Christophe Candolfi

Martin Misek

Patrick Gougeon

Rabih Al

Rahal Al Orabi

Philippe Gall

Regis Gautier

Sylvie Migot

Jaafar Ghanbaja

Jiri Kastil

Petr Levinsky

Martin Míšek

Régis Gautier

Jiří Kaštil

Petr Levinský

Jiří Hejtmánek

Anne Dauscher

Bernard Malaman

Bertrand Lenoir

Coexistence of a charge density wave and superconductivity in the cluster compound K2Mo15Se19

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

I. INTRODUCTION

Periodic modulations of the conduction electrons, referred to as charge density wave (CDW), are often observed in low-dimensional systems such as quasi-1D chain-like and 2D layered materials 1-3. While phonon-mediated superconductivity is favored by a high density of states at the Fermi level 𝑁(𝐸 𝐹 ), the coupling between the electronic and lattice degrees of freedom gaps 𝑁(𝐸 𝐹 ) through the appearance of a different structural periodicity at the Fermi wave vector below the transition temperature 𝑇 CDW . Because of these two antagonistic characteristics, CDW and superconductivity are widely thought of as being two competing electronic orders 3,4. In many CDW systems such as the canonical layered dichalcogenides 1,5, suppressing the long-range CDW order through intercalation, substitution or external pressure often leads to the emergence of superconductivity 6-9. In some specific cases, these tuning parameters continuously drive the transition temperature 𝑇 CDW towards 𝑇 = 0 K, marking the onset of a structural quantum critical point (SQCP) near which superconductivity also develops 10-13. Because the appearance of unconventional superconductivity is often furnished by destroying such an ordered phase, the study of the interplay between these two electronic phenomena remains a central theme in condensed matter physics, reinforced by the recent discovery of CDW order in high-temperature cuprate superconductors 14-17. Despite the discovery of many CDW materials to date, only a handful of them shows a coexistence of CDW and superconductivity under ambient pressure or without manipulation of their chemical composition 18-26. In addition to Heusler phases such as LuPt2In 10 or the ternary rareearth dicarbides RNiC2 (R = rare earth) 27-30, the ternary stannides crystallizing with a Yb3Rh4Sn13-type structure have recently attracted attention due to the interplay between CDW and superconductivity and the possibility to induce a SQCP 11,13,31-33. Identifying novel systems supporting both phenomena would provide interesting platforms to systematically investigate their interplay via various probes, such as external pressure or intentionally introducing weak disorder by electron irradiation or substitutions 34,35, and possibly induce SQCP through chemical substitutions. Furthermore, the coexistence of both electronic orders LnMo6S8 (Ln = Tb, Dy, Er) and HoMo6S8, respectively 54-57. The low dimensionality of the crystal structure of some of these compounds make them prone to various structural or electronic instabilities and hence, a fertile area for searching novel materials experiencing symmetry-breaking density wave transitions.

Among the numerous families of Mo-based cluster compounds known to date, the compounds A2Mo15Se19 (A = Li, K, Na, Rb, Ba, Sn, In, Pb or Tl) have been shown to become superconducting with a critical temperature 𝑇 c of up to 2.8 K that depends on the nature of the A atoms 58,59. Here, we consider the K2Mo15Se19 compound and demonstrate that the previously reported superconducting state emerging at 𝑇 c = 2.8 K 59 coexists with a CDW transition that sets in upon cooling below 𝑇 CDW = 114 K. The CDW transition is accompanied by a weak commensurate modulation that doubles the c axis of the room-temperature hexagonal crystal lattice shown in Figure 1. Preliminary low-temperature measurements performed on other A2Mo15Se19 compounds (A = In, Rb, Tl) further indicate that the coexistence of superconductivity and CDW may be commonly observed in this family. Due to the large variety of A atoms that can be accommodated by this crystal structure, the A2Mo15Se19 family is a novel intriguing system in which the interplay between these two electronic orders can be systematically investigated.

II. EXPERIMENTAL AND COMPUTATIONAL DETAILS

All the elements and precursors were handled in an argon-filled glovebox to avoid contamination by oxygen and moisture. Prior to use, Mo powders were reduced under H2 flowing gas at 1000°C during 10 h to eliminate possible traces of oxygen. MoSe2, K2Mo6Se6 and Mo were used in powder form as starting materials for the solid-state syntheses. MoSe2 was prepared by direct reaction of stoichiometric quantities of the elements in an evacuated, sealed silica tube, heated to 700°C and maintained at this temperature for two days. K2Mo6Se6 was synthesized by a two-step process. As a first step, In2Mo6Se6 was prepared from a stoichiometric mixture of InSe, MoSe2 and Mo at 1000°C for 36 h in an evacuated, sealed silica tube. The second step consisted in an ion-exchange reaction of In2Mo6Se6 with KCl at 800°C. Powders of In2Mo6Se6 and KCl were mixed in a ratio 1:2.5 before being cold pressed into cylindrical pellets. The pellets were subsequently sealed under vacuum in a long silica tube.

The end of the tube containing the pellet was placed in a furnace, leaving about 5 cm of the other end out of the furnace. The furnace was heated at 800°C for 48 h. After reaction, white crystals of InCl were observed at the cold end of the tube. All starting reagents were determined to be monophasic according to their powder X-ray diffraction patterns. Phase-pure polycrystalline samples of K2Mo15Se19 were finally prepared from stoichiometric quantities of K2Mo6Se6, Mo and MoSe2 heated at 1300°C during 40 h in a Mo crucible, sealed under a low pressure of argon using an arc-welding system. The obtained ingot was ground into fine powders that were subsequently consolidated by hot uniaxial pressing under vacuum (about 10 - 2 mbar). To this end, the powders were introduced into a graphite die (Ø 12mm), previously coated with boron nitride. The consolidation process was realized at 1300°C under a uniaxial pressure of 85 MPa with a dwell time of 2h. The density of the consolidated cylindrical pellet, determined from the weight and geometric dimensions, was calculated to be around 98% of the theoretical density from crystallographic data.

Single crystals of K2Mo15Se19 were grown by heating a polycrystalline sample at 1750°C for 1h, cooled to 1000°C at 100°C h -1 and finally furnace-cooled to room temperature. Cuboidshaped single crystals of typical dimensions of 300200200 m 3 with grey metallic luster could be isolated.

The purity of the polycrystalline sample was checked by room temperature powder x-ray diffraction (PXRD) using a Bruker D8 Advance (CuK1 radiation,  = 1.54056 Å) diffractometer. Additional PXRD measurements were performed at 100 K using an Xpert Pro MPD diffractometer (CuK1 radiation,  = 1.54056 Å and CuK2 radiation,  = 1.54439 Å, ratio K2/K1 = 0.5).

The crystal structure of single crystals was verified by XRD at 300 K and further measured down to 100 K on a Bruker APEX-II diffractometer equipped with a CCD detector using MoK radiation ( = 0.71073 Å). The structure solution was determined by direct methods and refined by full-matrix least-squares techniques with the Sir97 and SHELX-2013 software, respectively, implemented through the WinGX program package 60. Details of structure refinements and refined crystallographic parameters are given in Tables S1 andS2, respectively, in Supplemental Material 61.

For transmission electron microscopy experiments, a thin slice was prepared from a grown single crystal of K2Mo15Se19 by the dual focused ion beam (FIB)scanning electron microscope system using the in situ lift-out technique. At 300 K, scanning TEM (STEM) measurements in high-angle annular dark-field (HAADF), annular bright-field (ABF) and high resolution (HRTEM) modes were carried out on a JEOL ARM 200F-cold FEG microscope operating at 200 keV. This system is equipped with a spherical aberration (Cs) probe and image correctors. In situ cooling TEM experiments were performed using a double-tilt liquidnitrogen-cooling TEM sample holder (Gatan Inc., USA), which can reach temperatures of about 100 K. The sample was slowly cooled down to the lowest temperature and the experiments were conducted after a significant waiting time in the absence of the electron beam to achieve thermal equilibrium. Selected area electron diffraction (SAED) and HRTEM images were collected at about 100 K along various zone axes.

Measurements of the transport properties were performed on samples cut from the consolidated polycrystalline pellet using a diamond-wire saw. At low temperatures (2 -300 K), a Quantum Design Physical Property Measurement System (PPMS) was used to measure simultaneously the temperature dependence of the electrical resistivity, thermopower and thermal conductivity with the thermal transport option (TTO). For this measurement, four copper leads were mounted on the samples using a silver-containing conductive epoxy. Specific heat measurements under zero magnetic field were performed in the temperature range 0.35 -4 K using the 3 He option of the PPMS system. A small polycrystalline piece of approximately 20 mg was glued with a tiny amount of Apiezon N grease to ensure a good thermal contact between the sample and the platform of the specific heat puck. Further measurements with magnetic field sweeps from 0 up to 3 T were performed. For all measurements, the contribution of the grease has been systematically measured under the same magnetic fields prior to gluing the sample.

Magnetic property measurements were carried out between 2 and 300 K using a Magnetic Property Measurement System (MPMS) Superconducting Quantum Interference Device (SQUID) magnetometer. The magnetic susceptibility 𝜒 was determined from the slope of the magnetization curves, 𝜒 = 𝜕𝑀/𝜕𝜇 0 𝐻 for magnetic fields 𝜇 0 𝐻 → 0, measured as a function of temperature. These measurements were performed on a piece of the polycrystalline bar-shaped sample used for TTO measurements of approximately 90 mg. The sample was loaded in a gelatin capsule which was mounted in a plastic straw. Further 𝑀(𝜇 0 𝐻) curves were measured below 3 K to determine the lower and upper critical fields 𝐻 𝑐1 and 𝐻 𝑐2 , respectively. Zerofield-cooled (ZFC) susceptibility data were collected below 10 K with an applied field of 10 Oe. The ZFC data have been corrected for a demagnetization factor 𝑁 = 0.456 estimated using the formula derived in Ref. 62 for a rectangular cuboid shape.

Density functional theory (DFT) calculations were carried out using the WIEN2k program package 63. The generalized gradient approximation (GGA) exchange-correlation functional within the parameterization of Perdew, Burke and Ernzehof (PBE) was used for calculations 64. A plane-wave cutoff corresponding to 𝑅 𝑀𝑇 𝐾 𝑚𝑎𝑥 = 7 was used with the radial wave-functions inside the nonoverlapping muffin-tin spheres surrounding the atoms expanding up to 𝑙 𝑚𝑎𝑥 = 12. The charge density was Fourier expanded up to the reciprocal vector 𝐺 𝑚𝑎𝑥 = 14 Å -1 . Total energy convergence was achieved with respect to the Brillouin zone (BZ) integration using a mesh of 500 k-points. We used 3000 k-points in the BZ to compute the electronic band structure and Fermi surface. Views of the Fermi surface were obtained using the software XCrySDen 65. The dispersion curves were shifted so that the Fermi level 𝐸 𝐹 lies at 0 eV. Because spin-orbit coupling (SOC) hardly modifies the electronic band structure of K2Mo15Se19 (see Figure S1 in Supplemental Material 61), SOC was not taken into account.

III. RESULTS

A first clear experimental evidence for a CDW transition in K2Mo15Se19 is provided by the temperature dependence of the electrical resistivity 𝜌(𝑇) shown in Figure 2a. Upon cooling, a metallic-like dependence prevails down to about 115 K where an upturn in 𝜌(𝑇) can be observed, followed by a broad maximum. These two characteristics are typical signatures of a CDW transition with an onset temperature of 𝑇 CDW = 114 K 10. Below the broad maximum, 𝜌(𝑇) decreases upon further cooling to 3 K.

The anomaly seen in 𝜌(𝑇) leads to corresponding signatures in the magnetic susceptibility 𝜒(𝑇), specific heat 𝐶 𝑝 (𝑇) and thermopower 𝛼(𝑇) (Figures 2b, 2c and2d). Above 𝑇 CDW , 𝜒(𝑇) is practically temperature-independent as expected in a non-magnetic metal. The positive values measured are due to the Pauli susceptibility 𝜒 𝑃 of the conduction electrons that outweighs the diamagnetic 𝜒 𝑑𝑖𝑎 and Van Vleck 𝜒 𝑉𝑉 contributions. A drop in 𝜒(𝑇) is observed as the temperature decreases and passes through 𝑇 CDW . The decrease in the 𝜒 values is consistent with a gap opening in the electronic density of states at the Fermi level. However, the fact that the 𝜒(𝑇) values remain positives below 𝑇 CDW implies that 𝜒 𝑃 is still the main contribution indicating only a partial gapping. These conclusions are further corroborated by the evolution of the thermopower (Figure 2d) which represents a sensitive probe of changes in the Fermi surface. The linear variation upon cooling, characteristic of a diffusive regime in a metallic compound, shows a sharp increase at 𝑇 CDW , peaks near 70 K before decreasing down to the lowest temperature measured. The CDW transition is not accompanied by a change in the dominant carrier type which remains hole-like over the entire temperature range. The slight but visible increase in 𝛼 below 𝑇 CDW is consistent with a decrease in the hole concentration as a result of the partial gapping at 𝑁(𝜀 𝐹 ).

The anomalies seen in the magnetic and transport coefficients are not marked by a sudden, step-like variation suggesting a second-order phase transition in the present case. Although unusual for this type of transition, which is usually first-order, the second-order nature of the present CDW is not unprecedented and has been observed, for instance, in the ternary Ce3Rh4Sn13 33 or in the Lu(Pt1-xPdx)2In alloys for which similar temperature dependences of 𝜌 and 𝜒 have been reported 10. Measurements of 𝜒 performed on cooling and warming across 𝑇 CDW did not reveal any hysteretic behavior to within experimental uncertainty, further suggestive of a second-order transition (Figure S2 in Supplemental Material 61).

One important characteristic of the present CDW transition is the absence of a welldefined peak in the specific heat at 𝑇 CDW , should it be measured upon cooling or warming (Figure 2c). In the present case, the 𝐶 𝑝 values plateau from 114 down to 110 K, the temperature at which 𝐶 𝑝 (𝑇) decreases again upon cooling. In contrast, most of the CDW systems studied so far displays a large specific heat jump at 𝑇 CDW associated with a non-negligible entropy variation across the transition 10,66,67. Such a commonly behavior has been observed for both first-order and second-order CDW transitions and is therefore rather tied to the periodic lattice distortion that accompanies the CDW 10,66,67. The absence of significant anomaly in 𝐶 𝑝 (𝑇) suggests that the CDW in K2Mo15Se19 is not strongly coupled synonymous with a small amplitude of the lattice distortion.

The presence of a modification of the crystal lattice accompanying the CDW transition in K2Mo15Se19 has been directly observed by electron diffraction (Figure 3). While the roomtemperature STEM images (Figures 3a and3b) are consistent with the hexagonal crystal structure of K2Mo15Se19, electron diffraction images, taken along the 101 ̅ 0 zone axis (equivalent to the 011 ̅ 0 zone axis in the present case) at 300 K (Figure 3c) and below 𝑇 CDW (Figure 3d), show that additional satellite reflections appear in addition to the main Bragg reflections. The superlattice reflections appear solely along the 0001 direction corresponding to a modulation of the crystal structure along the c axis. The absence of modulation along the a axis is further demonstrated by the image taken along the 123 ̅ 0 zone axis (Figure S3 in Supplemental Material 61), which does not evidence the presence of satellite reflections along the 21 ̅ 1 ̅ 0 direction. Within the accuracy of these measurements, the spacing between the main and satellite reflections leads to an integer modulation vector 𝑞 𝐶𝐷𝑊 ⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ = 0.5𝑐 * indicative of a modulation commensurate with the crystal lattice. These measurements further suggest that the modulated structure retains a hexagonal symmetry, which is supported by electron diffraction images showing reflections in the first-order Laue zone (Figure S4 in Supplemental Material 61), for which the shift observed between the zero-order and the first-order Laue zones are consistent with the symmetries of the 𝑅3 ̅ 𝑐 space group observed above 𝑇 CDW . Thus, this lattice distortion contrasts with the rhombohedral-to-monoclinic or rhombohedral-to-triclinic distortions observed at low temperatures in some Chevrel phases, which are often of first-order nature and result in a strong hysteric behavior of the transport properties around the transition temperature 68-75. Because the modulation of the basic unit cell of K2Mo15Se19 is weak, leaving fingerprints neither in low-temperature laboratory single-crystal X-ray diffraction nor in powder X-ray diffraction, and due to the large number of atoms in the low-temperature unit cell (432 atoms, that is, twice as high as at room temperature), further synchrotron diffraction experiments on single crystal will be necessary to ascertain the superspace group and develop a detailed structural model of the modulated structure.

Upon further cooling below 𝑇 CDW , superconductivity emerges with a critical temperature 𝑇 c of 2.8 K as revealed by the drop of 𝜌(𝑇) to zero (Figure 4a). This 𝑇 c value is in agreement with an early study on this compound which had reported a critical temperature of 2.45 K 59.

The bulk nature of the superconducting transition is further confirmed by measurements of the specific heat and magnetic susceptibility (Figures 4b and4c, respectively). The clear meanfield-type anomaly observed in the 𝐶 𝑝 (𝑇)/𝑇 data at 2.6 K, as determined by an isentropic construction, is consistent with the transition temperature observed in 𝜌(𝑇). Further evidence for a bulk phenomenon is provided by the 𝜒(𝑇) data which show a Meissner effect of nearly 100% indicating a full superconducting volume fraction.

A more detailed characterization of the superconducting state has been carried out by 𝐶 𝑝 measurements under magnetic fields 𝜇 0 𝐻 ranging between 0 and 3 T. As expected, applying an K2Mo15Se19 are summarized in Table 1.

IV. DISCUSSION

One important question raised by these results is the underlying physical mechanism driving the CDW ordering. While Fermi surface nesting has for long been considered as the driving force dictating the CDW wave vector 𝑞 CDW and the corresponding lattice distortion 77,78, a growing body of evidence acquired over the last years suggests that this simple scenario fails in most of the known CDW systems 78-81. This failure led to a recentlyproposed new classification scheme of CDW with the momentum dependence of the electron-phonon coupling matrix playing a central role in determining the main traits of the CDW order 82.

In order to shed light on the possible origin of the CDW in K2Mo15Se19, we computed the electronic band structure and Fermi surface using density functional calculations (Figure 5a and Figure S6 in Supplemental Material 61). The Fermi level, hosted by the valence bands, crosses four distinct electronic bands giving rise to a complex Fermi surface, represented in Figure 5b.

The Fermi surface comprises a hole-like cylinder at the zone center  surrounded by a second distorted hole-like cylinder, small electron-like elongated ellipsoids near the M and L points, and a large electron-like surface (sometimes referred to as a "monster" in the literature, see, e.g., Refs. 83 and 84) spanning most of the Brillouin zone. While 2D-like Fermi surfaces are often considered as favoring Fermi surface nesting, no significant portions of the present Fermi surface seem to be clearly connected through the experimentally observed wave vector 𝑞 CDW , as it would be expected within a nesting scenario. Hence, these results do not seem to confirm a clear, dominant Fermi surface nesting at the observed wave vector suggesting that K2Mo15Se19 may be another example of system where the CDW is driven by q-dependent electron-phonon coupling. Further direct measurements of the phonon dispersions and their linewidths across 𝑇 CDW by inelastic X-ray scattering will be important in better understanding the origin of the CDW transition in K2Mo15Se19.

The coexistence of CDW and superconductivity reported herein is likely not limited to K2Mo15Se19 but may be also observed in other A2Mo15Se19 compounds for various A atoms. The comparison of low-temperature measurements performed on the A = Rb, In and Tl compounds (Figure S7 in Supplemental Material 61) show that all of them exhibit a similar CDW signature between 100 and 150 K suggesting that CDW may be ubiquitous in this family of compounds. Interestingly, combined with the 𝑇 c reported for several A2Mo15Se19 compounds (Table 2) 58, these results show that both 𝑇 CDW and 𝑇 c are not equivalent and depend on the nature of the A atoms. Although the highest 𝑇 c values seem to be obtained for the lightest elements, the absence of superconductivity in the Na compound suggests a more complex dependence of 𝑇 c with the nature of the A atoms. Furthermore, the presence of a more stronglycoupled CDW transition may naturally explain the absence of superconductivity in some of these compounds. Substitutions or application of high-pressure may be used as external parameters to tip the balance towards superconductivity by suppressing the CDW, thereby possibly driving the electronic system towards a QCP.

Finally, the large anisotropic thermal displacement parameters of the K atoms (see Table S2 in Supplemental Material 61), and more generally of the A atoms, adds another degree of complexity in this family of compounds. As observed in several cage-like compounds, large vibration amplitudes are associated with low-energy optical modes in the phonon spectrum, which are believed to be the key ingredient that limits the propagation of acoustic phonons.

Meanwhile, the presence of these localized, often anharmonic, modes may also play a role in determining the superconducting transition temperature 85-88. These so-called rattling modes may strongly couple with the conduction electrons contributing to enhance the electron-phonon coupling constant 𝜆 𝑒-𝑝ℎ and hence, 𝑇 c . This effect has been discussed in several families of cage-like compounds or in compounds where the rattling atoms reside in tunnels, similarly to the A2Mo15Se19 compounds 85-88. In these studies, the anharmonic character of the low-lying optical modes was found to correlate with 𝑇 c indicating that highly-anharmonic modes tend to reinforce superconductivity. In A2Mo15Se19, the low-energy modes associated with the dynamics of the A atoms might therefore contribute to strengthen the electron-phonon coupling, partially counterbalancing the removal of electronic states at the Fermi level upon crossing the CDW transition. Because calculations of the phonon spectra are challenging due to the large number of atoms in the unit cell above 𝑇 CDW , and a fortiori below 𝑇 CDW , inelastic neutron or X-ray scattering experiments would be worthwhile to determine and compare the degree of anharmonicity of the dynamics of the A atoms in these compounds and possibly unveil a correlation between this characteristic and 𝑇 c .

IV. CONCLUSION

Low-temperature physical property measurements combined with electron diffraction revealed the coexistence of CDW and superconductivity in the Mo-based cluster compound K2Mo15Se19. Electron diffraction performed below and above 𝑇 CDW = 114 K evidenced a weak commensurate modulation along the c axis of the crystal structure which is observable neither by laboratory PXRD nor by single-crystal XRD. The characterization of the superconducting state that develops below 𝑇 c = 2.8 K indicates that this compound can be classified as a weaklyto-intermediate-coupled type-II superconductor. Electronic band structure calculations suggest that K2Mo15Se19 provides another example of CDW system where the transition is probably not driven by Fermi surface nesting but is more likely due to the q-dependence of the electronphonon coupling. The similarities between the transport properties measured in K2Mo15Se19 and those observed in other A2Mo15Se19 compounds strongly suggest that the coexistence of a CDW state and superconductivity may be commonly observed in this family of compounds.

Future systematic studies will be important to better understand their interplay and ascertain the physical origin of the CDW transition. In this respect, the chemical tuning afforded by this family and the application of external pressure or introduction of weak structural disorder will be important tools for tipping the balance between CDW and superconductivity in these compounds. S1 andS2, respectively, in Supplemental Material 61. (right panel) Crystal structure projected along the c axis. In both panels, the hexagonal unit cell is represented by the black solid lines.

Tables

  in a single material may open the possibility to observe the Higgs mode in Raman spectra below the superconducting critical temperature 𝑇 c 36-39. Mo-based cluster compounds are a large class of materials that were recently shown to display interesting thermoelectric properties 40-50. Their crystal structures are built up by Mo3nX3n+2 (X = S, Se or Te) clusters units in which the nuclearity of the Mo3n clusters can vary from Mo3 up to Mo36 51-53. Many of them show superconductivity with some rare examples of emergence of superconductivity within the antiferromagnetic state or of ferromagnetic reentrant superconductivity (the so-called Jaccarino-Peter effect) in the Chevrel phases

  external magnetic field tends to shift down 𝑇 c . The normal-state specific heat has been analyzed via the conventional Fermi-liquid relation 𝐶 𝑝 = 𝛾𝑇 + 𝛽𝑇 3 where 𝛾𝑇 represents the contribution of the free charge carriers and 𝛽𝑇 3 the phonon contribution. A fit of the data plotted as 𝐶 𝑝 (𝑇)/𝑇 versus 𝑇 2 yields a Sommerfeld coefficient 𝛾 = 63 mJ mol -1 K -2 . After subtraction of the phonon contribution, the superconducting energy gap Δ(0) was estimated from a plot on a logarithmic scale of the electronic contribution to the specific heat 𝐶 𝑒𝑙 /𝛾𝑇 𝑐 versus 𝑇 𝑐 /𝑇. The comparison of this curve to the Bardeen-Cooper-Schrieffer (BCS) formula predicting 𝐶 𝑒𝑙 (𝑇) below 𝑇 𝑐 , 𝐶 𝑒𝑙 /𝛾𝑇 𝑐 = 8.5exp[-0.82Δ(0)/𝑘 𝐵 𝑇], where 𝑘 𝐵 is the Boltzmann constant, yields an energy gap of 0.40 meV corresponding to a ratio 2Δ(0)/𝑘 𝐵 𝑇 𝑐 = 3.64, that is, very close to the weakcoupling BCS value of 3.53. Further assuming a phonon-mediated pairing mechanism, the electron-phonon coupling constant 𝜆 𝑒-𝑝ℎ can be inferred from the McMillan relation 𝑇 𝑐 = 04(1+𝜆 𝑒-𝑝ℎ ) 𝜆 𝑒-𝑝ℎ -𝜇 * (1+0.62𝜆 𝑒-𝑝ℎ ) ] 76. With the Debye temperature 𝜃 𝐷 = 183 K determined from the 𝛽 parameter (𝜃 𝐷 = ( 3 with 𝑅 the gas constant and 𝑁 the number of atoms per formula unit), and assuming a Coulomb repulsion parameter 𝜇 * of 0.13 as commonly used in the literature, a coupling constant 𝜆 𝑒-𝑝ℎ of 0.61 is obtained, placing this compound among the weakly-to-intermediate-coupled superconductors.Measurements of the field dependence of the magnetization 𝑀 performed at 2 K (FigureS5in Supplemental Material 61) give estimates of the lower 𝜇 0 𝐻 𝑐1 and upper 𝜇 0 𝐻 𝑐2 critical fields of 4 mT and 0.6 T, respectively. In conventional superconductors, the BCS theory predicts that 𝜇 0 𝐻 𝑐2 should exhibit a temperature dependence following the law 𝜇 0 𝐻 𝑐2 ≈ 𝜇 0 𝐻 𝑐2 (0)[1 -(𝑇/𝑇 𝑐 ) 2 ]. This expression gives an upper critical field at 0 K, 𝜇 0 𝐻 𝑐2 (0), of approximately 2.9 T. Using a similar expression for the lower critical field yields an estimate for 𝜇 0 𝐻 𝑐1 (0) of 12 mT. Note that the critical field values should be taken only as estimates of the actual values due to the strongly anisotropic crystal structure of K2Mo15Se19 which likely gives rise to anisotropic critical fields. Thus, the polycrystalline nature of the sample used herein only yields average values of 𝜇 0 𝐻 𝑐1 , 𝜇 0 𝐻 𝑐2 and 𝜇 0 𝐻 𝑐2 (0). Nevertheless, considering these values as reasonable approximations of the actual values, the coherence length 𝜉 GL and the penetration depth 𝜆 GL are estimated to be 110 Å and 2000 Å using the two Ginzburg-Landau formulas 𝐻 𝑐2 (0) = Φ 0 /2𝜋𝜉 GL 2 and 𝐻 𝑐1 (0where ℎ is the Planck constant, 𝑒 is the elemental charge and Φ 0 = ℎ/2𝑒 is the quantum flux. From these values, the Ginzburg-Landau parameter 𝜅 𝐺𝐿 = 𝜆 GL / 𝜉 GL is equal to 18, which further confirms the classification of K2Mo15Se19 as a type-II superconductor. The thermodynamic critical field 𝜇 0 𝐻 𝑐 (0) can be obtained from the above-mentioned values using the relation 𝐻 𝑐1 𝐻 𝑐2 = 𝐻 𝑐 ln𝜅 𝐺𝐿 , yielding an estimate of 106 mT. All the superconducting parameters determined for

Figure 1 .

 1 Figure Captions

Figure 2 .

 2 Figure 2. Experimental signatures of the charge density wave order in K2Mo15Se19 observed in

Figure 3 .

 3 Figure 3. a) HRTEM image obtained on single-crystalline K2Mo15Se19 taken along the 101 ̅ 0

Figure 4 .

 4 Figure 4. Experimental signatures of bulk superconductivity in K2Mo15Se19 observed in the

Figure 5 .

 5 Figure 5. (a) Magnification around the Fermi level 𝐸 𝐹 of the electronic dispersion curves of

  Figure 1

Table 1 .

 1 Summary of Superconducting Parameters of K2Mo15Se19.

	Parameter	K2Mo15Se19
	𝑇 𝑐	2.8 K
	𝜇 0 𝐻 𝑐2 (0)	2.9 T
	𝜇 0 𝐻 𝑐1 (0)	12 mT
	𝜇 0 𝐻 𝑐 (0)	106 mT
	𝜉 GL	110 Å
	𝜆 GL	2000 Å
	𝜅 𝐺𝐿	18
	𝛾	63 mJ mol -1 K -2
	Δ(0)	0.40 meV
	2Δ(0)/𝑘 𝐵 𝑇 𝑐	3.64
	𝜆 𝑒-𝑝ℎ	0.61
	𝜃 𝐷	183 K

Table 2 .

 2 Superconducting temperature 𝑇 c and possible charge-density-wave temperature 𝑇 CDW of the A2Mo15Se19 (A = K, Rb, Tl and In) compounds reported herein (see FigureS7in Supplemental Material 61) and gathered in Ref. 58 (A = Li, Na, Sn and Pb). No transport property measurements have been reported to date for these four last compounds. The symbol / corresponds to properties that have not yet been measured.

	Compound	𝑇 c (K)	𝑇 CDW (K)
	K2Mo15Se19	2.8	114
	Rb2Mo15Se19	/	125
	In2Mo15Se19	1.4	140
	Tl2Mo15Se19	< 2	140

Acknowledgements C. C. acknowledge the financial support of the Pôle M4 of the University of Lorraine. R. A. R. A. O. acknowledges the Région Bretagne for a PhD grant. 3 He specific heat measurements were performed in MGML (www.mgml.eu), which is supported within the program of Czech Research Infrastructures (project no. LM2018096).


Jérome Long

Dmitry M Lyubov

Tatyana V Mahrova

Konstantin A Lyssenko

Alexander A Korlyukov

Yury V Fedorov

Ekaterina Yu Chernikova

Yannick Guari

Joulia Larionova

Alexander A Trifonov

Jérôme Long

Heteroleptic lanthanide complexes coordinated by tripodal tetradentate ligand: synthesis, structure, magnetic and photoluminescent properties

Keywords: Synthesis, Structure, and Magnetic and Photoluminescent Properties

published or not. The documents may come

Introduction

Single-molecule magnets (SMMs) based on lanthanide ions have recently emerged as new exciting molecular objects due to their fascinating magnetic properties offering interesting potentialities in high-density information storage or quantum computing. [START_REF] Luzon | Lanthanides in molecular magnetism: so fascinating, so challenging[END_REF][START_REF] Woodruff | Lanthanide single-molecule magnets[END_REF][START_REF] Troiani | Molecular spins for quantum information technologies[END_REF][START_REF] Bogani | Molecular spintronics using single-molecule magnets[END_REF] The occurrence of a slow relaxation of the magnetization, which is usually associated with a magnetic bistability at the molecular scale is highly dependent of the congruous combination between a lanthanide ion and the surrounding ligands to allow maximizing the magnetic anisotropy inherent to the 4f ion.

Hence, this synergistic effect creates a large crystal-field splitting that generates an anisotropic barrier, , separating two opposite magnetic states with a subsequent superparamagnetic-like behavior. An efficient SMM aims a high multistep energy barrier involving as many doublet states as possible by its multiplet structure and suppression of other undesirable relaxations, such as Quantum Tunneling of Magnetization (QTM) and Raman. [START_REF] Woodruff | Lanthanide single-molecule magnets[END_REF][START_REF] Tang | Lanthanide Single-Ion Molecular Magnets[END_REF][START_REF] Layfield | Lanthanides and Actinides in Molecular Magnetism[END_REF][START_REF] Ungur | Strategies toward High-Temperature Lanthanide-Based Single-Molecule Magnets[END_REF][START_REF] Lunghi | The role of anharmonic phonons in underbarrier spin relaxation of single molecule magnets[END_REF][START_REF] Escalera-Moreno | Spin states, vibrations and spin relaxation in molecular nanomagnets and spin qubits: a critical perspective[END_REF] With this in hands, numerous lanthanide complexes with considerable anisotropic barriers have been recently reported, [START_REF] Ungur | Strategies toward High-Temperature Lanthanide-Based Single-Molecule Magnets[END_REF][START_REF] Liu | A Stable Pentagonal Bipyramidal Dy(III) Single-Ion Magnet with a Record Magnetization Reversal Barrier over 1000 K[END_REF][START_REF] Gupta | An air-stable Dy(iii) single-ion magnet with high anisotropy barrier and blocking temperature[END_REF][START_REF] Guo | A Dysprosium Metallocene Single-Molecule Magnet Functioning at the Axial Limit[END_REF][START_REF] Ding | On Approaching the Limit of Molecular Magnetic Anisotropy: A Near-Perfect Pentagonal Bipyramidal Dysprosium(III) Single-Molecule Magnet[END_REF] with the milestones belonging to the dysprosium metallocene family showing a high temperature hysteresis, [START_REF] Guo | A Dysprosium Metallocene Single-Molecule Magnet Functioning at the Axial Limit[END_REF][START_REF] Goodwin | Molecular magnetic hysteresis at 60 kelvin in dysprosocenium[END_REF][START_REF] Randall Mcclain | High-temperature magnetic blocking and magneto-structural correlations in a series of dysprosium(iii) metallocenium single-molecule magnets[END_REF] that for some complexes even exceed liquid nitrogen's boiling temperature. [START_REF] Guo | Magnetic hysteresis up to 80 kelvin in a dysprosium metallocene single-molecule magnet[END_REF] However, the examples of non-organometallic high performance lanthanide SMMs based on classical ligands involving nitrogen and/or oxygen atoms are still relatively scarce. [START_REF] Liu | A Stable Pentagonal Bipyramidal Dy(III) Single-Ion Magnet with a Record Magnetization Reversal Barrier over 1000 K[END_REF][START_REF] Ding | On Approaching the Limit of Molecular Magnetic Anisotropy: A Near-Perfect Pentagonal Bipyramidal Dysprosium(III) Single-Molecule Magnet[END_REF][START_REF] Chen | Symmetry-Supported Magnetic Blocking at 20 K in Pentagonal Bipyramidal Dy(III) Single-Ion Magnets[END_REF][START_REF] Meng | Low-Coordinate Single-Ion Magnets by Intercalation of Lanthanides into a Phenol Matrix[END_REF][START_REF] Long | Dysprosium Single-Molecule Magnets with Bulky Schiff-base Ligands: Modification of the Slow Relaxation of the Magnetization by Substituent Change[END_REF][START_REF] Ding | A Study of Magnetic Relaxation in Dysprosium(III) Single-Molecule Magnets[END_REF][START_REF] Canaj | Insight into D6h Symmetry: Targeting Strong Axiality in Stable Dysprosium(III) Hexagonal Bipyramidal Single-Ion Magnets[END_REF][START_REF] Li | Air-Stable Hexagonal Bipyramidal Dysprosium(III) Single-Ion Magnets with Nearly Perfect D6h Local Symmetry[END_REF] In this sense, we have recently reported the synthesis and magnetic properties of a series of original heteroleptic lanthanides complexes [Ln(Tpm)X 3 ] (Ln = Tb, Dy, Er; Tpm = tris (3,5dimethylpyrazolyl)methane; X  = NO 3  , Cl  ) based on a tripodal tris(pyrazolyl)methane ligand. [START_REF] Long | Synthesis, structure and magnetic properties of tris(pyrazolyl)methane lanthanide complexes: effect of the anion over the slow relaxation of the magnetization[END_REF] These complexes exhibit a field-induced slow relaxation of the magnetization which is highly dependent on both, the lanthanide ion's and counter anion's nature. Such complexes and others obtained by utilizing others six electron donor ligands such as the negatively charged tris(pyrazolyl)borate (Tp) [START_REF] Xu | A promising new route towards single-molecule magnets based on the oxalate ligand[END_REF][START_REF] Xu | MIIIDyIII3 (M = FeIII, CoIII) Complexes: Three-Blade Propellers Exhibiting Slow Relaxation of Magnetization[END_REF][START_REF] Guégan | Teaching an old molecule new tricks: evidence and rationalisation of the slow magnetisation dynamics in [DyTp2Acac[END_REF][START_REF] Zhang | Exchange coupling and single molecule magnetism in redox-active tetraoxolene-bridged dilanthanide complexes[END_REF][START_REF] Norel | A Terminal Fluoride Ligand Generates Axial Magnetic Anisotropy in Dysprosium Complexes[END_REF] or tris(pyrazolyl)methane (Tpm), [START_REF] Long | Synthesis, structure and magnetic properties of tris(pyrazolyl)methane lanthanide complexes: effect of the anion over the slow relaxation of the magnetization[END_REF][START_REF] Machado | Polymer encapsulated scorpionate Eu3+ complexes as novel hybrid materials for high performance luminescence applications[END_REF] constitute an interesting family of SMMs where the steric hindrance and the electronic density of ligands may be adjusted through the substituent groups of pyrazolyl and the central atom (carbon or boron). In comparison, the family of tris(pyrazolyl)methylamine ligands (Tpma) based on the tripodal tris(2-aminoethyl)-amine (tren) has received far less attention in coordination chemistry of transition metals and lanthanide ions, [START_REF] Driessen | Transition metal coordination compounds of two pyrazole-substituted ammonia ligands. X-ray structure of [Biss(1-pyrazolylmethyl)aminecobalt(II)] bis(nitrate)[END_REF][START_REF] Blackman | The coordination chemistry of tripodal tetraamine ligands[END_REF][START_REF] Haldón | Synthesis, Structural Characterization, Reactivity, and Catalytic Properties of Copper(I) Complexes with a Series of Tetradentate Tripodal Tris(pyrazolylmethyl)amine Ligands[END_REF][START_REF] Goswami | Tuning the Magnetoluminescence Behavior of Lanthanide Complexes Having Sphenocorona and Cubic Coordination Geometries[END_REF] despite their several interesting features. In fact, the peripheral functionalization of the tren moiety may allow to finely tune both, the steric and electronic features. Moreover, its strongly chelating ability associated with the rigid backbone may reduce the molecular vibrations (metal-ligand modes) that are known to enhance Raman relaxation and in turn decrease the magnetic potentialities. [START_REF] Escalera-Moreno | Spin states, vibrations and spin relaxation in molecular nanomagnets and spin qubits: a critical perspective[END_REF][START_REF] Goodwin | Molecular magnetic hysteresis at 60 kelvin in dysprosocenium[END_REF][START_REF] Guo | Magnetic hysteresis up to 80 kelvin in a dysprosium metallocene single-molecule magnet[END_REF][START_REF] Guo | A Dysprosium Metallocene Single-Molecule Magnet Functioning at the Axial Limit[END_REF] For these reasons, such ligands should afford a different coordination environment with the respect to the previously considered ones and could in turn alter the slow relaxation features. In addition, these ligands present also an interesting potential with the aim to introduce others functionalities besides magnetism, such as luminescence or redox activity. [START_REF] Kachi-Terajima | Luminescence tuning of imidazole-based lanthanide(iii) complexes [Ln = Sm, Eu, Gd, Tb[END_REF][START_REF] Molloy | Sevencoordinate lanthanide complexes with a tripodal redox active ligand: structural, electrochemical and spectroscopic investigations[END_REF] However, to the best of our knowledge, examples of lanthanide complexes based on such ligands are relatively scarce, [START_REF] Goswami | Tuning the Magnetoluminescence Behavior of Lanthanide Complexes Having Sphenocorona and Cubic Coordination Geometries[END_REF] while only a few lanthanide SMMs based on the tripodal C 3 tren ligand derivatives such as Schiff bases have been published. [START_REF] Pedersen | Design of Single-Molecule Magnets: Insufficiency of the Anisotropy Barrier as the Sole Criterion[END_REF][START_REF] Lucaccini | Beyond the anisotropy barrier: slow relaxation of the magnetization in both easy-axis and easy-plane Ln(trensal) complexes[END_REF][START_REF] Perfetti | Determination of Magnetic Anisotropy in the LnTRENSAL Complexes (Ln = Tb, Dy, Er) by Torque Magnetometry[END_REF][START_REF] Craze | Synthesis and characterisation of new tripodal lanthanide complexes and investigation of their optical and magnetic properties[END_REF] Following this, we report in this article the synthesis, structures, magnetic and photoluminescent properties of a series of lanthanide ion complexes based on an original tripodal tetradentate Tmpa ligand. Depending on the nature of the lanthanide ion, slow relaxation of the magnetization could be evidenced for all Kramers ions, while typical lanthanide luminescence is exhibited for the europium, terbium and dysprosium analogues.

Experimental Section

Materials and Methods

The synthesis of complexes 1-5 was performed under aerobic conditions. Acetonitrile was distilled prior to use. Tris((1H-pyrazol-1-yl)methyl)amine) (Tpma) was prepared according to the literature procedure. [START_REF] Driessen | Transition metal coordination compounds of two pyrazole-substituted ammonia ligands. X-ray structure of [Biss(1-pyrazolylmethyl)aminecobalt(II)] bis(nitrate)[END_REF] IR spectra were recorded as Nujol mulls on a Bruker-Vertex 70 spectrophotometer. The C, H, N elemental analyses were carried out in the microanalytical laboratory of the IOMC by means of a Carlo Erba Model 1106 elemental analyzer with an accepted tolerance of 0.4 unit on carbon (C), hydrogen (H), and nitrogen (N). Lanthanide metal analysis was carried out by complexonometric titration. [START_REF] Lyle | Complexometric titration of yttrium and the lanthanons-I[END_REF] Thermogravimetric analysis was performed on a Perkin Elmer Pyris-6 TGA equipment with a heating speed of 5 °C.min -1 for all complexes 1-5.

Synthesis of [Ln(Tpma)(NO 3 )

Single-Crystal X-ray diffraction studies

The data collection was with Bruker APEX-II CCD [(Mo K )= 0.71072 Å, -scans). The structures were solved by direct method and refined by the full-matrix least-squares technique against F 2 in the anisotropic-isotropic approximation. Analysis of the Fourier density synthesis.

Hydrogen atoms were placed in calculated positions and were refined in the "riding" model with U(H) iso = 1.2U eq of their parent atoms. All calculations were performed using SHELXTL-2017.

Crystallographic data for the structures 1-5 were deposited at the Cambridge Crystallographic Data Centre as supplementary no. CCDC 1989944-1989947 and 2006866. Crystal data and structure refinement parameters are listed in Table S1.

Powder XRD studies.

All powder XRD measurements were carried out with Bruker D8 Advance II diffractometer (sealed Cu tube, [CuKα = 1.5418 Å], motorized slits) equipped with Lynx Eye detector at room temperature.

Magnetic Measurements

Magnetic susceptibility data were collected with a Quantum Design MPMS-XL SQUID magnetometer working in the range 1.8-350 K with the magnetic field up to 7 Tesla. The samples were prepared in a glove box. The data were corrected for the sample holder and the diamagnetic contributions calculated from the Pascal's constants. The AC magnetic susceptibility measurements were carried out in the presence of a 3 Oe oscillating field in zero or applied external DC field.

Photoluminescence measurements

The solid sample was sealed in a quartz tube in an argon atmosphere (glove box) using a Fluorolog-3-221 (HORIBA Jobin Yvon) spectrofluorimeter. The emission spectra were corrected for detection and optical spectral response of the spectrofluorimeter. The luminescence lifetime for solid complexes 1, 2 and 3 was determined at 295 K using a Fluorolog-3-221 spectrofluorimeter with a UV xenon flash lamp as an excitation source. The data was analyzed with OriginPro 2015 using single order exponential decay fitting.

Results and discussion

Synthesis and crystal structures. The equimolar reaction of rare-earth metal nitrates Analysis by single-crystal X-ray diffraction indicates that among the investigated compounds, 1-4 are isomorphous and crystallize in the triclinic space group (Table S1) as a solvate with a unique lanthanide complex and half of an acetonitrile molecule in the asymmetric unit. For clarity, only the structure of the dysprosium analogue 3 will be described in details, while bond length and angles for 1-5 are listed in Table 1 for comparison. The structure could be viewed as a mononuclear complex in which the Dy 3+ ion is coordinated by four nitrogen from Tpma and six oxygens arising from three bidentate nitrate moieties giving a coordination number of 10 (Figure 1a). Quantitative analysis of the Dy 3+ site's geometry using the SHAPE software [START_REF] Casanova | The Rich Stereochemistry of Eight-Vertex Polyhedra: A Continuous Shape Measures Study[END_REF] indicates that it could be described as a distorted spherocorona (Table S2). [Dy(Tpm)(NO 3 ) 3 ] (Tpm = tris(3,5-dimethylpyrazolyl)methane). [START_REF] Long | Synthesis, structure and magnetic properties of tris(pyrazolyl)methane lanthanide complexes: effect of the anion over the slow relaxation of the magnetization[END_REF] The crystal packing analysis indicates that the shortest intermolecular Dy-Dy distance is equal to 8.268 Å, while hydrogen bonds involving the nitrates and hydrogens from the ligands as well as stacking interaction between pyrazoles (with the shortest C•••C contact ca. 3.39Å) could be detected (Figure 1b andc).

In contrast to 1-4, the Yb complex 5 crystallizes without any solvate molecules in the monoclinic P2 1 /c space group (Table S1). Most likely, due to the lanthanide contraction and the consequent decrease of intramolecular distances leading to steric repulsions in the coordination sphere, only two nitrate anions are bidentate, while the remaining one is monodentate. As a result the coordination number of the Yb 3+ ion in 5 is 9, with a geometry of the Yb site which could be best described as a muffin with C s symmetry (Table S3). The geometry of Tpma ligand found in using the free-ion approximation. [START_REF] Van Vleck | The theory of electric and magnetic susceptibilities[END_REF] Upon cooling, all compounds exhibit the typical decrease of T caused by the thermal depopulation of the m J levels (Figure 3a) to reach the values at 1.8 K of 5.25, 7.35, 4.98 and 1.08 cm 3 .K.mol 1 for 2, 3, 4, and 5, respectively. The field dependences of the magnetization at 1.8 K show values of 6.80, 6.87, 5.12 and 1.91 Nβ under a 70 kOe field for 2, 3, 4, and 5, respectively (Figure 3b). For all complexes, the lack of saturation for the magnetization curves point out the presence of magnetic anisotropy as expected for such lanthanide ions. 6 AC Magnetic properties. Alternate currents (ac) measurements were conducted to probe the occurrence of a slow relaxation of the magnetization. Under a zero-dc field, no strong out-ofphase susceptibility (") components could be detected for all samples (Figure S4), which may be ascribed to occurrence of the fast QTM.

In order to decrease the QTM contribution, ac measurements were performed under various dc-fields. Except compound 2, for which the out-of-phase component remains weak, all the other complexes exhibit a strong " component upon applying dc fields (Figure S4). Note that for both complexes 3 and 4, the appearance of a second low frequency peak could be observed at high magnetic fields. Due to the complexity in the slow relaxation in lanthanide complexes, the origin of these second peaks remains difficult to explain but might originate from different relaxation mechanisms actuated by large magnetic fields. From these data, extraction of the relaxation time related to the main peak, , allows to study its field dependence (Figure S5) which could be modeled with the equation  1 = DH 4 T + B 1 /(1+B 2 H²) + K (Eq. 1) and for which the first term accounts for the direct process (for Kramers-ion), the second one stands for the QTM, while the K constant accounts for the Raman and thermally activated processes. [START_REF] Orbach | Spin-Lattice Relaxation in Multilevel Spin Systems[END_REF][START_REF] Shrivastava | Theory of Spin-Lattice Relaxation[END_REF][START_REF] Meihaus | Influence of pyrazolate vs N-heterocyclic carbene ligands on the slow magnetic relaxation of homoleptic trischelate lanthanide(III) and uranium(III) complexes[END_REF] The fitting parameters could be found in Table S4. Note that it was however not possible to obtain a pertinent fitting for the erbium analogue 4, which might be due to the presence of the second relaxation process that appear at high magnetic fields. The optimum field corresponding to the greatest relaxation time value for each complex is estimated at 500, 250 and 1000 Oe for 3, 4 and 5, respectively. The frequency dependence of the ac susceptibilities under these fields (Figure 4, Figure S6) reveals the presence of a single peak for all investigated samples for which the maximum shifts towards higher temperature upon increasing the temperature, reflecting a field-induced slow relaxation.

The fitting of the Cole-Cole plots with a generalized Debye model confirms the occurrence of a reasonable distribution of relaxation times with moderate values of the  parameter (i.e.< 0.2;

Table S5-S7, Figure S7). Based on these data, the relaxation time could be estimated to study the relaxation dynamics (Figure 5). CT n + AT (Eq. 2, Table S8) has been performed. [START_REF] Meihaus | Influence of pyrazolate vs N-heterocyclic carbene ligands on the slow magnetic relaxation of homoleptic trischelate lanthanide(III) and uranium(III) complexes[END_REF] The first term accounts for a thermally activated process, while the second and third ones stand for two-phonon Raman and direct process, respectively. To avoid over-parameterization, the n coefficient was fixed to different values until getting the best correlation coefficient. For 3, the best fitting gave unrealistic  0 value suggesting that the Orbach process may not be involved. Hence, considering only Raman and direct processes ( 1 = CT n + AT, Eq. 3) gave a pertinent fit of the data (Table 2). As regards 4, while using Eq. 2 allows fitting correctly the data, a strong contribution from the Raman process suggests that the Orbach process is also not dominant. Finally, the fitting of the temperature dependence of the relaxation time for 5 could not be achieved using Eq. 2 while letting the n parameter free in Eq. 3 leads to unrealistic parameters. Consequently, the n parameter was fixed to different values until getting the best correlation coefficient (Table 2). Hence, it turns out that for all investigated compounds, the magnetization relaxes through a combination of Raman and direct processes. However, one can note different Raman coefficient exponent values suggesting different acoustic and optical phonons distributions. [START_REF] Liddle | Improving f-element single molecule magnets[END_REF] Table 2: Fit parameters of the temperature dependence of the relaxation time for 3-5. The solid line represents the fit with Eq. 3.

Compound n C (s 1 .K n ) A(s -1 .K -1 )
Magneto-structural correlations. The observed results could be rationalized by simple electrostatic considerations and by analyzing the angular dependence the 4f electronic density for

T -1 / K -1
the Ln 3+ ions. [START_REF] Rinehart | Exploiting single-ion anisotropy in the design of f-element single-molecule magnets[END_REF] The terbium analogue 2 is the only sample which does not exhibit a strong outof-phase component that would account for a slow relaxation of the magnetization, even in the presence of a dc field. This could be easily explained by its non-Kramers nature ( 7 F 6 ground state) which requires a particular high symmetry in order to observe a slow relaxation of the magnetization. [START_REF] Ishikawa | Mononuclear lanthanide complexes with a long magnetization relaxation time at high temperatures: A new category of magnets at the single-molecular level[END_REF] In contrast, Dy 3+ , Er 3+ and Yb 3+ are all Kramers ions, which ensure that the ground states will be doubly degenerate and allowed the possibility of the observation of the slow relaxation of the magnetization in not strictly axial or high symmetry complexes. However for compounds 3-5, in zero-dc field, the presence of the QTM prevents the observation of a slow relaxation of the magnetization. This QTM could be however suppressed by applying dc fields to permit the appearance of a field-induced SMM behavior. Indeed, Dy 3+ presents an oblate electronic density requiring an axial crystal-field to enhance its anisotropy while Er 3+ has a prolate one that necessitates an equatorial crystal-field. On the other hand, Yb 3+ exhibits a prolate or oblate electronic density depending on the nature of the ground doublet. [START_REF] Rinehart | Exploiting single-ion anisotropy in the design of f-element single-molecule magnets[END_REF] It appears therefore that the neutral tetradentate Tpma ligand in association with nitrates and the resulting lowsymmetry of the lanthanide site do not provide the requirements to maximize the anisotropy for either oblate or prolate lanthanide ions in zero dc field. Thus, the use of a static dc field appears necessary to reveal the slow relaxation. Such fact is confirmed by the determination of the anisotropic axes direction for the ground doublet with the MAGELLAN package for the dysprosium analogue 3. [START_REF] Chilton | An electrostatic model for the determination of magnetic anisotropy in dysprosium complexes[END_REF] Hence, this axis is not found collinear to any specific dysprosiumligand bond and passes between the three nitrate moieties and the Tpma ligand (Figure S8).

Interestingly, these results could be compared with those obtained from our recent investigations of the magnetic properties of nine-coordinate heteroleptic complexes [Ln(Tpm)X 3 ] (X  = NO 3  , Cl  ) based on the tridentate ligand tris(pyrazolyl)methane ligand (Tpm). [START_REF] Long | Synthesis, structure and magnetic properties of tris(pyrazolyl)methane lanthanide complexes: effect of the anion over the slow relaxation of the magnetization[END_REF] In these systems, an anion dependence has been demonstrated depending on the oblate/prolate character of the lanthanide ion since a field-induced slow relaxation was only observed for the [Dy(Tpm)(NO 3 ) 3 ]

and the [Er(Tpm)Cl 3 ] analogue. In contrast, the Tpma ligand in association with nitrate moieties is able to provide a field-induced slow relaxation for both oblate (Dy 3+ ) and prolate (Er 3+ ), as well as for Yb 3+ ion. Hence, the additional coordination of the tertiary amine with a quite long Ln-N distance (ranging from 2.709 to 2.774 Å) may affect the overall geometry and nitrate distribution favoring the slow relaxation of the magnetization with all Kramers ions.

Photoluminescence. The presence of Tpma, which might act as an antenna, directly coordinated to Ln 3+ ions opens the possibility to observe a lanthanide-based luminescence and to design bifunctional luminescent SMMs. [START_REF] Pointillart | Lanthanide Ion and Tetrathiafulvalene-Based Ligand as a "Magic" Couple toward Luminescence, Single Molecule Magnets, and Magnetostructural Correlations[END_REF][START_REF] Jia | Luminescent single-molecule magnets based on lanthanides: Design strategies, recent advances and magneto-luminescent studies[END_REF][START_REF] Long | Recent advances in luminescent lanthanide based Single-Molecule Magnets[END_REF] The luminescence properties of Eu 3+ (1), Tb 3+ (2) and Dy 3+

(3) based complexes were investigated in the solid state at 295 K. Complexes 1, 2 and 3 exhibit emission in red (1), green and yellow color regions respectively, when excited at 380 nm (Figure 6). As regards 1, emission peaks centered at 581, 593, 615 (621, 628), 650, and 682 (691, 698, 703) nm could be assigned to the infra 4f transitions 5 D 0 → 7 F J (J = 0, 1, 2, 3, 4), respectively (Figure 6). The weak 5 D 0 → 7 F 0 Eu 3+ ion transition observed at 581 nm confirms the low symmetry of the coordination environment of the Eu 3+ site which is also reflected by the splitting of the 5 D 0 → 7 F 2 and 5 D 0 → 7 F 4 electronic dipole transitions, which are sensitive to the symmetry of the coordination environment. [START_REF] Bünzli | Basics of Lanthanide Photophysics[END_REF] Similarly, the characteristic terbium-based luminescence could be observed for 2 with emission peaks centered at 491, 544 (551), 582 (596), 624, 653, 668 and 681 nm, assigned to 5 D 4 → 7 F J (J = 6, 5, 4, 3, 2, 1, and 0) transitions, respectively. On the other hand, the characteristic dysprosium-based emission peaks centered at 480, 574 and 663 nm for 3 could be assigned to 4 F 9/2 → 6 H 15/2 , 4 F 9/2 → 6 H 13/2 and 4 F 9/2 → 6 H 11/2 transitions (Figure 6).

For all complexes, the absence of a broad emission originating from the Tpma ligand suggests either an efficient energy transfer from this ligand to the lanthanide ion or a direct excitation of the lanthanide through the f-f transitions. Analysis of the excitation spectra for complexes 1, 2 and 3 (Figures S9-S11) monitored into the emission bands at 593, 543 and 574 nm, respectively, shows the typical sharp lines related to direct intra f-f transitions. [START_REF] Carnall | Electronic Energy Levels of the Trivalent Lanthanide Aquo Ions. IV. Eu3+[END_REF][START_REF] Carnall | Electronic Energy Levels of the Trivalent Lanthanide Aquo Ions. III. Tb3+[END_REF][START_REF] Kumar | Bright green emission from f-MWCNT embedded co-doped Bi3+ + Tb3+:polyvinyl alcohol polymer nanocomposites for photonic applications[END_REF] Consequently, the absence of broad bands related to the organic ligand in the excitation spectra clearly points out a direct sensitization of the lanthanide ion through f-f transitions rather than an energy transfer involving the Tpma moieties. [START_REF] Hänninen | Lanthanide luminescence: photophysical, analytical and biological aspects[END_REF][60][START_REF] Bünzli | Lanthanide Luminescence for Biomedical Analyses and Imaging[END_REF] Thus, although it does not act a sensitizer, the Tpma ligand does not quench the lanthanide luminescence.

In order to get further details, the photoluminescence emission decay of complexes 1, 2 and 3 in the solid state at 295 K was performed to determine the luminescence lifetimes. These later are found equal to 1.51, 1.77 and 0.022 ms for 1, 2 and 3 respectively (Figs. S12-S14), which are typical values for europium, terbium and dysprosium complexes. [START_REF] Bünzli | Basics of Lanthanide Photophysics[END_REF][START_REF] Bunzli | Taking advantage of luminescent lanthanide ions[END_REF][START_REF] Eliseeva | Lanthanide luminescence for functional materials and bio-sciences[END_REF][START_REF] Bunzli | Handbook on the Physics and Chemistry of Rare Earths: Including Actinides[END_REF][START_REF] Carlos | Lanthanidecontaining light-emitting organic-inorganic hybrids: A bet on the future[END_REF] Figure 6. Solid-state luminescence spectra ( exc = 380 nm) for 1, 2 and 3 at room temperature.

Conclusions

A series of heteroleptic lanthanide complexes based on an original tetradentate tripodal ligand has been described. While the Eu 3+ , Tb 3+ , Dy 3+ and Er 3+ complexes are found isostructural, the Yb 3+ analogue presents a different structure due to the reduction of the coordination number through different coordination of one of the nitrates. Among this series, the complexes based on Kramers ions (Dy 3+ , Er 3+ and Yb 3+ ) exhibit a field-induced SMM behavior involving a relaxation through Raman and direct processes. On the contrary, the complex based on the non-Kramers Tb 3+ ion does not present a slow relaxation. In contrast to the complexes based on a related tridentate trispyrazolylmethane ligand, [START_REF] Long | Synthesis, structure and magnetic properties of tris(pyrazolyl)methane lanthanide complexes: effect of the anion over the slow relaxation of the magnetization[END_REF] ions. This clearly highlights that the fine tuning of the coordination allows the adjustment of the slow relaxation features. In addition, the Tmpa ligand does not act as luminescence quencher since a clear lanthanide-based photoluminescence could be detected for the Eu 3+ , Tb 3+ and Dy 3+ complexes through a direct f-f excitation. The dysprosium complex 3 could be therefore viewed as a bifunctional complex showing a field-induced slow relaxation and lanthanide luminescence. The great tunability of these tripodal ligands, in terms of steric and electronic factors, could be taken as an advantage to in fine enhance the slow relaxation and photoluminescence features.

ASSOCIATED CONTENT Supporting Information.

Additional crystallographic parameters, magnetic and photoluminescence data. The following files are available free of charge.

  40. IR (Nujol, KBr) ν/cm -1 : 500 (w), 530 (m), 605 (s), 650 (s), 745 (s), 770 (s), 785 (m), 795 (m), 810 (s), 865 (m), 890 (s), 910 (m), 920 (s), 945 (m), 980 (s), 1005 (w), 1025 (s), 1045 (w), 1065 (s), 1105 (m), 1150 (s), 1200 (m), 1210 (m), 1285 (s), 1505 (m), 1685 (w), 1735 (m), 1775 (m), 1985 (m), 2055 (m), 2260 (m), 2305 (m), 2330 (m), 2535 (m). Characterization of [Yb(Tpma)(NO 3 ) 3 ] (5): Elemental analysis calcd. (%) for C 12 H 15 N 10 O 9 Yb: C, 23.38; H, 2.45; N, 22.72; Yb, 28.08; found (%): C, 23.61; H, 2.36; N, 22.92; Yb, 27.85. IR (Nujol, KBr) ν/cm-1: 505 (w), 525 (m), 605 (s), 650 (s), 750 (s), 770 (s), 785 (m), 800 (m), 815 (s), 865 (m), 895 (s), 910 (m), 920 (s), 945 (m), 980 (s), 1005 (w), 1030 (s), 1045 (w), 1065 (s), 1105 (m), 1145 (s), 1200 (m), 1215 (m), 1280 (s), 1510 (m), 1680 (w), 1740 (m), 1780 (m), 1990 (m), 2055 (m), 2260 (m), 2300 (m), 2330 (m), 2535 (m).

  Scheme 1. Synthesis of complexes 1-5.

Figure 1 .

 1 Figure 1. Molecular structure for complex 3. Colour code: orange, Dy; blue, N; red, O; grey, C.

5 differs significantly to those found in 1 - 4 .

 14 Thus, two pyrazolyl rings of the Tpma ligand in 5 are almost coplanar (the dihedral angle between N Pyr -Yb-N Amino is 170.0°), while the third pyrazolyl moiety is almost orthogonal relatively to two others (dihedral angles are 94.3 and 104.3°). For complexes 1-4, the dihedral angles between N Pyr -Yb-N Amino planes lie within 114.1 -129.0°. Moreover, the Yb-O and Yb-N distances in 5 are expectedly shorter with respect to the other analogues due to the lanthanide's contraction and the decrease of the coordination number. In particular, the Yb-N distance with the central tertiary amine's nitrogen atom is equal to 2.708(3) Å. In contrast to 1-4, the only type of intermolecular interaction in the crystal is weak CH•••O hydrogen bonds, while the shortest intermolecular Yb-Yb distance is much shorter and equal to 7.425 Å. The change of the crystal symmetry from triclinic space group to monoclinic P2 1 /c space group also induces a reduction in the volume accessible for solvents molecules.

Figure 2 .

 2 Figure 2. a) Molecular structure of 5. Colour code: orange, Yb; blue, N; red, O; grey, C.

Figure 3 .

 3 Figure 3. a)Temperature dependence of T measured under a 1000 Oe dc field for 2-5. b) Field

Figure 4 .

 4 Figure 4. Frequency dependence of" for compounds 3-5 under the corresponding optimal dc

Figure 5 .

 5 Figure 5. Temperature dependence of the relaxation time for 3-5 using the ac susceptibility data.

  3

Complexes [Tb(Tpma)(NO 3 ) 3 ]•0.5MeCN (2), [Dy(Tpma)(NO 3 ) 3 ]•0.5MeCN (3), [Er(Tpma)(NO 3 ) 3 ]•0.5MeCN (4), and [Yb(Tpma)(NO 3 ) 3 ] (

	solution was concentrated by evaporation of the solvent at 80 °C approximately to half of the
	initial volume. Cooling the solution to room temperature resulted in the formation of 1 as
	colorless crystals. The mother liquid was decanted and the crystals were washed twice with
	MeCN (5 mL). Crystals of 1 were dried in vacuum for 10 min.
	5) were prepared following the
	similar procedure and were isolated as colorless (2, 3, and 5) or pale pink (4) crystals.
	Complexes 1-5 were isolated in 70, 76, 83, 72 and 65% yields respectively. Complexes 1-4
	crystallize as solvates [Ln(Tpma)(NO 3 ) 3 ]•0.5MeCN or as [Yb(Tpma)(NO 3 ) 3 ] for 5
	]•nMeCN
	[Eu(Tpma)(NO 3 ) 3 ]•0.5MeCN (1): MeCN (30 mL) was added to a mixture of Eu(NO 3 ) 3 (H 2 O) 6

(0.410 g, 0.919 mmol) and slight excess of Tpma (0.260 g, 1.011mmol; 1.1 equivalent) and the resulting suspension was heated at 80 °C for 10 min to give a clear pale yellow solution. The

. Characterization of [Eu(Tpma)(NO 3 ) 3 ]•0.5MeCN

	(1): Elemental analysis calcd. (%) for
	C 13 H 16.5 N 10.5 O 9 Eu: C, 25.36; H, 2.70; N, 23.88; Eu, 24.68; found (%): C, 25.54; H, 2.93; N,
	24.07; Eu, 24.42. IR (Nujol, KBr) ν/cm -1 : 505 (m), 525 (m), 610 (s), 650 (s), 745 (s), 770 (m),
	780 (m), 795 (s), 815 (s), 860 (w), 880 (s), 900 (w), 920 (s), 955 (s), 980 (s), 1005 (w), 1030 (m),
	1045 (m), 1065 (m), 1100 (w), 1150 (s), 1195 (m), 1210 (w), 1295 (s), 1515 (m), 1680 (m), 1740
	(w), 1775 (w), 1985 (m), 2050 (m), 2260 (m), 2300 (m), 2325 (m), 2525 (m).

Characterization of [Tb(Tpma)(NO 3 ) 3 ]•0.5MeCN

	1045 (m), 1065 (m), 1100 (w), 1150 (s), 1195 (m), 1210 (w), 1295 (s), 1515 (m), 1680 (m), 1740
	(w), 1775 (w), 1985 (m), 2050 (m), 2260 (m), 2300 (m), 2325 (m), 2525 (m).
	(2): Elemental analysis calcd. (%) for
	C 13 H 16.5 N 10.5 O 9 Tb: C, 25.07; H, 2.67; N, 23.62; Tb, 25.52; found (%): C, 25.18; H, 2.59; N,
	23.47; Tb, 25.45. IR (Nujol, KBr) ν/cm -1 : 505 (m), 525 (m), 610 (s), 650 (s), 745 (s), 770 (m),
	780 (m), 795 (s), 815 (s), 860 (w), 880 (s), 900 (w), 920 (s), 955 (s), 980 (s), 1005 (w), 1030 (m),

Characterization of [Dy(Tpma)(NO 3 ) 3 ]•0.5MeCN

	(3): Elemental analysis calcd. (%) for
	C 13 H 16.5 N 10.5 O 9 Dy: C, 24.93; H, 2.66; N, 23.48; Dy, 25.94; found (%): C, 25.03; H, 2.69; N,
	23.39; Dy, 25.80. IR (Nujol, KBr) ν/cm -1 : 505 (m), 525 (m), 605 (s), 650 (s), 745 (s), 765 (s),
	780 (m), 795 (m), 815 (s), 865 (w), 890 (s), 910 (w), 920 (s), 940 (m), 980 (s), 1005 (w), 1030
	(s), 1045 (w), 1065 (s), 1100 (m), 1145 (s), 1200 (m), 1210 (w), 1290 (s), 1500 (m), 1690 (w),
	1735 (m), 1775 (m), 1985 (m), 2055 (m), 2250 (m), 2300 (m), 2320 (m), 2525 (m).

Characterization of [Er(Tpma)(NO 3 ) 3 ]•0.5MeCN

(4)

: Elemental analysis calcd. (%) for C 13 H 16.5 N 10.5 O 9 Er: C, 24.74; H, 2.64; N, 23.30; Er, 26.50; found (%): C, 24.83; H, 2.70; N, 23.39; Er, 26.

  The Tpma ligand coordinates to Dy 3+ from 2.494(2) to 2.533(2) Å and are significantly shorter than the Dy-N distance from the central tertiary N amine of 2.767(2) Å. On the other hand, the Dy-O distances have the values spanning from 2.433(2) to 2.502(2) Å. Note that this Dy-N(tertiary amine) distance in 3 is significantly longer than that of 2.680(2) in similar Dy(NO 3 ) 3 tris(benzimidazol-2ylmethyl)amine complex.[START_REF] Goswami | Tuning the Magnetoluminescence Behavior of Lanthanide Complexes Having Sphenocorona and Cubic Coordination Geometries[END_REF] Moreover, the average values of Dy-N (2.510 Å) for heterocyclic nitrogen atoms and Dy-O (2.469 Å) in 3 are longer that those (2.498 and 2.423 Å) in

ion in a symmetrical fashion. The Dy-N distances involving the heterocyclic nitrogen atoms are a) b) c) ranging

Table 1 .

 1 Selected bond distances and angles in complexes 1-5. At the exception of the europium analogue 1, which is known to be diamagnetic at low temperature, the static (direct current, DC) and dynamic magnetic properties (alternate currents, AC) of other complexes were investigated by using a SQUID MPMS-XL.

		1	2	3	4	5
	M-N Pyr	2.539(2)-2.567(2) 2.507(2)-2.542(2) 2.494(2)-2.533(2) 2.470(2)-2.513(2) 2.365(3)-2.453(3)
	M-N Amine	2.793(2)	2.774(2)	2.767(2)	2.756(2)	2.708(3)
	M-O (κ 2 -NO 3 ) 2.470(2)-2.524(2) 2.443(2)-2.509(2) 2.433(2)-2.502(2) 2.408(2)-2.493(2) 2.346(3)-2.425(3)
	M-O (κ 1 -NO 3 )	-	-	-	-	2.266(3)
	N Pyr -M-N Pyr	94.9(1)-104.5(1) 95.4(1)-104.8(1) 95.5(1)-104.8(1) 95.8(1)-104.9(1)	95.8(1)-104.9(1)
	N Pyr -M-N Amine 61.1(1)-61.5(1)	61.4(1)-61.8(1)	61.4(1)-62.0(1)	61.6(1)-62.2(1)	61.6(1)-62.2(1)

  these results suggest that the Tmpa ligand in association with nitrate moieties stabilize either the prolate or oblate electronic densities of the lanthanide

				5 D 4 	5 D 4 	7 F 2,1,0
	3		7 F 9/2 	6 H 13/2
	7 F 9/2 	6 H 15/2
					7 F 9/2 	6 H 11/2
	450	500	550	600	650		700
			Wavelength/ nm

The authors thank the University of Montpellier, CNRS for financial support, PAC of ICGM for magnetic measurements. The Russian co-authors thank the Russian Science Foundation (grant 17-73-30036).

AUTHOR INFORMATION

Corresponding Author

*E-mail: jerome.long@umontpellier.fr *E-mail: trif@iomc.ras.ru

Author Contributions

The manuscript was written through contributions of all authors. All authors have given approval to the final version of the manuscript.

ACKNOWLEDGMENT

For Table of Contents Use Only


M Debiossac

P Pan

P Roncin

Inelastic diffraction of fast atoms on crystal surfaces a Lamb-Dicke regime in shallow atomic collisions

Synopsis Grazing incidence fast atom diffraction (GIFAD) uses keV atoms to investigate properties of crystal surfaces. It exists in two flavors; The elastic diffraction is identified by bright spots sitting on the Laue circle of energy conservation. We will focus on inelastic diffraction with a model where the surface atoms are quantum harmonic oscillators and where the successive collisions of the projectile take place in the Lamb-Dicke regime. The model explains observed lines shapes and predicts a drastic dependence of the energy loss ∆E ∝ θ 7 .

GIFAD is a surface sensitive technique providing a rapid characterisation of crystalline surfaces. Not only the atoms do not penetrate below the surface but the grazing incidence is also very sensitive to defects. The large kinetic energy allows efficient detection by imaging detectors collecting at once all the diffracted beams [1] as illustrated on fig. 1. So far the main focus was to the elastic signal for which a simple interpretation could be given. Crudely speaking, the analysis of the elastic diffracted intensity is a |Fourrier| 2 transform of the surface were the atoms are bouncing (in blue on fig. 1.). Inside a molecular beam epitaxy chamber GI-FAD was able to to follow in situ and online the growth of the successive layers on a GaAs crystal [2,4] and to provide a few pm accuracy on the atomic arrangement [3,5].

The Debye-Waller factor (DWF) describes the amount of decoherence due to the thermal motion of surface atoms; Diffraction will be lost if the amplitude of thermal motion becomes larger than to projectile wavelength. This spatial approach was adapted to GIFAD [6,7] to account for the fact that the momentum transfer is accumulated in several successive shallow collisions along the grazing projectile trajectory but no prediction could be made on the line-shape.

Taking a momentum approach and simplifying assumptions, we describe [8,9] these binary encounters as sudden collisions with harmonic oscillators. For each surface atom, the momentum transfer δk to the vibrational wave function Ψ is evaluated. The probability that δk induces a vibrational excitation is | Ψ|e iδkz |Ψ | 2 and calculated as e -Er/hω were hω is the vibrational energy and E r = δk 2 /2m the recoil energy that would be transferred if the atom would be free. In spectroscopy and in the cold atom community, it is known as the Lamb-Dicke probability of Recoil-less emission. The effect of an inelastic collisions would require summing over all initial and final vibrational states, instead we assume a quasi classical behaviour where the position of the surface atom is replaced by its Gaussian probability distribution P(z)= Ψ|z 2 |Ψ . Each inelastic collision contributes to a angular broadening taking a log-normal profiles in the polar direction and an associated energy loss equal to the transferred recoil energy E r (now real). Combining the statistics of successive collisions with the associated probabilities one can reproduce the inelastic line shapes and predict a drastic dependence of the energy loss ∆E ∝ θ 7 .

Figure 1 .

 1 Figure 1. In GIFAD, the elastic intensity is seen as tiny spots on the Laue circle of energy conservation while the inelastic intensity extends beyond.