Supporting a policy of success for disadvantaged schools:

Identification of contradictions around hierarchical power issues between activity systems

Yannick Lémonie, yannick.lemonie@lecnam.net, Cnam, Paris, France Vincent Grosstephan, vincent.grosstephan@univ-reims.fr, ESPE Champagne-Ardennes, Reims, France Tomàs, Jean-Luc, jean-luc.tomas@lecnam.net, Cnam, Paris, France French educational system as an instrument of hegemony Schools have an important responsibility in reproducing social inequalities. More than 50 years ago, [START_REF] Bourdieu | La reproduction. Éléments pour une théorie du système d'enseignement[END_REF] highlighted the latent ideological functioning of the French educational system, which, in the name of its openly democratic recruitment, makes a social selection based on the cultural criteria of the dominant class. For them, this social selection is also made acceptable to the excluded and disadvantaged by a process of ideological production in which pupils from the working and middle classes find it necessary to legitimize their low social status through their failure at school. Half a century later, where do we stand? In 2012, the international evaluation PISA1 point out that 'the correlation between socio-economic background and performance' is much more pronounced in France, 'than in most other OECD countries'. The French education system 'is more unequal in 2012 than it was 9 years earlier and social inequalities worsened mainly between 2003 and 2006. In France, when you belong to a disadvantaged environment, you clearly have less chance of success today than in 2003'.

The school remains a powerful tool of hegemony in the sense that this concept was developed by [START_REF] Gramsci | Selections from the Prison Notebooks of Antonio Gramsci[END_REF]. The concept of hegemony refers to 'a particular way of conceptualizing power which amongst other things emphasizes how power depends upon achieving consent or at least acquiescence rather than just having the resources to use force, and the importance of ideology in sustaining relations of power' (Fairclough, 2003: 45). The observation of serious inequalities in access to educational success for pupils from disadvantaged backgrounds could lead to a pessimistic vision suggesting that any possibility of transformation of the system, not only to improve it [START_REF] Engeström | Improvement versus Transformation[END_REF], is doomed to failure. However, for Sannino & Vainio (2015), who point out the importance of the concept of hegemony in a study on gender inequality in Finnish universities, 'Dominance based on consensus is a contradictory phenomenon that is historically constructed and thus bound to transformations' (p. 508).

Accordingly, Kontinen, 2013, p. 118) argue that 'the concept of hegemony is relevant to the contemporary developmental work research as well. First, it suggests paying attention not to economic relations but to political, institutional, ideological and cultural forms of power.' More important for us, [START_REF] Sannino | Gendered hegemony and its contradictions among Finnish university physicists[END_REF] offer a key to methodologically articulate the concept of hegemony to the theory of activity: 'When historicized, local hegemonic struggles identified in discourse can direct the analysis towards the origins of clashes and problems occurring in everyday activities. This allows a deeper level of understanding of hegemonic relations by pointing at historical contradictions from which [inequalities of success in school] originates ' (p.508).

The purpose of this paper is to identify, through the problems encountered by school actors in shaping Priority Education (PE) reform, the contradictions but also the power, control and authority relationships they express. We will proceed as follows. First, we will present the priority education policy reform implemented following PISA 2012. In a second step, we will present the methods implemented as part of a research-intervention project to support the implementation of this reform. The results will focus on identifying contradictions at work through the identification of problems and conflicts that arise in daily practices. Finally, we will briefly discuss the results in terms of the relationship between power, hierarchy and authority.

The law on the rebuilding of the school and the reform of priority education

Following the results of PISA 2012, the French government undertook, in the space of three years, a series of reforms aimed at improving the ranking of schools in international rankings: curriculum reforms, priority education reforms, reform of school rhythms, etc. For the purpose of this paper, we will present briefly only two of them: the reform of primary and secondary education cycles and the reform of PE.

The reform of primary and secondary education cycles

National educational programs have traditionally been organized by year. In the law 'on orientation and programming for the rebuilding of the school of the republic' (July 2013), schooling is now organized in a 3-year cycle of education. The programs refer to these cycles and aim to allow greater flexibility in the work of curriculum development teams. The particularity of this reform is that Cycle 3 is shared between primary and secondary schools (see Figure 1). In this context, the monitoring and establishment of a dialogue between primary schools and the secondary school seems important in the context of monitoring student performance. The reform of 'Priority Education'

The second important reform concerns PE. The PE policy was initially set up following the accession to power of a socialist government in 1981. If it has known different periods [START_REF] Rochex | L'éducation prioritaire en France. Les paradoxes d'une politique incertaine[END_REF], several innovations are attributed to it [START_REF] Frandji | De la lutte contre les inégalités à l'adaptation aux « besoins spécifiques[END_REF]: (1) Working with local partners in a network (local administrations of urban policies, cultural institutions, social services, etc.); (2) Deconcentrating of regulation and policy instruments: the policy of devolution of powers and administration is supposed to allow adaptation to local specificities;

(3) Positive discrimination: to give more to those who have less.

The 2015 reform aims to institutionalize the networking that emerged from the first PE policies.

It is concretized by the setting up of Networks of Disadvantaged Schools (NDS). A NDS consists of a secondary school and several primary and nursery schools that are part of its recruiting sector (see Figure 2). The NDS is 'managed' jointly by an inspector in charge of primary schools and the director of the secondary schools. The NDS's animation policy is carried out within a steering committee whose geometry is variable. The actors of this steering committee, in addition to the two institutional pilots, are the network coordinator, the network trainer, the nursery and sometimes, primary school directors. The design of these NDS aims to the 'development of educational and teaching practices adapted to students' needs' and to enable evolution of educational organizations, specifically working better and more collectively as a team.

Methods

The request was initiated by the Créteil school region. The Créteil school region is the secondlargest school region in France in terms of school population and is often cited for the social, economic and academic difficulties of its inhabitants. Seven NDS are involved in this project which concerns a population of about 10,000 pupils.

The initial request was for assistance in rebuilding a collective activity on each of the NDS. In this context, the methodology was based on developmental methods aimed at enabling each NDS to build their future collective work on the basis of the identification of contradictions.

The protocol is strongly inspired by Change Laboratory [START_REF] Virkkunen | The change Laboratory. A tool for Collaborative Dévelopmental Work and Education[END_REF], although it combines 'clinic of activity' methods, e.g. individual self-confrontation [START_REF] Clot | Dialogue, activité et développement[END_REF] and inter-network groups that also allow for intra-trade collective work (for example, grouping of secondary school principals).

The data collection consists initially of an audio-video recording of steering committee meetings within each NDS. The participants are then self-confronted to this audio-video data.

A third type of data concerns the recording of sequences of debate and discussions within the steering committee organized on the basis of tools produced by the participants, such as diagrams or a network map.

We mobilized a methodological framework to identify symptoms of contradictions in discursive data [START_REF] Engeström | Discursive manifestations of contradictions in organizational change efforts: A methodological framework[END_REF]. We thus articulate self-confrontation interviews in order to understand the contradictions between activity systems behind the problems, conflicts, disturbances that occur during the steering committee.

Results

For the purpose of this paper, we choose to present only the analysis of a conflict case that emerges during an NDS steering committee. From this, we will present in a first part, the main contradiction that has become salient through the implementation of the PE reform. In a second step, we will interpret from this case the genesis of this conflict from the questions of power, authority and control.

Conflict during a steering committee

The conflict that emerges within the framework of the steering committee concerns the setting up by the network coordinator of a pedagogical action involving primary school teachers and secondary school mathematics teachers (see table 1). The director of the secondary school opposes the participation of secondary school teachers because, although informed, she must plan and organize the work of teachers. The inspector of the first level, who is the hierarchical superior of the coordinator, seeks to be the mediator of this conflict and points out several times that the rules are different between the first and second levels: if it is possible to deal directly with teachers in the first level, it is not possible to do so with the second level: it is necessary to discuss in advance with the management.

Table 1: The case of conflict during the steering committee.

Coordinator (1):

We received an email from the mathematics teacher. So five teachers want to participate in the math challenge that concerns all schools of the town. And so the next steering committee will be on December 4, all day and so I will send at the same time as the teachers, to the teachers of your school (address to the director). (…) They seemed very interested, very involved, so they will be invited to this working time on December 4. So yes, they are invited. They are given time slots and they come whenever they want. Director: Oh yes…but no!. You have to check with us before. It's not that we're against this kind of meeting, but we need to know as soon as possible to know when they won't be there in the establishment. Inspector: I think the idea there was about their off-class hours. But, it is indeed necessary that the direction is in the loop so that you can't it organize yourself. Director: (…) so that you understand, not every teacher is a free electron. Coordo (1): so it is up to you to define their availability that day? Director: That's not what I'm saying either. You talked about the fact that you're going to offer them several slots....

Coordo (1): But it's all day long…

Inspector: But in fact you have to talk with the director before. Director: Yes! Inspector: It is... it is absolutely essential that you (the director) be in the loop, that you be solicited for these questions. Because so that this does not disrupt the secondary school and at the same time it is a joint institutional work, you (the coordinator) really need to be aware of it. Coordo (1): But it seems to me that you were in the loop of the email that the math teacher coordinator sent. Director: (…) I was in that email, but then. Inspector: In fact, you should not deal directly with the teacher. In the first degree at least, you can do it without going directly through the directors. But for the secondary school, you have to talk with the director before. Coordo (1): But....

Director:

Listen, send the email and then let us handle it Coordo (1): Okay. Director: Because our goal will be to make this connection possible.

All the actors involved spoke about this episode in their self-confrontation interviews. The coordinator and the principal of the secondary school do not share the same object of work on the network. For the director, it is a question of working on 'the articulation between the levels of education'. More specifically, it is a question of working on the articulation between primary and secondary schools, which explains why the agenda of the steering committee concerns almost exclusively cycle 3. For the coordinator, on the contrary, it is a question of 'working on new pedagogical modalities with primary and secondary school teachers who are not used to working together'. While the coordinator is trying to create a different space for teachers to collaborate, the secondary school principal is trying to control the actions implemented on the network insofar as they must mainly concern cycle 3.

Authority, control of change and power to act

This conflict illustrates the resistance inherent in the interaction between different activity systems to the networking of disadvantaged institutions. The actors (the director of the secondary school and the inspector) who are institutionally the managers of the NDS seek to maintain the authority of which they are institutionally responsible. In doing so, the network coordinator finds it impossible to carry out her work: it is a question of organizing school work in new forms in a dedicated collaborative space where teachers work in horizontal collaboration. In this example, the director of the secondary school has a direct hierarchical authority over teachers, an authority he does not intend to grant: 'At our level, we are the ones who make the changes in secondary school. Mastery of change (…) and work materials are essential to be able to control this change. At the network level, we direct the change, that is to say the inspector and me, but after it is not we, who will implement the actions that we suggest.'

The question of control of change seems very important for the managers and they try to maintain their authority to enable this control in the network. This explains why the director exercises a form of authority over the coordinator, although she is not hierarchically in her vertical division of labour. The coordinator has to develop the collaborative work in the network without having authority on her work: 'We, coordinators, are the converging center of conflicting interests'. In this diagram, it is possible to note that this relation of authority is in a relation of contradiction with the object that it makes of the network. For her, it is "to offer teachers of different levels of school different working modalities that engage them together." And projects, it's always a different modality. So suddenly, things have a meaning for students ". This authority relations diminish the power to act of the coordinator.

Object of work and vertical management of change

In an another steering comitte, the vice director of the secondary schools ask for the harmonization of the student skills assessment sheet which enables transmission of information between primary and secondary schools.

During interview he said that he needs to harmonize form between schools to be able to constitute homogeneous classes at the beginning of the year.

For the directors of primary schools this request is badly felt: « as usual it is a demand from the secondary school »; « we have to do the job three times: first for the institution, second for the student's parents, third for the college». They explain that is it difficult for them to force their colleagues to harmonize their assessment practices, because, in the French system, director of a primary school is a function and not a job: they don't have the authority to order their colleagues to change their assessment practices.

The inspector of primary schools understood that « all teachers do not respect the institutional rules of evaluation: some continue to evaluate with grades whereas they should evaluate skills ». Director of nursery schools are looking for people to join a conversation that is not directed at them: «It's always like this: no demand comes to us. While we have competencies to participate to such task.

The fact that the object for the direction of the secondary school is to work between the level of teaching implies that all the meeting is focus on the cycle 3 (see figure 1). This articulation is also problematic because of institutional differences between the first One of the participants explain that: "In terms of contradiction, they are trying to steer us towards a 6-year-old school of 16 years old with completely different status, the culture of the second degree will never be able to function like the first degree. We pilot with opposing energies that are institutional ". Moreover, because the director of the secondary school is the designed manager of the steering committee (with also the inspector of the primary schools), it involves that the director of the nursery schools tries to find their place in the meeting. However, the early enrolment of socially disadvantaged children appears to be an important lever in the fight against inequalities in school success.

From symptom to contradictions: implementing a network in a hierarchical system Figure 3, based on the work of [START_REF] Spinuzzi | Toward a Typology of Activities:Understanding Internal Contradictions in Multiperspectival Activities[END_REF], represents the contradiction between the implementation of the reform embodied, on the one hand, by the activity of the coordinator (whose function was created in the reform) and the activity of the principals who seek to maintain control. The reform that promotes network collaboration comes up against a system that is hierarchically and vertically organized, on the one hand, by the student's path, and, on the other hand, by the way in which it is managed. [START_REF] Spinuzzi | Toward a Typology of Activities:Understanding Internal Contradictions in Multiperspectival Activities[END_REF] Expansion of object without transformation of power relations?

In Activity-Theory, the identification of contradiction is of crucial importance: "If actors are able to identify and analyze the secondary contradictions of their activity system, they may focus their energy on the crucial task of resolving those contradictions by means of expanding the object and reorganizing the activity accordingly, instead of being victimized by changes that roll over them as it forces of a natural disaster (Engeström, 2018, po. 171).

The expansion of the object of activity need to be distinguish from quantitative increase of extension but is a qualitative and transformative reorganization of the object [START_REF] Engeström | Spatial and Temporal expansion of the object as a challenge to reorganizing work[END_REF]. Engeström (2000) identify tree dimensions of the expansion of the object: the social-spatial dimension (who else should be included); the anticipatory-temporal dimension (what previous and forthcoming steps should be considered) and the moral ideological dimension (who is responsible and who decides?).

In our study, we identify three potential dimensions of expansion of the object:

(1) The socio-temporal expansion of the object: this means that the object become the manage the network in order to resolve the problems teachers faces. In that way, the coordinator become the more relevant person to manage the network because he/she is also involved in vocational training of teachers of the network.

(2) The temporal expansion of the object: In that way, the object of the network is to build a coherent curriculum for student. The instrument the collective choosen to discuss and to build is four educational path which are a part of other reform of educational system: health education courses, citizenship educational pathway,

(3) The ethical-political expansion of the object. Network were chosen by the French ministry of National Education because of social difficulties. They are implanted on territory which are involved in in the politics of town because the population cumulate social, economic and cultural difficulties. In that expansion of the object, the object of the network is to begin to be an actor of the territory in which it is implemented.

However, in the discussion, the collective always start to expand the temporal dimension of the activity. This temporal expansion let intact the division of work and the relations of power that matters in the intervention. This may be not the case in the future reforms of educational system which will involve more actors outside the school (see below).

Discussion and conclusion

Power and authority

The question of power and authority has been very rarely formalized in AT as in research on organizational learning (Blacker & McDonald, 2000[START_REF] Engeström | Spatial and Temporal expansion of the object as a challenge to reorganizing work[END_REF]. In view of the possible link between the hegemonic tendency towards social reproduction and its inclusion in structures (such as the school institution) in which the actors act, the question of power and authority seems crucial to us in order to nourish the development of emancipatory activity within intervention. The role of activity theory in this context is indeed to collectively overcome the contradictions identified, or in the words of Sannino & Vaino (2015, p.519) 'to give rise to wider transformations in work and organizations'. In other words, the question of Williams, Ferholt, Jornet, Nardi & Vadeboncoeur (2018, p. 190) seems to us at the heart of the development of a forth theory of activity theory: "When we analyze an activity system, help resolve some of its contra-dictions, and claim to have made the system more efficient, have we changed, transformed, or transgressed the power relations and structural relations that matter? Has there been "development" from a critical, or any perspective?"

The interest of this paper is to make a modest contribution to the development of a fourth generation of activity theory. Indeed, [START_REF] Engeström | Spatial and Temporal expansion of the object as a challenge to reorganizing work[END_REF] evokes the idea of a 'runaway object' as a largely fragmented object, out of control and involving the linking of multiple interacting heterogeneous activity systems. We believe that the fight for equal student achievement is such a form of object in that it involves the collaboration of multiple activity systems: primary and secondary schools, of course, but also associations, local authorities, the regional political level, the Ministry of National Education, as well as researchers engaged in developmental work research. However, grasping these objects probably implies taking into account the power relations that are expressed in local situations, to enable actors to overcome them by reinventing their practices and relationships.

Future development: expansion of the network in future political reforms

In the continuity of priority education networks, the current government intends to develop the idea of reorganizing the education system around the networking of heterogeneous actors. The Educational Cities aim to intensify the educational care of children from 3 years old and young people up to 25 years old, before, during, around and after the school setting. They consist of a broad alliance of educational actors working in the priority districts of urban policy: State services, local authorities, associations, residents. Through the Cities of Education, the Government wants to federate all the actors in school and extracurricular education in the territories that need it most and where public resources will be concentrated.

Starting from the work on network in priority education, the future challenge will be to involve more heterogenic actors concerned with education in a specific territory.

Figure 1 :

 1 Figure 1: The organization of education cycles in French School since September 2016

Figure 2 :

 2 Figure 2: A network of disadvantaged schools

Figure 3

 3 Figure3illustrates the configuration of authority relationships during this meeting. A relationship of authority is expressed by the principal on the coordinator so as to enforce the

Figure 3 :

 3 Figure 3: Relation of authorithy during the steering comittee

Figure 4 :

 4 Figure 4: Contradiction related to the implementation of the reform based on[START_REF] Spinuzzi | Toward a Typology of Activities:Understanding Internal Contradictions in Multiperspectival Activities[END_REF]

Figure 5 :

 5 Figure 5: The educative city. Reform starting in September 2019

PISA is an internationally standardized evaluation of educational systems in the 30 OECD countries. The purpose of PISA is to assess the extent to which 15-year-olds completing compulsory schooling have the knowledge and skills required to play their full role in society(Figazzolo,

2009, p.3).


L Ghizzo

D Trémouilles

F Richardeau

S Vinnac

F Jamin

G Guibaud

Over-Voltage and Cross-Conduction Hard Switching Stress on Schottky Gate-Type p-GaN HEMT in Half-Bridge Operation Experimental and Physical Approaches

The GaN HEMT power device emerges as a promising wide-bandgap component for obtaining ultra-compact and very high efficiency power converters. Nevertheless, its long-time switching operation capability in high overload regimes near or just above its maximum ratings remains unknown. The physical degradation mechanisms specific to GaN HEMT as well as the associated electrical signatures must now be established and better understood. In this article a stress protocol is proposed combining or not functional drain-source over-voltage switching and transient low dead-time cross-conduction stresses. A no-loaded Half Bridge (HB) is used. All stress parameters studied have been associated to electric changes such as drain-source, gate-source leakage currents, and threshold voltage drifts of intrinsic values and even some physical degradation by LIT compound analysis. After analysis, it appears that both the electric field and the current are very stressful for the devices and lead to current leakage caused by physical degradation. The aging observed might be generated from the injection of hot electrons and located depending upon the drain voltage value which influences the depletion region and thus the electric field peak.

Introduction

The HEMT GaN transistor has several advantages over its silicon counterpart in power applications where high efficiency, high power density, and high frequency are important. However, its functional reliability is still a concern for some even though significant results have been obtained in hard switching regimes in [START_REF] Minetto | Hot electron effects in AlGaN/GaN HEMTs during hard-switching events[END_REF] [START_REF] Rossetto | Evidence of Hot-Electron Effects During Hard Switching of AlGaN/GaN HEMTs[END_REF]. To extend the functional reliability of the p-GaN HEMT from a use within its nominal ratings to overload regimes (near or just above its maximum ratings), a no-loaded halfbridge inverter topology is considered in this study. In addition to the VDS over-voltage stress, the 2DEG switching energy stress applied through COSS discharge can be greatly intensified using a small dead-time value and therefore leading to an additional cross-conduction stress. The effect of the over-voltage magnitude (120% of the bus voltage Vdc) under hard-switching is studied in correlation with the cross conduction stress. Intrinsic values (Crss, Coss, Ciss, Igss, Idss, Vth, etc.) are monitored to understand the electrical phenomena responsible for the aging in each case. This paper presents the no-load halfbridge and cross conduction test setup, the drift of device's intrinsic values and gives a possible explanation based on experimental results and physical mechanisms analysis.

Experimental details

The device under test (DUT) is a commercial, 100V, 90A, 7mΩ packaged p-GaN gate AlGaN/GaN HEMT. 26 devices were installed on 13 half-bridges integrated on dedicated daughter Printed Circuit Boards (PCBs) which were developed and used for the proposed stress as illustrated in Fig. 1(a) and (b) and plugged-in on a main power motherboard including its FPGA-based control set as illustrated in Fig. 1(c). Results presented in this paper are based on a stress duration of 35 hours (7 hours/day). Characterization measurements are performed with a Keysight B1505A power device analyzer/curve tracer. The switching frequency Fsw is set to 2kHz, this very low value allows to avoid heating up the components during the stress to only focus on electrical stress and not consider the thermal stress that could complicate the interpretation. A higher switching frequency will degrade more rapidly the DUTs and can change the degradation mechanisms with the temperature stress added. This stress will be studied with a very precise monitoring of temperature in a future study. The absence of load leads to negligible losses (25mW/transistor at 2kHz) and makes it unnecessary to add a heatsink. With a nominal dead-time of typically 120ns, the 2DEG is only activated by the controlled transistor during the discharge of its COSS and to recharge the opposite transistor's COSS under Vdc with a high dv/dt and low losses. Now considering a low deadtime (Td) value, a cross-conduction arises quickly over a few nanoseconds and exhibits a 2DEG through the two transistors with a lower dv/dt extra losses, as explained hereafter. The waveforms of Fig. 2 illustrate the main parameters of interest. All stress parameters are matched to their respective daughterboards' numbers in table 1. Two main stress parameters are studied. The switched voltage Vdc, on the transistors of boards #4 and #5 for Vdc chosen voluntarily high (120 % of the rating) and on boards #7 and #11 for a smaller value (Vdc = 67V: 2/3 of the rating). The cross-conduction will be studied (with a small Td and high RG,OFF) on boards #1,#2,#3 for a high Vdc value and on #9 and #10 for a smaller value. When the LS transistor is off, the turn-on of the HS transistor through the small RG,ON produces a strong positive dv/dt on the LS drain. The negative dv/dt applied to the CGD of the LS transistor generates a current peak from the gate to the driver through RG,OFF that induces the spike ∆V in Fig. 2(a). A high RG,OFF value of 50Ω produces a large VGS spike and combined with a short Td it generates a cross-conduction phenomenon. In fact, if Td ≈ 20 -30ns, the spike will appear earlier on the decreasing slope of VGS at the turnoff, and could exceed the threshold voltage [START_REF] Perera | Hard-Switching Losses in Power FETs: The Role of Output Capacitance[END_REF]. The active turn-on of one transistor forces the re-triggering on the opposite transistor that is turning off (or that has just turned off), as shown in Fig. 2(d)(e)(f). For the duration of the spike (≈ 10 to 20ns) LS and HS 2DEGs are simultaneously fully activated and a significant current flows through both channels under a smaller dv/dt than in a long dead-time case (26 kV/µs without cross-conduction and 11kV/µs with cross conduction). Indeed, as illustrated in Fig. 2(d) the extra 2DEG formed in the LS transistor can be assimilated to a transient load subtracted from the active HS 2DEG and therefore reducing the resulting current injected into COSS across LS and HS devices. The energy at turn-on Eturn-on is then increased (×10) with an additional cross-conduction energy Ecross-cond and mo re stressful and could be expressed as [START_REF] Jadli | A Simple Equation for the Energy Stored by Voltage-Dependent Capacitances[END_REF][5]:

Eturn-on = VdcQCOSS+ Ecross-cond [START_REF] Minetto | Hot electron effects in AlGaN/GaN HEMTs during hard-switching events[END_REF] The turn-on energy with cross-conduction is four times higher than the one in the case of a 400µH (≈40A) Half-Bridge without cross-conduction. Boards #6,#7,#8 and #11 will serve as references for the stress applied on the other boards. A DC voltage of 67V was applied on the transistors of #12 and #13 for 35h at off state (VGS = -3V) to compare with dynamic stress as well.

Drift and degradation after the stresses applied to

the transistors.

Experimental results

The comparison between the characterization measurements at t0 (before stress) and t35h (after a stress duration of 35 hours, 7h/day) shows some irreversible degradation and drifts of electrical characteristics as highlighted in Fig. 3,4,5,6. Initial curves at t0 are displayed in cool colors (blue, green, violet) and after stress in warm colors (orange, red and brown). Forward and backward sweeps are performed to take into account a possible hysteresis. The results of the high voltage stress are shown first and the low voltage stress results are shown next. The influence of cross-conduction is shown is each case. The transistors of boards #4 and #5 were stressed on the drain with a switched voltage of 120% of their rating (i.e. Vdc = 120V). The irreversible degradation is visible on both transistors of board #4 as shown in Fig. 3(c) on the drain leakage Idss = f(VDS ). This degradation is visible starting at VDS = 90V for the LS, and around VDS = 98V for the HS. It is characterized by a steep increase of the drain leakage current. Below this voltage, the leakage current mainly overlaps with the curve at t=t0 and the drain leakage is around 10nA at this voltage. When VDS is reduced again (backward sweep), the leakage current gets back to a normal value as in the forward sweep as shown in Fig. 3(d). This leakage current seems to be between drain and source, as the gate current during this measurement is not changed (similar curves to Idss of board #2 in Fig. 3 The devices of boards #2 and #3 were also stressed under Vdc = 120V (like boards #4 and #5) with an additional cross-conduction phenomenon. They also exhibit a change of the drain current leakage above 80V but stronger than in the previous cases. This leakage can reach the mA range. The capacitance CGD = Crss has significantly changed in this case and becomes constant just before or near the abnormal current leakage.

High voltage stress, no cross-conduction

High voltage stress with cross-conduction

Low voltage stress with cross-conduction

To evaluate the impact of cross-conduction alone, a stress under VDS = 67V (2/3 of the voltage rating) with a short deadtime of 20ns is conducted on boards #9 and #10. The degradation of the HS transistors identified by a high leakage current between gate and drain is depicted in Fig. 5(c) and (d). The capacitance Crss knee's drift to higher voltage (Fig. 5(b)) and the gate leakage current Igss = f(VGS) tends to increase especially at low voltage (Fig. 5(a)). The LS switches exhibit the same degradation patterns, yet in a less pronounced manner. The devices of board #1 that underwent cross-conduction at higher VDS share the same degradation patterns on Igss between t0 and t35h in addition to a constant Crss starting at 60V. The Idss curve has both the electrical changes observed before: a gate drain leakage current and a steeper increase of the drain-source leakage current above VDS = 90V.

Low voltage stress, no cross-conduction

The transistors of the other boards #7,#11,#6,#8 tested under a Vdc voltage within the ratings and no crossconduction show a similar change but less significant than the low voltage cross-conduction case (see Fig. 7 for #8). A high gate-drain leakage current also appears and the Crss knee is shifted to higher VDS values for all transistors. The stress was done with a smaller RG,OFF value for #7 and #11 in order to reduce the switching energy but no change has been observed compared to transistors #6 and #8. Most of the transistors were degraded after 35 hours (leakage current above the ratings) yet all devices are still working post-stress. Despite being physically degraded after t35h, the transistors are still functional, they still switch and no visible change (dv/dt, δVGS, oscillations...) has been observed, even for the degraded ones. The less degraded transistors have leakage levels within the ratings given in the datasheet despite being physically and electrically "changed". With the hard switching stress or high VDS voltage stress, 100% of the devices tested drifted or were degraded. Unexpectedly, for the "non-stressed" transistors (VDS 67V, long deadtime conditions) that were not supposed to drift as well as for the transistors that were under DC voltage at off-state for the same duration, around 30% of these devices were physically degraded (for example board #8 as it can be seen in Fig. 7).

Table 2: Summary of the most significant drifts of intrinsic parameters observed on the stressed devices. The arrows stand for increase or decrease, the symbol ≈ stands for "less significant" for the devices that exhibit the same drift but to a lesser degree, "× Dpin" means that the Dpin effect is not present anymore due to the degradation.

Table 2 summarizes the most important degradations observed during the stresses. The high voltage stress (120V) seems to change the drain leakage current at high VDS. The transistors stressed at low voltage shows a gatedrain leakage current and a Crss-knee shift. The crossconduction increases the level of leakage and degradation in both cases (Fig. 3 vs Fig. 4). The high voltage stress with cross-conduction can combines the two changes (see Fig. 6) , a high drain leakage current appears as well as a GD leakage and capacitance knee shift. A physical interpretation of these results is given in the next part. board, all the board layers (PCB and copper layers), leadframe and transistor package were mechanically removed by parallel lapping. The silicon substrate was thinned to a thickness of 100µm. The view of the chip on Fig. 8 is an upside-down view of the chip's GaN layers through the silicon substrate. Around 20 hot spots appear on the chip with one hotter than the others (10x magnification zoom in small window).

Hypothesis on degradation

A localization of physical degradation has been conducted with infrared Lock-In Thermography (LIT) (Fig. 8) indicating the location of possible physical degradations in the transistor. Many hotspots were revealed on the whole active surface of the device. As shown in Fig. 9, that uses another type of localization of defect method called Photon Emission Microscopy (PEM), the emission spots (defects zones) are on two different locations on the metal pattern of the transistor represented on Fig. 9c. One of them is near the gate, between the gate and the field plate (FP) (Fig. 9a) while the other is closer to the drain contact, between the field plate and the drain metal (Fig. 9b). The most stressed devices (#1,#2,#3: High voltage and cross-conduction), have numerous spots in both LIT and PEM (more than 20) and are on the two locations that have been described before, but mainly near the drain edge. In general, it has been observed that spots are near the drain for devices that have been stressed at VDS=120V and near the gate for 67V stress. A non-stressed device was also tested and presented no such hot spots.

Degradation near the gate metal (Low voltage/crossconduction)

The spots located between gate and field plate could be related to the gate-drain leakage current observed on transistors of boards #1,#8,#9 (HS only), #10 and #11. It could be the failure of the Dpin diode in the gate stack see Fig. 10a or may correspond to a leakage path on the sidewall of the p-GaN layer as illustrated in blue in Fig. 11. This zone is a weak spot in terms of insulation, the form of the gate metal makes a spike and a cavity below, reducing the insulation distance. During the FIB Slice&View it has been observed that this distance could be much smaller than the one represented in Fig. 11. Electrically, it shorts the equivalent Dpin diode between the p-GaN/AlGaN and AlGaN/GaN interfaces. This leakage current is correlated with the Crss-knee shift to higher VDS (Fig. 7c). The preconditioned threshold voltage (see the full protocol in [START_REF] Ghizzo | Preconditioning of p-GaN power HEMT for reproducible Vth measurements[END_REF]) is also affected with an increase up to 30% as shown in figure 7. The CGD-knee shift to higher VDS is an indicator of degradation in the field plate (FP) area.

This degradation is ascribed to charge trapping in defects caused by hot electrons generation. These traps could be located in the oxide layer (injected via tunneling over the AlGaN barrier, the same traps can also be in the GaN layer) and change the polarization around the 2DEG. The new polarization shifts the Crss knee to a highe r voltage (shift up to 20V) which means it affects the depletion behavior of the2DEG under VDS at off-state as explained in [START_REF] Ghizzo | Preconditioning of p-GaN power HEMT for reproducible Vth measurements[END_REF]. The 2DEG under the FP is still activated at a higher VDS than at t0. The presence of the 2DEG in this range of VDS voltage acts against the field plate effect. Indeed, the FP purpose is to protect the gate stack from the electric field by moving away the main peak to the edge of the FP with the depletion of the 2DEG. Then during the electrical stress VDS generates a high electric field which eventually leads to a local gate-drain breakdown. In another study the negative breakdown voltage (BV) value of the gate was measured around -38V which meets the voltage value of the shifted Crss-knee after stress. It indicates that this voltage may have been applied on the gate during the stress. Note that the "drain" in this leakage path is actually the 2DEG connected to the drain metal.

The nature of defect for the accumulated degradation might either be cracks formation through thermomechanical stress, generated by hot electrons scattering the lattice [START_REF] Chaudhuri | On the Channel Hot-Electron's Interaction With C-Doped GaN Buffer and Resultant Gate Degradation in AlGaN/GaN HEMTs[END_REF] or the hypothesis stated in [START_REF] Pozo | GaN Reliability and Lifetime Projections Phase 14, Efficient Power Conversion[END_REF]. The report of [START_REF] Pozo | GaN Reliability and Lifetime Projections Phase 14, Efficient Power Conversion[END_REF] explained it might be hot electrons that generate electron-hole pairs with impact ionization, injecting holes in the oxide layer, locally increasing the electric field too and leading to dielectric failure. A metal filament can be formed from the metal gate for instance leading to the GD leakage. A Transmission Electronic Microscope lamella would be necessary to verify the hypothesis made in this paper.

For the transistors that were less stressed (67V, 120 ns and DC stress: #6,7,8,11,12,13) the higher electric field on the gate stack due to the Crss-knee shift due to trapping reducing the FP effect is also the cause of failure. However, in these cases the trapping phenomenon is far less significant and might not be related to hot electrons and degradation of layers but is only due to the electric field of VDS [START_REF] Chen | OFF-State Drain-Voltage-Stress-Induced VTH Instability in Schottky-Type p-GaN Gate HEMTs[END_REF] this work might present a technology that may not have a sufficient quality regarding of the high hysteresis encountered, the low duration before degradations and the significant instability of Vth in [START_REF] Ghizzo | Preconditioning of p-GaN power HEMT for reproducible Vth measurements[END_REF].

Degradation near the drain metal (High voltage)

The second spot location is a defect near the drain edge (or field plate drain side edge but less likely). This defect localization is correlated with a change of Idss and CGD at high voltage (above VDS = 80V). The capacitance after the knee is reduced and for the strongest stresses (#2, #3 for instance) it even becomes constant at a given VDS value. The depletion of the 2DEG is faster after the field plate(Vds>20V) for a degraded device and the surface of the 2DEG does not even vary (after 65V on Fig. 4b for instance) for the devices that show a constant capacitance from a given VDS value at t35h. It means that the 2DEG is fully depleted or that the leakage current is entirely vertical through the defect. Indeed, the drain-source leakage current has no influence on the gate current during the Idss measurement as seen on the characterization results (see Fig 4c and4d).

This leakage path is expected to be at the edge of the drain metal and to be vertical across the GaN layer as illustrated on Fig. 10(b) and Fig. 11 in yellow. The leakage current then reaches the conductive epitaxial layers (AlGaN or GaN, the epitaxy is a stack of AlGaN and GaN layers) below the GaN buffer and connected to the source electrode. This leakage is not across the whole silicon substrate as the thinning of the silicon substrate (10 µm to 2µm) and backside package removal did not make the spot disappear.

A drain-source leakage current between the field plate and the 2DEG across the AlGaN and oxide layer under the field plate hypothesis can be ruled out thanks to another study we made on this part-number device (study of static VDS breakdown voltage). This defect location leads to the melting of the FP and to a filament path (around 150V) see Fig 11, the current leakage path is a direct short and not an abnormal current leakage appearing only above 80V like in this case. A Focused Ion Beam (FIB) Slice and View verified this failure mechanism (Fig. 11 in red).

The current and high electric field distribution might alter the GaN layer quality through the emission of hot electrons that transfer energy to the lattice and change its quality as the stress is an aggravated semi-ON/Hard switching stress [START_REF] Rossetto | Evidence of Hot-Electron Effects During Hard Switching of AlGaN/GaN HEMTs[END_REF]. The high VDS chosen in the stress of boards #1-5 eases this location degradation compared to 67V stress. With a same drain bias of 120V, crossconduction accelerates this alteration as it increases the number of electrons and energy. The layer is then sufficiently degraded to completely deplete the 2DEG from VDS = 60V during the off-state as shown in Fig. 4(b). The completely depleted 2DEG during the stress increases the electric field peak amplitude at the edge of the drain metal edge. The peak location will locally accentuate the degradation caused by hot electrons. As in the 67V stress scenario the physical defect is suspected to be a crack or a cavity due to thermomechanical energy generated by hot electrons.

This defect is located in a thick layer and does not entirely break the electrical isolation as the leakage appears only at high VDS (a leakage threshold voltage value). The defect reduced the thickness of insulation, as VDS increases the electric field peak amplitude also increases especially under the drain edge metal. At a given value (around VDS=90V) the electric field amplitude is sufficient to reach the threshold leakage value and the leakage current Idss can follow the new vertical leakage path across the GaN layer. This mechanism might be similar to Stress Induced Leakage Current (SILC). a small cavity which reduces the dielectric insulation above (blue circles). This particular zone seems to be a weakness of this structure, in other pictures of the FIB Slice&View, it has been observed that the spike and the cavity could be bigger. In red the leakage path formed between drain and source in the previous study about static breakdown voltage. The AlGaN barrier is not visible in this picture due to its small thickness.

Conclusion

This original no-load Half-Bridge test is particularly stressful and degrades the transistor very rapidly compared to its expected lifespan. A high drain leakage current occurs through physical defects that may originate from hot electrons's damages. The degraded areas are around the gate or in the GaN buffer, depending on the electric field distribution, which depends on the 2DEG depletion and thus on the bus voltage. The crossconduction stress seems to accelerate and amplify the degradations as it increases the quantity of electrons. It has been observed that high voltage stress generates a high drain-source leakage current at high VDS. This leakage current might come from a degradation in the GaN buffer near the drain metal. A stress at low voltage or a crossconduction stress creates a Crss knee shift and a high gate drain current leakage. This electrical change is supposed to result from a physical degradation near or the gate stack. Further failure analysis will be conducted to verify the degradation mechanisms.

  (P) applied to each couple of transistors of the daughterboard number (#N). L. Ghizzo et al., LAPLACE -LAAS-CNRS, Université de Toulouse, CNRS, INPT, UPS, Toulouse, France -THALES SIX France 34th European Symposium on Reliability of Electron Devices, Failure Physics and Analysis (ESREF) from October 2nd to 5th, Toulouse -France, 2023. Postprint Accepted Author Manuscript : https://hal.science/ . Postprint Accepted Author Manuscript :

Figure 1 :

 1 Figure 1: (a) Schematic of the half bridge daughterboard composed of a decoupling ceramic capacitor connected to two HEMTs, one high-side and one low-side , and their gate control circuit with gate resistors RG,ON and RG,OFF. The capacitor can be easily disconnected to perform characterization measurements (b) daughterboard. (c) Test bench composed of the main motherboard (12 legs in parallel) and 2 daughterboards. Gate driver boards are connected to feed gate voltages by a FPGA board with a user interface where stress parameters can be selected.

Figure 2 :

 2 Figure 2: (a) LTspice simulated waveforms of the driver voltage Vdrv and gate voltage VGS at the turn-off of the LS transistor. b) driver voltage Vdrv and gate voltage VGS at the turn-on of the HS transistor, the threshold voltage Vth is in dashed line. (c) LS and HS transistors' VDS and drain current. (d) Vdrv, VGS at the turnoff of the LS transistor with a small Td that generates crossconduction. (e) driver voltage Vdrv and gate voltage VGS at the turn-on of the HS transistor, (e) LS and HS transistors' VDS and drain current (multiplied by 10) in the case of cross-conduction.

Figure 3 :

 3 Figure 3: Characterization of the stress board #4 before (t0) and after 35 hours of stress (t35h) at 120V with no cross-conduction (Td = 120ns). (a) Gate leakage current Igss = f(VGS ) at VDS = 0V. (b) Crss and Coss under VDS at F = 1 MHz. (c) Drain leakage current Idss= f(VDS ) at VGS = 0V. (d) Display of (c) with log scale. The main degradation to observe is the high drain source leakage current that appears around VDS = 90V for LS and VDS = 98V for HS.

  (a)). L. Ghizzo et al., LAPLACE -LAAS-CNRS, Université de Toulouse, CNRS, INPT, UPS, Toulouse, France -THALES SIX France 34th European Symposium on Reliability of Electron Devices, Failure Physics and Analysis (ESREF) from October 2nd to 5th, Toulouse -France, 2023. Postprint Accepted Author Manuscript : https://hal.science/ 3/8

Figure 4 :

 4 Figure 4: Characterization of board #2 before (t0) and after (t35h) stress at 120V with cross-conduction (Td = 30ns). (a) Gate leakage current Igss = f(VGS) (b) Crss and Coss. (c) Drain leakage current Idss = f(VDS ) at VGS = 0V. (d) Gate current during the Idss measurement depicted in (c). The main degradation to observe is the high drain-source leakage current that appears after a threshold voltage for around VDS = 95V for the LS and VDS = 97V for the HS.

Figure 5 : 8 Figure 6 :

 586 Figure 5: Characterization of the stress board #10 before (t0) and after 35hours of stress (t35h) under 67V with cross-conduction (Td =20ns). (a) Gate leakage current Igss = f(VGS). (b) Crss and Coss. (c) Drain leakage current Idss = f(VDS). (d) Gate current during the Idss measurement. The main degradation to observe is the drainsource high leakage current that appears for LS and the gatedrain leakage current for HS.

Figure 7 :

 7 Figure 7: Characterization of the stress board #8 before (t0) and after 35hours of supposedly no stress (t35h) under 67V without cross-conduction (Td = 130ns). (a) Drain leakage current Idss = f(VDS). (b) Gate current leakage during Idss measurement of (a) showing the gate drain leakage (c) Crss and Coss. (d) Preconditioned measurement of Vth (for obtaining a stable value independent from the history of the device), a shift up to 25% can be observed.

Figure 8 :

 8 Figure 8: Infrared light LIT of transistor #2 LS (stitching of several images) with a 5x magnification under an electric polarization between 75V and 95V (lock-in frequency 25Hz), applied with probes landed on the electrical vias. The acquisition time is 3 to 5s. The LS transistor of board 2 was cut-off from the

Figure 9 :

 9 Figure 9: InGaAs camera PEM image of the emission spots. (a) Defect near the gate that might be a GD leakage current through the AlGaN barrier (b) defect near the drain metal that is expected to be a DS leakage current through the GaN layer. (c) optical zoom of the metal structure seen from the backside. The blur in these pictures cannot be avoided as it depends on the camera resolution and the remaining GaN thickness of 10 µm we look through with the IR camera.

  . The traps accumulated with time increased the shift and the electric field near the gate stack leading, for some of the devices, to failure. The samples lot used in L. Ghizzo et al., LAPLACE -LAAS-CNRS, Université de Toulouse, CNRS, INPT, UPS, Toulouse, France -THALES SIX France 34th European Symposium on Reliability of Electron Devices, Failure Physics and Analysis (ESREF) from October 2nd to 5th, Toulouse -France, 2023. Postprint Accepted Author Manuscript : https://hal.science/ 6/8

Figure 10 :

 10 Figure 10: (a) Formation of the gate leakage path through degradation caused by the emission of hot electrons at (a) VDS = 67V and (b) VDS = 120V. The yellow "explosions" symbols illustrate the defect heating/light emission captured with LIT and PEM on Fig. 9 through the GaN layer.

Figure 11 :

 11 Figure11: FIB cross-section of the active area showing the location where the defects are expected to be according to our hypothesis. In yellow the drain-source leakage which corresponds to Fig.4celectrical signature. In blue the gate-drain leakage (Fig.7ab) which is the failure of Dpin diode shown in Fig.10. Also, the form of the metal contact is not perfect due to fabrication maybe thermal relaxation. It makes a small spike and

Table 1 :

 1 Table of the stress parameters

Postprint Accepted Author Manuscript : https://hal.science/

  Ghizzo et al., LAPLACE -LAAS-CNRS, Université de Toulouse, CNRS, INPT, UPS, Toulouse, France -THALES SIX France 34th European Symposium on Reliability of Electron Devices, Failure Physics and Analysis (ESREF) from October 2nd to 5th, Toulouse -France, 2023.

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L.

Acknowledgements

This is part of a PhD study on power GaN reliability at Thales Toulouse, LAAS-CNRS and Laplace laboratory in Toulouse and the follow-up of Centre National d'Etudes Spatiales (CNES).

This work was supported by the LAAS-CNRS PROOF platform, partly financed by the Occitanie region.


Jan Stępień

Dung Di Caprio

Janusz Stafiej

Dung Di Caprio

3D simulations of the metal passivation process in potentiostatic conditions, using discrete lattice gas automaton

Keywords: Corrosion, Modelling, Cellular Automata, Passivation, parallel

published or not. The documents may come

3D simulations of the metal passivation process in potentiostatic conditions using discrete lattice gas automaton

Introduction

Most of metals and alloys of practical importance are thermodynamically unstable in contact with water, oxygen and aqueous electrolytes. As a result, corrosion processes initiate at their surface in contact with an electrolyte solution. They owe their relative metastability to passivation. The passive layer forms at the metal surface, and it is composed of hardly soluble corrosion products -metal oxides or insoluble salts. It can slow corrosion down to practically negligible rates. In our study we try to address the following problems: When and how does this phenomenon occur? How precisely can we control the layer's thickness and morphology, e.g. by applying electric current or potential to a piece of metal? Can we apply the knowledge based on simulation to obtain interesting or useful nanostructures in real experiments? The nanopores appearing in surface oxide layers of most valve metals (Al, Ti. . . ) [START_REF] Yang | The precise preparation of anodic aluminum oxide template based on the current-controlled method[END_REF] are an example of such structures.

Computer simulations are our method of choice to study passivation. They can provide valuable information for the design of experiments and provide a better understanding of the experimental results [START_REF] Bartosik | Simulation of nanostructured surfaces obtained by passivity and growth[END_REF]. Recent experimental work in the field of corrosion and passivation is described e.g. in [START_REF] Pagitsas | Localized passivity breakdown of iron in chlorate-and perchlorate-containing sulphuric acid solutions: A study based on current oscillations and a point defect model[END_REF][START_REF] Sazou | Potential oscillations induced by localized corrosion of the passivity on iron in halide-containing sulfuric acid media as a probe for a comparative study of the halide effect[END_REF][START_REF] Sazou | Intrinsic coherence resonance in the chloride-induced temporal dynamics of the iron electrodissolutionpassivation in sulfuric acid solutions[END_REF][START_REF] Pavlidou | Potential oscillations induced by the local breakdown of passive iron in sulfuric acid media. an evaluation of the inhibiting effect of nitrates on iron corrosion[END_REF].

In this paper, we present the simulation results for passivation of a metal electrode in electrolyte solution in potentiostatic conditions (see fig. 1). A 2D mesoscopic discrete lattice asynchronous stochastic automaton model for simulation of such systems has been proposed. Its results are presented in our previous papers [START_REF] Di Caprio | Simulations of passivation phenomena based on discrete lattice gas automata[END_REF][START_REF] Di Caprio | The role of adsorption in passivation phenomena modelled by discrete lattice gas automata[END_REF]. The main idea behind the model is to make it as simple as possible, and still keep the most important features of passivation. Our goal for this paper is to create a 3D version of the model, in order to be closer to the physical reality. We are interested especially in the passive layer morphology and the transitions between active and passive regime.

Even for our simplified model the simulations appear to be computationally expensive. We use parallel processing on NVIDIA graphics processing units, such as Tesla C2050, and GTX Titan, which feature 448 and 2688 processor cores respectively. They are programmed using CUDA technology [9]. In the case of very strong interaction between the layer particles, manifested as a weak layer solubility, the computation can take more than a week for a single run.

Since our model is based on cellular automata (CA for short), let us recall that CA are a class of discrete space and discrete time abstract machines. A definition of a cellular automaton is composed of:

• finite or infinite set of cells, usually corresponding to nodes of a spatial lattice;

• finite set of possible cell states;

• specification of the neighborhood of a cell;

• transition rule that specifies a cell's state at the next time step as a function of its own state and the states of the cells in its neighborhood at the current time step.

In a traditional, synchronous CA all of the cells undergo the transition simultaneously according to the transition rule at each time step. The initial states of all cells have to be specified to start an evolution of the CA. A common feature of many CA is that complex behavior and patterns can result from a simple transition rule. This phenomenon is called emergence -see [START_REF] Gotts | Emergent phenomena in large sparse random arrays of Conway's Game of Life[END_REF] for an example with a detailed discussion.

CA are widely used in simulation of reaction-diffusion systems. The system considered in this paper belongs to this category. In particular, threedimensional CA are used for corrosion processes [START_REF] Di Caprio | 3D cellular automata simulations of intra and intergranular corrosion[END_REF][START_REF] Bartosik | 3D simulations of ordered nanopore growth in alumina[END_REF], materials science [START_REF] Lhuissier | Geometrical control of lattice structures produced by EBM through chemical etching: Investigations at the scale of individual struts[END_REF][START_REF] Chen | Three-dimensional cellular automaton-finite element modeling of solidification grain structures for arcwelding processes[END_REF][START_REF] Popova | Coupled crystal plasticity -probabilistic cellular automata approach to model dynamic recrystallization in magnesium alloys[END_REF][START_REF] Li | A three-dimensional cellular automata-crystal plasticity finite element model for predicting the multiscale interaction among heterogeneous deformation, DRX microstructural evolution and mechanical responses in titanium alloys[END_REF], electrode processes [START_REF] Pérez-Brokate | Cyclic voltammetry simulations with cellular automata[END_REF] and biology [START_REF] Wcisło | PAM: Particle automata model in simulation of fusarium graminearum pathogen expansion[END_REF], among others. An important feature of the CA is the fact that they often translate into parallel algorithms with no major difficulty.

In some cases, it is sensible to modify the traditional CA formula outlined above. One of the alternatives is the stochastic CA whose transition rule uses not only the states of the cells in the neighborhood but also a random variable. Another one is the asynchronous automaton where the cells change their states in an order, for instance random order, rather than simultaneously. Asynchronous CA are more tricky to implement on parallel systems than their synchronous counterparts. A way of doing this is discussed later. The cellular automaton considered in this article is both stochastic and asynchronous.

The rest of the paper is structured as follows: In the Model section the basic assumptions are exposed and the 3D cellular automaton model is presented.

The section Results and Discussion is what its title indicates -a presentation of the simulation results with an attempt to find their rationalization. Finally, in the Conclusions we summarize the paper and outline our future plans.

Model

Physicochemical basis of the model

Metal immersed in an electrolyte solution containing aggressive anions corrodes by electrochemical oxidation. Depending on the corrosion rate, solubility of the corrosion products and the strength of their adhesion to the metal surface, a passive layer may form. It protects metal from further corrosion by separating it from the solution. The corrosion products (further called oxide for simplification) should have a low solubility. But they can still dissolve, diffuse and precipitate back on the surface. Thus the passive layer morphology is subject to change. We also assume that some oxide mass is being permanently lost by ultimate dissolution in the bulk solution.

Even when the surface is completely covered by oxide the corrosion can still proceed, albeit at a much lower rate owing to a mechanism of transport through the passive layer. This is explained in the literature using the concept of ionic vacancies [START_REF] Sikora | Defining the passive state[END_REF]. In our model, instead, the solution diffuses in the oxide symmetrically to the oxide diffusing in solution. On the level of our mesoscopic description this is equivalent to the ionic vacancies model.

In this and our previous papers on passivation, the system is controlled by a potentiostat that sets a constant electric potential of the metal with respect to the solution by means of an appropriate reference electrode. The potential controls the rate of the metal surface electrooxidation in contact with the solution.

In the original 2D model from [START_REF] Di Caprio | Simulations of passivation phenomena based on discrete lattice gas automata[END_REF] the formation of the passive layer is a result of the oxide's low solubility only (high tendency to aggregate). Oxide's adsorption on the metal is absent, as the layer material does not adhere to the metal. Nevertheless, the oxide has a strong tendency to stay on the surface. It is sufficient for passivation, that the oxide is produced at the metal surface, and that the oxide layer prevents the formation of a solution layer in between the metal and the oxide layer. The model simulates the properties of passivation reasonably well. The revised model from [START_REF] Di Caprio | The role of adsorption in passivation phenomena modelled by discrete lattice gas automata[END_REF] introduces adsorption. The layer material sticks to the metal surface. This results in a significant change in the system behavior. Namely, in the no-adsorption version of the model, depending on the potential, we observe two regimes -bare metal surface at sufficiently cathodic potentials, and a thick oxide layer at higher potentials. In the adsorption version there appears a third, intermediate regime of a thin, but compact oxide layer.

In this paper, we employ a 3D version of that model in both no-adsorption and adsorption versions. More attention is given to the adsorption version as more realistic and resulting in a more complex behavior. As previously, in this paper we define and use our model for qualitative, generic features of a larger class of passivating metal systems and simulate the general properties of metal passivation for this class rather than for a specific system.

Discrete lattice model specification

The model presented in this paper is based on asynchronous, or more precisely -block synchronous stochastic automaton. The lattice is cubic, and the von Neumann neighborhood of radius one is used. Every cell has six neighbors: upper, lower, left, right, forward and backward. There are three possible cell states corresponding to chemical species: metal MET, metal oxide OXI and solution SOL. The initial state of the lattice is a metal block with a flat surface, covered with solution. No oxide is present yet. There are three parameters that control the system's behavior:

• Electrode potential V, influencing the corrosion rate;

• Oxide bond breaking probability P break , influencing the passive layer redistribution -P break is related to the oxide cohesion strength;

• Oxide ultimate dissolution probability P die .

There are three transitions corresponding to physical and chemical processes occuring in the system (see fig. 2; 2D illustrations are used for clarity). They occur stochastically with given probabilities:

1. Metal oxidation: MET + SOL → OXI + OXI, with probability

P corr = exp(V )/(1 + exp(V ))
for each MET cell neighboring solution (the SOL neighbor is chosen at random); 2. Oxide diffusion -swap with a neighboring SOL cell: OXI + SOL → SOL + OXI (SOL is chosen at random as well), with probability depending on the neighborhood, discussed in detail below; 3. Oxide dissolution: OXI → SOL with probability P die , for each OXI cell having only SOL neighbors.

Let us note that the basic parameter in our model is P corr , not V , while in experiments we deal with the potential V . The hyperbolic tangent formula used here is an arbitrary selection to convert the P corr scale to V scale in arbitrary units. We may ignore its real form, as the only important feature of this relation When calculating the probability of an oxide diffusion event p swap , we need to account for the adhesion of the oxide to itself and to the metal surface. Shortrange attractive interaction is modelled by introducing a bond between each two neighboring oxide cells, and optionally between oxide-metal neighboring cell pairs. Those do not necessarily correspond to chemical bonds, because the cells do not correspond to single atoms or molecules (the model is mesoscopic in scale). Two versions of the model have been used for the simulations:

• without adsorption -only OXI-OXI bonds are counted;

• with adsorption -MET-OXI bonds are included as well.

Each bond has an energy E bond which is proportional to -ln P break . Therefore, if the swap is going to increase the number of existing bonds, or leave it unchanged, then it is energetically favorable. It occurs with probability p swap = 1. If, however, the swap would decrease the number of bonds, then p swap = P n broken break , where n broken is the net number of bonds that would be broken during the transition, that is -the difference between the number of bonds existing before and after the swap. See fig. 3 for examples.

Metal and corrosion products form totally immiscible phases. Therefore, we expect that MET-OXI bond is weaker than OXI-OXI bond. We consider two limiting cases. First is no adsorption, where MET-OXI bond energy vanishes. The second is with strong adsorption -MET-OXI bond energy equals that of OXI-OXI bond. In the simulation, the same P break value is used for both types of bonds. This assumption is also plausible from a coarse-grained point of view regarding the metal surface. We assume that it is covered with a submesoscopic layer of metal oxides, especially at anodic potentials. Here, submesoscopic means much thinner than the cell size.

In this paper, we adopt such a coarse-grained perspective. Owing to the simplification of having just one P break parameter, we introduce adsorption of oxide on metal without an increase in the model's complexity. In a more finegrained point of view, we would have to admit that P break need not be the same for OXI-OXI and MET-OXI pairs.

Parallel implementation

If we try to formulate our model as a synchronous CA, then we run into the problem of collisions whenever two oxide cells would simultaneously move into the same solution cell; see fig. 4. Similarly, it is easy to imagine an event, when an OXI moves into a cell which is going to be occupied by a new OXI coming from a simultaneous corrosion reaction. It is not clear how to process the collisions simultaneously. The solutions are different for sequential or parallel computation. In the sequential case [START_REF] Di Caprio | Simulations of passivation phenomena based on discrete lattice gas automata[END_REF], we can replace the classic CA with an asynchronous automaton, which processes the cells one by one, in a random sequence. This is not suitable for parallel computation since the sequential update by its very nature cannot be distributed between multiple processors. In this case an option is to use a block-synchronous automaton (BSA for short). Here, the system is divided into small blocks. The size of such unit block should be as small as possible, but large enough to prevent the collisions -in this case it is 4×4×4. Then, every single time step of the BSA is executed using the following algorithm:

• choose a random cell within the unit block;

• do a CA transition for that cell's translated image in every block;

• Repeat those two steps n times, where n = volume of the block (here, 64 times)

An example of lattice division into blocks is shown in fig. 5.

This CA parallelization method is very similar to that described in [START_REF] Bandman | Parallel simulation of asynchronous cellular automata evolution[END_REF]. The only difference is that in our case, random sampling with replacement is used to select the cell in the unit block. This gives every cell the same probability of being chosen. It may happen that during a single time step some cells are selected more than once while other cells are omitted. Nevertheless, the diffusion process is very accurately reproduced by this method [START_REF] Bartosik | Simulation of nanostructured surfaces obtained by passivity and growth[END_REF][START_REF] Lucas | Microscopie electrochimique en milieu sel fondu: experience, modelisation et application à l'étude de la corrosion[END_REF].

The simulation box is a finite size cuboid with periodic boundary conditions parallel to initial metal surface. For the direction perpendicular to this surface, the simulation program uses a scrolling mechanism. As the block of metal becomes thinner due to oxidation, the solution is removed from the top of the lattice and metal is added at the bottom. This allows for simulating corrosion of metal thickness larger than the box size. For the selected box sizes, the systems considered reach their steady state within a reasonable number of time steps.

After that, the stationary thickness of the passive layer results in a balance between corrosion and oxide dissolution.

Results and Discussion

The simulations cover the range of potential values from -15 to +10. In this range we observe the transition from bare active surface to passive, potential independent region for P break values of 0.125 to 0.8. In all cases considered we set a constant P die = 0.01. With this value we obtain reasonable heights of the passive layer. The focus is on the steady-state properties of the system.

The data collected include values of corrosion current, passive layer thickness, roughness and morphology (rendered as images). The current is defined as the number of oxidation events per time step, divided by the horizontal (x and y) dimensions of the simulation box. The plots in fig. 9 show the dependence of a) corrosion current I, b) passive layer thickness h and c) surface roughness N/N 0 on the model parameters.

When comparing the 2D [8, figs 1-3] and 3D version (fig. 9) results, it is visible, that the current-potential curves for 3D, P break = 0.2 and 2D, P break = 0.01, look similar. This is understandable, as in three dimensions an oxide cell has, on average, two more neighbors than in 2D, so the same P break value effectively corresponds to much stronger passive layer cohesion. Another striking features in 3D are the stronger current maximum at low P break values, and the weaker current minimum (compare: [8, fig. 3a]).

At this stage, our simulations are able to reproduce the shape of experimental polarization curves, where the ratio of the maximum current I max to passivation current I passiv is not much higher than 1. As we can see, this ratio increases with lowering P break . It is also expected to increase after decreasing P die . In most experimental studies on passivation, I max /I passiv is substantially higher than in our results. However, the range of P break values considered in this paper is limited by the computational resources. For very low P break , the simulations become impractically long. This limitation is likely to be overcome by the rapid development of computing technologies.

We claim at least qualitative agreement of our simulations with the experimental results for passivation of: various stainless steel types [22, fig 1.2], iron in H 2 SO 4 solutions, [START_REF] Georgolios | On the mechanism initiating bursting oscillatory patterns during the pitting corrosion of a passive rotating iron-disc electrode in halide-containing sulphuric acid solutions[END_REF][START_REF] Pagitsas | Localized passivity breakdown of iron in chlorate-and perchlorate-containing sulphuric acid solutions: A study based on current oscillations and a point defect model[END_REF][START_REF] Sazou | Potential oscillations induced by localized corrosion of the passivity on iron in halide-containing sulfuric acid media as a probe for a comparative study of the halide effect[END_REF][START_REF] Sazou | Intrinsic coherence resonance in the chloride-induced temporal dynamics of the iron electrodissolutionpassivation in sulfuric acid solutions[END_REF][START_REF] Pavlidou | Potential oscillations induced by the local breakdown of passive iron in sulfuric acid media. an evaluation of the inhibiting effect of nitrates on iron corrosion[END_REF], lead in Na 2 SO 4 [24, fig. 1.a)], zinc [START_REF] Davies | Passivation and pitting characteristics of zinc[END_REF][START_REF] Abd El Rehim | Pitting corrosion of zinc in alkaline medium by thiocyanate ions[END_REF], and also salt passivation of iron and nickel [27, fig. 7]. Our model reproduces active-passive transition to a large extent. To our knowledge, this study is the first of its kind. However, the model is too simple to predict some other phenomena, like transpassive regime -where the current starts increasing again at sufficiently anodic potentials. Another phenomenon is the multistage passivation. It is visible in Fe|H 2 SO 4 systems (especially with added chlorates, perchlorates, chlorides, iodides or bromides) [START_REF] Georgolios | On the mechanism initiating bursting oscillatory patterns during the pitting corrosion of a passive rotating iron-disc electrode in halide-containing sulphuric acid solutions[END_REF][START_REF] Pagitsas | Localized passivity breakdown of iron in chlorate-and perchlorate-containing sulphuric acid solutions: A study based on current oscillations and a point defect model[END_REF][START_REF] Sazou | Potential oscillations induced by localized corrosion of the passivity on iron in halide-containing sulfuric acid media as a probe for a comparative study of the halide effect[END_REF][START_REF] Sazou | Intrinsic coherence resonance in the chloride-induced temporal dynamics of the iron electrodissolutionpassivation in sulfuric acid solutions[END_REF][START_REF] Pavlidou | Potential oscillations induced by the local breakdown of passive iron in sulfuric acid media. an evaluation of the inhibiting effect of nitrates on iron corrosion[END_REF]]. There we can see a current maximum followed by a shallow minimum and a plateau when passing to more anodic potential, as predicted by our model. However, a much steeper fall of the current occurs at even more anodic potential as a next stage of passivation. This stage is absent in our results.

Minimal model for active-passive transition

The mechanism of active-passive transition can be studied using a mean-field model. Here, we calculate the steady-state coverage r of the metal surface by the oxide. First, let us assume that the metal surface is flat. In the present study, we model oxidation using MET + SOL → OXI + OXI. Thus, the corrosion product is continuously forming at the MET-SOL interface, with the rate 2P corr , as oxidation of metal produces twice as large a volume of oxide. This rate is then multiplied by 1 -r , the ratio of exposed metal area to the total electrode surface area. Further, the oxide leaves the surface at the rate I out determined by the average strength of attraction between oxide cell and its nearest OXI and MET neighbors. The probability P det of OXI detaching from the surface in a unit time is 1 6 P N b break . Here, Nb is the average number of bonds formed by an OXI. This number equals 4r and 4r + 1, for the no-adsorption and withadsorption model respectively. To obtain I out , we multiply P det by the coverage

Conclusions

Our 3D simulations reproduce qualitative features of passivation known from passivation phenomenology. The polarization curves for sufficiently low P break display a clearly marked maximum and the current drop when passing from active to passive regions. The phenomenon resembles closely the analogous behaviour in 2D. The simulated polarization curves are remarkably similar to the experimental ones. This is a nontrivial result, as a priori it is harder to block a 2D surface in 3D than 1D surface in 2D and the extrapolation of 2D results to 3D is by no means obvious. The quantitative impact of higher dimensionality in 3D seems to result from an increased number of nearest neighbors as reflected in the fact that polarization curve maximum appears for higher P break values than in 2D. They are estimated as P break = e -1 in 3D and P break = e -2 in 2D, which is roughly reflected in the simulation data.

The simulations in 3D give a possibility to relate the simulated layer morphology in 3D with experimental findings. Some examples are presented in the form of the snapshots, but more work is needed to characterize the passive layer topology at various stages of passivation. The work in this direction is in progress.

We also aim to go beyond the limitations of our current research, so that we can observe significantly lower passivation current values related to I max .

This might involve an optimization allowing for faster simulations in case of low

P break .
We are also working on a different mechanism of passive layer material loss.

In the present version, the mechanism is implemented as OXI → SOL transition with probability P die . In the future, the effect of the presence of aggressive halide ions can be included explicitly. Further, the solution-dissolution equilibria for the oxide material can be introduced. The present approach accounts for the irreversible anodic oxidation of metal as the only potential-driven process. The inverse cathodic process can be included for a more realistic description with corrosion current and corrosion potential appearing naturally in such a model.

Additionally, we could withdraw the assumption that P break is equal for MET-OXI and OXI-OXI bonds.

Finally, we might consider changing the lattice to a more isotropic one.

Currently we are using the cubic primitive lattice, in the future we can use the hexagonal or face-centered cubic lattice. In both cases, every cell has 12 neighbors. The change of lattice could result in a stronger active-passive transition.

We have already advanced some research on passivation in galvanostatic conditions and observe potential oscillations when a constant current is forced.

We plan to report on these results soon.

Figure 1 :

 1 Figure 1: The experimental setup of the system being simulated.

  is the continuous 1-1 monotonous mapping of the [0, 1] range of P corr values to [-∞, +∞] range of potential values and vice-versa. All qualitative features of the V dependence remain unchanged under a continuous monotonous transformation of this scale. To convert our arbitrary potential scale to volts we can use the nF/RT factor, where n, F , R and T are reaction valence, Faraday constant, gas constant and temperature respectively. For, say, n = 1 and T = 295.15 K it transforms our potential range of -15 to 10 in arbitrary units onto -0.38 V to 0.25 V. Additionally, this functional reproduces the Tafel law for sufficiently low potential with the slope 1 and corresponds in this limit to the form adequate for rapid, Nernstian response electrochemical systems[START_REF] Pérez-Brokate | Cyclic voltammetry simulations with cellular automata[END_REF].

1 2 3 Figure 2 :

 32 Figure 2: The examples of lattice states in the beginning of the CA evolution and after consecutive transitions -1) metal oxidation, 2) oxide diffusion and 3) oxide dissolution. The color codes are as follows: oxide is represented by dark rusty, metal -by medium grey and solution -by light greenish.

Figure 3 :

 3 Figure 3: Modelling of oxide adhesion -example OXI cell moves; below are the numbers of bonds broken in each move, in the adsorption and no-adsorption version respectively. The bonds are marked with dashes -red for OXI-OXI and gray for MET-OXI.

Figure 4 :

 4 Figure 4: Two oxide cells moving simultaneously into the same empty (solution) cell. An example of a collision that could occur if the model was formulated as a synchronous cellular automaton.

Figure 5 :

 5 Figure 5: Block-synchronous automaton principle -the lattice is divided into blocks and one cell in the unit block is selected, marked with circle.

Fig. 6

 6 Fig. 6 shows the difference between the no-adsorption and adsorption version of the model. Flat, thin passive layer is visible, for intermediate potentials, only in the adsorption case. Fig. 7 explores the dependence of the surface morphology on the potential in more detail. It is clearly visible how it affects the oxide layer thickness and coverage ratio, as well as the surface roughness. Fig. 8 shows the influence of the P break parameter on the passive layer morphology.

Figure 6 :Figure 7 :

 67 Figure 6: Comparison of fragments of steady-state systems for the no-adsorption (top) and adsorption (bottom) version of the model, V = { 0, -8, -12 } (left to right), P break = 0.15.Dark rusty is oxide, bright grey is metal surface.

Figure 8 :

 8 Figure 8: Dependence of the surface morphology on oxide cohesion strength, Which is inversely proportional to P break . Fragments of steady-state systems for V = -4 (top), -10 (bottom), P break = {0.1, 0.125, 0.15, 0.40} (left to right), with adsorption.

Fig. 10

 10 Fig. 10 shows dependence of passivation current on oxide cohesion strength. The passivation current is measured for V = 10, and it is, in this case, equal to the current measured at any potential value that corresponds to the passive regime. Only values for the with-adsorption version are plotted, because they are the same as for the no-adsorption version -see also fig. 9, the ln(I) plot. The most interesting surface morphology examples are obtained for relatively low P break (strong adhesion) and for potential values close to the maximum of current. Figures 11, 12 show examples of such morphology. Fig. 11 is a snapshot of rough, corroding metal with small islands (clusters) of oxide that appear and disappear with time. This phenomenon occurs in the no-adsorption version. With adsorption however, as seen in fig. 12, we can obtain an almost completely smooth surface, with the top layer of cells being partially oxidized. The picture resembles a real corroded surface of a metal. It is worth repeating that a smooth surface with thin oxide layer is obtained only in the adsorption version (see fig. 6 again).

Figure 9 :Figure 10 :

 910 Figure 9: Steady state characteristics of investigated system -a) corrosion current I (logarithmic scale); b) passive layer thickness h; and c) chemical roughness (defined as logarithm of the ratio of real N and geometric N 0 surface area) -as functions of the electrode potential V and bond breaking probability P break . ADS -with adsorption, NOADS -without adsorption.

Figure 11 :

 11 Figure 11: Zoomed-out view for V = -9, P break = 0.15, no adsorption -rough, bare metal surface with several small oxide isles.

Figure 12 :Figure 13 :

 1213 Figure 12: Zoomed-out view for V = -8, P break = 0.125, with adsorption -smooth, corroding metal surface partially covered with oxide.

Acknowledgements

This work is done within the research grant no. 2015/19/B/ST4/03753 obtained from the National Science Center (NCN) in Poland.

Under the assumption that the dissolving oxide does not return to the surface, the steady state occurs for I in = I out . Altogether, it is possible to establish the dependence between r, P break , and P corr . Instead of P corr , we can use

First, let us consider P corr or V as a function of the other two variables:

V (r) functions for several fixed P break values are plotted on fig. 13. For high P break values, the functions are monotonous and reversible -thus, it is possible to consider r as a function of V. For lower P break values, the V (r) function stops being reversible. The critical P break value is the one, for which the V (r) function stops being monotonous in the entire domain, i.e. dv dr reaches zero for any r in