E A Tereshina
email: tereshina@physics.msu.ru

M D Kuz'min

Y Skourski

M Doerr

J Wosnitza

I S Tereshina

Forced-ferromagnetic state in a Tm 2 Fe 17 H 5 single crystal

Keywords: single crystal, hydride, forced-ferromagnetic state, rare-earth-iron intermetallic compound

Forced-ferromagnetic state in a Tm2Fe17H5 single crystal

Introduction

Magnetism of the rare-earth (RE)-iron intermetallic compounds is of special importance in view of applications. Suffice it to mention the celebrated Nd2Fe14B magnet, in which light boron stabilizes the crystal structure. The Sm2Fe17Nx system, interstitially modified with nitrogen (here x ranges between 2 and 3), has some advantages over Nd2Fe14B due to its high fundamental characteristics (such as magnetic polarization, Curie temperature, anisotropy constant [START_REF] Fujii | Interstitially modified intermetallics of rare earth and 3d elements Handbook of Magnetic Materials[END_REF][START_REF] Nakayama | Correlation between microstructure and magnetic properties in Sm2Fe17N3 magnet prepared by pulsed current sintering[END_REF]), but its poten-tial has not been realized yet [START_REF] Gutfleisch | Magnetic materials and devices for the 21st century: stronger, lighter, and more energy efficient[END_REF]. Improved characteristics can rationally be reached only with a better understanding of the magnetic properties of such interstitially modified RE2Fe17 materials and for that the use of bulk, nondecrepitated single crystals is highly desirable [START_REF] Tereshina | Strong room-temperature easy-axis anisotropy in Tb2Fe17H3: an exception among R2Fe17 hydrides[END_REF].

The RE and Fe sublattices are ordered ferro-or ferrimagnetically depending on the rare-earth atom [START_REF] Duc | Intersublattice exchange coupling in the lanthanide-transition metal intermetallics Handbook on the Physics and Chemistry of Rare Earths[END_REF][START_REF] Campbell | Indirect exchange for rare earths in metals[END_REF][START_REF] France | Magnetic properties of binary rare-earth 3d-transition-metal intermetallic compounds Handbook of Magnetic Materials[END_REF][START_REF] Buschow | Permanent magnet materials based on 3d-rich ternary compounds Handbook of Magnetic Materials[END_REF][START_REF] Brooks | Density functional theory of molecular fields in R-M systems[END_REF][START_REF] Belorizky | Evidence in rare-earth (R)-transition metal (M) intermetallics for a systematic dependence of R-M exchange interactions on the nature of the R atom[END_REF][START_REF] Verhoef | Multiple field-induced transitions in Er2Fe17 in fields up to 50[END_REF]. The total magnetic moment of light rare earths (LRE) is usually small (~3 µB/atom for Nd, ~0.7 µB/atom for Sm) and is parallel to the total magnetic moment of the iron, which results in a mod-erate enhancement of the net polarization. It is for this reason that LRE-Fe intermetallics are used as magnetic materials [START_REF] Gutfleisch | Magnetic materials and devices for the 21st century: stronger, lighter, and more energy efficient[END_REF][START_REF] Campbell | Indirect exchange for rare earths in metals[END_REF][START_REF] Skomski | Permanent Magnetism[END_REF][START_REF] Herbst | R2Fe14B materials: intrinsic properties and technological aspects[END_REF][START_REF] Coey | Magnetism and Magnetic Materials[END_REF]. The situation in heavy rare earth (HRE)-Fe com-pounds is different. The total magnetic moment of HREs is as large as 10 µB/atom (for Dy or Ho). Unfortunately, the large total moment is of no immediate benefit, since it is antiparal-lel to the moment of the iron sublattice. From a fundamental point of view, the magnetic behavior of HRE-Fe compounds (ferrimagnets) is more diverse than that of ferromagnets. While a ferromagnet simply saturates in external magnetic field, a ferrimagnet demonstrates a significant growth of mag-netization. It goes through a sequence of spinreorientation phase transitions until it reaches the fieldinduced ferromagn-etic state.

Sufficiently high external magnetic fields have to be employed in order to observe a complete magnetization pro-cess along different crystallographic directions of the sample. Mutual orientation of the RE spin and orbital moments is gov-erned by the spin-orbit coupling, H so = λ SRE • LRE, where the coupling constant λ is largely an atomic property [START_REF] Skomski | Simple Models of Magnetism[END_REF]. All HRE have λ negative and large, |λ| ~ 10 3 K. As against that, the RE-Fe exchange interaction, H ex = JRE-Fe SRE • SFe, is two orders of magnitude weaker, JRE-Fe ~ 10 K [START_REF] Liu | On the 4f-3d exchange interaction in intermetallic compounds[END_REF]. In a favorable case, a magnetic field of several dozen Tesla should suffice to align SRE and SFe parallel. More importantly, JRE-Fe is not an atomic constant. It varies from compound to com-pound, depending in particular on interatomic distances [START_REF] Campbell | Indirect exchange for rare earths in metals[END_REF].

Here, we report the attainment of the ferromagnetic state in HRE-Fe single-crystalline hydrides, observed in high magn-etic fields. It is advantageous to perform such experiments on Tm-Fe compounds. Since thulium is the HRE with a Landé factor closest to unity, it takes the smallest applied magnetic field to reach the ferromagnetic state (for a given JRE-Fe). For the same reason, the field in our experiments was always applied along the easy magnetization direction of single-crystalline samples.

Earlier, a Tm2Fe17 single crystal and its polycrystalline deuteride Tm2Fe17D3.2 were studied in fields up to 74 and 60 T, respectively [START_REF] Isnard | High magnetic field study of the Tm2Fe17 and Tm2Fe17D3.2 compounds[END_REF]. Unfortunately, Isnard et al did not reach the ferromagnetic state in both compounds. In the present work, samples with various hydrogen content, including the maximum possible amount of 5 at.H/f.u., were used. We showed that Tm2Fe17H5 reaches the ferromagnetic state in a magnetic field of 57 T and attains an enviably high polarization of 2.25 T. The study is performed on single-crystalline Tm2Fe17Hx samples that allowed us to obtain reliable data on the type and behavior of the magnetocrystalline anisotropy in these materials.

Materials synthesis and experimental details

Initial samples were prepared by arc-melting of 99.9% pure elements under helium atmosphere. To stimulate growth of single-crystalline grains, the ingots were re-melted and cooled down slowly in a resistance furnace. Several 1 mm-large grains were extracted and checked by x-ray Laue tech-nique. Usually single crystals decrepitate when exposed to hydrogen. To prevent this from happening, the hydrogenation was carried out under particularly mild conditions-elevated temperature and low partial pressure of H2-both of which slow down the reaction. The higher hydrogen contents were achieved by subsequently raising the pressure. The amount of absorbed hydrogen in Tm2Fe17Hx was determined from the pressure drop in the reaction chamber. Phase composi-tion of the samples was controlled by powder x-ray diffrac-tion. Magnetization measurements were performed by using a pulsed-field induction magnetometer [START_REF] Skourski | High-field magnetization of Ho2Fe17[END_REF] at the Dresden High Magnetic Field Laboratory. The maximum field was 60 T with a total pulse duration of 25 ms [START_REF] Zherlitsyn | Status of the pulsed-magnet-development program at the Dresden high magnetic field laboratory[END_REF]. A commercial magnetom-eter (Quantum Design PPMS-14) was used for magnetization measurements in steady magnetic fields up to 14 T at temper-atures between 2 and 500 K.

Results and discussion

Pure Tm2Fe17 has the hexagonal Th2Ni17-type crystal structure (space group P63/mmc) with lattice parameters a = 8.409(3) Å and c = 8.300(3) Å. Hydrogenation increases the unit-cell volume of Tm2Fe17 without changing its crystal structure. The lattice parameter a of Tm2Fe17Hx grows linearly with hydrogen content x, see figure 1. However, c remains practically constant as hydrogen fills the octahedral interstitial sites (at x < 3), and increases in the interval 3 x 5, as the tetrahe-dral sites are filled [START_REF] Isnard | A structural analysis and some magnetic properties of the R2Fe17Hx series[END_REF]. For the composition Tm2Fe17H5, the lattice parameters are a = 8.540(3) Å and c = 8.381(3) Å; the relative volume change upon hydrogenation V/V is ~4.2%. Tm2Fe17 is a ferrimagnet with a Curie temperature of TC = 295 K and a spontaneous moment of 22 µB/f.u. (µ0Mferri = 1.0 T) at T = 4.2 K. Figure 2 shows the magnetic phase diagram of Tm2Fe17, as well as of its hydrides Tm2Fe17Hx, deduced from magnetization data measured on single crystals along different crystallographic directions in a wide temperature range up to 500 K (not shown here). At low temperatures, the anisotropy of pure Tm2Fe17 is of the 'easy-axis' type; this changes to 'easy plane' as temperature increases. The change proceeds via an intermediate 'easy-cone' phase. Our results agree well with data presented in [START_REF] Isnard | High magnetic field study of the Tm2Fe17 and Tm2Fe17D3.2 compounds[END_REF].

Hydrogenation is known [START_REF] Tereshina | Strong room-temperature easy-axis anisotropy in Tb2Fe17H3: an exception among R2Fe17 hydrides[END_REF] to affect strongly (i) the type of magnetic anisotropy (all investigated Tm2Fe17Hx hydrides are easy-plane magnets), and (ii) the Curie temperature (seen to rise from 295 to 450 K as x grows from 0 to 5). It also modi-fies significantly the magnetization curves of Tm2Fe17Hx, see figure 3. Here, the field was applied along the easy magnetization direction, namely, along the c axis for Tm2Fe17 (figure 2(a)) and perpendicular to the c axis for Tm2Fe17H2.5 (figure 2(b)) and Tm2Fe17H5 (figure 2(c)) (the basal-plane magnetocrystalline anisotropy of the studied compounds is close to zero). For Tm2Fe17, one can observe two rather sharp fieldinduced transitions, where the magnetization increases abruptly. As against that, the magnetization of Tm2Fe17H2.5 and Tm2Fe17H5 shows a continuous growth above a certain threshold field. For the final composition with x = 5 the magn-etic field of 57 T was sufficient for attaining the ferromagnetic state (figure 3(c)).

Let us assume for the ground state of Tm2Fe17 an antiparallel configuration of the Tm and Fe moments (Mferri = MFe - MTm = 22 µB/f.u.) and adopt the free-ion value for the magnetic moment of Tm, MTm = 14 µB/f.u. This yields for the iron-sublattice moment MFe = 36 µB/f.u., which is in good agreement with what is known for RE2Fe17 [START_REF] Herbst | R2Fe14B materials: intrinsic properties and technological aspects[END_REF]. In the forced ferromagnetic state, Tm2Fe17 should have Mferro = MFe + MTm = 50 µB/f.u. (µ0Mferro = 2.29 T). For Tm2Fe17H5, Mferri = 23 µB/f.u. and Mferro = 51 µB/f.u., that is, µ0Mferri = 1.02 T and µ0Mferro = 2.25 T. In both compounds, the magnetization more than doubles as a result of the ferrito-ferro transition. Since hydrogenation weakens the Tm-Fe exchange we are able to observe the ferromagnetic state for Tm2Fe17H5 already at 57 T (figures 3(c) and 4). Neglecting the weak Tm-Tm interaction, one can consider the Tm-Fe exchange integral JTmFe to be proportional to the molecular field on Tm. The latter can be determined by several meth-ods [START_REF] Liu | On the 4f-3d exchange interaction in intermetallic compounds[END_REF][START_REF] Campbell | Indirect exchange for rare earths in metals[END_REF][START_REF] Isnard | High magnetic field study of the Tm2Fe17 and Tm2Fe17D3.2 compounds[END_REF] In [START_REF] Isnard | High magnetic field study of the Tm2Fe17 and Tm2Fe17D3.2 compounds[END_REF], it was shown that the molecular field is just given by the abscissa of the midpoint of the sloping part of the experimental magnetization curve. Since the pro-portionality coefficient is not accurately known, it is convenient to describe the effect of hydrogen on JTmFe by means of the reduced quantity,-JTmFe(x)/JTmFe(0), i.e. the exchange integral or the molecular field normalized to unity for the parent compound. This is plotted for Tm2Fe17Hx versus the relative volume change, V/V, as determined from the lattice parameters (figure 5). In agreement with earlier observations [START_REF] Hong | Volume dependence of magnetic interactions in R2Fe17 and R2Fe14B and their interstitial compounds with H[END_REF], JRE-Fe(x)/JRE-Fe(0) decreases as a result of hydrogenation. Moreover, in figure 5 we present similar data for Gd2Fe17(D,N,C)x compounds, whose molecular fields were determined by use of inelastic neutron scattering [START_REF] Isnard | A high energy inelastic neutron scattering investigation of the Gd-Fe exchange coupling in Gd2Fe17Dx (x = 0, 3 and 5[END_REF][START_REF] Loewenhaupt | Nitrogen-and carbon-induced reduction of Gd-Fe exchange coupling in Gd2Fe17 doped samples[END_REF]. All data can be approximated by a straight line.

Interestingly, the JRE-Fe data plotted against relative volume change in HREFe2-H systems (figure 5) also demonstrate a linear trend for the HRE Ho, Er and Tm [START_REF] Mushnikov | Magnetization process and magnetic structure of the TmFe2H3.4 hydride in ultrahigh magnetic fields up to 100[END_REF][25][START_REF] Mushnikov | Magnetovolume effects and magnetic anisotropy in the itinerant and localized subsystems of f-d intermetallics Thesis of Doctor of Sciences Ekaterinburg[END_REF]. Hydrogenation of the cubic Laves phases REFe2 can well pro-duce values of V/V between 25 and 30%, while in RE2Fe17 by way of hydrogenation, nitrogenation, and/or carbonation a volume increase of just under 9-10% appears [START_REF] Fujii | Interstitially modified intermetallics of rare earth and 3d elements Handbook of Magnetic Materials[END_REF]. For REFe2, the exchange fields weaken by up to about 70%, whereas for RE2Fe17, it weakens by 20-30%. Despite the considerable decrease of JTmFe, high magnetic fields are required in order to attain the forced-ferromagnetic state in TmFe2Hx due to the increased magnetocrystalline anisotropy upon hydrogenation caused by the structural phase transition from the parent cubic to the rhombohedral structure.

Conclusions

The study of the field-induced transitions from a ferri-to a ferromagnetic state in the intermetallic compounds is an interesting fundamental problem. Up to now very few forcedferromagnetic states in rare-earth intermetallic compounds with iron were reported [START_REF] Mushnikov | Magnetization process and magnetic structure of the TmFe2H3.4 hydride in ultrahigh magnetic fields up to 100[END_REF][START_REF] Kato | Field-induced phase transitions in ferrimagnetic R2Fe14B in ultra-high magnetic fields[END_REF]. In the present work, the measurements were carried out on single-crystalline Tm2Fe17 hydrogenated to the maximum hydrogen content of 5 H atoms per formula unit along the easy magnetization direction in fields up to 60 Т. Hydrogenation allowed us to decrease the Tm-Fe exchange and thus to lower the transition field to the forcedferromagnetic state down to 57 T as compared to about 80 T expected for the parent compound Tm2Fe17 [START_REF] Isnard | High magnetic field study of the Tm2Fe17 and Tm2Fe17D3.2 compounds[END_REF]. In the ferromagnetic state, Tm2Fe 17H5 attained an enviably high polarization of 2.25 T. The larger value µ0Mferro = 2.65 T can be obtained for Ho2Fe17 in the even higher applied magnetic field (84 T [START_REF] Skourski | High-field magnetization of Ho2Fe17[END_REF]). (Other numbers for comparison: the celebrated hard magnetic material Nd2Fe14B has 1.85 T at T = 0 K while this value in the industrial magnets does not surpass 1.5 T).

Although it may take a lower magnetic field to stabilize ferromagnetism in Co-based compounds, e.g. 39 T sufficed for Tm2Co17 [START_REF] Andreev | High-field magnetization study of a Tm2Co17 single crystal[END_REF], polarization in the ferromagnetic state is lower, too (µ0Mferro = 1.96 T for Tm2Co17). The search for new multicomponent compounds with substitutional or/and interstitial atoms where a transition into a high-magnetization state takes place at even lower magnetic field is of practical importance. Further progress in this field can be anticipated.

Figure 1 .

 1 Figure 1. The lattice parameters a and c and the unit cell volume V of Tm2Fe17Hx (lines are guides to the eye).

Figure 2 .

 2 Figure 2. Magnetic phase diagram demonstrating the orientation of the magnetic moment with respect to the crystal axis c for Tm2Fe17 and some hydrides Tm2Fe17Hx. The width of the bars corresponds to the experimental error in x.

Figure 3 .

 3 Figure 3. Magnetization curves of Tm2Fe17, Tm2Fe17H2.5 and Tm2Fe17H5 measured at T = 4.2 K in pulsed fields applied along the easy axes.

Figure 4 .

 4 Figure 4. Ferrimagnetic (top) and field-induced ferromagnetic (bottom) states of an HRE-Fe compound.

Figure 5 .

 5 Figure 5. Reduced exchange integral JRE-Fe(x)/JRE-Fe(0) for RE2Fe17(H, N, C)x (RE = Gd [22, 23], Tm) and REFe2Hx (RE = Ho, Er, Tm [25-27]) as a function of the relative volume change V/V.

Acknowledgments

We wish to thank Prof D Givord and Prof N V Mushnikov for useful discussion, O Chistyakov and A Hackemer for help in producing single-phase parent samples. This work was supported by RFBR, project No. 16-03-00612. We gratefully acknowledge the support of the HLD at HZDR, member of the European Magnetic Field Laboratory (EMFL). Part of the measurements has been performed in MLTL (http://mltl. eu) and was supported by Material Growth and Measurement Laboratory MGML (http://mgml.eu/).


A Pivano

V O Dolocan
email: voicu.dolocan@im2np.fr

Systematic motion of magnetic domain walls in notched nanowires under ultra-short current pulses

Keywords: numbers: 75.60.Ch, 75.10.Hk, 75.40.Mg

The precise manipulation of transverse magnetic domain walls in finite/infinite nanowires with artificial defects under the influence of very short spin-polarized current pulses is investigated. We show that for a classical 3d ferromagnet material like Nickel, the exact positioning of the domain walls at room temperature is possible only for pulses with very short rise and fall time that move the domain wall reliably to nearest neighboring pinning position. The influence of the shape of the current pulse and of the transient effects on the phase diagram current-pulse length are discussed. We show that large transient effects appear even when α=β, below a critical value, due to the domain wall distortion caused by the current pulse shape and the presence of the notches. The transient effects can oppose or amplify the spin-transfer torque (STT), depending on the ratio β/α. This enlarges the physical comprehension of the DW motion under STT and opens the route to the DW displacement in both directions with unipolar currents.

I. INTRODUCTION

Current induced magnetic domain wall motion (CIDWM) in nanowires or nanostrips is a highly active research field 1,2 with applications in high-density and ultrafast nonvolatile data-storage devices like the racetrack memory 3 or for logic devices 4 . In the racetrack memory, the data processing is based on the controlled displacement between precise distinct positions of the domain walls (DWs) due to the transfer of angular momentum (spin-transfer torque) from a spin-polarized electric current. To achieve precise positioning of DW, artificial constrictions or others patterned geometrical traps are usually used, which create an attractive pinning potential for the DW. Different types of traps were studied in cylindrical or flat/strip nanowires [5][6][7][8] , along with the possible interaction between the DWs 9,10 . In some cases, depending on the pinning potential, the DW displacement between pinning sites can display a chaotic behavior 11 or a stochastic resonance 12 under harmonic excitation.

The required currents for STT based DW movement are usually high (∼ 1A/µm 2 ) which limits the applicability due to Joule heating. To displace accurately the DWs between the pinning sites, the current density should be kept at relatively low values and/or very short current pulses should be applied. Experimentally, it was observed that an efficient DW motion is reached for pulses in the nanosecond regime 13 and that the resonant excitation of the DW by a short train of current pulses decreases the depinning current 14 . More recently, the effect of the temporal and spatial shape of the current pulse was highlighted [15][16][17] . It was shown that a fast changing current with an ultra-short pulse rise time decreases the critical current density due to the dependence of the DW motion on the time derivative of the current 18 and leads to high DW velocity 19 . Another aspect of the CIDWM under short pulses is the existence of important transient effects related with the DW inertia-like behavior 13,[20][21][22][23] due to deformation of the wall. The consequences are a delayed response at the current onset and at the end of the current pulse. The theoretical and experimental results show that the distance traveled by a DW is almost proportional to the current pulse length and that the transient motion depends on the variation of the generalized angle of the wall, the wall width and the ratio of the damping (α) and nonadiabatic (β) parameters 20,21 . For very short pulses (one nanosecond), the transient displacement is comparable with the steady-state motion 24 . A DW that propagates without deformation should display no inertia 25 like in cylindrical nanowires 26 or in certain perpendicular magnetic anisotropy systems 27 . The absence of inertia will allow a fast response to external forces while the transient DW displacement after the application of the pulse limits the application in fast devices. In the main time, it was recently demonstrated that inertia-like behavior of a DW can be also an advantage when ultra-short optical pulses are used 28 with applications in the optical recording. Moreover, in systems with strong spin-orbit coupling where additional contributions from spin-Hall effect complicate the DW dynamics, a DW tunable inertia was proposed 29 . Before studying more complex systems, the influence of the transient effects on the systematic DW movement under ultra-short spin-polarized current pulses should be completely understood in a classical 3d ferromagnet like Nickel, which is the aim of this paper.

In this paper, we address the systematic motion of a magnetic transverse DW between fixed artificial constrictions (notches), when submitted to a series of ultra-short spin-polarized current pulses (transient regime) at low and room temperature. The artificial constrictions are situated at regular positions in a flat finite or infinite nanowire with only in-plane (shape) anisotropy like in a classical 3d ferromagnet. We determine the influence of the current pulse shape (rise and fall time) on the mo-tion of the DW. We show that even at zero temperature, there is a transition region between the different bands in the current-pulse time phase diagram, each band corresponding to the positioning of the DW at a well-defined notch. Our results show that at room temperature, the precise positioning can be achieved only by very short pulses with very short rise and fall times that displace the DW by only one notch at a time. Therefore, to move a DW several notches reliably, a sequence of very short pulses should be used. By examining the influence of the damping and nonadiabatic parameters, we show that when β=0 the transient effects (automotion) of the DW are very large and oppose the STT, being observed in the phase diagrams as predicted 20 . The transient effects are related with the change in the DW structure that is due to a combination of factors: pinning potential of the notches which induces a sufficient variation of the DW angle 30 , position of the DW inside the potential well (different restoring force), low damping parameter and shape of the current pulse. Contrary to expectations, the transient effects also appear when α = β, below a critical value. For β > α, the transient effects can oppose or amplify the STT, thus explaining the oscillatory DW depinning at higher currents observed experimentally 13 . This brings new physical insight into CIDWM under STT and paves the way for systematically displacing DW in nanowires in both directions using only unipolar current pulses.

This article is organized as follows. In Sec. II, we present the micromagnetic and the stochastic 1D model used to calculate the pulsed DW dynamics. In Sec. III, we compute and investigate the phase diagram of the DW dynamics for a finite and infinite nanostrip at T=0K and room temperature. Discussion and concluding remarks are presented in Sec. IV.

II. MODEL

We study numerically the systematic motion of a pinned transverse domain wall in a finite or infinite Ni nanostrip with symmetric rectangular notches. The finite strip has a length L x =1µm, a cross section of L y ×L z = 60×5nm 2 and has ten rectangular symmetric double notches separated by 80nm. The results presented below are for notch dimensions of 20 × 9 × 5 nm 3 . The variation of length and depth of the notches does not influence much the physics of phase diagrams presented in Section III. The notch depth influences the depinning current as the potential barrier increases, while the notch length influences lightly the depinning current and mostly the slope of the potential wells.

Fig. 1(a) shows the equilibrium position of a headto-head transverse DW in the notched nanostrip using the parameters of Nickel: saturation magnetization M s =477kA/m, exchange stiffness parameter A = 1.05 × 10 -11 J/m, spin polarization P=0.7 and damping parameter α=0.05. The DW is moved by a series of spin polarized current pulses applied along the x-axis. The geometry of the current pulse is described in Fig. 1(b), with t r , t s , t f and t z the rise, stable, fall time and zerocurrent time respectively. The nonadiabatic parameter is set to β=2α, if not specified otherwise.

The DW dynamics was computed using 3D micromagnetic simulations with the MUMAX3 package 31 and with the one-dimensional DW model 32,[START_REF] Thiaville | Spin Dynamics in Confined Magnetic Structures III[END_REF] . In both cases, the magnetization dynamics is determined from the Landau-Lifschitz-Gilbert (LLG) equation with adiabatic and nonadiabatic spin-transfer torques [START_REF] Zhang | [END_REF] :

Ṁ = -γ 0 M×H ef f +α(M× Ṁ)-(u•∇)M+βM×(u•∇)M
(1) where γ 0 is the gyromagnetic ratio, u = j e P µ B /eM s is the spin drift velocity, P the spin polarization of conductions electrons, µ B the Bohr magneton and j e the applied current density. No additional exotic torques (like the ones due to the spin-Hall or Rashba effect) were considered.

For the micromagnetic computations, the strip was discretized into a mesh with a cell size of 2×3×2.5nm 3 , inferior to the exchange length (∼5nm). The DW dynamics in a finite wire is compared with the one of an infinity long wire where the magnetic charges at the ends of the nanostrip are compensated. The average position of the DW center (X) is extracted for each simulation (in the axial x direction) along with the azimuthal angle (ψ) of magnetization in the yz plane. No magnetocrystalline anisotropy is considered, the shape anisotropy insures that the easy axis is in-plane. The effect of the temperature is studied both micromagnetically and with the 1D model. The 1D model of the DW (collective coordinates X and ψ) supposes that the DW is rigid and gives a quasi-quantitative understanding of the motion of the DW. The Langevin equations of motion of the DW 2,35 are detailed in Ref. 36.

The pinning potential energy is determined from quasistatic micromagnetic simulations and is shown in Fig. 1(c) and (d) for the finite case and the infinite case respectively. The pinning potential determined by fitting the micromagnetic results is harmonic inside the notches and sinusoidal between them:

V p (x) = 1 2 k i (x + x i ) 2 , for x i -L ≤ x ≤ x i + L V 0 cos(2πf x + φ i,i+1 ), for x i + L < x ≤ x i+1 -L
(2) with k i the stiffness constant and x i the DW stable equilibrium position of the site i, φ i,i+1 the phase between the i site and its nearest neighbor. V 0 and f correspond respectively to the effective height of the potential and its spatial frequency. For the finite case, the stiffness constant varies from 7.07×10 -5 N/m to 6.77×10 -5 N/m, when moving from the center of the nanowire to its ends. In the infinite case, the stiffness constant is equal to 7.16 ×10 -5 N/m for all the pinning sites and L = 16.5nm. The expression (Eq. 2) was used in the equations of the 1D model through the pinning field H p included in H ef f .

The pinning energy is controlled by the dimensions and distance between the notches. A clear difference is observed between the two potentials due to edge dipolar energy. In the finite strip, the depinning field decreases from 39 Oe, in the central wells, to 26 Oe when the DW is closed to the two ends of the strip. This is due to the attractive interaction between the DW and magnetic surface charges located at the sides 37 . As a result, the potential wells are asymmetric in energy along the strip and their energy minima decrease when the distance between the notches and the ends of the strip is reduced. In contrast, for the infinite case, each well has the same depinning field and energy barriers.

III. RESULTS

Our analysis of the DW dynamics begins with the study of the differences between a finite and an infinite nanostrip at T=0K. Afterwards, the influence of the pulse shape is discussed and the particularities of the DW motion at room temperature. The last subsection details the results when the damping and nonadiabatic parameters are varied and their influence on the transient displacement.

A. Phase diagrams at T = 0K

To characterize the systematic motion of the DW between the notches, we computed point-by-point phase diagrams for all systems with the 1D DW model for a large range of pulse duration and current amplitude. We compare our 1D results with phase diagrams computed micromagnetically on less points than the 1D calculus. A similar micromagnetic computation will require an enormous execution time. The control parameters are the amplitude, the duration and the shape of the current pulses. The range of the current amplitude (≤10 A/µm 2 ) is chosen to have only viscous motion (no precession) for the pulse duration used ( 1.5ns). The pulse duration range is selected to be on the same order of magnitude with access or reading/writing time in possible magnetic memories based on DW.

The phase diagram, in the parameter space stable time -current amplitude, which characterizes the DW dynamics in the finite nanostrip at T=0K, is shown in Fig. 2(a). The diagram represents 200×400 point-by-point integration with a fourth order Runge-Kutta scheme. The micromagnetic results (scattered symbols) are compared with the 1D results (colored regions). The DW is initialized in the left central well and a series of periodic bipolar current pulses are applied during 350ns to move periodically forth and back the DW between two desired notches. The pulse characteristics are t r =t f =5ps and t z =10ns. The influence of t r and t f is discussed below. The value of t z (10ns) was chosen to correspond to the return to equilibrium time of the DW at room temperature. The t z can be reduced to 3ns for T=0K, without a change of the phase diagrams. In Fig. 2(a), several regions are visible, each region corresponding to one state of the DW oscillation. The first state which appears is the pinned state, noted state 0 in the micromagnetic simulation, and corresponds to the DW being pinned in the initial notch. After the pulse ends, the DW behaves like a damped harmonic oscillator. The state 0 is observed for all t s , when the external current j e is inferior to the depinning current 2.31 A/µm 2 . This state is also ob- served at higher currents (until 10 A/µm 2 ), when the stable time t s is low (between 0ps and 55ps). The diagram displays others bands, where the DW oscillates periodically between the same two potential wells, which can be next-neighbors (noted as band 1) or not, until the fifth state that corresponds to the periodic oscillation between the initial notch and the fourth notch to the right (band 4) hoping the three notches in between. The number of bands is given by the considered finite size of the nanostrip. The second state (next-neighbors notches noted band 1) is observed up to t s =0.96ns at j e =2.3308 A/µm 2 , while the others bands continue above t s =1ns. Thus, the DW can cross severals notches back and forth for a given current pulse characteristic. We observe that the interband transitions are characterized by an unintended state (state u), where the DW oscillation does not take place between the desired positions. This transition is more pronounced between the last two bands. The micromagnetic results, which are superimposed on the 1D results, give quantitatively the same results until the fourth state, after which a small shift appears in the t s and j e values, but the bands are qualitatively the same. The upper right region, which correspond to an unwanted state, is due to the finite size of the nanowire, here the DW is pinned at the nanowire end.

In the infinite case, the DW is initialized in the first well from the left. A phase diagram similar to the finite case is shown in Fig. 2(b). This diagram contains three more bands than the finite case, which correspond to additional states where the DW oscillates between two notches, starting from the initial one, separated by four intermediate notches (band 5) until six intermediate notches (band 7). To have a better visibility over these new bands we computed the DW dynamics for t s up to 1.5ns on 300×400 points. We observe that the band 1 and 2 exist until t s =1.045ns and t s =1.27ns respectively at j e =2.3308 A/µm 2 , while the other bands continue above 1.5ns. The interband transition (band u) is observed between the bands zero and one around j e =3 A/µm 2 and t s =250ps. For the superior bands, the interband transition at boundaries is quasi-nonexistent, which shows that the infinite case is more stable than the finite case. In both cases, the 1D model gives quantitatively the same results as the micromagnetic simulations in the three first bands and quasi-quantitatively in the others (a shift in values is visible). As for the finite case, the upper right region appears due to the finiteness of the simulated length of the nanowire, even if the end charges are suppressed. If a longer simulated window is considered, other superior bands will follow as seen for example in Fig. 5 or Fig. 6.

The influence of the pulse shape is detailed in Fig. 3 for the infinite wire as calculated with the 1D model. In panel (a), the pulse shape is symmetric as t r = t f = 5ps and t z = 10ns, while in panel (b) and (c) the pulse shape is asymmetric, t r = t f with t z kept constant at 10ns. The values of t r and t f were varied between 5ps and 300ps. In panel (b), the case with t r = 5ps and t f = 300ps is shown, while in panel (c), the one with t r = 300ps and t f = 5ps. In the three panels, the total time (t tot = t r + t s + t f ) is shown starting from 300ps, to be able to compare the diagrams evolution with the pulse shape. We observe that the first depinning current depends mainly on the rise time as depicted in panels (a) and (b), where only the fall time is varied. In these cases, the first depinning current is the same and equal to 2.31 A/µm 2 . The influence of the t f is an offset of the bands along the total time axis, therefore if t f is decreased the second band is shifted to shorter times and almost disappears from the shown phase diagram. The influence of the rise time manifests itself also as an offset of the bands to larger times, but also to larger currents, therefore a higher first depinning current equal to 2.69 A/µm 2 . The first depinning current for a total time of 0.3ns is 2.78 A/µm 2 for the symmetric pulse, raising to 4.54 A/µm 2 (panel (b)) and 5.03 A/µm 2 (panel (c)) in the asymmetric case. The micromagnetic calculation (not shown) gives similar re- sults as the ones shown in Fig. 2, meaning a small offset of the bands compared with the ones calculated with the 1D model starting from the fourth band.

The dependence of the depinning current on the rise time was deduced from the linearized equation of motion in the 1D approximation, as the force on the DW can be written as 15,18

: Ẍ = - Ẋ τ d - 1 m dE dX + β ατ d u + 1 + αβ 1 + α 2 u ( 3
)
where m = 2αSµ0Msτ d ∆γ0 the DW mass, τ d = 1+α 2 αγ0H k is the damping time, with H k the anisotropy field, ∆ the DW width and E the pinning energy. The force on the wall depends on the current and its derivative, therefore a shorter rise time increases the derivative term which leads to a decreasing of the depinning current and vice versa. For the present results, the damping time is 0.27ns (0.68ns for α = 0.02) therefore the DW is in the transient regime for the pulse duration used.

B. Temperature dependence

The temperature influences the systematic motion of the DW by modifying the DW relaxation in a potential well after an applied current pulse. The oscillations during the DW relaxation could be sustained by the thermal noise, which could lead to a jump to the wrong well while the following pulse occurs, or on the contrary, the thermal noise could counter the effect of the current pulse and the DW could stay pinned in the non-desired potential well.

To carry out this study, we computed the DW dynamics at T=293K for finite/infinite nanostrip and different pulse shapes. The results are shown in Fig. 4. In all cases, panels (a) to (d), the DW first oscillates freely (relaxation) in the presence of the thermal noise in its initial well during 10ns and afterwards a current pulse is applied to push the DW to another well (corresponding to one of the bands in Figs. 2 and3), followed by a DW relaxation during another 10ns. The damping parameter α is taken as 0.02, lower than the one at T=0K 38 , and the non-adiabatic parameter is taken as β = 2α = 0.04. The same phase diagrams, as presented in Figs. 2 and3, were recalculated with α = 0.02 and β = 0.04 at T=0K and these bands, from one to four, are indicated by dotted lines in Fig. 4 The stochastic motion of the DW was computed for a number of bands with the stochastic 1D model, and only on a certain number of points micromagnetically. The Fig. 4(a) shows the probability of the DW motion in the first four bands for a finite nanowire, when a symmetric current pulse (t r = t f = 5ps) is applied after a relaxation time of 10ns. At the end of the pulse, the DW is relaxed another 10ns before its position is considered acquired. A certain number of realizations was computed for a quarter of the phase diagram points of each band: 2700 realizations for each point shown from the first band and 500 realizations for each point of the superior bands. The maximum of probability (100%) for the precise positioning of the DW to the nearest notch is found only for 0.76% of the first band's calculated points (17 points), while on 32.18% of the points the probability is superior to 95%. The maximum of probability decreases rapidly with increasing the band number, being 98.6% (for 3 points) for the second band, 67.4% for the third band and 71% for the fourth band. These results are to be compared with panel (b), where the same pulse is applied in an infinite nanowire and the same number of realizations were computed for each band. The maximum probabilities are similar for the first two bands, for similar band point number density, indicating that at room temperature only few current pulse characteristics give 100% probability of precise positioning. The points in the first band, that correspond to 100% probability of desired mo-tion, appear for an applied current superior to 7.7 A/µm 2 and a pulse length between 100ps and 130 ps for the finite strip and superior to 8.1 A/µm 2 and a pulse length between 90ps and 120 ps for the infinite case respectively. For superior bands, starting with the fourth, the maximum of probability and band point number density are increased in the infinite nanostrip compared to the finite case, as in the latter the potential barrier is weaker for the more distant notches (see Fig. 1(c)). Micromagnetically, we computed the probability for the phase diagram points corresponding to the 100% values found with the 1D model (which appear only in the first band) on 200 realizations/point. These probabilities vary between 92% and 97%. The small difference between the probability calculations of the micromagnetic and 1D model is attributed to the small shift in the phase diagrams that was shown to exist between the two.

Fig. 4(c) and (d) show the probability of DW motion when an asymmetric current pulse is applied after 10ns initial and final relaxation. The pulse shape is t r = 5ps, t f = 300ps in panel (c) and t r = 300ps, t f = 5ps in panel (d). The maximum of probability in the first band of panel (c) is 96.77%, with only 2.31% of calculated band points having a probability superior to 95% (from 2700 realizations). For the superiors bands, the maximum of probability diminishes with only 2.93% of points having a probability superior to 90% in the second band and 2.32% of points having a probability superior to 80% in the third band. For panel (d), the maximum of probability in the first band is found to be 98.07%, with 7.69% of points having a probability superior to 95%. The maximum of probability diminishes faster in the superior bands as compared with the panel(c) results, with less points having maximum bands probability. For example, only 0.79% of points have a probability superior to 90% in the second band and 0.54% of points have a probability superior to 80% in the third band. The difference in probability between the two asymmetric pulses is due to the fact that the long rise time of the pulse (panel (d)) produces the shift up and right of the bands compared with panel (c), therefore more points with maximum probability are found in the first band (as these points are usually close to the band center) and lower points with maximum probability in the superior bands as there are less points on these bands.

These results suggest that to achieve a well defined DW positioning at room temperature by STT alone, individual pulses should be applied with very short rise and fall time.

C. Influence of α and β on the DW dynamics

The influence of the damping parameter α and of the non-adiabatic parameter β on the phase diagram for an infinite strip at T=0K is detailed in Fig. 5. The damping parameter α was varied between the 0.02 and 0.05, which correspond to the zero and room temperature values 38 . The non-adiabatic parameter was varied between zero and 2α. The case with β = 2α is shown in Fig. 2(b) and Fig. 4(b) for α = 0.05 and 0.02 respectively. The computations with β = 0 are presented in panels (a) and (b) of Fig. 5, while the ones with β = α are displayed in panels (c) and (d). We observe that for α = 0.05, when β is diminished from 2α to zero, important changes appear in the phase diagram only when β is less than α. In this case (panel (a)), the results obtained with the 1D model (colored regions) exhibit only the first band (band +1) with large pockets of negative numbered bands. In all cases, the DW is pushed initially by the current pulse in the positive direction (to the right), so a negative band express a DW position at the end of the pulse to the left of the initial notch, in the opposite direction of the STT. The micromagnetic computations confirm this behavior, which was predicted and observed before 13,20,21,27 . Due to the pinning potential, the DW deforms and can change its internal structure giving rise to a transient motion as-sociated with DW inertia. The transient DW movement is proportional to the variation of the generalized angle of the wall:

δX = - ∆ α 1 - β α δψ (4)
with ψ the azimuthal angle of the wall. For β = 0, the transient motion is increased for large DW width or small damping parameter. This effect is displayed in Fig. 5(b) for α = 0.02, as compared with panel (a), where an increased number of negative bands are visible. For the ultra-short current pulses used, which are comparable with the DW damping time τ d , the transient effects dominate at low damping parameter.

A discrepancy is found between the micromagnetic results (symbols) and the 1D results in the upper right quadrant (high current, longer pulse) for the phase diagrams with β = 0. This discrepancy is due to the large angle variation at large current and longer pulse, that leads to the transformation of the transverse DW and the creation of an antivortex close to the initial notches before depinning 39 (see figure and movie in Ref.36). DW velocity boosting was predicted through antivortex generation at a singular notch in a nanostrip 40 . The antivortex disappear quickly after the end of the current pulse, and in certain cases can reverse the orientation of the magnetization at the center of the transverse DW leading to DW motion in opposite direction. The antivortex does not appear in all the computed micromagnetic results in the upper right quadrant of these phase diagrams. In the 1D model, the antivortex nucleation is not taken into account.

As is obvious from Eq.( 4), for β = α the transient motion is somehow blocked and the DW travels rigidly. This seems to be the case for β = α = 0.05, as computed in the phase diagram shown in Fig. 5(c). However, when β = α = 0.02, the negative bands are still present at low currents (panel (d)) without the apparition of an antivortex. This effect was verified micromagnetically on a number of points (empty symbols), which compare very well with the 1D results. The pocket form of the -1 band is respected with a small shift, even a -2 band was observed micromagnetically corresponding to the small -2 pocket inside the -1 band. The DW motion in the case β = α can be well explained analyzing the 1D equations of motion of the DW:

(1 + α 2 ) Ẋ = - αγ∆ 2µ 0 M s S ∂E ∂X + γ∆ 2 H k sin 2ψ + (1 + αβ)u (1 + α 2 ) ψ = - γ 2µ 0 M s S ∂E ∂X - γα 2 H k sin 2ψ + β -α ∆ u (5)
where S is the section of the wire, γ the gyromagnetic ratio, and H k the DW demagnetizing field. E is the pinning potential energy which is assumed parabolic inside the notch. The DW width variation is given by ∆(t) = ∆[Ψ(t)] = π 2A µ0MS 2 sin 2 ψ+µ0MS H k . From the above equations, one can notice that the azimuthal DW angle ψ and DW position X depend on the pinning potential (restoring force) created by the symmetric notches. When a current pulse is applied, the DW is first compressed and distorts on the potential barrier moving in the direction of the STT, while the azimuthal angle decreases in some cases below -10 • (dotted line), as is displayed in Fig. 5(h) (videos and additional figures in Ref.36). Initially, the STT pushes the DW in the positive direction, resulting in a positive DW velocity and a negative restoring force (negative DW angle). If the DW does not have enough velocity to surpass the potential barrier in this direction, it goes down the potential well towards the center of the notch and the restoring force still stays negative, while the velocity and angle continue to decrease with the velocity becoming negative. When the DW starts to mount the potential well in the other direction, the restoring force becomes positive so the velocity and DW angle start to increase (Fig. 5(f)). When the current pulse ends, the velocity decreases abruptly (increases in absolute value) and ψ increases lightly do to the short fall time. At this moment, the DW position inside the potential well is important as it imposes a restoring force in the direction of movement or in the opposite direction. However, the velocity and the DW angle are equally important for the jump to the previous notches to happen 36 The jump occurs therefore due to a combination of factors, even if the restoring force is positive or negative. The presence of the notch pinning potential induces the DW distortion and therefore the transient motion, as for the same pulse parameters no observable angle variation is determined for a perfect strip. As the variation of the DW angle is directly related to the variation in position, the automotion is pos-sible in this case triggered by the pinning induced DW distortion, restoring potential force and smallness of the damping parameter.

In Fig. 5(g), several angular variations are presented corresponding to the -1 and -2 bands from panel (b). We observe that the amplitude of angular variation is directly proportional with the spatial displacement. When an antivortex appears, the DW angle rotates out of plane and the DW position does not correspond anymore to the 1D results and the motion opposite to the STT can be completely blocked. In general, we observe the automotion in all the cases when the DW angle increases above 10 • (in absolute values) during or after the current pulse and the maximum DW velocity is close or superior to 400m/s (details in Ref. 36). Exactly at the boundary between the -1 band and the zero band (pinned state), a small increase in the azimuthal angle of 0.2 • and of the DW velocity by 5m/s at the end of the current pulse, between two points in different bands, is enough to promote a DW jump to a previous notch.

The transient effects also appear when β > α, as depicted in Fig. 4 for β = 2α = 0.04. A low value of the damping parameter α is required to obtain observable consequences. A particularity of the case β > α is that the transient effects oppose or amplify the STT, as negative bands are determined and a re-entrant transition is seen at higher currents and pulse length in panels (b) and (d). The transient effects depend on the pulse shape 36 , as for the asymmetric pulse with t r = 5 ps, t f = 300 ps, the band -1 is barely visible and no re-entrant transition of the bands is seen, while the -1 band increases when the pulse is symmetric and continues to increase, with even a second -1 band appearing at larger t s for the asymmetric pulse with t r = 300 ps, t f = 5 ps. The fall time t f plays an important role in the value of the DW velocity and DW angle at the end of the current pulse, as a short t f leads to a very high DW velocity and a higher DW angle at the end of the pulse increasing the impact of transient effects and inducing a DW depinning. Contrary to earlier beliefs 13 , the DW depinning does not necessarily result from a large DW angle or the DW position at the end of the pulse. A maximum DW angle and DW velocity large enough during the current pulse suffice to ensure for example the jump to the previous notch, even if the angle is not that large (and at its maximum) at the pulse end, as observed in the case of the asymmetric pulse with t r = 5 ps, t f = 300 ps (Ref.36). This is purely an transient effect due to a combined action of the pinning potential, low α and pulse shape.

The influence of the rise time t r on the phase diagrams is presented in Fig. 6 for t s = t f = 5 ps and several parameters α and β. The rise time is varied between 0 and 1.5 ns (case of a very asymmetric pulse). A first observation is that the first depinning current is increased as the force term that depends on the current derivative in Eq.( 3) diminishes. The first depinning current actually oscillates with t r due to resonants effects, as the resonant frequency of the potential well is around 1.75 GHz. This resonant effect are more important for the bands with negative numbers for low α, as is depicted in panels (a) and (b) for α = 0.02 and β = 0.4 or 0.2 respectively. The 1D model gives good quantitative results as compared with the micromagnetic calculations, as shown in panel (a), although a small shift is again found at high currents and more unwanted states. When β ≤ α, the positive bands are shifted to very high current, with only negative bands remaining for β = 0 (image not shown). The phase diagram for α = 0.05 and β = 0.1 is also shown in panel (c), that still does not presents negative bands but an the same oscillation of the first depinning current and of the interband transitions.

The influence on the phase diagram is less dramatic when varying the fall time t f (images shown in Ref.36). For α = 0.02 and β > α no negative bands appear and the positive bands are shifted to larger currents. When β ≤ α = 0.02, only a small pocket of the -1 band appears and the positive bands are shifted as compared with Fig. 5(d).

IV. DISCUSSION AND CONCLUSION

The presence of artificial constrictions in a nanostrip influences drastically the movement of a magnetic DW. When ultra-short current pulses are applied, the DW can exhibit an important distortion at the notch depending on the pulse characteristics. Thereby, the DW displays inertia-like effects, which can have dramatic consequences on its transient displacement. These effects depend largely on the damping parameter α and on the non-adiabatic parameter β. For β < α, these effects generally oppose the STT effect after the pulse end and DW motion in the direction of the electric current is possible. If β > α, these effects oppose or amplify the STT effect and jumps to the left or the right notches are possible after the end of the pulse. The transient effects in this case depend on the pulse characteristics. This could constitute another way of experimentally compare the two parameters α and β for a ferromagnetic material.

At room temperature, the jump probability to the desired notches decreases with increasing band number, each band number corresponding to a positioning to desired notches in the direction of the STT. Maximum positioning probability is reached only for very short rise and fall time to the nearest neighbors notches only. Therefore, to shift reliably the DWs between notches, current pulses corresponding to displacement from one symmetric notch to the closest neighbors should be used. The shape of the current pulse influences the depinning current and shifts the bands. The phase diagram for the case of two domain walls situated at different symmetric notches (images not shown), that are displaced by the same current pulse in the same direction, is very similar with the ones presented in Sec.III, but the bands are narrower and the interband depinning is larger between the first bands.

The main drawback of the classical DW displacement under STT alone compared with more exotic torques (of spin-orbit origin) is the high current values necessary. The current induces Joule heating in the nanostrip, that can largely increase the temperature and could even destroy the ferromagnetic state. The increase in temperature is even more important at the constrictions in the nanostrip. Several theoretical studies were dedicated to Joule heating in nanowires [41][42][43][44] , usually considering a standard Py nanostrip on a Si/SiO 2 substrate. We evaluated the temperature increase for our Ni strips on different substrates like pure Si, SiO 2 or Ni 3 Si 4 membranes for a current pulse length of 1ns. On pure Si, considering an infinite 3D substrate 41 , the temperature increase is negligible being of 4K for j = 5 A/µm 2 (17K for j = 10 A/µm 2 ). However, on SiO 2 substrate of 300nm thickness 42 , the temperature increase is larger being 26K for j = 5 A/µm 2 (103K for j = 10 A/µm 2 ). If 100nm thick Ni 3 Si 4 membranes are used 15,42 , the temperature increase is of 41K for j = 5 A/µm 2 (163K for j = 10 A/µm 2 ). The variation in temperature depends on the material parameters of the substrate (like thermal conductivity) and on the conductivity of the nanostrip. In the above estimations, the bulk Ni room temperature conductivity was used (σ -1 = 7.3 µΩ•cm) 45 . If the dimensions are reduced to nanometers 46 , the conductivity of Ni can vary drastically (a factor four) and the temperature increase can be more important, but in the main time the length of the current pulse that amounts to the maximum probability at room temperature is around 0.1ns which reduces significantly the temperature increase.

In conclusion, systematic DW motion between precise artificial pinning constriction by very short current pulses is possible at room temperature in a classical ferromagnet. The constrictions induce DW distortions and important transient effects can be observed. Depending on the ratio β/α, the inertia-like effect can oppose or amplify the STT effect on the DW motion after the pulse end. As the value of β is still under debate, this could constitute another way of determining its relative value. Our results open the path to DW motion in both directions by unipolar current pulses.

  FIG. 1. (Color online) (a) Simulated structure: planar nanowire with ten symmetric double notches. The equilibrium position of a pinned DW is shown in the finite case. (b) Definition of current pulse with its temporal characteristics. Two successive current pulses with opposite polarity are shown. The normalized potential pinning energy for finite and infinite case as determined by micromagnetic simulations (symbols) are shown in (c) and (d) respectively. The line is a fit as described in the text.

FIG

  FIG. 2. (Color online) Phase diagram for a DW at T=0K in a finite (a) and infinite (b) nanostrip in the parameter space stable-time -current amplitude with tr=t f =5ps and tz=10ns. The total time of the periodic pulses is 350ns. The micromagnetic results (scattered symbols) are compared to the 1D model (colored regions). The diagrams show only a few bands due to the finite size of the wire or due the number of notches used. The upper right region is due to the finite size of the nanowire or of the simulated window (infinite case).

FIG. 3 .

 3 FIG. 3. (Color online) Influence of the pulse shape on the phase diagram for a DW at T=0K in an infinite nanostrip. The parameter space is the total time (tr + ts+ t f ) vs. current amplitude. (a) Symmetric pulse with tr = t f = 5 ps and tz = 10 ns, Asymmetric pulse with (b) tr = 5 ps, t f = 300 ps and tz = 10 ns, and (c) tr = 300 ps, t f = 5 ps and tz = 10 ns. The upper right region is due to the finite size of the simulated window.

  (a) for the finite nanostrip, while the bands from one to seven are indicated in panels (b) to (d) for the infinite nanostrip. Starting from the fourth band, the shape of the bands changes showing a reentrant transition (except panel (c)), and the phase diagrams from panels (b) to (d) show band pockets to the left corresponding to negative DW displacement of one notch (noted as band -1) even though the STT pushes the DW in the positive direction. These features are discussed in Sec. III C.

FIG. 4 .

 4 FIG. 4. (Color online) Probability of DW motion in different bands for the finite strip (a) and the infinite strip (b)-(d) at T=293K. (a) and (b) A symmetric pulse is applied with the characteristics tr = t f = 5 ps, after an initial and final tz of 10 ns. (c) An asymmetric pulse is applied with pulse shape tr = 5 ps, t f = 300 ps, after an initial and final tz of 10 ns. (d) An asymmetric pulse is applied with pulse shape tr = 300 ps, t f = 5 ps, after an initial and final tz of 10 ns. The damping parameter is taken as α = 0.02 and β = 2α. The band pockets, which appear in the panels (b) to (d) on the left, correspond to the band -1. The dotted lines are guide to the eyes and represent the bands edges.

  FIG. 5. (Color online) Influence of the damping parameter α and non-adiabatic parameter β on the phase diagram for an infinite strip at T=0K: (a) α = 0.05 and β = 0, (b) α = 0.02 and β = 0, (c) α = β = 0.05, (d) α = β = 0.02. A sequence of symmetric pulses with same characteristics as in Fig. 2 are applied. The scattered symbols represent the micromagnetic results, while the colored regions are calculations with the 1D model. Trajectories in the phase space (X,ψ) corresponding to the -1 band in panels (b) respectively (d): (e) ts = 250 ps, je = 3.4 A/µm 2 , α = 0.02, β = 0 and (f) ts = 300 ps, je = 3.6 A/µm 2 , α = β = 0.02. The scattered symbols and the full line represent the micromagnetic and the 1D results respectively. The dotted part of the full line indicates the applied pulse duration. (g) DW angle variation for α = 0.02, β = 0 for several points in the panel (b). (h) Comparison of the DW angular variation for α = β = 0.02 and ts = 300 ps, je = 3.6 A/µm 2 for the infinite strip with notches (band -1 in panel (d)) and strip without notches.

FIG. 6 .

 6 FIG. 6. (Color online) Influence of the pulse raise time tr on the phase diagram for a DW at T=0K in an infinite nanostrip for different parameters α and β. The parameter space is the raise time vs. current amplitude. In all cases, ts = t f = 5 ps and tz = 10 ns. (a) α = 0.02 and β = 0.04, (b) α = β = 0.02, and (c) α = 0.05 and β = 0.1. In (a) the micromagnetic results (scattered symbols) are compared to the 1D model (colored regions), while in (b) and (c) only 1D model results are shown. At high currents, the micromagnetic results give several unintended states (the empty scattered symbol region to the right of (a)).


Tatiana Prosvirnova

Michel Batteux

Pierre-Antoine Brameret

Abraham Cherfi

Thomas Friedlhuber

Jean-Marc Roussel

Antoine Rauzy

Thomas Friedl- Huber

The Altarica

The AltaRica 3.0 Project for Model-Based Safety Assessment

Keywords:

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

The AltaRica 3.0 Project for Model-Based Safety Assessment

Tatiana Prosvirnova * Michel Batteux * Pierre-Antoine Brameret * * Abraham Cherfi * Thomas Friedlhuber * Jean-Marc Roussel * * Antoine Rauzy *

INTRODUCTION

The Model-Based approach for safety and reliability analysis is gradually wining the trust of engineers but is still an active domain of research. Safety engineers master "traditional" risk modeling formalisms, such as "Failure Mode, Effects and Criticality Analysis" (FMECA), Fault Trees (FT), Event Trees (ET), Markov Processes. Efficient algorithms and tools are available. However, despite of their qualities, these formalisms share a major drawback: models are far from the specifications of the systems under study. As a consequence, models are hard to design and to maintain throughout the life cycle of systems. A small change in the specifications may require revisiting completely safety models, which is both resource consuming and error prone.

The high-level modeling language AltaRica Data-Flow [START_REF] Rauzy | Mode automata and their compilation into fault trees[END_REF][START_REF] Boiteau | The altarica data-flow language in use: Assessment of production availability of a multistates system[END_REF] has been created to tackle this problem. AltaRica Data-Flow models are made of hierarchies of reusable components. Graphical representations are associated with components, making models visually very close to Process and Instrumentation Diagrams. AltaRica Data-Flow is at the core of several Integrated Modeling and Simulation Environments: Cecilia OCAS (Dassault Aviation), Simfia (EADS Apsys), and Safety Designer (Dassault Systèmes). Successful industrial experiments were held using AltaRica Data-Flow (e.g. certification of the flight control system of the aircraft Falcon 7X) [START_REF] Bernard | Experiments in model-based safety analysis: flight controls[END_REF][START_REF] Bieber | Integration of formal fault analysis in assert: Case studies and lessons learnt[END_REF]. In a word, AltaRica Data-Flow has reached an industrial maturity. However, more than ten years of experience showed that both the language and the assessment tools can be improved. AltaRica 3.0 is an entirely new version of the language. Its underlying mathematical model -Guarded Transition Systems [START_REF] Rauzy | Guarded transition systems: a new states/events formalism for reliability studies[END_REF][START_REF] Prosvirnova | Guarded transition systems: Pivot modelling formalism for safety analysis[END_REF] -makes it possible to design acausal components and to handle looped systems. The development of a complete set of freeware authoring and assessment tools is planned, so to make them available to a wide audience.

The aim of this article is to present the AltaRica 3.0 project. It is organized as follows. Section 2 gives an overview of the project. Section 3 introduces the new version of the language. Section 4 presents Guarded Transition Systems. Section 5 presents the current state of the development of authoring and assessment tools. Section 6 presents related works. Finally Section 7 concludes the article and outlines directions for future works.

OVERVIEW OF THE PROJECT

The objective of the AltaRica 3.0 project is to propose a set of modeling and assessment tools to perform preliminary safety analyses. Figure 1 presents the overview of the project.

AltaRica 3.0 is in the heart of the project. It significantly increases the expressive power of AltaRica Data-Flow without decreasing the efficiency of assessment algorithms. Models are compiled into a low level formalism: Guarded Transition Systems (GTS). GTS is a states/transitions Fig. 1. Overview of the AltaRica 3.0 project formalism generalizing classical safety formalisms, such as Reliability Block Diagrams and Markov Chains. It is a pivot formalism for Safety Analyses: other safety models can be compiled into GTS to take benefits from assessment tools. The assessment tools for GTS already include a Fault Tree compiler to perform Fault Tree Analysis (FTA), a Markov Chain generator, a stochastic and a stepwise simulators. Other tools are under specification or implementation: a model-checker and a reliability allocation module. These tools will be distributed under a free license in order to make them available to a wide audience, especially in the academic community. They enable users to perform virtual experiments on systems, to compute reliability indicators and, also, to perform cross check calculations.

3. ALTARICA 3.0 MODELING LANGUAGE AltaRica is an event based modeling language. The state of the system is described by means of variables. The system changes its state when, and only when, an event occurs. The occurrence of an event updates the value of variables. Events can be associated with deterministic or stochastic delays so to obtain (stochastic) timed models. Models of components can be assembled into hierarchies, their inputs and outputs can be connected and their transitions can be synchronized.

Two main variants of AltaRica have been designed so far. These two variants differ essentially with respect to the way variables are updated after each transition firing. In the first version, variables are updated by solving constraints [START_REF] Point | AltaRica: Constraint automata as a description language[END_REF][START_REF] Arnold | The altarica language and its semantics[END_REF]. This mechanism, although very powerful, is too resource consuming for industrial scale applications. To be able to assess industrial scale models, in the second version, AltaRica Data-Flow [START_REF] Rauzy | Mode automata and their compilation into fault trees[END_REF][START_REF] Boiteau | The altarica data-flow language in use: Assessment of production availability of a multistates system[END_REF], variables are updated by propagating values in a fixed order. This order is determined at compile time, which imposes strong constraints on the way assertions are written. However, some problems stay: located synchronizations cannot be captured, bidirectional flows, circulating through a network, cannot be modeled in a natural way, and looped systems remains difficult to model.

The new version is currently under specification. It improves AltaRica Data-Flow into two directions:

(1) Its semantic is based on the new underlying mathematical model: Guarded Transition Systems (GTS).

(2) It provides new constructs to structure models.

The new underlying formalism, i.e. GTS, makes it possible to handle systems with instant loops and to define acausal components, i.e. components for which the input and output flows are decided at run time. It is much easier to model systems with bidirectional flows, such as electrical systems.

AltaRica 3.0 is a prototype oriented modeling language, see e.g. [START_REF] Noble | Prototype-Based Programming: Concepts, Languages and Applications[END_REF] for a discussion on objects versus prototypes. Prototype orientation makes it possible to separate the knowledge into two distinct spaces: the stabilized knowledge, incorporated into libraries of onthe-shelf modeling components; the sandbox in which the system under study is modeled. In the sandbox, many components are unique and some others are instances of reusable components. With prototype-orientation, models can be reused in two ways: at component level by instantiating off-the-shelf components; at system level by cloning and modifying a model designed for a previous project.

Example

Consider a part of a system, named TwoValves (see Figure 2), composed of two valves connected in series. rightFlow and leftFlow are bidirectional flows circulating through this sub-system. The event ccf represents the common cause failure of the two valves. It is expressed by means of the synchronization of the events v1.failure and v2.failure. The event ccf may occur when at least one of two valves is working. Operator "!"/"?" means that the event is mandatory/optional for the synchronization.

To be able to describe the sub-system TwoValves with the previous version of the language, the class Valve should include four flow variables: two inputs and two outputs for the left and right hand sides (i.e. inputRightFlow, input-LeftFlow for inputs and outputRightFlow, outputLeftFlow for outputs). Thus, the instruction of the assertion should be replaced by a block connecting inputs and outputs: When considering a complex system with a significant number of similar components, connecting them together may be a tedious and error prone task.

GUARDED TRANSITION SYSTEMS

Before any assessment, AltaRica models are "flattened" (i.e. reduced to a single class without sub-classes). This "flattened" class produces a GTS model. The new semantics of instructions [START_REF] Prosvirnova | Guarded transition systems: Pivot modelling formalism for safety analysis[END_REF]) makes it possible to represent components with bidirectional flows.

Definition

A Guarded Transition Systems is formally a quintuple V, E, T, A, ι , where:

• V = S F is a set of variables, divided into disjoint sets S of state variables and F of flow variables. • E is a set of symbols, called events. • T is a set of transitions.

• A is an assertion (i.e. an instruction built over V ).

• ι is the initial (or default) assignment of variables of V .

GTS is thus a states/transitions formalism where states are implicit, i.e. given by variables assignments σ. A transition is a triple e, G, P , also denoted e : G → P , where e ∈ E is an event, G is a guard, i.e. a Boolean formula built over V , and P is an instruction built over V , also called an action or a post-condition. A transition e : G → P is said fireable in a given state σ if its guard G is satisfied in this state.

Instructions

Both assertions and actions of transitions are described by means of instructions. There are basically four types of instructions:

• The empty instruction noted skip.

• The assignment v := E, where v is a variable and E is an expression built over variables from V . • The conditional assignment if C then I, where C is a Boolean expression and I is an instruction. • The block {I 1 , . . . , I n }, where I 1 , . . . , I n are instructions.

State variables can be presented as the left member of an assignment only in the action of a transition. Flow variables can be presented as the left member of an assignment only in the assertion. Instructions are interpreted in a slightly different way depending they are used in the actions or in the assertion. Let σ be the variable assignment before the firing of the transition e : G → P .

Applying the instruction P to the variable assignment σ consists in calculating a new variable assignment τ . The right hand side of assignments and conditional expressions are evaluated in the context of σ. Thus, the result does not depend on the order in which instructions of a block are applied. In other words, instructions of a block are applied in parallel. Let denote by U pdate(P, σ) the variable assignment τ resulting from the application of the instruction P to σ. In case when the same state variable is affected several times and receives different values, an error is raised.

Let A be the assertion and τ the variable assignment obtained after the application of the action of a transition. Applying A consists in calculating a new variable assignment (of flow variables) π as follows. We start by setting all state variables in π to their values in τ : ∀v ∈ Sπ(v) = τ (v). Let D be a set of unevaluated flow variables, we start with D = F . Then,

• If A is an empty instruction, then π is left unchanged. • If A is an assignment v := E, then if π(E) can be evaluated in π, i.e. all variables of E have a value in π, then π(v) is set to π(E) and v is removed from D.
An error is raised if the value of v has been already modified and is different from the calculated one. • If A is a conditional assignment if C then I and π(C) can be evaluated in π and is true, then the instruction I is applied to π. Otherwise, π is left unchanged. • If A is a block of instructions {I 1 , . . . , I n } then instructions I 1 , . . . , I n are repeatedly applied to π until there is no more possibility to assign a flow variable.

If after applying A to π there are unevaluated variables in D, then all these variables are set to their default values ∀v ∈ Dπ(v) = reset(v) and A is applied to π in order to verify that all assignments are satisfied. If that is not true an error is raised. Let denote by P ropagate(A, σ) the variable assignment resulting from the application of the instruction A to σ.

Reachability graph

Guarded Transition Systems are implicit representations of labeled Kripke structures, i.e. of graphs whose nodes are labeled by variable assignments and whose edges are labeled by events. This graph is constructed in the following way.

Assume that σ is the variable assignment just before the firing of a transition. Then, the firing of the transition transforms σ into the assignment F ire(e : G → P, A, σ) defined as follows:

F ire(e : G → P, A, σ) = P ropagate(A, U pdate(P, σ))

The so-called reachability graph Γ = (Σ, Θ) is the smallest Kripke structure verifying:

(1) σ 0 = P ropagate(A, ι, ι) ∈ Σ. σ 0 is the initial state of the Kripke structure.

(2) If σ ∈ Σ and ∃t = e, G, P ∈ T , such that the guard G is verified in σ then the state τ = F ire(P, A, ι, σ) ∈ Σ and the transition (σ, e, τ ) ∈ Θ,

In special cases, the calculation of Γ = (Σ, Θ) may raise errors, when GTS are not well designed. Currently assessment tools detect modeling errors at the execution of the model.

Timed/Stochastic Guarded Transition Systems

A probabilistic time structure can be put on top of a Guarded Transition System so to get timed/stochastic models. The idea is to associate to each event:

• A delay which can be deterministic or stochastic and may depend on the state. When a transition labeled with the event becomes fireable at time t, a delay d is calculated and the transition is actually fired at time t + d if it stays fireable from t to t + d. • a weight, called expectation, used to determine the probability that the transition is fired in case of several transitions are fireable at the same date.

In the example given in Section 3, the transition f ailure is a timed stochastic one, obeying to the exponential distribution with a failure rate 0.0005.

Example

The GTS model of the sub-system TwoValves, introduced in Section 3 and pictured Figure 2, is as follows: This GTS is obtained by flattening the corresponding AltaRica 3.0 model. All variables, parameters and events are prefixed by the name of the instantiated class. The synchronized transition ccf is flattened into a simple transition. Each bidirectional assignment x :=: y is transformed into two simple assignments x := y and y := x. A compiler automatically transforms AltaRica 3.0 models to GTS.

ASSESSMENT TOOLS

As shown Figure 1, the AltaRica 3.0 project provides a set of assessment tools. Some of them are already available and others are under specification or implementation. In the sequel, we will present first versions of four of them. All assessment tools take a GTS model as input.

Compiler to Fault Trees

Fault trees are widely used to perform Safety Analyses and some regulation authorities require to use them to support the certification process. From a GTS model, it is possible to generate corresponding Fault Trees (FT), i.e. to transform a states/transitions model into a set of Boolean formulae. It may seem inefficient at a first glance to use a states/transitions formalism to end up with a Fault Tree. However in practice, it is of great interest. It is easier and less time consuming to automatically generate FT from high-level models rather than create them from scratch. High-level models improves greatly the design, the sharing and the maintenance of models. The algorithm of compilation to Fault Trees for AltaRica Data-Flow, described in [START_REF] Rauzy | Mode automata and their compilation into fault trees[END_REF], can be extended to a general case of GTS. As illustrates Figure 3, this algorithm includes 3 steps:

(1) The GTS model is partitioned into independent GTS and an independent assertion. (2) Reachability graphs of each independent GTS are calculated.

(3) Reachability graphs and the assertion are separately compiled into Boolean equations.

Partitioning is a key point of the algorithm that ensures its efficiency. In practice, components of a system fail in general in a relatively independent way. In that case a partitioning is possible. If the GTS is combinatorial, its compilation to Fault Trees is efficient and does not lose information.

The generated Fault Tree could be assessed with any Fault Tree calculation engine supporting Open-PSA format [START_REF] Hibti | Overview of the open psa platform[END_REF]. For example, XFTA [START_REF] Rauzy | Anatomy of an efficient fault tree assessment engine[END_REF] can be used to calculate minimal cutsets, events probabilities, importance factors, etc.

Stepwise simulator

The stepwise simulator enables to perform an interactive step by step simulation of a GTS model. This interactive tool can be very useful to debug models, to play different failure scenarios, etc. The stepwise simulator can be coupled with a graphical simulator as illustrated in [START_REF] Perrot | Expériences de couplages de modèles AltaRica avec des interfaces métiers[END_REF]. Graphical simulation of models can be used to perform virtual experiments on systems, via models, helping to better understand the system behavior.

The first version of the interactive stepwise simulator supports commands to display information about the simulated model (all variables and observers and their values, all transitions fireable in the current state, execution history, etc.) and to perform "actions" (to fire a transition, to cancel the last action, to restart the simulation, etc.).

Stochastic simulator

Stochastic simulation is a basic tool for safety analyses of systems. It provides fine results to calculate reliability indicators, even with complex systems. The principle is to run many pseudo-random histories of the behavior of the system and to make statistics on them. The stochastic simulator of the AltaRica 3.0 project has been designed by taking into account original features.

A generic mechanism is implemented to define specific delays. In general, simulation tools offer a lot of stochastic delays. Users have to choose the best matching one and fill out parameters. A few ones are widely used for safety/reliability analyses (e.g. exponential, Weibull, etc.), but others are generally difficult to use: difficult to understand or to fill out parameters. Due to this fact, the stochastic simulator implements the widely used delay functions, previously mentioned. For other specific ones, a generic mechanism allows to describe them by means of a set of points interpolated in a triangular way.

Compilation techniques are used to reduce computation time of simulations. In fact, the only limit of stochastic simulation is the number of histories, and their length, necessary to stabilize the measures. But with current computing technology, it is relatively easy to perform up to several million histories (of reasonable length). Beyond, the computation time gets an issue and with that respect, compilation technique may be of a great help. Thus the considered GTS model is translated into C++ classes, representing a set of instructions for the simulator engine.

Then, they are compiled, with this simulator engine, to constitute the stochastic simulator of the system to study.

Markov chains Generator

The different reachable states of a system can be built from its GTS model. The state space of the system can be transformed into a Markov chain to compute the sojourn times / steady state probabilities of the different states of the system. The Markov chain formalism can be efficiently assessed by numerical methods such as developed in [START_REF] Rauzy | An experimental study on iterative methods to compute transient solutions of large markov models[END_REF]. It is a very straightforward method to compute mean values of the observers defined in the AltaRica 3.0 model. For instance, the availability of the system can be build from an observer which takes value 1 when system is working and 0 otherwise. Figure 4 illustrates the process. This computation method has two limits:

(1) the Markov hypothesis must hold for the system, ie. the transition rates between states must be constant, (2) the size of the Markov model is subject to explode exponentially.

The hypothesis ( 2) is difficult to overcome. A method has been developed to limit the construction of the state space. It consists in selecting the most influential states toward the given reward to assess, thus giving accurate results while drastically limiting the size of the state space. The influence of the hypothesis (1) really depends on the system modeled. It is usually valid for safety assessment, and it gives good results to quickly assess an AltaRica model with the Markov chain generator. It is particularly valid while designing the system architecture, when engineers need to assess and compare several models.

RELATED WORKS

Two approaches for (high-level) Model-Based Safety Assessment can be found in literature. The first one consists in creating extensions of high-level modeling languages used in other domains. The second approach consists in defining domain specific languages, dedicated to Safety Analyses.

In the first category, we can find [START_REF] Feiler | Dependability modeling with the architecture analysis & design language (aadl)[END_REF] who added an Error Model annex to the modeling formalism for embedded real-time systems AADL. In the same way the HiP-HOPS workbench [START_REF] Pasquini | Hierarchically performed hazard origin and propagation studies[END_REF] enables to add reliability data to models imported from different modeling tools: Matlab/SIMULINK, Eclipse-based UML tools, etc., and then to automatically generate Fault Trees and FMEA tables.

Similarly, translations have been defined from specialized UML/SysML models to Fault Trees or Petri nets [START_REF] Bernardi | From uml sequence diagrams and statecharts to analyzable petri net models[END_REF]. In [START_REF] David | Reliability study of complex physical systems using sysml[END_REF], the functional design phase, using SysML, is combined with commonly used reliability techniques (i.e. FMEA and construction of Al-taRica Data-Flow models).

In the second category, we can find Figaro [START_REF] Bouissou | Knowledge modelling and reliability processing: presentation of the figaro modelling language and associated tools[END_REF], developed by EDF R&D. It is a textual modeling language dedicated to dependability assessment of complex systems. It combines object-orientated languages features, such as inheritance, and first order production rules (interaction and occurrence rules). It is used as a description language to create knowledge bases for the workbench KB3 [START_REF] Bouissou | Automated dependability analysis of complex systems with the kb3 workbench: the experience of edf r&d[END_REF], to automatically perform systems dependability assessment: Monte-Carlo simulation, Markov Chain generation, quantification and generation of critical sequences, etc.

In between the two categories, we can find SAML (Safety Analysis Modeling Language) [START_REF] Güdemann | A framework for qualitative and quantitative model-based safety analysis[END_REF], which is a synchronous language. It expresses a model in terms of finite stochastic state automata and its semantics is defined as Markov decision process. S3E is a design and verification environment focused on SAML models. It provides a stochastic simulator and translators to the input languages of the model-checkers PRISM and NuSMV.

CONCLUSION

In this article, we presented the AltaRica 3.0 Project, which aims to develop a complete set of tools, under a free license, to create, edit, check, simulate, debug and assess models. The new version of AltaRica modeling language increases the expressive power of the previous one without decreasing the efficiency of assessment algorithms. It makes it possible to model looped systems and components with bidirectional flows. First versions of assessment tools include: a compiler to Fault Trees, a compiler to Markov chains, a stepwise and a stochastic simulators.

Forthcoming works will focus, amongst others, on the implementation of other assessment tools (i.e. a modelchecker and a reliability allocation module) and the development of modeling methodology and pedagogical materials, including benchmark tests.

Fig. 2 .

 2 Fig. 2. The component TwoValves. A valve is represented by the class Valve. This valve has two states: closed or not. When the valve is opened, the right flow and the left flow are the same. In functional mode, the states change according to the command events open and close. For this class of valves, we consider only a failure behavior: the valve is blocked in the current state. The AltaRica 3.0 model of a valve is the following: class Valve Boolean closed (init = true), blocked (init = false); Real rightFlow (reset = 0), leftFlow (reset = 0); event open, close, failure(delay = exponential(0.0005)); transition open: closed and not blocked -> closed := false; close: not closed and not blocked -> closed := true; failure: not blocked -> blocked := true; assertion if not closed then leftFlow :=: rightFlow; end The class Valve contains two state variables closed and blocked ; and two flow variables rightFlow and leftFlow. Events open, close and failure are declared and then used to define transitions. The behavior of a valve is thus defined by transitions (to update its state) and the assertion (to propagate flows). The operator ":=:" expresses a bidirectional flow circulating through the valve. The sub-system TwoValves is represented by the class TwoValves, containing two instances of the class Valve. rightFlow and leftFlow are bidirectional flows circulating through the sub-system when both valves are open. Each valve can fail individually. Both valves can fail simultaneously due to a common cause failure. The corresponding AltaRica 3.0 model is given below:

  if not closed then {outputRightFlow := inputLeftFlow; outputLeftFlow := inputRightFlow;} Finally, the connection between the two valves should be done by connecting inputRightFlow of v1 to outputLeft-Flow of v2 and inputLeftFlow of v2 to outputRightFlow of v1 :v1.outputRightFlow := v2.inputLeftFlow; v2.outputLeftFlow := v1.inputRightFlow;

Fig. 3 .

 3 Fig. 3. Algorithm of compilation of GTS to FT

Fig. 4 .

 4 Fig. 4. Illustration of the Markov chain generation process