Chen Hui Li

Cristiano Poltronieri

Grégory Abadias

Philippe Djemia

Computational approach to identify structural and elastic relationship in metastable crystalline and amorphous alloy thin films: Mo1-xNix and Mo1-xSix case studies

Keywords: ab initio molecular dynamics, density functional theory, special quasi-random structures, elastic constants, metallic glass

Previously well-established experimental trends of ground-state properties of crystalline and amorphous Mo1-xSix and Mo1-xNix alloys with 0 x are predicted by first-principles calculations and inter-relationship with sound velocities and elastic properties are identified. The free energy of mixing, calculated at 300K provides accurate values of the critical Si and Ni concentrations leading to the crystalline-to-amorphous transition. Specifically, a transition from BCC to amorphous state is predicted for xSi ~ 0.14 and xNi ~ 0.32, while FCC to amorphous transition is observed for xNi ~ 0.77. These structural transitions are accompanied by modifications of the out-of-plane longitudinal modulus C33, and out-of-plane shear modulus C44. In the crystalline region, a pronounced softening of C44 and C33 is predicted for both solid solution alloys. While for amorphous Mo-Ni, the elastic constants do not undergo significant changes, for amorphous Mo-Si, they exhibit two distinct behaviors depending on the electronic properties and bonding character. For 0.2 xSi 7, the metallic character of the amorphous alloys is maintained and the elastic properties are remarkably stable. For xSi > 0.7, a progressive increase in the atomic volume is observed and the amorphous alloys acquire a covalent character and a reduced coordination number. Surprisingly, during this transition, both the longitudinal and shear acoustic velocities continuously increase, despite a progressive softening of the elastic stiffness constants. These calculations provide deep insight at the atomic-scale and reproduce satisfactorily the earlier experimental works of magnetron co-sputtered Mo-Si films, while some disagreement on elastic properties remain for more energetic ion beam sputtered Mo-Ni films, likely partially attributed to point-defects.

Introduction

In nanoscale multilayered structures, the physical properties and more specifically the elastic behavior have been shown to exhibit deviations compared to the bulk properties of the constitutive materials [START_REF] Schuller | Elastic anomalies in superlattices[END_REF]. Elastic anomalies have been observed in superlattices with small modulation wavelengths (a few nm) highlighting the role of interfaces versus bulk properties. Surprisingly, such effects have been also reported in systems made of immiscible elements [START_REF] Perrin | Picosecond ultrasonics study of metallic multilayers[END_REF][START_REF] Clemens | Relationship between interfacial strain and the elastic response of multilayer metal films[END_REF][START_REF] Abadias | Experimental evidence for the role of supersaturated interfacial alloys on the shear elastic softening of Ni/Mo superlattices[END_REF][START_REF] Martin | Strain, interdiffusion, and microstructural evolution under ion irradiation in Ni (111) Mo (110) multilayers: Interdependence with elastic properties[END_REF]. From a phenomenological point of view, the observed softening can be ascribed to an interfacial zone exhibiting anomalous elastic properties. Stress/strain, topological and chemical disorder, enhanced solubility or amorphization are possible sources of such interfacial softening. Experimentally, Brillouin light scattering (BLS) [START_REF] Abadias | Experimental evidence for the role of supersaturated interfacial alloys on the shear elastic softening of Ni/Mo superlattices[END_REF][START_REF] Martin | Strain, interdiffusion, and microstructural evolution under ion irradiation in Ni (111) Mo (110) multilayers: Interdependence with elastic properties[END_REF] and picosecond ultrasonics (PU) [START_REF] Belliard | Elastic properties and phonon generation in Mo/Si superlattices[END_REF][START_REF] Rossignol | Elastic properties of ultrathin permalloy/alumina multilayer films using picosecond ultrasonics and Brillouin light scattering[END_REF] techniques have proven to be complementary in measuring the elastic properties of various metallic multilayers. The softening of some effective elastic constants has been reported in various systems, such as in Cu/W [START_REF] Perrin | Picosecond ultrasonics study of metallic multilayers[END_REF], Mo/Ni [START_REF] Clemens | Relationship between interfacial strain and the elastic response of multilayer metal films[END_REF][START_REF] Abadias | Experimental evidence for the role of supersaturated interfacial alloys on the shear elastic softening of Ni/Mo superlattices[END_REF][START_REF] Martin | Strain, interdiffusion, and microstructural evolution under ion irradiation in Ni (111) Mo (110) multilayers: Interdependence with elastic properties[END_REF] and Pt/Ni or Ti/Ni [START_REF] Clemens | Relationship between interfacial strain and the elastic response of multilayer metal films[END_REF]. In the metal/oxide Ni80Fe20/Al203 multilayers [START_REF] Rossignol | Elastic properties of ultrathin permalloy/alumina multilayer films using picosecond ultrasonics and Brillouin light scattering[END_REF], with no intermixing at the interface, Rossignol et al. also found an anomalous behavior of C33 and C44 elastic constants, measured by PU and BLS, respectively, as function of the bilayer period, which could also be ascribed to a weak interfacial bonding but not to a modified structure at the interface.

By contrast, there also exist systems with a modified interface layer, for which no anomalous behavior of elastic constants has been observed, such as for Mo/Si multilayers studied by PU [START_REF] Belliard | Elastic properties and phonon generation in Mo/Si superlattices[END_REF]. In that case, high-resolution transmission electron microscopy observations of the samples confirmed the strong asymmetry of the interfaces with the existence of an amorphous silicide interfacial layer at the Mo-on-Si interface, identified as MoSi2 and extending over up to ~2 nm [START_REF] Slaughter | Structure and performance of Si/Mo multilayer mirrors for the extreme ultraviolet[END_REF][START_REF] Krause | Direct observation of the thickness-induced crystallization and stress build-up during sputterdeposition of nanoscale silicide films[END_REF], whereas the Si-on-Mo interface is mainly characterized by a rough profile.

In binary alloys, phase transitions are also accompanied by strong variations of the elastic properties. This is the case of Mo1-xNix thin films produced by ion-beam sputtering [START_REF] Abadias | Experimental evidence for the role of supersaturated interfacial alloys on the shear elastic softening of Ni/Mo superlattices[END_REF][START_REF] Martin | Strain, interdiffusion, and microstructural evolution under ion irradiation in Ni (111) Mo (110) multilayers: Interdependence with elastic properties[END_REF], for which the variations of C44 and C33 elastic constants have been reported from PU and BLS measurements [START_REF] Laborde | Acoustique picoseconde dans les multicouches métalliques[END_REF]. X-ray diffraction (XRD) results, also presented in these works, evidence the formation of solid solutions at both ends of the phase diagram and an extended amorphous region is observed for 0.27 < xNi < 0.73. This range matches well the one (0.3 < xNi < 0.6) observed by Zhao et al. [START_REF] Zhao | Formation of metastable alloy films in the Ni-Mo binary system by ion-beam-assisted deposition[END_REF], which is a little wider than that obtained from their calculated free-energy diagram (0.48 < xNi < 0.62). Glass-forming range of MoNi was derived from molecular dynamics simulations by Zhang et al. [START_REF] Zhang | Glass-forming range of the Ni Mo system derived from molecular dynamics simulation and generalized lindemann criterion[END_REF] with a satisfactory agreement 0.25 < xNi < 0.79. The alloying of Mo and Ni is found to decrease the intrinsic stacking fault energy of the solid solution from 150 mJ/m 2 (pure Ni) to 50 mJ/m 2 (82.5 at.% Ni) almost linearly [START_REF] Zhao | Tuning the plasticity of Ni-Mo solid solution in Ni-based superalloys by ab initio calculations[END_REF]. The first-principles calculations of composition-dependent thermal expansion, thermal electronic excitation and the magnetic transition are shown to be the main factors rendering the complex variations in the elastic properties and twinning behavior of Ni-Mo solid solution with temperature [START_REF] Zhao | Thermo-mechanical properties of Ni-Mo solid solutions: A first-principles study[END_REF], with Mo addition increasing the solid solution ductility.

A detailed and extensive literature review of the Mo Ni binary system thermodynamic modeling can be find in Ref. [START_REF] Yaqoob | Thermodynamic modeling of the Mo Ni system[END_REF], with a discussion of the existence of several stable intermetallics such as MoNi2 and MoNi8.

A complementary study of elastic properties on sputter-deposited Mo1-xSix films was also performed, combining PU and BLS [16], with XRD results, showing formation of solid solutions and an amorphous region extending above xSi~0.19. In this system, there exist three intermediate ordered phases [START_REF] Okamoto | Mo-Si (Molybdenum-Silicon)[END_REF], Mo3Si, Mo5Si3, and MoSi2, but not observed in amorphous compositional range of these films. For both systems, Mo-Ni and Mo-Si, prior experimental works [START_REF] Abadias | Experimental evidence for the role of supersaturated interfacial alloys on the shear elastic softening of Ni/Mo superlattices[END_REF][START_REF] Belliard | Elastic properties and phonon generation in Mo/Si superlattices[END_REF][START_REF] Laborde | Acoustique picoseconde dans les multicouches métalliques[END_REF]16] have therefore provided accurate data covering the whole compositional range, among them: phase analysis retrieved from XRD, mass density and atomic volume measured by X-ray reflectivity (XRR), sound velocities and elastic constants obtained by PU and BLS. However, the atomistic-scale understanding of these established trends, in particular the link between structural and mechanical evolutions on either side of the crystalline-toamorphous transition, remains to be investigated theoretically.

In the present study, we investigate by first-principles calculations the structural (interatomic distance dav, atomic volume Vat and mass density ) and elastic properties (shear and longitudinal elastic moduli, C44 and C33, respectively) of Mo1-xNix and Mo1-xSix alloys in the entire chemical composition range, covering both BCC or FCC crystalline and amorphous phases. Ab initio molecular dynamics (AIMD) and special quasi-random structures (SQS) method are used to design the amorphous alloys, and BCC or FCC crystalline random solid solutions, respectively. All numerical results are compared to the existing experimental data [START_REF] Abadias | Experimental evidence for the role of supersaturated interfacial alloys on the shear elastic softening of Ni/Mo superlattices[END_REF][START_REF] Belliard | Elastic properties and phonon generation in Mo/Si superlattices[END_REF][START_REF] Laborde | Acoustique picoseconde dans les multicouches métalliques[END_REF]16] and discussed in relation with topological analysis of the amorphous state to provide insights on local atomic configurations region and its role. In addition, metal-toinsulator transition in the Mo-Si system is also discussed based on density of electronic states calculations. Overall, the computed data show a good agreement with experimental trends, shedding light on the inter-relationships between atomic scale structural and physical properties.

Numerical methods

Special quasi-random structures (SQS) for the crystalline cubic phases

First-principles calculations based on density functional theory (DFT) were performed utilizing the Vienna Ab initio Simulation Package (VASP) [START_REF] Kresse | Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set[END_REF]. Special quasi-random supercells [START_REF] Van De Walle | Efficient stochastic generation of special quasirandom structures[END_REF] with 54 and 108 atoms, were built as input atomic structures, to mimic the random crystalline BCC-and FCC-alloys in Mo-Si and Mo-Ni systems. The calculations were done with the standard projector augmented wave (PAW) pseudopotentials provided by the code. The exchange and correlation interaction between electrons were described by the Perdew-Burke-Ernzerhof (PBE) functional [START_REF] Perdew | Generalized gradient approximation made simple[END_REF]. The cutoff energy of 350 eV for the plane-wave-basis set was fixed for all the calculations to ensure well-converged results. For Ni-rich compositions, spin-collinear option was employed to take into account the ferromagnetic properties. The conjugate gradient algorithm was utilized for geometry optimization. The optimization stopped when the norms of all the forces are smaller than 0.01 eV/Å.

The three independent single crystal elastic constants c11, c12 and c44 were computed at 0 K using the stress-strain method ( = c ) [START_REF] Page | Symmetry-general least-squares extraction of elastic data for strained materials fromab initiocalculations of stress[END_REF], considering both relaxed and unrelaxed atomic positions after each applied strain. We used here the quasi-Newton algorithm to relax atoms, in this case, the stopping force criterion was 0.05 eV/Å. Effective elastic constants were also computed for direct comparison with experiments. For polycrystalline materials with a preferential crystal orientation (in the present case <110>-textured Mo-rich BCC alloys (Mo-Ni and Mo-Si systems) and <111>-textured Ni-rich FCC alloys), the elastic symmetry is hexagonal, such that five independent elastic constants C11, C12, C33, C13, C44, are needed. These effective Cij were calculated by Voigt-Reuss-Hill averages [START_REF] Hill | The elastic behaviour of a crystalline aggregate[END_REF] considering the experimentally observed preferred crystal orientations [START_REF] Abadias | Experimental evidence for the role of supersaturated interfacial alloys on the shear elastic softening of Ni/Mo superlattices[END_REF][START_REF] Martin | Strain, interdiffusion, and microstructural evolution under ion irradiation in Ni (111) Mo (110) multilayers: Interdependence with elastic properties[END_REF]16]. Then longitudinal and shear wave velocities could be determined using the relation and , with C33

and C44 corresponding to the out-of-plane longitudinal and shear effective modulus. These are the relevant parameters to calculate for comparison to experimental data measured by PU ( ) and BLS ( ). In the simpler case of amorphous materials (isotropic case),

. To validate most of the used parameters, we checked the good agreement between our elastic constant calculated for Mo and Ni with their bulk ones (Mo: C33 = 459 GPa, C44 = 111 GPa; Ni: C33 = 347 GPa, C44 = 66 GPa). Similarly, we compared our elastic constant calculated for a-Si with previous calculations (C11 = 139 GPa, C44 = 34 GPa) [START_REF] Schall | Elastic constants of silicon materials calculated as a function of temperature using a parametrization of the second-generation reactive empirical bond-order potential[END_REF] and experiments (C11 = 156 GPa, C44 = 48.8 GPa) [START_REF] Sandre | Calculation of elastic constants in defected and amorphous silicon by quantum simulations[END_REF]. Similarly, the elastic constants cij of a given structure were investigated at 0 K using the stress-strain method ( = ) [START_REF] Page | Symmetry-general least-squares extraction of elastic data for strained materials fromab initiocalculations of stress[END_REF], considering both relaxed and unrelaxed atomic positions after each applied strain. The same geometry optimization algorithms and force convergence criteria were used as for the crystalline phases.

Ab initio molecular dynamics (AIMD) and cluster analysis for the amorphous phase

The Voronoi tessellation method, implemented through the OVITO software package [START_REF] Stukowski | Visualization and analysis of atomistic simulation data with OVITO the Open Visualization Tool[END_REF],

was employed to analyze the last thousand-time steps of the quenched alloys. In this analysis, each polyhedral cell was assigned five indices < n3 n4 n5 n6 n7 >, where ni (i 3) represents the number of faces with i-edges in each Voronoi cell. Of particular interest were icosahedral structures denoted by the Voronoi index <0 0 12 0 0>, with the cluster-central atom [START_REF] Guo | Quasi-Icosahedral Clusters in Zr-Based Metallic Glasses[END_REF].

Results and discussion

Energetics and phase stability

The phase stability of the crystalline and amorphous alloys were firstly evaluated from DFT total energy calculations at 0 K for both systems. Based on the obtained formation enthalpies, the following compositional range for stable amorphous Mo-Ni (0.45 < xNi < 0.95) and Mo-Si (xSi > 0.32) alloys was determined. These boundary regions do not perfectly match the reported experimental values (0.27 < xNi < 0.73 and xSi > 0.20, for MoNi and MoSi systems, respectively). Additionally, we employed a simple analytical thermodynamic model with a finite temperature, in which the limits of the glass-forming compositional range (which is linked to the glass forming ability) of the alloys were defined by the compositions at which the free energy of formation of amorphous phase ( ) is lower than that of the crystalline counterpart ( ) [START_REF] Lopez | Determination of the glass-forming concentration range in binary alloys from a semiempirical theory: Application to Zr-based alloys[END_REF]. The free energy of mixing of a solid solution was calculated as while, the amorphous one is defined as where , and , are the enthalpies and entropies of mixing of the crystalline and amorphous phase, respectively, and is the absolute temperature. The enthalpies of mixing, or formation enthalpies, of amorphous and solid solution phases were evaluated following the model proposed by Van der Kolk et al. [START_REF] Van Der Kolk | On the composition range of amorphous binary transition metal alloys[END_REF] with pure crystalline solid elements as a common reference. between 14 and 90 at. % Si (see Fig. 1b), while for Mo-Ni this compositional interval appears smaller and between 32 and 77 at. % Ni (see Fig. 1a). These predicted values are in closer agreement with the experimental results comparatively to DFT predictions [START_REF] Abadias | Experimental evidence for the role of supersaturated interfacial alloys on the shear elastic softening of Ni/Mo superlattices[END_REF]16]. Voronoi tessellation analysis was carried out to better investigate the short-range order (SRO) related to the atomic packing. Higher glass forming ability (GFA) has been observed in several metallic glasses [START_REF] Sha | Statistical composition-structure-property correlation and glass-forming ability based on the full icosahedra in Cu Zr metallic glasses[END_REF][START_REF] Chang | Icosahedral clusters in Cu100-xZrx (x=32,34,36,38.2,40 at.%) metallic glasses near the peak of glass-forming ability (x=36): A balance between population and encaging strength[END_REF][START_REF] Peng | Effect of local structures and Letters[END_REF][START_REF] Houska | Molecular dynamics and experimental study of the growth, structure and properties of Zr Cu films[END_REF] containing a maximum population fraction of full-icosahedra < 0 0 12 0 0 >. In particular, Abbasi et al., [START_REF] Abbasi | On the glass-forming ability of (Zr0[END_REF] proved that in ZrCuAl metallic glasses, low- Figure 4 shows the predominance of clusters with a coordination number of 12-15 for both systems (with the exception of Si-rich alloys, as it will be discussed hereafter). This common peculiar atomic coordination of 12-15 for metallic glasses (see Figure 4) was also related to the maximum shear resistance behavior [START_REF] Arman | glass to high-strain-rate shock loading: Plasticity, spall, and atomic-level structures[END_REF][START_REF] Park | Effect of the Atomic Packing Density on the Structural Change Rate of Amorphous Alloys under Elastostatic Stress[END_REF] and thermal stability [START_REF] Park | Atomic packing density and its influence on the properties of Cu Zr amorphous alloys[END_REF] of metallic glasses.

Additionally, Guo [START_REF] Guo | Quasi-Icosahedral Clusters in Zr-Based Metallic Glasses[END_REF] suggested to consider also other slightly distorted icosahedra (quasiicosahedra) identified by the indices < 0 2 8 1 0 >, < 0 2 8 2 0 >, < 0 1 10 2 0 > which are the most similar full-icosahedra among all polyhedra. both Mo-and Ni-centered clusters shows a constant value around 5% and 25%, respectively.

The maximum in Ni-centered cluster, associated to the maximum in GFA, was found between 10 and 70 at. % Ni which is consistent with the experimental compositional range of amorphization [START_REF] Abadias | Experimental evidence for the role of supersaturated interfacial alloys on the shear elastic softening of Ni/Mo superlattices[END_REF][START_REF] Laborde | Acoustique picoseconde dans les multicouches métalliques[END_REF]. However, Mo-Si alloys show a monotonic decrease in the population fraction of both Si-and Mo-centered clusters from 40 and 15%, respectively, to ~0% when the Si content is higher than 70 at. %. The absence of full-and quasi-icosahedra for Si content higher than 70 at.% is linked to a change in atomic-packing configuration and a reduction of atomic coordination, from 12-15 for metallic glasses to 4 for covalent a-Si (see Figure 4). This variation is correlated to a transition visible on atomic volume (discussed in next Section, see b)

The measured [START_REF] Abadias | Experimental evidence for the role of supersaturated interfacial alloys on the shear elastic softening of Ni/Mo superlattices[END_REF][START_REF] Belliard | Elastic properties and phonon generation in Mo/Si superlattices[END_REF][START_REF] Laborde | Acoustique picoseconde dans les multicouches métalliques[END_REF]16] and calculated mass density ( ) and atomic volume (Vat) of Mo-Ni and Mo-Si alloys are shown in Figure 6 as black circles. It can be observed that the numerical simulations consistently and quantitatively reproduce the experimental data. The mass density is found to continuously decrease from ~10.2 to ~8.9 g/cm 3 and to ~2. the amorphous alloys acquire a covalent character and a reduced coordination number down to 4 (as shown in Figure 4). One can notice that for xSi~0.80 the experimental atomic volume is higher than the calculated one. It is not excluded that such a high value could be attributed to an excess of a-Si having higher volume with formation of depleted sub-silicides such as MoSi2, Mo3Si or Mo5Si3 [START_REF] Govindarajan | Synthesis of molybdenum disilicide on molybdenum substrates[END_REF].

Sound velocities and elastic constants

The measured [START_REF] Abadias | Experimental evidence for the role of supersaturated interfacial alloys on the shear elastic softening of Ni/Mo superlattices[END_REF][START_REF] Belliard | Elastic properties and phonon generation in Mo/Si superlattices[END_REF][START_REF] Laborde | Acoustique picoseconde dans les multicouches métalliques[END_REF]16] The observed differences can be ascribed to the presence of point defects (of interstitial types) introduced by energetic bombardment during growth of MoNi alloys by ion beam sputtering process, as was reported by Debelle et al. for pure Mo layers [START_REF] Debelle | Stress field in sputtered thin films: Ion irradiation as a tool to induce relaxation and investigate the origin of growth stress[END_REF], and affecting more elastic properties of the crystalline phase than of the amorphous one. Intrinsic stress induced during film growth can also affect elastic properties [START_REF] Abadias | Stress in thin films and coatings: Current status, challenges, and prospects[END_REF]. Quantitative assessment of such complicate effects on sound velocities and elastic constants, as we performed in the case of defectivenitrides materials such as TaN [START_REF] Abadias | Large influence of vacancies on the elastic constants of cubic epitaxial tantalum nitride layers grown by reactive magnetron sputtering[END_REF] and MoAlN [45], is out the scope of this work. while the Si-centered atomic volume first does not change (see Figure 3), an increase in the atomic volume is progressively observed and the amorphous alloys acquire a covalent character and a reduced coordination number down to 4. We are again witnessing a progressive softening of the elastic stiffness constants while, surprisingly, both the longitudinal and shear acoustic velocities increase continuously. All these behaviors are intrinsic consequences of the high Si, Ni and Mo supersaturation, leading to lattice instability and accompanying the phase transitions.

Metal to insulator transition in Mo-Si

In supplementary materials , we report the calculated total (TDOS) and partial (PDOS) electronic density of states N(E) in the amorphous compositional range of the Mo-Si system, as well as the previously measured electrical resistivity of Mo-Si alloys [16]. TDOS displays metallic and semi-metallic behavior as evidenced by the absence of a gap at the Fermi energy EF. According to Mott al. [START_REF] Mott | Electronic processes in non-crystalline materials[END_REF], static conductivity can be defined by the following expression:

with the reduced Planck constant, dav the average interatomic distance, the electron mass, e the electron charge, N(EF) the density of states at Fermi energy. We calculated the electrical resistivity e = 1/ dc shown in figure 9, that can satisfactory reproduce the trends of the measured electrical resistivity at 300K in the amorphous region [16]. In the range 0.20 < xSi < 0.40, the resistivity increases only slightly with the Si content x, pointing out the metallic character of these amorphous alloys. This behavior is commonly observed in highly resistive amorphous metallic alloys, in good agreement with the Mooij correlation [START_REF] Cote | Glassy Metals I ed HJ Guntherodt and H Beck[END_REF], with an electron mean-free path of the order of the interatomic spacing. There is a transition for 0.9 > xSi > 0.4, as the electrical resistivity is increasing continuously, as well as atomic coordination is decreasing and atomic volume increasing. A sudden jump above xSi~0.9 to a-Si is observed, when tetravalent atomic coordination (Figure 4) is reached with the highest atomic volume (Figure 6), showing that the metal to insulator transition is expected for xSi~0.9.

Summary and conclusions

SQS and AIMD methods were used to design the supercells of crystalline and amorphous structures of Mo-Ni and Mo-Si alloys, together with DFT, to calculate their ground state properties (atomic volume, mass density, average interatomic distance, polyhedral cluster analysis and electronic density of states), their sound velocities and elastic properties. We could elucidate the complex interrelationships between short-range atomic order properties and measured macroscopic properties in their amorphous states. While a quantitative agreement is found for BCC Mo-Si alloys between our calculated elastic constants and the measured values [START_REF] Abadias | Experimental evidence for the role of supersaturated interfacial alloys on the shear elastic softening of Ni/Mo superlattices[END_REF][START_REF] Belliard | Elastic properties and phonon generation in Mo/Si superlattices[END_REF][START_REF] Laborde | Acoustique picoseconde dans les multicouches métalliques[END_REF]16], it is not the case for BCC Mo(Ni) and FCC Ni(Mo). Any wrong or approximate assumption about crystallographic texture is not the cause of the discrepancy, leading to the consideration that the incorporation of point-defects are likely responsible for such deviation, in the case of Mo-Ni films obtained under highly energetic deposition process (ion-beam sputtering).

For both amorphous alloys, we calculated elastic constants in two ways, considering the possibility of atoms to relax (softer values) or not relax (stiffer values) after the application of elastic strains. The spread between upper and lower values is much larger in comparison to the polycrystalline case, but the trends are similar and account for the experimental data [START_REF] Abadias | Experimental evidence for the role of supersaturated interfacial alloys on the shear elastic softening of Ni/Mo superlattices[END_REF][START_REF] Belliard | Elastic properties and phonon generation in Mo/Si superlattices[END_REF][START_REF] Laborde | Acoustique picoseconde dans les multicouches métalliques[END_REF]16]. C33 and C44 exhibit only a limited variation in the amorphous compositional range.

Nevertheless, an additional softening is confirmed by our calculations for the Mo-Si system at the transition from metallic-to-covalent bonding character observed at xSi~0.7, with a reduction of the atomic coordination from 12-15 for the polyhedral clusters in the metallic glass, to 4 for covalent a-Si.

For Mo-Si amorphous alloys, electronic density of states shows no gap at the Fermi level, confirming their metallic character. Their electrical resistivity is increasing with Si addition and the metal to insulator transition is found to occur for higher content of Si, xSi > 0.9.

Overall, these numerical results provide an additional and comprehensive illustration of computational efficiency, relying on the combination of first-principles calculations (DFT and AIMD) and special quasi-random structures methods for designing amorphous structure and random alloys supercells, and the prediction of their structural and elastic properties

  Amorphous alloys were designed by AIMD calculations, to examine the local atomic arrangement in relation to chemical composition and its impact on structural and mechanical properties in both Mo-Ni and Mo-Si systems. A cubic box with periodic boundary conditions was used, containing 256 atoms of the desired alloy composition. Initially, all alloys were heated to 3500 K and equilibrated for a minimum of 10 picoseconds (ps) under isobaricisothermal (NPT) ensemble, providing homogeneous molten alloys with relaxed atomic positions. Amorphous solid state was obtained from cooling down the equilibrated melt from 3500 K to 300 K at a cooling rate of 1.06 × 10 15 K/s (time step of 1.5 fs and 2000 steps, total cooling time of 3 ps), using the isovolumic-isothermal (NVT) ensemble and the Nose thermostat, followed by a relaxation of the cell with NPT ensemble and Langevin thermostat for at least 15 ps by steps of 1.5 fs (more than 10 000 steps). The atomic volume and mass density, together with pair distribution function of atoms (PDF), were subsequently determined. Our criterion of successful production of the amorphous supercell, was the limited variation of averaged atomic volume (less than 0.1 Å 3 /at.) that we can satisfactorily compare to experiments (even more stringent for MoSi case) and of averaged total energy (less than a few meV/at.), during last NPT relaxation runs. To check the influence of the cooling time c on the resulting amorphous structures, different AIMD simulations were run for Mo40Si60 alloy, varying the cooling time c from 0.3 to 3 ps with a common 0.3 fs step, and comparing the final total energy and atomic volume obtained after the NPT runs. It was noticed by Houska et al.[START_REF] Houska | Ab initio modeling of complex amorphous transition-metal-based ceramics[END_REF] for ZrSiN case (a transition metal alloyed with light elements) that the duration of the cooling should not be much lower than 2 ps for producing structures independent of the cooling rate. Our results showed only a moderate influence of c, with a lower total energy ~-19 meV and lower volume ~-1% for the 0.3 ps cooling time, relative to the chosen parameter ( c =3 ps).

Figure 1 :

 1 Figure 1: Free energy of formation of amorphous and crystalline phases calculated at 300 K

Figure 1

 1 Figure 1 represents the free energies of mixing of the amorphous and crystalline phases

Figure 2 :

 2 Figure 2: Calculated pair distribution function for a) Mo1-xNix, and b) Mo1-xSix amorphous

volume

  Al-and Cu-centered clusters are responsible for the GFA, increasing the mass density and stabilizing the amorphous structure. Hence in this study, Ni-and Si-centered clusters are expected to be responsible for the increase in GFA of these alloys because of their lower volume. As shown in Figure 3, Si-centered clusters volume is decreasing from ~15 to 11 Å 3 above xSi~0.2 while Ni-centered clusters volume remains roughly constant ~11 Å 3 . Higher volumes are found for Mo-centered clusters, 18-16 Å 3 and 15.5-16 Å 3 , for Mo-Si and Mo-Ni amorphous alloys, respectively.

Figure 3 :

 3 Figure 3: Mo-, Ni-and Si-centered clusters volume analysis for amorphous a) Mo1-xNix, and

Figure 4 :

 4 Figure 4: Mo-, Ni-and Si-centered clusters coordination analysis for amorphous a) Mo1-xNix,

Figure 5

 5 Figure5shows the polyhedral population of full-and quasi-icosahedra found in Mo-Ni and

Figure 5 )

 5 Figure5) and elastic response (see section 3.5, further softening of C33 and C44). Indeed, the

Figure 5 :

 5 Figure 5: Voronoi full-and quasi-icosahedral population of a) Mo1-xNix and b) Mo1-xSix

  1 g/cm 3 , for both Mo-Ni and Mo-Si alloys when Ni or Si is added to Mo, respectively. While the atomic volume of Mo-Ni is decreasing monotonously from ~15.5 to ~11 Å 3 , the one of Mo-Si is first decreasing until xSi ~0.5 before increasing rapidly beyond xSi ~0.70 and reaching the highest value of ~20 Å 3 for a-Si.

Figure 6 :

 6 Figure 6: Comparison between experimental (black plain symbol) [4, 6, 10, 16] and first-

  and calculated longitudinal (VL) and shear (VS) sound velocities of Mo-Ni and Mo-Si alloys are shown in Figure 7 as black circles. The structural transitions from BCC to amorphous state observed for xSi ~ 0.19 and xNi ~ 0.27, and from FCC to amorphous state observed for xNi ~ 0.73, are all accompanied by a reduction of sound velocities, when Ni or Si solute element is added to Mo or when Mo is added to Ni or Si. VL is reduced from 6400 to 5700 m/s in Mo(Ni) and from 6700 to 5600 m/s in Ni(Mo), while VS is reduced from 3100 to 2200 m/s in Mo(Ni) and from 2800 to 2100 m/s in Ni(Mo). In Mo(Si), VL is reduced from 6300 to 5700 m/s, while VS is reduced from 3300 to 2500 m/s. The trends in the crystalline or the amorphous region are well captured by our calculations, whereas the spread between relaxed (lower value) and non-relaxed (upper value) calculated data is much wider for the amorphous structure than for the crystalline ones (BCC and FCC), due to nonequivalent atomic positions and related local relaxations, in the supercell of amorphous materials. One should notice that a quantitative agreement between experiments and calculations is achieved in the BCC-Mo(Si) (0.19 xSi 0) with a <011>-texture, while it is not the case for the BCC-Mo(Ni) (0.27 xNi 0) with <011>-texture and FCC-Ni(Mo) (1 xNi 0.75) having <111>-texture.

Figure 7 :

 7 Figure 7: Comparison between experimental (black plain symbol) [4, 6, 10, 16] and first

Figure 8 :

 8 Figure 8: Comparison between experimental (black plain symbol) [4, 6, 10, 16] and first

Figure 9 :

 9 Figure 9: Calculated dc-electrical resistivity of Mo-Si amorphous alloys versus Si