C De Melo

C Guillemard

A M Friedel

V Palin

J C Rojas-Sánchez

S Petit- Watelot

S Andrieu

Unveiling transport properties of Co2MnSi Heusler epitaxial thin films with ultra-low magnetic damping

Keywords: Heusler alloys, Co2MnSi, electrical resistivity, anisotropic magnetoresistance, Hall effect, magnetic damping

The family of Co-based Heusler compounds contains promising candidates for spintronic applications regarding their predicted Half-Metal-Magnetic nature, ultra-low magnetic damping coefficients, high curie temperatures and tunable electronic properties. Here we focused on the transport properties of Co2MnSi thin films with thickness in the range of 4-44 nm exhibiting magnetic damping in the 10 -4 range. The goals of this study are to examine the impact of the peculiar electronic band structure on the transport properties, to identify the temperature-dependent scattering process, and to extract robust conduction parameters to exploit this material in magnetoelectric devices. In order to undoubtedly correlate all results, the full study has been performed on the same series of samples. Scanning transmission electron microscopy experiments were performed to check the chemically-ordered L21 phase in our films, and also allowed us to identify misfit dislocations generated at the interface with the substrate. The variation of the resistivity with film thickness was measured at different temperatures. The results are examined under the Fuchs and Sondheimer model which allowed us to extract the electron mean free path in Co2MnSi in the temperature range 5 -300 K. Values for the residual resistivity, Debye temperature, and distance between the Fermi energy and the conduction band for minority spins were obtained from the fit of the resistivity versus temperatures curves. A negative AMR ratio was measured for all the samples which confirmed the Half-metallic nature of the Co2MnSi films. The determination of the ordinary Hall coefficient alloys allowed us to extract the carrier concentration and carrier mobility and their dependency on the temperature. Finally, scaling of the anomalous Hall coefficient with the longitudinal resistivity was performed indicating that skew scattering is the dominant temperature-dependent scattering mechanism in our films.

INTRODUCTION

X2YZ Heusler intermetallic compounds, where X and Y are transition or rare-earth metals while Z is a main-group element (s or p-block in the periodic table), have attracted a lot of attention since their discovery by F. Heusler in 1903 [START_REF] Heusler | Über die Synthese ferromagnetischer Manganlegierungen[END_REF]. The broad family of Heusler compounds find applications in a variety of fields like spintronic, magneto-optics, thermoelectricity, magnonics, superconductivity, among others [START_REF] Galanakis | Spin-polarization and electronic properties of half-metallic Heusler alloys calculated from first principles[END_REF][START_REF] Graf | Simple rules for the understanding of Heusler compounds[END_REF][START_REF] Mallick | Tunability of Domain Structure and Magnonic Spectra in Antidot Arrays of Heusler Alloy[END_REF]. Even today, these materials constitute an active field of research with the emergence of new functionalities and prospective applications, for example, the discovery of the Half-Heusler topological insulators [START_REF] Lin | Half-Heusler ternary compounds as new multifunctional experimental platforms for topological quantum phenomena[END_REF][START_REF] Chadov | Tunable multifunctional topological insulators in ternary Heusler compounds[END_REF] or the recent observation of single pulse all-optical switching in the Mn2RuxGa Heusler compound [START_REF] Banerjee | Single pulse all-optical toggle switching of magnetization without Gd: The example of Mn2RuxGa[END_REF].

An interesting class of Heusler compounds are the half metal ferromagnets (HMF), that were discovered by Groot et al. in 1983 [START_REF] De Groot | New Class of Materials: Half-Metallic Ferromagnets[END_REF]. In these materials, the majority spin band has a typical metallic behavior while the minority spin band possesses an insulating or semiconducting character due to the existence of a bandgap around the Fermi energy. Therefore, HMF are 100% spin polarized at the Fermi energy with the potential to generate fully spin-polarized currents, which is highly desirable to improve the efficiency and functionalities of magnetoresistive devices [START_REF] Sakuraba | Giant Magnetoresistive Devices with Half-Metallic Heusler Compounds[END_REF]. Furthermore, due to the lack of minority spin density of states at the Fermi energy, ultra-low magnetic damping coefficients have been predicted (~ 10 -4 -10 -5 ) for these materials [START_REF] Sakuma | First-principles study of the Gilbert damping constants of Heusler alloys based on the torque correlation model[END_REF][START_REF] Liu | Origin of low Gilbert damping in half metals[END_REF], which are highly sought for magnonic applications, spin-transfer-torque and spin-orbit-torque-based devices [START_REF] Gabor | Spin-orbit Torques and Magnetization Switching in Perpendicularly Magnetized Epitaxial Pd/Co2FeAl/MgO Structures[END_REF][START_REF] Hirohata | Heusler Alloy Films for Spintronic Devices[END_REF].

Co-based Heusler compounds are a very promising class of HMF since together with the halfmetallicity and low magnetic damping coefficients [START_REF] Guillemard | Ultralow Magnetic Damping in Co2Mn-Based Heusler Compounds: Promising Materials for Spintronics[END_REF][START_REF] Trudel | Magnetic anisotropy, exchange and damping in cobalt-based full-Heusler compounds: an experimental review[END_REF][START_REF] Husain | Growth of Co2FeAl Heusler alloy thin films on Si(100) having very small Gilbert damping by Ion beam sputtering[END_REF][START_REF] Nembach | Perpendicular ferromagnetic resonance measurements of damping and Landé g-factor in sputtered (Co2Mn)1-xGex thin films[END_REF] they possess high Curie temperatures well above room temperature, tunable band structure [START_REF] Guillemard | Ultralow Magnetic Damping in Co2Mn-Based Heusler Compounds: Promising Materials for Spintronics[END_REF] and lattice constants close to some commercial semiconductors like GaAs and Si, which favor their possible integration in semiconductors industry [START_REF] Chen | Heteroepitaxy of Co-Based Heusler Compound/Muscovite for Flexible Spintronics[END_REF]. Among the 3d-transition-elements-based Heusler compounds, Co2MnSi is a promising candidate for spintronic applications due to its large bandgap for minority spins (0.5 to 0.8 eV) [START_REF] Guillemard | Ultralow Magnetic Damping in Co2Mn-Based Heusler Compounds: Promising Materials for Spintronics[END_REF]. The existence of a minority spin gap and 100% spin polarization has been demonstrated for this compound by spin-resolved photoemission spectroscopy [START_REF] Jourdan | Direct observation of halfmetallicity in the Heusler compound Co2MnSi[END_REF][START_REF] Andrieu | Direct evidence for minority spin gap in the Co2MnSi Heusler compound[END_REF]. Furthermore, we measured ultra-low magnetic damping coefficients with a record value equal to 4.6 * 10 -4 at 300 K, which is the lowest value reported so far for a conductive material [START_REF] Guillemard | Ultralow Magnetic Damping in Co2Mn-Based Heusler Compounds: Promising Materials for Spintronics[END_REF]. Tunnel magnetoresistance (TMR) ratios as high as 1995 % and 354 % have been reported for epitaxial Co2MnSi/MgO/Co2MnSi magnetic tunnel junctions at low and room temperature, respectively [START_REF] Liu | Giant tunneling magnetoresistance in epitaxial Co2MnSi/MgO/Co2MnSi magnetic tunnel junctions by halfmetallicity of Co2MnSi and coherent tunneling[END_REF]. Large magnetoresistance ratios over 30% at room temperature and over 60 % at low temperatures have also been reported for current perpendicular-to-plane giant magnetoresistance (CPP-GMR) devices with Co2MnSi as ferromagnetic electrode [START_REF] Sakuraba | Mechanism of large magnetoresistance in Co2MnSi/Ag/Co2MnSi devices with current perpendicular to the plane[END_REF]. However, the electronic properties and the half-metallicity seem to be very sensitive to the structure and composition of the films [START_REF] Guillemard | Issues in growing Heusler compounds in thin films for spintronic applications[END_REF], which can negatively impact the physical properties and the performance of these films when implemented in CPP -GMR or TMR magnetoresistive devices.

In this work, we report on the transport properties of Co2MnSi epitaxial thin films deposited on MgO substrates by molecular beam epitaxy, which allows a precise control of films' stoichiometry [START_REF] Guillemard | Issues in growing Heusler compounds in thin films for spintronic applications[END_REF]. This allows us to get systematically magnetic damping in the 10 -4 -10 -3 range for this compound, at least for film thicknesses above 5 nm. The crystalline quality of the films was monitored by electron diffraction, X-ray diffraction and transmission electron microscopy. Particular attention was paid to the possible presence of dislocations due to the mismatch between the film and the substrate since such defects may play a role in transport properties. The electronic transport properties in these films were then explored by performing resistivity measurements on a series of films with thickness varying in the range 4 -44 nm. The resistivity versus thickness measurements were examined using the Fuchs-Sondheimer analysis in order to extract important parameters like the electron mean free path. The resistivity dependence with temperature in the 5 -300 K range was also analyzed and compared to models taking into account the spin gap in this compound. Magneto-transport measurements were also systematically performed on the sample series. Negative Anisotropic Magneto-Resistance (AMR) values were obtained and explained as a consequence of the half-metallic nature of the films. Hall measurements were done to determine the carrier mobility and carrier concentration and to study their dependency on temperature and film thickness. The temperature dependence of the Anomalous Hall Effect (AHE) was analyzed to go deeper in understanding the scattering process in these films.

STRUCTURAL AND MAGNETIC PROPERTIES OF THE FILMS

The Co2MnSi (CMS) films were deposited on MgO (001) substrates by molecular beam epitaxy. The 3 elements were co-evaporated and each atomic flux was controlled with a  1 % accuracy (see [START_REF] Guillemard | Issues in growing Heusler compounds in thin films for spintronic applications[END_REF] for more details). The Co2MnSi layers grow on (001) MgO substrates with (001) [110] Co2MnSi || (001) [100] MgO epitaxial relationship (Fig. 1). The thickness of the films was tuned in the range 4 -44 nm and the films were protected from oxidation by a 2 nm Au-capping layer. X-ray diffractograms of the samples are presented in Fig. 1a, showing the Co2MnSi (002) and (004) peaks and the (111) crystallographic planes of the Au-capping layer.

To extract the in-plane lattice parameter of Co2MnSi, (422) crystallographic planes were measured in a non-symmetrical configuration (see supplementary information Figure S1). The in-plane and out-of-plane lattice parameters were thus extracted from (004) and (422) peaks and a similar lattice parameter was measured for all the thickness series except for the 4 nm sample for which a small deviation is noticeable (Fig. 1b).

Since X-ray diffraction at a fixed wavelength is not sufficient to unambiguously determine the chemical ordering [START_REF] Guillemard | Issues in growing Heusler compounds in thin films for spintronic applications[END_REF], Scanning Transmission Electron Microscopy experiments were performed using the High-Angle Annular Dark-Field technique (STEM-HAADF) along the [110] zone axis. A special pattern is thus expected for the L21 structure [START_REF] Guillemard | Issues in growing Heusler compounds in thin films for spintronic applications[END_REF] along this zone axis which is actually observed in the whole thickness of the films as shown in Fig. 1c. This technique was also exploited to look at possible structural defects like misfit dislocations expected in this kind of hetero-epitaxial Co2MnSi on MgO system presenting a -4.9 % lattice misfit. According to heteroepitaxial theory [START_REF] Frank | One-dimensional dislocations. I. Static theory[END_REF]25], the distance between interfacial dislocations is given by the relationship 𝑎 MgO (100) * 𝑚 ≈ 𝑑 CMS (110) * (𝑚 + 1), where 𝑚 is an integer, 𝑑 CMS (110) is the interplanar distance of the (110) crystallographic planes of Co2MnSi. This gives 𝑚 ≈ 19, meaning that the distance between misfit dislocations should be observed around this value. In order to evidence the dislocations, a Fast Fourier transform (FFT) of the TEM image was performed at the interface between Co2MnSi and MgO (violet frame in Fig. 1c). The (220) and (2 ̅ 2 ̅ 0) Co2MnSi diffraction spots were thus selected (Fig. 1d) and inverse-FFT of the masked planes was performed, obtaining the image in Fig. 1e. The misfit dislocations marked by dotted circles in Fig. 1e are clearly visible. These dislocations were observed along the Co2MnSi/MgO interface, with a similar distribution for all Co2MnSi thickness series. From the analysis of several interface images, the distance between dislocations was observed to vary from 𝑚 = 17 to 21, in good agreement with misfit dislocation theory. The presence of this dislocations network at the interface explains why the lattice spacing slightly deviates from the bulk value for low thicknesses as shown in Fig. 1b. Such structural defects have to be considered when interpreting conduction properties as presented in the following. The magnetic damping and effective magnetization of the Co2MnSi films were extracted from ferromagnetic resonance (FMR) measurements. This is a powerful technique for verifying the quality of the films and for ruling out the existence of multiple phases and/or stoichiometry issues [START_REF] Guillemard | Issues in growing Heusler compounds in thin films for spintronic applications[END_REF]. Ferromagnetic resonance (FMR) measurements were performed in a Physical Properties Measurement System (PPMS) equipped with a PhaseFMR-40 system. The measurements were performed at 300 K with the magnetic field applied perpendicular to the film plane to avoid two-magnon scattering contributions. The microwave frequency was kept fixed and the magnetic field was swept, starting from the highest field to guarantee the saturation of the films' magnetization. Multiple measurements were performed at different frequencies ranging from 10 to 40 GHz (see supplementary information Figure S2). The effective magnetization and the effective magnetic moment per formula units (f.u.) were extracted from the fit of the microwave frequency vs. resonance field, and the Gilbert damping (𝛼) and inhomogeneous linewidth (∆𝐻 0 ) from the fit of resonance linewidth (∆𝐻) vs. microwave frequency. The inhomogeneous linewidth ∆𝑓 0 can be obtained from ∆𝐻 0 by the Larmor equation:

𝑓 = 𝛾 2𝜋 𝜇 0 𝐻, where 𝛾 2𝜋
⁄ is the gyromagnetic ratio. The obtained values are reported in Table I.

As can be seen, the effective magnetization of the films increases with increasing film thickness, reaching the highest value of 1.33 T for the 44 nm film. In general, the effective magnetization of the films fits well to the theoretical expected value for Co2MnSi of 5 μ B f. u. ⁄ (1.28 T) except for the thinnest films where slightly smaller values were obtained. The evolution of the effective magnetization is discussed in supplementary information section 2.

Most of the samples showed ultra-low magnetic damping coefficients in the 10 -3 -10 -4 range (see Table I). It is worth to mention that the damping coefficients reported here are the raw effective values containing the intrinsic and extrinsic contributions (see details in the supplementary information section 2). There exist some approaches to separate the different damping contributions and to extract the intrinsic damping coefficient [START_REF] Schoen | Ultra-low magnetic damping of a metallic ferromagnet[END_REF][START_REF] Shaw | Magnetic damping in sputter-deposited Co2MnGe Heusler compounds with A2, B2, and L21 orders: Experiment and theory[END_REF]. However, we prefer to report here the global values, which represent upper limits, but most realistic ones for comparisons. The magnetic damping of the Co2MnSi thickness series increases with the increase of film thickness from 8 to 44 nm (Table I). We believe that this is due to the extrinsic damping contributions, that originates from Eddy currents induced in the sample (𝛼 𝑒𝑑𝑑𝑦 ) and Eddy currents induced in the waveguide, known as radiative damping (𝛼 𝑟𝑎𝑑 ), since both contributions increase with increasing film thickness (𝛼 𝑒𝑑𝑑𝑦 ~𝑡2 𝜌(𝑡) ⁄ , and 𝛼 𝑟𝑎𝑑 ~𝑡 [START_REF] Schoen | Radiative damping in waveguide-based ferromagnetic resonance measured via analysis of perpendicular standing spin waves in sputtered permalloy films[END_REF]). This hypothesis is supported by the increase of FMR lineshape asymmetry that was observed with the increase of film thickness which is a signature of the presence of Eddy currents in the samples [START_REF] Flovik | Eddy current interactions in a ferromagnetnormal metal bilayer structure, and its impact on ferromagnetic resonance lineshapes[END_REF] (see supplementary information Figure S2). It is worth to mention that the damping coefficients reported here are slightly higher than the values reported in our previous work for the sample of the same thickness [START_REF] Guillemard | Ultralow Magnetic Damping in Co2Mn-Based Heusler Compounds: Promising Materials for Spintronics[END_REF]. However, the measurements reported here were performed in a different experimental setup with a waveguide of larger dimensions implying a higher sensitivity to inhomogeneities, that can also contribute to an increase of the intrinsic damping since this technique is very sensitive to even small deviations from the 2:1:1 stoichiometry [START_REF] Guillemard | Issues in growing Heusler compounds in thin films for spintronic applications[END_REF]. The fact that the extrinsic contributions depend on the waveguide geometry may also affect the measured damping coefficients. The inhomogeneous linewidth (FWHM) of the films, ∆𝑓 0 , increases with the decrease of film thickness (Table I). This parameter is related with inhomogeneities of the magnetic phase and with the existence of local defects in the films. Therefore, the misfit dislocations observed in the STEM micrograph (Fig. 1e) are a source of inhomogeneous linewidth. Since these defects were only observed at the Co2MnSi/MgO interface and not in the bulk of the Co2MnSi layer, their impact should become more relevant with the decrease of film thickness due to the increase of the relative amount of defects, and this is what we observe from FMR measurements. Finally, the Landé factor (𝑔) was obtained from the linear fit of the microwave frequency vs. resonance field mentioned above, through the relation:

𝑔 = ℎ 𝛾 2𝜋 𝜇 𝐵 ⁄ ,
where ℎ is the Plank constant, 𝛾 2𝜋 ⁄ is the gyromagnetic ratio, and 𝜇 𝐵 is the Bohr magneton. The 𝑔 values were practically constants through all the thickness series and they are close to the free electron value (𝑔 = 2.0023) as expected for 3d compounds (Table I).

TRANSPORT PROPERTIES

For the transport measurements the films were patterned into Hall bars of lateral dimension measurements were performed by applying an in-plane magnetic field between ± 100 mT, which is enough to saturate the magnetization of the Co2MnSi films in-plane (see supplementary information Figure S4a). The AMR ratio was calculated from the difference in the film resistivity in the parallel (𝐻 || 𝑖) and perpendicular (𝐻 ⊥ 𝑖) configurations by rotating the magnetic field by 90° in-plane. Hall measurements were performed by sweeping the magnetic field applied perpendicular to the film plane, from -6 T to 6 T, to extract the ordinary and anomalous Hall coefficients.

Electrical Resistivity

Due to the Au-capping layer, the measured resistance is the equivalent resistance of the bilayer system composed by Au and Co2MnSi. Considering the bi-layer system as two parallel conductors the equivalent sheet resistance (R sheet ) can be expressed as the individual sheet resistance in each CMS (R CMS ) and Au (R Au ) layer:

1 R sheet ⁄ = 1 R CMS ⁄ + 1 R Au ⁄ , with R CMS = ρ CMS t ⁄ (1)
where ρ CMS and 𝑡 are the resistivity and thickness of the CMS films. In the following, the resistivity measurements are analyzed into two ways. First, the R sheet variation with the film thickness at a fixed temperature was studied through conduction models that take into account scattering in the film and at the interface / surface. Second, we analyzed the R sheet variation with temperature for a fixed thickness, by considering the different resistivity contributions coming from intrinsic resistivity (defects), magnonic and phononic processes.

The R sheet (and 1 R sheet ⁄ in inset) is plotted as a function of film thickness in Fig. 2c for two fixed temperatures, 300 K and 5 K. The increase of resistance at low thicknesses is due to the scattering of conduction electrons at the film surface and interface, which is enhanced when the film thickness becomes smaller than the mean free path of electrons in the material. The bulk resistivity (𝜌 𝐵 ) and the electron mean free path (𝜆) were thus extracted by fitting the sheet resistance vs. thickness (𝑡). According to the Fuchs and Sondheimer model [START_REF] Sondheimer | The mean free path of electrons in metals[END_REF], the Co2MnSi resistivity dependence with thickness can be expressed as :

𝑅 𝐶𝑀𝑆 = 𝜌 𝐵 𝑡 [1 + 3 4 (𝑞 - 𝑞 3 12
) * 𝐼(𝑞) - )

𝑒 -𝑞 ] -1 (2)
where 𝑞 = 𝑡 𝜆 ⁄ and 𝐼(𝑞) is the exponential integral

(𝐼(𝑞) = ∫ 𝑒 -𝑥 𝑥 𝑑𝑥 ∞ 𝑞
). We thus proceeded in a fitting of R sheet using eq. ( 1) and eq. ( 2), by varying 3 parameters, i.e. 𝜌 𝐵 , 𝜆 and R Au . The fitting curves are in very good agreement with experimental data, as shown in Fig. 2c. The extracted parameters are listed in Table II. It should be noted that the extracted 𝜌 𝐵 values are in good agreement with the residual resistivity reported for epitaxial Co2MnSi thin films [START_REF] Wang | Magnetic and transport properties of epitaxial Co2MnSi films[END_REF][START_REF] Prestigiacomo | Hall effect and the magnetotransport properties of Co2MnSi1-xAlx Heusler alloys[END_REF]. Our analysis allows us to go further by determining the electron mean free path in Co2MnSi in the temperature range 5 -300 K.

Eq. ( 2) can be modified to account for partial elastic scattering at the film surface/interface [START_REF] Sondheimer | The mean free path of electrons in metals[END_REF]. Fitting of the experimental data including a factor p, that represents the fraction of specular surface collisions, results in an increase of the electron mean free path when increasing p. Since the estimation of this parameter is not straightforward, we only consider the simple case of purely inelastic scattering (𝑝 = 0), which means that the values of 𝜆 obtained here represent a lower limit for the electron mean free path in Co2MnSi. We now focus our analysis on the temperature effect for a given thickness. The electrical resistivity of a ferromagnetic metal can be separated in three main contributions: the residual resistivity (𝜌 𝑟 ) independent of temperature, coming from the scattering of electrons with the impurities and imperfections of the crystal lattice, the phononic contribution (𝜌 𝑝ℎ ) arising from the scattering of electrons with the lattice vibrations, and the magnonic contribution due to scattering of electrons with the magnetic spin system (𝜌 𝑚 ). In general, the magnonic contribution has a quadratic dependency with temperature. However, the electron-magnon scattering is expected to decrease exponentially with temperature in HMF due to the suppression of spin-flip as a result of the existence of a gap in one of the spin bands. As a consequence, the following expression is typically employed to account for the magnonic contribution in HMF [START_REF] Bombor | Half-Metallic Ferromagnetism with Unexpectedly Small Spin Splitting in the Heusler Compound Co2FeSi[END_REF][START_REF] Barry | Evidence for a gap in the excitation spectrum of CrO2[END_REF]:

𝜌 𝑚 = 𝐴 * 𝑇 2 𝑒 -∆ 𝑘 * 𝑇 ⁄ ( 3
)
where 𝐴 is a constant that represents the strength of electron-magnon scattering, 𝑇 the temperature, 𝑘 the Boltzmann constant, and ∆ the separation between the Fermi energy and the first available states at the minority spin band. The phononic contribution can be described by the Bloch-Grüneisen (𝑛 = 5) or Bloch-Wilson (𝑛 = 3) formula, as follows:

𝜌 𝑃ℎ (𝑇) = B * ( 𝑇 𝛩 𝐷 ) 𝑛 ∫ 𝑥 𝑛 (𝑒 𝑥 -1) * (1-𝑒 -𝑥 ) 𝛩 𝐷 𝑇 ⁄ 0 𝑑𝑥 ( 4
)
where B is a constant, 𝛩 𝐷 the Debye temperature, and 𝑛 indicates the electron-phonon scattering mechanism contributing to the resistivity of the films: n=5 for intraband 𝑠 -𝑠 scattering and n=3

for interband 𝑠 -𝑑 scattering [START_REF] Wilson | The Electrical Conductivity of the Transition Metals[END_REF]. Therefore, the final expression for the resistivity of the films is:

𝜌 𝐶𝑀𝑆 = 𝜌 𝑟 + 𝜌 𝑚 (𝑇) + 𝜌 𝑝ℎ (𝑇, 𝑛) = 𝜌 𝑟 + 𝐴 * 𝑇 2 𝑒 -∆ 𝑘𝑇 ⁄ + B * ( 𝑇 𝛩 𝐷 ) 𝑛 ∫ 𝑥 𝑛 (𝑒 𝑥 -1) * (1-𝑒 -𝑥 ) 𝛩 𝐷 𝑇 ⁄ 0 𝑑𝑥 (5)
We can now analyze our experimental results by considering these different mechanisms. To get experimental 𝜌 𝐶𝑀𝑆 we subtracted the Au contribution obtained from the fit with eq. ( 1). Then we proceed to fit the 𝜌 𝐶𝑀𝑆 by varying 6 parameters, i.e. 𝜌 𝑟 , 𝐴, ∆, B, 𝛩 𝐷 , and 𝑛 was allowed to take the values 3 or 5. Fig. 3a shows the Co2MnSi resistivity, 𝜌 𝐶𝑀𝑆 , as a function of the temperature from 5 to 300 K, the dotted lines being the fit to eq. ( 5). The 𝜌 𝑟 , 𝛩 𝐷 and ∆ extracted values are reported in Table III for the entire thickness series. (c) Distance between the Fermi energy and the bottom of the minority spin conduction band, ∆, obtained from the fit of 𝜌 𝐶𝑀𝑆 vs. T with eq. ( 5) for the entire thickness series. A schematic of the DOS for both spins is shown as an inset.

Table III.

Residual resistivity 𝜌 𝑟 , energy ∆ of the minority spin conduction band relative to the Fermi energy, and Debye temperature 𝛩 𝐷 extracted from the fits shown in Fig. 3a using eq. ( 5) for n=3. This analysis considering eq. ( 5) leads to several conclusions: (i) The fits of Co2MnSi resistivity vs. temperature without taking into account the magnonic contribution already reproduce the experimental data fairly well, which means that the principal contribution to the resistivity of the films is coming from electron-phonon scattering, at least in the range of temperature studied here.

(ii) A superior fit was obtained with 𝑛 = 3 for all the samples indicating that 𝑠 -𝑑 scattering is the predominant contribution to the films' resistivity.

(iii) In this analysis, the thickness dependence is included in the residual resistivity 𝜌 𝑟 . It is thus not surprising to observe similar 𝜌 𝑟 (𝑡) and R sheet (𝑡) curves (respectively in Fig. 3b and Fig. 2c). Nevertheless, this term is certainly also affected by the dislocations network observed in the STEM micrographs (Fig. 1e) for the entire thickness series. Since these defects were only observed at the Co2MnSi and MgO interface and not in the bulk of the Co2MnSi layer, their impact on the residual resistivity should be considerably reduced when the thickness of the film increases due to the reduction of the relative amount of defects. [START_REF] Anderson | A simplified method for calculating the debye temperature from elastic constants[END_REF] using the Co2MnSi bulk modulus obtained from DFT calculations by Amari et al. [START_REF] Amari | First-principle study of magnetic, elastic and thermal properties of full Heusler Co2MnSi[END_REF] (see the supplementary information section 4 for more details). This result is mandatory to validate the model and demonstrates the robustness of this analysis.

(v) Better fitting was obtained when including the magnonic term which allowed us to extract ∆ values, i.e. the distance between 𝐸 𝐹 and the minority-spin conduction band minimum (𝐸 𝐶𝐵𝑀 ). We obtain ∆ ≅ 0.3 ± 0.1 eV. Since the half spin gap from the minority spin valence band to 𝐸 𝐹 was measured around 0.3 -0.4 eV in similar samples by using Spin Resolved Photoemission Spectroscopy (SR-PES) [START_REF] Guillemard | Ultralow Magnetic Damping in Co2Mn-Based Heusler Compounds: Promising Materials for Spintronics[END_REF], this leads to a spin gap roughly equal to 0.6 eV, which is in excellent agreement with ab initio calculation values around 0.35 -0.82 eV [START_REF] Galanakis | Spin-polarization and electronic properties of half-metallic Heusler alloys calculated from first principles[END_REF][START_REF] Pradines | First-principles calculation of the effects of partial alloy disorder on the static and dynamic magnetic properties of Co2MnSi[END_REF][START_REF] Picozzi | Co2MnX (X=Si, Ge, Sn) Heusler compounds: An ab initio study of their structural, electronic, and magnetic properties at zero and elevated pressure[END_REF].

It is worth to mention a particular feature that was only observed in the resistivity of the 4 nm Co2MnSi film. Below 20 K, an upturn in the resistivity was observed for this sample (inset of Fig. 3a). This increase of resistivity at low temperature has already been reported in Co2MnSi1-xAlx Heusler compounds, and has been ascribed to a weak localization phenomenon, due to the enhanced coherent backscattering of conduction electrons as a result of disorder in this system [START_REF] Prestigiacomo | Hall effect and the magnetotransport properties of Co2MnSi1-xAlx Heusler alloys[END_REF]. Such disorder can originate from chemical disorder or/and the presence of crystallographic defects. Indeed, dislocation cores were observed on our samples at the Co2MnSi/MgO interface (Fig. 1e) as expected due to the lattices misfit [START_REF] Markov | Crystal Growth For Beginners: Fundamentals Of Nucleation, Crystal Growth And Epitaxy[END_REF]. These dislocations cores thus induce some local disorder mainly at the interface, that is expected to become more relevant with the reduction of the film thickness. Nevertheless, other contributions relevant at low temperatures for very low thicknesses cannot be excluded such as enhanced electron-electron interaction (EEI) [START_REF] Lee | Disordered electronic systems[END_REF]. To disentangle these contributions for this particular thickness is beyond the scope of this study.

Anisotropic Magnetoresistance

Anisotropic magnetoresistance (AMR) is a phenomenon observed in ferromagnetic materials as a result of spin-orbit interaction. It manifests as a variation in the resistivity of the film with the relative orientation of the magnetization and current, and is generally defined as:

AMR = ∆𝜌 𝜌 = 𝜌 ∥ -𝜌 ⊥ 𝜌 ⊥ (6)
where 𝜌 ∥ and 𝜌 ⊥ are the resistivities of the film when the magnetic field is parallel to the current direction and perpendicular to the current direction, respectively.

While most common ferromagnets like Co, Ni, Fe and their alloys possess positive AMR ratios, in the case of half-metallic ferromagnets negative values have been typically reported. This can be understood under the theoretical model developed by Kodako et al. [START_REF] Kokado | Anisotropic Magnetoresistance Effects in Fe, Co, Ni, Fe4N, and Half-Metallic Ferromagnet: A Systematic Analysis[END_REF] where general expressions are obtained for the resistivity and AMR by specifying the spin state (𝜎 =↑ or ↓) of conduction sstates in s-d scattering. In this model, a relation is obtained between the sign of AMR and the dominant s-d scattering process. When s-d scattering occurs predominantly between states of different spin (s ↑ to 𝑑 ↓ or 𝑠 ↓ to d ↑ ) the sign of AMR tends to be positive (𝜌 ∥ > 𝜌 ⊥ ) and when it occurs predominantly between states of the same spin (s ↑ to 𝑑 ↑ or 𝑠 ↓ to d ↓ ) AMR tends to be negative (𝜌 ∥ < 𝜌 ⊥ ). Since HMF possess a null density of states (DOS) at the Fermi energy here for spin-down electrons (but it is also true for HMF with a spin gap in the majority spin bands), AMR should be always negative. Therefore, AMR measurements can provide a signature for half metallicity.

An example of a magneto-resistivity measurement performed on a 13 nm thick Co2MnSi film is shown in Fig. 4a with the magnetic field being applied parallel to the current direction (black line) or perpendicular to the current direction (red line). The AMR curves for the entire thickness series can be found in the supplementary information section (Figure S5). Two peaks were observed at the coercive field (𝐻 𝑐 ) when the magnetic field was reversed back and forward for all the samples [START_REF] Hong | Quantifying electric-field control of magnetization rotation in Ni/SiO2/Ti/(011)-PMN-PT multiferroic heterostructures via anisotropic magnetoresistance measurements[END_REF]. AMR ratios measured on the Co2MnSi series are reported in Fig. 4b, together with some reference values for comparison. Negative AMR ratios were obtained for the whole thickness range, which confirms the half-metallic nature of our Co2MnSi films (see Table IV). The AMR values obtained here can be compared with the best values reported in the literature for Co2MnSi proving the excellent quality of our films (Fig. 4b). An increase of |∆𝜌| = |𝜌 ∥ -𝜌 ⊥ | is observed when the film thickness is reduced below the electron mean free path (see Fig. 4c) in agreement with the study of Dieny et al. [START_REF] Dieny | Effect of interfacial specular electron reflection on the anisotropic magnetoresistance of magnetic thin films[END_REF]. However, the increase of film resistivity due to the increase of electron diffuse scattering at the outer film surfaces dominates causing a reduction of the AMR ratio at lower thicknesses.

Hall Effect Measurements

The field dependence of Hall resistivity (𝜌 𝑥𝑦 ) for Co2MnSi films is shown in Fig. 5a. At high magnetic fields (applied perpendicular to the film), above saturation a small negative slope is observed as a result of the Lorentz force experienced by the carriers under the action of an external magnetic field, that is, the ordinary Hall effect. On the other hand, a steeper linear dependency is observed in the low field region, arising from the anomalous Hall effect (AHE).

The origin of AHE is the spin-orbit coupling of conduction electrons in ferromagnetic materials which produces an additional deviation of electron trajectories perpendicular to the current and the external magnetic field. The total Hall resistivity for a ferromagnet taking both terms into account can be described by the empirical formula [START_REF] Obaida | Magnetotransport properties of Cu2MnAl, Co2MnGe, and Co2MnSi Heusler alloy thin films: From nanocrystalline disordered state to long-rangeordered crystalline state[END_REF]:

𝜌 𝑥𝑦 = 𝜇 0 ℛ 𝐻𝐸 𝐻 + 𝜇 0 ℛ 𝐴𝐻𝐸 𝑀 𝑍 (𝐻) (7)
where 𝜇 0 is the vacuum permeability, ℛ 𝐻𝐸 the ordinary Hall coefficient, ℛ 𝐴𝐻𝐸 the anomalous Hall coefficient, 𝐻 the external magnetic field along 𝑧 direction and 𝑀 𝑍 (𝐻) the magnetization of the film along 𝑧 direction (see supplementary information section 6 for more details).

At high magnetic fields above saturation (>1.5 T) the second term in eq. ( 7) is constant which allows us to extract the ordinary Hall coefficient ℛ 𝐻𝐸 from the linear fit of 𝜌 𝑥𝑦 (𝜇 0 𝐻) (see Table IV). From the ℛ 𝐻𝐸 values we can determine the carrier concentration (𝑛) by using the single parabolic band model 𝑛 = 1 𝑒 ℛ 𝐻𝐸 ⁄ . The negative slope is consistent with electrons being the main charge carriers in this material. The carrier concentration is close to 1 * 10 23 𝑐𝑚 -3 and no significant variation was observed within the sample series. The carrier mobility was determined using 𝜇 = ℛ 𝐻𝐸 𝜌 𝑥𝑥 ⁄ with values ranging from 0.6 to 1.5 𝑐𝑚 2 𝑉 𝑠 ⁄ (see Table IV). The decrease of electron mobility for lower thickness results from the increase of inelastic electron scattering as the film thickness is reduced below the electron mean free path. IV. As can be seen, the anomalous Hall coefficient is two orders of magnitude higher than the ordinary Hall coefficient and it decreases with the increase of film thickness. To quantify the anomalous Hall effect, we calculated the anomalous Hall angle, 𝜃 𝐴𝐻𝐸 = 𝜌 𝑥𝑦 𝐴𝐻𝐸 𝜌 𝑥𝑥 ⁄ , where 𝜌 𝑥𝑥 is the longitudinal resistivity and 𝜌 𝑥𝑦 𝐴𝐻𝐸 is the anomalous Hall resistivity (see details in supplementary information section 8). The anomalous Hall angles obtained at saturation for all the thickness series are reported in Table IV.

Small anomalous Hall angles were obtained compared with the Co2MnGa Heusler compound, which has been widely studied in the literature and for which a large 𝜃 𝐴𝐻𝐸 of ~10 % have been reported [START_REF] Zhang | Berry curvature origin of the thickness-dependent anomalous Hall effect in a ferromagnetic Weyl semimetal[END_REF][START_REF] Manna | From Colossal to Zero: Controlling the Anomalous Hall Effect in Magnetic Heusler Compounds via Berry Curvature Design[END_REF][START_REF] Chen | Large anomalous Hall angle in a topological semimetal candidate TbPtBi[END_REF]. This can be explained by the weaker spin-orbit coupling in the Co2MnSi compound linked to its lower magnetocrystalline anisotropy compared to Co2MnGa. These measurements were also performed in the 5 -300 K temperature range to test and better understand the underlying physics in these films. The evolution of Hall coefficients, carrier concentration and carrier mobility with temperature are shown in Fig. 6. When the temperature rises, the carrier concentration slightly increases while the carrier mobility decreases (Fig. 6 b,c).

The decreasing carrier mobility can be explained by the increase of electron-phonon scattering with increasing temperature. On the other hand, the increase of the carrier concentration is puzzling since it is expected to be constant for a metal. As these HMF materials are in between normal metals and semiconductors due to the existence of the minority spin gap, a rough explanation should be the increase of the minority spin carrier concentration with temperature. However, the thermal energy at 300 K is considerably smaller than the Co2MnSi spin gap (0.35 -0.8 eV [START_REF] Galanakis | Spin-polarization and electronic properties of half-metallic Heusler alloys calculated from first principles[END_REF][START_REF] Pradines | First-principles calculation of the effects of partial alloy disorder on the static and dynamic magnetic properties of Co2MnSi[END_REF][START_REF] Picozzi | Co2MnX (X=Si, Ge, Sn) Heusler compounds: An ab initio study of their structural, electronic, and magnetic properties at zero and elevated pressure[END_REF]). A better explanation would be based on the complex Fermi surfaces with the coexistence of electrons and holes in these Heusler HMF materials [START_REF] Prestigiacomo | Hall effect and the magnetotransport properties of Co2MnSi1-xAlx Heusler alloys[END_REF]. In that way, the parabolic single-band model employed here to calculate 𝑛 and 𝜇 cannot completely describe the situation of Heusler compounds. Therefore, the values of 𝑛 and 𝜇 reported in this study represent effective values associated to the predominant type of carriers in this material. Contrary to the ordinary Hall coefficient, the anomalous Hall effect coefficient increases with temperature. In fact, this temperature dependence arises from the temperature dependence of longitudinal resistivity since ℛ 𝐴𝐻𝐸 (𝑇) ∝ 𝜌 𝑥𝑥 𝛼 (𝑇), where the 𝛼 exponent gives information on the main scattering mechanism. An 𝛼 = 1 exponent is usually assigned to skew scattering, i.e. an extrinsic effect resulting from the asymmetry of the electron scattering cross-section due to spin-orbit interaction. On the other hand, 𝛼 = 2 can be the signature of another extrinsic effect known as side-jump scattering or to the intrinsic Berry phase-related mechanism [START_REF] Nagaosa | Anomalous Hall effect[END_REF]. The longitudinal resistivity of the entire Co2MnSi thickness series places Co2MnSi in the good-metal regime (1 < 𝜌 < 10 2 𝜇Ω 𝑐𝑚) [START_REF] Nagaosa | Anomalous Hall effect[END_REF]. Studies performed on common ferromagnets like Co and Fe have suggested that side-jump and intrinsic scattering mechanisms predominate in this regime at low temperatures [START_REF] Nagaosa | Anomalous Hall effect[END_REF]. In this work, the change of the films' resistivity with temperature and thickness provides an ideal scenario to explore the AHE scaling in Co2MnSi and to identify the dominant scattering mechanism. In Fig. 6d, Δℛ 𝐴𝐻𝐸 = ℛ 𝐴𝐻𝐸 (𝑇) -ℛ 𝐴𝐻𝐸 (5 𝐾) versus Δ𝜌 𝑥𝑥 = 𝜌 𝑥𝑥 (𝑇) -𝜌 𝑥𝑥 (5 𝐾) is plotted and a linear trend (𝛼 = 1) was observed suggesting that skew scattering is the dominant temperature-dependent scattering mechanism taking place in the Co2MnSi films, even in the goodmetal regime.

DISCUSSION AND CONCLUSIONS

In summary, the investigated Co2MnSi films with ultra-low magnetic damping allowed us to extract important and robust information on the electronic transport properties in this Half-Metal-Magnetic material. The analysis of film resistivity versus thickness allowed us to estimate the inelastic electron mean free path in these films to be equal to 14 nm at room temperature. The increase of resistivity for low thicknesses thus arises from interface scattering, but possibly also from misfit dislocations. This mean free path is an important parameter to consider towards the fabrication of magnetoresistive devices with Co2MnSi electrodes. The temperature dependence of the films' resistivity indicates that s-d electron-phonon scattering is the main scattering mechanism contributing to the film resistivity from 50 to 300 K. This analysis is enforced by the excellent agreement between the Debye temperature extracted from the resistivity curve fits (478 ± 9 K) with the calculated one based on Co2MnSi elastic constants in the bulk (479.4 K). In addition, the introduction of an exponential decrease to the typical T 2 dependence from electronmagnon scattering allows us to determine the distance between the Fermi energy and the first available states at the minority-spin conduction band, i.e. the half-spin gap above the Fermi energy. As the half-spin gap between the minority spin valence band and the Fermi energy was known from SR-PES measurements, a total gap around 0.6 eV is obtained in agreement with ab initio calculations [START_REF] Galanakis | Spin-polarization and electronic properties of half-metallic Heusler alloys calculated from first principles[END_REF][START_REF] Pradines | First-principles calculation of the effects of partial alloy disorder on the static and dynamic magnetic properties of Co2MnSi[END_REF][START_REF] Picozzi | Co2MnX (X=Si, Ge, Sn) Heusler compounds: An ab initio study of their structural, electronic, and magnetic properties at zero and elevated pressure[END_REF]. This half-metallicity is confirmed by the negatives AMR ratios obtained for the entire thickness series. This demonstrates that performing transport measurements is a good method to verify the half-metallicity of Heusler compounds.

Furthermore, Hall measurements confirm that electrons are the main charge carrier in Co2MnSi, and allowed us to determine the electron concentration and electron mobility. The electron mobility is observed to decrease when decreasing film thickness, due to an enhancement of inelastic scattering at the film surface (and probably also at dislocation defects) when the thickness is reduced below the electron mean free path; while the electron concentration seems not to change appreciably. The electron mobility was observed to decrease with temperature as a result of the increase in phonon-electron scattering. However, a slight increase of the carrier concentration with temperature is observed and probably needs ab initio calculations to be explained. Finally, the evolution of the films' resistivity with thickness and temperature provides the ideal scenario for studying the scaling of the Hall resistivity with the longitudinal resistivity, indicating that skew scattering is the predominant temperature-dependent scattering mechanism taking place in Co2MnSi. Finally, we may go as far as to say that all the transport parameters reported here are robust and characteristic of HMF Co2MnSi thin films with ultra-low magnetic damping. This may be very useful in views of the implementation of Co2MnSi films in magnetoresistive devices.

Fig. 1 .

 1 Fig. 1. Structural characterization of the Co2MnSi films. (a) X-ray diffraction patterns of the Co2MnSi thickness series in the Bragg-Brentano configuration. (b) Lattice parameter of the Co2MnSi thickness series: inplane (orange stars) and out-of-plane (blue diamonds), obtained from (004) and (422) reflections, respectively. (c) STEM micrograph of the 17 nm Co2MnSi film along the [11 ̅ 0] zone axis. A filtered STEM-HAADF micrograph performed in the region enclosed by a white frame is shown as inset. (d) Fast Fourier transform (FFT) image performed at the interface between CMS and MgO (violet frame in c). (e) Inverse-FFT image of the masked planes (cyan circles) in the FFT pattern shown in (d). The dislocations at the interface are highlighted by dotted circles.

4

  Fig. 2a, the current (𝑖) was injected along the [110]-CMS direction between contacts pad A-B

Fig. 2 .

 2 Fig. 2. Patterning and electrical characterization of the films. (a) Top-view optical image of the sample: Hall bar geometry used for transport measurements. (b) Schematic of the device and contact geometry. (c) Equivalent sheet resistance of the samples (CMS + Au) versus CMS thickness at 300 K (triangles) and 5 K (circles). Solid lines correspond to the fitting with the Fuchs and Sondheimer model (eq. 1, 2). Inverse of the equivalent sheet resistance is shown in the inset.

Fig. 3 .

 3 Fig. 3. Evolution of CMS resistivity with temperature. (a) Resistivity versus temperature for the Co2MnSi thickness series. Solid lines are the fits to experimental data with eq. (5). In the inset, a zoom of the 𝜌 𝐶𝑀𝑆 (T) curve in the low-temperature range for the 4 nm Co2MnSi film is shown. (b) Co2MnSi residual resistivity versus thickness.

Fig. 4 .

 4 Fig. 4. Anisotropic magnetoresistance measurements. (a) Magnetoresistance vs. magnetic field for a 13 nm thick Co2MnSi film, with the magnetic field oriented perpendicular (red line) or parallel (black line) to the current direction. (b) AMR ratio vs. CMS thickness for our samples (blue circles) and some reference values taken from the literature for comparison (other symbols) [45-48]. (c) ∆𝜌 vs. thickness. The dotted vertical line indicates the CMS electron mean free path determined in this study.

Fig. 5 .

 5 Fig. 5. Hall effect measurements. Field dependence of the Hall resistivity (𝜌 𝑥𝑦 ) for the Co2MnSi thickness series.

Fig. 6 .

 6 Fig. 6. Transport properties of the CMS films. (a) Normal and anomalous Hall coefficients, (b) CMS carrier concentration and (c) carrier mobility versus temperature for samples of different thickness. (d) Scaling of AHE coefficient for CMS films of different thickness derived from the Hall measurements performed at different temperatures from 5 -300 K.

Table I .

 I Effective magnetization (relative error of ~1%), magnetic damping (𝛼), inhomogeneous linewidth in field (𝜇 0 ∆𝐻 0 ) and frequency (∆𝑓 0 ) and Landé g-factor, extracted from FMR measurements performed on the Co2MnSi thickness series.

	Co2MnSi thickness (nm)	𝝁 𝟎 𝑴 𝒆𝒇𝒇 (T) / (𝛍 𝐁 f. u. ⁄ )	𝜶 (10 -4 )	𝝁 𝟎 ∆𝑯 𝟎 (𝒎𝑻) ∆𝒇 𝟎 (𝐌𝐇𝐳)	𝒈
	4	1.15 / 4.51	20.9 ± 0.6	3.12 ± 0.11	88 ± 3	2.01 ± 0.01
	8	1.19 / 4.64	7.8 ± 0.3	1.73 ± 0.07	49 ± 2	2.03 ± 0.02
	13	1.26 / 4.92	8.9 ± 0.1	0.88 ± 0.03	24.9 ± 0.7	2.02 ± 0.01
	21	1.22 / 4.77	9.3 ± 0.2	0.46 ± 0.05	13.1 ± 1.4	2.02 ± 0.01
	44	1.33 / 5.20	15.6 ± 0.4	0.43 ± 0.07	12.2 ± 1.9	2.03 ± 0.01

Table II .

 II Bulk resistivity, 𝜌 𝐵 (error ~ 8 %), electron mean free path, 𝝀 (error ~ 15 %), Au sheet resistance, 𝑅 𝐴𝑢 (error ~ 17 %), for Co2MnSi at different temperatures from 5 to 300 K.

	Temperature (K)	𝝆 𝑩 (𝛍𝛀 𝐜𝐦)	𝝀 (𝐧𝐦)	𝑹 𝑨𝒖 (𝛀)
	5	14.8	36	489
	50	15.6	32	510
	100	18.1	26	650
	150	25.3	21	936
	250	22.0	20	900
	300	31.5	14	1116

  (iv) The extracted Debye temperature values (Θ 𝐷 ) reported in Table III are in excellent agreement with the 479.4 K theoretical value calculated from the Debye model

	Co2MnSi thickness (nm)	𝝆 𝒓 (µ*cm)	∆ (𝐞𝐕)	𝜣 𝑫 (𝐊)
	4	60.8 ± 0.1	0.23 ± 0.02	477 ± 3
	8	43.2 ± 0.2	0.36 ± 0.30	493 ± 8
	13	29.4 ± 0.2	0.32 ± 0.10	477 ± 6
	21	25.3 ± 0.1	0.21 ± 0.04	457 ± 1
	26	23.4 ± 0.1	0.29 ± 0.03	483 ± 2
	44	22.8 ± 0.1	0.18 ± 0.15	485 ± 2

Table IV .

 IV Review of the AMR and Hall measurements performed on the Co2MnSi thickness series at room temperature: films thickness (t), AMR values, ordinary (ℛ 𝐻𝐸 ) and anomalous (ℛ 𝐴𝐻𝐸 ) Hall coefficients, carrier concentration (𝑛), carrier mobility (μ) and anomalous Hall angle (𝜃 𝐴𝐻𝐸 ) (relative error ~ 5 %).

	t (nm)	AMR (%)	𝓡 𝑯𝑬 (𝟏𝟎 -𝟑 𝛍𝛀 𝐜𝐦/𝐓)	𝒏 (10 23 cm -3 )	𝝁 (cm 2 /V s)	𝓡 𝑨𝑯𝑬 (𝟏𝟎 -𝟏 𝛍𝛀 𝐜𝐦/𝐓)	𝜽 𝑨𝑯𝑬 (%)
	4	-0.14 ± 0.03	5.97 ± 0.05	1.05 ± 0.02	0.67 ± 0.05	4.7 ± 0.05	0.65
	8	-0.15 ± 0.02	7.20 ± 0.05	0.87 ± 0.01	1.21 ± 0.03	3.5 ± 0.02	0.75
	13	-0.190 ± 0.008	6.21 ± 0.08	1.01 ± 0.01	1.38 ± 0.03	2.25 ± 0.03	0.64
	17	-0.190 ± 0.02	6.03 ± 0.02	1.03 ± 0.03	1.61 ± 0.03	2.22 ± 0.12	0.74
	21	-0.20 ± 0.01	6.5 ± 0.3	0.96 ± 0.04	1.70 ± 0.08	2.19 ± 0.08	0.52
	26	-0.23 ± 0.01	5.93 ± 0.2	1.05 ± 0.04	1.54 ± 0.06	1.99 ± 0.03	0.64
	44	-0.24 ± 0.009	6.72 ± 0.3	0.93 ± 0.04	1.84 ± 0.09	1.85 ± 0.01	0.60

Acknowledgments

We would like to thanks Sylvie Migot and Jaafar Ghanbaja for the FIB preparation and TEM imaging.

Funding sources

This work was carried out in the context of the research project CHIPMuNCS, founded by the Agence Nationale de la Recherche (Contract No. ANR-17-CE24-0008). This work was also partially supported from the French PIA project "Lorraine Université d'Excellence", reference ANR-15IDEX-04-LUE. Devices in the present study were patterned at MiNaLor clean-room platform which is partially supported by FEDER and Grand Est Region through the RaNGE project.

Data availability

The raw/processed data required to reproduce these findings cannot be shared at this time as the data also forms part of an ongoing study.


Fateh Guenab

Jean-Louis Boulanger

Walter Schön

Walter Schön Safety

Safety of railway control systems: A new Preliminary Risk Analysis approach

Keywords: Railway safety, Preliminary risk analysis (PRA), risk, potential accident, feared events, Automatic Train Control

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

I. INTRODUCTION

Ensuring railway systems safety requires knowledge of all life cycle phases of these systems. Upstream and downstream phases (Concept, Definition of the System and its Application's Conditions, Acceptance of the System, Operating and Maintenance, Monitoring of the Performances, Modifications) are essentially the responsibility for the owners and the Official Authorities. The central phases of the lifecycle (Risks Analysis, System Requirements, Safety Requirements Allocation, Design and Realization, Manufacture, Installation, and Validation) are essentially the responsibility of the railway systems suppliers. Within the framework of export businesses, manufacturers take also more and more responsibilities in the phases upstream and downstream. The phases corresponding to responsibilities transfers between actors are obviously crucial.

Current railway standards [START_REF]Railway Applications -The specification and demonstration of dependability -reliability, availability, maintainability and safety (RAMS)[END_REF], [START_REF]Railway Applications -Software for railway control and protection systems[END_REF] and [START_REF]Railway Applications -Safety related systems for signalling[END_REF] were regularly revised to take into account the permanent technological projections in the electronic materials fields and in the data-processing techniques. Those have an important impact on the railway systems design. However the standards did not formalize yet well the process of distribution of the safety regulations of the railway system on its subsystems, hardware and software which supports them. In other words, if the risks are well identified and followed through demonstrations and safety justifications, it remains to improve the Risks Analysis and Safety Requirements Allocation steps. Authors of [START_REF] Jo | Investigation of risk analysis methods for safety assurance in the train control system[END_REF] have examined the methods for risk analysis and assessment of safety activities and proposed optimized one method for risk estimation.

The objective of this work is to propose a method and standard contents for a PRA in the context of railway signalling and command and control systems. This method will integrate the impact of the NICT on safety in terms of risks induced on the whole system. The evolution of the systems design passes by the integration of the NICT. The NICT are considered as Components Off The Shelf (COTS). The COTS allow controlling the cost of system realization, but on the other hand there is a loss of safety control [START_REF] Boulanger | Reference systems and standards for safety assessment of railway applications[END_REF]. Thus the proposed method will have to take into account the inherent risks in this type of component. Another constraint relates to the taking into account of the human errors. This paper is organized as follows. Section 2 is dedicated to the columns definitions of the PRA and a PRA method is proposed in section 3. Section 4 is devoted to the description of the phases of PRA method presented previously. Finally, concluding remarks and perspectives are given in the last section.

II. COLUMNS DEFINITIONS OF PRELIMINARY RISK ANALYSIS

Knowing that the results of the Preliminary Risk Analysis are presented in a worksheet and the various definitions of the terms and concepts related to the PRA, we dedicate this section to detail these concepts. Basing on CENELEC standards [START_REF]Railway Applications -The specification and demonstration of dependability -reliability, availability, maintainability and safety (RAMS)[END_REF], [START_REF]Railway Applications -Software for railway control and protection systems[END_REF], [START_REF]Railway Applications -Safety related systems for signalling[END_REF] and [START_REF]Functional Safety of electrical / electronic / programmable electronic safety related systems -Part 4: Definitions and abbreviations[END_REF] we propose definitions of the columns of PRA applied to the railway control systems. A standardization of the principal concepts and associated terminology is indeed proved to be necessary, after study of several railways PRA of various sources (manufacturers, owners...), where

Safety of railway control systems: A new Preliminary Risk Analysis approach

F. Guenab, J.L. Boulanger, W. Schön Heudiasyc laboratory UMR CNRS 6599, Technology University of Compiegne, Compiègne -France important inconsistencies could be noted (for example the same term is used for different concepts).

-Operating mode (exploitation mode, or working mode, or phase or context): knowing that the analysis of hazards and risks is carried out for all the reasonably foreseeable situations, which justifies this column. It is useless to consider a safety measurement « the platform doors must be strictly closed » for a metro which is on line, even in station where the platform doors don't exist… In some cases, it is not mandatory to specify the operating mode.

-Dangerous Entity (or Dangerous element, or hazardous entity/element): a subset of the studied system which is at the origin of the feared event, it is the initial cause of the studied scenario. The function of the dangerous entity can be specified if necessary.

-Feared event (or event causing a dangerous situation, or undesirable event, or error): is a dangerous event, it is the event affecting the dangerous entity or its function leading the system in a dangerous situation.

-Dangerous situation (or danger): undesirable state of system following the feared event, may lead to a potential accident.

-Damage (or consequence): it is result of an accident given in terms of death, physical wound, injuries, attack to people health or environment damage.

-Severity level: a classification on several levels, it allows to evaluate and estimate the consequences of potential accidents. According to [START_REF]Railway Applications -The specification and demonstration of dependability -reliability, availability, maintainability and safety (RAMS)[END_REF], Table I describes typical hazard severity levels and the consequences associated with each severity level for railway systems. In order to avoid bad interpretations of the qualitative terms, some manufacturers and/or owners use numbers to describe severity levels (from 1 to 4 in order of increasing severity, 4 = catastrophic).

-Frequency of Occurrence: probability of the sequences of events. As in [START_REF]Railway Applications -The specification and demonstration of dependability -reliability, availability, maintainability and safety (RAMS)[END_REF], qualitative evaluation of probability or frequency of occurrence of a hazardous event and a description of each category is proposed in Table II.

-Measurements (safety measures, safety constraints or safety requirements): suitable actions to reduce or eliminate risk. They can be preventive measures which allows reducing the probabilities of occurrences of the dangerous events or protection measures by using safety devices allowing reducing the severity of damage. This column could describe the subsystem (equipment, system…) charged to ensure the safety measures -Event causing a potential accident: it is a dangerous event that transforms a dangerous situation into a potential accident. It does not exist in the case of scenarios of order 1. Columns could appear in the PRA, for the sequences of dangerous events which allow passing from a dangerous situation to an accident, this depends on the studied scenario order.

-Potential accident: a potential accident could be an accident or quasi accident. The effective occurrence of damages (e.g. collision) determines the accidental identity of the potential accident else it is an incident (e.g. crossing over a restrictive signal without effective collision).

III. PROPOSED METHOD

The objective of Preliminary Risk Analysis method is [START_REF] Villemeur | Sûreté de fonctionnement des systèmes industriels : fiabilité-acteurs humains informatisation[END_REF]:

-To determine the dangers (hazards) and their causes (dangerous entities, dangerous situations and potential accidents). -To evaluate the severity of the consequences of situations and accidents previously determined. -To deduce the measurement and the suitable actions to eliminate or reduce dangerous situations and the potential accidents. Fig. 1 summarizes these steps. The identification of dangerous entities, dangerous situations and the potential accidents rests at the beginning on the experiment and the judgment of the specialists, helped by guide lists which are updated by the experience feedback throughout the lifecycle of the system. The PRA is generally considered as an inductive approach (proceed from causes to identify consequences), however some actors consider it as a deductive approach. A deductive or inductive analysis does not relate to the general method described by Fig. 1 but only to the first step (1): identification of dangerous entities, dangerous situations and potential accidents. During this step, some experts determine the set of potential accidents (consequences) by induction on the basis of the dangerous entities (causes); other experts proceed by deduction to identify the dangerous entities or the dangerous events (causes) from the potential accidents (consequences). When we dispose of a complete list of all potential accidents (respectively dangerous entities / feared events) the deductive approach (respectively inductive approach) alone is valid and lead to acceptable results covering all possible scenarios. On the other hand for systems having a significant number of scenarios and if we don't dispose of complete lists of potential accidents (respectively dangerous entities / feared events), using only deductive approach (respectively inductive approach) is not efficient. In these cases, it is possible that the used approach does not take into account the potential accidents (respectively dangerous entities / feared events) not included in the generic list. To avoid this problem, a deductive-inductive approach should be used. -ER 0 , D 0 and A 0 are the preliminary lists of dangerous entities /feared events, dangers and potential accidents respectively. They are defined from the generic lists analyzed by experts; the latter can remove incoherent scenarios or add other missing scenarios. These lists are obtained from the initialization step (step (0) of Fig. 1). -ER, D and A represent the lists of dangerous entities/feared events, dangers and potential accidents respectively. At the beginning of the analysis they are initialized at ER 0 , D 0 and A 0 and they contain the final lists at the end of the analysis. -The step contains several cyclic phases: two inductive phases (solid line), two deductive phases (broken line) and a phase to generate the list of the consequences (damages) generated by the potential accidents. The phases are noted i j At the beginning of a cycle « j », the sets of the dangerous entities/feared events, dangers and potential accidents are indexed by « j-1 », ER j-1 , D j-1 and A j-1 . For example at the beginning of cycle 1, the sets ER, D and A are equal to ER 0 , D 0 and A 0 respectively.

IV. PROCEDURE FOR PROPOSED PRA METHOD

In this paragraph we will present the various phases of PRA method presented in the previous paragraph.

At the beginning of cycle 1, the lists of feared events, dangers and potential accidents are initialized at ER 0 , D 0 and A 0 Phase 1 (Fig. 3) allows determining the dangers starting from the feared events (inductive phase). Note that the analysis could also be performed from the potential accidents to the feared events (opposite direction of the cycle). This by reversing the order of the phases in the following way: the cycle starts with phases 4 and 3, phase 5, then the phase 1 and finished by phase 2. Note that this description is not known a priori, the guide lists ER 0 , D 0 and A 0 are the only known information.

Let us suppose that the guide lists (preliminary lists) are given by the following sets:

Fig. 1 .

 1 Fig. 1. Steps of Preliminary Risk Analysis

Fig. 2 .

 2 Fig.2. first step of PRA method.

Fig. 2

 2 Fig. 2 represents the first step (1) of the PRA illustrated by Fig. 1.-ER 0 , D 0 and A 0 are the preliminary lists of dangerous entities /feared events, dangers and potential accidents respectively. They are defined from the generic lists analyzed by experts; the latter can remove incoherent scenarios or add other missing scenarios. These lists are obtained from the initialization step (step (0) of Fig.1). -ER, D and A represent the lists of dangerous entities/feared events, dangers and potential accidents respectively. At the beginning of the analysis they are initialized at ER 0 , D 0 and A 0 and they contain the final lists at the end of the analysis. -The step contains several cyclic phases: two inductive phases (solid line), two deductive phases (broken line) and a phase to generate the list of the consequences (damages) generated by the potential accidents. The phases are noted i j

  the index « i » indicates the number of phase and the index « j » indicates the current cycle. i.e. 4 1 , φ is the 4 th phase of the 1 st cycle.-The cycles start with index 1 (j > 0), in other words, the first cycle of this step corresponds to « j = 1 ».

Fig. 3 .D- 0 DPhase 2 (A- 0 A

 3020 Fig. 3. Phase 1 During this phase, the list ER 0 can generate a list of dangers that corresponds to: -A part of the preliminary list noted ' 0 0 D D ⊆. We

Fig. 4 . 2 Phase 3 (

 423 Fig. 4. Phase 2 Phase 3 (Fig. 5) is used to identify the damages from the list of the accidents obtained at the end of the previous phase ( ' " ' n 0 0 0 A A A ∪ ∪ ). It is noted Dom 1 .

Fig. 5 . 3 Phase 4 (Fig. 6 . 4 Phase 5 (Fig. 7 .

 5346457 Fig. 5. Phase 3 Phase 4 (Fig.6): Knowing that the only sub-list of the potential accidents which does not have correspondence in the dangers list is " 0 A , this phase allows determining the possible dangers from the sub-list " 0 A (deductive phase).

TABLE I HAZARD

 I

		SEVERITY LEVEL
	Severity	Consequence to Persons or	Consequence
	Level	Environment	to Service
	Catastrophic Fatalities and/or multiple severe injuries
		and/or major damage to the environment.
	Critical	Single fatality and/or severe injury and/or	Loss of a major
		significant damage to the environment.	system
	Marginal	Minor injury and/or significant threat to	Severe system
		the environment	damage
	Insignificant Possible minor injury	Minor system
			damage

ANR-PREDIT-SECUGUIDE is a project financed by the National Agency for Research -France (http://www.agencenationale-recherche.fr) which began in

January 2006 per 3 years duration. The project aims to study the impact of introducing the NICT into the railway systems on safety.

{ }

A

, a a =

2) Phases of the analysis

Cycle n°1

The preliminary lists ER 0 , D 0 and A 0 are represented by dashed square. The damages set Dom is initially empty. , , er er er . The conditions to stop analysis are not sufficient, thus new cycle must be performed.

Cycle n°2

The same phases are repeated during the second cycle. At the end of this cycle, the obtained lists are given by: { }

Cycle n°3

During this cycle, phases 1 3 , φ and 2 3 , φ do not generate anymore new dangers and accidents respectively. It is sufficient condition to stop the analysis. Note that the analysis has covered all accidents scenarios contrary if only deductive approach (or inductive approach) was used.

VI. CONCLUSION

This paper has presented a Preliminary Risks Analysis method in the context of the railway systems. After presenting the definitions of the used terms in a PRA, we proposed a PRA method using the two approaches: deductive and inductive. Then an example is proposed to illustrate the approach.

Our research perspectives are articulated around two points: -To propose standard contents of the Preliminary Risks Analysis and to develop a method of analysis regarding the command-control and signalling systems. -To formalize links between the functionalities and/or techniques of the NICT and the standard PRA, in order to identify the new risks induced by the NICT and the safety measures to be taken to reduce these risks levels.


Fateh Guenab

Jean-Louis Boulanger

Walter Schön

Safety of railway control systems: A new Preliminary Risk Analysis approach

Keywords: Railway safety, Preliminary risk analysis (PRA), risk, potential accident, feared events, Automatic Train Control, New Information and Communication Technologies (NICT)

Preliminary risk analysis (PRA) is a methodology used in critical systems safety studies. It is primarily used at the preliminary stage of the system's design so as to determine the scenarios of potential accidents, to evaluate their probabilities of occurrence (frequency) as well as the severity of the resulting consequences and to propose solutions (preventive and/or mitigative safeguards) in order to reduce the risk level in terms of severity/occurrence (to reduce the frequency of the contributors or reduce the severity of the accident). The preliminary risk analysis was largely used in several industrial fields (aeronautics, weapons systems, chemistry, railway...) in order to study the safety of the systems. From one field to another, from one expert to another, many extremely different approaches and methods are used to carry out this analysis. Moreover, the formats representing the results of the PRA are often varied as well as the terminology and the concepts related to the PRA. The main goal of this paper, completed within the framework of ANR-PREDIT-SECUGUIDE project (project financed by the National Agency for Research -France. It aims to study the impact of introducing the NICT into the railway systems on safety), is to propose a PRA method and to determine standard contents of PRA to be used in the context of the railway control systems by taking into account the impact of New Information and Communication Technologies (NICT).

Introduction

Ensuring railway systems safety requires knowledge of all lifecycle phases of these systems. Upstream and downstream phases (Concept, Definition of the System and its Application's Conditions, Acceptance of the System, Operating and Maintenance, Monitoring of the Performances, Modifications) are essentially the responsibility for the owners and the Official Authorities. The central phases of the lifecycle (Risks Analysis, System Requirements, Safety Requirements Allocation, Design and Realization, Manufacture, Installation, and Validation) are essentially the responsibility of the railway systems suppliers. Within the framework of export businesses, manufacturers take also more and more responsibilities in the phases upstream and downstream. The phases corresponding to responsibilities transfers between actors are obviously crucial.

Current railway standards EN 50126 [START_REF]Railway Applications -The specification and demonstration of dependability -reliability, availability, maintainability and safety (RAMS)[END_REF], EN 50128 [START_REF]Railway Applications -Software for railway control and protection systems[END_REF] and EN 50129 [START_REF]Railway Applications -Safety related systems for signalling[END_REF] were regularly revised to take into account the permanent technological projections in the electronic materials fields and in the data-processing techniques. Those have an important impact on the railway systems design. However the standards did not formalize yet well the process of distribution of the safety regulations of the railway system on its subsystems, hardware and software which supports them. In other words, if the risks are well identified and followed through demonstrations and safety justifications, it remains to improve the Risks Analysis and Safety Requirements Allocation steps.

The objective of this work is to propose a method and standard contents for a PRA in the context of railway signalling and command and control systems. This method will integrate the impact of the NICT on safety in terms of risks induced on the whole system. The evolution of the systems design passes by the integration of the NICT. The NICT are considered as Components Off The Shelf (COTS). The COTS allow controlling the cost of system realization, but on the other hand there is a loss of safety control [START_REF] Boulanger | Reference systems and standards for safety assessment of railway applications[END_REF]. Thus the proposed method will have to take into account the inherent risks in this type of component. Another constraint relates to the taking into account of the human errors. This paper is organized as follows. Section 2 is dedicated to the columns definitions of the PRA and a PRA method is proposed in section 3. Section 4 is devoted to the description of the phases of PRA method presented previously. Finally, concluding remarks and perspectives are given in the last section.

Columns definitions of Preliminary Risk Analysis

Knowing that the results of the Preliminary Risk Analysis are presented in a worksheet and the various definitions of the terms and concepts related to the PRA, we dedicate this section to detail these concepts. Basing on CENELEC standards EN 50126 [START_REF]Railway Applications -The specification and demonstration of dependability -reliability, availability, maintainability and safety (RAMS)[END_REF], EN 50128 [START_REF]Railway Applications -Software for railway control and protection systems[END_REF], EN 50129 [START_REF]Railway Applications -Safety related systems for signalling[END_REF] and EN 61508-4 [START_REF]Functional Safety of electrical / electronic / programmable electronic safety related systems -Part 4: Definitions and abbreviations[END_REF] we propose definitions of the columns of PRA applied to the railway control systems. A standardization of the principal concepts and associated terminology is indeed proved to be necessary, after study of several railways PRA of various sources (manufacturers, owners...), where important inconsistencies could be noted (for example the same term is used for different concepts).

-Operating mode (exploitation mode, or working mode, or phase or context):

knowing that the analysis of hazards and risks is carried out for all the reasonably foreseeable situations, which justifies this column. It is useless to consider a safety measurement « the platform doors must be strictly closed » for a metro which is on line, even in station where the platform doors don't exist… In some cases, it is not mandatory to specify the operating mode. -Dangerous Entity (or Dangerous element, or hazardous entity/element): a subset of the studied system which is at the origin of the feared event, it is the initial cause of the studied scenario. The function of the dangerous entity can be specified if necessary. -Feared event (or event causing a dangerous situation, or undesirable event, or error): is a dangerous event, it is the event affecting the dangerous entity or its function leading the system in a dangerous situation. -Dangerous situation (or danger): undesirable state of system following the feared event, may lead to a potential accident. -Damage (or consequence): it is result of an accident given in terms of death, physical wound, injuries, attack to people health or environment damage. -Severity level: a classification on several levels, it allows to evaluate and estimate the consequences of potential accidents. According to EN 50126 [START_REF]Railway Applications -The specification and demonstration of dependability -reliability, availability, maintainability and safety (RAMS)[END_REF], table 1 describes typical hazard severity levels and the consequences associated with each severity level for railway systems. In order to avoid bad interpretations of the qualitative terms, some industrialists and/or owners use numbers to describe severity levels (from 1 to 4 in order of increasing severity, 4 = catastrophic). -Measurements (safety measures, safety constraints or safety requirements): suitable actions to reduce or eliminate risk. They can be preventive measures or protection measures. This column could describe the subsystem (equipment, system…) charged to ensure the safety measures (responsible entity of risk reduction) and its reference. -Event causing a potential accident: it is a dangerous event that transforms a dangerous situation into a potential accident. It does not exist in the case of scenarios of order 1. Columns could appear in the PRA, for the sequences of dangerous events which allow passing from a dangerous situation to an accident, this depends on the order of the studied scenario.

-Potential accident: a potential accident could be an accident or quasi accident. The effective occurrence of damages (e.g. collision) determines the accidental identity of the potential accident else it is an incident (e.g. crossing over a restrictive signal without effective collision).

Proposed Method

In Villemeur [START_REF] Villemeur | Sûreté de fonctionnement des systèmes industriels : fiabilitéacteurs humains informatisation[END_REF], the objective of Preliminary Risk Analysis method is: -To determine the dangers (hazards) and their causes (dangerous entities, dangerous situations, potential accidents). -To evaluate the severity of the consequences of situations and accidents previously determined. -To deduce the measurement and the suitable actions to eliminate or reduce dangerous situations and the potential accidents.

Figure 1 summarizes these steps. The identification of dangerous entities, dangerous situations and the potential accidents rests at the beginning on the experiment and the judgment of the specialists, helped by guide lists which are updated by the experience feedback throughout the lifecycle of the system.

Figure 1: Steps of Preliminary Risk Analysis

The PRA is generally considered as an inductive approach (proceed from causes to identify consequences), however certain actors consider it as a deductive approach. A deductive or inductive analysis does not relate to the general method described by figure 1 but only to the first step (1): identification of dangerous entities, dangerous situations and potential accidents. During this step, some experts determine the set of potential accidents (consequences) by induction on the basis of the dangerous entities (causes); other experts proceed by deduction to identify the dangerous entities or the dangerous events (causes) from the potential accidents (consequences). When we dispose of a complete list of all potential accidents (respectively dangerous entities / feared events) the deductive approach (respectively inductive approach) alone is valid and lead to acceptable results covering all possible scenarios. On the other hand for systems having a significant number of scenarios and if we don't dispose of complete lists of potential accidents (respectively dangerous entities / feared events), using only deductive approach (respectively inductive approach) is not efficient. In these cases, it is possible that the used approach does not take into account the potential accidents (respectively dangerous entities / feared events) not included in the generic list. To avoid this problem and to have complete results, a deductive-inductive approach should be used. -ER 0 , D 0 and A 0 are the preliminary lists (initial) of dangerous entities /feared events, dangers and potential accidents respectively. They are defined starting from the generic lists analyzed by experts; the latter can remove incoherent scenarios or add other missing scenarios. We consider that we obtained these lists resulting from the initialization step (step (0) of figure 1).

-ER, D and A represent the lists of dangerous entities/feared events, dangers and potential accidents respectively. At the beginning of the analysis they are initialized at ER 0 , D 0 and A 0 and they contain the final lists at the end of the analysis.

-The step contains several cyclic phases: two inductive phases (solid line), two deductive phases (broken line) and a phase to generate the list of the consequences (damages) generated by the potential accidents. The phases are noted φ i,j :

Where the index « i » indicates the number of phase and the index « j » indicates the current cycle. i.e. φ 4,1 is the 4 th phase of the 1 st cycle.

-The cycles start with index 1 (j > 0), in other words, the first cycle of this step corresponds to « j = 1 ».

-At the beginning of a cycle « j », the sets of the dangerous entities/feared events, dangers and potential accidents are indexed by « j-1 », ER j-1 , D j-1 and A j-1 . For example at the beginning of cycle 1, the sets ER, D and A are equal to ER 0 , D 0 and A 0 respectively.

In this paragraph we will present the various phases of PRA method presented in the previous paragraph.

At the beginning of cycle 1, the lists of feared events, dangers and potential accidents are initialized at ER 0 , D 0 and A 0 . Phase 1 (inductive phase) allows determining the dangers starting from the feared events (Fig. 3). During this phase, the list ER 0 can generate a list of dangers that corresponds to:

-A part of the preliminary list noted From these new lists, we start a new cycle with the same described phases.

The necessary and sufficient condition to stop analysis if during a given cycle defined by index j = f:

-Phase φ 1,f does not generate anymore new dangers, And -Phase φ 2,f does not generate anymore new potential accident.

In another manner: Note that the analysis could also be performed from the potential accidents to the feared events (opposite direction of the cycle). This by reversing the order of the phases in the following way: the cycle starts with phases 4 and 3, phase 5, then the phase 1 and finishes by phase 2.

Figure 2 :

 2 Figure 2: first step of PRA method.

Figure 2

 2 Figure 2 represents the first step (1) of the PRA illustrated by figure 1.

-.

  And a new list of dangers noted 'n 0 D which is added to the preliminary list.At the end of this phase, the new list of dangers is The only sub-list of dangers which does not have correspondence in the list of the feared events is

Figure 3 : Phase 1 Phase 2 - 0 A

 3120 Figure 3: Phase 1 Phase 2 allows determining the potential accidents from dangers (inductive phase). During this phase, list of dangers generated from the previous phase ( 'n 0 0 D D ∪ ) allows generating a list of potential accidents that corresponds to: -A part of the preliminary list noted ' 0 0 A A ⊆ . We note by

Figure 4 : Phase 2 Phase 3

 423 Figure 4 : Phase 2 Phase 3 is used to identify the damages (consequences) from the list of the accidents obtained at the end of the previous phase ( ' " 'n 0 0 0 A A A ∪ ∪). It is

Figure 5 : Phase 3 Phase 4 : 0 A 0 A

 53400 Figure 5 : Phase 3 Phase 4: Knowing that the only sub-list of the potential accidents which does not have correspondence in the dangers list is " 0 A , this phase allows determining the

Figure 6 : Phase 4 Phase 5 :

 645 Figure 6 : Phase 4 Phase 5: this phase allows determining the feared events list from dangers. The sub-list of dangers which does not have correspondence in the feared events list is " "n 0 0 D D ∪ . We note by

Figure 7 : Phase 5

 5 Figure 7 : Phase 5 At the end of cycle 1, lists ER, D and A are given as follows: 'n 1 0 0 ER ER ER = ∪ 'n "n ' " 'n "n 1 0 0 0 0 0 0 0 D = D D D = D D D D ∪ ∪ ∪ ∪ ∪ 'n ' " 'n 1 0 0 0 0 0 A = A A A A A ∪ = ∪ ∪

Table 1 :

 1 Hazard Severity Level.

	Severity	Consequence to Persons or	Consequence to
	Level	Environment	Service
	Catastrophic Fatalities and/or multiple severe injuries
		and/or major damage to the environment.
	Critical	Single fatality and/or severe injury and/or	Loss of a major
		significant damage to the environment.	system
	Marginal	Minor injury and/or significant threat to	Severe system(s)
		the environment	damage
	Insignificant Possible minor injury	Minor system damage

Table 2 :

 2 Frequency of Occurrence of Hazardous Events.

	Category	Description
	Frequent	Likely to occur frequently. The hazard will be continually experienced
	Probable	Will occur several times. The hazard can be expected to occur often
	Occasional	Likely to occur several times. The hazard can be expected to occur
		several times
	Remote	Likely to occur sometime in the system life cycle. The hazard can
		reasonably expected to occur
	Improbable	Unlikely to occur but possible. It can be assumed that the hazard may
		exceptionally occur.
	Incredible	Extremely unlikely to occur. It can be assumed that the hazard may not
		occur.