# On-Chip Ferromagnetic Resonance for van der Waals Heterostructures: Anisotropy and Damping of Cobalt Interfaced with Exfoliated 2D Materials

Karen Sobnath, Roberto Bellelli, Mehrdad Rahimi, Linsai Chen, Guillaume

Wang, Roméo Bonnet, Pascal Filloux, Martial Nicolas, Batiste Janvier,
Philippe Lafarge, et al.

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## ► To cite this version:

Karen Sobnath, Roberto Bellelli, Mehrdad Rahimi, Linsai Chen, Guillaume Wang, et al. On-Chip Ferromagnetic Resonance for van der Waals Heterostructures: Anisotropy and Damping of Cobalt Interfaced with Exfoliated 2D Materials. ACS Applied Electronic Materials, 2025, 7 (19), pp.8821-8827. 10.1021/acsaelm.5c00702. hal-05300390

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# On-chip ferromagnetic resonance for van der Waals heterostructures: anisotropy and damping of Cobalt interfaced with exfoliated 2D materials

Karen Sobnath,† Roberto Bellelli,† Mehrdad Rahimi,† Linsai Chen,† Guillaume

Wang,† Roméo Bonnet,† Pascal Filloux,† Martial Nicolas,† Batiste Janvier,†

Philippe Lafarge,† Maria Luisa Della Rocca,† Pascal Martin,‡ Clément

Barraud,† and François Mallet*,†;¶

†Université de Paris, CNRS, Laboratoire Matériaux et Phénomènes Quantiques, F-75013 Paris,
France.

‡Université de Paris, CNRS, Laboratoire ITODYS, CNRS, F-75013 Paris, France.

¶Sorbonne Université, F-75005 Paris, France.

E-mail: francois.mallet@sorbonne-universite.fr

## Abstract

Designing the next generation of ultra-low-power spintronic devices hinges on identifying suitable combinations of materials and interfaces that effectively process spin information.

Two-dimensional (2D) materials and their associated van der Waals heterostructures offer a promising platform, thanks to their ideal interfaces and exceptional tunability. However, the typical lateral dimensions of state-of-the-art exfoliated crystals are constrained to a few tens of micrometers. Addressing this limitation, we introduce a fully on-chip ferromagnetic resonance (FMR) setup optimized for these constraints. Our setup demonstrates remarkable sensitivity, capable of probing the magnetization dynamics of single cobalt (Co) patches with surfaces
below $10^2 \ \mu m^2$ and thicknesses in the 10 nm range at room temperature. Furthermore, the versatility of our prototype towards studying stacked van der Waals heterostructures spintronics is demonstrated by shedding light onto Co surface anisotropy and magnetic damping when proximitized by multilayers thick 2D materials such as graphene, hBN, and $WSe_2$.

## Keywords

Spintronics, Van der Waals heterostructures, on-chip ferromagnetic resonance, proximity effect and magnetic anisotropy, Gilbert damping and two magnon scattering

## Introduction

Since their discovery in 2004,¹ two-dimensional (2D) materials have provided an extraordinary platform for solid-state physics, enabling advancements across diverse domains such as flexible electronics,² optoelectronics,³ quantum electronics⁴,⁵ and spintronics.⁶,⁷ A wide range of conventional (metallic, semiconducting, insulating) to more exotic electronic phases (Mott insulating, superconducting, magnetic, etc.) is spanned depending on the elemental composition of the 2D materials and related van der Waals heterostructures,⁸,⁹ on the number of layers¹⁰ and on the twist angle between the layers.¹¹,¹² From a spintronics perspective, graphene stands out as an ideal medium for spin propagation. Its weak spin-orbit coupling (SOC) and hyperfine interactions, derived from the $\pi$-delocalized orbitals, coupled with its exceptionally high electronic Fermi velocity, enable spin diffusion lengths exceeding several tens of $\mu m$, as experimentally demonstrated.¹³⁻¹⁵ For spin-charge interconversion, 2D materials with stronger SOC, such as transition metal dichalcogenides (TMDs),¹⁶,¹⁷ and even topological insulators¹⁸ offer substantial promise. Furthermore, the recent experimental realization of 2D magnets¹⁹ has unlocked exceptional opportunities to explore, manipulate, and utilize their magnetic properties.²⁰⁻²²

Consequently, integrating 2D materials into spintronic devices remains a vibrant area of research,²³⁻²⁵ for computational platforms,²⁶ spin sources,²⁷ or spin detectors.²⁸ The intrinsic quality
of the materials and their interfaces is critical to achieving optimal performance. Accordingly, cur-
rent experimental efforts to fully investigate the physics of van der Waals heterostructures predom-
inantly rely on the pick-up/drop-down techniques<sup>29</sup> of mechanically exfoliated flakes.<sup>30</sup> However,
this approach imposes significant constraints on device size, typically limiting it to a few tens of
micrometers. As a result, while traditional broadband FMR measurements have historically been
considered challenging to implement on ultra-thin 2D materials and their heterostructures due to
sensitivity limitations, spin-torque ferromagnetic resonance<sup>31</sup> (ST-FMR), a related technique, has
been used to explore such nanoscale devices.<sup>32,33</sup> However, these two techniques are not strictly
equivalent. Unlike conventional FMR, which probes the magnetization dynamics via inductive
coupling and microwave absorption methods, ST-FMR leverages the direct electrical detection
through spin-orbit torques generated by spin Hall effect. Thus, it necessitates to interface the FM
under study with an auxiliary material with strong SOC in order to generate the spin current re-
quired to trig the magnetization dynamics. However, it is well established, and actually this article
will illustrate it, that proximity effects can strongly influence the magnetization dynamics of a fer-
romagnet.<sup>34,35</sup> As a result, ST-FMR, unlike FMR, does not give access to the intrinsic properties
of the ferromagnet alone, but rather to those of the entire HM/FM heterostructure.

Experiment

In this letter, we present an innovative on-chip design, illustrated in Fig.1, which challenges these
commonly accepted limitations and enables broadband ferromagnetic resonance (FMR) detection
for single cobalt (Co) patches with surface dimensions as small as 10² µm² and thicknesses of
10 nm, while, as shown in Fig.1 (c), maintaining easy galvanic accesses to the structure to go be-
yond simple material characterization and allow spintronics devices implementation. We further
demonstrate the versatility of this design and its compatibility with stacked van der Waals het-
erostructures by growing Co patches on various commonly utilized 2D material stacks, including
few-nanometer-thick hBN, hBN/Gr(aphene), and hBN/WSe₂. Furthermore this approach provides
new experimental insights into the interplay between growth, interfaces, surface anisotropy and the
magnetic damping dynamics of nm-thick cobalt, a key material in spintronics industry.

<figure><img src="image_3.png" /><figcaption>Figure 1: (a) Colorized titled SEM image (color coding specified in (b)) of an on-chip FMR device. The Co patch, measuring 6 µm × 10 µm, is deposited on a van der Waals heterostructure comprising 10-nm-thick graphene (Gr) and 10-nm-thick hBN flakes. The Co magnetization undergoes precession induced by inductive coupling with the Oersted-like rotating microwave magnetic field $\vec{h}_{RF}$, generated by the microstrip transmission line energized at microwave frequencies. (b) (c) Left: (yellow) Layout of the RF microstrip line and (grey) Layout of additional voltage and current lines illustrating the versatility of our on-chip microstrip design to implement beyond FMR spintronics experiments. The RF microstrip antenna has a nominal width of 220 µm ensures wideband impedance matching. Right: The microstrip width is reduced to 10 µm at the 2D/Co heterostructure location to enhance $\vec{h}_{RF}$.</figcaption></figure>

All samples were fabricated on highly resistive intrinsic Si wafers covered with a 280 nm thick
layer of thermal SiO2. As shown in Fig.1 (b), fabrication begins with an on-chip RF antenna made
of a 50 nm thick Au track patterned by e-beam lithography and evaporated in a high vacuum (base
pressure below 10⁻⁶ mbar). The actual design is of a microstrip transmission line, as presented in
Fig.1 (a). It has been chosen over a coplanar waveguide for its ability to forego extensive lateral ground planes, thus facilitating easier access to the heterostructures for additional DC lines, as shown in gray in Fig.1 (c). The heterostructure under investigation is positioned directly atop the microstrip, maximizing inductive coupling. This coupling is further enhanced by narrowing the strip width from 220 µm (nominal width for 50 Ω impedance matching) to 10 µm. Supported by electromagnetic simulation, Maxwell-Ampère's law indicates that RF magnetic field $\mu_0 \vec{h}_{RF}$ amplitude of the order of 1 mT can be achieved at the maximum power of 18 dBm from the microwave source. Additionally, the narrow strip width allows for the use of single hBN flakes exfoliated from pristine commercial crystals as dielectric spacers, providing galvanic isolation from the microwave antenna while maintaining a smooth surface with minimal added roughness. These flakes are deposited on the RF line via a hot pick-up technique employing PDMS drops coated with a PPC film.³⁶ Depending on the sample requirements, few-layer graphene or $WSe_2$ flakes (with typical thicknesses ranging from 5 to 20 nm) are deposited in a subsequent hot pick-up step. Finally, e-beam lithography and metallic thin-film deposition via evaporation are employed to fabricate the single Co patches (6 µm × 10 µm, variable thickness), which are capped with 4 nm of Au to prevent oxidation.

Interface's roughness and quality are critical parameters to control in spintronics. Surface topography characterization was performed using AFM, as illustrated in Fig.2 (a) and (b). Analysis revealed a slight but consistent increase in the root-mean-squared (RMS) roughness as a function of Co thickness $t_{Co}$ and the number of stacking layers. Measured RMS roughness values are reported in Table 1. The hBN deposited on the Au microstrip line exhibits a typical RMS value of 0.6 nm, Table 1: Measured AFM RMS surface roughness in nm.¹ is before Co deposit, ² is after Co deposit.

<table><thead><tr><th>Roughness (nm)</th><th>hBN<sup>1</sup></th><th>hBN/Co<sup>2</sup></th><th>hBN/WSe<sub>2</sub><sup>1</sup></th><th>hBN/WSe<sub>2</sub>/Co<sup>2</sup></th></tr></thead><tbody><tr><td>Co(8)</td><td>0.55</td><td>0.62</td><td>0.49</td><td>1.0</td></tr><tr><td>Co(15)</td><td>0.59</td><td>0.69</td><td>1.2</td><td>1.9</td></tr><tr><td>Co(30)</td><td>0.63</td><td>0.78</td><td>1.3</td><td>1.8</td></tr></tbody></table>

similar to that initially observed on the bare Si substrate. On top of it, the Co roughness increases
from 0.6 to 0.8 nm as $t_{Co}$ increases from 8 to 30 nm. Interestingly, WSe$_2$ atop hBN displays higher RMS roughness values, ranging from 1.0 to 1.5 nm prior to Co deposition, which correlates to a denser distribution of bubbles, as shown by comparing Fig.2 (a) and (b). To investigate the quality of the interfaces, several devices were examined post-measurement by cutting them with FIB at the IEMN Lab to perform TEM slice observations using a JEOL ARM 200F TEM. A representative image is displayed in Fig.2 (c). The atomic planes of the 2D materials are clearly visible, confirming the monocrystalline nature of the 2D materials. In contrast, evaporated metals like Co appears distinctly polycrystalline. TEM analysis further confirms that the 2D materials/Co interfaces are

<figure><img src="image_4.png" /><figcaption>Figure 2: (a) (resp. (b)) 2 µm × 2 µm AFM image of the Co surface for the hBN/Co(8) (respectively hBN/WSe$_2$Co(8)) heterostructure. The estimated RMS roughness from these scans are reported in table 1. c) TEM scan performed on a JEOL ARM 200F TEM of an hBN/WSe$_2$/Co(13) heterostructure.</figcaption></figure>

clean, with no evidence of chemical contamination such as resist residues or interdiffusion between
Co and the 2D materials.

In-plane field swept ferromagnetic resonance (FMR) spectra has been recorded at room temperature across RF frequencies *f* ranging from 4 to 14 GHz and external DC magnetic fields $\mu_0H_{ext}$ ranging from ±0.2 T. A small modulation (≈17 Hz) of a few mT is superimposed for lock-in detection. The total microwave power output is rectified by a diode, and the in-phase component of the resulting voltage is plotted as a function of *f* and $H_{ext}$ for each sample. Fig.3 (a) provides
an example of this data for a hBN/WSe₂/Co(13) heterostructure, where the number in parentheses refers to the Co thickness $t_{Co}$ in nanometers. All thicknesses were determined post-fabrication via AFM, with a typical uncertainty of ±0.5 nm.

A cut at $f = 14$ GHz from the FMR data trace in Fig.3 (a) is shown in Fig.3 (b), alongside cuts at the same frequency for other heterostructures: hBN/Gr/Co(16) and hBN/Co(16). Two key observations can be made from these results. First, our fully on-chip setup resolves FMR spectra for a single Co patch with dimensions 10 µm × 6 µm × 13 nm, corresponding to a magnetic volume of approximately a µm³. This demonstrates our capability to detect the dynamics of magnetic moments as small as $10^{-9}$ emu (around $10^{11}$ spins) at room temperature. Such a sensitivity for room temperature broadband FMR is a significant achievement and places our setup already among the more sensitive ones.³⁷ Improved sensitivity, down to $10^6$ spins,³⁸,³⁹ can be achieved at room temperature with micro resonators and cavities with optimized filling factor, but thus comes at the expense of renouncing to the broadband character.

Second, remarkably, the proximity of Gr and WSe₂ exhibits opposing effects on the Co FMR line-width compared to hBN: it is broadened for the former but reduced for the latter. Such FMR line-width reduction due to proximity effects has previously been observed for Co only interfaced with a conventional 3D metals such as Ti,⁴⁰ this is the first reported instance involving any 2D material.

Due to the field modulation and synchronous detection, the measured lock-in voltage is directly proportional to the field-derivative of the absorbed microwave power evaluated at $H_{ext}$, which writes as:

$$ \left(\frac{dP}{dH}\right)_{H_{ext}} = A_{sym} \frac{(\delta H / \Delta H)}{\left(1 + (\delta H / \Delta H)^2\right)^2} + A_{asym} \frac{1 - (\delta H / \Delta H)^2}{\left(1 + (\delta H / \Delta H)^2\right)^2} \quad (1) $$
<figure><img src="image_5.png" /></figure>

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

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

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

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

<figure><img src="image_10.png" /><figcaption>Figure 3: Magnetization dynamics in hybrid "2D materials"/Co heterostructures. (a) FMR data map of the derivative of the absorbed microwave power ($dP/dH$) (in arbitrary units) as a function of the applied magnetic field $H_{ext}$ and the excitation microwave frequency $f$ for the hBN/WSe$_2$/Co(13) heterostructure. (b) Square symbol scatter plot (purple) is a cut at $f=14$ GHz of the previous map centred around the positive field resonance $H_{res}$. Also reported here, at the same $f=14$ GHz, are similar cuts of FMR map but resp. with in triangle (green) a hBN/Gr/Co(16) heterostructure and with circle (blue) a hBN/Co(16) heterostructure. Superposed continuous lines are fits to eq. 1. (c) (resp. (d)) Extracted $\mu_0H_{ext}$ (resp. $\mu_0\Delta H$) as a function of $f$. Continuous lines are fits to eq.2 (resp. eq.4). (e) Effective magnetization $\mu_0M_{eff}$ as a function of $t_{Co}^{-1}$ with their linear fit from eq.3. (f) Total Gilbert damping as a function of $t_{Co}$ in the main panel and $t_{Co}^{-1}$ in the inset. The different continuous lines are fits whose parameters are explained in the main text.</figcaption></figure>
here $\delta H = H_{ext} - H_{res}$ is the difference between the applied field $H_{ext}$ and the FMR field $H_{res}$. The first term, with amplitude $A_{sym}$, represents the derivative of the Lorentzian FMR absorption centred at $H_{res}$ with a Half Width at Half Maximum (HWHM) $\Delta H$, while the second term, with amplitude $A_{asym}$, accounts for dispersive eddy currents effects driven by the $\vec{h}_{rf}$'s out of plane component.⁴¹

Fig.3 (c) shows $\mu_0H_{res}$ extracted by fitting Eq. (1) for the three investigated heterostructures with similar $t_{Co}$ as a function of excitation frequency $f$. The Kittel relation in the thin-film limit is given by:

$$f = \frac{\gamma\mu_0}{2\pi} \sqrt{H_{res}(H_{res} + M_{eff})} \quad (2)$$

where $\gamma$ is the gyromagnetic ratio and $M_{eff}$ is the effective magnetization. By fitting $H_{res}$ vs. $f$, as shown by the continuous lines in Fig.3 (c), $M_{eff}$ values are extracted, assuming $\gamma/2\pi = 28$ GHz/T ($g \approx 2$) for all devices. For a continuous polycrystalline film of thickness $t_{Co}$, $M_{eff}$ is related to the saturation magnetization $M_S$ and surface magnetic anisotropy $K_S$ as follows:

$$\mu_0 M_{eff} = \mu_0 M_S - 2 \frac{K_S}{M_S t_{Co}} \quad (3)$$

This expected linear dependence of $\mu_0 M_{eff}$ as a function of $t_{Co}^{-1}$ is indeed observed in Fig.3 (e), which allows to extract, for each heterostructure, the $M_S$ and $K_S$. They are summarized in Table 2. $K_S$ for hBN/Co is larger than that for Gr, consistent with previous studies.⁴²,⁴³ Surprisingly, WSe₂

<table><caption>Table 2: Co saturation magnetization $M_S$ and magnetic surface anisotropy energy $K_S$ for our different hybrid heterostructures extracted from data fitting to eq. (3). In all cases, Co has been capped with 4 nm of Au.</caption><thead><tr><th></th><th>μ<sub>0</sub>M<sub>S</sub> (T)</th><th>K<sub>S</sub> (mJ/m<sup>2</sup>)</th></tr></thead><tbody><tr><td>hBN/Co</td><td>2.4</td><td>8.6</td></tr><tr><td>hBN/Gr/Co</td><td>2.2</td><td>5.9</td></tr><tr><td>hBN/WSe<sub>2</sub>/Co</td><td>2.15</td><td>2.3</td></tr></tbody></table>

proximity causes a significant decrease in $K_S$. This result contrasts with common expectations, as
transition-metal dichalcogenides like WSe₂ typically exhibit strong spin-orbit coupling, often leading to enhanced magnetic anisotropy, as reported for MoS₂ for instance.⁴⁴ We anticipate that these findings will motivate *ab initio* theoretical investigations to further substantiate and elucidate the underlying mechanisms which explain how the proximity of WSe₂ makes the Co orbital moments at its interface more isotropic.

Similarly, the fitted FMR HWHM $\mu_0\Delta H$ as a function of $f$ is presented in Fig. 3 (d). A plateau is observed below 7 GHz, above which a linear frequency dependence becomes apparent for all heterostructures. This allows for the quantification of each heterostructure’s total dimensionless Gilbert damping constant $\alpha_{tot}$ using:

$$ \mu_0\Delta H = \mu_0\Delta H_0 + \frac{2\pi f}{\gamma}\alpha_{tot} \quad (4) $$

where $\mu_0\Delta H_0$ is the inhomogeneous linewidth, primarily attributed to static magnetic disorder at low polarization fields. Fig. 3 (f) summarizes our results for $\alpha_{tot}$ for '2D'/Co heterostructures as a function of $t_{Co}$ in the main panel and its inverse $t_{Co}^{-1}$ in the inset. First, no clear distinction can be made between the Gilbert damping behaviors of hBN/Gr/Co and hBN/Co heterostructures within the experimental uncertainties. However, when comparing both hBN and Gr interfaces to the WSe₂ case, two clearly distinct behaviours emerge. Co damping on Gr and hBN shows divergence as $t_{Co}$ decreases, whereas Co damping on WSe₂ remains largely independent of FM thickness down to below 10 nm, locked at a value lower than 0.01. Such a small value indicates the excellent quality of our Co evaporated on 2D materials stacks.

In ferromagnetic/normal metal (FM/NM) heterostructures, an increase in FM magnetization damping is typically attributed to spin pumping, where a pure spin current is generated in the NM as angular momentum is transferred via exchange interaction from the precessing FM spins.⁴⁵ Another dissipation mechanism is two-magnon scattering, where uniform FMR magnons scatter off material imperfections into degenerate non-uniform short-wavelength magnons.⁴⁶ In broad-
band transmission line FMR experiments on electrically conductive FMs, two additional damping mechanisms might contribute: radiative damping $\alpha_{rad}$, caused by inductive coupling between the sample and the waveguide, and eddy current damping $\alpha_{eddy}$, induced by magnetization changes op-posing eddy currents.⁴⁷ However, $\alpha_{rad}$ and $\alpha_{eddy}$ are negligible in our devices, evaluated at values below $10^{-3}$ and $10^{-5}$, respectively.

Thus, total Gilbert damping $\alpha_{tot}$ in our devices is given as:⁴⁸

$$ \alpha_{tot} = \alpha_{int} + \beta_{SP} t_{Co}^{-1} + \gamma_{TMS} t_{Co}^{-2} \quad (5) $$

where $\beta_{SP}$ and $\gamma_{TMS}$ account for damping due to spin pumping and to two-magnon scattering, respectively. More specifically, $\beta_{SP} = g_{eff} g\mu_B/\mu_0 M_s$ where $g_{eff}$ is the effective spin-mixing conductance which quantifies the amount of pure spin current potentially generated in the NM,⁴⁹ while the TMS coefficient, $\gamma_{TMS}$, depends on the factor $(K_S/M_S)^2$ and on the density of magnetic defects on the FM surfaces.⁵⁰ It is well established that the TMS contribution to FMR damping is highly sensitive to surface anisotropy. In thin ferromagnetic films, structural imperfections —such as surface roughness, interfacial defects, or spatial inhomogeneities in surface anisotropy— break the in-plane translational symmetry, thereby providing the necessary momentum transfer for scattering from the uniform FMR mode into non-uniform spin wave modes. Enhanced surface anisotropy further amplifies these spatial fluctuations at the film boundaries, increasing both the density and effectiveness of scattering centres and thus promoting TMS-mediated damping. Moreover, it is also known that this TMS contribution to the FMR line-width is generally not linear in frequency,⁵¹⁻⁵³ which is consistent with a careful observation our data in Fig.3 (d). First, this observation indicates that the damping is dominated by the extrinsic TMS contribution. Second, it should also be noted that we observe a difference in the bulk contribution to the FMR relaxation of Co between the WSe₂ and hBN/Gr interfaces: their respective $\alpha$ do not converge in the large $t_{Co}$ limit as seen in Fig.3 (f). It is likely attributable also to the known non-linear frequency dependence of the
TMS-induced line-width broadening which renders our analysis based on eq. (4) and eq. (5) only approximate.

Assuming negligible spin pumping contributions for hBN and WSe₂ interfaces ($\beta_{SP}^{hBN} = \beta_{SP}^{WSe_2} = 0$), fits to Eq. 5 yield the continuous lines in Fig. 3 (f), with $\gamma_{TMS}^{hBN} = 1.8 \pm 0.4 \text{ nm}^2$ and $\gamma_{TMS}^{WSe_2} = 0.09 \pm 0.02 \text{ nm}^2$, reflecting a 20-fold reduction. This reduction aligns with the predicted TMS suppression due to the decreased $K_S/M_S$ ratio reported in Table 2. Reciprocally, this finding independently corroborates the proximity effect of WSe₂ on Co surface magnetic anisotropy. Applying the same scaling for hBN/Gr gives $\gamma_{TMS}^{Gr} \le 0.85 \text{ nm}^2$, whose upper limit is represented by the dashed green line in Fig. 3 (f). This model fails to explain experimental data within error bars. Adding a spin-pumping term $\beta_{SP}^{Gr} = 0.084 \text{ nm}$, corresponding to an effective spin-mixing conductance $g_{eff} \approx 100 \text{ nm}^{-2}$ for the Gr/Co interface, provides a better match to observations, as depicted by the continuous green line in Fig. 3 (f). Similar high $g_{eff}$ values have been reported for Gr/Py interfaces.⁵⁴

## Conclusion

In conclusion, this study showcases an on-chip design capable of resolving at room temperature the FMR of micrometer-square, 10 nanometer-thick ferromagnets. Thus, the ultimate monolayer limit of 2D materials and their van der Waals heterostructures necessitates just an order of magnitude sensitivity improvement, which is we are convinced within reach by a careful combined optimization of the microwave design and detection scheme. Additionally, the versatility of this design is demonstrated by comparing the FMR of 2D materials /Co heterostructures, where '2D' encompasses hBN, hBN/Gr, and hBN/WSe₂. While we could not conclusively establish spin pumping in 2D materials due to two-magnon scattering's dominance in in-plane Co FMR, we observed that Co surface anisotropy —a reference property of 3D spintronics materials— is highly sensi-
tive to the underlying 2D material on which it is grown. Notably, this study reports for the first
time that hBN/WSe₂/Co heterostructures exhibit a substantial damping reduction correlated with
a decreased Co surface anisotropy, locking the total Gilbert damping at its 3D limit of below
0.01 for Co thicknesses down to below 10 nm. These findings open avenues for advancements in
STT-MRAM⁵⁵ and opto-magnonics.⁴⁰ Furthermore, the versatility of our design opens up direct
prospects in injection/propagation/detection of spin current experiments in 2D spintronics van der
Waals heterostructures, as pioneered in.⁵⁴,⁵⁶,⁵⁷

Acknowledgement

This work is partly supported by France 2030 government investment plan managed by the French
National Research Agency under grant reference PEPR SPIN – SPINMAT ANR-22-EXSP-0007
and by the NANOFUTUR Equipex+ Program (ANR- 21-ESRE-0012) overseen by the French Na-
tional Research Agency as part of the “Programme d’Investissements d’Avenir”. We acknowledge
the use of clean-room facilities from the Groupement des Plateformes de Micro et Nanotechnolo-
gies de Paris and support from Renatech+ for micro and nanofabrication. FM thanks also the
physics UFR of Université Paris Cité for "Bonus Qualité Recherche" and the CNRS for the "INP-
TREPLIN" fundings. FM & CM acknowledges Bruno Dluback, Alexandra Mougin and Laurent
Vila and for enlighting discussions. Data acquisition has been conducted with the PyMoDAQ free
and open-source CNRS software developed by Sebastien Weber who is greatly acknowledged for
his support.⁵⁸

**Supporting Information Available**

Estimations of the amplitude of the RF magnetic field, of the heating due to microwave excitation and of the radiative and Eddy contributions to Gilbert damping.
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Supporting Information on "On-chip
ferromagnetic resonance for van der Waals
heterostructures: anisotropy and damping of
Cobalt interfaced with exfoliated 2D materials"

Karen Sobnath,† Roberto Bellelli,† Mehrdad Rahimi,† Linsai Chen,† Guillaume

Wang,† Roméo Bonnet,† Pascal Filloux,† Martial Nicolas,† Batiste Janvier,†

Philippe Lafarge,† Maria Luisa Della Rocca,† Pascal Martin,‡ Clément

Barraud,† and François Mallet*,†,¶

†Université de Paris, CNRS, Laboratoire Matériaux et Phénomènes Quantiques, F-75013 Paris,
France.

‡Université de Paris, CNRS, Laboratoire ITODYS, CNRS, F-75013 Paris, France.

¶Sorbonne Université, F-75005 Paris, France.

E-mail: francois.mallet@sorbonne-universite.fr

**Amplitude of the RF magnetic field**

We report 2D electromagnetic simulations of the magnetic field generated around the microstrip
constriction. They have been performed with the HFSS software through its 2D extractor design.
Taking into account both propagation reflection due to impedance mismatches and propagation
losses, we estimate that a current $I_{RF}$ in the 10 to 20 mA range flows through the antenna at the Co
location at the maximal output power of 18 dBm of our microwave source. The following figure
<figure><img src="image_11.png" /><figcaption>Fig. S1: 2D plot of the RF magnetic field amplitude above the microstrip constriction. Its is composed of a 10 µm wide Au line on a substrate of SiO₂(280 nm)/Si(300 µm). A ground plane made of an Au layer is added at the back of the substrate. The height reference is taken at the upper surface of the microstrip and the x axis is bounded by the microstrip edges. In this simulation, the microwave frequency is set to 10 GHz.</figcaption></figure>

Fig.S1 presents the results of these simulations for $I_{RF}=10$ mA as a function of lateral position and height above the antenna. Close to the surface, we recover an amplitude $\mu_0h_{RF}$ of the order of 0.6 mT, which consistent with the analytical estimate based on Maxwell-Ampere equation applied to an infinite sheet, which writes:

$$ \mu_0h_{RF} = \frac{\mu_0I_{RF}}{2W} \qquad (1) $$

## Heating due to microwave excitation

Given the level of RF power delivered to the chip, it is pertinent to consider the possibility of uncontrolled heating that could affect the validity of our analysis or, at least, limit the generalizability of our design and restrict its applicability in low-temperature operating conditions. Let's first bound the temperature rise of the Au antenna in the absence of any hBN nor Co layers deposited on top. Based on RF transmitted power measurements, the maximum transmission loss at our highest operating power is estimated to be approximately 20 mW. For the sake of this analysis, we make the following assumptions:

* All of this power loss is converted into Joule heating, thereby neglecting RF reflection and
radiative losses

* The only cooling channel for the antenna is through 1d thermal conduction through the underlying SiO₂/Si substrate, neglecting convective and radiative heat transfer as well as lateral heat dissipation.

* The backside of the SiO₂/Si is a reservoir maintained at the ambient temperature and the contact surface for Au/SiO₂ is 400 µm × 10 µm and the contact surface for SiO₂/Si is 400 µm × 400 µm.

Under these assumptions, the temperature rise can be estimated by calculating the total thermal resistance from the Au constriction to the heat sink. In a one-dimensional heat conduction model, the thermal resistance *R* of a material with cross-sectional area *A* and thickness *t* is given by:

$$R = \frac{t/k + G_{int}^{-1}}{A} \qquad (2)$$

where *k* is the bulk thermal conductivity and *G*<sub>int</sub> is the thermal boundary Kapitza conductance of the interface. Using tabulated values, we take *k*<sub>SiO₂</sub> = 1.4 W/m.K and *G*<sub>Au/SiO₂</sub> = 2 × 10⁷ W/K.m² on the one hand,¹ and *k*<sub>Si</sub> = 150 W/m.K and *G*<sub>SiO₂/Si</sub> = 10⁸ W/K.m² on the other hand.² Combining these contributions yields a total thermal resistance of the order of 60 K/W range for the SiO₂ and 20 K/W range for the Si. Consequently, the resulting temperature increase due to Joule heating of the Au antenna and the Co patch on top of it is evaluated below 2 K and thus negligible under our experimental conditions.

Considering operating at low temperatures, the outcome depends on the specific cooling setup and target temperature. For instance, a pulse tube cryocooler typically provides a cooling power in the Watt range at 4 K, rendering the heating effect still insignificant. At even lower temperatures, our design remains applicable since RF dissipation predominantly arises from the skin effect within the 50 nm × 10 µm cross-section of the 400 µm long Au constriction. Power loss can be easily
reduced by increasing the cross-sectional area, reducing the constriction length, or alternatively, replace the Au conductor with a superconducting material.

## Radiative and Eddy contribution to Gilbert damping

From Faraday's law, a time-varying magnetic field produces eddy currents through the waveguide. The field can also couple inductively to the magnetization precession of the adjacent magnetic layer, generating microwave currents flowing through it. These two effects contributes to the dissipation of input power away from the waveguide respectively known as radiative damping and eddy current damping.

Assuming an homogeneous magnetization over the sample, the radiative damping arising in a magnetic film of length $l$, of width $W$ and thickness $\delta$ can be written as given in$^3$ as:

$$ \alpha_{rad} = \frac{\eta\gamma M_s \mu_0^2 \delta l}{2Z_0 W} \qquad (3) $$

where $\eta$ is a dimensionless parameter associated with the spin wave mode, $\gamma$ the gyromagnetic ratio, $M_s$ the saturation magnetization, $Z_0$ the characteristic impedance of the waveguide. To estimate the radiative damping in our samples, let us consider a sample with a ferromagnetic layer thickness $\delta = 10$nm, of length $l = 10\mu m$ and of width $W = 10\mu m$ on top of an ideal lossless rf waveguide with a characteristic impedance $Z = 50\Omega$. Given that $\mu_0 M_s$ extracted experimentally from our heterostructures is in the range of $2T$, $\alpha_{rad} \sim 10^{-5}$.

The calculated $\alpha_{rad}$ corresponds to the damping coming from a ferromagnetic layer directly deposited on the waveguide. Indeed, it is important to note that the radiative damping decreases as the distance between the ferromagnet and the waveguide increases as reported in$^3$. In our samples, as the ferromagnetic layer is separated from the waveguide by the insulating layer of tens of nm thick, this should further lower the effective value of $\alpha_{rad}$.
To compute the eddy current damping, we used the model proposed by Scheck et al. in.⁴ For a circular magnetization precession, the eddy current damping scales quadratically with the conductive layer thickness, expressed in⁵ as follows:

$$ \alpha_{eddy} = \frac{\sigma}{12} \gamma \mu_0^2 M_s \delta^2 \qquad (4) $$

where $\sigma$ is the conductivity of the conductive layer. With $\sigma$ for Au of the order of $10^7 \Omega^{-1} m^{-1}$ extracted from literature, we evaluate $\alpha_{eddy} \sim 10^{-3}$.

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