzyaloshinskii-Moriya Interaction, Chiral spin textures, Fe₅GeTe₂
Ferromagnetic van der Waals (vdW) materials present an interesting new class of materials for the study of nanomagnetism. Their structure, made of 2D layers with relatively weak inter-layer bonding, can produce very thin films with flat and clean surfaces, either by exfoliation from macroscopic crystals or epitaxial growth. This structure also lends itself naturally to the fabrication of heterostructures with good interfacial quality with a large variety of other vdW materials (semiconductors, metals, insulators, topological insulators, ...) — ideal for studying interfacial magnetic effects that are the basis of spintronic devices. Among the growing list of vdW ferromagnets of interest to spintronics, the Fe<sub>x</sub>GeTe<sub>2</sub> family stands out due to their high Curie temperature,<sup>1</sup> metallic character, and large magneto-transport effects.

Moreover, there have been reports of chiral magnetic structures in Fe<sub>x</sub>GeTe<sub>2</sub>, opening the way to their use in the many proposed devices based on chiral textures such as skyrmions. Homochiral spin spirals were observed in exfoliated flakes of Fe<sub>3</sub>GeTe<sub>2</sub><sup>2</sup> (thickness *t* = 185 nm), and Néel skyrmions were reported in Fe<sub>3</sub>GeTe<sub>2</sub> (70 nm)/[Co/Pd]<sup>3</sup> and WTe<sub>2</sub>/Fe<sub>3</sub>GeTe<sub>2</sub> (32 nm)<sup>4</sup> heterostructures. In other members of the Fe<sub>x</sub>GeTe<sub>2</sub> family, Néel skyrmions were observed in Fe<sub>5-x</sub>GeTe<sub>2</sub><sup>5,6</sup> (on 61 and 100 nm thick flakes), as well as in (Fe,Co)<sub>5</sub>GeTe<sub>2</sub> (110 nm)<sup>7</sup> or in the related Fe<sub>3-x</sub>GaTe<sub>2</sub>.<sup>8,9</sup> On the other hand, other groups working also on flakes of Fe<sub>5</sub>GeTe<sub>2</sub> observed only non-chiral Bloch bubbles.<sup>10,11</sup>

The observation of homo-chiral textures suggests a significant Dzyaloshinskii-Moriya interaction (DMI). Indeed, a large DMI parameter (*D* = 1.0 mJ/m<sup>2</sup>) was estimated in ref. 4. A nonzero DMI requires a symmetry-breaking mechanism along the out-of-plane direction. However, both Fe<sub>3</sub>GeTe<sub>2</sub> and Fe<sub>5</sub>GeTe<sub>2</sub> are presumed to have inversion-symmetric crystal structures incompatible with DMI.<sup>12</sup> Nevertheless, several experimental studies suggest that the actual crystalline structure of Fe<sub>x</sub>GeTe<sub>2</sub> may lack inversion symmetry. Possible sources of symmetry breaking include Fe split-site ordering,<sup>13</sup> Fe self-intercalation in the van der Waals gaps,<sup>14</sup> ordered Te vacancies,<sup>11</sup> and asymmetric Fe vacancy distribution.<sup>15</sup> Additionally, deviations from the expected vdW stacking sequence (ABCABC in Fe<sub>5</sub>GeTe<sub>2</sub> and ABAB in Fe<sub>3</sub>GeTe<sub>2</sub>) could also break inversion symmetry.

DMI can also arise from asymmetric interfaces in thin magnetic layers. This mechanism has
been well established in metallic systems, such as nanometric Co films in a AlOx/Co/Pt triplet film (up to D = 2 mJ/m²¹⁶,¹⁷). Although this interfacial mechanism could be expected to be weaker in vdW materials due to their weak interlayer coupling, first-principle calculations by Park et al.¹⁸ show that the interface of Fe₃GeTe₂ with its natural oxide can induce a significant DMI (≈ 2 mJ/m² for a monolayer of Fe₃GeTe₂). Interfacial mechanisms, however, are only significant in ultra-thin systems, since their effects are progressively diluted with increasing thickness, while all the cited experimental observations involve relatively thick (30-200 nm) exfoliated flakes.

Characterizing the DMI solely from the observation of magnetic textures is challenging, as these are influenced by multiple interactions, including dipolar effects, magnetocrystalline anisotropy, and exchange. In some cases, dipolar interactions may even create seemingly chiral textures at the surface of thick samples.³,¹⁹ To determine the origin of chiral magnetic textures and disentangle interfacial and bulk contributions, a more direct measurement of the DMI is essential. In this letter, we investigate the static and dynamic magnetic properties of epitaxial Fe₅GeTe₂ thin films of varying thickness using Brillouin light scattering (BLS) spectroscopy. By directly measuring the DMI, we unambiguously demonstrate the presence of a significant DMI of bulk origin. Furthermore, we show that the magnetic damping in these films is low for a metallic ferromagnet. These properties make Fe₅GeTe₂ a promising material for the study of the propagation of chiral magnetic textures.

# Results

## Sample growth & structure

Fe₅GeTe₂ films (t = 12, 8 and 3 nm) were epitaxially grown on Al₂O₃(0001) at 350°C by coevaporation of elemental Fe, Ge and Te, with a Fe:Ge flux ratio of 5 and with a slightly overstoichio-metric Te flux, and then annealed ~ 500°C for 10 min. The 12-nm-thick film was characterised in depth in Ribeiro et al.,²⁰ and all films were grown following the protocol reported in that work. The films were capped by 3 nm of Al deposited at room temperature, then naturally oxidized in air. This capping layer prevents the oxidation of Fe₅GeTe₂ and provides similar top and bottom
<figure><img src="image_2.png" /><figcaption>Figure 1: a) Sample structure and diagrams of ordered split sites Fe<sub>5</sub>GeTe<sub>2</sub> with broken centrosymmetry: full ordering (“DDD”) or 1/3 (“UUD”). Open circles represent empty sites. b) XRD spectrum for the three films. c) RHEED along the (100) and (110) reciprocal directions, obtained after growth (3-nm thick film). d) RHEED intensity profiles along the dashed line in c. The profiles are normalized to the (11) intensity and shifted vertically for clarity. Inset: Top view schematic of the “UUD” √3 × √3 reconstruction.</figcaption></figure>

inter interfaces (Fig. 1a).

The X-ray diffraction (XRD) specular scans (Fig.1b) show single-phase Fe<sub>5</sub>GeTe<sub>2</sub> with a con-
stant c-axis lattice parameter (0.98 nm per monolayer) across all thicknesses (see S.I.), consis-
tent with the good crystalline quality previously observed for the 12-nm film by XRD and scan-
ning transmission electron microscopy (STEM).<sup>20</sup> The RHEED patterns of Fe<sub>5</sub>GeTe<sub>2</sub> (8 nm) in
the (100) and (110) reciprocal directions, measured after growth, are shown in Fig. 1c. Simi-
lar anisotropic diffraction patterns are observed at all thicknesses and demonstrate the good epi-
taxy on sapphire. However, faint (10) diffraction rods are visible in the (110) direction (peak at
$q/q(11) = 1/\sqrt{3}$ in Fig. 1d). They denote the presence of a small proportion of domains that are
rotated by 30° from the main crystal orientation. This feature is more apparent for the thinnest layer
and disappears for the 12 nm film, indicating an improvement of the crystal in-plane orientation
quality with increasing thickness.
In addition, the RHEED patterns display a (3 × 1) reconstruction characteristic of a (√3 × √3) R30° superstructure (peaks at q/q(11) = 1/3 and 2/3 in Fig. 1d). This superstructure is observed in all three films, and is more apparent at larger thicknesses. The nature of this feature will be discussed later.

<figure><img src="image_3.png" /><figcaption>Figure 2: a) $M_s(T)$ of the three films under in-plane field (SQUID measurement). b) LMOKE hysteresis loops of the three films close to 150 K. c) In-plane magnetic domains in the 12-nm-thick sample imaged by LMOKE (the contrast orientation is left/right).</figcaption></figure>

## Magnetic properties

The spontaneous magnetization ($M_s$) was measured by SQUID magnetometry versus temperature (Fig. 2). Thinner films showed lower $M_s$, with the 3-nm film showing about one-third of the $M_s$ of the 12-nm film at 150 K (see Table 1). The Curie temperature ($T_C$) of the thicker films (297–299 K) is similar to bulk values (270–310 K$^1$), whereas the 3-nm film had a lower $T_C$ (258 K). The evolution of $M_s$ and $T_C$ indicates that a thickness of a few layers is needed to fully develop the bulk magnetic properties of the films. The $M_s(T)$ curves showed a monotonic trend, unlike previous reports of various transitions in $M_s(T)$.10,13,21,22 However, those studies focused on virgin films, whereas our samples had been previously magnetized. These transitions may also be specific to
bulk samples rather than epitaxially-grown films, suggesting structural or chemical differences between these two classes of materials.

Hysteresis loops measured by longitudinal magneto-optical Kerr effect (LMOKE) (150 K; Fig. 2b) show in-plane magnetization reversal with a small coercive field of a few mT. The sharp transitions in the loops indicate reversal by nucleation and domain wall propagation, typical of homogeneous films. LMOKE imaging of the 12-nm film (Fig. 2c) confirms domain wall propagation during reversal and reveals large domains (10–100 µm), further indicating film homogeneity.

<table><caption>Table 1: Material parameters at 150 K for the three films. The $T_C$ and $\mu_0M_s$ were determined from SQUID magnetometry, $B_K$ and the gyromagnetic g-factor were taken from fits of the BLS frequency versus field curves, the $\alpha$ (an upper bound; see text) from the peak width, and $v_{DMI}$ from fits of the Stokes/anti-Stokes frequency difference. The negative $B_K$ represents an easy-plane anisotropy.</caption><thead><tr><th>t<br>nm</th><th>T<sub>C</sub><br>K</th><th>μ<sub>0</sub>M<sub>s</sub><br>T</th><th>B<sub>K</sub><br>T</th><th>g</th><th>α</th><th>v<sub>DMI</sub><br>m/s</th><th>D<br>μJ/m<sup>2</sup></th></tr></thead><tbody><tr><td>3</td><td>258</td><td>0.19</td><td>-0.59</td><td>2.3</td><td>&lt; 0.08</td><td>18 ± 4</td><td>22 ± 5</td></tr><tr><td>8</td><td>299</td><td>0.36</td><td>-0.84</td><td>2.2</td><td>&lt; 0.03</td><td>20.5 ± 1.5</td><td>47 ± 3</td></tr><tr><td>12</td><td>297</td><td>0.57</td><td>-1.0</td><td>2.1</td><td>&lt; 0.02</td><td>10.9 ± 1.3</td><td>41 ± 5</td></tr></tbody></table>

The samples were further characterised by studying the spin wave frequency as a function of field and wave vector using BLS spectroscopy (analysed using the laws of dispersion determined in refs. 16,23,24; see S.I.). The negative anisotropy field $B_K$ extracted from the BLS measurements of frequency versus field (Fig. S2) confirms the easy-plane (i.e., ab plane) anisotropy (at 150 K; Tab. 1). Its magnitude is significantly larger than the expected contribution from shape anisotropy (given by $-\mu_0M_s$), a behavior previously observed in $Fe_5GeTe_2$.20,21 This suggests either the presence of an intrinsic easy-plane magnetocrystalline anisotropy or that the magnetisation is inhomogeneous across the vdW layers and gaps, leading to a stronger shape anisotropy than expected from the measured $M_s$ (see S.I.).

The g-factor obtained from the BLS measurements consistently falls in the range $g = 2.1-2.3$, showing no significant temperature dependence, in close agreement with previous reports.21,25 The notable deviation from the free-electron value of $g = 2.0$ suggests a substantial orbital moment and significant spin-orbit coupling in the material. This is consistent with the orbital/spin moment ratio
measured in ref. 20.

The Gilbert dissipation parameter ($\alpha$) can be inferred from the BLS peak width ($\Gamma$), similar to its extraction from ferromagnetic resonance measurements. However, $\Gamma$ is also broadened by inhomogeneities in the film's magnetic properties, which are typically disentangled from the effect of $\alpha$ with high magnetic fields taking advantage of their distinct field dependence. Since the available field was insufficient for this separation, we estimated an upper bound, $\alpha_{eff}$ (see Fig. S3 and the S.I.). The lowest $\alpha_{eff}$ values for each film are listed in Table 1. The $\alpha_{eff}$ is larger for thinner films, which could be due to a higher dissipation, as is often observed in other magnetic films (e.g. ref. 26). It may also be due to a higher inhomogeneity of the magnetic properties, as the properties of thinner films are more affected by variations of the thickness or of the interfaces. Overall, $\alpha_{eff}$ as low as 0.02 were observed, lower than reported measurements in bulk $Fe_5GeTe_2$.²¹

This low $\alpha$ is noteworthy, as metallic systems containing heavy elements typically exhibit large damping. This behavior is commonly interpreted using the breathing Fermi surface model,²⁷ which predicts that $\alpha$ scales quadratically with the spin-orbit coupling (SOC) strength, linearly with the density of states (DoS) at the Fermi level, and with the electron relaxation time $\tau$ (itself proportional to the conductivity). The DoS of $Fe_5GeTe_2$ is reported to be comparable to that of iron.²⁸ Conversely, its low conductivity suggests a short $\tau$. In addition, its weak magnetocrys-talline anisotropy points to a modest SOC. Indeed, Te primarily bonds to the weakly magnetic Fe1 sublattice,²⁰ limiting its contribution to damping. These factors may explain the observed $\alpha$, but they also highlight the need for an *ab initio* study of damping in these materials. Moreover, such low magnetic dissipation facilitates the motion of spin textures, such as skyrmions, further adding to the interest of $Fe_5GeTe_2$ as a material for Spintronics.

## DMI

The Stokes and anti-Stokes peaks in the BLS spectrum (Fig. 3a) correspond to cycloidal spinwaves of opposite rotation sense (Fig. 3b), and are subject to an opposite energy contributions from the DMI. In the limit of thin films ($kt \ll 1$, where $k$ is the wave vector of the spinwave determined by
<figure><img src="image_4.png" /><figcaption>Figure 3: a) A BLS spectrum of the 12 nm film ($\phi$ = 45°, 150 K, B = 130 mT). The green lines show the Lorentzian fit (fitted centres are written above the peaks). The central peak corresponds to non-scattered light and shows the instrumental width. b) Schematic of the BLS geometry. c) Stokes/anti-Stokes frequency difference, $\Delta f$, versus wave vector $k$ for the three films (points; taken at B = 211, 250, and 130 mT, respectively.) The linear fits (lines) are used to extract $v_{DMI}$ and D. d) Extracted D versus sample thickness t (points) and fits assuming a uniform bulk origin (constant blue line) and an interfacial origin ($\propto 1/t$, green line).</figcaption></figure>

the incidence angle of the light), as is the case here, the difference in frequency between the Stokes and anti-Stokes peaks ($\Delta f \equiv |f_{\text{anti-Stokes}}| - |f_{\text{Stokes}}|$) can be fully attributed to this effect, and can be used to determine the DMI parameter D:¹⁶,²⁹

$$ \Delta f = \frac{2\gamma}{\pi M_s} Dk, \qquad (1) $$

where $\gamma = g\mu_B/\hbar$ is the gyromagnetic ratio (and $\mu_B$ and $\hbar$ are the Bohr's magneton and the reduced Planck constant, respectively.) As there may be a small constant $\Delta f$ due to an imperfect alignment of the spectrometer, several spectra were measured and the slope of $\Delta f(k)$ (which we dub $v_{DMI}$) was determined. The expected reversal of the sign of $\Delta f$ with a reversal of the field direction was also verified. Fig. 3c shows the measured $\Delta f(k)$ for the 3, 8 and 12 nm films at 150 K. We observe that there is clearly a linear slope and thus a finite D in all three films. All three show a slope, and thus a DMI, of the same sign. With the geometry of our setup ($k_{\text{Stokes}} \times \mathbf{B}$ pointing into the film surface, Fig. 3b), this positive slope corresponds to favoured right-handed spinwaves, and thus a
positive DMI parameter. The calculated *D* parameters (eq. 1) are shown in Tab. 1 for the three films. As a check, the DMI of the 8-nm film was also measured at 100 K and 300 K (Fig. S4 in S.I.), yielding a similar value at 100 K and no detectable DMI at 300 K, as expected near *T*<sub>C</sub>.

Discussion

The measured DMI (0.02 to 0.05 mJ/m²) is comparable to values in many spintronic systems, though lower than the strongest known effects (e.g., ~0.2 mJ/m² for Co/Pt at 8 nm thickness¹⁶). Micromagnetic simulations³⁰ (see S.I.) show that this DMI suffices to induce chiral magnetic textures if the anisotropy is tuned to favour out-of-plane magnetisation.

The observation of DMI that favours cycloidal textures requires broken inversion symmetry along the *c*-axis, either in the bulk of the material or due to asymmetric interfaces. This symmetry breaking must persist over a large lateral scale, as BLS measurements average over the 30-µm laser spot and are consistent between points several 10s of µm apart. The dependence of *D* on film thickness *t* can help distinguish between bulk and interfacial origins: a bulk contribution would result in a constant *D*, whereas an interfacial contribution should scale as *D* ∝ 1/*t*.¹⁶ Figure 3d shows *D*(*t*) alongside best-fit curves for both scenarios. Clearly, *D*(*t*) deviates from the 1/*t* trend expected for interfacial DMI, in alignment with the rather electronically similar interfaces of the films (Al<sub>2</sub>O<sub>3</sub>/Fe<sub>5</sub>GeTe<sub>2</sub> and Fe<sub>5</sub>GeTe<sub>2</sub>/AlO<sub>x</sub>). Instead, the *D*(*t*) data is more compatible with the constant behaviour of a bulk origin, with a *D* = 0.04 to 0.05 mJ/m², as measured in the thickest films. The weaker DMI in the 3-nm-thick film, which correlates with lower *M*<sub>s</sub> and *T*<sub>c</sub>, can be attributed to weaker ferromagnetism in the ultrathin regime.

Given these observations, the symmetry-breaking mechanism extends both laterally over large
scales and through the film thickness (up to 12 nm). The consistent DMI sign observed in all
three films indicates a robust and reproducible effect, likely associated with the growth process,
which is inherently asymmetric. We now discuss some of the possible mechanisms. The ideal
structure of Fe<sub>5</sub>GeTe<sub>2</sub> is a rhombohedral stacking of centrosymmetric monolayers in the nonpolar
ABCABC sequence (space group R3̄m). Disruptions to this sequence by stacking faults could locally break inversion symmetry.³¹ Indeed, cross-sectional STEM images reveal a high density of stacking faults in our films,²⁰ consistent with observations in bulk crystals that were slowly cooled to room temperature after growth.³² However, we did not identify any specific stacking sequence that would lead to a net nonzero DMI across the film thickness or a consistent DMI sign across different samples.

Chakraborty et al.¹⁵ attributed a possible bulk DMI in Fe₃GeTe₂ to intercalated Fe in the vdW gaps and Fe vacancies. In our films, however, intercalated Fe was not detected²⁰ (see Fig. S1 in S.I.); it would also raise the Curie temperature,¹⁴ which was not observed. Furthermore, Rutherford backscattering (RBS) analysis show a stoichiometric Fe content (Fe:Ge = 5.00±0.08),²⁰ indicating at most a very low concentration of Fe vacancies.

A second class of mechanisms involves symmetry breaking within the Fe₅GeTe₂ vdW layers. This is supported by our RHEED observations of a √3 × √3 superstructure, which cannot derive from a specific vdW stacking. A non-centrosymmetric phase with a √3 × √3 superstructure was reported by Gopi et al.,¹¹ who ascribed it to a pattern of 1/3 Te vacancies ordered along the perpendicular direction. Since this phase represented only a small fraction (< 10⁻²) of the studied bulk crystals, it was deemed to have a negligible effect on DMI. While this phase might be more prevalent in epitaxial films, RBS showed no significant Te depletion in our films²⁰ (Te:Ge = 2.0 ± 0.08 for the 12-nm-thick film), suggesting that this mechanism is unlikely to explain our results.

Fe₅GeTe₂ is expected to have two partially-filled Fe sites (“split sites”) at different vertical positions within each monolayer, “D” and “U” (Fig. 1a). Several groups interpreted the √3 × √3 superstructure as the ordering of Fe split sites.¹,¹³ In the centrosymmetric structure, these sites are equally and randomly populated. However, *ab initio* calculations showed that the “UDD” and “UUD” configurations are energetically more favorable.³³ While our RHEED measurements cannot determine whether this ordering extends beyond the top layer, electron microscopy analyses in Ref. 1 showed a similar short-range order in the bulk of Fe₅GeTe₂. Three groups have calculated the DMI assuming a perfectly ordered “UUU” arrangement in a monolayer, using different
computational methods.⁵,³⁴,³⁵ Their predicted $|D|$ values vary significantly (0.76, 0.6, and 0.06 mJ/m²), despite all employing commonly used techniques. Our measured $|D|$ (0.04 – 0.05 mJ/m²) is comparable to the lowest theoretical estimate, and much lower than the highest prediction. This quantitative difference may be due to several reasons. First, the calculations do not account for temperature, which reduces the macroscopic *D*. Second, our films are not ordered in the “UUU” configuration since this would be inconsistent with the observed $\sqrt{3} \times \sqrt{3}$ superstructure. According to Ghosh et al.,³⁵ the DMI of the “UUD” or “UDD” configurations is only slightly lower than that of the “UUU” configuration ($|D| = 0.05$ instead of 0.06 mJ/m²), contrary to the naïve expectation of a 2/3 reduction. Finally, by symmetry, the “UUD” and “UDD” configurations contribute equally but with opposite sign DMI. While one of the two must be dominant to account for the observed DMI, the other may be present in a smaller amount and partially compensate the net DMI.

Assuming that the DMI arises from an ordering of the Fe split sites, the sign of *D* can be used to infer the dominant split-site configuration. Both refs. 34 and 5 found that a fully-ordered “UUU” configuration favours counterclockwise spinwaves, corresponding to a negative DMI. Therefore, the observed positive DMI suggests that the predominant ordering in our films is “UDD.”

## Conclusion

In conclusion, we have used BLS to quantitatively measure a significant DMI favouring cycloidal chiral textures in several epitaxial Fe<sub>5</sub>GeTe<sub>2</sub> films with nominally symmetric interfaces. The DMI values were consistent across films of different thicknesses and exhibited the same sign, indicating a bulk origin for the DMI. Although our measurements cannot reveal the microscopic origin of the DMI, they are consistent with a partial ordering of the Fe split sites during film growth, a mechanism supported by published *ab initio* calculations showing that such ordering induces DMI.

These results confirm that Fe<sub>5</sub>GeTe<sub>2</sub> can host a significant DMI and suggest that the reported magnetic textures in Fe<sub>5</sub>GeTe<sub>2</sub> are likely chiral in nature. If the mechanism for the non-centrosymmetric
structure is indeed induced during the growth as we suggest, the DMI may be enhanced by adjust-
ing the growth conditions or by employing chemical substitutions which preferentially affect the
split-site sublattice, such as with Co³⁶ or Ni.³⁷ Additionally, DMI might be stronger in Fe₅GeTe₂
films synthesized or processed using other techniques. The bulk DMI mechanism observed here
may coexist with interfacial contributions, such as those involving oxide layers as considered in
previous studies, offering opportunities to further enhance the DMI.

With its significant bulk DMI, low magnetic dissipation, and the potential to engineer its inter-faces, Fe₅GeTe₂ emerges as a promising material for exploring the dynamics of chiral magnetic textures and enabling spintronic applications.

**Supporting Information Available**

The supporting information (S.I.) includes more details and figures relating to the experimental methods, the structural analysis of the films, the fitting and analysis of the BLS spectra, additional BLS measurements, and micromagnetic simulations of the effects of the measured DMI.

**Acknowledgement**

The authors thank V. Jeudy for his support with cryogenic measurements and H. Okuno for the
STEM image of Fig. S1. Magnetometry measurements were kindly performed by E. Rivière at
ICMMO (CNRS) and at the Physical Measurements Platform at the LPS. This work was sup-
ported by the French National Research Agency (ANR) through the projects ELMAX (ANR-20-
CE24-0015), ESR/EQUIPEX+ 2D-MAG (ANR-21-ESRE0025), NEXT (ANR-23-CE09-0034),
the FLAG-ERA grant MNEMOSYN (ANR-21-GRF1-0005-01), the LANEF framework for mu-
tualised infrastructure (ANR-10-LABX-51-01), and the France 2030 government investment plan
(PEPR SPIN: SPINMAT ANR-22-EXSP-0007 and CHIREX ANR-22-EXSP-0002).
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# Dzyaloshinskii-Moriya interaction in Fe<sub>5</sub>GeTe<sub>2</sub> epitaxial thin films

João Sampaio *,† Antoine Pascaud †, Edgar Quero †, André Thiaville †, Vincent Polewczyk ‡,
Alain Marty ‡, Frédéric Bonell ‡, Alexandra Mougin †

†Université Paris-Saclay, CNRS, Laboratoire de Physique des Solides, 91405 Orsay, France

‡University Grenoble Alpes, CNRS, CEA, IRIG-Spintec, F-38000 Grenoble, France

* joao.sampaio@cnrs.fr

## Structural analysis

An in-depth analysis of the structure of these films is reported in the reference 20.

Figure S1 a-c shows RHEED measurements taken during the growth process. It can be seen that the Fe<sub>5</sub>GeTe<sub>2</sub> lattice parameter is fully relaxed to the bulk value from the first monolayer, indicating van der Waals epitaxy and negligible strain.

Figure S1d shows a high-angle annular dark-field (HAADF) cross-section image of the 12 nm-thick film in the [100] direction. No signs of intercalated Fe can be seen.

## Shape anisotropy of films with depth-varying magnetisation

Assuming a homogeneous $M_s$ when it actually varies along the film thickness, $M_s(z)$, results in an underestimation of the shape anisotropy $K_d$. $K_d$ is the difference of dipolar energy density between the in-plane and perpendicular magnetisation states. The average dipolar energy density is $-\langle \frac{\mu_0}{2} \mathbf{H}_d(z) \cdot \mathbf{M}(z) \rangle$, where $\mathbf{H}_d(z)$ is the local dipolar field and $\langle \cdots \rangle$ denotes an average along the thickness ($\frac{1}{t} \int_0^t \cdots dz$). In the in-plane state, $\mathbf{H}_d = 0$, while in the perpendicular state $\mathbf{H}_d(z) =$
<figure><img src="image_5.png" /><figcaption>Figure S1: RHEED measurements during the growth and STEM analysis of the Fe₅GeTe₂ films. (a) Intensity profiles in the (100) direction for different thicknesses and after post-growth annealing. (b) RHEED pattern in the (100) direction illustrating the intensity profile across (10) reflections. (c) Measured in-plane lattice parameter from profiles in (a), assuming a = 4.758 Å for sapphire. The Fe₅GeTe₂ lattice parameter is fully relaxed to the bulk value from the first monolayer, indicating vdW epitaxy and negligible strain. (d) High-angle annular dark-field (HAADF) cross-section image of the 12 nm-thick film in the [100] direction.</figcaption></figure>

$-M_s(z)\hat{z}$. This leads to

$$K_d = -\frac{\mu_0}{2} \langle M_s^2(z) \rangle.$$

Since it is always the case that $\langle M_s^2(z) \rangle \ge \langle M_s(z) \rangle^2$, $K_d$ is always larger than the value calculated using the average magnetisation: $|K_d| \ge \frac{\mu_0}{2} \langle M_s(z) \rangle^2$ (where $\langle M_s(z) \rangle$ is the magnetisation given by macroscopic magnetometry measurements.)

## Determination of magnetic parameters from BLS spectroscopy

BLS spectroscopy was used to analyse the spinwave frequency as a function of field or wavevector. A laser beam ($\lambda = 532$ nm) is directed onto the sample at an angle $\phi$ (Fig. 3b), with the sample placed inside a cryostat. A magnetic field is applied using permanent magnets to align the equilibrium magnetization in the film plane and perpendicular to the incident light, following the Damon-Eshbach geometry. The backscattered light is analysed with a double, three-pass Fabry-Perot spectrometer (TFP2-HC, JRS Instruments). Some of the light is inellastically backscattered by spinwaves, with a wave vector $k = 4\pi \sin\phi/\lambda$ defined by the incidence angle. The re-
sulting spectrum exhibits a redshifted (Stokes) peak from forward-propagating spin waves and a blueshifted (anti-Stokes) peak from backward-propagating ones. The incident and analysed light are cross-polarised, which suppresses most peaks from phonon scattering. The confinement of the Damon–Eshbach mode to the film interfaces restricts this method to sufficiently thin samples (thinner than the laser wavelength). In this regime, BLS has become a widely used technique for the quantitative determination of DMI, as well as of other magnetic parameters. Peak parameters — average frequency ($f_0$), frequency difference ($\Delta f$), and linewidth ($\Gamma$) — are extracted via Lorentzian fits. The variation of these frequencies with field or wavevector allows the extraction of static ($D, B_K, M_s, ...$) and dynamic ($g$ and $\alpha$) magnetic parameters.

The effective anisotropy field $B_K$ and g-factor $g$ are extracted from $f_0(B)$ by using the law of dispersion of magnetostatic spinwaves determined by Kalinikos & Slavin:¹⁶,²³

$$f_0 = \frac{\gamma}{2\pi} \sqrt{B + \mu_0 M_s P + \frac{2A}{M_s} k^2} \sqrt{(B - B_K) - \mu_0 M_s P + \frac{2A}{M_s} k^2}, \quad (2)$$

where $P \approx kt/2$ for thin films ($kt \ll 1$) and $A$ is the exchange stiffness (neglected in our analysis). Fig. S2a shows BLS spectra of the 8-nm sample taken at different $B$. All spectra clearly show the Stokes and anti-Stokes peaks, which are of different amplitude, typical of magnon BLS (but not of phonons). $B_K$ and $g$ parameters were extracted from fits of $f_0(B)$ (lines in Fig. S2b) and are shown in Tab. 1 for all three films at 150 K.

## Determination of the Gilbert dissipation parameter

The Gilbert dissipation parameter $\alpha$ can be obtained from the peak width $\Gamma$. The main contributions to $\Gamma$ are the instrumental resolution ($\Gamma_0$), the magnetic dissipation, and the inhomogeneity of the magnetic properties such as the anisotropy ($\delta B$):

$$\Gamma = \Gamma_0 + \frac{\gamma}{2\pi} \alpha(2B - B_K) + \gamma \delta B F(B, B_K), \quad (3)$$
<figure><img src="image_6.png" /><figcaption>Figure S2: a) BLS spectra at different applied fields (8 nm film, 150 K, $\phi = 30^{\circ}$). The instrumental linewidth is evaluated from the elastic peak ($f = 0$). b) Spin-wave frequency versus applied field for the three film thicknesses (150 k) ($\phi = 20^{\circ}$ for the 3 and 12 nm films, $30^{\circ}$ for the 8 nm). The lines are fits using the model in ref. 23.</figcaption></figure>

where $F()$ is a function that tends towards a constant for $B \gg B_K$ (see SI of ref. 24; a $2\pi$ factor is missing in the definitions of $f_x$ and $f_z$ in that reference.)

$\Gamma_0$ is measured from the elastic peak at $f = 0$ (seen in Fig. S2a). $\delta B$ may be disentangled from the dissipation by applying a large field such that $\alpha(2B - B_K) \gg \delta B F(B, B_K)$, however this was not possible with our setup. An upper bound of the dissipation parameter, $\alpha_{eff}$, can be determined by neglecting $\delta B$:

$$ \alpha_{eff} = \frac{2\pi \frac{\Gamma - \Gamma_0}{\gamma}}{2B - B_K} \quad (4) $$

The measured $\Gamma$ vs. $B$, along with the calculated $\alpha_{eff}$, is shown in Fig. S3, and the minimum values of $\alpha_{eff}$ are included in Tab. 1.

## Additional BLS measurements at different temperatures

Some additional measurements of $f(k)$ were performed at 100 K and near $T_C$ (300 K) for the 8 nm film, and the Stokes/anti-Stokes frequency difference $\Delta f(k)$ is shown in Fig. S4. At 100 K,
<figure><img src="image_7.png" /></figure>

Figure S3: Peak FWHM $\Gamma$ versus field $B$ for the three films. The grey lines are the expected variation assuming different values of $\alpha_{\text{eff}}$ (see text for details on the calculation of $\alpha_{\text{eff}}$).

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

Figure S4: Stokes/anti-Stokes frequency difference, $\Delta f$, versus wave vector $k$ for the 8 nm film at 100 and 300 K, showing a finite slope (and thus DMI) at 100 K. At 300 K, the slope is negligible (the fit p-values, i.e. the probability of a zero slope, is 0.85 and 0.29).

a finite $v_{\text{DMI}} = \Delta f/k$ of similar magnitude to that at 150 K is observed ($17 \pm 3$ m/s compared to $20.5 \pm 1.5$ m/s). At 300 K, two series of $f(k)$ with opposite applied field were measured. If there were DMI, the two series should present slopes of opposite sign. Instead, the fits are compatible with no slope ($v_{\text{DMI}} = 1 \pm 5$ and $10 \pm 7$ m/s, with the statistical tests of the fits giving a high probability for no slope). This is expected, as, like the conventional exchange interaction, the DMI energy should scale with $M_S^2$ (and $v_{\text{DMI}}$ with $M_S$), which drops sharply near $T_C$ but varies little between 100 K and 150 K.
Micromagnetic simulations of stabilised chiral textures

To evaluate whether the observed DMI magnitude is sufficient to stabilise chiral spin textures, we
performed micromagnetic simulations³⁰ of the domain wall structure for varying values of D, as-
suming a small induced perpendicular anisotropy (200 J/m³) for the parameters of the three films.
Such a modest anisotropy could be through interfacial effects (a common approach in thin-film
systems) or by electric field bias. The simulations, shown in Figure S5, reproduce the behaviour
described by Thiaville et al.:¹⁷ a continuous transition from Bloch to chiral Néel walls as D in-
creases, up to a saturation value. In the simulations, the 3-nm-thick film exhibits fully Néel-type
DWs, while the thicker films display hybrid chiral Bloch–Néel configurations. Based on the param-
eters of the 3-nm film, we also simulated a skyrmion lattice state under various values of induced
perpendicular anisotropy and applied bias magnetic field, Fig. S6. We found that small isolated
skyrmions can be stabilised for higher effective anisotropy (> 1 kJ/m³), while larger skyrmions
appear for lower anisotropy values.

<figure><img src="image_9.png" /></figure>

Figure S5: Micromagnetic simulations of the domain wall core angle as a function of DMI strength D, showing the continuous transition from Bloch to Néel walls. Simulations assume an effective perpendicular anisotropy of 200 J/m³ and exchange stiffness A = 2 pJ/m. Insets: examples of DW profiles. Vertical lines indicate experimentally measured values of D.
<figure><img src="image_10.png" /><figcaption>Figure S6: Simulated stable spin textures for the 3-nm-thick film as a function of the induced perpendicular anisotropy *K* and applied magnetic field. Simulations were run in a 2.05 × 2.05 µm² square, starting from two positive-core skyrmions (white) in a uniform background *m* = −*z* (black). Isolated skyrmions are stabilised for *K* > 600*J/m*³; at lower *K*, the stable skyrmion diameter likely exceeds the simulation area.</figcaption></figure>