M Lazar
email: lazar@cegely.insa-lyon.fr

G Cardinali

C Raynaud

A Poggi

D Planson

R Nipoti

J P Chante

The role of the ion implanted emitter state on 6H-SiC power diodes behavior. A statistical study

Keywords: ion implantation, annealing, bipolar diodes, JTE

Two device runs of high voltage bipolar 6H-SiC p + n diodes were fabricated. Emitter and JTE (Junction Termination Extension) periphery were both realized by Al ion implantation, at room temperature (RT) for JTE, and both RT and 300°C for emitters. All diodes Current-Voltage characteristics were systematically measured in forward (0 ® 5V) and reverse bias (0 ® -50V). Breakdown voltages were also measured on a part of diodes. The emitter quality after implantation and annealing seems to be important for the diode behavior in forward and in reverse polarization also. For the 300°C implanted diodes higher current densities (200 A.cm -2 at 5V) are obtained in forward bias, with a more homogeneous distribution, showing better doping activation and layer quality. Compared to the RT implanted diodes, a less number of 300°C implanted diodes with high leakage currents is obtained. Those with the smallest emitter and the largest JTE length sustain more than 1100V. The breakdown voltage value (VBR) is close to the calculated value and independent on the testing ambient. This indicates a breakdown in the bulk volume, confirmed by a lower VBR for the RT implanted diodes. This lower VBR value is due to the presence of an amorphous layer after implantation and thus residual defects at the emitter-epilayer junction interface.

Introduction

Realization of silicon carbide (SiC) power devices and integrated power systems, requires local doping. Due to the very low diffusion coefficients of the dopants in SiC, ion implantation is needed, followed by a high temperature annealing. To avoid amorphous layer creation and thus the presence of residual defects, ion implantation has to be performed at high temperature (300-600°C) which is an impediment in industrial SiC device development, utilizing silicon standard equipment.

In this paper performances of two fabricated runs of 6H-SiC bipolar power p + n diodes implanted at 300°C and room temperature (RT) are compared.

Experiment

Two commercial 6H-SiC n-type epilayers (~10 16 cm -3 , 10 µm) grown on n-type substrates were Al ion implanted, to realize the p-type emitter and the Junction Termination Extension (JTE) which acts as the periphery protection (Fig. 1). The parameters of implantations (energies, doses) were preliminary determined by Monte-Carlo simulations [1]. Then the obtained doping profiles were introduced in Medici software, a finite element electrical simulator for device structures. The diode design, the JTE geometry (doping, length and depth), were thus optimized to obtain a maximum breakdown voltage (VBR) in the bulk with usefulness lateral protection by the JTE.

1.3x10 13 cm -2 total doses were implanted at room temperature simultaneously in the two wafers to form the JTE. For the emitter a 1.75x10 15 cm -2 total dose was implanted forming box profile of ~4x10 19 cm -3 on 0.45µm, using different temperatures for the two wafers (300°C and RT). The postimplantation annealing (1700°C/30min) was performed in a rf-induction furnace specially dedicated to this SiC technological step [2].

Capacitance-Voltage measurements on layers implanted with almost the same JTE dose (10 13 cm - 2 ) and annealed in the same conditions have proved the Al implanted dopant migration on active Si sites by the post-implantation annealing [3]. The dopant profile was also conserved after post-Fig. 1. Bipolar power diodes cross-section structure.

implantation annealing.

The ohmic contacts were formed employing an Al/Ti alloy on the p-type emitters and Ni on the whole n-type backside.

Current-voltage characteristics of all fabricated diodes were systematically measured in forward (0 ® 5V) and reverse bias (0 ® -50V) using an automatic probe-stage and 4 K237 Source Measurements Units

Results and Discussion

Forward current densities (J) superior to 200A.cm -2 are obtained at 5V for the diodes with the emitter implanted at 300°C whereas a limit value of ~160A.cm -2 is obtained for the RT implanted ones(Fig. 2a). In reverse bias at -50 V leakage currents as low as 10 -8 A.cm -2 are measured (Fig. 2b).

a) b)

Fig.2. J-V spectra in forward (a) and reverse bias (b) of p-n diodes with the emitter implanted at 300°C (full square) and RT (open circle). The presented spectra are selected from the best diodes in terms of low leakage currents and high current densities in forward bias.

The emitter sheet resistances were measured on Van der Pauw (VdP) structures distributed on the whole surface of the two wafers. A lower resistance is found for the 300°C implanted layers (Fig. 3) that explains the higher current density in forward bias on these diodes. The lower residual damage in the implanted layers is responsible for the higher doping and so the higher injection levels of the emitter. The emitter layer resistances have a radial distribution decreasing from the center to the periphery due to the temperature gradient in the post-implantation furnace [2]. Nevertheless the diode current densities do not follow this distribution, the series resistance of the implanted layer being a negligible term in the total measured series resistance dominated by the contact [4] and epilayer series resistance. Fig. 4(a,b) presents current density histograms obtained by measurements under 5V in forward bias, on 160 µm diameter emitter diodes from the two wafers. A more homogeneous distribution with higher current densities is obtained for the 300°C implanted diodes, which indicates a better dopant activation and layer qualities. Rutherford Backscattering Spectroscopy in the Channeling mode measurements [3] on similar implanted layers have shown a recrystallized state close to the virgin non-implanted state, after annealing in the case of non-amorphous as-implanted layers. The layers implanted at RT, amorphous on 0.3µm depth, are recrystallized by the annealing but with a residual damage quantified to 6% dechanneling yield. Increasing the emitter size (Fig. 4c) translates the histograms towards lower current density range, the emitter surface becoming more sensitive to the material quality.

This last remark is also valid for the histograms obtained in reverse bias at -50V (Fig. 5(b,c)). Increasing the emitter size and so the probability to the eventual presence of structural defects (micropipes, dislocations,…), increases the number of diodes with high leakage currents (Abs(J)>10 - 4 A.cm -2 ). Comparing the two wafers (Fig. 5(a,b)), the diodes have almost the same distribution. A less number of diodes with Abs(J)>10 -4 A.cm -2 is obtained for the 300°C implanted wafer. However current densities as low as 10 -9 A.cm -2 are attained, only for the diodes implanted at room temperature. Measurements in reverse bias up to -1100 V with a K238 are presented in Fig. 6. The role of the JTE is proved comparing the diodes with and without JTE. The breakdown voltage also increases with the JTE length. All the diodes with the smallest emitter implanted at 300°C and the largest JTE length can block more than 1100V. VBR on these diodes measured with a 12.5kV FUG power supply, increases up to 1270 V which is very close to the value calculated by Medici software (1300 V), using ionization coefficient published by Konstantinov [5] Although for the two wafers the JTEs are realized in the same conditions and the epitaxial doping is slightly superior for the 300°C implanted wafer (Cree TM data sheet), more diodes are able to sustain more than 1100V in the 300°C implanted wafer. The emitter state after implantation and annealing seems to be also important for the diodes behavior in reverse polarization. The VBR dependence on the emitter quality state indicates a breakdown in the bulk volume.

We may relate the lower VBR value for the RT implanted diodes to the residual defects present at the emitter-epilayer junction interface, due to the presence of an amorphous layer before the annealing of the RT implanted emitters.

The breakdown in the bulk volume is confirmed also by the non-variation of the VBR by testing in different ambient (air, oil, SF6) the diodes with the largest JTE. No arcs occurs in surface in this case.

These results could not be obtained without useful JTE lateral protection against breakdown. The efficiency of our JTE were also proved by cartography on diodes surface with Optical Beam Induced Current Measurements up to 1kV [6].

Conclusion

The role of the implanted emitter state in the diode behavior was demonstrated both in forward and reverse bias. Avoiding amorphous layer formation by implanting the emitter at high temperature increases the injection level in forward bias. The lack of the defects related to the recrystallized amorphous layers allows also to obtain blocking voltages close to the theoretical ones obtained by finite element simulation. The efficiency of the JTE lateral protection, also realized by Al ion implantation, was a positive factor to obtain these results. Thus the breakdown in reverse voltage bias for the diodes with the largest JTE occurs in the volume, that is confirmed also by the non-variation of the measured breakdown voltage in different testing ambient.

Fig. 3 .

 3 Fig.3. Emitter sheet resistance measured on VdP structures distributed on the whole wafers surfaces. (full square for the wafer diodes with emitter implanted at 300°C and open circle for the RT ones.

  structure with respect to the center of the wafer [mm]

Fig. 4 .

 4 Fig.4. Current density histograms of diodes forward biased at 5V. The y-axis range is proportional to the number of each type of diodes: a) RT implanted, 160 µm diameter emitter, b) 300°C implanted, 160 µm diameter emitter, c) 300°C implanted, 250 µm diameter emitter

Fig. 5 .

 5 Fig.5. Current density histograms of diodes reverse biased at -50V. The y-axis range is proportional to the number of each type of diodes: a) RT implanted, 250 µm diameter emitter, b) 300°C implanted, 250 µm diameter emitter, c) 300°C implanted, 160 µm diameter emitter

Fig. 6 .

 6 Fig.6. JTE length dependence blocking voltage measured with K238 up to 1100 V on diodes from the wafer with the emitter implanted at RT (a) and 300°C (b). Open circles correspond to diodes with 160 µm diameter emitter size and full squares to 250 µm diameter.


Pierre Brosselard

Thierry Bouchet

Dominique Planson

Sigo Scharnholz

Gontran Pâques

Mihai Lazar

Christophe Raynaud

Jean-Pierre Chante

Emil Spahn

A 3.5 kV thyristor in 4H-SiC with a JTE periphery

Keywords: 4H-SiC, thyristor, GTO, mesa, JTE

Introduction :

Today, at ISL, there are pulsed power applications under investigation which handle electric energies up to 10 MJ using spark gaps or silicon based high power semiconductors as switching elements [1]. Since silicon based technology is reaching its physical limits concerning blocking and power handling capability, GTO thyristors based on SiC are under investigated for compact future pulsed power systems we are investigating [2]. Reaching breakdown voltages of 19 kV [START_REF] Sugawara | Proceedings of 2001 International Symposium on Power Semiconductor Devices & Ics[END_REF], the potential of SiC based devices appears very interesting. According to preparatory device simulations using the finite element code MEDICI TM , the developed GTO-thyristors should be able to block voltages up to 6 kV. The models and parameters used by the simulator are described by Brosselard and al. in [START_REF] Brosselard | Proc of Int. Semiconductor Conference (CAS)[END_REF]. After a short introduction about structure and the operating mode of the thyristor in 4H-SiC, simulations will be presented on the peripheral protection of mesa/JTE type of the component. The stress will be laid on the parameters of the JTE (length and implanted dose). The passivation of the component is ensured by a deposition of SiO2. This layer was taken into account in simulation and the results will be presented. Technological unfolding will be presented in order to show the various technological stages necessary for the manufacture of the thyristor. The electric characterization of the component will be to divide into two parts. First of all, the forward state of the thyristor will be presented and followed better results of the breakover voltage. This paper will end in a cartography of the behaviour in tension of the components before passing to the conclusion and the prospects for the various components.

Structure and operating mode of the 4H-SiC thyristor :

The n-type 4H-SiC wafer material was purchased from Cree Research TM , including a PP -NP + (from the wafer up to the top) epitaxial layer structure (figure 1). Three junctions determine the thyristor. If an anode cathode voltage is applied, the junctions J1 and J3 are into direct and J2 in reverse. To put in conduction the thyristor, it is necessary to polarize the J2 junction on line by injecting electrons in the gate zone. To block the thyristor, the voltage VAK should be cancelled. So the holding current is determined at the switch-off of the thyristor by IH. The thyristor can be put in conduction starting from the state blocked by the mechanism of impact ionization. Thus, the latch-up current is determined by the symbol IL. The reverse mode is characterize by a negative voltage VAK. This mode is very often destroying for the component. In this paper, no simulation and electric characterization will be presented for this case. This part showed the various modes of the thyristor, the phase of design of the thyristor will be to approach.

Simulation results :

The first stage of this work rests on the design of the component using simulation with finite elements code. In this part, a detailed attention will be given on the behaviour in tension of the component and in particular on peripheral protection and passivation. The characteristics of the various layers were given during a preceding work. [START_REF] Brosselard | [END_REF]. It is only the J2 junction which is responsible for the behaviour in voltage. To this end, only the diode NP -P was simulated. Figure 2 illustrates the structure simulated with the various parameters to optimize for protection (depth of engraving, length and amount of the JTE). An angle of 10° was retained bus actually a perfectly perpendicular engraving is very difficult to obtain. The characteristics of the various layers from SiC are those of the plate which will be process. A first group of simulations was carried out for a etching depth Wetch of 1 µm and a profile of doping of the JTE obtained by a software of ion implantation in SiC [6]. Figure 3 represents the breakdown voltage of the diode according to the implanted dose of the JTE (NJTE) for various lengths of JTE. An optimal dose (dop) can be retained for the various lengths of JTE (lJTE) and it equals at 9×10 12 cm -2 . If the dose is lower than dop, the voltage varies little and the electric field is maximum in the JTE on the left of the JTE. On the contrary, if the dose is higher than dop, the voltage varies brutally. The electric field peak is in end of JTE. Thus, the JTE plays the role of the anode and the problem of periphery is deferred. The parameters retained for the JTE periphery are a length higher or equal to 150 µm (lJTE) and an optimal dose of 9×10 12 cm -2 . With these optimized parameters, simulations were carried out by holding account a thickness of SiO2 (Wox) equalizes with 1,5 µm. The density of fixed charge to the SiC/SiO2 interface was defined. According to, the conditions of growth of oxide and the nature of the equipment used, the sign and the value of this density vary. Figure 4 illustrates the results of simulation. Charge of the positive type do not have an influence on the behaviour voltage. On the other hand for the negative charges of which the density of loads is higher than 10 12 cm -2 , the behaviour in voltage decreases. If the charge are negative in oxide, they are positive in SiC (neutrality). The spreading out of the space charge is thus disadvantaged even in the presence of the JTE what leads to the fall of the behaviour in voltage. In conclusion, the state of SiC/SiO2 interface can be responsible for a bad behaviour in voltage of the component. These various results will be confronted with the experimental characteristics. The process sequence has been divided in five photolithographic levels. The realization began with two different dry etchings, the first allowing to make the gate contact. The etch depth equals to 1.4 µm. The second one necessary for the mesa protection. Within sight of the results of simulation, a etching depth of 1 µm was requested from the technology center and it was reached. A nitrogen implantation followed by 1700°C/30mn annealing were needed to realize the JTE. This establishment required 4 different energies and 4 amounts since the mechanism of diffusion in SiC is non-existent. After a 1.5 µm oxide deposition by LTO (Low Temperature Oxidation) process to passivate the thyristor structure, a nickel metallization has been realized. With a thickness of 500 nm only, the realization of bonding is not possible. It will be to consider after one on-metallization and the first metallization show some good ohmic contact.

Electrical characterizations :

First of all, an electric characterization of setting in conduction of the thyristors was carried out. The voltage source used is one curve tracer Tektronix 370 whose control signal was

} device n°1 device n°2

Q it = -3 10 12 cm -2 Q it = -2 10 12 cm -2 I anode [A] U AK [V]
formatted. Figure 5 represents the results in the form of current density and current. The threshold voltage lies between 5 and 6 V. These values are higher than those obtained by simulation (3 V). This is due to non-ohmicity of the anode contacts. This voltage can decrease by improving to a significant degree metallization of anode. The blocking forward state was characterized first once on 46 components. The best held in tension is equal to 3500 V. A comparison is carried out between the experiment and simulation and it represents by figure 6. Thus, charge are of the negative type to the SiC/SiO2 interface and the density lies between 2 and 3 10 12 cm -2 . Moreover, statistics were carried out and they are represented by figure 7. This reveals that in fact the components of low size have the best held in voltage. This is due to the various defects in material. All devices on the sample have been electrically characterised with a 1100 V Keithley K2410. Figure 8 shows the location of the breakover voltage on the wafer surface. The dark ones hold a voltage of up to 1kV. This figure clearly shows that the repartition of the better ones, is not uniform. We can see that most of them are placed in the upper left side of the sample, whereas nearly no thyristor, able to hold a higher voltage, are placed in the left border region. That illustrates the non uniform distribution of wafer defects.

Conclusion :

We conclude, that the results are encouraging for the use of SiC-thyristors in power applications. The turn-on characterisations are in progress and will be described is the final paper. The setting in conduction of the components show limits from a point of view of threshold voltage. Confrontation experiment and simulation for the blocked state made it possible to highlight which the stage of passivation was to be optimized in particular in term of preparation of surface.

References :

[1] : E. Spahn, G. Buderer, V. Wegner, F.

Fig. 1 :

 1 Fig. 1 : Cross sectional view of the 4H-SiC GTO-thyristor

Fig. 3 :Fig. 4 :

 34 Fig. 3 : Breakdown voltage of the device describe versus implanted dose (NJTE) for different length of the JTE (lJTE),

Fig. 5 :Fig. 6 :

 56 Fig. 5 : Electrical characterisations in forward state. T40 and T90 is two different devices

Fig. 7 :

 7 Fig. 7 : Statistic of the breakover voltage as a function of the device surface (in percents) i

  Jamet 9 th IEEE Pulsed Power Conference Albuquerque/NM, USA, 18-24 June 1993

[2] : S. Scharnholz, V. Zorngiebel, P. Brosselard, and E. Spahn; in IEE Proc. of the 1st European Pulsed Power Symposium (EPPS) , 22-24 October 2002, Saint Louis, France; 2002; pp. 14/1-14/6.


Lionel Gélébart

Michel Bornert

Thierry Bretheau

Daniel Caldemaison

Jérôme Crépin

André Zaoui

Lamellar grains distribution and plastic strain heterogeneities in TiAl cast samples. Experiments and modelling

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

DISTRIBUTION DES GRAINS LAMELLAIRES ET HÉTÉROGÉNÉITÉ DE DÉFOR-MATION PLASTIQUE DANS DES ALLIAGES DE FONDERIE À BASE TiAl. EXPÉRIENCES ET MODÉLISATION

RÉSUMÉ : Pour certaines conditions de vitesse de refroidissement, les alliages de fonderie à base TiAl présentent une microstructure biphasée [Ti 3 Al (α 2 ) + TiAl (γ)] constituée de petits grains monophasés γ et de grains lamellaires biphasés (α 2 + γ); la distribution radiale des grains lamellaires génère alors une "macrostructure" dont la taille est commensurable à celle des barreaux. D'autre part, à cause de la forte anisotropie de comportement plastique des grains lamellaires (due au peu de systèmes de déformation activable et à l'influence de la morphologie des lamelles), la distribution de la déformation plastique du matériau est fortement influencée par une telle "macrostructure". L'hétérogénéité de déformation au sein du matériau est clairement mise en évidence au moyen d'une mesure de champ de déformation par corrélation d'images appliquée à un essai de compression uniaxial réalisé sur une éprouvette à microstructure radiale. Il découle de cet essai qu'un échantillon avec une telle microstructure ne permet pas de définir un volume élémentaire représentatif du matériau. Il est alors difficile d'évaluer directement des paramètres matériaux tels que cissions critiques, limite d'élasticité, écrouissage, etc. Afin de remédier à ce problème de non séparabilité des échelles, une approche basée sur un calcul de structure, qui prend en compte la description géométrique de la microstructure radiale ainsi que l'anisotropie de comportement mécanique au moyen d'un critère de Hill isotrope transverse, est présentée. L'optimisation des paramètres du critère de Hill par comparaison avec la réponse expérimentale à la fois à l'échelle macroscopique (courbe de comportement) et à l'échelle locale (champ de déformation) conduit à une meilleure adéquation entre expérience et simulation. Cette étude met en évidence le risque qu'il y a de ne pas prendre en compte l'agencement de la microstructure : cela conduit à sous-estimer les niveaux de déformation plastique, ce qui pose problème pour des alliages aussi peu ductiles que les alliages à base TiAl. MOTS CLEFS : Macrostructure, mesure de champ de déformation, TiAl, grains lamellaires ABSTRACT : During the cooling of TiAl cast alloys, the distribution of the dual-phase lamellar grains [Ti 3 Al (α 2 ) + TiAl (γ)] generates a so-called 'macrostructure'. Because of the highly anisotropic plastic behaviour of the lamellar grains, the distribution of the plastic strain is strongly influenced by such a macrostructure. This heterogeneity is clearly observed by means of a strainfield measurement technique applied to a compression test performed on a cast sample having a 'radial' macrostructure. As a result, a sample having such a macrostructure is not representative of the material and can not be used directly to evaluate 'material parameters'. Moreover the mechanical simulations of cast components should take into account such heterogeneities at the scale of the component. This strong influence is experimentally demonstrated, and a way to take it into account in a mechanical simulation is proposed. KEYWORDS : Macrostructure, strainfield measurement, TiAl, lamellar grains D uring the two last decades, the aeronautic and automotive industry has shown an increasing interest for titanium aluminides (TiAl intermetallic alloys). The main reasons for such an interest are a low density (~3.8), a good stiffness and a high temperature strength up to 750 °C. On the other hand, these alloys exhibit, in their polycrystalline form, a poor ductility (4 % seems to be an upper bound at room temperature) [1][2][START_REF] Guédou | 14 ème Congrès Francais de Mécanique[END_REF] with widely scattered values. One way to improve this type of material could be to optimize the microstructure; thus it seems important to understand its influence on the occurence of plastic strain heterogeneities leading to crack initiations. The material under study is a cast alloy with a nearly fully lamellar microstructure. Three different scales can be used to describe and study this material: the scale of the lamellae (a few µm), the scale of the lamellar grains (a few mm), and the scale of the sample (a few cm). The last scale describes the organisation of the lamellar grains created during the cooling of the billets. In the following, these scales are used to describe the material. Then, the study is focused on the "scale of the sample" in order to show that the organisation of the lamellar grains has an important effect on the plastic strain distribution. A way of modelling this type of microstructural organisation is finally proposed and applied to a finite element simulation.

Description of the microstructure

The material has a TiAl 48 Cr 2 Nb 2 (at.%) composition and has been cast in billets with a length of 100 mm and a diameter of 20 mm. The resulting microstructure is nearly fully lamellar [Ti 3 Al (α 2 ) + TiAl (γ)] with a low volume fraction (~10 %) of small equiaxed monophasic γ grains. Thus, the following description is mostly focused on the lamellar phase and its distribution.

The scale of the lamellae

This scale has been extensively investigated. The reported studies, performed by means of Transmission Electron Microscopy, address the different crystalline phases [START_REF] Denquin | [END_REF], the orientation relationships between lamellae [START_REF] Zghal | [END_REF], and the deformation mechanisms [6].

Structure and deformation mechanisms of the lamellar phase

The lamellar phase forms during the cooling of the α phase. It is the result of the decomposition of the α phase (an hexagonal compact structure, type A 3 ) in α 2 and γ phases. In one lamellar grain, all the α lamellae have the same orientation, and the γ lamellae have 6 possible orientations. The γ phase [START_REF] Denquin | [END_REF] has a L1 0 structure. It is a tetragonal structure (with c/a ratio ~1.015) very close to an F.C.C. structure, with an alternating stacking of titanium and aluminium atomic planes. Many studies on the deformation mechanisms of the lamellar phase have been realised on so-called Poly-Synthetically Twinned (PST) crystals, which can be considered as "single crystals" of lamellar phase. Such crystals have been submitted to tension and compression for different loading axis orientations [6,7]. The observed mechanisms in γ lamellae are: • 4 systems {111}<1-10] of ordinary dislocations, • 4 twinning systems of type {111}<11-2],

• and 8 systems {111}<-101] of superlattice dislocations. The α 2 phase [START_REF] Denquin | [END_REF] has a D0 19 structure. It results in the ordering of the α phase and the c/a ratio is equal to 0.8. A single crystal of a D0 19 compound with an off-stoichiometric composition (Ti-36.5at.Al), close to the composition of α 2 phase present in lamellar structures, has been studied [8] ; the deformation mechanisms are the following: • 3 prismatic slip systems {1-100}<11-20>, • 3 basal slip systems (0001)<11-20>, • and 6 pyramidal slip systems {11-21}<-1-126>. A lamellar grain can be described (fig. 1) as the stacking of seven different types of lamellae : the α 2 lamellae with the basal plane parallel to the lamellar interface and 6 γ lamellae in crystallographic relation with the α 2 lamellae : the [110] γ direction is parallel to a <11-20>α 2 direction and the (111) γ plane is parallel to the basal plane (0001)α 2. The 6 γ lamellae deduce from each others by a rotation of kπ/3 (k = 0,...,6) around [0001] α2 .

The strong anisotropy of the lamellar phase

Tensile tests performed on PST crystals [7] for different loading axis orientations have shown a very strong plastic anisotropy (fig. 2). Such an anisotropy can not be only reproduced by choosing different appropriate values for the CRSS (Critical Resolved Shear Stress) of the different types of mechanisms. The role of the interfaces as barriers for the transmission of glide [9] must be accounted for, similarly to what is done with the Hall-Petch law. A "morphological" classification (fig. 3) of the slip systems is then derived [10]: a mechanism is called Longitudinal if its slip plane is parallel to the interfaces, Transverse if its slip direc- Fig. 2 -Comportement anisotrope de la structure lamellaire [7] tion and its slip plane are not parallel to the interfaces, and Mixed if its slip direction is parallel to the interfaces but its slip plane is not. Of course, for a same type of slip system, the CRSS will be the smallest for a Longitudinal morphology, and the highest for a Transverse one.

The scale of the lamellar grains

This scale will be used to describe the misorientations between lamellar grains and to study the interactions between misoriented grains. As the plastic anisotropy is important, high plastic strain incompatibilities can be expected, that could generate strong stress concentration and explain, partly, the initiation of cracks and the poor ductility of the polycrystalline material. This question which is not the purpose of the present work is presented in [11].

The scale of the sample

While cooling a cast billet, lamellar grains grow radially from the periphery towards the center along the direction of the heat flux. In this radial organisation, the lamellar interfaces are tangentially distributed (fig. 4). Such an heterogeneity, at the scale of the sample, has as consequence that the sample tested is not representative of the 'material' behaviour and all the macroscopic results obtained on such a sample cannot be used directly to evaluate material parameters. Moreover the mechanical simulations of cast components should take into account such heterogeneities at the scale of the component. This strong influence is experimentally demonstrated in the next part, and a way to take it into account in a mechanical simulation is proposed in the last one.

Experimental study

In order to show the influence of the radial organisation of the lamellar phase, a compressive test is performed with the compression axis along a radial direction of the billet. The strain distribution is investigated by means of a strain field measurement technique. [12] The procedure is based on the comparison between digital images of the surface, before and after deformation, corresponding to the reference and the deformed configuration. A virtual grid of reference points is defined on the reference configuration and using classical correlation techniques [12][13][14][START_REF] Soppa | Proceedings of IWCM9[END_REF] the position of each reference point is found in the deformed configuration. The correlation relies on a contrast provided by a black on white speckle realised on the surface of the specimen (fig. 5). So, the in-plane displacement components and then the in-plane strain field components can be evaluated. The spatial resolution of the method is defined by the choice of the perodicity of the virtual grid of reference points.

Strain field measurement technique

Sample

The sample is cut such that the compression axis is along a radial direction of the billet, and such that the center of the macrostructure nearly corresponds to the center of the sample (fig. 6). As the natural contrast of the surface is not sufficient for the correlation method to work efficiently, an artificial marking of the surface has to be done.

Image acquisition and local strain measurement

A conventionnal CCD camera (Sony SSC M370CE) associated with a 50 mm focal length lens (Pentax) and appropriate extension rings, a computer with an acquisition card (Data Translation) and the software NIH image (from the National Health Institute), are used for the acquisition of digital images. To ensure a sufficient local resolution two images are taken to cover the full sample. The images are digitized in 738 × 512 pixels on a grey scale of 256 levels. To reduce the background noise, the image is the result of the averaging of 16 frames. The virtual grid of points defined on the images (fig. 5) has a periodicity of 0.5 mm, leading to a gage length of 1 mm for the local strain measurement, since the deformation gradient components at a given point are estimated from the displacement of the 8 neighbouring points [13]. The strain components are computed under the assumption that the out-of-plane displacement can be neglected, which is a reasonable assumption for the small strains under consideration. In the following, the intensity of the strain will be given by the equivalent strain ε eq () , where dev(ε = ) is the deviatoric part of the strain tensor.

Compression test

The compression test is performed with an Instron machine under a macroscopic strain rate of 10 -4 s -1 . The force transducer has a full scale of ± 50 kN with a gain of 5 kN/V. As the microstructure presents a strong heterogeneity at the scale of the sample, it has been decided to use a displacement transducer between the crosshead and the fixed platen of the testing machine, rather than a strain gauge. The displacement transducer (LVDT) has a full scale of 10 mm with a gain of 4.09 mm/V. The test is performed sequentially, images being registered at each stage. Four stages are considered : three during the loading, and the fourth one after the elastic unloading.

The experimental response

The macroscopic behaviour is presented on figure 7 and the strain distribution and its evolution from one stage to another, is shown on figure 8, where the equivalent strain is represented on a grey scale from 0 % to 10 %. From the analysis of these strain maps different conclusions can be drawn: The strain distribution is very heterogeneous at the scale of the sample. As a consequence, the macroscopic response given on figure 7 has to be considered as the response of a 'structure' and can not be directly used to evaluate 'material properties' such as the yield stress, the hardening coefficient or any other one. The strain distribution is strongly linked to the radial organisation of the microstructure. Actually, it can be schematically described by two strain localisation bands and four blocks which are nearly undeformed. The intersection of these bands is exactly positionned at the center of the radial microstructure, which is not exactly the center of the specimen. The position of the two bands can be linked to the orientation of the lamellar grains which have an angle between the compression axis and the lamellar interfaces below 60 o and above 30 o . Because of the strong anisotropy of the lamellar phase, these orientations are easy to deform (see orientations B1 and C1 on figure 2). On the contrary, the other orientations are harder to deform, and the areas where the grains have such hard orientations correspond to the nearly undeformed four blocks.

The strain distribution remains unchanged when the global deformation increases; only the contrast between the bands and the blocks evolves.

A possible model for such 'structures'

As clearly established in the experimental study, the organisation of the microstructure at the scale of the sample has a very strong effect on plastic strain distribution. As a consequence, the question is now to propose a way of modelling such a microstructural organisation to reproduce the local strain distribution as well as the global response obtained experimentally.

The model

The modelling of such a material with an heterogeneity at the scale of the sample is based on two steps of representation. The first one is the definition of local reference axes at each point of the sample and the second is the definition of the behaviour in these local axes. The first step relies on a schematization of the microstructural organisation. In our case, the microstructure is radial what allows us to use cylindrical coordinates with e1 parallel to the axis of the billet, e3 the radial direction and e2 the tangential direction (fig. 6). The second step deals with the local behaviour definition which should take into account the description of the microstructure. A rough description of the microstucture can be used as a first trial : it is assumed to be fully lamellar with the lamellar interfaces exactly tangentially distributed, so that the local behaviour is given by the behaviour of the lamellar phase. It is assumed elasto-plastic, with isotropic elastic properties and the elastic domain defined by a transversal isotropic Hill's criterion (e1, e2) being the plane of transverse isotropy. The hardening is supposed to be linear and isotropic. This behaviour is then given by the following relations:

σ = = K(ε = -ε = p ) f (σ = ) = R ˙= hε ˙p eq
where σ 2 eqE = -(2σ 33 -σ 11 -σ 22 ) 2 , σ 2 eqT = -[(σ 22 -σ 11 ) 2 + 4 σ 2 12 ), and σ 2 eqL = σ 2 13 + σ 2 23 . The elastic coefficients, measured by Schafrik [START_REF] Schafrik | [END_REF], are E = 180 GPa and ν = 0.28. The three coefficients of the Hill criterion are evaluated from the three yield stresses (380 MPa, 120 MPa and 520 MPa) obtained on tensile tests performed on PST crystals [7] with the orientations (0°, 45°and 90°) (fig. 2). With this set of parameters (N = 6.49, L = 73.24, and σ 0 = 520 MPa), the evolution of the yield stress as a function of the angle between the tensile axis and the lamellar interface is given figure 9. The hardening coefficient is chosen as h = 7 000 MPa so that the hardening of the simulated global response is in good agreement with the experimental one.

Application to a finite element calculation

In order to show that the contact between the platen and the sample can not lead to the experimental strain distribution, a compression test is simulated assuming that the material is isotropic and homogeneous. Then, the heterogeneous and anisotropic model is taken into account, and finally the modelling is discussed and its parameters are modified in order to improve the comparison between simulation and experiment.

Simulation with an homogeneous and isotropic model

In order to minimize the calculations, and by accounting with the different symmetries of the problem, only one eighth of the specimen is simulated. The dimensions are those of the compression specimen. Quadratic elements with 20 nodes and 27 integration points are used. The boundary conditions (fig. 10) are : symmetry conditions for the faces A, B and C, free surface condition for the faces E and F, the displacements of face D are imposed to zero in X and Y directions, and imposed to -0.25 mm (or ∆l/L 0 = -3%) in Z direction. These boundary conditions for face D represent the contact between the specimen and the platens of the testing machine, where the strong friction locks any X, Y displacements. As the aim of this first calculation is to evaluate the effect of the contact platens/specimen in the case of a compression test, the material is chosen homogeneous and isotropic. The elasto-plastic behaviour is described by a Von Mises criterion associated to a linear isotropic hardening, the yield stress is chosen equal to 400 MPa and the other parameters are those already mentionned. These values are fitted on the experimental global response (fig. 7). The obtained equivalent plastic strain distribution are presented on figure 11. Two different effects can be attributed to these boundary conditions : the first is a localisation near the edges of the specimen, and the second one is an heterogeneity centered in the middle of the specimen. However such a distribution is too smooth compared to the experimental one (fig. 8). So, the boundary conditions can not explain the observed experimental strain distribution, even if they give a small contribution to it.

σ eqE 2 + Nσ eqL 2 + Lσ eqT 2 -(σ 0 + R) = 0

Simulation with an anisotropic behaviour radially distributed

For this second calculation, the behaviour of the material is the one described in section "The model". For simplicity, the center of the microstructure is supposed to be exactly at the center of the specimen, so that the same symetry conditions as above can be used. The obtained equivalent plastic strain distribution is shown on figure 12. This distribution is qualitatively close to the experimental strain distribution. In order to have a quantitative comparison between the simulation and the experimental results, the equivalent plastic strain is averaged in three particu-lar elements shown on figure 12. Element N is supposed to be representative of the block where the lamellae are almost normal to the compression axis, element L is representative of the block where the lamellae are almost parallel to the compression axis, and element B is representative of the strain localisation band. The equivalent plastic strains averaged on these elements for ∆l/L 0 = -3 % are compared to the same averages of the experimental equivalent strain measured after elastic unloading (Table I). This quantitative analysis performed at local scale shows that, despite of a qualitative agreement, the simulation still has to be improved. The same conclusion can be drawn from the global response (fig. 7), which is too soft compared to the experiment. It seems that the localisation in this second calculation is too strong compared to the experiment, especially at the center of the sample.

Discussions and improvements of the simulation

The previous modelling can be seen as the result of a deductive approach where the simulation at the scale of the sample directly takes into account the behaviour of the local constituents, the lamellar grains in our case. The quantitative disagreement observed can find its explanation in the drastic simplification done in the definition of the local behaviour. First, the Hill's criterion used to described the yield surface of the lamellar phase can be different from the real yield surface, even if, at least, it describes correctly the response for tensile tests. Secondly, the local behaviour is fitted on the behaviour of the single lamellar crystal, so it cannot take into account : the misorientations of the lamellar grains compared to the tangential position, the interactions between these misoriented grains, and the role of the small γ grains that can be seen at the boundaries of the lamellar grains. Finally, a fully deductive approach should take into account all these considerations via a polycristalline modelling, leading to a local behaviour that is more isotropic than the behaviour of the lamellar phase. In the following, this fully deductive approach is given up and a more inductive approach is used. To improve the comparison between simulation and experiment, and because of the qualitative agreement of the previous simulation, we suggest to keep the previous modelling (the tangential distribution of a transversal isotropic Hill's criterion), but to modify the parameters to get a more isotropic behaviour. The new set of parameters σ 0 = 500 MPa, N = 4.71, L = 9.68 and h = 6000 MPa) is evaluated so that the global simulated response as well as the local one (the averaged equivalent plastic strains in the elements N, B and L), are in a quantitative good agreement with the experimental responses. The global response is shown figure 7, the equivalent plastic strain distribution is shown figure 13, and the average values on the elements N, L and B are given in table I. With this new set of parameters, the evolution of the yield Element N Element L Element B ε eq (Experimental value) 2.2 % 1.6 % 4.6 % ε eq (numerical value) 0.01 % 0.3 % 2.2 % ε eq (optimized numerical value) 2.0 % 1.9 % 4.0 % stress as a function of the angle between the tensile axis and the lamellar interface is shown figure 14. It can be seen that the new local behaviour is more isotropic than the behaviour of the lamellar phase (figure 9).

Conclusions

Thanks to a strain field measurement technique the influence of the distribution of the lamellar phase in a cast billet, has been shown. This radial distribution leads to a very strong heterogeneity of the plastic strain distribution. As a consequence : a tensile test realised on a sample having such a microstructural distribution cannot be used directly to evaluate material properties, a mechanical simulation realised on a cast component, should take into account such a microstructural distribution. The strong heterogeneity of plastic strain can be related to the strong anisotropy of the lamellar phase. A simple way of modelling this macrostructure has been proposed. It relies on a transversal isotropic Hill's criterion tangentially distributed. First, the parameters are fitted on the behaviour of the lamellar phase, but the obtained results are just in a qualitative agreement with experiment, the obtained strain field being too heterogeneous. Then, the parameters have been modified to obtain a quantitative agreement. This new set of parameters leads to a more isotropic behaviour, which could be due to the polycristalline interactions which were ignored in the first test. ❑

Fig. 1 -

 1 Fig. 1 -Description of the lamellar structure.

Fig. 1 -

 1 Fig.1 -Description de la structure lamellaire.

Fig. 2 -

 2 Fig. 2 -Anisotropic behaviour of the lamellar structure [7].

Fig. 3 -

 3 Fig. 3 -"Morphological" classification of the strain mechanisms.

Fig. 3 -Fig. 4 -

 34 Fig. 3 -Classification "morphologique" des mécanismes de déformation

Fig. 4 -

 4 Fig. 4 -Micrographie d'une section transverse du lingot (diamètre = 20 mm).

Fig. 5 -

 5 Fig. 5 -Black and white speckle and virtual grid of reference points.

Fig. 5 -

 5 Fig. 5 -Speckle en noir et blanc avec grille virtuelle des points de référence.

ε eq = 2 /Fig. 8 -Fig. 8 -

 288 Fig. 8 -Equivalent strain distribution (1) ∆ ∆l/L 0 = -1.5 %, (2) ∆ ∆l/L 0 = -2.2 %, (3) ∆ ∆l/L 0 = -3.0 %, (4) ∆ ∆l/L 0 = -2.6 %.Fig. 8 -Distribution de la déformation équivalente: (1) ∆l/L 0 = -1.5 %, (2) ∆l/L 0 = -2.2 %, (3) ∆l/L 0 = -3.0 %, (4) ∆l/L 0 = -2.6 %.

Fig 6 -

 6 Fig 6 -Description of the sample geometry and axes definition.

Fig. 6 -Fig. 7 -Fig. 7 -

 677 Fig. 6 -Description de la géométrie de l'échantillon et définition des axes.

Fig. 9 -

 9 Fig. 9 -Hill criterion fitted on behaviour of the lamellar structure.

Fig. 9 -Fig. 10 -

 910 Fig. 9 -Critère de Hill ajusté sur le comportement de la structure lamellaire.

Fig. 10 -

 10 Fig. 10 -Description des conditions aux limites.

Fig. 11 -Fig. 11 -Fig. 12 -

 111112 Fig. 11 -Homogeneous microstructure and isotropic modelling. Equivalent plastic strain distribution.Fig. 11 -Modélisation isotrope pour microstructure homogène. Distribution de la déformation plastique équivalente.

Fig. 12 -

 12 Fig. 12 -Modélisation anisotrope et microstructure radiale. Distribution de la déformation plastique équivalente.

Fig. 13 -Fig. 13 -Fig. 14 -

 131314 Fig. 13 -Radial microstructure and anisotropic modelling. Equivalent plastic strain distribution. Optimized parameters. Fig. 13 -Modélisation anisotrope et microstructure radiale. . Distribution de la déformation plastique équivalente. Paramètres optimisés.

Fig. 14 -

 14 Fig. 14 -Critère de Hill avec paramètres optimisés.

Acknowledgements

This work is realised in a common project between the french scientific institute CNRS, two industrial firms (SNECMA Moteurs and Turboméca), and six french laboratories (LSG2M in Nancy, CEMES in Toulouse, LTPCM in Grenoble, LMPM in Poitiers, GMP in Rouen, CECM in Vitry and LMS in Palaiseau).