Thibault Broux

François Fauth

Nikita Hall

Yohann Chatillon

Matteo Bianchini

Tahya Bamine

Jean-Bernard Leriche

Emmanuelle Suard

Dany Carlier

Yvan F Reynier

Loïc Simonin

Christian Masquelier

Laurence Croguennec
email: laurence.croguennec@icmcb.cnrs.fr

High rate performance for carbon-coated Na3V2(PO4)2F3 in Na-Ion Batteries

Keywords: Na-ion prototype batteries, polyanionic positive electrode material, oxygen defects, carbon coating, phase diagram

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

The development of large scale electrical energy storage systems is the greatest challenge for the storage and the integration into the grid of renewable energies. Concerns about the availability and thus the prize of lithium sources [1] recently promoted an intense research activity towards Na-ion batteries, that rapidly became the most appealing short term alternative to Li-ion batteries. [START_REF] Johansson | Prospects for Li-ion Batteries and Emerging Energy Electrochemical Systems[END_REF][START_REF] Yabuuchi | [END_REF][4] Two families of compounds turn out to be promising on the positive electrode side: layered transition metal oxides of general formula NaxMO2 [5] and phosphate-based materials [6] . The latter have three-dimensional frameworks built of MO6 octahedra and PO4 tetrahedra that offer a large number of atomic arrangements and possible compositions. Layered oxides have concentrated considerable research efforts, [7][8][9][10][11][12][13][14] from the demonstration of the electrochemical sodium deintercalation and re-intercalation in NaxCoO2 [15] to the release of a prototype Na-ion battery using an air-stable electrode Na0.90[Cu0.22Fe0.30Mn0.48]O2 free of critical raw elements versus hard carbon. [16] Polyanionic-based materials appear as promising alternatives to layered oxides despite the weight penalty generated inherently by the replacement of the anion O 2-by the anionic group PO4 3-. The sodium superionic conductors (called NASICON) of general formula AxMM'(PO4)3 were the first ones to draw attention, among which Na3V III 2(PO4)3 was shown to be the most attractive with a reversible capacity of 120 mAh/g at an average potential of 3.4 V versus Na + /Na and the involvement of the V III/IV redox couple. [17,18] Among polyanionic materials, the most promising to date turns out to be sodium-vanadium fluorophosphate compositions Na3V2(PO4)2F3-yOy where y can vary from 0 to 2. [19][20][21][22][23][24][25][26][27][28][29][30] This whole range of compositions, from the V III -rich one Na3V2(PO4)2F3

(y = 0) to the V IV -rich one Na3(VO)2(PO4)2F (y = 2), can exchange reversibly two Na + ions. For instance, Na3V2(PO4)2F3 (y = 0) delivers a reversible capacity of 128 mAh/g at an average voltage close to 3.9 V versus Na + /Na, corresponding to a high intrinsic energy density of 507 Wh/kg. [26,31,32] Note that this energy density is competitive with that delivered by LiFePO4 in Li-ion batteries, and much higher than that delivered by the system just mentioned before, i.e.

Na0.90[Cu0.22Fe0.30Mn0.48]O2, which is limited to an energy density of 325 Wh/kg (with a reversible capacity of 100 mAh/g at an average voltage of 3.25V).

As demonstrated by some of us, [23] the stoichiometric compound Na3V III 2(PO4)2F3 (y = 0) crystalizes in the Amam orthorhombic space group at room temperature. This structure is described in Figure 1a and consists of a three dimensional framework with V2O8F3 bi-octahedra connected by PO4 tetrahedra and of large tunnels where Na + ions are mobile upon extraction/insertion reactions. The structural evolution of Na3-xV2(PO4)2F3 (x ≤ 2) has been extensively characterized operando during the electrochemical cycling of Na//Na3V2(PO4)2F3 cells: the phase diagram stabilized within the composition range Na3V III 2(PO4)2F3 -Na1V IV 2(PO4)2F3 is a complex succession of biphasic and solid solution-type reactions as summarized in Figure 1b. [24] Recently we have also studied the impact of small amounts of oxygen defects on the structural, transport and electrochemical properties of Na3V2(PO4)2F3-yOy. [27] The structural framework remains similar to that of Na3V2(PO4)2F3 but with an increasing disorder in the Na + ions distribution, changing from three to four crystallographic sites with increasing y. [27] The richer the content in oxygen the higher the electronic conductivity, in relation with the mixed valence state V 3+ /V 4+ within the bi-octahedral units V2O8F3-yOy, and the easier the diffusion of Na + ions in these bare (i.e. not coated) Na3V2(PO4)2F3-yOy type compounds. Nevertheless, we have also reported that carbon-coated Na3V2(PO4)2F3 shows excellent electrochemical performance at high rate, with only 20% loss of the capacity at 20C. [33] In this paper, a series of three different samples of so-called NVPF materials (i.e. Na3V2(PO4)2F3) are compared. They were obtained with different synthesis methods, either as bare NVPF powder (without any carbon coating), or as composites with carbon, either as carbon nanoparticles within the grain boundaries or as a conductive thin layer covering the particles. The impact of the microstructure (morphology of the particles, nature of the carbon coating) and of a small amount of oxygen defects will be discussed considering the structure of the material and its electrochemical performances in Na-ion batteries. Especially, changes within the phase diagram were carefully scrutinized operando (during the cycling of the battery) using Synchrotron X-ray diffraction, depending on the electrode formulation, cycling rate and temperature. From all these findings the most optimized material was thoroughly studied in "extreme" conditions, at low temperature and/or at high cycling rate, and in 18650 prototypes versus hard carbon.

Experimental

Polycrystalline samples of Na3V2(PO4)2F3 (NVPF) were synthetized by three different synthesis routes, the three of them being two-step solid-state reactions. The first step consists in the synthesis of the vanadium phosphate (VPO4) and the second step in the sintering under argon at 750 °C for 2 h (heating/cooling rate of 3 °C/min) of a stoichiometric mixture of VPO4 and NaF (Sigma-Aldrich, ≤ 99%). The difference between the three synthesis routes lies in the nature of the vanadium phosphate used as a precursor and/or of the carbon coating:

 VPO4, used as the precursor to obtain the bare Na3V2(PO4)2F3 sample (called bare NVPF in the following), was prepared by mixing stoichiometric amounts of V2O5 (Sigma-Aldrich, ≥ 99.6 %) and NH4H2PO4 (Sigma-Aldrich, ≥ 99.99 %) in a high energy ball miller for 90 min. This mixture was then continuously heated under Ar/H2 (95%/5%) to ensure a complete reduction of V V to V III at 300 °C for 5 h (with a heating ramp of 0.5 °C/min) and at 800 °C for 5 h (with a heating ramp of 2 °C/min), before being cooled down to room temperature at 3 °C/min.

 A VPO4carbon composite was used as a precursor to obtain a first carbon composite sample (called C-VPO4 NVPF in the following). It was prepared by mixing stoichiometric amounts of V2O5 and (NH4)2HPO4 with an excess of divided carbon soot (Csp) as described in ref 34 . The role of Csp is first to ensure the carbothermal reduction of vanadium from the oxidation state 5+ to 3+, and then to be a source of carbon for the formation of the composite C-VPO4 with carbon nanoparticles in the grain boundaries. [23]  VPO4 used as a precursor for the formation of the carbon coated sample (called coated NVPF in the following), is the same as the one used to prepare the bare NVPF sample. The peculiarity of coated NVPF lies in the nature of its carbon coating, obtained as described in ref [34] from the decomposition of an organic compound. In that case, the covering of the particles and of the grain boundaries is homogeneous, as a thin layer.

Note that the partially oxidized bare composition Na3V2(PO4)2F2.8O0.2 (NVPF2.8O0.2), whose synthesis has been described elsewhere, [27] has been used for comparison. The chemical composition (stoichiometry in Na, V, P, i.e. within the accuracy equal to 3/2/2) of the three synthesized samples was confirmed by inductively coupled plasma-optical emission spectroscopy (ICP-OES) using a Varian Model 720-ES spectrometer after complete dissolution of the powder into a hydrochloric acid (HCl) solution.

High angular resolution Synchrotron X-ray powder diffraction (SXRPD) data were collected using the powder diffractometer BL04-MSPD beamline of ALBA synchrotron (Barcelona, Spain). The powders were placed in a 0.5 mm diameter capillary and the data recorded in Debye-Scherrer geometry with wavelengths around 0.95 Å in the 2θ angular range 1 -70 ° with an accumulation time of 6 minutes. For sake of clarity, due to differences in wavelengths, the comparison of the SXRD patterns has been done using Q range (Å -1 ). Diffraction data treatment was performed using the FullProf Suite. [35] 23 Na magic angle spinning nuclear magnetic resonance (MAS NMR) spectra were recorded on a Bruker Avance 500 MHz spectrometer (132.3 MHz resonance frequency for 23 Na) using a standard Bruker 2.5 mm MAS probe with a 30 kHz typical spinning frequency. A short pulse length of 1 µs corresponding to a selective π/12 pulse determined using an aqueous 0.1 mol/L NaCl solution was employed. The spectral width was set to 1 MHz and the recycle time to D0 = 0.5 s, long enough to avoid T1 saturation effects. The baseline distortions resulting from the spectrometer dead time

(5-10 µs) were removed computationally using a polynomial baseline correction routine. The 0 ppm external reference was a 0.1M NaCl aqueous solution. 31 Scanning electron microscopy (SEM) analysis of metallized samples (Pd-deposited) was performed using a Hitachi Model S-4500 microscope.

Operando SXRPD data were collected on the beamline BL04-MSPD at ALBA using a transmission in situ electrochemical cell already described in details elsewhere. [36] The sodium battery was obtained from the stacking of the positive electrode, glass fiber separators (Whatman)

wetted with a 1 M solution of NaPF6 (Strem Chemical; 99%) in a mixture of ethylene carbonate and dimethyl carbonate (EC:DMC = 1:1) and sodium metal as the counter electrode. The internal pressure within the cell was applied through a spring to ensure an optimal electrical contact. This stacking was inserted between two beryllium windows acting as X-ray-transparent current collectors. The electrochemical behavior and the corresponding phase diagram were studied using different electrode formulations (relative percentages of the active material, carbon conductive additive and polytetrafluoroethylene binder within the electrode), different cycling rates (C/n with n corresponding to the exchange of 2 Na + in n hours) and different electrochemical cycling temperatures (0°C versus 25°C). The in situ cell was cooled down using an external serpentine fixed at the height of the electrochemical stacking and the effective temperature of the sample was estimated from the comparison of its cell parameters with those of NVPF as a function of the temperature. For the micro-diffraction experiment performed on the BL04-MSPD beamline, the data were collected at the Sn K-edge wavelength (λ = 0.4246 Å), with 2 s of effective integration time using a 2 dimensional (2D) Rayonix CCD165 detector. The detector was positioned at 445mm from the sample, hence avoiding the strong Bragg peaks of the Be windows, but allowing to cover down to 2Å d-spacing, which is sufficient for accessing the most relevant (110)/(002) and

(220)/(113) group of reflections characteristic of NVPF. 2D to 1D azimuthal integrations were performed using the Fit2D software while masking strong spots arising from the sodium foil used as negative electrode. The background induced by the electrolyte, the carbon and the cell was modelled for the first pattern collected for each rate and position, it was then subtracted from all successive data.

Cyclability and rate capability electrochemical tests were performed in full cells versus hard carbon, either in coin cells or in 18650 cells. The positive electrode contained 91 wt.% active material and the loading was 12 mg/cm 2 for the cyclability tests and 6 mg/cm 2 for the rate capability tests, whereas the hard carbon negative electrode contained 92 wt.% active material and 6 mg/cm 2 and 3 mg/cm 2 respectively. Both electrodes were double side coated on aluminum foil using a roll to roll machine with PVDF binder. After a drying overnight under vacuum, the electrodes were slitted and calendared to 35% porosity, then wound together with a standard trilayer polypropylene polyethylene separator. Finally the jelly rolls were filled with EC/DMC (1/1) NaPF6 1M electrolyte. Cycle life was carried out at constant current between 2 and 4.25V at room temperature. Discharge rate capability test was evaluated using slow rate C/5 charge up to 4.25V and increasing discharge rate from C/10 to 150C down to 2V, while charge rate capability was tested the opposite way up to 40C.

Results

Pristine materials

The SXRPD patterns of the three NVPF samples obtained and discussed in this paper are compared in Figure 2. All the main diffraction lines can be indexed in unit cells described in the Amam space group as determined previously for Na3V2(PO4)2F3. [23] Close inspection of the SXRD data shows small shifts towards higher Q values as well as a slight right asymmetry for all the l-dependent reflections for the sample C-VPO4 NVPF, highlighting a lower c cell parameter which is the propagation direction of the bi-octahedral units V2O8F3, and thus very sensitive to the presence of oxygen defects. A comparison of all the cell parameters determined from the structural Rietveld refinements is given in Table 1. It confirms the lower c cell parameter for the C-VPO4

NVPF sample, and thus a volume cell of 877.778(1) Å 3 vs. 878.066(1) and 878.389(1) Å 3 for the coated and bare NVPF samples respectively. As already discussed in [27] , oxygen substitution for fluorine in Na3V2(PO4)2F3-yOy can easily be detected from the close comparison of their cell unitcell parameters and volumes. Nevertheless, here, a variation of less than 0.1% is observed suggesting the presence of localized defects but in very small quantities. Note that the b/a ratio (i.e. the parameter sensitive to the orthorhombic distortion induced by the distribution of the Na + ions within the channels) is similar for the three phases, close to 1.0016.

Solid-state MAS NMR was identified as being extremely sensitive to oxygen defects in vanadium fluoride phosphate materials (LiVPO4F [29] and Na3V2(PO4)3F3 [27] ). Besides the SXRD data analysis, the NMR spectra are thus displayed in Figure 3 with their main signals at 138 ppm for 23 Na NMR and at 6097 ppm for 31 P NMR. These features are associated with Na + ions located in the direct vicinity of V III 2O8F3 bi-octahedral units and with P 5+ ions in interaction with four V III O4F2 octahedra sharing one oxygen with PO4, respectively. As previously reported, [27] the appearance of additional signals below 120 ppm for 23 Na NMR and 5000 ppm for 31 P NMR is caused by the presence of V IV defects in Na3V2(PO4)2F3: the more V 4+ in the vicinity of the probed nucleus, the more shifted to lower chemical shifts the signals are observed. They can be ascribed to a partial oxidation of V III to V IV as a charge compensation for the presence of oxygen defects (O 2-) in substitution for fluorine (F -). [27] The NMR spectrum of the oxidized material Na3V2(PO4)2F2.8O0.2, with 10% of vanadium at the tetravalent state as vanadyle-type defects, is given for comparison. It is clearly much more oxidized than the three NVPF samples studied in this paper, with higher intensities for the signals associated to the mixed valence environments V 3+ -V 4+ and smaller intensities for the V 3+ only environments. In very good agreement with the analysis of the cell parameters, the C-VPO4 NVPF sample is the one which presents the "highest" amount of defects among the three NVPF samples, whereas the bare NVPF and coated NVPF samples are very similar in cationic and anionic compositions. Bare NVPF is characterized by particles of 300 nm in size in average, C-VPO4 NVPF by a few micrometer sized particles with carbon nanoparticles in the grain boundaries, and coated NVPF by 100 to 500 nm sized particles with a homogeneous carbon coating at their surface.

Changes in the phase diagram observed upon cycling at a rate of C/2

In order to precisely compare the electrochemical behavior of the three Na3V2(PO4)2F3 samples, operando SXRPD patterns have been collected during a charge performed at a rate of C/2, and are compared in Figure 5. The corresponding electrochemical data are given in Figure 6. The overall evolution of the SXRPD patterns upon Na + deintercalation from Na3V2(PO4)2F3 is nearly the same for the three samples and essentially similar to that previously reported. [24] Starting from gathered and compared in Table 1. As expected, there is a continuous decrease of the cell volume (V/Z) upon Na + extraction as the vanadium transition metal ion is oxidized for charge compensation. Its first coordination sphere VO4F2 is thus contracted thanks to the stronger covalency of the V-O(F) bonds. Furthermore, the comparison with the results reported in reference 24 reveals that the compositions we associate to Na2VPF and Na1VPF are very close to the compositions expected with a deviation being less than 0.27 and 0.18 % respectively. Since the main difference between these three samples lies in the Na2.4VPF composition that displays an almost extinguished orthorhombic distortion for the C-VPO4 NVPF sample (i.e. b/a = 1.002, vs.

1.008 for bare NVPF and 1.006 for coated NVPF) (Figure 7 and Table 1), the big size of the particles appears to be more limiting for the kinetics of the reaction than the electronic percolation.

In the following, as being the most affected by the kinetics of the reaction, only the composition range 2 ≤ x ≤ 3 in the various NaxVPF samples will be discussed in details. Indeed, as in the composition range 1 ≤ x ≤ 2 Na + deintercalation occurs mainly as a solid solution reaction, it is as expected less sensitive to the kinetics.

In order to visualize the impact of a small oxygen substitution for fluorine, Figure 8 to V 4+ in the oxidized phase. In fact the solid solution could be rather seen/described as a series of phases close in structure and in composition that evolve together towards the same Na2VPF2.8O0.2 target, which is the composition corresponding to the deintercalation of 1 Na + per formula unit. As displayed in Figure S2, pattern #21 that corresponds to an average composition Na~2.67VPF2.8O0.2 can be refined by the Rietveld method from a two-phase model: the first and main one is the pristine composition Na3VPF2.8O0.2 (60%); the second one is found to be Na2.4(1)VPF2.8O0.2 (40%).

Whereas the structure of Na3VPF2.8O0.2 is described in an orthorhombic unit cell (Amam, with a small distortion such as a/b = 1.001), [27] the structure of the de-intercalated phase is described in a tetragonal unit cell (I4/mmm) similar to that stabilized for Na3V2(PO4)2F3 above 400K, [23] with cell parameters a = 6.3710 phases. [24,37] Beyond the pattern #51 the growth of the intermediate phase Na2VPF2.8O0.2 is observed concomitantly with the voltage step observed in the electrochemical curve and corresponding to the transition between the first voltage domain at 3.75 V vs. Na + /Na and the second at 4.2 V.

As highlighted in Figure 7, the phase diagram observed for the C-VPO4 NVPF sample around the composition Na2.4VPF is thus closer to that determined for the partially oxygen substituted phase Na3-xVPF2.8O0.2 than to that determined for the bare and coated NVPF compounds. Indeed, the presence of small amounts of oxygen defects induces locally, for charge compensation, the oxidation of V 3+ to V 4+ ions, and thus the formation of strong covalent vanadyle type bonds {V=O} 2+ , very short versus the V 3+ -F bonds observed along the bi-octahedral units. These defects obviously prevent the formation of the charge and/or cation ordering required to stabilize the intermediate phase Na2.4VPF, which corresponds to an average oxidation state around +3.25 and thus to an ordering between 3V III :1V IV . The defects limit the extension of the ordered domains, in good agreement with the global broadening of the diffraction lines and with the almost disappearance of the orthorhombic distortion (b/a = 0.998 versus 0.992 and 0.994 for the bare and coated NVPF compounds respectively). Oxygen and vanadyle-type defects have thus a significant impact on the phase diagram stabilized upon sodium deintercalation from NVPF, with the disappearance of the intermediate phases and the associated bi-phased reactions, to the benefit of a solid solution type reaction. Note that changes in the structural mechanism observed upon cycling have also been observed for the V IV -rich Na3V2-y(VO)y(PO4)2F3-y composition (y  2) depending on the concentration of V III defects in the V IV -rich compound. [38,39] Beside the impact of oxygen defects on the phase diagram, it is also interesting to highlight the influence of the carbon coating. From this point of view, pristine bare and coated NVPF samples can be compared since they have similar diffraction peak shapes (Figure 2), similar structure and negligible amounts of oxygen defects. Despite these similarities, they exhibit small differences upon Na + deintercalation as shown in Figure 5 and Figure 7. As already discussed, the first one is bare (i.e. without any carbon coating) and the second one is such that the active compound is covered by a thin layer of carbon at the surface of the primary particles. Here no difference is observed regarding the symmetry of the stabilized phases. For instance the orthorhombic split is clearly observed (Figure 7). However, a significant difference is seen in the widths of the diffraction lines observed for the bare NVPF, which tends to indicate that the crystalline coherent domains were reduced, despite they were similar, in their pristine state, to the coated NVPF. A carbon coating allows to significantly improve the electrochemical performance as it was demonstrated for LiFePO4 [START_REF] Armand | Patent priority n°CA2423129C[END_REF] and more recently for Na3V2(PO4)3 [START_REF] Li | [END_REF][42] : a thin conductive layer leads to an easier diffusion of the electrons and thus of the alkali ions within the whole electrode, and thus to a decrease of the internal resistance and to a reaction closer to the equilibrium with extended crystalline and ordered domains.

Performance of the optimized coated NVPF

As just discussed and revealed by an in-depth comparison of three NVPF samples, the control of the amount of oxygen defects and of the nature of the composite carbon -active material is essential for an optimization of the transport properties, and thus for the fast stabilization of the complex phase diagram which consists in a succession of phase transitions implying either alkali or charge ordering. In the following, we report on the impact of cycling in extreme conditions (low temperature, very high rate) on the phase diagram. We will demonstrate that a well optimized NVPF sample shows excellent transport properties and make it a material of choice for batteries operating at low temperature and in high rate (power) conditions. This will be also demonstrated within 18650 prototypes, in excellent agreements with results already obtained a few years ago by some of us in laboratory full cells versus hard carbon. [33]

Impact of extreme conditions on the phase diagram

The Coated NVPF sample has been tested in extreme conditions such as low temperature (at ~ 0°C) and high cycling rate (from C/10 to 25C, through 5C), in order to evaluate the impact of these parameters on both the phase diagram and the electrochemical behavior. The first electrochemical cycles obtained in different conditions for Na//coated NVPF cells are gathered and compared in electrochemical cell developed and modified by Leriche et al. [36] that was cooled down with a conventional antifreeze agent refrigerated at -10 °C. Due to heat dissipation in the whole line from the chiller to the cell and through the cell stainless steel body, the actual temperature within the battery was estimated to be near 0°C. The electrochemical performance obtained at C/10, at room S1 for comparison with those determined at RT for the same material, i.e. coated NVPF.

As expected, a contraction of the unit cells is observed, but very limited (less than 0.04%) all along the Na + deintercalation from the structure. The orthorhombic distortion tends to slightly increase, up to 0.09 %. The results of this experiment performed at low temperature highlight again that the mobility of Na + ions is very fast in these materials, [28] with the ability to stabilize attractive performance at low temperature.

In order to check for the stability of the phase diagram upon cycling at very different rates, from low rates at C/10 to very high rates at 25C, the electrode formulation was modified. To prevent possible heterogeneity due to limited electronic percolation within the electrode for experiments performed at 5C and 25C, the proportion of carbon conductive additive was increased to 55 wt.% for 35 wt.% of active material and 10 wt.% of PTFE as binder. The corresponding electrochemical data are given in Figure 9, in comparison with those obtained at C/10, either at RT or at ~0°C. The first cycle obtained at 5C is very similar to that obtained at C/10. It demonstrates the ability of the optimized coated NVPF to deliver an attractive reversible capacity of ~105 mAh/g within 12 mins.

Obviously, higher polarization and thus lower reversible capacity are observed at a charge rate of 25C, but only 15% of the capacity is lost. Na3VPF and Na2VPF which reveal that the system, even if strongly exposed to non-equilibrium conditions can stabilize a phase diagram exactly identical to the one previously observed at lower rates. This high rate diffusion ability and stability of the phase diagram are also in good agreement with the absence of metastable phases previously observed for the olivine LiFePO4 (LFP) in Lithium-ion batteries cycled at high rate. [43,44] In the case of LFP, lithium deintercalation occurs according to a bi-phasic reaction and the solubility limits change at high cycling rates, by-passing the first-order phase transition between the end-members LiFePO4 and FePO4 with the formation of an intermediate phase close in composition to Li2/3FePO4. In the case of coated NVPF, we did not observe such phenomena, showing the ability of NVPF to reach its equilibrium upon cycling even at high rates.

One issue when collecting operando diffraction, spectroscopy or real space imaging data, especially at very high cycling rates, is the possibility for spatial heterogeneities. In that case, different parts of the electrode may behave differently and the part studied (i.e. irradiated by the beam) can be not representative of the overall sample due to a state of charge or of discharge different from the average. Such an inhomogeneous behavior has already been reported for XAS experiments performed at a rather low rate of 1C for the system Li//LiFePO4. [36,45] It was thus worth to check that point, considering especially that we did consider cycling rates as high as 25C

in electrochemical cells similar to those used in [45] . All the SXRPD patterns presented here were collected with the detector MYTHEN on a 2mm x 0.7mm area of the electrode corresponding to the size of the beam. A 10 degrees oscillation around the direction of the beam further contributes to improve the averaging of the sample probed (surface and bulk). In order to track for possible spatial inhomogeneity during cycling, similar experiments were performed on the High Pressure/ MicroDiffraction station of the BL04-MSPD beamline. In that case, the X-ray beam was focused down to 20 µm x 20 µm and the powder diffraction patterns were collected using a 2D detector.

For different cycling rates (as shown here in Figure 13 for C, 10C and 20C rates), SXRPD patterns were collected at three positions of the electrode separated by 1mm, which correspond thus to the central and extreme positions of the standard experiments just mentioned. The comparison of these data allow to discriminate between an SXRPD pattern being representative of an average due to heterogeneity and an SXRPD pattern being representative of all the parts of the electrode. As highlighted in Figure 13 this experiment demonstrates that the three probed regions exhibit the same state of Na deintercalation for all measured rates. Typically, the characteristic feature of the Na2.4V2(PO4)2F3 phase is observed, as well as the "crossing" between the (220) and (113) Bragg reflections, at exactly the same voltage. Hence beam-induced artefacts or unequal pressures within the cell can be ruled out, and this micro-diffraction experiment reveals that all the results presented here are representative of the global electrode.

Electrochemical performance in 18650 prototypes versus hard carbon

These very attractive results motivated the preparation of 18650 industry representative cells aimed at testing the ability of NVPF materials to sustain long cycle life and high cycling rates.

Figure S3 given in supplementary information shows the first cycle of a full Na-ion coin cell, with hard carbon at the negative electrode and the optimized coated NVPF at the positive electrode.

The electrochemical cycling was performed in the potential window 2 -4.3 V, yielding a first charge capacity of ~130 mAh/g in good agreement with the theoretical capacity corresponding to the exchange of 2 Na + per formula unit. The reversible capacity is then close to 100 mAh/g at an average potential of 3.4 V. Note that the comparison with a full Na-ion coin cell made with hard carbon at the negative electrode and C-VPO4 NVPF at the positive electrode reveals again the better performance of the coated compound, with a higher first charge capacity, a higher discharge capacity and a smaller irreversible capacity at the end of the first cycle.

The performance obtained in 18650 cells are given in Figure 14. Figure 14a S4) . Coated NVPF appears thus as an already very much optimized material for high power density Na-ion cells.

Conclusion

The comparison of three Na3V2(PO4)2F3 samples, prepared according to different synthesis routes and with or without carbon coating, has demonstrated that despite subtle differences between them New progress is quickly expected, for instance with the optimization of the electrolyte and electrode formulation. Indeed, the appealing results presented in this paper were obtained in nonoptimized cells. 31 P MAS NMR spectra of the three NVPF samples and of a slightly oxidized sample Na3VPF2.8O0.2. The shift is given in ppm for each signal, as well as its environment in vanadium. As an example, the 31 P MAS NMR signal observed at 6100 ppm corresponds to the P 5+ ions with 4 V 3+ in their close vicinity. (a) 23 Na NMR (b) 31 P NMR (please on a full column) 1 . 9 5

V III -V III 3 x V III 1 x V IV 2 x V III 2 x V IV 4 x V III 4 x V IV ~ 0 1 x V III 3 x V IV
V III -V IV ~50 V IV -V IV
1 . 9 6 1 . 9 7 1 . 9 8 1 . 9 9 2 . 0 0 1 . 9 5 1 . 9 6 1 . 9 7

1 . 9 8 1 . 9 9

2 . 0 0 1 .9 5 1 .9 6 1 .9 7 1 .9 8 1 .9 9 2 .0 0

Table captions

Figure 4

 4 Figure 4 displays scanning electron micrographs of bare NVPF, C-VPO4 NVPF and coated NVPF.

Na3V III 2 (

 2 PO4)2F3 (Na3VPF) the sodium extraction leads to a succession of reactions involving all the key compositions mentioned in Figure 1b, with the formation of the end-member Na1V IV 2(PO4)2F3 (Na1VPF) at the end of charge for Vmax = 4.4 V. As highlighted in Figure 7 the orthorhombic splitting between the (200) and (020) diffraction peaks characteristic of the composition Na2.4VPF is clearly visible for the bare NVPF and coated NVPF samples, whereas only a significant broadening of these lines is observed for C-VPO4 NVPF. This suggests that at this C/2 rate the corresponding alkali and/or charge ordering cannot be properly established within large and non-coated particles of C-VPO4 NVPF for which the diffusion length is larger and electronic percolation limited. The key SXRPD patterns shown in Figure 7 were analyzed by Rietveld refinements while examples of such refinements are given in supplementary information in Figure S1. The cell parameters determined for the representative intermediate phases NaxVPO4F (x = 2.4, 2 and 1) are

  shows a comparison of the phase diagrams observed upon Na + de-intercalation from the samples C-VPO4 NVPF (the most oxidized as seen from NMR) and bare NVPF2.8O0.2 at a cycling rate of C/2. As highlighted, C-VPO4 NVPF allows to deintercalate towards clearly identified Na2.4VPF and Na2.2VPF compositions, showing three well-defined biphasic reactions, successively between Na3VPF, Na2.4VPF, Na2.2VPF and Na2VPF (note that the orthorhombic distortion is almost not observed for Na2.4VPF). For the oxygen substituted material, Na3VPF2.8O0.2, the phase diagram observed upon Na + deintercalation is clearly different: it undergoes a biphasic domain between Na3VPF2.8O0.2 and Na3-xVPF2.8O0.2 (0 ≤ x ≤ x1), a solid solution domain for x1 ≤ x ≤ x2 and a biphasic domain between Na3-xVPF2.8O0.2 and Na2VPF2.8O0.2 (x2 ≤ x ≤ 1). The progressive shift of the (200) reflection ascribed to a Na3-xVPF2.8O0.2 solid solution and its significant broadening (Figure 8b) suggest the continuous evolution of the charge/alkali ordering and the existence of locally-ordered crystalline domains, very limited in size. This tends to minimize the electrostatic repulsions and the strains induced by the progressive formation of vanadyle-type bonds in relation with the oxidation of V 3+

( 2 )

 2 Å and c = 10.7522(3) Å and a distribution of the Na + ions on a ring within the tunnels of the host structure. These cell parameters evolve down to a = 6.3502(2) Å and up to c = 10.7572(3) Å (pattern #51) in the solid solution domain Na3-xVPF2.8O0.2, with a c parameter expansion of 0.05% concomitant with a basal plane contraction of 0.65% in agreement with previous observations made on Na3V2(PO4)2F3 and other related

Figure 9 .

 9 Figure 9. First, operando experiments have been carried out at low temperature in the in situ

  temperature (RT) and at low temperature (~ 0°C), are very close to each other, with no extra polarization resistance for the battery cycled at low temperature despite a higher irreversible capacity at the end of the first cycle. Moreover, the SXRPD patterns recorded upon cycling at low temperature indicate that the phase diagram remains unchanged as displayed in Figure 10. The cell parameters determined for the intermediate phases Na2.4VPF, Na2VPF and Na1VPF are given in Table

Figure 11

 11 compares two sets of XRD data collected while charging the electrochemical cell at 5C (charge in 12 mins and 12 s per pattern) and at 25C (charge in 2 mins 24 s and 5 s per pattern). It can be noticed that the intermediate member Na2VPF is also clearly identified at the voltage step between the two domains or just after at high rates, whereas Na1VPF is unambiguously identified through the diffraction line (220) at 17.63°. Once again the main differences are observed in the composition range between Na3VPF and Na2VPF since the intermediate phases corresponding to alkali and charge orderings are not stabilized in conditions far from the equilibrium. Nevertheless, as revealed by the galvanostatic intermittent titration technique-like experiment shown in Figure12, the intermediate ordered phases can be recovered as soon as the relaxation is long enough (here 20 mins) for the system to reach the equilibrium. Approximatively a thousand of SXRPD patterns have been collected between

(

  morphology and nature of the carbon coating, amount of oxygen defects) the phase diagram observed during Na + extraction is only slightly altered at moderate cycling rates (here C/2). An optimized carbon-coated Na3V2(PO4)2F3 was shown to deliver promising electrochemical performance in in situ electrochemical cells, at room temperature and at 0°C, at very high rates (up to 25C), with no major loss of reversible capacity and no significant modification of the phase diagram observed within the composition range Na3V2(PO4)2F3 -Na1V2(PO4)2F3. 18650 prototypes of 75 Wh/kg, Na3V2(PO4)2F3 versus hard-carbon, have demonstrated exceptional rate and electrochemical cycling capabilities with 4000 charge and discharge cycles performed at 1C rate.

Figure 1 :

 1 Figure 1: Adapted and reproduced with permissions rom refs [23 and [24 . 2018 ACS a) On the left, the three dimensional framework of NVPF3-yOy where the dark blue atoms emphasize the substitution site of oxygen for fluorine. In red are highlighted the (a,b) planes where the Na + ions are contained. On the right, distribution of the Na + ions assuming a small amount of oxygen defects, the fraction of filling in red of the Na atom highlights the partial occupancy of the crystallographic site. b) The potential-composition electrochemical curves obtained upon Na + extraction from Na3V2(PO4)2F3 (NVPF). The first cell was cycled (GITT) between the compositions Na3VPF and Na2VPF; the second one (galvanostatic cycling) between the compositions Na2VPF and Na1VPF. Single-phase compositions are highlighted by colored circles.

Figure 2 :

 2 Figure 2: Comparison of the experimental SXRPD patterns recorded for the three NVPF samples. The indexation of the diffraction lines is given in the Amam space group. The inset emphasizes the subtle shift of the (006) reflection for the C-VPO4 NVPF compound, and more generally of all the (00l) diffraction lines.

Figure 3 :

 3 Figure 3: Comparison of the a) 23 Na and b) 31 P MAS NMR spectra of the three NVPF samples and

Figure 4 :Figure 5 :

 45 Figure 4: Scanning electron micrographs obtained for the three NVPF samples (bare NVPF, C-VPO4 NVPF and coated NVPF).

Figure 6 :

 6 Figure 6: Comparison of the electrochemical curves obtained for the three NVPF samples studied within the in situ cell and cycled at C/2 up to 4.4 V vs. Na + /Na. A floating (i.e. potential maintained constant) was performed at 4.4 V at the end of the charge during approximately 10 minutes. The corresponding SXRPD patterns are given in Figure 5.

Figure 7 :

 7 Figure 7: Comparison of the SXRPD patterns obtained operando, at the composition close to Na2.4, for the three NVPF samples and for the partially oxidized sample Bare NVPF2.8O0.2.

Figure 8 :

 8 Figure 8: SXRPD patterns recorded operando during Na + de-intercalation from C-VPO4 NVPF and Bare NVPF2.8O0.2 at a rate of C/2, between the compositions Na3VPF (resp. Na3VPF2.8O0.2) and Na2VPF (resp. Na2VPF2.8O0.2).

Figure 9 :Figure 10 :

 910 Figure 9: Voltage-composition curves obtained for coated NVPF in in situ electrochemical cells cycled in different conditions: C/10, 5C and 25C rates at room temperature (in blue, grey and yellow respectively), and C/10 rate at 0°C (in red).

Figure 11 :

 11 Figure 11: 2D plot of a selected angular domain of the SXRPD patterns recorded operando during Na + deintercalation at 5C (top) and 25C (bottom) from coated NVPF, between the compositions Na3VPF and Na1VPF.

Figure 12 :Figure 13 :Figure 14 :

 121314 Figure 12: 2D plot of selected angular domain of the experimental operando SXRPD patterns recorded for coated NVPF, between the compositions Na3VPF and Na2VPF, during the GITT experiment performed with 25C current pulses followed by 20 minutes relaxation.

  Cycle life at RT, 1C and 100% DOD In 75 Wh/kg 18650 Na-ion cell Rate capabilities, at 100% DOD In 55 Wh/kg 18650 Na-ion cell

  reveals first that a long term cyclability can be demonstrated at a 1C rate upon charge and discharge for a first batch of prototypes (higher loading, obtained energy density of 75 Wh/kg). Over 3750 cycles were performed in the potential window 2 -4.25 V (100% of depth of discharge) before reaching the limit of 20% capacity loss. This result compares favorably with best state of the art Li-ion technology. Then, as shown in Figure14b, the rate capabilities were tested in a second batch of prototypes (lower loading, obtained energy density of 55 Wh/kg) with discharge and charge rates ranging between C/2 to 150C, i.e. from 2h to 24s. Interestingly, outstanding capacity retentions were observed even at very high rates: more than 75% of the capacity is recovered up to 10C (i.e.

	6 mins) in charge and up to 100C (i.e. 36 s) in discharge. At this rate and taking into account an
	average discharge voltage of 2.55 V the power cell achieves more than 3.8 kW/kg continuous
	power in discharge. These first results on un-optimized cells are very encouraging for the
	technology. These experiments reveal thus that Na + ions diffusion is significantly faster in
	discharge than in charge, both being fast (as highlighted in Figure

Acknowledgments

The authors acknowledge Philippe Dagault, Cathy Denage and Laëtitia Etienne from ICMCB for their technical assistance, Sébastien Launois from CEA-LITEN for his help in the NVPF synthesis development, Nelly Martin, Côme Leys and Christophe Vincens from CEA-LITEN for electrode development and prototyping, as well as Long H.B. Nguyen, Paula Sanz-Camacho, Rénald David and Mathieu Morcrette from ICMCB and LRCS for discussions. RS2E and Alistore-ERI are also acknowledged for the funding of TB's postdoctoral fellowship, as well as ALBA (Barcelona, Spain) for Synchrotron X-ray diffraction experiments on the MSPD beamline (proposal numbers 20150221236 and 2016021618). This project has received funding from Région Nouvelle Aquitaine, the French National Research Agency (STORE-EX Labex Project ANR-10-LABX-76-01 and SODIUM Descartes project ANR-13-RESC-0001-02), and the European Union's Horizon 2020 research and innovation program under grant agreement No 646433-NAIADES. Received Revised Published online

Supporting information

Supporting information is available from the Wiley Online Library.

Table 1:

The cell parameters determined for the intermediate phases NaxVPO4F (x = 2.4, 2 and 1) formed upon Na + deintercalation from the pristine phase Na3V2(PO4)2F3 are compared for the three NVPF samples (bare NVPF, C-VPO4 NVPF and coated NVPF).

Compounds


J-C Crivello

V Paul-Boncour

Relation between the weak itinerant magnetism in A 2 Ni 7 compounds (A = Y, La) and their stacked crystal structures

Keywords: intermetallic, electronic structure, weak itinerant magnetism, antiferromagnetism, first principle calculation

The weak itinerant magnetic properties of A 2 Ni 7 compounds with A = {Y, La} have been investigated using electronic band structure calculations in the relation with their polymorphic crystal structures. These compounds crystallizes in two structures resulting from the stacking of two and three blocks of [A 2 Ni 4 + 2 ANi 5 ] units for hexagonal 2H-La 2 Ni 7 (Ce 2 Ni 7 type) and rhombohedral 3R-Y 2 Ni 7 (Gd 2 Co 7 type) respectively. Experimentally, 2H-La 2 Ni 7 is a weak itinerant antiferromagnet whereas 3R-Y 2 Ni 7 is a weak itinerant ferromagnet. From the present first principles calculation within non-spin polarized state, both compounds present an electronic density of state with a sharp and narrow peak centered at the Fermi level corresponding to flat bands from 3d-Ni. This induces a magnetic instability and both compounds are more stable in a ferromagnetic (FM) order compared to a paramagnetic state (ΔE ≈ -35 meV/f.u.). The magnetic moment of each of the five Ni sites varies with their positions relative to the [A 2 Ni 4 ] and [ANi 5 ] units: they are minimum in the [A 2 Ni 4 ] unit and maximum at the interface between two [ANi 5 ] units. For 2H-La 2 Ni 7 , an antiferromagnetic (AFM) structure has been proposed and found with an energy comparable to that of the FM state. This AFM structure is described by two FM unit blocks of opposite Ni spin sign separated by a non-magnetic layer at z = 0 and ½. The Ni (2a) atoms belonging to this intermediate layer are located in the [La 2 Ni 4 ] unit and are at a center of symmetry of the hexagonal cell (P6 3 /mmc) where the resultant molecular field is cancelled. Further noncollinear spin calculations have been performed to determine the Ni moment orientations which are found preferentially parallel to the c axis for both FM and AFM structures.

Introduction

Weak itinerant ferromagnets (wFM) and antiferromagnets (wAFM) are not very common and have raised a large and ongoing interest as they are near the onset of magnetism, present large spin-fluctuations and are often close to quantum critical point and even superconductivity [START_REF] Moriya | Antiferromagnetic spin fluctuations and superconductivity in high-Tc oxides[END_REF][START_REF] Ishigaki | On the theory of spin fluctuations around the magnetic instabilities-effects of zero-point fluctuations[END_REF][START_REF] Moriya | Antiferromagnetic spin fluctuation and superconductivity[END_REF]. Among the wFM, several studies have been performed on ZrZn 2 [START_REF] Mazin | Spin fluctuations and the magnetic phase diagram of ZrZn 2[END_REF][START_REF] Kabeya | Phase diagram of the itinerant-electron ferromagnet ZrZn 2[END_REF], Ni 3 Al [START_REF] Morkowski | Theory of magnetic and thermal-properties of the weak itinerant ferromagnet Ni 3[END_REF][START_REF] Guo | First-principles and experimental studies of the electronic structures and magnetism in Ni 3 Al, Ni 3 Ga and Ni 3[END_REF][START_REF] Steiner | Non-Fermi liquid form of the low temperature resistivity in the low moment ferromagnets Ni 3 Al and YNi 3[END_REF][START_REF] Aguayo | Why Ni 3 Al is an itinerant ferromagnet but Ni 3 Ga is not[END_REF][START_REF] Velasco | Short range order fluctuations and itinerant ferromagnetism in Ni 3[END_REF], Fe 2 N [START_REF] Chen | Spin fluctuation effect in the ordered Fe 2 N alloy[END_REF][START_REF] Naganuma | Magnetic properties of weak itinerant ferromagnetic ζ-Fe 2 N film[END_REF] and AsNCr 3 [START_REF] Lin | Critical behavior in the itinerant ferromagnet AsNCr 3 with tetragonal-antiperovskite structure[END_REF]. wAFM are even less common than wFM and have been observed in some compounds like TiBe 2 [START_REF] Matthias | Itinerant antiferromagnetism of TiBe 2[END_REF][START_REF] Povzner | Spin fluctuation theory for temperature-induced ferromagnetism in itinerant antiferromagnets and paramagnets[END_REF][START_REF] Torun | Origin of weak magnetism in compounds with cubic Laves structure[END_REF], TiAu [START_REF] Svanidze | An itinerant antiferromagnetic metal without magnetic constituents[END_REF][START_REF] Goh | A mechanism for weak itinerant antiferromagnetism: mirrored van Hove singularities[END_REF][START_REF] Goh | Competing magnetic instabilities in the weak itinerant antiferromagnetic TiAu[END_REF][START_REF] Mathew | Probing hole-doping of the weak antiferromagnet TiAu with first principles methods[END_REF] and UN [START_REF] Solontsov | Weak itinerant-electron antiferromagnetism of uranium nitride[END_REF]. Weak itinerant magnets are generally characterized by high transition temperatures and low magnetic moments related to their particular electronic properties.

Recently Singh [START_REF] Singh | Electronic structure and weak itinerant magnetism in metallic Y 2 Ni 7[END_REF] has compared wFM classical facedcentered-cubic Ni substituted by Al forming the Ni 3 Al (75% Ni) with wFM Y 2 Ni 7 which is also a Ni rich compound (78% Ni). Y 2 Ni 7 is an interesting itinerant magnet which has deserved both detailed theoretical and experimental studies [START_REF] Buschow | Magnetic-properties of Y 2 Ni 7 and its hydride[END_REF][START_REF] Tolkachev | Spin fluctuation mechanism for the temperature-induced magnetic moment (TIMM) in the electron liquid of Y 2 Ni 7 Alloy[END_REF][START_REF] Bhattacharyya | Investigation of weak itinerant ferromagnetism and critical behavior of Y 2 Ni 7[END_REF].

With the same stoichiometry than Y 2 Ni 7 , La 2 Ni 7 has been the subject of many experimental studies revealing a weak itinerant antiferromagnetic ground state, whose origin has not been clearly solved until now. Due to the very few number of wAFM compounds, it appears of particular interest to describe the AFM structure of La 2 Ni 7 and to understand the origin of its specific behavior.

La 2 Ni 7 crystallizes preferentially in the hexagonal Ce 2 Ni 7 structure type and Y 2 Ni 7 in the rhombohedral Gd 2 Co 7 structure type. Hexagonal La 2 Ni 7 is a wAFM with T N = 50 K and undergoes a metamagnetic transition toward a wFM state [START_REF] Buschow | Magnetic properties of La 2 Ni 7 and its hydride[END_REF][START_REF] Parker | Magnetic properties of La 2 Ni 7[END_REF][START_REF] Fukase | Successive metamagnetic transitions in hexagonal La 2 Ni 7[END_REF][START_REF] Fukase | Itinerant electron magnetism of hexagonal La 2 Ni 7[END_REF][START_REF] Tazuke | Metamagnetic transitions in hexagonal La 2 Ni 7[END_REF], whereas rhombohedral Y 2 Ni 7 is a wFM with T C = 53 K [START_REF] Singh | Electronic structure and weak itinerant magnetism in metallic Y 2 Ni 7[END_REF][START_REF] Buschow | Magnetic-properties of Y 2 Ni 7 and its hydride[END_REF][START_REF] Bhattacharyya | Investigation of weak itinerant ferromagnetism and critical behavior of Y 2 Ni 7[END_REF][START_REF] Nakabayashi | Itinerant Electron Weak Ferromagnetism in Y 2 Ni 7 and YNi 3[END_REF][START_REF] Tazuke | Magnetism of R 2 N i7 and RNi 3 (R = Y[END_REF]. Both compounds have in common low value of the mean Ni moments (0.06 to 0.11 µ B /Ni) [START_REF] Bhattacharyya | Investigation of weak itinerant ferromagnetism and critical behavior of Y 2 Ni 7[END_REF][START_REF] Tazuke | Metamagnetic transitions in hexagonal La 2 Ni 7[END_REF] and relatively large ordering temperatures, characteristic of the weak itinerant magnetism. They have also effective Ni moments around µ eff = 0.8-1 µ B and positive paramagnetic Curie temper atures θp between 50 and 70 K derived from the analysis of their magnetic susceptibility in the paramagnetic range [START_REF] Buschow | Magnetic-properties of Y 2 Ni 7 and its hydride[END_REF][START_REF] Bhattacharyya | Investigation of weak itinerant ferromagnetism and critical behavior of Y 2 Ni 7[END_REF][START_REF] Parker | Magnetic properties of La 2 Ni 7[END_REF][START_REF] Tazuke | Metamagnetic transitions in hexagonal La 2 Ni 7[END_REF]. This is rather surprising for La 2 Ni 7 , as a negative θp is expected for an antiferromagnetic compound.

Several experimental studies have confirmed the wAFM behavior of La 2 Ni 7 at low field, but the attempts to perform neutron powder diffraction (NPD) experiments at low temperature [START_REF] Tazuke | Magnetic properties of La-Ni system[END_REF] did not allow to determine its microscopic magnetic structure. The absence of magnetic peaks in the neutron patterns below T N was attributed to the itinerant character of the antiferromagnetism and the low values of the Ni moments.

In order to solve the origin of the weak antiferromagnetic ground state of La 2 Ni 7 as well as its metamagnetic transition toward a ferromagnetic structure, we have performed first principle calculations using collinear and non-collinear spin polarized density-functional theory (DFT). Taking into account the particular geometry of the La 2 Ni 7 crystal structure, we will propose an antiferromagnetic structure which can explain, for the first time, many of the magnetic experimental particularities of this compound.

The magnetic and electronic properties of La 2 Ni 7 will be compared with those of Y 2 Ni 7 all along this paper. A more general discussion on the specific itinerant magnetic behavior of La 2 Ni 7 compared to other intermetallic compounds with large spin fluctuations will be introduced. The electronic structure was calculated for the ordered La 2 Ni 7 and Y 2 Ni 7 compounds in both hexagonal and rhombohedral structures. In the frame of the DFT, the pseudo-potential approach using the VASP package [START_REF] Kresse | Ab initio molecular dynamics for open-shell transition metals[END_REF][START_REF] Kresse | From ultrasoft pseudopotentials to the projector augmented-wave method[END_REF] was considered using projector-augmented wave [START_REF] Blöchl | Projector augmented-wave method[END_REF] with a 600 eV cut-off energy and a high k-mesh density (at least 250 k-point in the irreducible Brillouin zone). Several exchange and correlation (XC) functionals have been considered, such as local density approximation (LDA) with parametrization from Perdew and Zunger [START_REF] Perdew | Self-interaction correction to density-functional approximations for many-electron systems[END_REF], the generalized gradient approximation with the PBE functional [START_REF] Perdew | Generalized gradient approximation made simple[END_REF][START_REF] Perdew | Generalized gradient approximation made simple[END_REF] and a meta-GGA functional, the strongly constrained and appropriately normed semilocal density functional (SCAN) [START_REF] Sun | Strongly constrained and appropriately normed semilocal density functional[END_REF], including the second derivative of the electron density. In addition, an empirical coulomb interaction (Hubbard) was introduced (U = 5eV, J = 0) to test the influence of GGA + U calcul ation [START_REF] Rohrbach | Electronic correlation effects in transition-metal sulfides[END_REF]. Preserving the original crystal symmetry, each structure has been fully relaxed within several magnetic ordering such as ferromagn etic and antiferromagnetic structure using the electronic collinear and non-collinear spin-polarization. For the latter, the most stable spin moments orientation has been investigated in GGA-PBE (several moment directions have been tested such as parallel and perpendicular to c axis). Final static calculation was done using the linear tetrahedron method with Blöchl corrections on relaxed structures [START_REF] Blöchl | Improved tetrahedron method for Brillouin-zone integrations[END_REF]. The charge distribution on the atoms was investigated using Bader topological analysis within the 'Bader' code developed by Henkelman et al [START_REF] Bader | Atoms in Molecules. A Quantum Theory[END_REF][START_REF] Henkelman | Fast and robust algorithm for Bader decomposition of charge density[END_REF].

Structural description and methodology

Results and discussion

The non-spin polarized (NSP) density of state (DOS) of hexagonal La 2 Ni 7 (2H-La 2 Ni 7 ) and rhombohedral Y 2 Ni 7 (3R-Y 2 Ni 7 ) are compared in figure 2(left). The electronic DOS results in a large main structure dominated by the 3d states from Ni. A charge transfer from the A element to Ni is observed (1.2 e-per A atom according to the Bader method), but is not sufficient to completely fill the Ni-bands and both compounds present a DOS with a sharp and narrow peak centered at the Fermi level (E F ). This result is in agreement with previous studies [START_REF] Singh | Electronic structure and weak itinerant magnetism in metallic Y 2 Ni 7[END_REF][START_REF] Werwinski | Effect of substitution La by Mg on electrochemical and electronic properties in La 2-x Mg x Ni 7 alloys: a combined experimental and ab initio studies[END_REF][START_REF] Werwiński | Effect of Gd and Co content on electrochemical and electronic properties of La 1.5 Mg 0.5 Ni 7 alloys: a combined experimental and first-principles study[END_REF] and was also observed in wAFM orthorhombic TiAu [START_REF] Goh | A mechanism for weak itinerant antiferromagnetism: mirrored van Hove singularities[END_REF][START_REF] Goh | Competing magnetic instabilities in the weak itinerant antiferromagnetic TiAu[END_REF] and H 3 S superconductor [START_REF] Quan | Van Hove singularities and spectral smearing in high-temperature superconducting H 3[END_REF]. This peak corresponds to the flat d-bands in the electronic band structure as shown in figures 2(right, zoom scale) and S1(full scale) (stacks.iop.org/ JPhysCM/32/145802/mmedia) for 2H-La 2 Ni 7 . Such flat bands were also visible for the paramagnetic calculation of 3R-Y 2 Ni 7 in [START_REF] Singh | Electronic structure and weak itinerant magnetism in metallic Y 2 Ni 7[END_REF]. It can be noticed that the peak maximum is located at E F for Y 2 Ni 7 , while it is slightly shifted to higher energy for La 2 Ni 7 . The main contribution to this peak is thus due to the 3d states of Ni and leads to a high DOS at E F ; N(E F ) = 14.6 states/eV per f.u. for 2H-La 2 Ni 7 , smaller than that calculated for 3R-Y 2 Ni 7 : N(E F ) = 21.7 states/eV per f.u. (close to the 24 states/eV per f.u. from [START_REF] Singh | Electronic structure and weak itinerant magnetism in metallic Y 2 Ni 7[END_REF]). Both values are large enough to follow the Stoner criterion [START_REF] Stoner | Collective electron ferromagnetism II. Energy and specific heat[END_REF] (N(E F ) * I > 1 for I ≈ 0.8 eV/at usually taken for transition metals) and induce a lowering of the total energy by a shift of the minority and majority bands.

The calculated spin polarized DOS of 3R-Y 2 Ni 7 and 2H-La 2 Ni 7 , assuming a ferromagnetic ground state, are displayed in figures 3(a) and (b) respectively. The calculated moments for each atomic site as well as the magnetic energies (ΔE) are reported in table 1. The high density peak position is shifted to lower and higher energy for spin up and spin down bands respectively and N(E F ) decreases due to its position located in a valley of the density of states. As a consequence the total energy of ferromagnetic state is lowered by 5 meV/Ni atom compared to NSP calculated value for both compounds. This magnetic energy corresponds to a temperature of 58 K, which is close to the experimental ordering temperature of Y 2 Ni 7 (T C = 53 K) [START_REF] Bhattacharyya | Investigation of weak itinerant ferromagnetism and critical behavior of Y 2 Ni 7[END_REF].

The calculated moment for each atomic site in ferromagnetic state are reported in table 1. The magnetic moments in Y 2 Ni 7 are in good agreement with those calculated by Singh using a linear augmented plane-wave method [START_REF] Singh | Electronic structure and weak itinerant magnetism in metallic Y 2 Ni 7[END_REF]. It is noticeable, that the total calculated moments of 1.08 µ B /f.u. for La 2 Ni 7 and 1.24 µ B for Y 2 Ni 7 are larger than experimental values (0.77 µ B /f.u. and 0.43 µ B /f.u. measured at 4.2 K and extrapolated from high field for A = La and Y respectively) [START_REF] Bhattacharyya | Investigation of weak itinerant ferromagnetism and critical behavior of Y 2 Ni 7[END_REF][START_REF] Parker | Magnetic properties of La 2 Ni 7[END_REF]. This overestimation is due to the limit of our DFT calculations, which do not take into account the large spin fluctuations present in such compounds. The Ni moments contribute mainly to the total magnetization, whereas the induced A moments remain weak and are, as expected, of opposite sign compared to Ni moments.

Further non-collinear spin calculations have been performed to determine the preferential orientation of the Ni moments in the FM structure. The Ni moments are stabilized in the direction parallel to the c axis: the energy difference between moments parallel and perpendicular to the c axis is about -30 meV/f.u.. The coordination polyhedra have been represented with their calculated magnetic vectors for the Ni1 to Ni4 sites in figure 4 and for the Ni5 site in figure 5 In a simple molecular field approximation, the Ni-Ni exchange interactions can be described by an internal field H exch acting on the Ni site and resulting from the closest Ni neighbor moments (the influence of the Ni-La interactions can be neglected as they are much smaller):

H exch = λ Ni-Ni • m Tot (1)
where λ Ni-Ni is the Ni-Ni molecular field coefficient and m tot = j z j • m j is the sum of the z j Ni neighbor moments m j . Assuming that the Ni-Ni short range interaction is close for all Ni atoms and that the exchange integral are not very different for the Ni-Ni distances between 2.45 and 2.62 Å, it is possible to verify if there is any proportionality between the Ni moments on each site and the sum of its first neighbor moments. The variation of each Ni moment (m i ) versus m Tot shows two different behaviors for the Ni atoms related to the [AB 2 ] units and those belonging to the [AB 5 ] units (figure 6(a)) with a dm i /dm Tot slope of 0.58 and 0.125 respectively. To explain this difference of behavior, which cannot be only due to the variation of Ni-Ni distances as they are comparable for Ni1 and Ni2 atoms and their next Ni neighbors, one should also consider the influence of the first La neighbors (2.93 Å ⩽ d Ni-La ⩽ 3.33 Å).

The electronic charge transfer CT from La to Ni atoms is calculated using the Bader method and is given as a function of m Tot in figure 6 As a consequence, the magnitude of the Ni moments can be related to their positions relative to the [AB 2 ] and [AB 5 ] units. For both structure types, it appears that the Ni moment is maximum for the Ni2 atoms located between two [AB 5 ] units and minimum for the Ni5 (2a) atoms belonging to the [AB 2 ] units, at the z = 1/2 plane containing the center of symmetry of the cell. This is illustrated in figure 7, where the hexagonal structure of La 2 Ni 7 and the corresponding magnitude of the Ni moments along the c axis are presented. Similar Ni moment distribution can be obtained for 3R-Y 2 Ni 7 . It is also noticeable that the distribution of Ni magnetic moments along half the hexagonal cell forms a triangle. This is repeated twice, because the stacking [A 2 B 4 + 2 AB 5 ] is repeated two times in this hexagonal lattice, whereas the same sequence appears three times in 3R-Y 2 Ni 7 . This emphasizes a kind of modulation of the Ni moments along the c-axis.

The dependence of the Ni magnetic moments versus their localization in [AB 2 ] and [AB 5 ] units in hexagonal La 2 Ni 7 can be partly related to the difference of magnetic properties of the corresponding binary LaNi 2 and LaNi 5 compounds: LaNi 2 is a Pauli paramagnet with a weak susceptibility (χ = 1.5 10 -4 emu/mole) [START_REF] Tazuke | Magnetic properties of La-Ni system[END_REF] whereas LaNi 5 is an enhanced Pauli paramagnet with χ = 2.0 10 -3 emu/mole [START_REF] Gignoux | Onset of ferromagnetism and spin fluctuations in rare earth (or actinide)-3d compounds[END_REF]. In general, ANi 5 compounds are close to the onset of ferromagnetism, but the N(E F ) is not large enough to stabilize a ferromagnetic state. The weak itinerant magnetism observed in A 2 Ni 7 as well as ANi 3 compounds has been attributed to the existence of the sharp and narrow peak near E F providing a large enough N(E F ) contribution to stabilize a SP state [START_REF] Shimizu | Electronic-structure and magnetic-properties of Y-Ni intermetallic compounds[END_REF].

However, all the experimental studies have shown that the magnetic ground state of 2H-La 2 Ni 7 is characteristic of an AFM structure (T N = 50 K). In addition, 2H-La 2 Ni 7 undergoes a metamagnetic transition towards a FM state with a transition field of 4.6 T and a stabilization of a wFM behavior above 6 T at 4.2 K [START_REF] Parker | Magnetic properties of La 2 Ni 7[END_REF]. Up to now, all the studies performed on La 2 Ni 7 did not allow to propose an AFM structure filling these criteria. The only observation obtained from the DOS calculation was the smaller N(E F ) value of La 2 Ni 7 compared to Y 2 Ni 7 [START_REF] Werwinski | Effect of substitution La by Mg on electrochemical and electronic properties in La 2-x Mg x Ni 7 alloys: a combined experimental and ab initio studies[END_REF].

Taking into account the symmetry of the hexagonal structure (2 blocks of [AB 2 + 2 AB 5 ]) and assuming that the Ni5 moments, which belong to the [AB 2 ] unit and are already very small in the calculated FM structure (m Ni5 = 0.023 µ B /Ni) are equal to zero, the AFM structure presented in figure 7 can be proposed. The spins of the Ni atoms are of same sign in the same block (between layers at z = 0 and 1/2) whereas they have opposite sign in the next stacking (between z = 1/2 and 1), and they are all aligned parallel to the c axis. In other words, Ni atoms in the same block are ferromagnetically aligned while the magnetic coupling between adjacent blocks is antiferromagnetic (table 1 and figure 7). The DFT calculation converges to the proposed AFM structure with a net magnetization equal to zero on the Ni5 site and an antiferromagnetic coupling between the Ni1 atoms which are its neighbors and belonging to the different blocks (z < ½ or z > ½). Due to the particular point symmetry of the Ni5 site ( 3m.), the molecular field generated by the six Ni1 neighbor atoms forming two tetrahedra with antiparallel spin cancels exactly at this 2a position (figure 5(left)). This can explain that these Ni5 atoms have no ordered moments in the AFM structure. On the contrary in the FM configuration, where all the Ni moments are parallel to the c axis a small Ni moment is induced on the 2a position by the magnetic moments of the 6 Ni1 neighbors as explained above (figure 5(right)).

Similar geometric environment with a central atom surrounded by six neighbors, which can be coupled ferromagnetically or antiferromagnetically, depending on the presence of an ordered moment or not on the central atom position, has been observed in other Laves phase systems: C14 hexagonal TiFe 2 , Hf 0.825 Ta 0.175 Fe 2 and monoclinic YFe 2 H 4.2 derived from a C15 cubic parent compound [START_REF] Brown | Magnetic structure of the Laves phase compound TiFe 2[END_REF][START_REF] Diop | Neutron diffraction study of the itinerant-electron metamagnetic Hf 0.825 Ta 0.175 Fe 2 compound[END_REF][START_REF] Paul-Boncour | Interplay between crystal and magnetic structures in YFe 2 (H α D 1-α ) 4.2 compounds studied by neutron diffraction[END_REF].

In La 2 Ni 7 , the distance between two next Ni1 neighbors of different blocks connected via a Ni5 atom is 5.20 Å and these distances are large enough to favor negative J Ni1-Ni1 interactions in the absence of a moment on Ni5 site. Except for Ni5 atoms, all the other Ni atoms carry Ni moments whose magnitudes are close to those calculated in the wFM state (table 1 and figure 4). As their environment remains similar in both FM and AFM structures, they are less affected by the magnetic ordering contrary to the Ni5 atoms.

The DOS for this AFM structure shows that the narrow peak observed in NSP DOS is shifted below E F for the majority spin and N(E F ) falls in a valley at E F (figure 3(c)). However, only from the energy calculated for the proposed AFM and the FM structures, it is not possible to distinguish the most stable state since the difference is negligible with less of 1 meV/at., whereas the DFT accuracy is known to be few meV/at.. Thus, whatever the XC functional or method, the DFT calculation indicates that both magnetic structures present the same stability at 0 K. In table 2, the total energy difference of A 2 Ni 7 compounds is compared in NSP, FM and AFM structures for both the rhombohedral and hexagonal symmetries. It shows that the most stable state is the magnetic state (AFM or FM) for both La 2 Ni 7 and Y 2 Ni 7 without a clear difference between both symmetries, as already discussed in previous paper about stacking phases [START_REF] Crivello | Structural stability of AB(y) phases in the (La,Mg)-Ni system obtained by density functional theory calculations[END_REF].

The DFT calculation allows to propose an hexagonal AFM structure, whereas it become more challenging to propose an AFM structure for the rhombohedral cell. In fact, the AFM ordering requires an even number of blocks to cancel the total magnetic moment, as observed in the Ce 2 Ni 7 type structure. As the Gd 2 Co 7 type structure contains three blocks, it will require a doubling of the unit cell along the c axis to obtain a comparable AFM order. As the c parameter of 3R-Y 2 Ni 7 is already 36.19 Å, this will generate a very anisotropic magnetic cell with c/a ≈ 15, compared to only c/a ≈ 5 for 2H-La 2 Ni 7 . Although we have not calculated such large magnetic cell, it was observed experimentally that 3R-La 2 Ni 7 should have a ferromagnetic ground state. Indeed, the magnetization curves of La 2 Ni 7 sample annealed at 873 K and containing a mixture of hexagonal and rhombohedral phases, were analyzed by a superposition of FM and AFM states [START_REF] Parker | Magnetic properties of La 2 Ni 7[END_REF]. This proposed AFM structure for 2H-La 2 Ni 7 allows explaining several experimental features of the literature as detailed below:

(i) it confirms the high uniaxial anisotropy determined from the magnetization curves of 2H-La 2 Ni 7 by Parker et al [START_REF] Parker | Magnetic properties of La 2 Ni 7[END_REF]. They calculated an anisotropy field of at least 12 T and an uniaxial anisotropy energy greater than 160 kJ m -3 . (ii) It can explain the positive value of θ P above T N [START_REF] Buschow | Magnetic properties of La 2 Ni 7 and its hydride[END_REF][START_REF] Tazuke | Metamagnetic transitions in hexagonal La 2 Ni 7[END_REF][START_REF] Tazuke | Magnetic properties of La-Ni system[END_REF], as locally the FM interactions remain larger than the AFM ones. Parker et al [START_REF] Parker | Magnetic properties of La 2 Ni 7[END_REF], already suggested that the large value of θ P was characteristic of a metamagnetic behavior of a compound which has uniaxial anisotropy and with ferromagnetic intralayer and antiferromagnetic interlayer exchange interactions. (iii) The study of 2H-La 2 Ni 7 by NPD did not allow to observe magnetic lines at low temperature [START_REF] Tazuke | Magnetic properties of La-Ni system[END_REF]. A simulation of the NPD pattern was therefore done for FM and AFM structures assuming a wavelength of 2.43 Å, typical of a spectrometer dedicated for magnetic structure determination (figure S2). This reveals how it will be difficult to observe the magnetic structures experimentally. As the magnetic cell has a [0 0 0] propagation vector, it would only increase very weakly few nuclear line intensities. The most intense AFM magnetic line will contribute to only 1.2% of the (1 0 0) Bragg peak and even less considering the experimental moments and not the calculated one.

Although the DFT calculations cannot observe a clear difference of stability between the AFM and FM structures, all the experimental magnetic studies clearly indicates that the ground state of 2H-La 2 Ni 7 is AFM. But the difference of stability between the two types of magnetic order is clearly small: it was observed experimentally that the substitution of only 3 at% of Cu for Ni stabilizes a FM state, whereas the AFM ground state is maintained up to 10 at% Co [START_REF] Fukase | Itinerant electron magnetism of hexagonal La 2 Ni 7[END_REF]. This difference was attributed to a critical number of d electrons as the total number of d-electron increases upon Cu substitution and decreases upon Co substitution. It confirms that we are very close to the limit of stability of the AFM structure, which can be favored by the existence of spin-fluctuations. The low value of the saturation moment and the large Rhodes-Wohlfarth ratio observed for both 2H-La 2 Ni 7 [START_REF] Parker | Magnetic properties of La 2 Ni 7[END_REF][START_REF] Fukase | Itinerant electron magnetism of hexagonal La 2 Ni 7[END_REF][START_REF] Tazuke | Metamagnetic transitions in hexagonal La 2 Ni 7[END_REF][START_REF] Tazuke | Magnetic properties of La-Ni system[END_REF] and 3R-Y 2 Ni 7 compounds [START_REF] Bhattacharyya | Investigation of weak itinerant ferromagnetism and critical behavior of Y 2 Ni 7[END_REF] as well as other experimental results confirms their weak itinerant character and the existence of large spin fluctuations in both compounds. In addition, the metamagnetic behavior of the magnetization curves of La 2 Ni 7 was analyzed within the Moriya and Usami theory using a Landau-type free energy equation, used in the frame of systems with large spin fluctuations [START_REF] Tazuke | Metamagnetic transitions in hexagonal La 2 Ni 7[END_REF][START_REF] Moriya | Recent progress in the theory of itinerant electron magnetism[END_REF][START_REF] Moriya | Theory of itinerant electron magnetism[END_REF].

For weak itinerant magnetic systems, the existence of large spin fluctuations was able to explain the difference between results obtained by DFT calculations and those observed experimentally. For example, Ni 3 Ga is a strongly renor malized paramagnet, but the DFT calculation predicts a weak ferromagnetic ground state similar to isostructural Ni 3 Al, with even larger Ni moment [START_REF] Aguayo | Why Ni 3 Al is an itinerant ferromagnet but Ni 3 Ga is not[END_REF]. This difference has been explained by taking into account the low-frequency spin fluctuations present in this compound which is close to a ferromagnetic quantum critical point. The theoretical study of TiAu confirmed also the influence of spin fluctuations for the stabilization of the weak antiferromagnetic state, as well as the overestimation of the Ti moment determined by DFT calcul ations [START_REF] Goh | Competing magnetic instabilities in the weak itinerant antiferromagnetic TiAu[END_REF]. Such overestimation of magnetic moment is often observed for compounds which lies close to quantum critical point.

In the case of 2H-La 2 Ni 7 the magnetic order is driven by the magnitude of the Ni5 moment which is predicted by DFT calcul ation to be 0.023 µ B in the FM structure and 0 in the AFM structure. But it was found, that the total magnetic moment of 2H-La 2 Ni 7 in FM state is overestimated of at least 30% compared to the experimental value. In this case, the influence of spin fluctuations on the particular Ni5 site should be large enough to stabilize a geometric environment with the next Ni1 neighbors adopting an antiparallel coupling.

Conclusions

In this work, we have proposed for hexagonal La 2 Ni 7 an AFM structure with a stability comparable to that of the FM structure. This AFM structure of 2H-La 2 Ni 7 results from the geometric conditions presented by its stacked hexagonal structure: the molecular field is canceled on the Ni5-2a site at z = 0 and z = 1/2 which belong to the [AB 2 ] units and acts as in inversion center for the first neighbors Ni at the interface between the [AB 2 ] and [AB 5 ] units (Ni1-12k). These nearest Ni atoms adopt an AFM coupling and the symmetry of the structure on this position allows an inversion of the sign of the other Ni spin orientation belonging to different blocks. This generates a modulated AFM structure with two ferromagnetic slabs of opposite directions separated by a non-magnetic layer. Such magnetic structure type was already observed in some AFe 2 Laves phases or their hydrides. The metamagnetic transition from AFM towards a FM structure is explained by a non-zero Ni moment on the Ni5 site induced by the application of a magnetic field.

The stabilization of this AFM structure, observed experimentally, should be explained by the existence of large spin fluctuations, not taken into account by simple DFT calculation, as observed for other weak itinerant magnets as Ni 3 Ga or TiAu. Further works using Moriya and Usami theory of spin fluctuation, could be very interesting to consider for La 2 Ni 7 .

Further experimental and theoretical studies will be performed on pseudo-binary La

La 2

 2 Ni 7 and Y 2 Ni 7 crystallize in a Ce 2 Ni 7 type hexagonal structure (P6 3 /mmc space group) and Gd 2 Co 7 -type rhombohedral structure (trigonal symmetry, R-3m space group) respectively as presented in figure 1. These phases result from the stacking of [A 2 B 4 ] and [AB 5 ] units according to the rule [A 2 B 4 + n • AB 5 ], where n is an integer. If n = 2, the stacking leads to the A 2 B 7 formation in two polymorphic forms: the hexagonal Ce 2 Ni 7 cell [2H] contains two blocks of [A 2 B 4 + 2 AB 5 ], whereas the rhombohedral Gd 2 Co 7 cell [3R] contains three blocks of [A 2 B 4 + 2 AB 5 ] in the hexagonal description as shown in figure 1 [34]. For the sake of simplicity, the [A 2 B 4 ] subunit will be renamed as [AB 2 ] in the following. Both hexagonal and rhombohedral structures are strongly anisotropic, with large values of the c parameter compared to the a parameter (a = 5.062 Å, c = 24.71 Å for 2H-La 2 Ni 7 ; a = 4.946 Å and c = 36.26 Å for 3R-Y 2 Ni 7 ). Both cell contains 2 A = {Y, La} and 5 B = Ni different Wyckoff sites (table 1) which are characterized by their positions along the c axis forming A and Ni layers. Each A site is located either in one [AB 5 ] or one [AB 2 ] units, whereas each Ni site is located either in one of these units or at the interface between 2 units. Regarding the Ce 2 Ni 7 type structure, it displays a mirror inversion of the [AB 2 ] units at z = 0 and z = 1/2.

Figure 1 .

 1 Figure 1. Comparison of the rhombohedral and hexagonal structures of A 2 Ni 7 compounds showing the stacking of [AB 2 ] and [AB 5 ] units and the A and Ni atom positions.

FFigure 2 .

 2 Figure 2. Non-spin polarized density of state of 3R-Y 2 Ni 7 and 2H-La 2 Ni 7 with a zoom near E F in the inset (left) and corresponding electronic band structure of 2H-La 2 Ni 7 zoomed near E F (right) in GGA-PBE. The full scale of the latter is given in supplementary materials.

Figure 3 .

 3 Figure 3. Calculated spin polarized DOS with partial 3d-Ni contribution for (a) ferromagnetic 3R-Y 2 Ni 7 (M = 1.24 µ B /f.u.), (b) ferromagnetic 2H-La 2 Ni 7 (M = 1.08 µ B /f.u.) and (c) antiferromagnetic 2H-La 2 Ni 7 (M = 0 µ B /f.u.).

Figure 4 .

 4 Figure 4. Coordination polyhedra around the Ni1, Ni2 and Ni3 (Ni4) atoms. For the Ni3 (Ni4) atoms two types of polyhedra have been represented for interatomic distances below 2.65 Å (left) and below 2.95 Å (right) The magnetic vectors have been represented with length proportional to their magnetic moments.

Figure 5 .

 5 Figure 5. Coordination polyhedra of Ni5 atoms surrounded by Ni1 atoms with magnetic vectors in AFM and FM ordering perpendicular (top) and parallel (bottom) to the c axis.

  (b). Similar to the m i individual moments, the CT presents two different behaviors for Ni in [AB 2 ] and [AB 5 ] units: a sharp decrease from Ni5 to Ni3,4 and an almost constant CT value for Ni3,4 and Ni2 atoms. The CT values are related to the number of La first neighbors (table 1). For Ni atoms with the largest CT values (Ni5 and Ni1) the additional electrons transferred from La atoms contribute to a progressive filling of the d-Ni bands. It yields a reduction of their local magnetic moment compared to the values extrapolated from Ni3,4 or Ni2 moment versus m Tot .

mFigure 6 .

 6 Figure 6. Ni moment and charge transfer CT from La to Ni versus the sum of the Ni moments in the first coordination sphere m Tot for each Ni site in 2H-La 2 Ni 7 .

Figure 7 .

 7 Figure 7. Antiferromagnetic structure of La 2 Ni 7 and Ni moment distribution along the c axis in both antiferromagnetic (red) and ferromagnetic (blue) structures of La 2 Ni 7 . (Color online).

  2-x Y x Ni 7 compounds to observe the evolution from wAFM 2H-La 2 Ni 7 towards wFM 3R-Y 2 Ni 7 , and verify what the critical geometric and electronic parameters are. First principles calculations of other compounds with stacking structures like ANi 3 or A 5 Ni 19 compounds from [A 2 Ni 4 + n • ANi 5 ] with n = 1 and 3 respectively, could be also very interesting in order to analyze the relation between their crystal structures and magnetic properties.

Table 1 .

 1 Calculated magnetic moments in hexagonal La 2 Ni

	La1	4f 1	[AB 2 ]	4 A + 12 Ni	-0.026	±0.015
	La2	4f 2	[AB 5 ]	2 A + 18 Ni	-0.041	±0.041
	Ni1	12k	[AB 2 /AB 5 ]	5 A + 7 Ni	0.102	±0.095
	Ni2	6h	[AB 5 ]	4 A + 8 Ni	0.287	±0.293
	Ni3	4f 3	[AB 5 ]	3 A + 9 Ni	0.198	±0.186
	Ni4	4e	[AB 5 ]	3 A + 9 Ni	0.197	±0.197
	Ni5	2a	[AB 2 ]	6 A + 6 Ni	0.023	0
	M -0.035	-0.033
	3R-Y 2 Ni 7				FM
	Atom	Site	Unit	Neighbors	m (µ B /at.)
	Y1	6c 1	[AB 5 ]	2 A + 18 Ni	-0.043
	Y2	6c 2	[AB 2 ]	4 A + 12 Ni	-0.034
	Ni1	18h	[AB 2 /AB 5 ]	5 A + 7 Ni	0.153
	Ni2	9e	[AB 5 ]	4 A + 8 Ni	0.315
	Ni3	6c 3	[AB 5 ]	3 A + 9 Ni	0.208
	Ni4	6c 4	[AB 5 ]	3 A + 9 Ni	0.216
	Ni5	3b	[AB 2 ]	6 A + 6 Ni	0.071
	M total (µ B /f.u.)				1.238
	ΔE (eV/f.u.)				-0.035

7 (FM, AFM) and rhombohedral Y 2 Ni 7 (FM). ΔE = E -E 0 (NSP). 2H-La 2 Ni 7 FM AFM Atom Site Unit Neighbors m (µ B /at.) m (µ B /at.) total (µ B /f.u.) 1.08 0 ΔE (eV/f.u.)

Table 2 .

 2 Heat of formation of A 2 Ni 7 compounds in both allotropic hexagonal and rhombohedral structures in different magnetic states. For hexagonal La 2 Ni 7 the calculations were done with different functionals and method as described in the text.

					ΔE = E(i) -E(NSP)
	Compound	XC-method	Structure	Magnetic state	(meV/f.u.)
	Y 2 Ni 7	GGA	Rhombohedral	NSP	0
				FM	-35
			Hexagonal	NSP	0
				FM	-35
				AFM	-33
	La 2 Ni 7	GGA	Rhombohedral	NSP	0
				FM	-31
			Hexagonal	NSP	0
				FM	-35
				AFM	-33
		Non-colinear	M parallel to z	FM	-34
				AFM	-33
		Non-colinear	M parallel to x, y	FM	-3
		LDA	Hexagonal	NSP	0
				FM	-22
				AFM	-20
		SCAN	Hexagonal	NSP	0
				FM	-75
				AFM	-72
		GGA + U	Hexagonal	NSP	0
				FM	-35
				AFM	-32

J. Phys.: Condens. Matter 32 (2020) 145802

Acknowledgments

The authors are very thankful to M Gupta for her advice and fruitful discussion. DFT calculations were performed using HPC resources from GENCI-CINES (Grant 2019-96175).

ORCID iDs

J-C Crivello https://orcid.org/0000-0002-4849-2556 V Paul-Boncour https://orcid.org/0000-0002-0601-7802