Dr Anton A Ivanov
email: ivanov338@niic.nsc.ru

Dr Tatiana N Pozmogova

Dr Anastasiya O Solovieva

Prof Tatiana S L V Frolova

Dr Olga I Sinitsyna

Dr Olga V Lundovskaya

Dr Alphiya R Tsygankova

Dr Mohamed Haouas

David Landy

Enrico Benassi
email: enrico.benassi@sns.it

Lidiya V Shestopalova

Dr Clément Falaise

Emmanuel Cadot

Dr Michael A Shestopalov

Dr Pavel A Abramov
email: abramov@niic.nsc.ru

Maxim N Sokolov

Prof. M N Sokolov Nikolaev

Dr A R Shestopalova

Dr O I Tsygankova

Prof. M N Sinitsyna

Dr E Sokolov

Benassi

Prof E Cadot

Prof D Landy

From Specific γ-CD/[Nb 6 Cl 12 (H 2 O) 6 ] 2+ Recognition to Biological Activity Tuning

teaching and research institutions in France or abroad, or from public or private research centers.

Introduction

Specific molecular recognition occurs in a wide range of biological processes playing a key role at different levels of supramolecular interactions. One of the main types of molecular recognition results in formation of host-guest complexes. The most widely studied class of host molecules are cyclodextrins (CDs), cyclic oligosaccharides consisting of six, seven or eight glucopyranose units, respectively shorthanded as α-CD, β-CD and γ-CD. This ability to form host-guest complexes has led to the use of CDs as material in a number of industries. [START_REF] Easton | Modified Cyclodextrins : Scaffolds and Templates for Supramolecular Chemistry[END_REF] [START_REF] Hedges | [END_REF] For example, CDs are used in pharmaceutical industry, [3][4] [5] [6] in analytical applications [7][8][9] [10] and in the chemical technology as catalysts or catalyst additives, phase-transfer agents and stabilizer of functional species such as azo dyes. [START_REF] Szejtli | Comprehensive Supramolecular Chemistry[END_REF][12] [13] [14] Medicinal use of CDs, focusing on the complexation of problematic drugs, such as poorly soluble, unstable, irritating, or difficult to formulate substances, [15] resulted in improved wettability, dissolution, and solubility. Other positive features include improved stability, reduced side effects, and alleviation of undesired properties, e.g., bitter taste or repelling odor. Cyclodextrins can be used practically in any drug forms -oral, rectal, pulmonary, external, ocular, etc. [16] [17] There is a variety of important interactions involved in the complexation with CDs. [1][18][19] [START_REF] Tabushi | [END_REF] [21] Bender with co-workers, as well as Saenger, assume that water molecules captured inside the cavity of CD torus are highly energetic, i.e., enthalpically frustrated: these molecules are unable to undergo the favorable four-fold hydrogen bindings characteristic in bulk water. [22][23] [24] This concept of enthalpically frustrated water molecules has found support in recent theoretical studies, which also revealed the role of the concave geometry of the apolar binding pocket with respect to the nature and shape of the guest. [25][26] [27][28] [29] Evaluation of metal-cluster complexes, including M6-based clusters in terms of anticancer therapy is one of the promising areas of research. Currently, luminescent octahedral molybdenum, tungsten, or rhenium clusters are actively studied as promising photosensitizers for photodynamic therapy. [30][31] [START_REF] Krasilnikova | Nanomedicine Nanotechnology[END_REF] On the other hand, Echeverría with co-workers recently demonstrated selective cytotoxicity of the Re6-cluster complex [Re6Se8I6] 3-toward different cell lines. [33] They have shown that this cluster induces selective tumor cell death without affecting normal cells, which turns it into promising anticancer and cytostatic agent. The authors explain this behavior by the ability of [Re6Se8I6] 3-to penetrate through the cell membrane until ssDNA bonding in the nucleus. Recently we have discovered the formation of a new type of host-guest supramolecular complexes based on the specific recognition between γ-CD and cationic [Ta6Br12(H2O)6] 2+ cluster, used in the design of soft nanomaterials. [34] The key role in the step-by-step formation of such kind of assemblies lies in the specific recognition of building units at the molecular level supported by solvent effect arising from chaotropic behavior of the cluster ion (water structure breaking). Moreover, formation of the host-guest complexes based on anionic octahedral rhenium chalcogenide clusters [Re6Q8(CN)6] 4-(Q = S, Se, Te) has been demonstrated. [35][36] Hence occurrence of such adducts shows the ability of the CD to form energetically stable supramolecular complexes with both cationic or anionic clusters. In this research we focus on the complexation behavior and biological activity of hexanuclear niobium complex [Nb6Cl12(H2O)6] 2+ in the presence of γ-CD. Herein we report the synthesis and crystal structure of a new host-guest compound {[Nb6Cl12(H2O)6]@(γ-CD)2}Cl2. Since the biological properties of group 5 hexanuclear clusters {M6X12} n+ (M = Nb, Ta; X = Cl, Br, I) have never been studied, we evaluated the cytotoxicity, genotoxicity, cellular uptake and localization of free hexaniobium cluster complex [Nb6Cl12(H2O)6]Cl2 and influence of γ-CD acting as a properties tuning agent e.g. changes in cytotoxicity, penetration or intracellular localization, on larynx carcinoma (Hep-2) and normal fibroblastic (DK-4) cell lines. Additionally, we demonstrated overproducing of reactive oxygen species (ROS) in cells that contributes to the cluster cytotoxicity.

Results and Discussion

Synthesis and structure Starting Nb6Cl14•6H2O material is a brown solid synthesized from NbCl5 [37] which solubilizes slowly after reflux producing green aqueous solutions of a [Nb6Cl12(H2O)6] 2+ aqua complex (1). Typically, after dissolution a small amount of insoluble materials remains. Due to this fact we determined the effective concentration of niobium cluster solution using thermogravimetry technique to calculate correct molar absorption coefficients ε (Fig. S1). Aqueous solutions of [Nb6Cl12(H2O)6] 2+ are air stable and were used for further investigations as stock solutions. Addition of two or more equivalents of γ-CD to hot solution leads to the formation of green cubic crystals of {[Nb6Cl12(H2O)6]@(γ-CD)2}Cl2•20H2O (2) upon cooling. The complex crystallizes in a tetragonal (I422) space group (Table S1). The main building block of the crystal structure is the supramolecular cation {[Nb6Cl12(H2O)6]@(γ-CD)2} 2+ (see Fig. 1), in which the hexanuclear [Nb6Cl12(H2O)6] 2+ cation appears encapsulated closely by two γ-CD molecules. Supramolecular complex 2 is isostructural to the previously characterized {[Ta6Br12(H2O)6]@(γ-CD)2}Br2•14H2O. [34] This cationic association can be classified as a host-guest complex, when the cluster unit is a cationic guest inside the cage formed by two γ-CD host molecules. The subunits are linked together through hydrogen bonding. The four coordinated water molecules in the belt part of the cluster participate in the formation of hydrogen bonds with both host molecules, giving O•••O distances within 2.75 -3.30 Å intervals (Figure 1b). In the crystal structure the supramolecular cations form dimeric tubular structures in [001] crystal direction (Fig. S2 andS3), which can be classified as secondary building blocks (SBB). Due to high positional disordering of Cl -counter anions and crystallization water molecules in the structure, the Cl -positions cannot be precisely localized. Considering that {Ta6Br12}-and {Nb6Cl12}-cluster derivatives are isostructural, the Cl -anions have been then located in the positions occupied by the Br -ions in the analog structure. We thus assume localization of anions inside the dimeric tubular SBB, ensuring contacts through H•••Cl interactions. Besides, dimeric tubular associates form a bcc sub-lattice with SBB forming infinite columns running along [001] crystallographic direction.

Solution studies

Compound 2 was studied in solution by means of Isothermal Titration Calorimetry (ITC), mass spectrometry and nuclear magnetic resonance (NMR) spectroscopy. The interaction between γ-CD and the cationic cluster [Nb6Cl12(H2O)6] 2+ was probed by 1 H NMR titration experiment. Upon increasing the amount of the cationic cluster in D2O solution of 1 mM γ-CD (Fig. 2), 1 H NMR spectra showed significant line broadening of all signals together with slight chemical shift changes of some specific resonances. Surprisingly, the most affected signals were those corresponding to the outer peripheral H1 and H4 protons of the CD tori (Δδ ≈ -0.08 ppm) (Fig. S4), while among the inward directed protons, H3 appears the most altered by Δδ ≈ -0.02 ppm). Such unexpected observations suggest that the cationic cluster is able to interact with both the external or inner wall of the γ-CD. Otherwise, the CD macrocycle could also undergo conformational distortion able to alter mostly the external C-H bonds. In all cases, this NMR fingerprint is drastically different to that previously observed with the strong inclusion complex {[Ta6Br12(H2O)6@(γ-CD)2]} 2+ . Indeed, the NMR analysis was fully consistent with the retention of the 2:1 supramolecular adduct in aqueous solution, [34] where the most affected resonance corresponded to the signal attributed to the inward-directed H3 protons. Furthermore, NMR study evidenced a frozen inclusion complex arising from a slow host-guest exchange. In the current case of the {Nb6Cl12}-based cluster, the H3 signal exhibited only tiny change for large excess of cluster, indicating a moderate complexation in solution. Consistently, Job plots (Fig. S5) indicate 1:1 CD:cluster stoichiometry as prominent species. Stability of 2 in aqueous solution has been studied with ESI-MS technique. The ESI-MS of the 2:1 complex aqueous solution consists in a set of species including 2:1 and 1:1 adducts as well as free species, observed as desolvated [Nb6Cl12] 2+ species (Fig. S6). This result appears in fair agreement with quantumchemical calculations which showed favorable association for only 1:1 complex in aqueous solution (see below). ITC (Table S2 and Fig. S7) allows quantification of interactions between the γ-CD and [Nb6Cl12(H2O)6] 2+ by measuring binding constants, enthalpy ΔrH* and entropy ΔrS* changes, which are summarized in Table 1 and compared to previously obtained data with [Ta6Br12(H2O)6] 2+ . [33] While the ITC data of [Ta6Br12(H2O)6] 2+ /γ-CD system were analyzed using a two-site binding model involved in a sequential process, those of [Nb6Cl12(H2O)6] 2+ are rather consistent with a 1:1 stoichiometry model. This means that recognition processes are significantly dependent on the nature of the cluster, where the aggregation process leads consecutively to the 1:1 and the 2:1 adducts with {Ta6Br12}-based cluster and restricted to the 1:1 stoichiometry with the {Nb6Cl12} cluster. Consistently, the ITC results confirm the difference in host-guest association affinity with a K11 decrease of two magnitude orders from {Ta6Br12} cluster to Nb type-cluster. Furthermore, large association constants found in both cases result of enthalpy driven processes, each compensated more or less by entropy penalty. Such thermodynamic fingerprints featured by a large enthalpy gains counterbalanced by the strong entropic energy appear characteristic for supramolecular assembly involving super chaotrope species which release high energy water molecules upon the recognition process. [38] Overall results are rather consistent with weak interactions between [Nb6Br12(H2O)6] 2+ and γ-CD in solution as shown by [START_REF] Easton | Modified Cyclodextrins : Scaffolds and Templates for Supramolecular Chemistry[END_REF] H NMR in comparison to the analogous [Ta6Br12(H2O)6] 2+ /γ-CD system. This difference in behavior can be explained on the basis of size-matching effect. Calculated cluster volumes give 407 and 460 Å 3 for {Nb6Cl12} and {Ta6Br12} clusters, correspondingly. Such a difference appears significant enough to explain the solution behavior as a similar correlation were previously established for rhenium clusters {Re6Q8} with Q = S, Se or Te. [35][36] On the other hand, large negative values of both ΔrH* and TΔrS* parameters should be also related to the nature of the supramolecular adducts arising from dynamic interactions between the {Nb6Cl12}-cluster and the external / internal part of the γ-CD rather than the typical 1:1 and 1:2 host-guest species observed in the case of the {Ta6Br12}-based cluster. [a] data from reference [33].

Quantum-chemical calculations

The differences in Gibbs' free energy at T = 298.15 K and p = 1 atm were computed for the following five equilibria:

γ-CD + 40 H2O = (γ-CD)•(H2O)40 (A1) (γ-CD)2 + 48 H2O = (γ-CD)2•(H2O)48 (A2) 2 γ-CD = (γ-CD)2 (B) [Nb6Cl12(H2O)6] 2+ + γ-CD = {[Nb6Cl12(H2O)6]@(γ-CD)} 2+ (C1) [Nb6Cl12(H2O)6] 2+ + (γ-CD)2 = {[Nb6Cl12(H2O)6]@(γ-CD)2} 2+ (C2)
Processes (A1) and (A2) describe solvation of γ-CD and (γ-CD)2, respectively.For eq. (B), the head-to-head dimerization mode of γ-CD is chosen, since it was shown to be preferred with respect to other aggregation types. [39] Processes (C1) and (C2) describe inclusion of the {Nb6Cl12}-cluster inside the torus of γ-CD and (γ-CD)2, respectively. The results obtained both in the gas phase and aqueous solution are summarized in Table 2. As shown in Table 2, dimerization of γ-CD according to process (B) seems to be energetically more favorable in the gas phase than in aqueous solution. Such a scenario appears quite reasonable since the stabilization is mainly due to hydrogen bonding. In the dimer, the two γ-CD units interact mutually through 16 hydrogen bonds. From values given in In aqueous solution, solvation of either the monomeric or the dimeric γ-CD species is evidenced as a highly favorable process (see reactions A1 and A2).

Regarding the inclusion process of the {Nb6Cl12}-based cluster inside the γ-CD torus, formation of the 1:1 adduct (equation C1) reveals to be the most stable species in aqueous solution (ΔG = -291kJ.mol -1 ) compared to the other scenario depicted in Table 2. This supports the experimental evidences that the 1:1 adduct corresponds to the predominant species in aqueous solution. Furthermore, additional evidences are provided by computed 1 H-NMR shifts. Considering the isolated γ-CD and (γ-CD)2, and the related adducts {[Nb6Cl12(H2O)6]@(γ-CD)} 2+ and {[Nb6Cl12(H2O)6]@(γ-CD)2} 2+ , computation results predict a shift for the signals of H1 and H4 protons of CD. In the case of 1:1 adduct, a quite good agreement was found between calculated and experimental shifts. Variations are about -0.074 ppm for both H1 and H4. To investigate the presence and nature of possible interactions, the non-covalent interaction (NCI) index combined with the second derivative of the reduced density gradient (RDG) along the second main axis of variation were employed. RDG analysis (Fig. S8 To summarize, solution studies and quantum-chemical calculations suggest several types of supramolecular assemblies based on 1:1 stoichiometry resulting of either hostguest complex or "outer complexes" built from external interactions. Such a scenario is schematically depicted in Figure 3, where the "outer" 1:1 adduct resulting from γ-CD•••cluster lateral interactions can be proposed as the simplest model while more complicated aggregates could be envisioned.

Biological properties

The initial stage in the evaluation of the biological properties of new compounds is the determination of their cytotoxicity. This parameter plays crucial role in understanding the effect of CD molecules on biological properties of the clusters. In order to evaluate the influence of γ-CD on cytotoxicity of [Nb6Cl12(H2O)6] 2+ the MTT colorimetric assay on cancer Hep-2 and normal DK-4 cell lines was used. It is important to note that, hereinafter, studies of compound 2 in fact reflect mainly the influence of the 1:1 form on cell viability, because, as shown earlier previously, this form corresponds to the prominent species mainly exists in aqueous solution. According to the data given in Fig. 4, the half maximal inhibitory concentration (IC50) for cluster complex 1 after 72 h of incubation (74 ± 7 µM for Hep-2 and 594 ± 24 µM for DK-4) were lower than those for the adduct with γ-CD (2) (222 ± 20 µM for Hep-2, 953 ± 86 µM for DK-4). At the same time, the cytotoxicity of both compounds is higher for cancer Hep-2 cells than for normal DK-4. It is important to note, that the cytotoxic evaluation experiments were carried out at the micromolar concentration range of the cluster and CD. Since the binding constant of the 1:1 adduct is 1.3x10 3 M -1 (at 308K), its presence in the experimental medium drastically depends on the concentrations used at this range. The calculated content in the 1:1 adduct from dissolution of 2 in the experimental medium is presented on the Figure 4 as a gray curve with asterisks. As it can be seen from the data the differences in the MTT curve pattern for Hep-2 cells appear at the concentration 25.31 µM that corresponds to an about 6% content of 1:1 adduct in solution. Subsequent increase in concentration of the complexes leads to the greater difference in their cytotoxicity. The significant difference in the cytotoxicity of 1 and 2 for DK-4 cells was observed only at the concentrations when the 1:1 adduct content exceeds 50% These findings suggest that i) the CD adduct (when appears) has lower cytotoxicity due to the shielding effect of γ-CD and ii) both free and encapsulated clusters have selective cytotoxic effect for tumor cells in comparison with normal cells. A possible reason for such behavior can be the difference in the cellular uptake of 1 and 2 by Hep-2 and DK-4 cells. Unlike previously investigated octahedral cluster complexes in biological systems [40][41][42] [43] the hexaniobium cluster complexes do not exhibit luminescence, making their cellular uptake unavailable for studies by fluorescent microscopy or flow cytometry. However, we were able to quantify the accumulation of 1 in Hep-2 and DK-4 cells by inductively coupled plasma mass spectrometry (ICP-MS).

The results obtained show that the cluster [Nb6Cl12(H2O)6]Cl2 is able to penetrate through cell membrane and amounts of niobium atoms accumulated by cells of both lines are actually relatively similar (Table S3). Note, that the presence of CD in the case of 2 at investigated concentration (1.5 µM) does not influence on its cellular uptake (Table S2). Moreover, it is most likely that after cellular uptake, even at high concentration of 2, the real content of 1:1 adduct inside the cells will be extremely small (if at all). Therefore, it is clear that the difference in the cytotoxicity between cell cultures is not related to the difference in the cellular uptake of cluster complexes, but in the mechanism of toxicity.

Transmission electron microscopy assay with elemental mapping on Hep-2 cells incubated with 1 indicated cellular uptake and subcellular localization of niobium atoms (Fig. S9).

According to the TEM images, the cluster is located inside the nucleus and in mitochondria. It is also important to mention that no signs of abnormal changes of cells ultrastructure were observed. This ability to penetrate into the cell nucleus and mitochondria can lead to the binding with DNA molecules. The possibility of interaction between [Re6Se8I6] 3-cluster complex and ssDNA has already been referred. [33] The ability of the cluster to enter the nucleus and mitochondria of Hep-2 cells is an unexpected finding, revealing the potential use of this cluster as a targeting deliverer of various substances, such as cytostatics, in tumor cells. To evaluate the binding of the niobium cluster with DNA, the Hep-2 cells with 1 were incubated and the DNA was then isolated. The niobium quantity determinations in the isolated DNA by ICP-MS clearly demonstrate the presence of Nb in the samples DNA and correspond to 1.0 ± 0.3 µg per 1 mg of DNA. The Nb amount in the control DNA (without incubation with cluster compounds) was 0.020 ± 0.005 µg(Nb)/mg(DNA). This data allows us to assume that hexaniobium clusters can really bind to DNA in some way. This binding of the cluster to DNA can lead to DNA damage, and as a result, to replication blocking (genotoxicity). To assess the contribution of genotoxicity into the toxicity of 1, the SOS chromotest on E. coli strain PQ37 was used. Cluster compound showed slight genotoxic effects (Table S4), indicating some interactions between the clusters and DNA, regardless of the CD presence. Descending trend of I(c) (induction factor, considered as the ratio of the activity of two enzymes, beta-galactosidase to alkaline phosphatase) with increasing complex concentration is due to the cell toxicity of the cluster: with decreasing number of bacteria (decreased activity of alkaline phosphatase) the SOS response weakened. Moreover, the similar genotoxicity of the niobium cluster correlates with the similar quantity of the Nb in the samples. Since 1 has slight genotoxicity it does not seem to be a main reason of its cytotoxicity. Moreover, it does not explain the difference in the toxicity of the compounds on different cell lines. One of the common mechanisms for cellular death is overproducing of reactive oxygen species (ROS). In order to monitor the presence of intracellular ROS after incubation with 1 and 2, a cell-permeable fluorescent probe, 5,6-carboxy-2′,7′dichlorofluoresceindiacetate (DCFH-DA), sensitive to oxidation by different ROS including singlet oxygen, was used. A 100 µM H2O2 solution was used as a ROS-positive control reference.

According to the data obtained (Fig. S10, S11) incubation both with 1 and 2 increased the ROS level in the cells.

In general, the ROS level was higher for compound 1 than for 2 in both cell cultures. Indeed, the presence of cyclic oligosaccharide γ-CD can inhibit oxidative stress caused by cluster complex. [44] Nevertheless, both compounds show a tendency to generate ROS in Hep-2 cancer cells more efficiently. These results may explain the difference in the toxicity of 1 and 2 on Hep-2 and DK-4 cells, and indicate contribution of ROS production to the cluster cytotoxicity.

Conclusion

The specific molecular recognition between [Nb6Cl12(H2O)6] 2+ and γ-CD has been demonstrated. According to the NMR and ITC data, 1:1 complex is preferable in solution, while in solid state 2:1 complex formation was obtained. Dissolution of 2:1 complex crystals leads to formation of an equilibria set including mainly the 1:1 adduct and free [Nb6Cl12(H2O)6] 2+ species. The first quantum-chemical calculations of octahedral cluster/CD assemblies have been performed. Calculated data appeared rather consistent with the formation of host-guest assemblies both in the gas phase and in aqueous solution. Especially, comparison of the computed values of ΔG for the 1:1 and 2:1 complexes gives more negative value for the 1:1 complex in aqueous solution which supports the experimental evidences for the presence of 1:1 adduct as dominant form in solution.

Host-guest complex formation between [Nb6Cl12(H2O)6] 2+ and γ-CD changes biological properties of the cluster. In addition to significant differences in cytotoxicity, we observed distinction in the producing of reactive oxygen species, as well as revealed selective toxicity of 1 and 2 for the Hep-2 tumor cell line. Another interesting result was the ability of hexaniobium cluster to accumulate in the nucleus of tumor cells. According to the experimental data, one can suggest [Nb6Cl12(H2O)6] 2+ cluster to be useful for the development of novel metal-based anticancer drugs. In context, cyclodextrin plays the role of protective nanocontainer decreasing cytotoxic effect of the clusters allowing possible release inside the cancer cells.

  Hydrogen bonding are shown also between primary faces of two host molecules involving the CH2OH groups through shorter O•••O distances of 2.70 -2.90 Å. Furthermore, inner chloroligands are involved in hydrogen bond interactions with inwarddirected protons H3 of CD with Cl•••H distances of 3.15 -3.20 Å. Such bonds are much longer than those observed with Tacluster (2.83 Å) indicating weaker host-guest connection.

Figure 1 .

 1 Figure 1. Structures of a) γ-CD showing the six types of protons in the glucopyranose unit, and b) the inclusion complex {[Nb6Cl12(H2O)6]@(γ-CD)2} 2+ . Blue dashed lines indicate hydrogen bonds between water ligands and OHgroups of γ-CD.

Figure 2 . 1 H

 21 Figure 2. 1 H NMR spectra of 1 mM γ-CD in the presence increasing amount of [Nb6Cl12(H2O)6] 2+ cluster.

Table 1 .

 1 Binding constants K involving {Nb6Cl12}-and {Ta6Br12}-based octahedral cluster cations with γ-CD and associated thermodynamic parameters (in kJ.mol -1 ) at T = 298 K.

  ) of ground state optimized geometry of {[Nb6Cl12(H2O)6]@(γ-CD)} 2+ shows the presence of several noncovalent interaction regions (see panel (a)). In particular, chlorine atoms in the equatorial plane can interact with γ-CD via Cl•••H (panel (b)) or Cl•••O interactions (panel (c)), both of them having attractive character.

Figure 3 .

 3 Figure 3. Schematic representation of equilibrated species in the aqueous solution containing γ-CD and [Nb6Cl12(H2O)6]2+ cationic cluster.

Figure 4 .

 4 Figure 4. Effect of 1 and 2 on viability of DK-4 (blue and green lines) and Hep-2 (red and violet lines) cells measured by MTT. Gray line indicates calculated 1:1 adduct content (%) in solution at 308 K according to ITC data.

  This research was applied in France-Russia international collaboration (IRP-CLUSPOM CNRS). This work was supported by the Russian Science Foundation (grant No. 15-15-10006). Tatiana N. Pozmogova are grateful to the Russian Foundation for Basic Research (grant No. 18-315-00235). The NIIC team thanks the Ministry of Science and Education of the Russian Federation. The RICEL team thanks the state task of RICELbranch of the ICG SB RAS № 0324-2019-0046-C-02. Keywords: niobium • cluster complex • γ-cyclodextrin • cytotoxicity • genotoxicity

Table 2 .

 2 Differences in Gibbs' free energy, ΔG, at T = 298.15 K and p = 1 atm, computed at DFT M06-2X level (basis set specified in the Computational Details). Data are listed in units of kJ/mol.

		gas	Water
	A1	-	965.2
	A2	-	1156.9
	B	381.2	378.6
	C1	207.8	291.4
	C2	255.9	275.4

Table 2

 2

, the average contribution per H-bond can be roughly estimated to about ΔG / n ~ 23.8 kJ/mol, which is in reasonable agreement with the typical mean value of O-H•••:O strength (~ 21 kJ/mol).

Acknowledgements

Entry for the Table of Contents

Specific molecular recognition of γ-cyclodextrin toward [Nb6Cl12(H2O)6] 2+ niobium cluster complex in aqueous solutions results in a 1:1 supramolecular assembly {[Nb6Cl12(H2O)6]@γ-CD} 2+ . Crystallization results in a 1:2 host-guest complex [Nb6Cl12(H2O)6@(γ-CD)2]Cl2•20H2O. The molecular association affects the biological activity of [Nb6Cl12(H2O)6] 2+ , decreasing its cytotoxicity despite enhanced cellular uptake. Quantum chemical calculations were performed to model host-guest interaction.


M Sushruth

M Grassi

K Ait-Oukaci

D Stoeffler

Y Henry

D Lacour

M Hehn

U Bhaskar

M Bailleul

T Devolder

J.-P Adam

Electrical spectroscopy of forward volume spin waves in perpendicularly magnetized materials

We study the potential of all-electrical inductive techniques for the spectroscopy of propagating forward volume spin waves. We develop a one-dimensional model to account for the electrical signature of spin-wave reflection and transmission between inductive antennas and validate it with experiments on a perpendicularly magnetized Co/Ni multilayer. We describe the influence of the antenna geometry and antenna-to-antenna separation, as well as that of the material parameters on the lineshape of the inductive signals. For a finite damping, the broadband character of the antenna emission in the wave vector space imposes to take into account the growing decoherence of the magnetization waves upon their spatial propagation. The transmission signal can be viewed as resulting from two contributions: a first one from propagating spin-waves leading to an oscillatory phase of the broadband transmission coefficient, and another one originating from the distant induction of ferromagnetic resonance because of the long-range stray fields of realistic antennas. Depending on the relative importance of these two contributions, the decay of the transmitted signal with the propagation distance may not be exponential and the oscillatory character of the spin-wave phase upon propagation may be hidden. Our model and its experimental validation allow to define geometrical and material specifications to be met to enable the use of forward volume spin waves as efficient information carriers.

I. INTRODUCTION

The eigenexcitations of magnetic materials -the spin waves (SWs) [1] -are attractive for future wave-based-computing applications [1][2][3][4][5][6] because they combine small wavelengths, GHz frequencies, and tunability [7]. Information can be stored in the amplitude and phase of the SWs [8] and transported through a conduit made from a thin magnetic material. In in-plane magnetized thin films, the SWs are categorized depending on the orientation of their wave vector k with respect to the magnetization M . The most popular configuration is the Damon-Eshbach (DE) one, where M ⊥ k; these waves can be efficiently generated by standard inductive or rf-spin-orbit-torque (SOT) techniques [9,10]. Beside, they possess large group velocities and can therefore propagate over long distances before being attenuated. Unfortunately, the DE configuration is somewhat inadequate for transmitting spin waves in a curved conduit, because the conduit would need to be magnetized transversely by a non-uniform applied field [11]. The slower backward volume (BV) spin waves, where M k, are far less often used [12] because their weak coupling with inductive antennas and SOT antennas renders them difficult to excite and detect.

To enable spin-wave based transmission of information in any arbitrary direction, one can rather harness isotropic spin waves like the forward volume spin waves [13] (FVSW), whose wave vector lies in the plane of a film magnetized in the out-of-plane direction. Such waves with isotropic propagation recently enabled logic operations [6]. Unfortunately, most past studies of FVSW [2] relied on materials with easyplane anisotropy and therefore required the application of an * jean-paul.adam@u-psud.fr unpractical strong perpendicular magnetic field, substantially diminishing the benefits of the FV configuration. In addition, most of the studies on FVSW [6,14,15] used ferrites, i.e. materials which are hardly portable to the silicon platform and suffer from a low saturation magnetization M s , synonymous of a modest spin-wave group velocity, limiting the ability to transport information in a fast manner. A priori, materials with both a perpendicular magnetic anisotropy (PMA) and a large magnetization-thickness product would be much preferred for FVSW applications [16]. This qualitative speculation needs however to be backed up by predictive models.

In this paper, we study whether transition-metal-based materials with perpendicular magnetic anisotropy (PMA) are adequate for forward volume spin-wave based information transport. We first develop a one-dimensional analytical model to account for the electrical signature of spin-wave reflection and transmission between inductive antennas. This analytical model sheds light on two particular limiting cases of interest when manipulating spin waves in inductive transceivers configuration: i) when the spin-wave attenuation length is much longer than the characteristic size of the antennas and ii) when it is much shorter. We then discuss the exact influence of the geometry (dimensions and separation between the antennas) and of the material damping parameter on the spin-wave signals expected in the general case. We finally report on experimental investigations of Co/Ni multilayers, which validate the model. Our findings promise to be insightful for the edition of the geometrical and material specifications to be met to enable the use of FVSW as efficient information carriers.

II. ANALYTICAL STUDY OF THE SPIN-WAVE SIGNALS IN INDUCTIVE TRANSCEIVERS CONFIGURATIONS

In this section, we develop a simple analytical model to describe the electrical signals encountered when exciting and detecting FVSW with inductive antennas of canonical geometries. This model can be viewed as a simplified version of the formalism developed by Vlaminck and Bailleul [2,17]. Here, our objective is to maintain the formalism at a sufficiently didactic level so as to ease the interpretation of the lineshapes of the spin-wave inductive signals. We have systematically checked that the present model yields the same results as the more complicated model established in Refs. 2 and 17.

In the present study, we assume that the width of the spinwave conduit is much larger that any other dimension of the system and that the inductive antennas are infinitely long in the transverse direction y. Consequently, we consider that all relevant magnetic fields, including the applied rf field and the demagnetizing field, are sufficiently uniform in the transverse direction so that we can restrict our analysis to the sole spin waves with a wave vector purely oriented in the longitudinal direction x. We shall omit the x subscript and systematically write k instead of k x . We examine the response to a continuous-wave excitation at the working frequency ω 0 and we aim to describe the lineshapes of the transmission signal between two inductive antennas, T (ω 0 ), and that of the reflection signal of each individual antenna, R(ω 0 ).

A. Susceptibility for each wave vector

In spin-wave spectroscopy experiments, we work in the forced oscillation regime and impose a harmonic stimulus at the working frequency ω 0 . In response to this stimulus, the magnetic system is excited within a broad range of wave vectors: It naturally responds at the wave vector ±k 0 (with k 0 conventionally chosen positive) determined by the spin-wave dispersion relation but it is also susceptible to respond at wave vectors k different from ±k 0 . For the sake of clarity, we shall refer to a forced oscillation response at |k| = k 0 as a magnetization wave, in opposition to the (resonant) spin-wave at k 0 . The amplitude and phase of each magnetization wave {ω 0 , k} created in response to the inductive excitation torque are governed by a complex tensorial susceptibility χ(ω 0 , k). For FVSW with weak damping and negligible exchange, the in-plane susceptibility to an in-plane harmonic field h x reads (see appendix I)

χ xx (ω 0 , k) ≈ γ 2 0 M s H eff z (ω 2 k -ω 2 0 ) + ıω 0 ∆ω k . (1)
Here, γ 0 is the gyromagnetic ratio, H eff z is the equilibrium effective field, and ω k is the spin-wave dispersion relation describing the frequency of the free oscillations of magnetization at a given wave vector

ω 2 k = γ 2 0 H eff z (H eff z + n(k)M s ), (2)
with

n(k) = 1-1-e -|k|t |k|t
. Finally, ∆ω k is the mode linewidth, which can be expressed as

∆ω k = αγ 0 (2H + n(k)M s ). (3)
The longitudinal susceptibility χ xx (ω 0 , k) is essentially a Lorentzian distribution with a finite width determined by damping. At fixed wave vector, its full width at half maximum in the frequency space is, by definition, ∆ω k . At fixed frequency, its pendant in the wave vector space is ∆k Gilbert = 2/L att , where L att = 2|v g (k)|/∆ω k is the attenuation length and v g ≡ ∂ω k ∂k is the group velocity. For FVSW, in the small wave vector limit (n(k) ≈ |k|t 2 ), the latter writes

|v g (k)| ≈ γ 0 H eff z M s t 4 H eff z H eff z + Mst|k| 2 ≈ γ 0 M s t 4 , (4)
from which it comes [18] L

att (k) ≈ 1 α γ 0 M s 4ω FMR t. (5)
Equation 4 recalls that, at reasonably small wave vectors, the FVSW are essentially non-dispersive and that materials with large saturation magnetization are desirable to achieve large group velocities.

B. Efficiency of inductive antennas in reciprocal space

Since each magnetization wave {ω 0 , k} has a finite susceptibility to the in-plane harmonic field components, it will only be excited if a stimulus with the appropriate spatial periodicity 2π/k is present; this depends solely on the antenna geometry. We consider below the three simplest antenna geometries (Fig. 1), namely single wire antenna (labelled S), U-shaped antenna (labelled GS) and coplanar waveguide antenna (labelled GSG). We assume that all antennas are infinitely thin and that each conductor carries a uniform current density. Furthermore, we assume that no Eddy current is induced in any other part of the sample. Under these conditions, the x component of the Oersted field for a single-wire antenna of width L, separated from the magnetic medium by a vertical spacing s, takes the form of a damped Sinc function in the wave vector space [START_REF] Stancil | Spin Waves: Theory and Applications[END_REF]:

h S x (k) ∝ sin(kL/2) kL/2 e -|k|s . (6)
This means that, in real space, h x is essentially constant under the antenna and almost vanishes everywhere else, decaying in a power-law manner.

The Oersted field created by composite antennas (GS and GSG) may be deduced from Eq. 6 using trivial summations. For a U-shaped antenna of gap g (Fig. 1), it is

h GS x (k) = 2ı sin k(g + L) 2 h S x (k), (7)
and for the coplanar waveguide antenna with the same gap, it becomes

h GSG x (k) = 2 sin 2 k(g + L) 2 h S x (k). (8)
Fig. 1 gathers the antenna efficiency functions (Eqs. 6-8) for representative experimental parameters. Three points are worth noticing. First, the efficiency of all antennas vanishes at k = 2π/L. Second, single-wire antennas can excite at FMR (k = 0) whereas composite antennas cannot. Third, the spectrum of the CPW antennas is comparable but sharper than that of U-shaped antennas. At this stage, it is useful to define a characteristic dimension L ant illustrating the spectral spread of the antenna. We define it as the inverse of the full width at half maximum of the main peak in the antenna efficiency function (Fig. 1).

C. Spin wave reflection and transmission signals

The magnetic sample responds to the field produced by the emitter at all wave vectors present in the excitation spectrum h x (k) and, assuming that the two antennas are identical, the magnetic flux leaking from the spin-wave conduit is detected inductively by the receiver with the same transduction efficiency h x (k). As a result, the spin-wave contribution to the self-inductance of an antenna, which governs its microwave reflection coefficient [START_REF] Stancil | Spin Waves: Theory and Applications[END_REF], writes

R(ω 0 ) = R norm +∞ -∞ h x (k) 2 χ xx (ω 0 , k) dk, (9)
where

R -1 norm = +∞ -∞ h x (k) 2
dk is a normalization factor. Let r be the center-to-center algebraic distance between the emitting and receiving antennas. Upon propagation, each magnetization wave undergoes a phase rotation e -ikr . The spin-wave contribution to the mutual inductance of a pair of antennas, which governs the microwave transmission coefficient, can therefore be written as the inverse Fourier transform of h x (k) 2 χ xx (ω 0 , k), i.e. as

T (ω 0 ) = R norm +∞ -∞ h x (k) 2 χ xx (ω 0 , k) e -ıkr dk. ( 10
)
We emphasize that the magnetic flux detected by the receiving antenna is the sum of contributions from all excitations, not only from the resonant k 0 spin wave but also from all other excited magnetization waves, among which some have wave vectors with sign opposite to that of r. We note also that the Gilbert losses are fully accounted for in the complex susceptibility (Eq. 1). Therefore, there is no need to introduce an explicit loss term of the form e -|r| L att in Eq. 10. As we shall see below, the latter appears naturally upon integration of Eq. 10 in cases where the inductive signal is dominated by the contribution from the resonant spin wave {ω 0 , k 0 }. This justifies the common practice [10,[START_REF] Ciubotaru | [END_REF]21] of introducing such a term in an ad-hoc manner when the choice is made of considering the sole resonant spin wave and neglecting off-resonant magnetization waves. Let us now examine how the antenna geometry affects the shape of the collected spin wave signals. Fig. 2 plots the frequency dependence of the reflection signal R (Eq. 9) for an hypothetical single wire antenna. For large L (L ≥ 100 µm), our calculation describes a typical vector-networkanalyzer(VNA)-FMR experiment using a waveguide with very wide central conductor [22]. In this case, the antenna efficiency function is essentially restricted to a Dirac distribution around k = 0 so that the spin-wave signal almost reduces to χ(ω 0 , k = 0). Upon decreasing L, the antenna starts to emit in a wider band, from k = 0 to typically δk antenna = π/L, which yields a substantial response above FMR frequency. This wave-vector spectral spread related to the finite width of the antenna corresponds to a linewidth enhancement δω antenna = δk antenna × ∂ω k ∂k , which reads

δω antenna = πγ 0 M s t 4L . (11)
Thus, the reflection signal R(ω 0 ) resembles the FMR susceptibility χ(ω 0 , k = 0) only when the spectral spread has much less impact than the Gilbert linewidth, i.e. when δω antenna ∆ω FMR . Equivalently, the lineshape of the reflected signal is given by the sole Gilbert damping only when α α antenna , where

α antenna = π 32 t L γ 0 M s ω FMR . ( 12
)
Note that this condition of Gilbert-dominated linewidth is not always satisfied in standard VNA-FMR characterization. For instance, if performing VNA-FMR on a 50 nm thick film at 6 GHz with antennas of width L = 50 µm, the lineshape of the response will be almost independent from the film damping if the latter is less than α antenna = 10 -4 . This indicates that Gilbert damping parameters smaller than a few 10 -4 , like those of some YIG films, cannot be properly quantified with such a setup. Conversely, for materials with larger damping, such as standard PMA transition metals (Table I), substantial distortions of the reflection signal R(ω) (or of the VNA-FMR signal) are not expected unless sub-micron antennas are used. We benefit from this in the experimental study of Co/Ni multilayered films, reported in section IV. FIG. 2. Modelled reflection signal R(ω) of single wire antennas of different widths. Real (a) and imaginary (b) parts of Eq. 9. The material parameters are that of table I with a damping chosen to be 0.01.

E. Lineshape of the transmission signal

According to Eq. 10, the transmission signal is built from the interferences of various magnetization waves with wave vectors spread over an interval of width ∆k, defined by the overlap of the susceptibility and antenna efficiency functions, χ xx (ω 0 , k) and h x (k), respectively. Upon propagation of these waves over a distance r, the spread in their phases increases by an amount ∆k r. Consequently, the interferences become more and more destructive as the propagation distance increases and the amplitude of the transmission signal decreases to eventually approach zero when the phase spread becomes much greater than 2π. Mathematically, this phase decoherence occurs because the propagation operator e -ıkr oscillates faster and faster and the integrand h x (k) 2 χ(ω 0 , k)e -ıkr in Eq. 10 averages out to a smaller and smaller value.

A priori, the transmission signal T involves three different lengths: the propagation distance r, the characteristic size of the antenna L ant , and the spin-wave attenuation length L att . If the conditions r L att and r L ant are both fulfilled, complete decoherence of the excited magnetization waves is achieved and the transmission signal completely vanishes. Besides the very peculiar case of zero damping, which yields a trivial oscillatory transmission extending to infinite distances (see Appendix II), one can identify two limiting cases in which the decoherence-induced decay of magnetization waves, hence the transmission signal, take very distinct forms: L att L ant and L att L ant .

Transmission signal in the limit of long attenuation length

Let us first consider the situation where the damping is small enough so that L att L ant and the spin waves can propagate far away from the emitting antenna. This corresponds to α < α antenna if single wire antennas are used. In this case, the integrand in Eq. 10 is dominated by the values of k very close to the poles of the susceptibility, ±k 0 . Linearizing the spin-wave dispersion and assuming a slow variation of v g and h x with k around these poles, the transmission signal may be approximated as

T (ω 0 ) ≈ R norm γ 2 0 M s H eff z 2ω 0 v g (k 0 ) h x (k 0 ) 2 +∞ -∞ e -ıkr |k| -k 0 + ı Latt dk.
(13) Applying the residue theorem and neglecting a fast decaying near-field contribution, the integral in Eq. 13 can be shown to amount to 2ıπe -ık0|r| e -|r| L att (See Appendix B in Ref. 23). It follows that, in the "resonant-spin-wave-dominated" regime, the transmission signal writes

T (ω 0 , L att L ant ) ∝ ı e -ık0|r| e -|r| L att h x (k 0 ) 2 . ( 14
)
This expression recalls the empirical one used to model various experiments conducted in the Damon-Eshbach configuration [10,[START_REF] Ciubotaru | [END_REF]21,24,25]. The factors e -|r| L att and e -ık0|r| , which appear naturally upon Fourier transforming the Lorentzian susceptibility function (Eq. 13), illustrate, respectively, the exponential attenuation [26] and the gradual phase rotation of the spin wave upon propagation. The factor ı arises from the fact that the response at the poles is solely related to the imaginary part of the susceptibility since the real part vanishes there. Finally the antenna efficiency h x (k 0 ) matters, as expected. As demonstrated in Appendix II, Eq. 14 naturally holds for α = 0.

2. Transmission signal in the limit of short attenuation length

Let us now consider the opposite situation where damping is large and the antenna is much wider than the attenuation length (L att L ant ). If single wire antenna are used, this corresponds to α α antenna , a situation where the reflection signal looks like the FMR susceptibility (see Sec. II D). In this limit, the susceptibility χ xx (ω 0 , k) is almost constant in the region where the antenna efficiency function h x (k) 2 is contributing most to the integrand of Eq. 10. Then, one can approximate the transmission signal by

T (ω 0 ) ≈ R norm χ xx (ω 0 , k = 0) +∞ -∞ e -ıkr h x (k) 2 dk. ( 15
)
Applying Parseval-Plancherel identity, the integral in the above equation can be rewritten as +∞ -∞ h x (x) h x (x + r) dx, which measures how the inductive fields generated by two antennas separated by r overlap in the volume of the magnetic medium. It thus expresses a form of inductive coupling, which does not involve the propagation of spin waves. As already noted in Ref. 27, near FMR, part of the magnetization waves existing far from the emitter can indeed be viewed as being directly excited by the long range rf field produced by the antenna. We will refer to this signal as the distant induction of FMR. The transmission signal in the wide antenna limit finally reads

T (ω 0 , L att L ant ) ∝ χ xx (ω 0 , k = 0) +∞ -∞ h x (x) h x (x+r) dx.
(16) This expression indicates that whenever the spin waves do not travel much, either because L att is too short or because the antennas are comparatively too wide, the transmission signal is mainly due to the distant induction of FMR. Therefore, in the L att L ant limit, the reflection and transmission spectra shall look alike and resemble both the FMR susceptibility.

III. NUMERICAL STUDY OF THE SPIN-WAVE TRANSMISSION SIGNALS

In the general case, the three different characteristic lengths controlling the transmission spectra, i.e. r, L ant , and L att , must all be taken into account and considered on an equal footing. Eq. 10 must then be integrated numerically. Below, we present results of such numerical integrations for material parameters typical of transition metal PMA films (Table I).

A. Evolution of the transmission spectra with damping

Let us first study how the transmission signal changes with the magnetic damping parameter, for a given propagation distance of 2 µm. Figure 3 shows the numerically determined transmission spectra T (ω 0 ) for four values of α and the three types of antennas introduced previously. At low damping [Fig. 3(a,e)], a broadband transmission is observed, in which the phase of T rotates within an envelope defined by the antenna efficiency function (Fig. 1), consisting of one (single wire antenna) or two lobes (composite antennas). These transmission spectra are dominated by the contribution of propagating spin-waves and they resemble the expectations from Eq. 14. As the damping increases [Fig. 3(b,c,f,g)], the transmission amplitude decreases rapidly (see the scaling factors) and its frequency span shrinks and progressively concentrate into the vicinity of FMR. Concomitantly, the oscillatory na-ture of the transmission signal above FMR is suppressed. The oscillations are lost at smaller damping values for single wire antennas than for composite antennas, which will be explained later in the paper. At large damping [Fig. 3(d,h)], the transmission spectra eventually become fully dominated by the contribution from the distant induction of the FMR and they resemble the expectations from Eq. 16.

B. Evolution of the transmission spectra with distance

Let us now examine how the transmission signal depends on the propagation distance, for a finite damping of 0.01. Figure 4 illustrates the case of single wire antennas, as well as CPW antennas separated by 2 to 14 µm.

Transmission signal below FMR

Below FMR frequency, the transmission signal does not contain oscillations related to phase rotation upon propagation. At these frequencies, indeed, there is no spin wave mode available to transport energy. The finite transmission is exclusively due to the lateral spread of non-resonant forced oscillations (magnetization waves) and the signal originates essentially from low k values, where the susceptibility is maximum. A direct consequence of this is that the transmission below FMR is extremely weak for composite antennas the efficiency of which is nil at k = 0 (Fig. 1).

Transmission signal above FMR

As the frequency increases above FMR, propagating spinwave modes become accessible. Their wave vector increases with increasing frequency, following the dispersion relation (Eq. 2), which results in a frequency-growing accumulated phase rotation -k 0 r after propagation over a distance r. The phase rotation translates into a cosine-like (resp. sine-like) variation of the real (resp. imaginary) part of T with frequency. The frequency spacing between two successive maxima, corresponding to a phase increase of 2π, may be identified with the ratio vg r [18]. Here, again, the amplitude of the oscillations follows an envelope, which, at low damping, is set by the antenna efficiency function h x [k 0 (ω 0 )].

Crossover between resonant-spin-wave-dominated transmission and distant FMR induction

Fig. 4(d) shows how the maximum amplitude of the transmission signal decays with increasing distance r. For CPW antennas, the decay is initially exponential. Beyond some distance [≈ 10 µm with our material parameters], however, it deviates from the commonly anticipated e -|r| L att law [24,25,28] and becomes significantly slower. For single-wire antennas, the decay is never truly exponential-like, even at the smallest distances. All of these deviations from a pure exponential behaviour occur because, as evoked before, the transmitted signal is generally made of two contributions, that of propagating spin waves and that of the distant induction of FMR. Since the rf field produced by the antennas decreases in a powerlaw manner with distance, while propagating spin waves decay exponentially, there will necessarily be a distance beyond which the contribution of the distant FMR induction (approximated by Eq. 16) will dominate over that of propagating spin waves (approximated by Eq. 14) and the maximum amplitude of the transmission signal will progressively tend to follow a power-law asymptote. This distance depends on the exponent of the power-law decay of the spreading rf field h x (r), hence on the type of antenna. This explains why the transition occurs at much longer propagation distance for the CPW antennas (h GSG x (r) ∝ r -4 ) than for single wire antenna (h S x (r) ∝ r -2 ) and why a much clearer crossover is observed for the former, see Fig. 4(d).

IV. EXPERIMENTAL SPIN WAVE SIGNALS

Let us now compare the results of our model with those of travelling SW spectroscopy experiments carried out in the FVSW geometry on an archetypal metallic PMA stack, namely a Co/Ni multilayer.

A. Sample

The studied device was fabricated from a multilayer of composition Pt(5)/Cu(2.5)/{Ni(0.6)/Co(0.2)} ×21 /Ni(0.6)/ Cu(2.5)/Ta (2.5), where the numbers are the thicknesses of the layers in nanometers. Several such stacks were deposited on natively oxidized intrinsic high resistance Si(100) substrates by DC magnetron sputtering. Damping parameters and effective anisotropy fields were determined by conventional Vector Network Analyser-FMR in perpendicularly applied magnetic fields (Fig. 5). The obtained values span from 0.013 to 0.02, and 3 mT to 280 mT, respectively. Then the multilayer with lowest damping (0.013) and anisotropy field (3 mT) was selected for microfabrication: it was patterned into a stripeshaped spin wave conduit by a combination of optical lithography and ion-beam etching.

The width of the patterned spin wave conduit (20 µm) was chosen much greater than any other dimension in the experiment so as to ensure quasi-translational invariance of the magnetic equilibrium configuration in the transverse direction and thereby promote the generation of spin waves with wave vectors directed primarily along the conduit length. The spinwave conduit was covered with 30 nm of SiO 2 for electric insulation. Finally, the above lying 90 nm thick coplanar waveguide antennas, made of aluminum, were patterned using electron-beam lithography. The CPWs consist of conductors of widths L = 200 nm and edge-to-edge gaps g = 250 nm [Fig. 6(a)]. The antenna center to antenna center distance was set to r = 2 µm.

B. Spin wave transmission signal

The spin wave transmission signal was measured using a 2port Vector Network Analyser (VNA). A standard phase and amplitude transmission calibration procedure of the VNA was applied in the reference planes of the device pads. Around the FMR frequency, the cross-talk between the input and output antennas was typically -70 dB. This field independent parasitic signal was larger than the spin-wave signal and we removed its contribution by subtracting an appropriate reference measured in zero field, while the sample was in a magnetic multidomain configuration and no clear spin-wave related signal could be detected.

A representative spin-wave transmission spectrum is plotted in Fig. 6(b,c). Its main qualitative features, namely its os- cillatory nature, the quadrature relation between its real and imaginary parts, the presence of a faint second lobe envelope at high frequency, and the quasi-absence of signal below FMR, are all in line with expectations. From a qualitative point of view, the agreement between experimental and theoretical spectra can be regarded as satisfactory, hence confirming the validity of the model. However, achieving a quantitative agreement between experiment and modeling required one to adapt the damping parameter α and/or the vertical spacing s between the antennas and the magnetic material. These two parameters have a similar influence on the lineshape of the transmission parameters (see appendix III) so that they cannot be evaluated independently from each other in a reliable manner from the fitting procedure. To adjust α, we have chosen to set s as the distance separating the middle of the magnetic film and the middle of the antenna. Under this assumption, the device damping needed to be increased to 0.03 to get a satisfactory fit. This is to be compared to the value of α = 0.013 extracted from the FMR data taken on the multilayer from which the device was fabricated [Fig. 5(b)]. We believe that the observed higher damping at device level could indicate that the fabrication process may have degraded the material damping parameter, which likely correlates to the formerly observed strong variability of the damping measured on multilayers from different batches. Alternatively, it is also possible that the inhomogeneous broadening of ferromagnetic resonance, which appears in VNA-FMR as an extrapolated non-zero linewidth at zero frequency, also contributes to the effective spin-wave damping. This could be the case if the magnetic inhomogeneities at play occurred on a length scale much smaller than the dimensions of our device.

V. CONCLUSION

In summary, we have investigated to what extent inductive techniques can be used for the spectroscopy of forward volume spin waves in metallic films with perpendicular magnetic anisotropy. We have developed and validated a simple model that describes the emission (Eq. 9) and the distant detection (Eq. 10) of plane waves in a one-dimensional magnetic conduit with out-of-plane magnetization. The excitation and collection efficiencies in wave vector space depend on the antenna geometry: a broadband emission starting from ferromagnetic resonance (k = 0) can be obtained with a single wire antenna, while U-shaped and coplanar waveguide antennas emit in a narrower band that excludes the vicinity of FMR. As a result of this wave vector spectral spread, the reflection signal of an antenna can have a complicated frequencylineshape. The lineshape mimics the classical Lorentzian-like macrospin susceptibility only when the damping contribution to the linewidth dominates the contribution related to the antenna spectral spread (Eq. 12). This criterion is generally not satisfied for YIG-based conduits with micron-sized antennas but it can be met easily for PMA transition metal magnets that possess a substantially larger Gilbert damping.

The antenna-to-antenna transmission signal (Eq. 10) can be understood as a combination of the wave-vector-resolved magnetic film susceptibility, the antenna spectral efficiency, as well as the magnetization-wave phase rotation upon propagation. For sufficiently low damping, the combination of the finite group velocity and the spin-wave phase rotation results in an oscillatory character of the transmission parameter versus the frequency. The damping decreases the susceptibility and mixes the contributions of the different wave vectors, which leads to the progressive decoherence of the propagating spin-wave packets and has dramatic qualitative and quantitative impacts on the transmission signal. In the limit of a moderate damping / moderate distance, this reduces to an exponential decay of the transmission signal, with a decay rate identified as the inverse of the spin-wave attenuation length. In other situations, however, the decay may be slower than exponential. Indeed, part of the transmission can be viewed as a distant induction of FMR, arising from the long-range stray fields produced by the antennas. Our findings offer physical insight that can help to engineer the geometrical and material specifications to be met to enable the use of forward volume spin waves as efficient information carriers.

In this first appendix, our aim is to determine the transverse susceptibility versus angular frequency ω ∈ R and wave vector k ∈ R of a PMA film in response to a transverse rf magnetic field. The calculated susceptibility is used to deduce the shape of the line expected in the forward volume wave spectroscopy models and experiments presented in this paper.

We calculate the dynamic response of the magnetization M in the presence of a magnetic field H starting from the Landau-Lifshitz-Gilbert (LLG) equation:

∂M ∂t = -γ 0 (M × H) + α M S M × ∂M ∂t . (17)
Here, γ 0 = γ µ 0 , where γ is the gyromagnetic factor, µ 0 is the permeability of vacuum, M s is the saturation magnetization, and α is the dimensionless damping parameter. In order to linearize the above equation around the equilibrium M //z, we may decompose the magnetization vector as the sum of static and dynamic components

M (t) = M s (e z + δm(t)), (18)
where the dynamic magnetization δm(t) is assumed to have the form of a plane wave δm(t) = δ m x e x + δ m y e y e ı(ωt-kx) . The tilde recalls that the transverse terms δ m x and δ m y , which obey | δ m i | 1 (i = x, y), are complex numbers.

Likewise, the total effective magnetic field H eff acting on M can be written as

H eff (t) = H eff z e z + h eff (t), (19)
H eff z is the static component of the total magnetic field. In our case, it is the sum of the out-of-plane applied field H z , the static demagnetizing field -M s , and the anisotropy field H ani . The dynamic magnetic field h eff has two contributions: the demagnetizing field produced by the periodic magnetization pattern -n(k) mx e x , with n

(k) = 1 -1-e -|k|t |k|t ≈ |k|t 2
and the rf applied field h = (h x e x + h y e y ) e ı(ωt-kx) , with h x and h y ∈ R.

Keeping only the terms up to first order in the dynamic components δm and h, the linearized LLG equation in the reciprocal space reads

ıωδm = -γ 0 M s (e z × h eff + δm × H eff z ) + ıωα(e z × δm). (20)
Introducing the specific plane-wave forms chosen for m and h in Eq. 20 and rearranging the terms adequately, we readily obtain a matrix expression for the rf applied field as a function of the dynamic magnetization

γ 0 M s h x h y = γ 0 (H eff z + n(k)M s ) + ı α ω -ı ω ı ω γ 0 H eff z + ı α ω δ m x δm y (21)
A simple matrix inversion finally yields the susceptibility tensor χ. Keeping only terms that are first order in α, we obtain

δ m x δ m y = χ h x h y ≈ γ 0 M s (ω 2 k -ω 2 ) + ıω∆ω k γ 0 H eff z + ı α ω ı ω -ı ω γ 0 (H eff z + n(k) M s ) + ı α ω h x h y , (22)
where

ω 2 k = γ 2 0 H eff z (H eff z + n(k) M s ) (23)
is the dispersion relation of the forward volume spin waves (Eq. 2) and ∆ω k = αγ 0 (2 H eff z + n(k) M s ) is the mode linewidth.

From Eq. 22, the transverse susceptibility of interest reads

χ xx (ω, k) ≈ γ 0 M s (γ 0 H eff z + ı α ω) (ω 2 k -ω 2 ) + ıω∆ω k . (24)
Considering that α is much smaller than ω γ0 H eff z ∼ 1, we can neglect the damping term in the numerator of the above expression, but not in the denominator since it dominates near the resonance.

For the discussion in the main text of this work it is useful to rewrite the denominator as linear function of the wave vector k for a given angular frequency ω 0 . At k ∼ k 0 , where ω 0 = ω(k 0 ), the dispersion relation can be written as ω 2 k = ω 2 + 2 v g (k 0 )(k -k 0 ). Here, v g (k 0 ) is the group velocity at k 0 , as defined in Eq. 4. Thus, knowing that ∆ω k 2 vg = L -1 att the susceptibility at ω 0 reads

χ xx (ω 0 , k) ≈ γ 0 M s γ 0 H eff z 2 ω 0 v g (k 0 ) |k| -k 0 + ı Latt . (25)
APPENDIX II: HYPOTHETICAL ZERO DAMPING CASE Let's test the case with no propagation loss, i.e. assuming α = 0 and an infinite attenuation length. We assume that we possess an hypothetical narrow band antenna that carries a sinc current partition with spatial periodicity 2π/k 0 . The antenna thus emits uniformly in the [k 0 -, k 0 + ] and [-k 0 -, -k 0 + ] intervals and does not emit for other wavevectors. We still consider a working frequency ω 0 > ω F M R with the corresponding spin wave wavevector k 0 , defined as positive.

The limit of the susceptibility for zero damping can be written as:

χ(ω 0 , k)| α→0 = 2 t 1 |k| -k 0 -ı 2π t δ k0 -ı 2π t δ -k0 ( 26
)
where δ is the Dirac distribution. The integration of the transmission parameter requires to split the integral in two parts:

T (ω 0 ) ∝ -k0+ -k0- + k0+ k0- χ(k) e -ıkr dk (27)
After some algebra, the real part can be written as:

( T (ω 0 )) α→0 = -4 t sin(k 0 r) π + 2 + r 0 sinc(q)dq (28) and the imaginary part is:

( T (ω 0 )) α→0 = - 4π t cos(k 0 r) (29)
Several points are worth noticing:

• If the antenna had an infinitely narrow band (i.e. = 0, i.e. if it was generating a perfectly spatially periodic field), the transmission signal in the absence of damping would simply reduce to:

T (ω 0 ) α→0, →0 = -ı 4π t e -ık0r (30)
which is a trivial monochromatic plane wave propagating without attenuation, as intuitively anticipated.

• If the antenna had a finite bandwidth (i.e. = 0), the integral in Eq. 28 would converge to π/2 for long propagation distances, i.e. when r → ∞, such that T (ω 0 ) α→0, r→∞ → -8π t sin(k 0 r). The amplitude of the transmission signal would not decay with the propagation distance but it would oscillate periodically because the amplitude of the real and imaginary parts differ. This beating is due to the finite spectral width of the wavepacket.

• Finally let's examine the case of an antenna that still has an finite spectral width (i.e. = 0), and look at the propagation in its vicinity. Noting that + r 0 sinc(q)dq ≈ r for r 1, implies that the real part of the transmission parameter would increase with the propagation distance at small distances. One could be surprised by this trend. This simply reflects the fact that one needs some distance so that the below-resonance wavevectors k ∈ [k 0 , k 0 + ] (with positive χ) and the above-resonance wavevectors k ∈ [k 0 -, k 0 ] (with negative χ) cease to have cancelling contributions thanks to their differential phase rotation e -ı(k-k0)r that increases with r.

APPENDIX III: EFFECT OF THE SPACING BETWEEN THE ANTENNA AND THE MAGNETIC FILM ONTO THE TRANSMISSION SPECTRUM Fig. 7 displays how the transmission parameter is expected to vary with the vertical distance s between the antenna and the magnetic film. The e -ks factor in Eq. 6 does not change the response at the FMR frequency but it induces a faster decay of the transmission signal versus frequency above FMR, and a progressive loss of the oscillatory character. As a result, s and α have qualitatively similar influences on the lineshape of the transmission signal, although α exponentially impacts the amplitude of the transmission. Note that in our electromagnetic model (Eqs. 6-8), the antenna and the film are approximated as infinitely thin sheets, such that there is no rigorous way of choosing the right value of s to fit the experiment.

FIG. 1 .

 1 FIG. 1. Square modulus of the antenna efficiencies (Eqs. 6-8) for three different types of antennas: single wire (blue), U-shaped (red) and CPW (black), where L = g = 200 nm and the antenna to spin wave conduit spacing is s = 0. The single wire curve is plotted after multiplication by a factor of 4.

FIG. 3 .

 3 FIG. 3. Damping dependence of the real part of the expected transmitted signal at the receiving antenna (Eq. 10) for single wire (a-d) and composite antennas (e-h). Parameters are L = 200 nm, g = 250 nm, s = 30 nm and r = 2 µm.

FIG. 4 .

 4 FIG. 4. Influence of the propagation distance on the imaginary part of the transmission signal (Eq. 10) with frequency for (a, b) CPW antennas (L = 200 nm, g = 250 nm and s = 80 nm) and (c) single wire antennas (L = 500 nm, s = 80 nm). The damping parameter is 0.01. Propagation distances are indicated in the legend. (d) Maximum of the modulus of the transmission coefficient versus antennato-antenna distance for the two antenna geometries. The blue line corresponds to an exponential decay with an attenuation length of 1.1 µm, as calculated from Eq.5.

FIG. 5 .

 5 FIG. 5. Magnetic properties of the [Co(2 Å)/Ni(6 Å)×21] selected multilayer. (a) Out-of-plane field dependence of FMR frequency. The inset shows the Mz vs Hz loop. (b) Variation of the half width at half maximum (HWHM) of the FMR line with frequency. The slope may be identified with Gilbert damping parameter. The linear fit yields α = 0.013 ± 0.002.

FIG. 6 .

 6 FIG. 6. (a) Colorised scanning electron micrograph of the device with CPW antennas on a Co/Ni multilayer. The scale bar is 1 µm. The nominal width of the conductor L is 200 nm and the gaps between the conductors are 250 nm. (b) Real and (c) imaginary parts of the experimental transmission parameter and fits thereof with a damping of 0.03, an antenna size of L = 180 nm, a gap of g = 200 nm, spacing of s = 80 nm and a propagation length of 1.8 µm.

FIG. 7 .

 7 FIG. 7. Imaginary part of the transmission parameter for α = 0.03, CPW antenna size of L = 180 nm and gap of g = 200 nm and a propagation length of 1.8 µm. The vertical separation between the antenna and the magnetic film is varied between s = 30 nm and s = 500 nm.

TABLE I .

 I Material parameters and SW characteristics SW linewidth ∆ω k /(2π) 230 MHz -2.3 GHz Attenuation length L att (k) 1.2 µm -0.12 µm D. Lineshape of the reflection signal versus size of the antenna

	Parameter		Value
	Effective field µ 0 H eff z		0.25 T
	Gyromagnetic ratio γ 0 /(2π)	30 GHz/T
	Magnetization µ 0 M s		0.89 T
	Magnetic thickness t		20 nm
	Group velocity v g (k)	105 m/s
	Damping constant α	0.01	-0.1

VI. ACKNOWLEDGEMENTS The authors acknowledge financial support from the French National Research Agency (ANR) under Contract No. ANR-16-CE24-0027 (SWANGATE). U.B acknowledges in addition the FETOPEN-01-2016-2017 [FET-Open research and innovation actions (CHIRON project: Grant agreement ID: 801055)]. The authors acknowledge the STnano cleanroom facility for technical support and Mikhail Kostylev for constructive discussions.