Hugues Goffman

Christian F Pothier

Urbina

C Janvier

L Tosi

Ç Ö Girit

M F Goffman

H Pothier

C Urbina

Superconducting Atomic Contacts inductively coupled to a microwave resonator

Keywords: break junctions, atomic contacts, superconductivity, Josephson effect, Andreev states

Superconducting atomic contacts inductively coupled to a microwave resonator

Introduction

Atomic-size contacts between metallic electrodes are routinely obtained using either scanning tunneling microscopes or break-junctions [1]. From the electrical transport point of view, atomic contacts are simple systems. First, as for any good metal, electronelectron interactions are strongly screened. Second, because their transverse dimensions are of the order of the Fermi wavelength (typically 0.2 nm) they accommodate only a small number of conduction channels. Moreover, as the transmission probability τ i for electrons through each conduction channel can be adjusted and measured in-situ [2], atomic contacts provide a test-bed to explore mesoscopic electronic transport [START_REF] Nazarov | Quantum transport : introduction to nanoscience[END_REF][START_REF] Cron | [END_REF][START_REF] Cron | Proceedings of the XXXVIth Rencontres de Moriond « Electronic Correlations: From Meso-to Nano-physics[END_REF]. In particular, when the metal is a superconductor atomic-contacts constitute elementary Josephson weak-links that allow probing the foundations of the Josephson effect [START_REF] Bretheau | [END_REF].

Josephson supercurrents [7] will flow through any barrier weakly coupling two superconductors, including a tunnel junction, a constriction, a molecule, or a normal metal [8]. Microscopically, weak links differ in their local quasiparticle excitation spectrum. For a non-interacting system, this spectrum is determined by the length of the weak link, compared to the superconducting coherence length, and its configuration of conduction channels as characterized by the set of transmission probabilities {τ i }. The excitation spectrum of a short single-channel weak link of arbitrary transmission τ contains, besides the continuum of states at energies larger than the superconducting gap ∆, a sub-gap spin-degenerate level, known as the Andreev doublet (Figure 1). Its energy E A = ∆ 1 -τ sin 2 (δ/2) [9,10] is a 2π-periodic function of the superconducting phase difference δ across the weak link. It is precisely this phase dependence that gives rise to the Josephson supercurrent. In the widespread case of Josephson tunnel junctions, for which all τ i 1, E Ai ∼ ∆ and the lowest-lying excitations conserving electron parity have a threshold energy only slightly lower than 2∆. By absorbing energy 2∆ a pair can be broken and two quasiparticles created at essentially the gap energy ∆, like in a bulk superconductor.

Two different spectroscopy experiments have recently probed this excitation spectrum for superconducting atomic contacts containing channels of high transmisssion. The first experiment [11,12] spotlighted the lowest energy excitation that conserves electron parity, the "Andreev transition" of energy 2E A , which leaves two quasiparticles in the Andreev level (red double arrow in Figure 1a). The second experiment [13] revealed a second type of excitation, with minimum energy E A + ∆. In this case, a localized Andreev pair is broken into one quasiparticle in the Andreev level and one in the continuum (green arrows in Figure 1a) [14,15,16], thus leaving the system in an "odd" state. These odd states had been previously detected through the spontaneous trapping of a single out-of-equilibrium quasiparticle in the Andreev doublet [17]. This ensemble of results firmly support the description of the Josephson effect in terms of Andreev bound states.

If parity is conserved, the ground state and the even excited state constitute a two-level system [18] that has been proposed as the basis for a new class of (a) superconducting qubits [19,20,21,22]. What is particularly interesting and novel is that in contrast with all other superconducting qubits based on Josephson junction circuits [23] an Andreev qubit is a microscopic two-level system akin to spin qubits in semiconducting quantum dots. Also, if one considers the odd states, despite the absence of actual barriers the system can be viewed as a superconducting "quantum dot" allowing manipulation of the spin degree of freedom of a single quasiparticle [24,25,26]. The coherence properties of Andreev doublets [20,22] are still to be addressed experimentally. The relaxation time of the excited state and the dephasing time of quantum superpositions of the two states have to be measured, understood, and if possible, controlled.

E A (G) ' ground state -E A (G) odd states 0 0 even excited state E A (G) (b) 2E A E A +' -E A +'
Both relaxation and dephasing mechanisms contribute in principle to the linewidth of the Andreev transition. In order to achieve coherent manipulation of these Andreev states one would need much narrower lines than those observed in the aforementioned experiments, where they were typically larger than 500MHz. This was most probably due to large superconducting phase fluctuations imposed by the dissipative measurement lines that were necessary to measure the current-voltage characteristics of the contacts, a key piece of information from which the {τ i } are extracted. Here, to isolate efficiently the contact from external perturbations, we follow a strategy that has been implemented successfully for superconducting qubits [27,28]. The idea is to include an atomic contact in a small superconducting loop to form an rf-SQUID inductively coupled to the electromagnetic field of a coplanar microwave resonator. The latter should act as a narrow-band filter to allow probing the Andreev transition at 2E A without excessive broadening. A similar setup was analyzed theoretically in [29], although here we have in addition avoided any galvanic connection of the SQUID loop with the rest of the circuit in order to minimize the probability of trapping out-of-equilibrium quasiparticles in the contact [17]. By varying the flux threading the SQUID loop the Andreev transition frequency can be brought into resonance with the resonator mode. This will result in hybridization of the Andreev levels and the cavity mode (see Figure 2). The goal of the experiment presented here is to perform spectroscopy of the Andreev levels of the contact as a first step towards coherent manipulation of the two-level system.

Experimental Methods

Sample fabrication

The samples are fabricated on a flexible 500 µm-thick Kapton substrate ( r 3.2, tan δ ∼ 1 × 10 -4 at 30 mK ), 50 mm in diameter. In a first step, a series of λ/4 Nb resonators is fabricated. The substrate is then cut into 7 mm × 16 mm chips which are individually processed afterward to fabricate the atomic SQUID.

The microwave resonator

A 200 nm thick Nb layer sputtered over the whole substrate is patterned via optical lithography, and then structured using reactive ion etching into a series of quarter-wave (λ/4) resonators in a coplanar waveguide geometry (see Figure 3). The total length of the 36 µm wide inner conductor is 5 mm. The gap between the inner conductor and the ground plane is 18 µm. The impedance of the coplanar waveguide is Z r ∼ 80 Ω. The resonator is coupled through an interdigitated capacitor C ∼ 5 fF to a 50 Ω line to measure its reflection coefficient S 11 ≡ 20 log ( Vout /V in ) . For the 5 fF coupling capacitor the external losses should dominate over the internal ones (arising essentially from dielectric losses in the kapton substrate) leading to a global quality factor on the order of 1000. 2.1.2. The atomic rf-SQUID. Using electron beam lithography we fabricate a 100 nmthick aluminum superconducting loop containing in one arm a micro-bridge, suspended over approximately 2 µm by reactive ion etching of a sacrificial polyimide layer (Figure 3). The bridge has a 300 nm constriction at the center. The width and the inner dimensions of the loop are 5 µm and 4 µm×90 µm, respectively, which lead to a geometrical inductance of around 100 pH , much smaller than the Josephson inductance of a typical atomic contact (a few nH). A magnetic flux φ through the loop is then used to impose a superconducting phase difference δ ∼ = 2π φ /φ 0 across the contact, where φ 0 = h /2e is the flux quantum. This allows adjusting the phase-dependent energy of the Andreev levels.

Setup

Figure 4 shows the break-junction setup. The ensemble is attached to the mixing chamber of a dilution refrigerator. A precision screw (not shown), driven by a room temperature dc motor, controls the vertical displacement of a spindle. A copper slab attached to the spindle pushes the free end of the sample, which is firmly clamped on the opposite side against a microwave SMA launcher. The elongation of the upper face of the substrate as it bends leads eventually to the bridge rupturing. Afterward, the distance between the two resulting electrodes varies by a few tens of picometers for every micrometer of vertical displacement of the pusher. The temperature of the ensemble is below 100 mK, and the cryogenic vacuum ensures that there is no contamination of the freshly exposed electrodes. The electrodes are gently brought back together, reforming the bridge and creating an atomic-size contact. Contacts can be made repeatedly in order to vary the number of channels and/or the transmission probabilities. An important feature of the microfabricated break junctions [30] is that a given contact can be maintained for weeks with changes in transmission below one part in a thousand. The sample holder is enclosed in a set of three cylindrical shields (Al, Cryoperm and Cu, innermost to outermost) attached to the mixing chamber of a dilution refrigerator. All shields are capped at both ends. The inside diameter of the Al shield is 76 mm. The intermediate Cryoperm shield diverts the ambient magnetic field to reduce flux fluctuations through the SQUID loop as well as to minimize the number of vortices trapped in the Nb superconducting film. The inner walls of the Al shield are covered with a 3 mm thick layer of epoxy loaded with bronze and carbon powder to damp cavity resonances and adsorb spurious infrared radiation [31,32]. A small superconducting coil, placed 2 mm above the sample inside a copper shield, allows controlling the flux through the loop. A single coaxial line enters this set of shields and connects to the SMA launcher. The overall microwave setup is sketched in Figure 5. There are two 8-12GHz circulators (Pamtech XTE0812KCSD) and one 0.1 -18 GHz directional-coupler (Clear Microwaves C20218) placed at the same temperature (25 mK) as the sample but outside the shields. A first microwave tone, injected at one circulator, probes the response of the resonator at a frequency close to ν 0 . The reflected signal from the resonator goes through the two circulators into a cryogenic amplifier (0.5-11 GHz LNA #265D from Caltech, gain 28 dB) placed at 1K. To minimize losses in the return signal a superconducting coaxial cable (Coax-Co SC-086-50-NbTi-NbTi) is used between the second circulator and the cryogenic amplifier. The output line from the cryogenic amplifier to room temperature is a CuNi coax, with a silver cladded inner conductor (CoaX-Co SC-086/50-SCN-CN). The two circulators prevent noise from the amplifier reaching the sample.

A second tone at frequency ν 1 can be injected through the directional-coupler (-20dB coupling) to drive the transition between the Andreev levels at the atomic contact. Each line has a series of attenuators placed at different stages of the refrigerator to prevent external noise from reaching the sample. The total attenuation of each of the two input lines (including losses in the cables) is 90dB. The two circulators divert the reflected signal towards a 28dB amplifier placed at 1K. A second tone of frequency ν 1 is used to drive the Andreev transition of the atomic contact.

Results

One-tone spectroscopy

After an additional 78dB room temperature amplification of the reflected signal, the reflection coefficient is directly measured using a vector network analyzer. As shown in Figure 6, three resonances appear in S 11 below 10K in the range 8 -10 GHz. Below the superconducting transition of aluminum, only the two lowest frequency lines (#1 and #2) depend on temperature and bending of the substrate, and are thus associated with on-chip modes. The coplanar mode resonance is the one at ν R ∼ 10.24 GHz, with a quality factor Q ∼ 300. This is three times lower than expected. The measurements clearly indicate an undercoupled regime with only 40°phase shift at resonance instead of the full 360°. We interpret this result as arising from the coupling of the coplanar resonator mode with a parasitic mode of the on-chip ground plane which is itself heavily damped by radiation to the enclosing dissipative cavity ‡. Despite the low quality factor, it is still possible to probe the atomic SQUID. Figure 7a shows the evolution of the resonance frequency of the resonator as the substrate is slowly bent at low temperature. As the coplanar resonator elongates, its resonance frequency decreases with the pusher vertical position at a rate of approximately 50 kHz/µm. The sharp frequency drop observed around a vertical deflection of 400 µm signals the last stage of rupture of the break junction. The frequency shift is in agreement with the change in inductance expected when opening the SQUID loop. As the vertical displacement of the pusher is actually not measured in-situ, but ‡ Because the sample must bent, the two outer electrodes of the coplanar resonator are actually grounded only at one end. As a result the ground plane behaves as an antenna. After carrying out the measurements we understood, through detailed electromagnetic simulations, that for the actual geometry of the resonator the quarter-wavelength mode had a similar resonance frequency than the antenna mode of the ground plane. The two modes hybridize and the resonator mode is also affected by radiation damping. Hybridization could be avoided by redesigning the resonator as a meander line (thus making the length of the ground plane electrode much smaller than the length of the resonator).

deduced from the measured number of turns of the motor and the pitch of the screw, the backlash of the mechanical driving setup leads to a hysteresis of around 25 µm between opening and closing directions. However, the position at which the abrupt frequency shift occurs is reproducible within a few microns for successive openings. In the region of this frequency drop the contact has atomic dimensions and its Josephson inductance becomes much larger than the geometrical inductance of the loop. In this limit, the resonator frequency evolves periodically with the magnetic flux threading the Al loop, as shown in Figure 7b. If the substrate is bent further, the loop opens completely and the flux modulation disappears. Figure 8 displays a spectrum of the reflection coefficient S 11 as a function of the probe frequency and the flux threading the loop. An anti-crossing between the resonator and an Andreev transition in the contact is clearly observed. As shown, the shape of the spectrum can be described, at least qualitatively, by considering a single channel of transmission (τ ∼ 0.995) coupled (g ∼ 120 MHz) to the coplanar resonator harmonic oscillator [29].

Two-tone spectroscopy

The spectrum can be explored over a much wider frequency range by using a two-tone technique. In this case the resonator is constantly probed at a fixed frequency ν 0 close to its resonance ν R , while sweeping the frequency ν 1 of a second tone that is applied through the directional coupler (see Figure 5). The amplitude of this second tone is chopped at an audio frequency ν a . The reflected signal at ν 0 is homodyne detected yielding the two quadratures I and Q which are then measured by lock-in amplifiers at ν a (see Figure 9). The pump signal at ν 1 is chopped at audio frequency. The reflected signal at ν 0 coming out of the cryostat is amplified and homodyne detected in a IQ mixer. The two phases are detected by the two lock-ins at the audio frequency at which the pump signal at ν 1 is chopped.

The spectra of the reflected signal as a function of the flux through the loop (horizontal axis) and the frequency ν 1 of the second tone (vertical axis) is shown in Figure 10 for two different contacts. Also shown in the figure are vertical cuts of each of the spectra at half flux quantum (δ = π), which are fitted using two lorentzian peaks, all with linewidths below 60 MHz. By fitting the spectra using the analytical expression for the Andreev transition frequency 2E A /h, one can extract the gap ∆ of the aluminum film and the transmission of the channels. Despite the apparently similar shapes of the multiple lines, the spectra do not correspond to contacts with several channels of slightly different transmissions, as shown by the continuous lines in 10(b). Instead, the appearance of several peaks is attributed to the high microwave power injected in order to acquire sufficient signal.

Conclusions

We have presented the first evidence for the efficient inductive coupling of a superconducting atomic contact to the electromagnetic field of a coplanar resonator. Using a two-tone setup, we have performed spectroscopy of the Andreev levels in the contact over several gigahertz. The observed linewidths are one order of magnitude smaller than in previous experiments and provide a lower bound of 10 ns for the coherence of the Andreev states. Although this is still too short for coherent manipulation of the Andreev doublet, we have identified a parasitic heavily damped resonance that loads the coplanar mode in the present design. A redesign of the resonator geometry to avoid this spurious resonance is expected to improve the lifetime by an order of magnitude. Pulsed pump and probe experiments should then allow performing Rabi oscillations of the state of the Andreev system.

Figure 1 .

 1 Figure 1. (a) The single particle excitation spectrum for each channel of a short Josephson weak link consists of a doubly-degenerate Andreev level at energy E A (δ) , and a continuum of states at energies larger than the superconducting gap ∆. Arrows indicate transitions that can be induced by microwave absorption. The four possible occupations of the Andreev level are shown in (b): they correspond to the ground state (Andreev level empty), the two odd states (a single quasiparticle in the Andreev level), and the excited pair state (doubly occupied Andreev level), with energies -E A , 0 and E A respectively.

Figure 2 .

 2 Figure 2. (a) An atomic rf-SQUID (in green) inductively coupled to a microwave resonator, represented here by a LC circuit (in red). The double triangle symbol represents the atomic contact. The spectrum of the combined system is probed through microwave reflectometry by weakly coupling the resonator to the external setup through a small capacitor. (b) example of the expected spectrum (blue full lines) as a function of the magnetic flux threading the SQUID loop. The resonance frequency of the bare resonator (red dashed line) is here 10.1 GHz; the Andreev frequency (green dashed line) corresponds to a channel with τ = 0.991; and the SQUID-resonator coupling energy is h × (120 MHz). The anti-crossing results from the hybridization of the two quantum systems.

Figure 3 .

 3 Figure 3. (a) Microphotograph of a coplanar quarter-wave Nb resonator. (b) Zoom on the shorted-end of the microwave Nb resonator (light gray). The 36 µmwide center line contains some small holes intended as pinning centers for eventual vortices trapped in the superconducting film. The aluminum loop (white), with a suspended microbridge in one arm, is placed within the 18 µm gap (black). (c) Scanning electron microscope image of suspended microbridge with a 300 nm-wide constriction. The bright v-shaped ridges on both sides correspond to the border of the underlying polyimide layer which was etched to free the bridge over ∼ 2 µm. (d) interdigitated coupling capacitor of the resonator.

Figure 4 .

 4 Figure 4. Break junction setup: a Kapton substrate is attached to SMA launcher (right side).The threaded spindle on the left side actuates the pusher vertically on the free end of the substrate with micrometer precision. The central cylindrical copper shield hosts a small superconducting coil that controls the flux through the SQUID loop. The whole system is enclosed in a set of three shields and anchored to the mixing chamber of a dilution refrigerator.

Figure 5 .

 5 Figure 5. Microwave setup: A first tone of frequency ν 0 is used to probe the resonator. The two circulators divert the reflected signal towards a 28dB amplifier placed at 1K. A second tone of frequency ν 1 is used to drive the Andreev transition of the atomic contact.

Figure 6 .

 6 Figure 6. Amplitude (a) and phase (b) of reflection coefficient of resonator as a function of frequency, measured at 30 mK. The coplanar mode resonance is the one at ν R ∼ 10.24GHz (labeled #1). There are two parasitic resonances, at 9.16 GHz and 10.45 GHz. Only resonances #1 and #2 change with temperature below 1K and when bending the substrate and are thus associated to on-chip modes. (c) and (d): zooms around coplanar mode resonance.

Figure 7 .

 7 Figure 7. (a) Resonance frequency of resonator ν R as a function of pusher position. Red (blue) curve correspond to the opening (closing) of the microbridge. (b) Phase of reflection signal as a function of the flux threading the SQUID loop (in reduced units) at the pusher position signaled by the blue dot in right panel. The modulation disappears when the microbridge is broken.

Figure 8 .Figure 9 .

 89 Figure 8. One-tone spectrum as a function of probe frequency (vertical axis) and phase across contact (horizontal). The gray level encodes the amplitude of the reflection coefficient. The dotted lines correspond to the bare frequencies for the resonator (red) and the Andreev transition of a single-channel of transmission τ ∼ 0.995 (white). The blue line is the prediction for the coupled system (coupling constant g ∼ 120 MHz).

Figure 10 .

 10 Figure 10. (a) and (b): Greyscale coded amplitude of the reflected signal at ν 0 , as a function of flux through the SQUID loop (horizontal) and frequency ν 1 of second tone (vertical) for two different contacts. Fits like red dashed line in (b) using the analytical expression for the Andreev transition frequency 2E A /h determine the channel transmissions (τ = 0.9906 in (a) and τ = 0.9934 in (b)). The white dashed line in (b) illustrates the fact that the additional features of the spectrum cannot be attributed to Andreev transitions of other channels with slightly different transmission coefficients. (c) Measured amplitude (black) of reflected signal at a flux φ = 0.5φ 0 for contact of panel (a). Red line: fit of measured data using two lorentzians of widths 17 MHz and 28 MHz. (d) Measured amplitude (black) of reflected signal at a flux φ = 0.5φ 0 for contact of panel (b). Red line: fit of measured data using two lorentzians of widths 61 MHz and 65 MHz.

Acknowledgments

We gratefully acknowledge technical support by P. Senat and P. F. Orfila, and the expert input and assistance of P. Bertet, D. Estève, P. Joyez and D. Vion. We have also greatly benefited from discussions with L. Bretheau, A. Levy Yeyati and J. M. Martinis. Financial support by ANR contracts DOCFLUC and MASH, C'Nano, the People Programme (Marie Curie Actions) of the European Union's Seventh Framework Programme (FP7/2007-2013) under REA grant agreement n°PIIF-GA-2011-298415, and ECOS-SUD (France-Argentina) n°A11E04 is acknowledged.


T Rouxel

Y Yokoyama

Elastic properties and atomic bonding character in metallic glasses

The elastic properties of glasses from different metallic systems were studied in the light of the atomic packing density and bonding character. We found that the electronegativity mismatch (e -) between the host-and the major solute -elements provides a plausible explanation to the large variation observed for Poisson's ratio ( among metallic glasses (MGs) (from 0.28 for Fe-based to 0.43 for Pd-based MGs) notwithstanding a similar atomic packing efficiency (C g ). Besides, it is found that ductile MGs correspond to e -smaller than 0.5 and to a relatively steep atomic potential well. Ductility is thus favored in MGs exhibiting a weak bond directionality in average and opposing a strong resistance to volume change.

I. INTRODUCTION

In comparison with oxide glasses, MGs are characterized by larger  and C g coefficients [START_REF] Wang | The elastic properties, elastic models and elastic perspectives of metallic glasses[END_REF][START_REF] Miracle | The influence of efficient atomic packing on the constitution of metallic glasses[END_REF] . Nevertheless, while a monotonic increase of  with C g was reported for oxide glasses [START_REF] Makishima | Calculation of Bulk Modulus, Shear. Modulus and Poisson's Ratio of Glass[END_REF][START_REF] Rouxel | Elastic Properties and Short-to Medium-Range Order in Glasses[END_REF] the situation seems much more complicated for MGs (Fig. 1). Indeed as far as ionic and/or covalent bonding are concerned, the atomic network connectivity has a direct incidence on C g (the better the connectivity is, the smaller  becomes), and the average atomic coordination for covalent glasses (chalcogenides) and the number of bridging anion per cation-centered tetrahedron (silicates) are relevant structural parameters to discuss the network connectivity. In the case of MGs the weak atomic bonding directionality and the lack of interconnected 1D-, 2D-, or 3D-structural units prevent from a simple analogy. However MGs are obtained from metalloids and transition metals with significantly contrasted physical and chemical properties, so that as soon as different chemical systems are under scrutiny large changes in the electronic bonding characteristics are observed. Here we focused on the e -and on the parameters (U o , m, n) of an empirical Lennard-Jones (LJ) type expression for the cohesion potential. While e -is a measure of the bond strength and directionality, LJ parameters reflect the sensitivity to a variation of the distance between atoms or structural units (clusters), and are related to the material stiffness. Combining e -and LJ parameters open new perspectives to promote ductility among MGs and to assess the plastic flow mechanism. In what follows, previously published elasticity and density data for MGs belonging to various chemical systems [START_REF] Wang | The elastic properties, elastic models and elastic perspectives of metallic glasses[END_REF][START_REF] Yokoyama | Compositional Dependence of Thermal and Mechanical Properties of Quaternary Zr-Cu-Ni-Al Bulk Glassy Alloys[END_REF][START_REF] Mattern | Structural evolution of Cu-Zr metallic glasses under tension[END_REF][START_REF] Park | Atomic packing density and its influence on the properties of Cu-Zr amorphous alloys[END_REF][START_REF] Yuan | On the correlation of Young's modulus and the fracture strength of metallic glasses[END_REF][START_REF] Tarumi | Low-temperature acoustic properties and quasiharmonic analysis for Cu-based bulk metallic glasses[END_REF] , including Cu-, Ni-, Pd-, Pt-, Fe-, Zr-, Ti-, Mg-, Ca-, La-, Ce-based glasses, were used to get insight into the relationship between the elastic properties, the atomic packing efficiency, and the atomic bonding character.

II. POISSON'S RATIO, ATOMIC PACKING DENSITY, AND ELECTRONEGATIVITY

In order to allow for a straightforward comparison with previous reports on oxide glasses, the atomic packing density is defined as the ratio between the minimum theoretical volume occupied by the atoms and the corresponding effective volume of glass

C g =  Σ f i V i /(Σ f i M i ) (1)
with for the i th constituent: V i =4/3  N r i 3 , where  is the specific mass, N is Avogadro number, r i is the metallic radius [START_REF]Handbook of Chemistry and Physics[END_REF][START_REF] Egami | Atomic size effect on the formability of metallic glasses[END_REF][START_REF] Inoue | Stabilization of metallic supercooled liquid and bulk amorphous alloys[END_REF] (i.e. half the shortest distance between two atoms in the pure metal), f i is the molar fraction and M i is the molar mass. The use of the experimental values for the interatomic distance [START_REF] Egami | Atomic size effect on the formability of metallic glasses[END_REF]13 to calculate C g instead of the classical values for pure metals was found to have only a minor incidence on C g in cases actual interatomic distances were available. For instance the values reported by Egami et al. [START_REF] Egami | Atomic size effect on the formability of metallic glasses[END_REF] and Inoue [START_REF] Inoue | Stabilization of metallic supercooled liquid and bulk amorphous alloys[END_REF] were found to alter the calculated C g values by less than 5% in all studied cases. For example, for Cu 50 Zr 50 and Ni 80 P 20 glasses, accounting for the actual average interatomic distances changes C g from 0.737 to 0.723 (-2%), and from 0.701 to 0.725 (+4%). Interestingly  is mostly larger for MGs than expected by means of a simple rule of mixture from the properties of the constituting elements. This suggests a better packing efficiency in average, in agreement with structural models and experimental observations. However, in contrast with silicate glasses where a one to one relationship was found between  and C g ( = 1/2-1/(7.2C g )) [START_REF] Makishima | Calculation of Bulk Modulus, Shear. Modulus and Poisson's Ratio of Glass[END_REF] , there is no straightforward correlation in the case of MGs. For instance, Fe-, Ti-, and Pd-based MGs have roughly the same C g (~0.63-0.65) but their  values spread from 0.28 to 0.43. A general trend among materials and structures which proved to be scale-independent is that  decreases as the connectivity increases [START_REF] Greaves | Poisson's ratio and modern materials[END_REF] . This rules holds for macrostructures such as construction frames or cellular systems, as well as for atomic-scale structures. In ionocovalent solids, 2D and 3D atomic networks are favored thanks to the strength and the directionality of the bonding. It is thus inferred that non transition metal host elements such as Ce, Ca, and Mg, develop more directional bonding through better localized f (for Ce) and sp (Ca,Mg) electrons giving rise to a relatively small . Some evidence for this is provided by the electronegativity difference between the host and the two major secondary elements (Table I), and the remarkable correlation found between  and e -(Fig. 2) which suggests that  primarily depends on the bond directionality and connectivity rather than on C g . Besides, this correlation provides an explanation for the variation of  within a given MG chemical system or for isostructural monoconstituent oxide glasses. For instance in Zr-based glasses,  is found to increase with the Al content in the ZrAlCu system, and e -~0.28 and 0.29 for the Zr-Al and Cu-Al pairs respectivily, to be compared with 0.57 for the Cu-Zr pair. In the case of "tetrahedral" oxide glasses such as a-SiO 2 , a-GeO 2 , a-TeO 2 , or "trihedral" glasses such as a-As 2 O 3 , a-B 2 O 3 , and a-P 2 O 5 [START_REF] Bridge | On the elastic constants and structure of the pure inorganic oxide glasses[END_REF] (where the double P=O bond confers a (P=O)O 3/2 trihedral structure) the change in which was not elucidated so far can also be interpreted in the light of the e -variation. Note that for MGs  provides an indication of the ductility that can be expected. Metallic glasses with >0.32 usually exhibit significant ductility at room temperature [START_REF] Gu | Ductility improvement of amorphous steels: Roles of shear modulus and electronic structure[END_REF][START_REF] Madge | Toughness, extrinsic effects and Poisson's ratio of bulk metallic glasses[END_REF] . A molecular dynamic study of amorphous solids using a two-body LJ potential to model atomic interactions with coordination numbers between 7 and 12 also corroborated this result and came to the conclusion that ductility is favored by reducing the covalent character [START_REF] Shi | Intrinsic ductility of glassy solids[END_REF] . >0.32 corresponds to e -<0.5. In the case of Pd-and Pt-based MGs, e -is as low as 0.01 and 0.09 respectively. For Fe-, Ce-, and Cabased MGs which are known to behave in a brittle manner, e -is typically larger than 0.5. In Febased glasses containing C and Cr, C is expected to react with Cr due to a large enthalpy of mixing [START_REF] De Boer | Cohesion in metals -Transition metal alloys[END_REF] Recalling that  scales with K/ where K and  are bulk and shear elastic moduli respectively ( =(3K/-2)/(6K/ +2)), as  increases, ductility is enhanced, and isochoric shear flow prevails over volume changes. While the e - parameter shows up as a measure of the ease for shear, i.e. for relative displacement of atoms or groups of atoms with respect to each other along paths normal to center-to-center direction, the shape, and especially the steepness of the interatomic potential comes into play as soon as local volume changes -or variation of center-to-center distances -are considered.

III. ELASTIC PROPERTIES AND DISSOCIATION ENERGY

In what follows we use a classical derivation of an elastic modulus from an empirical central force potential [START_REF] Grüneisen | 1 st rule in "Physical Properties of Solid Materials[END_REF][START_REF] Makishima | Direct calculation of Young's modulus of glass[END_REF][START_REF] Inaba | Young's Modulus and Compositional Parameters of Oxide Glasses[END_REF] , here a LJ potential. The metallic bonding is essentially collective and there is no such thing as an isolated "metallic bond", so that instead of looking for a rough estimation of the cohesive energy from the atomic pair interaction and a speculated and mostly irrelevant local coordination, an isotropic potential is considered. A volume density of energy, <U o >/<V o >, is estimated from the energy content <U o > and the corresponding volume <V o >, where <V o > is the molar volume of a gram atom, that is an equivalent atom accounting for the stoichiometric composition of the alloy (2) The external work to produce a small volume change in a reversible manner is expressed as (3) where P is the hydrostatic pressure, K is the bulk modulus and V the volume change.

Equating the internal energy change, U, to W and remembering that at equilibrium (i.e. for

V=V o ) leads to

Further considering a Lennard-Jones (LJ) type potential for the internal potential of the system where U(r) is not to be understood as an interatomic pair potential but as a central force potential aimed at describing essentially radial interactions between structural units (such as inter-cluster interactions), the classical expression known as the 1 st Grüneisen's rule is obtained [START_REF] Grüneisen | 1 st rule in "Physical Properties of Solid Materials[END_REF] It is noteworthy that this expression stems from the assumption that the cohesive energy can be differentiated with respect to the atomic volume. In the case of a pair-interaction potential, U o is the energy required to suppress the bond and the cohesive energy of the system is half the bonding energy for each bond and accounts for the local coordination number, Z <U o >=ZU o /2 (7) By analogy, considering a representative volume <V o > we may write where <U o > is the molar dissociation energy and was estimated from existing thermochemistry data [START_REF]Handbook of Chemistry and Physics[END_REF][START_REF] De Boer | Cohesion in metals -Transition metal alloys[END_REF] for the elements constituting the studied glasses. In eq. ( 8) the 2/9 pre-factor can hardly be given any physical meaning in the cases of glasses inasmuch as fine details of the atomic network structure are unknown and long-range interactions are disregarded. However some new insight into the structure and the peculiarities of MGs from different chemical systems can be gained by comparing the (m,n) values deriving from the experimental K values, <U o > and the molar volume <V o >.

In the case of a binary system, <U o > is expressed as

<U o > = xH at (A,g)+yH at (B,g)-H mixing (A x B y ) (9)
<U o > is the energy (standard pressure) necessary to obtained separate gaseous atoms from the solid material (a gram atom is considered: x+y=1), according to the schematic drawing shown in Fig. 4. Note that the enthalpy of mixing is rarely known for multi-constituent metallic systems. Furthermore it might not represent the situation in amorphous systems, inasmuch the glassy network exhibits some peculiar structural features such as chemical segregation, clustering etc. Nevertheless, in most cases, the enthalpy of mixture is much smaller than the atomization enthalpy in absolute values. The atomization (sublimation) enthalpy is typically of several hundreds kJ/mol (605, 326, 431, and 338 for Zr, Al, Ni and Cu respectively), whereas the enthalpy of mixing is less than the enthalpy of formation (about -34 kJ/mol for ZrCu, -57 kJ/mol for PdSi, -30 kJ/mol for Ni 5 P…) and is of few tens kJ/mol. Therefore in what follows, as long as different chemical systems are under scrutiny and for sake of comparison, the dissociation energy of a multi-constituent glass will be written (10) where f i is the atomic fraction of the i th constituent.

The dissociation energy, the molar volumes, and the corresponding volume density of energy of series of metallic glasses from different chemical systems were calculated. The global trend for glass is an increase of T g with <U o > (Fig. 4), as the binding energy of pure substances scales with the melting point. However, as soon as a particular system is under scrutiny, somewhat unexpected tendencies are observed, as is the case for in Zr-based glass systems, where <U o > may increase while T g decreases. This observation supports the hypothesis of an heterogeneous glass network structure where T g is primarily governed by a glassy sub-network having a composition differing from the stoichiometric one. For example, the addition of aluminum to glasses from the (Zr,Cu) system results in an important deviation from random mixing, with an excess of aluminum at the vicinity of zirconium leading to (Zr,Al) rich regions, embedded in a (Zr,Cu) rich matrix depleted in aluminum. While the increase in <U o > with the Zr/Cu ratio is somewhat consistent with Zr having a higher melting point (T m =2128 K) than Cu (T m =1358 K), the decrease of T g (from 706 to 671 K as the composition changes from Zr 50 Cu 40 Al 10 to Zr 60 Cu 30 Al 10 ) might be related to the deviation from random mixing and to the behavior of the percolating phase if any.

K is plotted as a function of <U o >/<V o > in Fig. 5. It is noteworthy that using the density of the pure constituent for an ab-initio calculation (open symbols) in lieu of the density of the glass to estimate <V o > (closed symbols) has only little -as for Pt, Pd and Ni-based glasses -or no incidence -other studied systems -on the volume density of energy. The bulk modulus of glasses belonging to a specific system depends almost linearly on the volume density of energy calculated for the given system. However the slope of the linear interpolation depends much on the chemical composition. The corresponding (m,n) exponents are given in Table I using the m=2n approximation, thus following previously published analysis on similar metallic systems [START_REF] Wakeda | Relationship between local geometrical factors and mechanical properties for Cu-Zr amorphous alloys[END_REF][START_REF] Lin | Relation between calculated Lennard-Jones potential and thermal stability of Cu-based bulk metallic glasses[END_REF] . The larger the (m,n) values are and the steeper the potential well is. Most glasses investigated in this study are associated to mxn≈10-11.5. The exponents are found to scale quite well with . Pd-and Pt-based glasses, with mxn >16, exhibit  values above 0.38, whereas La-and Ce-based glasses are associated to mxn values smaller than 10, and  smaller than 0.29. of 0.28 and 0.24 respectively. There is also a relatively good agreement (but for Zr and Mg) between the mxn product and the  value of the pure constituent (Table I): 0.38 and 0.39 for Pt and Pd respectively, and and 0.28 for Ce and La respectively. This is indeed what could be anticipated from changes in the potential steepness. A steep potential for the material compressibility leaves little room for changes in the inter-unit distance (a structural unit here can be either an atom, a group of atoms representative of the glass stoichiometry, or a cluster with a peculiar chemical composition, depending on the way the glass behave under mechanical loading at the atomic scale) so that shear is expected to predominate, in agreement with large  values. On the contrary for a relatively small mxn product, the structural units might move each with respect to the others with center to center distance variations and local volume contraction or expansion. Different atomic clusters may exist that also possess high local packing efficiency and may act as such structural units: The capped trigonal prism, and a coordination number of 9, and icosahedra, with a coordination number of 12, are two such solute-centered clusters that are known to exist in metallic glass structures [START_REF] Miracle | A structural model for metallic glasses[END_REF][START_REF] Hirata | Structure Analyses of Fe-based Metallic Glasses by Electron Diffraction[END_REF] .

The change in the composition for a given system leads to a progressive shift from a given interpolation line to another in Fig. 5. For instance in the Cu-based system (see the insert in Fig. 5), as the zirconium content increases, K decreases (consistently with the fact that K(Cu)=140 GPa>K(Zr)=71 GPa) from about 117 to 100, so that the data points are seen to meet the data corresponding to the Zr-based glasses. In the ZrCu system, the packing density increases with increasing the Cu content and plastic deformation becomes more difficult since a local structural change accompanied by a local volume expansion is needed for shear deformation to proceed. This is corroborated by the decrease of  ((Cu 46 Zr 54 )=0.391, (Cu 50 Zr 50 )=0.36, (Cu 65 Zr 35 )=0.33) [START_REF] Sheng | Atomic packing density and short-to-medium range order in metallic glasses[END_REF] and the increase of the "mxn" product with an increase of the Cu content. In fact the strong sensitivity of the LJ potential to the chemical system prevents from having a universal relationship between K and <U o >/<V o > allowing for an ab-initio estimation of the elastic modulus. The same situation was already observed in the case of oxide glasses [START_REF] Rouxel | Elastic Properties and Short-to Medium-Range Order in Glasses[END_REF] and reveals some structural complexity. It is noteworthy in this latter case that whatever the chemical system,  and C g are significantly smaller than for the metallic glasses, and so is the mxn product. C g for silicate glasses is typically between 0.4 and 0.55, and K/(<U o >/<V o >)≈3/4, so that mxn ≈3.4, consistently with the fact that oxide glasses are much more brittle than the metallic ones, and undergo significant volume change under mechanical loading (for instance pure SiO 2 glass, with ≈0.15, can experience up to 20 % densification under hydrostatic loading).

While e -provides an indication for the bond directionality -a property which is clearly related to the resistance to shear deformation -and is reflected in the three dimensional picture of the energy landscape, the two dimensional cross section depicted by the center to center potential profile determines the resistance to compression and stretching, and hence to volume change. Thus, the present analysis opens the perspective of possible correlations between structural characteristics such as e -and the potential well parameters, and the constitutive laws for plastic flow or for densification.

IV. ELASTIC MODULI AND GLASS TRANSITION RANGE

Young's modulus (E) and T g data for series of metallic glasses, taken from previously published papers [START_REF] Wang | The elastic properties, elastic models and elastic perspectives of metallic glasses[END_REF][START_REF] Yokoyama | Compositional Dependence of Thermal and Mechanical Properties of Quaternary Zr-Cu-Ni-Al Bulk Glassy Alloys[END_REF]13,[START_REF] Wang | Elastic moduli and behaviors of metallic glasses[END_REF][START_REF] Tanaka | Elasticity and anelasticity of metallic glass near the glass transition temperature[END_REF] , are plotted in Fig. 6. These data are discussed in the light of the atomic packing densities, which are given as a function of Poisson's ratio () in Fig. 1, and of the energy content, which is given as a function of T g in Fig. 4. To get some insight into a possible E-T g correlation, it is interesting to compare i) Metallic glasses with other glasses (silicates, chalcogenides…); ii) metallic glasses from different chemical systems; and iii) metallic glasses with different compositions from a given metallic system. In comparison with non-metallic glasses, metallic glasses are advantaged by mostly larger elastic moduli than oxide, as large as those of the oxynitride glasses although their T g is usually much smaller (Fig. 4). Metallic glasses being adavantaged by a significantly larger atomic packing efficieny, this comparison provides evidence for the effect of the atomic packing density on the elastic moduli. It is also noteworthy that there is a global increase of T g of the different glasses with the energy content which increases almost monotonically in the following order: chalcogenide<metallic<silicates<silicon oxynitrides<pure silica glass. As far as metallic glasses from different systems are concerned a general increase of E with T g is observed, with the lower values being observed for Ce-and Ca-based glasses while Feand W-based glasses are the stiffest and the most refractory. Nevertheless there is no one-to-one relationship and for a given E value, the transition temperature shift can be as high as 300 K. Interestingly a similar observation can be made when bulk modulus is plotted as a function of the melting temperature, as was done by Tanaka et al. [START_REF] Tanaka | Elasticity and anelasticity of metallic glass near the glass transition temperature[END_REF] . In this case, for a given value of bulk modulus -let us take 150 GPa for instance -the temperature interval where metallic and intermetallic materials melt extends from 1000 (PdCuNiP glass) to 2500 K (Niobium) (Fig. 5 in ref. [29]). Indeed E being given in Pa, i.e. in J/m 3 , whereas T g is comparable with an energy, the atomic packing density (C g ) needs to be invoked to describe the energy density of the network. Considering the horizontal segments labeled 1, 2 and 3 in Fig. 6, it turns out that C g decreases along these segments for compositions belonging to the same chemical system as illustrated by the insert in Fig. 6. Comparing metallic glasses from different chemical systems remains difficult though, probably because an average packing efficiency (Eq. 1) doesn't take into account the spatial distribution of the packing density [START_REF] Hirata | Structure Analyses of Fe-based Metallic Glasses by Electron Diffraction[END_REF] , as well as the incidence of the local packing geometry and coordination [START_REF] Miracle | A structural model for metallic glasses[END_REF] . For instance, in spite of a mostly smaller C g for Pt-based glasses than for Cu-based glasses (Fig. 1), Ptglasses along segment labeled 2 in Fig. 6 have about the same Young's modulus. With a smaller C g and a lower energy content (Figs. 1,4), Young's modulus (Fig. 6) and bulk modulus (Fig. 5) of Pdbased glasses are also surprisingly high. The calculation of the electronic structure of the characteristic clusters occurring in Pd-Ni-P glasses (system of concern in the present analysis) revealed dominantly covalent bonding at phosphorus atoms [START_REF] Takeuchi | Electronic Structure and Stability of the Pd-Ni-P Bulk Metallic Glass[END_REF] . The high stability of Pd-based glasses would find its source in this covalent nature of the bonding at phosphorus sites associated with the existence of more compliant inter-cluster regions enriched in palladium and nickel. These latter regions promote shear deformation and a large  value, while the covalent bonds might be responsible for the relatively low C g .

V. CONCLUSION

In conclusion, metallic glasses exhibits neither one to one relationship between C g and (as for silicate glasses) nor between elastic moduli and T g . Nevertheless,  is found to increase as the difference in electronegativity between the host metal and the major solute elements decreases, so that a ductile behavior is expected for e -<0.5 (corresponding to >0.33). This correlation also holds for monoconstituent oxide glasses and hence provides an explanation to the variation of  observed for seemingly "isostructural" glasses. The relationship between K and the volume density of energy <U o >/<V o > as estimated for series of glasses from thermochemistry data was modeled by means of a LJ type potential function. Largest LJ exponent values were observed for Pd-and Ptbased glasses, and smaller ones for Ce-and La-based glasses. Most glasses (Cu-, Zr-, Ti-, Fe-,…) exhibit similar values, much larger than those reported for oxide glasses. This is consistent with the fact that the efficient atomic packing in MGs leaves little room for volume change in comparison with oxide glasses, and is conducive to isochoric shear flow, inasmuch as e -is small. TABLE I. (m,n) power law exponents of the L-J potential as estimated by linear interpolation of the K vesrus <U o >/<V o > data (Fig. 5), and electronegativity (Pauling). Data obtained on the pure substances from their bulk modulus and density are given for comparison [START_REF]Handbook of Chemistry and Physics[END_REF] . Following earlier investigators on similar materials, the m=2n approximation is used.

Cubased

Nibased

Pdbased

Ptbased

Febased

Zrbased

Tibased

Mgbased

Cabased

La

  (in absolute value) and e -(Cr-C)=0.89, and e -(Fe-C)=0.72. For Ce-based MGs, containing Ni, Al and Cu, Ce preferentially reacts with Al (e -(Ce-Al)=0.49) and e -(Ce-Cu)≈e - (Ce-Ni)=0.79. For Ca-based MGs containing Mg and Cu, Ca is supposed to react with Cu (|H mixing (Ca,Cu)|> |H mixing (Ca,Mg)|) and e -(Ca-Cu)=0.9.

Figure captions : FIG. 1 .

 :1 Figure captions: FIG. 1. Poisson's ratio as a function of the atomic packing density (Eq. (1)).

FIG. 2 .

 2 FIG. 2.Poisson's ratio as a function of the electronegativity mismatch between the host metal and the major secondary solute elements (horizontal error bars show the interval with the two major solutes).

FIG. 3 .FIG. 4 .FIG. 5 .FIG. 6 .FIG. 1 FIG. 2 FIG. 3 FIG. 4

 34561234 FIG. 3.Schematic drawing of the enthalpy contributions which come into play in the determination of the dissociation energy <U o > in the case of a binary A x B y alloy.FIG. 4.Dissociation energy <U o > as estimated from thermochemistry data as a function of T g . Data on chalcogenide, phosphate, silicate, and oxynitride glasses were extracted from ref.[3].FIG. 5.Bulk modulus (K) as a function of the volume density of energy. Open symbols correspond to the volume density of energy obtained using the molar volume as calculated from the density of the individual constituents and their atomic fraction (rule of mixture), while the closed symbols derive from the molar volume of the glass as obtained from the density of the glass.FIG. 6.Young's modulus as a function of T g for metallic glasses from different chemical systems. Data from refs.[1,3,9,13,28,29].

Acknowledgments:

The European Research Council is greatly acknowledged for the Advanced Grant 320506 (DAMREG) of the 7th framework program "Ideas". This work was initiated during a study leave at IMR-Tohoku University (Sendai), thanks to Prof. A. Makino.

Correspondence and requests for materials should be addressed to Tanguy ROUXEL (e-mail: tanguy.rouxel@univ-rennes1.fr)

FIG. 5 FIG. 6


C Hahn

G De Loubens

M Viret

O Klein

V V Naletov

J Ben Youssef

Detection of the microwave spin pumping using the inverse spin Hall effect

We report on the electrical detection of the dynamical part of the spin pumping current emitted during ferromagnetic resonance (FMR) using the inverse Spin Hall Effect (ISHE). The experiment is performed on a YIG|Pt bilayer. The choice of YIG, a magnetic insulator, ensures that no charge current flows between the two layers and only the pure spin current produced by the magnetization dynamics is transferred into the adjacent strong spin-orbit Pt layer via spin pumping. To avoid measuring the parasitic eddy currents induced at the frequency of the microwave source, a resonance at half the frequency is induced using parametric excitation in the parallel geometry. Triggering this nonlinear effect allows to directly detect on a spectrum analyzer the microwave component of the ISHE voltage. Signals as large as 30 µV are measured for precession angles of a couple of degrees. This direct detection provides a novel efficient means to study magnetization dynamics on a very wide frequency range with great sensitivity.

One great expectation of spintronics regarding information technology is the promise that pure spin currents can be generated and manipulated without their charge current counterparts [1]. Pure spin currents correspond to the transport of angular momentum in a very wide range of materials including metals and insulators with or without magnetic order.

In ferromagnetic metals, charge currents are intrinsically associated to spin currents because electrons at the Fermi level are spin polarized. Using these as injection electrodes, pure spin currents can be generated into a non magnetic metal in a non-local geometry where charges are evacuated through one electrode whereas spin diffusion can be collected by another nearby electrode [2,3]. This lateral geometry is well suited to nanostructures, but it is limited by the required current densities and the short spin diffusion lengths [4]. Another option relies on using the spin Hall effect, a phenomenon based on the spin-orbit interaction of a charge current which generates a transverse spin current in a conductor [5,6]. Pure spin currents can also be generated in ferromagnetic insulators by the spin pumping mechanism [7][8][9] during magnetization precession. This effect is produced by the damping of spin-waves which transfer angular momentum across an interface to a neighbouring layer. The emitted pure spin current can be detected electronically in an adjacent layer by the inverse spin Hall effect (ISHE) using metals with strong spin-orbit coupling like Pt [10,11,13,14,34]. The novelty here offered by electrical detection of the spin pumping using the ISHE is that it can be used also on non-metallic ferromagnets, including Yttrium Iron Garnet (YIG) [9, 15-20, 32, 33], a magnetic insulator which has unsurpassed small damping in ferromagnetic resonance (FMR). But so far, only the dc component of the ISHE voltage induced by FMR has been measured, which is a second order effect in the precession angle. Here, we report on a direct measurement of its first order ac counterpart.

The experiments of the present study are performed at room temperature on a YIG|Pt bi-layer where the YIG is a 200 nm thick epitaxial film grown by liquid phase epitaxy. A 6 nm thick Pt layer is then sputtered on top and two contact electrodes are defined at each end. The sample is mounted on a stripline antenna generating a microwave field h oscillating at a frequency f p as sketched in Fig. 1. At resonance of the uniform mode, the YIG emits, perpendicularly to the YIG|Pt interface, a flow of angular momentum generated by the spin pumping effect,

J s = 2eM s 2 G ↑↓ M × ∂M ∂t . (1)
In this expression, M is the magnetization vector, whose norm is M s , is the reduced Planck constant, e the electron charge, and G ↑↓ the spin mixing conductance at the YIG|Pt interface in units of Ω -1 •m -2 . The spin current pumped into the adjacent Pt is then converted into a charge current by ISHE,

J c = 2e Θ SH [y × J s ] , (2)
where Θ SH is the spin Hall angle in Pt and y is the unit vector perpendicular to the interface (see Fig. 1).

Importantly, the flow J s (and hence J c ) has both dc and ac components [23]. The dc part of this signal is normally detected as a voltage, proportional to J c (dc), that is maximum in a transverse geometry (i.e. ⊥ H, see Fig. 1a). In contrast, J c (ac) is maximum in a parallel geometry (i.e. H, see Fig. 1b). It is interesting to note that for circular precession the dc signal is second order in the precession angle θ (∝ sin 2 θ, maximal for θ = 90 • ), while its ac counterpart is first order (∝ sin θ cos θ and maximal for θ = 45 • ). Thus the ratio of ac to dc scales as 1/ tan θ, which is large for small precession angles. However, the ac component is much harder to detect as it oscillates obviously at the same frequency as that of the microwave generator producing the FMR. This microwave excitation field h induces eddy currents in any closed circuit containing the sample. These spurious ac currents are generally rather large and dominate any other contribution at the same frequency. Therefore, a clear detection of the ac spin currents has not yet been successful as one has to carefully eliminate the large amplitude eddy currents. In this letter, we report on the unambiguous detection of these ac spin currents emitted at ferromagnetic resonance using a specially designed system leading to an ac signal totally unpolluted by any other contribution. The key strategy here is to generate the FMR at half the frequency of the excitation source. This phenomenon is known as parametric excitation [START_REF] Sparks | Ferromagnetic relaxation theory[END_REF]. It exploits the fact that due to the ellipticity of the inplane precession, the magnetization follows a clamshell trajectory. During a full revolution of M around its precession axis z, the z-component of the magnetization M z oscillates twice faster, see Fig. 1a. This is also illustrated in Fig. 1b using red and blue colors to code its xcomponent, M x . Therefore, by exciting parallel to M , one can trigger the precession at half the source frequency. One should note however that this parallel parametric excitation is only possible in systems with low damping, since the excitation power needs to exceed a minimum threshold corresponding to a fraction of linewidth (typically below one Oersted for YIG [START_REF] Sparks | Ferromagnetic relaxation theory[END_REF]) in order to drive the magnetization into oscillation.

Experimentally, since the technique depends sensitively on the respective orientations of the microwave excitation and the bias field, we shall measure the ratio of the dc and ac components of J c , by rotating H, the static bias magnetic field, in the film plane. For all practical purposes, the YIG slab can be considered as an infinite film, whose resonance conditions are independent of the orientation of H.

In order to characterize our sample, and in particular the electrical conversion of the pumped spin current at the YIG|Pt interface, we first perform standard FMR resonance, where the small microwave field h is perpendicular to H (φ H = 90 • ). It is indeed the most efficient configuration to excite the magnetization dynamics: in the case displayed in Fig. 2, the angle of precession induced in YIG at resonance by a microwave field h = 36 mOe

(P = -5 dBm) is θ = 1.1 • (see Supplementary Materials).
The FMR signal is detected simultaneously by probing the power transmitted through the microwave line using a diode (Fig. 2a) and by measuring the dc ISHE voltage transversally to the static magnetization (Fig. 2b). Both measurements yield the same evolution of the resonance field versus frequency following the Kittel law for an in-plane magnetized thin film, see Fig. 3a. By measuring the diode signal at low power (P = -20 dBm, corresponding to h = 6 mOe), one can also determine in the linear regime the dependence of the linewidth on frequency, which is reported in the inset of Fig. 2a. A linear fit yields the Gilbert damping, α G = (1.4±0.1)•10 -4 , highlighting the very small magnetic relaxation of our YIG film [33]. The inhomogeneous part to the linewidth, ∆H 0 = 1.7 ± 0.2 Oe, reflects sample imperfections specific to the growth process of this batch. We find that this contribution dominates the broadening below 10 GHz. The amplitude of V ISHE (dc) measured at resonance allows us to determine the transport parameters at play in the electrical conversion of the pumped spin current (see Supplementary Materials). We find that our dc ISHE data can be well explained using typical parameters of the YIG|Pt system [32,33]: spin diffusion length λ sd = 2 nm, spin

Hall angle Θ SH = 0.05, and spin mixing conductance

G ↑↓ = 10 14 Ω -1 •m -2 .
We now wish to excite parametrically the YIG magnetization at half the applied microwave frequency, f p /2, by taking advantage of the clamshell shape of the precession trajectory, which is sketched in Fig. 1. For this, the component of the microwave field h parallel to the bias field H should reach the excitation threshold for parallel parametric excitation [START_REF] Sparks | Ferromagnetic relaxation theory[END_REF]. To demonstrate this effect in our sample through dc ISHE voltage measurements, we set a finite angle φ H = 10 • between h and H (see inset of Fig. 3b). Compared to the previous perpendicular geometry, the resonance condition at f p has not changed, only the microwave field is less efficient to bring the magnetization out-of-equilibrium, thus a stronger excitation power should be used to reach the same precession angle. We move momentarily to lower frequency in order to insert in the microwave circuit an additional amplifier limited in bandwidth to 1.1 GHz (see Supplementary Materials). At f p = 1.092 GHz and P = +19.8 dBm we observe only one resonance peak at H = 80 Oe in Fig. 3b. The new feature here is that if we increase the power to P = +27.8 dBm, a second peak appears in the spectrum at H = 20 Oe. Looking at the Kittel law of Fig. 3a), we find that it corresponds to the uniform mode resonating at f p /2, which is thus parametrically excited [17,18,[START_REF] Kurebayashi | [END_REF]26]. Thanks to the quantitative analysis of V ISHE (dc) using the transport parameters determined previously, it is possible to quantify the angle of precession corresponding to this parametric excitation:

θ = 5.1 • (see Supplementary Materials).
The next step is to directly detect the ac ISHE voltage generated at f p /2 by the parametrically excited magnetization dynamics. For this, we now align the microwave field h with the bias field H (φ H = 0 • ) and excite the system at f p = 3.6 GHz and high power, P = +24 dBm. The two voltage leads which contact the Pt layer are connected directly into a spectrum analyzer (SA) without any preamplification scheme. By sweeping the frequency of the SA at fixed H = 205 Oe, we detect a large signal of amplitude 30 µV at exactly f p /2 = 1.8 GHz, as can be seen in Fig. 4a. We claim this signal to be the ac component of the pure spin current pumped from the YIG parametric excitation into Pt, and converted into a voltage by ISHE. To prove this, we plot in Fig. 4b the amplitude of the SA signal measured at f p /2 as a function of H. We find that the amplitude of the signal is maximum at H = 205 G, which is the resonance field determined by standard FMR at 1.8 GHz in Fig. 2, and dies out in a range of about ±1.5 Oe around this field. We have also checked that this ac voltage signal has a parametric excitation origin, by studying its amplitude as a function of the excitation power. One can observe in the inset of Fig. 4b that the peak at f p /2 suddenly appears on the SA above a critical power P c = +20.7 dBm, i.e., a critical microwave field h c = 0.7 Oe, in good agreement with the expected threshold for parallel parametric excitation in YIG [START_REF] Sparks | Ferromagnetic relaxation theory[END_REF]. We note that the envelope of the parametric excitation signal observed in Fig. 4b as a function of H has a shape close to that of the standard FMR peak of Fig. 2. Still, we observe abrupt jumps of the amplitude as H is varied. We find that the details of this variation are very sensitive to changes in the orientation φ H of the bias field. We attribute this to the excitation of spin-wave modes which are almost degenerate with the uniform precession mode [START_REF] Sparks | Ferromagnetic relaxation theory[END_REF]27]. We have repeated the same experiment for different excitation frequencies ranging from 0.8 GHz up to 6 GHz. Because the microwave power exceeds the threshold for parallel parametric excitation (P = +22 dBm), an ac ISHE voltage at half the excitation frequency is observed on the SA as a function of H when the condition for resonance is met at f p /2 (cf. Kittel law on Fig. 5). Lastly, in order to check experimentally that our method eliminates all spurious signals, we performed the same series of measurements on a reference sample where the Pt layer was replaced by 15 nm of Al. We found experimentally that no electrical signal is produced at f p /2, which allows to conclude that the layer with strong-spin orbit scattering like Pt is indispensable to observe the ac ISHE voltage.

To gain more insight into the ac ISHE voltage, we would like to comment on its amplitude. For that, we compare the ac and dc components using the φ H = 10 • configuration (see Fig. 3b). We measure an effective dc voltage V ISHE (dc) = 1.1 µV, while in the same conditions, the generated ac voltage is V ISHE (ac) = 6.2 µV (see Supplementary Materials). Thus, we obtain the experimental ratio [V ISHE (ac)/V ISHE (dc)] exp = 5.6. We have determined that the angle of precession corresponding to the parametric excitation observed at H = 20 Oe is θ = 5.1 • . Therefore, one would expect that [J s (ac)/J s (dc)] theo = ξ/ tan θ = 5.7, where ξ = 0.51 is the ellipticity correction factor for this case (see Supplementary Materials), in excellent agreement with our estimation above.

In conclusion, we have shown that the microwave part of the spin pumping current emitted by a ferromagnet driven at resonance can be detected by the inverse spin Hall effect using an adjacent metallic layer with strong spin-orbit scattering [28]. On our millimeter size YIG|Pt sample it leads to a micro-Volt range microwave signal measurable directly on a spectrum analyzer without any pre-amplification or impedance matching. We believe that this broad band direct detection provides a novel efficient means to study magnetization dynamics in a wide variety of ferromagnetic materials. Further analysis of the measured spectra in the parallel parametric geometry will provide new insights into the spin-wave competition in the nonlinear regime. The phenomenon will also allow dynamical studies of the process of spin transfer at the interface with strong spin orbit non-magnetic metals [29][30][31].

This research was supported by the French ANR Grant Trinidad (ASTRID 2012 program)

and by the RTRA-2011 grant Spinoscopy.

Analysis of the ac ISHE voltage

In order to calculate the ac spin current generated by an elliptical precession trajectory, we follow the analysis performed to derive the factor E in Eq.( 3) for the dc ISHE voltage [34]. To do so, we use the Landau-Lifschitz-Gilbert equation to compute at resonance the three components of the magnetization precessing around its equilibrium axis z and the expression of the pumped spin current flowing accross the FM|NM interface (see Eq.( 1) of main text).

In the case of a circular precession, this yields J circ s,x (t) = -ω 2e 2 G ↑↓ sin θ cos θ sin ωt and J circ s,z = ω 2e 2 G ↑↓ sin 2 θ. (In our geometry, J circ s,y (t) does not produce any charge current through ISHE because its spin polarization is parallel to the normal y of the FM|NM interface, see Eq.( 2) and Fig. 1 of main text). Hence, |J circ s (ac)|/J circ s (dc) = 1/ tan θ, where θ is the angle of precession.

In the case of an elliptical precession, the same approach yields the correction factors with respect to the circular case:

• J s (dc)/J circ s (dc) = 2ω(ω M + √ ω 2 M +4ω 2 ) ω 2 M +4ω 2
, for the dc component of the pumped spin current. This is nothing else than the factor E appearing in Eq.(3) [34].

• J s (ac)/J circ s (ac) = 2ω √ ω 2
M +4ω 2 for its ac component. In these expressions, ω M = γ4πM s . Hence, the ratio of the ac and dc components writes:

J s (ac) J s (dc) = ω 2 M + 4ω 2 ω M + ω 2 M + 4ω 2 1 tan θ = ξ tan θ . (4)
Using the same transport parameters as for the analysis of the dc ISHE voltage, one finds that the precession angle corresponding to the 30 µV ac ISHE voltage measured in Fig. 4 at

f p /2 = 1.8 GHz equals θ = 2.5 • .
In this experiment, φ H = 0 • , so that the corresponding dc voltage vanishes (see Eq.( 3)).

Setting φ H = 10 • , as in the experiment of At lower power (+19.8 dBm), the only peak detected in the spectrum occurs when the Larmor condition is met at f p . At higher power (+27.8 dBm), a new peak appears in the spectrum at f p /2, corresponding to the parametrically excited resonance.

  FIGURES

FIG. 2 .

 2 FIG. 2. (Color online) Standard FMR detected using dc ISHE voltage. The pumping field h oscillating at f p = 1.8 GHz is oriented perpendicularly to the static magnetic field H. (a) Larmor absorption peak of the uniform mode detected with a diode (the inset shows the dependence of the linewidth on frequency). (b) Corresponding dc ISHE voltage measured perpendicularly to H (the inset shows the geometry of the experiment).

FIG. 3 .

 3 FIG. 3. (Color online) Parametric excitation detected using the dc ISHE voltage. (a) Measured variation of the resonance frequency f res as a function of the applied field (dots) using the standard FMR geometry (Fig.2). The solid line is the Kittel law f res = (γ/2π) H(H + 4πM s ) with γ = 1.785 • 10 7 rad•s -1 •Oe -1 and M s = 139 emu•cm -3 . (b) Spin wave mode spectra detected using the dc ISHE voltage for two different power levels. The bias field H is oriented at 10 • from the pumping field h oscillating at f p = 1.092 GHz (the inset shows the geometry of the experiment).

FIG. 4 .

 4 FIG. 4. (Color online) Parametric excitation detected using the ac ISHE voltage. The large power (+24 dBm) pumping field h at f p = 3.6 GHz is oriented parallel to the static magnetic field H.(a) The ac ISHE voltage generated by the parametric excitation at H = 205 Oe is monitored on a spectrum analyzer (1 kHz resolution bandwidth): an oscillation voltage is detected at f p /2 = 1.8 GHz. (b) The amplitude of this oscillation is measured as the bias field is swept from 195 to 225 Oe. The envelope of the curve should be compared to that in Fig.2. The maximum parametric signal occurs at H res = 205 Oe. The inset shows the threshold behavior (P c +20.7 dBm) of the power dependence of the spectrum analyzer signal at 1.8 GHz to demonstrate the parametric excitation.

Supplementary Materials: Sample preparation and characterization

The 200 nm thick single crystal Y 3 Fe 5 O 12 (YIG) film was grown by liquid phase epitaxy on a (111) Gd 3 Ga 5 O 12 (GGG) substrate. A 6 nm thin Pt layer with conductivity σ = (2.3 ± 0.2) • 10 6 Ω -1 •m -1 was then sputtered on top. Vibrating sample magnetometry reveals that the YIG film does not exhibit any in-plane anisotropy and has both very low coercivity (0.5 Oe) and saturation field (2 Oe). Its saturation magnetization, M s = 139 emu•cm -3 , is the one of bulk YIG. This value was confirmed by in-plane broadband FMR (from 0.4 GHz up to 17 GHz). FMR also allows to extract the gyromagnetic ratio, γ = 1.785•10 7 rad•s -1 •Oe -1 .

Experimental setup

Experiments are performed at room temperature. The microwave cell is placed at the center of an electromagnet which can be rotated. The 500 µm wide, 2 µm thick Au transmission line cell is connected to a synthesizer providing frequencies up to 20 GHz and power up to +24 dBm (additional boost to +35 dBm is obtained by using an external amplifier for frequencies lower than 1.1 GHz). A standard diode provides a quadratic detection of the transmitted power. The calibration of the linearly polarized in-plane microwave field yields h = 60 mOe when the output power is 0 dBm. The YIG|Pt sample is cut into a slab with lateral dimensions of 1.7 mm × 7 mm and placed on top of the transmission line. The sample is mounted upside-down on the stripline with the Pt layer facing the Au antenna.

The two voltage leads are equidistant from the area of excitation and separated by 3 mm (the two-point resistance is R = 129 Ω). V ISHE (dc) is measured by a lock-in technique with the microwave power turned on and off at a frequency of 1.9 kHz. V ISHE (ac) is directly measured using a spectrum analyzer (input impedance Z 0 = 50 Ω). Due to impedance mismatch, the voltage transmission coefficient from the YIG|Pt sample to the spectrum analyzer is T = 1 -R-Z 0 R+Z 0 = 0.56. We have checked that this value holds below 2 GHz using a network analyzer.

Analysis of the dc ISHE voltage

The amplitude of the ISHE dc voltage generated in Pt by the magnetization precession with an angle θ at frequency f in YIG is given by [32,33]:

In this expression, G ↑↓ is the spin mixing conductance, σ the conductivity of the Pt layer, Θ SH its spin Hall angle, λ sd its spin diffusion length, t its thickness, and L the length of the sample excited by the excitation field. E is a factor close to unity, which depends on the ellipticity of the precession trajectory, hence on the frequency ω/2π [34]: in our case, E = 0.43 at 0.546 GHz and E = 1.06 at 1.8 GHz.

The dc ISHE voltage is maximal when the angle φ H between the direction along which the voltage drop is measured and the static magnetization equals 90 • , which is the case in our standard FMR geometry (Fig. 2). In this case, we have checked that the dc ISHE voltage is odd in applied magnetic field, which shows that the voltage generated at resonance is not due to a thermoelectrical effect. The measured V ISHE (dc) are in good agreement with Eq.( 3), using values estimated from high frequency susceptibility for the precession angle θ and the following typical transport parameters [33]: Θ SH = 0.05;

For instance, at f p = 1.8 GHz and P = -5 dBm, the angle of precession at resonance is estimated to be θ = 1.1 • , which yields V ISHE (dc) = 0.44 µV, to be compared to the value of 0.5 µV measured in Fig. 2b. The overall agreement between Eq.( 3) and the experimental measurements of V ISHE (dc) vs. power (from -20 dBm to +10 dBm) and frequency (from 0.4 to 17 GHz) is within 30%. The solid line is the Kittel law of the YIG|Pt bilayer (see Fig. 3a).


Rebecca Nakhoul
email: rebecca.nakhoul@mines-paristech.fr

Patrice Laure
email: patrice.laure@unice.fr

Luisa Silva

Michel Vincent
email: michel.vincent@mines-paristech.fr

Multiphase and multiscale approaches for modelling the injection of textured moulds

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Introduction

A micro-moulding process consists on transferring micro details of micro-structured mould to products having a shear-thinning behaviour in their liquid phases. By definition, a micro moulded part can be either a piece weighing less than 1 mg or a part exhibiting micro details with a wall thickness less than 100 µm. In the last decades, microinjection moulding was rated as the most used micro-moulding technique. It presents a wide range of applications like micro-mechanics, micro-optics, micro-medical and micro-chemical applications, with good cost-effectiveness ratio for mass production.

Modelling Micro-Injection moulding presents a lot of challenges as it can be difficult to scale down the macroscopic physical properties [START_REF] Tolinski | Macro challenges in micromolding[END_REF][START_REF] Attia | Micro-injection moulding of polymer microfluidic devices[END_REF][START_REF] Yang | Microinjection molding of microsystem components: new aspects in improving performance[END_REF]. In addition, several factor/phenomena commonly neglected in macroscopic moulding simulations have significant impact on micro-injection accuracy [START_REF] Hill | An investigation of computer modelling for microinjection moulding[END_REF] due to the greater surface-volume ratio. Some of these neglected components are the surface tension, the surface roughness, the heating of the melt by viscous friction and the fast cooling due to the thin wall thickness [START_REF] Yu | Flow and heat transfer simulation of injection molding with microstructures[END_REF].

A multiphase Eulerian approach is adopted where the different phases are described implicitly using a Level set method [START_REF] Coupez | Multidomain Finite Element Computations[END_REF]. The global properties are determined from local ones using mixture laws. One mesh is used all over the domain. It is anisotropically refined relocating nodes to zones where information is highly needed [START_REF] Coupez | Metric construction by length distribution tensor and edge based error for anisotropic adaptive meshing[END_REF]. The adopted strategy to pass from a macroscopic to a microscopic scale is described in the next sections.

Macroscopic computations using industrial software

Using the commercial software Rem3D R [START_REF] Francois | Multi Criteria Adaptive Meshing for Polymers Processing in Rem3D R[END_REF], we model the injection moulding on the macroscopic scale. The mould itself is represented by its internal boundaries on which both temperature and velocities are imposed. To clarify the adopted procedure, we begin by looking at an example at the macroscopic scale. The mould cavity illustrated in Figure 1 is injected by a viscous shear-thinning material (PolyPropylene, PP-PRO-FAX PD702 in Cadmould database) with a flow rate reaching Q = 25 cm 3 /s. Its initial temperature is T 0 = 205 o C. The mould surface has micro cavities in the central zone which are not detected in the macroscopic software due to mesh limitations. The Micro cavities have a depth of 10 µm and a length of 10 µm. The mould temperature does not vary with time and is denoted T m = 35 o C. It is imposed directly on the cavity boundaries. The melt follows a Carreau-Yasuda law where the thermo-dependency is defined by a William-Landel-Ferry model. Except close to the lateral walls, the flow is essentially 2D in the xy plane (x being the streamline direction, y being aligned along the thickness). Therefore, the flow motion and temperature profile would be obtained by a two dimensional computations. This is why a two dimensional computations will be performed in the sequel with the data collected before the textured zone.

Multiphase Computations

The depth of textured zone is around 10µm or 100 times smaller than the cavity thickness and thus cannot be detected using common industrial software. The micro details studied here are uniform slots in the transverse direction (perpendicular to the flow direction) positioned in the center of the bottom side. Therefore two dimensional computations can be enough to describe the melt behaviour in the textured zone.

They are then fed to the multiphase model as illustrated in Figure 3. The main parameter describing the lowest levels is the characteristic length L re f which gives the thickness of the computational domain equally divided between the mould and the cavity. In practice, we choose two characteristics lengths L re f equal to 1 mm and 0.1 mm and boundary conditions are recovered from previous computations.

Scaled equations and non-flow criterion

The computations are performed on a rectangular domain with an unitary height. Therefore, we choose to scaled the momentum and heat equations by taking L re f , V re f , a melt diffusivity k melt and a reference viscosity η 0 for scaling factors:

ρV ref L ref η 0 (∂ t v + v • ∇v) = ∇ • 2 η η 0 (v) -∇p (1)
∇ • v = 0 (2) ρ C p L ref V ref k melt ∂T ∂t + v∇T - k k melt ∆T = η 0 V 2 ref k melt 2 η η 0 (v) : (v) (3)
where v, p, T and (v) are the velocity, the pressure, the temperature and shear rate tensor respectively. The temperature dependence is usually described by Arrhenius or WLF laws. For simplicity, the Cadmould material database proposes that the PP behaviour is approximated by an amorphous polymer (the crystallisation is not taken into account) and its dependence is given a modification of WLF relationship which will depend on two temperatures T ref and T s . The temperatureT ref is the temperature chosen within the range of rheological measurements and the temperature T s is a temperature used to fit these measurements (namely T ref = 205 o C and T s = 62.5 o C for the PP). The shift factor has expression:

log a T = - C 1 (T -T s ) C 2 + (T -T s ) + C 1 (T ref -T s ) C 2 + (T ref -T s ) (4)
where C 1 and C 2 are universal constants obtained by averaging values for several polymers (C 1 = 8.86, C 2 = 101.6 in our study). As a consequence, the polymer should no longer progress as the temperature tends towards T s .

In our multi-domain approach [START_REF] Coupez | Multidomain Finite Element Computations[END_REF], the viscosities for mould and air are chosen with respect to the melt viscosity by taking into account the following constraints: for the numerical procedure, the matrix conditioning is better if the ratio between maximal and minimal viscosities through the computational domain is less than 10 10 ; the mould viscosity has to be important enough in order to have an almost null velocity inside the mould; the air viscosity should be small enough to guarantee the fountain flow at the air/melt interface. Thus, a non-flow criterion based on the high values of viscosity when the temperature is close to the solidification temperature has been already tested [START_REF] Nakhoul | Modeling the injection of textured molds, 22ème Congrès Fran cais de Mécanique[END_REF] and gives raise to ill conditioned matrix. As this criterion is especially crucial when modelling micro-cavity filling, we have checked another method based on a penalty method which can ensure no flow motion for a given temperature T ps .

The numerical method used to solve the above equations is based on the use of a stable mixed formulation, which consists of continuous piecewise linear functions enriched with a bubble function for the velocity and piecewise linear functions for the pressure. A brief presentation of our mixed finite element method is presented in [START_REF] Coupez | Multidomain Finite Element Computations[END_REF]. We can explain here how to impose the no-flow condition by penalty method. The weak formulation of Navier-Stokes equations is written on a suitable trial functional spaces Q and V and reads:

find (u, p) ∈ V × Q such that Ω ρ d t u • v + Ω 2η (u) : (v) dΩ - Ω p∇ • v dΩ = Ω A P (T ) u • v dΩ + Ω δ Γ [[σ • n]] • v dΩ (5) Ω ∇ • u q dΩ = 0 (6)
where the penalty factor A p (T ) is defined:

A p (T ) = 0 if T ≥ T ps 10 6 max(η melt ) if T < T ps (7)
and [[σ • n]] is the jump of the normal stress at the interface Γ, defined implicitly by T = T ps (δ Γ being the Dirac function associated to this interface). This latter term is added as the vanishing condition on the velocity and it is not compatible with the continuity of the normal stress involved in the classical Finite Element formulation.

Multiphase Computations at Macroscale: L re f = 1 mm

In the entrance, we impose a Poiseuille flow conserving the same flow rate (Q = 2.5 10 -5 /0.015/2 = 0.008333 m 2 /s). Thus, the reference velocity

V ref is equal to Q/L ref .
At the entrance of the cavity, a temperature profile is imposed: first the contact temperature T i between the mould and melt for a purely conductive heat transfer is computed thanks to an analytic formula [START_REF] Agassant | Mise en forme des polymres : Approche thermomcanique de la plasturgie[END_REF] by taking T mould = 35 o C and T melt = 210 o C; then a constant temperature T melt is put inside the melt expect in the thermal boundary layer at the mould/melt interface (the temperature smoothness between T melt and T i is made with an er f function); finally a linear profile between T m and T i is imposed inside the mould.

The results are presented in Figures 4 where the mould/melt interface is located at y = 0. For y = -0.5 (scaled vertical coordinate), the temperature is equal to the mould temperature 35 o C. As for y = 0.5, it is equal to the melt temperature 205 o C. The interface temperature corresponding to y = 0 is equal to 47.8 o C.

Studying the thermal transfers in the mould is achievable on this scale but the cavities are hardly visible. Thus, tracking the textured zone filling is impossible. To overcome this point, we take next L re f = 0.1 mm as explained earlier. This value seems suitable since the size of thermal boundary layer above the mould is around 0.05 mm. 1. The choice of a suitable solidification temperature T ps which can depend on the shear rate can be made by comparing experimental and numerical filling rates.

Multiphase Computations at

FIGURE 1 .

 1 FIGURE 1. The cavity used in computations: 50 mm by 15 mm piece with a thickness of 1 mm with a textured zone begining at 21 mm from the end wall

FIGURE 2 .

 2 FIGURE 2. The velocity and temperature in a transverse section at t = 0.0725 s.

•

  L re f = 1 mm (a macroscopic multiphase computation), the flow motion corresponds roughly to a half Poiseuille flow. The thermal transfers are properly represented. The upper velocity and initial temperature are deduced from Rem3D R computations. • L re f = 0.1 mm (a microscopic multiphase computation), the flow motion corresponds roughly to a Couette flow, and the cavity filling is analysed.

FIGURE 3 .

 3 FIGURE 3. The 2D scheme used for microscopic modeling.

  Miscroscale: L re f = 0.1 mm Taking L re f = 0.1 mm, it is equivalent to zooming 10 times. The boundary and initial conditions are recovered from the previous computation for L re f = 1 mm. The melt initial velocity and temperature are respectively equal to 0.56 m/s and 205 o C (for y = 0.05 mm or y = 0.55 (normalized coordinates with L ref = 1 mm) ). The new mould temperature is equal to 39.85 o C for y = -0.05 mm. This microscopic geometry allows a better observation of both the mould (a) (b)

FIGURE 4 .

 4 FIGURE 4. The temperature and velocity plotted on the transverse section for L re f = 1 mm and T m = 35 o C. The thermal boundary layer studied at microscale is delimited by vertical blue lines.

FIGURE 5 .TABLE 1 .

 51 FIGURE 5. Cavities filling for L re f = 0.1 mm and t = 0.1 s for various values of T s = 40 (green), 50 (pink), 80 (red), 100 (blue) o C by using a penalization method.