Laura Schuft

Sylvain Ferez

Fabienne Alessandro Bergamaschi

Keywords: fatigue, physical activity, exercise, chronic illness, illness experience

Introduction

Much literature has documented illness experience and identity adjustments among people living with the Human Immunodeficiency Virus (HIV) [START_REF] Baumgartner | Accepting Being Poz: The Incorporation of the HIV Identity Into the Self[END_REF][START_REF] Ciambrone | Illness and Other Assaults on Self: The Relative Impact of HIV/AIDS on Women's Lives[END_REF][START_REF] Thorpe | Doing" Chronic Illness? Complementary Medicine Use among People Living with HIV/AIDS in Australia[END_REF][START_REF] Wouters | Women's Experience of HIV as a Chronic Illness in South Africa: Hard-Earned Lives, Biographical Disruption and Moral Career[END_REF]. Whether constituting a major "biographical disruption" [START_REF] Bury | Illness Narratives: Fact or Fiction?[END_REF] or a "life-event" [START_REF] Goffman | Stigma: Notes on the Management of Spoiled Identity[END_REF] among social and economic hardships [START_REF] Ciambrone | Illness and Other Assaults on Self: The Relative Impact of HIV/AIDS on Women's Lives[END_REF][START_REF] Mcgrath | Challenging the Paradigm: Anthropological Perspectives on HIV as a Chronic Disease[END_REF][START_REF] Wouters | Women's Experience of HIV as a Chronic Illness in South Africa: Hard-Earned Lives, Biographical Disruption and Moral Career[END_REF], HIV diagnosis entails a certain amount of ongoing adaptations or identity work. These adaptations are in interaction with HIV stigma [START_REF] Geary | Gender Differences in HIV Disclosure, Stigma, and Perceptions of Health[END_REF][START_REF] Marsicano | Multiple Discriminations Experienced by People Living with HIV in France: Results from the ANRS-Vespa2 Study[END_REF][START_REF] Smith | A Meta-Analysis of Disclosure of One's HIV-Positive Status, Stigma and Social Support[END_REF] and the illness itself, including medical monitoring or the management of symptoms or treatment side effects [START_REF] Alexias | Embodiment and Biographical Disruption in People Living with HIV/AIDS (PLWHA)[END_REF][START_REF] Schuft | To Be or Not to Be Sick and Tired: Managing the Visibility of HIV and HIV-Related Fatigue[END_REF].

The experience of illness symptoms or health recommendations are also integral to the social construction of illness [START_REF] Conrad | The Social Construction of Illness: Key Insights and Policy Implications[END_REF] and the self living with chronic illness [START_REF] Baumgartner | Accepting Being Poz: The Incorporation of the HIV Identity Into the Self[END_REF][START_REF] Charmaz | The Self as Habit: The Reconstruction of Self in Chronic Illness[END_REF][START_REF] Clarke | The Radicalized Self: The Impact on the Self of the Contested Nature of the Diagnosis of Chronic Fatigue Syndrome[END_REF]. Chronic fatigue and physical activityviewed respectively as an illness symptom [START_REF] Schuft | To Be or Not to Be Sick and Tired: Managing the Visibility of HIV and HIV-Related Fatigue[END_REF]) and recommended health behaviour [START_REF] Duval | La construction de la valorisation de l'activité physique : le rôle des associations VIH[END_REF][START_REF] Ferez | L'activité Physique et Sportive Comme Outil Médicopsychologique ? Étude de l'offre de Pratiques En Direction Des Personnes Infectées Par Le VIH[END_REF] among people living with HIV (PLWH) in Francehave rarely been studied qualitatively or in relation to each other.

This study seeks to address the physical experiences and social significations of HIV-related fatigue and physical activity, their interrelations, and how they were woven into narratives of "a coherent self" [START_REF] Bury | Illness Narratives: Fact or Fiction?[END_REF].

Conceptual background

HIV has emerged as a "chronic" illness since the success of combination antiretroviral treatment (ART) in the late 1990s [START_REF] Mcgrath | Challenging the Paradigm: Anthropological Perspectives on HIV as a Chronic Disease[END_REF]. ART has allowed to keep viral loads low or even "undetectable", at the cost of regular treatments -or "pill burden", although the side effects of this burden have lessened with increasingly "simpler and better tolerated" ART regimens [START_REF] Costa | Adherence to Antiretroviral Therapy for HIV/AIDS in Latin America and the Caribbean: Systematic Review and Meta-Analysis[END_REF]. Chronic illness has long been considered transformative of the self [START_REF] Baumgartner | Accepting Being Poz: The Incorporation of the HIV Identity Into the Self[END_REF][START_REF] Charmaz | The Self as Habit: The Reconstruction of Self in Chronic Illness[END_REF][START_REF] Clarke | The Radicalized Self: The Impact on the Self of the Contested Nature of the Diagnosis of Chronic Fatigue Syndrome[END_REF], resulting in transformed identity narrativesor "illness narratives" [START_REF] Bury | Illness Narratives: Fact or Fiction?[END_REF][START_REF] Frank | The Rhetoric of Self-Change: Illness Experience as Narrative[END_REF]. Illness narratives can be considered "a significant means for studying the social construction of illness as a rhetorically bounded, discursively formulated phenomenon" [START_REF] Frank | The Rhetoric of Self-Change: Illness Experience as Narrative[END_REF]. As illness narratives account for "the altered relations of body, self and society brought about by illness" (Bury 2001: 276), they are viewed here as drawing fromand giving meaning tothe physical, psychological and social experiences of health and illness.

Narratives -or the performative act of "doing storytelling" (Smith & Sparkes, 2008) can be articulated with actions and agency, or "doing" chronic illness [START_REF] Thorpe | Doing" Chronic Illness? Complementary Medicine Use among People Living with HIV/AIDS in Australia[END_REF]. The latter concept considers that people have agency to "choose" to a certain extent how they live with or manage chronic illness. Recent research has suggested that such "choices" as well as the physical experience of HIV are facetted by social contexts of interaction, and an interplay between the sentiment of HIV (in)visibility [START_REF] Persson | Facing HIV: Body Shape Change and the (in)Visibility of Illness[END_REF][START_REF] Thomas | La perception de l'activité physique comme mode de prise de charge du VIH : effets de la visibilité du corps sur les stratégies de gestion d'une "maladie chronique[END_REF][START_REF] Alexias | Embodiment and Biographical Disruption in People Living with HIV/AIDS (PLWHA)[END_REF]) and the acceptability of disclosing HIV or physical symptoms (Mackworth-Young, Bond, and Wringe 2020; [START_REF] Schuft | To Be or Not to Be Sick and Tired: Managing the Visibility of HIV and HIV-Related Fatigue[END_REF]). These findings suggest that illness narratives are influenced by both physical experiencesregarding the body, treatments or symptomsand social experiences along illness trajectories. The embodiment of physical and social experiences is thus central to our consideration of how people "do" and narrate illness experiences, how they represent the bodyself, or how they "craft" new understandings of what it means to be an individual living with HIV.

Fatigue is a major physical symptom associated with HIV [START_REF] Zhu | Symptom Clusters in People Living With HIV: A Systematic Review[END_REF] and has been widely studied in terms of correlations with psychological and physiological variables.

Although the principal correlation appears to be depression, results have suggested that social factors such as socioeconomic conditions or social support are major predictors, while clinical variables such as date of diagnosis, treatment histories or viral load have yielded inconclusive or contradictory results [START_REF] Barroso | Fatigue in HIV and AIDS: An Analysis of Evidence[END_REF][START_REF] Bergamaschi | Perceived HIV-Related Physical Fatigue, Sociodemographic Characteristics and Physical Activity: A Cross-Sectional Study[END_REF][START_REF] Harmon | Demographic and Illness-Related Variables Associated with HIV-Related Fatigue[END_REF][START_REF] Hughes | A Biopsychosocial Approach to HIV Fatigue: A Cross-Sectional and Prospective Analysis to Identify Key Modifiable Factors[END_REF][START_REF] Jong | Predictors and Treatment Strategies of HIV-Related Fatigue in the Combined Antiretroviral Therapy Era[END_REF]. Recent work has complemented these studies with qualitative approaches to HIV-related fatigue and its social significations [START_REF] Schuft | To Be or Not to Be Sick and Tired: Managing the Visibility of HIV and HIV-Related Fatigue[END_REF], following work on fatigue related to chronic fatigue syndrome [START_REF] Clarke | The Radicalized Self: The Impact on the Self of the Contested Nature of the Diagnosis of Chronic Fatigue Syndrome[END_REF][START_REF] Olson | The Nature of Fatigue in Chronic Fatigue Syndrome[END_REF] and other chronic illnesses [START_REF] Olsson | Meanings of Fatigue for Women With Parkinson's Disease[END_REF]. In this study we focus on how the experience of fatigue is associated with physical activity in narratives about living with HIV. We refer to HIV-related fatigue as "chronic exhaustion" associated with HIV [START_REF] Jenkin | The Experience of Fatigue for Adults Living with HIV[END_REF], as opposed to exercise-induced "acute physical fatigue" [START_REF] Kluger | Fatigue and Fatigability in Neurologic Illnesses: Proposal for a Unified Taxonomy[END_REF].

Studies suggest that chronic fatigue among PLWH may be reduced by regular physical activity [START_REF] Bergamaschi | Perceived HIV-Related Physical Fatigue, Sociodemographic Characteristics and Physical Activity: A Cross-Sectional Study[END_REF][START_REF] Webel | Physical Activity Is Associated with Reduced Fatigue in Adults Living with HIV/AIDS[END_REF]. As among the general population, physical activity programmes for PLWH have been associated with other health benefits such as lower anxiety and depression, improved physical capacities or higher quality of life [START_REF] Quiles | Association Between Physical Activity, Depression, and Diabetes in Urban-Dwelling People Living with HIV[END_REF][START_REF] Voigt | Supervised Physical Activity and Improved Functional Capacity among Adults Living with HIV: A Systematic Review[END_REF]). Yet despite increasing knowledge and beliefs about the benefits of physical activity, few studies have considered the experiences and views of physical activity by PLWH [START_REF] Ferez | Continuer à s'engager Dans Des Loisirs Après Avoir Été Infecté Au VIH : Entre Quête de Normalité et Prescription Sociale[END_REF][START_REF] Gray | Perceived Barriers to and Facilitators of Physical Activity in People Living with HIV: A Qualitative Study in a French Sample[END_REF], or their relations with the experience of physical symptoms or with exercise recommendations from health professionals.

Public health messages have increasingly encouraged physical activity as well as other health-enhancing behaviours for the general public, all while particularly targeting people living with chronic illness [START_REF] Newman | Just Take Your Medicine and Everything Will Be Fine": Responsibilisation Narratives in Accounts of Transitioning Young People with HIV into Adult Care Services in Australia[END_REF] including HIV [START_REF] Duval | La construction de la valorisation de l'activité physique : le rôle des associations VIH[END_REF][START_REF] Ferez | L'activité Physique et Sportive Comme Outil Médicopsychologique ? Étude de l'offre de Pratiques En Direction Des Personnes Infectées Par Le VIH[END_REF]. Some doctors and clinical publications have recommended promoting health-enhancing behaviours like physical activity as symptom-management strategies, including for HIV-related fatigue [START_REF] Perazzo | Fatigue Symptom Management in People Living With Human Immunodeficiency Virus[END_REF]. Targeted messages promoting rationalized health behaviours can be considered what Foucault termed "biopedagogies": medical discourses which lead to forms of "governing" or disciplining certain bodies [START_REF] Rail | The Birth of the Obesity Clinic: Confessions of the Flesh, Biopedagogies and Physical Culture[END_REF][START_REF] Wright | Biopolitics and the 'Obesity Epidemic[END_REF]. Recently emerging as a useful concept to analyse public health policy regarding obesity in particular, "biopedagogies revolve around the moral regulation of bios or life (i.e., how to eat, how to move, how to live)" (Rail 2012, 229). As several authors have noted [START_REF] Newman | Just Take Your Medicine and Everything Will Be Fine": Responsibilisation Narratives in Accounts of Transitioning Young People with HIV into Adult Care Services in Australia[END_REF][START_REF] Wright | Biopolitics and the 'Obesity Epidemic[END_REF], these biopedagogies "draw on a neoliberal notion of individualism that constructs individuals as at once capable of and responsible for changing their lifestyles through a variety of disciplinary techniques" (Rail 2012, 229). In this light, experiences and interpretations of fatigue and physical activity may reflect ways in which PLWH negotiate self in the face of illness symptoms and management, or how they embody, resist or give meaning to biopedagogiesin terms of moral views about self-disciplinary measures surrounding health behaviours and physical activity.

Methods

Methodological underpinnings

We considered illness narratives to be rich data sources for social analysis when socially contextualized and considered systematically across participants' stories [START_REF] Atkinson | Illness Narratives Revisited: The Failure of Narrative Reductionism[END_REF]. Despite diverse or unique interpretations and meanings attributed to individual illness experiences during the performative act of narrating, we can agree with [START_REF] Atkinson | Illness Narratives Revisited: The Failure of Narrative Reductionism[END_REF] -within the "contested terrain of narrative methods" [START_REF] Thomas | Negotiating the Contested Terrain of Narrative Methods in Illness Contexts[END_REF])that the collective dimensions of illness experiences and meanings within sociocultural contexts can enhance understanding of the "social construction of illness" [START_REF] Conrad | The Social Construction of Illness: Key Insights and Policy Implications[END_REF]. As our research focused on discourses and themes across narratives, and less on storylines in terms of their individual chronology, uniqueness or performative delivery (Smith and Sparkes 2008), we combined the narrative approach to data collection with an analysis based on the principles of reflexive thematic analysis (Braun and Clarke 2021). This approach enabled us to first generate a rich dataset of in-depth individual experiences and interpretations through narrative, to situate these particularities within collective social dimensions across the dataset through thematic analysis, and to later re-think themes and meanings in relation to narrative "types". The methodological flexibility allowed reconciling between focus on the particular and the general, and between approaches stemming from inductive and deductive epistemologies.

Within a relativist ontological approach to data collection and rigor [START_REF] Sparkes | Judging the Quality of Qualitative Inquiry: Criteriology and Relativism in Action[END_REF], criteria for reliability was not viewed as necessarily improving the accuracy or trustworthiness of analyses, owing that "theory-free knowledge is unachievable and that realities are subjective" (Smith and McGannon 2018, 12). We nonetheless took steps to meet basic standards of methodological rigor to increase the representativity of data (data saturation) or the consistency of interpretation (triangulation), as discussed below, while recognizing from a relativist approach "that the goal of doing so is not to provide researchers with a more valid singular truth" (Tracy 2010, 844). In this light, we considered our analyses from an interpretivist approach, as co-constructed between participant and researcher understandings and subjectivities.

Participant recruitment

Participants were recruited through three major French hospital services specializing in HIV care as well as through three non-medical associations offering services such as leisure activities, social support and information for PLWH. The hospitals and associations were located in three regions in the centre, the southeast and the south of France. The aim was to obtain the perspectives of PLWH with diverse social and medical profiles in a nationwide sample, without imposing specific inclusion or exclusion criteria.

In each hospital, we contacted the department of HIV and infectious disease and organized a meeting to present the study to the practitioners. Each head of service then accepted to attribute a Clinical Research Assistant (ARC) from their service to recruit participants. Using a flyer and consent form, the ARC presented the study and its ethical guarantees, and proposed participation to every consecutive patient who came for their regular consultations over a set period of a month.

In the associations, we contacted staff and organized discussion sessions with the associations' beneficiaries to present the study and its ethical guarantees and to discuss the themes of HIV and a) physical activity and b) fatigue. At the end of each discussion session, which allowed us to become familiar with experiences and views regarding these topics, PLWH were invited to participate in the study.

Interviews

Volunteers recruited through these methods were later contacted by a researcher. Prior to their interview, participants signed a consent form which informed about: the project objectives; their liberty to decline or withdraw from the study; and the confidential and anonymous nature of data storage and analysis. On these ethical bases the study had previously obtained ethical approval with the French National Commission for Data Protection and Liberties (CNIL). Semi-structured interviews were conducted in spaces provided by the hospital services and associations, as well as in participants' personal homes, depending on the preference and availability of the participants.

Interviews lasted an average of two hours and were conducted in French. For the purposes of this publication, the selected excerpts were translated by a bilingual native English-speaker then verified by cross-translation by a native French-speaker.

Thirty-five interviews were conducted over six months, ending when researchers felt they had reached "data saturation", indicating a stage when researchers felt they had heard a "repetition of responses" or "the same comments again and again" [START_REF] Saunders | Saturation in Qualitative Research: Exploring Its Conceptualization and Operationalization[END_REF]. In line with relevant qualitative work outside of grounded theory, we considered data saturationas opposed to "theoretical" or "thematic saturation"as a helpful indicator for ending the collection of narratives at the level of data collection, in a manner unrelated to later phases of analysis or theory [START_REF] Saunders | Saturation in Qualitative Research: Exploring Its Conceptualization and Operationalization[END_REF]. Although "saturation" is a contested concept for reflexive thematic analysis (Braun and Clarke 2021), we viewed data saturation as coherent with an interpretivist approach. Rather than translating an assumption that "nothing new" could be garnered from further interviews, data saturation represents a subjective researcher perception of redundance, marking a turning point in phases of research.

The first section of the interview topic guide broached (dis)continuities in daily life following HIV diagnosis, with an aim to consider the overall illness experience and any mention of physical activity or fatigue. The second section concerned relations to and representations of physical activity, while the third focused specifically on the experience and interpretations of fatigue, including perceived causes and consequences. The interview ended with the completion of sociodemographic information including age, occupation, country of birth and year of diagnosis.

The interviews were transcribed verbatim and rendered anonymous. A table was created to keep track of pseudonyms, interview date and place, sociodemographic data and brief summaries of physical activity and fatigue levels as described by the participants.

Participant characteristics

The participants were diagnosed between 1983 and 2010, the median year of diagnosis being 1994.

The distribution of gender and sexual orientation among participants represented more women as compared with distributions in the national context (46% versus 33%), but also more men who have sex with women (MSW) (20% versus 28%) and less men who have sex with men (MSM) (34% versus 39%) [START_REF] Dray-Spira | État de Santé de La Population Vivant Avec Le VIH En France Métropolitaine En 2011 et Caractéristiques Des Personnes Récemment Diagnostiquées. Premiers Résultats de l'enquête ANRS-Vespa2[END_REF]. In terms of professional activity, a third of the participants qualified for handicap or unemployment benefits. This was comparable to available national data for handicap benefits, received by 21% of women and 14% of men living with HIV [START_REF] Lert | Socioeconomic Conditions of People Living with HIV Followed at Hospital in Metropolitan France. Results of the ANRS-VESPA2 Study[END_REF]. Lastly, France as a country of birth was over-represented in the interviewed population, as only 4 participants were born outside of France, while people born in Sub-Saharan Africa alone recently represented 38,7% of new diagnoses in France (ANRS 2018). All of the participants were following medical care and treatments. Although the recruitment process may have favoured participants with medical complications, medical history was not considered in a systematic way due to the great diversity in medical profiles and temporalities, as well as previous contradicting or inconclusive quantitative results linking fatigue to treatments or biological markers, as stated above.

Analysis

The first stage of analysis involved deductive thematic analysis in which all discourse pertaining simultaneously to fatigue and physical activity was coded within an umbrella category. This category was then subject to inductive thematic analysis, in which recurring concepts generated initial codes and then themes of analysis [START_REF] Braun | Reflecting on Reflexive Thematic Analysis[END_REF]. A triangulation of analyses by three researchers [START_REF] Nowell | Thematic Analysis: Striving to Meet the Trustworthiness Criteria[END_REF]) involved sharing one researcher's initial coding to critically discuss the themes of analysis and later reach a form of agreement on the overall interpretation of data via the themes of analysis and coding. In this light, the multiple researchers acted as "critical friends" (Smith and McGannon 2018, 24), seeking less to verify "intercoder agreement" or "trustworthiness" than to provide a process in which interpretations could be confronted, developed and then mutually agreed upon.

During a later phase of thematic analysis as defined by [START_REF] Braun | Reflecting on Reflexive Thematic Analysis[END_REF], where "researchers review the coded data extracts for each theme to consider whether they appear to form a coherent pattern" (Nowell et al. 2017, 9), the themes were re-organised to test the trend that themes varied by levels of fatigue and physical activity. This re-organization gave way to an original analytical framework in the form of a graph (Figure 1) in which coded meanings were considered in relation to individuals' physical experience, in terms of physical activity (whether physically active or not) and fatigue (whether experiencing chronic fatigue or not). This created four "ideal types" in the sense of Max Weber: exaggerated archetypal forms of classification that facilitate the categorization and analysis of otherwise elusive concepts or phenomena [START_REF] Swedberg | How to Use Max Weber's Ideal Type in Sociological Analysis[END_REF]. The ideal types were understood through an interpretivist paradigm, as an imperfect means of interpreting the diverse lived and recounted experiences. Because levels of physical activity or fatigue are also subjective and variable over time, the ideal type narratives were theorized as situated along intersecting continuums of reported levels of fatigue and physical activity (Figure 1). A continuum model of ideal types, as has previously been proposed in the domain of health and physical activity [START_REF] Christiansen | Outline of a Typology of Men's Use of Anabolic Androgenic Steroids in Fitness and Strength Training Environments[END_REF], allowed us to situate, theoretically, significations attributed to fatigue and physical activity in relation to their physical experience. It also allowed us to situate related themes, such as the importance accorded to "healthy living" or "preserving the vulnerable body", in relation to these social meanings and physical experiences.

[Figure 1 near here] [Table 1 near here]

Findings

The ideal type narratives are each described in terms of the related themes of analysis, samples of coded discourse and how people conveyed the body and self living with HIV.

Interpretation of these findings will be broadened in the discussion section.

"No sport due to fatigue"

The most predominant ideal-type profile involved those who reported chronic fatigue and physical inactivity. They viewed physical activity as ill-adapted to their state of health, or too fatiguing for the "sick" body. The body-self emerged as vulnerable and strongly associated with HIV.

Fatigue as a barrier to physical activity.

The topic of physical activity was met with reactions evoking fatigue: "I'm already too fatigued" or "I don't have the energy" (Marie, 48, unemployed with handicap benefits, diagnosed in 1986). In relation to the physical demands of her children, one woman recounted: "Playing ball, whatever, it's ok 5 minutes, but after, I just can't. Because I feel, after, tired and prefer to sit down on a bench" (Marion, 34, homemaker, diagnosed in 2008).

Physical fatigue or exhaustion were regularly cited as obstacles to physical activity. Yet the discourse surrounding such statements revealed other processes relating to illness and symptom experiences. For Janine, lack of energy was part of her description of living with HIV: "Even just going to the pool, it's… (heavy sigh). You say you're tired before even leaving for the pool. It requires an effort which… You know you won't be able to make that effort" (Janine, 44, unemployed, diagnosed in 1986).

When asked about physical activity, she replied that she was inactive, mentioning also cost as a barrier. However, Janine later termed cost an "excuse": "You look for the excuse, saying: 'Well, it's expensive'. […] So, it makes you feel less guilt. Because the doctors, the association, they push you to do things, to not stay isolated, and get out… but um… we know we're tired (laugh), that we won't be able to do it […] It's easier to say, 'I'm not going because it's expensive' […] . Psychologically, it brings you back to the illness to say, 'If I didn't have this illness […] I would do so many things, if I wasn't all the time exhausted!'". This excerpt demonstrates the tensions between (un)desired narratives of the self, involving recognizing or denying symptoms (fatigue) or the capacity to live up to expected health behaviours (physical activity) pushed by doctors and HIV associations. Although fatigue was described as an obstacle to physical activity, alternate "excuses" enabled her to portray herself as less handicapped by illness.

Physical activity as fatiguing or risky.

Physical activity also emerged as too tiring or risky for one's state of health and energy: "I don't do the whole session (of yoga at the HIV association), because it's too much for me! And I'm not ashamed to say it there" (Marie, 48, unemployed with handicap benefits, diagnosed in 1986).

"We fatigue too fast to follow others. It's not possible! Even yoga […] Too bad there aren't low-intensity fitness courses, just for us" (Carla, 56, unemployed with handicap benefits, diagnosed in 1983).

Many of these participants used the collective "we/us", anchoring illness-related difficulties in collective HIV identity or experience. This was compatible with an appeal for HIV-specific solutions such as adapted physical activity.

Physical activity was deemed "too tiring", the body too vulnerable. Three participants explicitly viewed HIV as "a sword of Damocles hanging over my head" (Marie, André, Christian), meaning a permanent threat to their well-being if not their lives. For many, this threat was linked to the need to "preserve" the body from over-exertion: "I don't smoke. Alcohol: I drink sentimentally, not abusively. And I preserve myself a lot.

So if I really need some time for rest, I rest. […] Outside of work, I come home, I stay at home, a small walk with the family and that's it" (Bernard, 50, skilled worker, diagnosed in 1997).

Avoiding smoking and alcohol but also many forms of physical activity (in terms of duration, intensity or rest time) were construed as measures to protect or preserve the body. This is in direct opposition with other ideal type narratives where physical activity emerged as a "healthy living" strategy to prevent or manage illness and its symptoms. The "No sport due to fatigue" narratives told a story in which the HIV-infected body and self were vulnerable, and could not or should not over-exert or engage in physical activity.

"Sporty to manage fatigue"

Among PLWH who felt regularly fatigued but engaged in some form of physical activity, physical activity was still strongly related to fatigue. It was presented as a part of "healthy living", and essential for managing illness and symptoms such as fatigue. "Healthy living" was also at times construed as particularly important for people who are "sick", whether for managing or actively combating illness or its symptoms.

Physical activity as a part of "healthy living" and illness management.

In these narratives, participants presented physical activity in light of its benefits in terms of illness or symptom management. Paulette (42, teacher, diagnosed in 1996) explained that physical activity "helps me" to manage fatigue, which she associated with both HIV and her work.

Similarly, Corentin specifically became a yoga instructor after his diagnosis, viewing yoga as a means of managing life with HIV. Physical activity represented "a boost of energy" and a means of countering illness symptoms, like reduced physical "endurance" for accomplishing everyday activities: "playing sports makes you less tired" (Corentin, 43, part-time yoga teacher with handicap benefits, diagnosed in 1998).

Physical activity was also viewed as part of "healthy living", presented as a set of behaviours that could both prevent and improve illness: "Sport or a healthy lifestyle will prevent disease from happening if you are not sick, and if the disease is there, it will improve things a lot" (Corentin).

Other participants described how this narrative reflected discourse relayed by HIV social and medical networks, where workers warned about the dangers of "unhealthy" lifestyle choices: "We're sure of [the importance of "healthy living"], too, because they were telling us about other people who had lives a little out of control, or things like that who didn't respond as well to treatment as we did" (Pauline, 40, sales assistant, diagnosed in 1999).

In this way, representations circulating in HIV social and medical networks linked physical activity and "healthy living" to health outcomes, encouraging/discouraging particular social behaviours deemed "(un)healthy"such as diet, levels of rest or intensity of exercise, but also "partying", drinking or smoking.

Physical activity as a means of combating illness.

Beyond managing illness and symptoms, physical activity was also at times represented as a means of actively combating fatigue: Others spoke similarly of forcing oneself "to move": "Whenever I was tired or coming out of major illnesses [...] even if it didn't last long, I had to go (to physiotherapy), I had to move!" (Isabelle, 51, unemployed with handicap benefits, diagnosed in 1990).

In these narratives, individuals emerged as taking control of their "fight" against illness and its symptoms by adopting behaviours promoted by health messages.

"Healthy because sporty"

Physically active participants who were not fatigued also construed physical activity as an essential part of healthy living and illness symptom management. Yet here physical activity was represented as their source of good health, the recipe for keeping illness and fatigue at bay. To reach this aim, the type and intensity of physical activity were specified as necessarily "moderate" in intensity and duration.

Illness and fatigue kept at bay "thanks to" physical activity and healthy living.

These participants tended to distinguish between "good fatigue" and "negative" or illnessrelated fatigue. "Good fatigue" was linked to regular, "normal" or chosen activities such as work or physical activity. This fatigue was described as one "where we recover quickly", as opposed to "fatigue due to HIV" where "there's nothing we can do about it" (Michel, 52, administrator, diagnosed in 1991). In the various descriptions opposing "good" versus illness fatigues, "energy fatigue" was compared with "a real physical fatigue", the two types being "not the same fatigue".

For some, physical activity represented an antidote to illness fatigue: sport "chases away the fatigue […] You feel good and it takes away fatigue" (Philippe, 35, massage therapist, diagnosed in 2008). More dramatically, for George, physical activity "saved" and "changed" his life, allowing him to improve his "quality of life" and "fight" off illness: "It gives me energy, it motivates me… […] It's also social. […] It saved my life! It's what… psychologically, physically, I think it's the most important things that I've done, to stay very active… For example, bodybuilding, for me, it's like a rebooting, rebooting a computer. […] I can be psychologically fatigued after work or whatever, I go to my sport, and… when I get home, everything's fine!" (George, 50, consultant, diagnosed in 1995).

Similarly, Giorgio depicted physical activity as an instrument of illness management, as part of his "cure for AIDS" (used here in lieu of HIV): "I discovered the true medicine for AIDS, […] healthy living! […] Life discipline, sport, going out, being active, never lazy, making oneself be active! […] That's my prescription.

You mustn't stay at home, in front of the television day and night" (Giorgio, 54, capital manager, diagnosed in 2002).

In these last excerpts, being energetic or active was not only associated with "good fatigue" and health, but also with a moral value of "effort" and a positive frame of mind. Conversely, lacking energy or activity was associated with illness but also laziness and a negative frame of mind.

Moderate physical activity as the appropriate form of health management.

Not just any physical activity, or any amount of it, was touted as beneficial. Many participants underlined an importance of carefully dosing the amount of physical activity in order to harvest energy rather than fatigue: "Sport is a domain which… […] is useful for relaxing, but […] I don't see the point of going every day, that's not relaxing, that's fatigue" (Charles, 41, associate director, diagnosed in 1998). Similarly, Raphael (76, retired teacher, diagnosed in 1987) spoke of avoiding "excessive" amounts of physical activity: doing "a little", "not too much", "not excessively", "not too fast", not too "violent".

Physical activity was also presented as part of normative health behaviours: "part of that healthy lifestyle, part of that regulated life. Not a wild lifestyle" (Giorgio, 54, capital manager, diagnosed in 2002). For many participants, this was considered particularly important for a person "sick with HIV" who "should not exaggerate with sport", and "must not come home exhausted" (Giorgio). In this sense, "moderate physical activity" was explicitly viewed as important means of preserving health for "people who have an illness", due to the precarious health status associated with HIV: "I had a very balanced life, which is very important in this sickness. […] A balanced life… moderate physical activity, well, that's ideal. I think that's the ideal for all illnesses!" (Michel, 52, administrator, diagnosed in 1991). This moderation was associated with "protecting" and "preserving" the body, "because your body is already fighting the illness. […] We shouldn't get exhausted" (Philippe, 35, massage therapist, diagnosed in 2008).

The surrounding medical and social environments reinforced these narratives about "healthy living" as involving moderate energy and physical activity levels: "Many around me say 'Maybe you should take it easy, since you're taking a treatment, it's poison for the body and […] the treatment on its own is exhausting'. […] Sometimes, the recommendations [from the general practitioner] are like 'Ok, maybe you should stop sport'; 'you should stop competition" (Gregory, 38, steward, diagnosed in 2007).

Philippe wondered if his doctor's recommendation "to not get exhausted" was due to his treatments, "seeing as I have an undetectable viral load". His question underlines a contradiction between the physiological reality for many who experience little to no HIV-related symptoms or even viral load, on the one hand, and representations in which HIV is associated with physical vulnerability and a need for protection and moderation, on the other hand.

"No sport, no fatigue, no problem"

Only four of the interviewed people spoke of little to no fatigue or physical activity. Physical activity was construed as foreign to their personality or lifestyle. In parallel, their body, health and, more largely, life were considered "normal". Physical activity was met with indifference, while inactivity emerged as consistent with identity and lifestyle, and dissociated from HIV.

Physical activity as foreign to one's personality or lifestyle.

These narratives included statements such as: "It isn't one of my usual habits"; "I'm not much of an athlete"; or "I'm lazy" (Fatima, 52, nursing auxiliary, diagnosed in 1992). When asked about physical activity, Fatima further elaborated on this statement, laughing that if she ever did her shopping by foot, "in the evening when I come home, I wouldn't even make dinner! I'd go to bed, because I wouldn't be used to it". Remy also made light of his physical efforts. When asked if he experienced fatigue, he laughed: "I'm tired by nature (laugh), because I don't go out! […] I don't make too much effort to do sport or things like that. […] No, no, I don't feel tired" (Remy, 55, employee, diagnosed in 1987).

These participants therefore recount no particular fatigue, but also no particular interest in physical activity.

Health and life as "normal".

These participants spoke of their health, fatigue and lives as "normal", "like everyone" (Fatima). Fatima sometimes wondered whether punctual fatigue was due to HIV or treatments, but generally did not view fatigue as a result of either: "I don't know if I'm tired because I work, because I'm taking care of the kids, I'm running here, running there, or if I'm tired because of the medicine or my immunosuppression. I don't know! […] I wonder, but I don't go beyond wondering… […] And then I forget, and it passes" (Fatima, 52, nursing auxiliary, diagnosed in 1992).

Overall, for Fatima and her husband, also living with HIV, she referred to their levels of fatigue and energy as "normal": "Compared to people around us who are the same age as us, they don't seem to be more fit than us. […] Maybe, if some at our age can do a sprint at 200 (km) per hour. We're not athletic so, of course we couldn't follow. But we walk, we work all the time; when there's something to be done, we do it. We don't feel out of breath. Know what I mean?" Describing themselves as "normal", "unathletic" individuals achieving their regular activities and everyday lives without a problem, they did not integrate illness experience or expected health behaviours into their overall identity narratives.

Discussion

One of few qualitative studies on HIV-related fatigue [START_REF] Perazzo | Fatigue Symptom Management in People Living With Human Immunodeficiency Virus[END_REF][START_REF] Schuft | To Be or Not to Be Sick and Tired: Managing the Visibility of HIV and HIV-Related Fatigue[END_REF], our results also corroborate quantitative work showing correlations with social and economic variables. All of the participants in our study receiving handicap or unemployment benefits spoke of chronic fatigue, as did nearly all of the participating women (see Table 1). These observations are consistent with results having shown that reported levels of HIV-related fatigue are related to class and gender, disproportionately impacting poorer women [START_REF] Jong | Predictors and Treatment Strategies of HIV-Related Fatigue in the Combined Antiretroviral Therapy Era[END_REF], although gender has not recently emerged as a correlating variable [START_REF] Bergamaschi | Perceived HIV-Related Physical Fatigue, Sociodemographic Characteristics and Physical Activity: A Cross-Sectional Study[END_REF]. The disproportion may be due in part to more ready admission to symptoms among women, while men may be more prone to "hegemonic" masculine gender prescriptions such as showing robustness and dissimulating fatigue or weakness [START_REF] Courtenay | Constructions of Masculinity and Their Influence on Men's Well-Being: A Theory of Gender and Health[END_REF][START_REF] Schuft | To Be or Not to Be Sick and Tired: Managing the Visibility of HIV and HIV-Related Fatigue[END_REF]. Similarly, women were less physically active in our study, which is consistent with trends regarding gender socialization and sport in France [START_REF] Mennesson | Socialisation familiale et investissement des filles et des garçons dans les pratiques culturelles et sportives associatives[END_REF]. Such ancillary observations can only suggest a need for greater study of fatigue in terms of social and economic contexts and hardships, which have already been shown to greatly impact the experience of living with HIV [START_REF] Ciambrone | Illness and Other Assaults on Self: The Relative Impact of HIV/AIDS on Women's Lives[END_REF][START_REF] Wouters | Women's Experience of HIV as a Chronic Illness in South Africa: Hard-Earned Lives, Biographical Disruption and Moral Career[END_REF]. As gender and socioeconomic conditions also affect social, physical and illness-related behaviours and experiences, they appear to play an indirect role in narratives about the latter.

More specifically our results showed the relations between the physical experiences of living with HIV, the meanings attributed to these, and narratives about the body and self. The ideal-type discourses and significations surrounding experiences of fatigue and physical activity particularly illustrated different "moral narratives", in which individuals "seek to account for and perhaps justify themselves in the altered relations of body, self and society" and (re)establish their moral status (Bury 2001, 276). This entailed underlining or justifying (non)conformity to normative health behaviours like physical activity. Due to transversally described pressure to engage in "healthy living" due to illness or HIV, those who described being inactive due to fatigue often evoked guilt for not following this moral order emanating from HIV and medical communities. Advice on physical activity was however ambiguous or contradictory, ranging from encouraging moderate activity to discouraging intensive activity, creating a context of ill-adapted practitioner support with regard to exercise as observed in other contexts such as arthritis [START_REF] Hunt | Being Physically Active through Chronic Illness: Life Experiences of People with Arthritis[END_REF]. "Healthy living" expectations also covered a wide range of normative behaviours including diet, alcohol, "partying" or levels of rest. In this way health messages attempted to influence a wide range of social activity in the everyday lives of PLWH, constituting biopedagogies regarding recommended or expected physical and social behaviours [START_REF] Rail | The Birth of the Obesity Clinic: Confessions of the Flesh, Biopedagogies and Physical Culture[END_REF][START_REF] Wright | Biopolitics and the 'Obesity Epidemic[END_REF]. Biopedagogies concerning health behaviours appeared as key elements of illness narratives, embodied and signified differently depending on a range of factors including physical experiences of fatigue or physical activity.

Physical activity was most often construed as either a moral responsibility or a means of health management, as observed with other chronic illnesses such as arthritis or multiple sclerosis, where exercise often emerged as a way to "preserve health" (Hunt and Papathomas 2020) and "control a chronic illness and keep physical deterioration at bay" [START_REF] Richardson | A Narrative Exploration of the Evolving Perception of Exercise among People with Multiple Sclerosis[END_REF]. In other cases physical activity constituted a mode of "battling" HIV, through action or positive energy. Since the pioneering work by Susan [START_REF] Sontag | AIDS and Its Metaphors[END_REF] noting the widespread use of military terminology to describe the "fight" against illnesses like HIV/AIDS, narratives of "combating" HIV have been documented [START_REF] Pierret | An Analysis over Time (1990-2000) of the Experiences of Living with HIV[END_REF]). In the case of bodybuilders, strength and muscle have been wielded as proof of "winning" over illness and illness stereotypes such as frailty [START_REF] Ferez | Ceux qui disent que je suis malade, je les invite à venir prendre l'entraînement avec moi." Devenir body-builder à l'épreuve du VIH[END_REF]. Their narratives were of empowerment, of a body and self at war with illness. This corresponds to a dominant illness narrative, present in media accounts, where "accounts of anyone suffering from a life threatening illness are almost invariably couched in terms of a courageous 'fight' by the sufferer" (Kelly andDickinson 1997, 268, cited in Bury 2001, 279).

In most narratives, however, the body emerged as vulnerable, whether weak and ill or tenuously robust thanks to exercise, or as needing protection or preservation from over-exertion.

These representations led to discourse promoting physical activity as necessarily moderate to maintain health and energy ("Healthy because sporty"), or the exclusion of physical activity altogether to manage fatigue and protect the body ("No sport due to fatigue"). Physical activity as "risky" corresponds to predominant social representations regarding physical activity for populations viewed as "fragile" such as older adults [START_REF] Emile | Aging Stereotypes and Active Lifestyle: Personal Correlates of Stereotype Internalization and Relationships with Level of Physical Activity among Older Adults[END_REF] or people living with cancer [START_REF] Falzon | Beliefs about Physical Activity in Sedentary Cancer Patients: An In-Depth Interview Study in France[END_REF] or HIV [START_REF] Gray | Exercise Stereotypes and Fatigue in People Living with HIV: Does Self-Efficacy Play a Mediating or a Moderating Role?[END_REF]. In contrast, public health messages in France promote physical activity for "fragile" populations to improve or preserve health [START_REF] Honta | Le prix de l'innovation : capacités des opérateurs et réception des programmes de lutte contre la sédentarité[END_REF]. In this regard, the chronically ill simultaneously endure normative pressure to engage in physical activity and stereotypes which hinder physical inactivity. These contradicting representations were reflected in the ambivalent social and medical discourses about physical activity: more or less encouraged, but not "too much". Certain profiles ("No sport due to fatigue") seemed particularly susceptible to such conflicting representations, feeling incapable of physical activity but guilty about inactivity.

Beyond its links with physical activity, fatigue emerged as more or less experienced, symptomatic of HIV or incapacitating. The distinctions between "normal" (chosen) and HIVrelated (borne) fatigue were coherent with distinctions that researchers have made between "acute" and "chronic" fatigue, associated respectively with exertion and illness [START_REF] Kluger | Fatigue and Fatigability in Neurologic Illnesses: Proposal for a Unified Taxonomy[END_REF]. This distinction also supports previous research on meanings attributed to chronic fatigue as symptomatic of illness/ HIV [START_REF] Schuft | To Be or Not to Be Sick and Tired: Managing the Visibility of HIV and HIV-Related Fatigue[END_REF].

Overall, the ideal types revealed four constellations of meanings and experiences attributed to fatigue and physical activity, but also four means of narrating the self, situated between opposing representations of physical health (healthy or ill) and the body-self (vulnerable or robust). For some, physical activity was part of dominant narratives about "winning" over illness or keeping it at bay, and distancing the body-self from incapacity, handicap or illness. For others physical activity was dissociated from normative behaviours for oneself, the body-self viewed as too ill, or else inactive due to alternative "excuses". Discourses on the self living with HIV have likewise been shown to vacillate between recognizing and denying diminished health or capabilities [START_REF] Ferez | Continuer à s'engager Dans Des Loisirs Après Avoir Été Infecté Au VIH : Entre Quête de Normalité et Prescription Sociale[END_REF]), denial being one adaptive response to living with chronic illness [START_REF] Telford | Acceptance and Denial: Implications for People Adapting to Chronic Illness: Literature Review[END_REF]. In this way, fatigue and physical activity constituted key dimensions of the way PLWH portray the body-self in varying shades of healthy or ill, capable or handicapped. In other words, the interpretations and portrayals of physical experiences illustratebut also participate in "crafting"understandings of the body-self living with HIV.

Despite providing an analytical framework to depict a diversity of discourses, ideal types also entail drawbacks, simplifying the view of complex social realities through an angle that isinevitablyinfluenced by researcher interpretation [START_REF] Smith | Developing Rigor in Qualitative Research: Problems and Opportunities within Sport and Exercise Psychology[END_REF][START_REF] Sparkes | Judging the Quality of Qualitative Inquiry: Criteriology and Relativism in Action[END_REF]). The theoretical continuums in terms of physical activity and fatigue levels sought to partly compensate for the limited frames of analysis in ideal types, and to add complexity to the understanding of the social construction of recounted experience. Further research might also seek to consider more diverse illness symptoms or health behaviorsbeyond fatigue and physical activitylongitudinally, and among a larger panel of participants. Lastly, further research may seek to better understand how social stratifications along lines of gender, socioeconomic status or handicap status impact illness experiences and narratives of the body-self.

Conclusion

Considered in relation to individuals' physical experiences in terms of fatigue and physical activity, the ideal type narratives exemplified diverse experiences and interpretations of chronic fatigue and physical activity, as well their interrelations. These physical dimensions of the HIV experience constituted key components of narratives about the body and self, situated between healthy and ill, vulnerable and robust, tired and active, needing and heeding expected "healthy living" behaviours. The results shed light on the importance of the body and physical experiences in identity work and the experience of chronic illness. More largely, they illustrate the role of socio-political discourses about health and illness, and different ways in which dominant biopedagogies are embraced, embodied or resisted. The results invite (re)consideration of how health messages or HIV-specific interventions could be adapted to lessen the predominance of chronic illness in discourses about the body-self, and limit the moral regulation of health behaviours and lifestyle choices among people living with chronic illness.

"

  I refuse to let fatigue take over, because otherwise I go to bed and it's a disaster! […] I really force myself to... I have to get busy! […] I have to not think about it [fatigue]. Because if I listen to myself… [...] then, I don't do anything!" (Joelle, 42, unemployed with handicap benefits, diagnosed in 1996).


Cuong

No-Arbitrage Condition and Existence of Equilibrium with Dividends

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


Anna Lisa Serra
email: annalisa.serra@cea.fr

Gauthier Lefevre
email: gauthier.lefevre@cea.fr

Olga Cueto
email: olga.cueto@cea.fr

Guillaume Bourgeois
email: guillaume.bourgeois@cea.fr

Marie Claire Cyrille
email: marie-claire.cyrille@cea.fr

Gabriele Navarro
email: gabriele.navarro@cea.fr

Etienne Nowak
email: etienne.nowak@cea.fr

Phase-change memory electro-thermal analysis and engineering thanks to enhanced thermal confinement

Keywords: Phase-Change Memory (PCM), thermal confinement, encapsulation layer, thermal conductivity, electro-thermal simulations

est destinée au dépôt

Introduction

Phase-Change Memory (PCM) is the most mature among the emerging non-volatile memories (NVM) and it represents a reliable alternative to Flash technology. Indeed, it recently entered the market addressing Storage Class Memory (SCM) applications [START_REF] Fong | Phase-change memory-towards a storage-class memory[END_REF]. Moreover, PCM demonstrated to fulfill the automotive market requirements thanks to the phase-change material engineering. In particular, the use of Ge-rich Ge-Sb-Te alloys (GGST) ensures a data retention of ten years for temperatures higher than 150 • C [START_REF] Cappelletti | Phase Change Memory for automotive grade embedded NVM applications[END_REF]. Nevertheless, the main challenge to be overcome in PCM to target ultra low power applications in advanced technology nodes is the programming current reduction. Main efforts in recent years have been oriented to the development of scaled architectures [START_REF] Kim | High performance PRAM cell scalable to sub-20nm technology with below 4F2 cell size, extendable to dram applications[END_REF][START_REF] Boniardi | Optimization metrics for Phase Change Memory (PCM) cell architectures[END_REF] and to the engineering of the active material [START_REF] Park | Reduction of reset current in phase change memory devices by carbon doping in GeSbTe films[END_REF][START_REF] Lung | A novel low power phase change memory using inter-granular switching[END_REF]. However, the main part of the power used during the programming operations of a PCM device is represented by the heat losses [START_REF] Fong | Phase-change memory-towards a storage-class memory[END_REF], making the PCM performance to be far from the adiabatic limit represented by a perfect thermal isolation and zero losses [START_REF] Xiong | Towards ultimate scaling limits of Phase Change Memory[END_REF]. Only in the last years, with the increasing in maturity of the PCM technology, the engineering of thermal barriers and of the thermal conductivity of the surrounding dielectrics became the object of preliminary studies confirming the huge improvement achievable already in analytic devices [START_REF] Ren | Carbon layer application in Phase Change Memory to reduce power consumption and atomic migration[END_REF][START_REF] Fong | Dual-layer dielectric stack for thermally isolated low-energy Phase Change Memory[END_REF] and demonstrating its even higher necessity in ultra scaled PCM [START_REF] Muneer | Vacuum-insulated self-aligned nanowire Phase Change Memory devices[END_REF]. In this framework, we propose the investigation of the thermal improvement in heater-based (i.e. "Wall") state-of-the-art PCM structures [START_REF] Boniardi | Optimization metrics for Phase Change Memory (PCM) cell architectures[END_REF] based on GGST alloy. Thanks to the introduction of an optimized SiC encapsulation layer featuring low thermal conductivity we highlight the benefit of an improved thermal isolation, with respect to standard SiN, by statistical electrical results obtained on 4kb arrays. In particular, we show the higher efficiency of the RESET operation (i.e. amorphization), which leads to a more uniform amorphous region and an improved data retention [START_REF] Serra | Outstanding Improvement in 4Kb Phase-Change Memory of Programming and Retention Performances by Enhanced Thermal Confinement[END_REF]. The electrical results are validated here by 3D electro-thermal simulations and Transmission Electron Microscopy (TEM) observations combined to Energy-dispersive X-ray (EDX) spectroscopy performed on programmed devices. Finally, we present a physical model to correlate the resistance-vs-current (RI) characteristic to the thermal conductivity of the dielectrics surrounding the cell, validating it thanks to the electrical and thermal data acquired. We provide a clear demonstration of the importance of the thermal properties engineering not only of the phase-change layer (PCL) but also of the dielectrics surrounding the heater, and we show the correlation of such properties to the final electrical characteristics of the PCM device.

Device fabrication and electrical characterization

The thermal conductivity of the SiN and SiC layers used as encapsulation were measured by 3ω method [START_REF] Cahill | Thermal conductivity of amorphous solids above the plateau[END_REF] giving a value of 0.41 Wm -1 K -1 for SiC and 1.39 Wm -1 K -1 for SiN [START_REF] Serra | Outstanding Improvement in 4Kb Phase-Change Memory of Programming and Retention Performances by Enhanced Thermal Confinement[END_REF]. The SiN and SiC encapsulation layers were deposited by Plasma Enhanced Chemical Vapor Deposition (PECVD) on the sides of the PCM device after the etching of the PCL (Fig. 1), based on a GGST alloy integrated by sputtering from single target. The PCM devices were fabricated in the Back-End-Of-Line (BEOL) process of our LETI Memory Advanced Demonstrator (MAD), based on 130 nm CMOS technology, used for statistical electrical parameters evaluation thanks to embedded 4kb arrays. The used BEOL process thermal budget is estimated in the order of about 30 minutes at 400 • C. For our electrical and physico-chemical analyses we considered devices with a wall width of about 80 nm and in the following we will refer to the devices by the material of their encapsulation (i.e. SiC or SiN).

The RI characteristics were measured for both SiC and SiN devices in 4kb arrays. After a seasoning procedure of ten SET/RESET cycles, the devices were initialized in the SET state (i.e. crystalline phase) with a pulse having a long fall time (SET-init) in order to achieve the lowest possible SET resistance state (i.e. rise/width/fall times equal to 10ns/300ns/10µs) for both SiC and SiN cells. The gradual resistance increase towards the RESET state is obtained with a staircase-up sequence of squared pulses (SP) of increasing voltage (i.e. current) amplitude, having rise/width/fall time equal to 10ns/300ns/10ns, pre-programming the device in the SET state with a SET-init pulse before each SP. The results are reported in Fig. 2. The different starting SET resistance, obtained by the specific procedure described above, is likely related to a different Ge expulsion during the forming operation [START_REF] Sousa | Operation fundamentals in 12Mb Phase Change Memory based on innovative Ge-rich GST materials featuring high reliability performance[END_REF] and the likely presence of residual amorphous regions due to the different thermal isolation, as it will be confirmed in following analyses. The different starting resistance, influences the "melting current" (I m represented by the first detection of resistance increase in the RI curve), as already reported in previous works [START_REF] Lacaita | The race of Phase Change Memories to nanoscale storage and applications[END_REF]. In particular, SiN based devices feature a melting current about 100 µA lower than the SiC based one. Both RI curves present two different slopes after reaching I m , identified as s 1 and s 2 and reported in Table 1, highlighting a steeper resistance increase in SiC (i.e. higher s 1 and s 2 ).

An isochronal annealing of one hour at increasing temperatures (from 150 • C up to 250 • C) was applied to test the data retention performances comparing SiC and SiN. The results are reported in Fig. 3 and are performed on a population of 2kb (half 4kb matrix) for both SET and RESET states. The SET state in both devices faces a known SET drift [START_REF] Ciocchini | Modeling resistance instabilities of set and reset states in phase change memory with Ge-rich Ge 2 Sb 2 Te 5[END_REF] up to 150

Electro-thermal simulations and TEM/EDX analyses

We performed 3D electro-thermal simulations in COMSOL Multiphysics in order to analyze the impact of the encapsulation on the PCM cell, introducing SiC and SiN thermal conductivities previously measured. Methods and parameters used are described in previous works [START_REF] Cueto | Coupling the phase-field method with an electrothermal solver to simulate phase change mechanisms in PCRAM cells[END_REF]. The different initialization is taken into account by tuning the electrical conductivity of the PCL to get a match between the simulated and the experimental SET resistance. Being the structure symmetric with respect to the z-axis in the zy-plane (see Fig. 1), only half of the structure is simulated. The temperature profile achieved in the two systems is reported in Fig. 4. The volume and the shape of the region reaching a temperature higher than 930 K (i.e. melting temperature of the phase-change material) is not the same in the two structures. In SiC the melted region in the PCL is larger and more uniform along the y axis, while in SiN the temperature decreases moving from the center (y = 0 a.u.) to the heater/encapsulation interface (y = 0.4 a.u.). This explains why SiC has a higher resistance when programmed at 1.2 mA in RESET state in Fig. 2, being the amorphized volume thickness more uniform in SiC than in SiN. A more specific analysis is presented in the next subsections for both SiN and SiC, describing in detail how the temperature profile changes in the PCM device increasing the programming current. In this investigation, the value of the simulated I m is represented by the current at which the PCL starts to melt.

SiN encapsulation

In order to understand the temperature profile achieved in the device, we analyzed the isotherms reaching 930 K in both the PCL and the heater and measured their distance from the PCL/heater interface (h 930K ). The evolution of h 930K in the PCL and in the heater at the center of the cell (i.e. y = 0 a.u.) and at the PCL/encapsulation interface (i.e. y = 0.4 a.u.) are reported as a function of the applied current in a.u. since the simulated device has a different width than the measured one (Fig. 5a). SiN based PCM starts to melt at the center of the heater width for an applied current of 0.4 a.u., whereas the melting at the PCL/encapsulation interface is achieved at 0.6 a.u.. Only at high current the melted thickness becomes homogeneous on the heater surface, and the heat propagation in the heater becomes really important. The position of the maximum temperature (T max ) along the z-axis during the programming operations in both previous analyzed regions of the cell (i.e. y = 0 a.u. and y = 0.4 a.u) is reported in Fig. 5b. At low current, T max is localized inside the PCL at both y positions, while at increasing current the scenario changes: in the center we observe a gradual localization of T max at the PCL/heater interface, while at the PCL/encapsulation interface T max is already inside the heater. TEM/EDX images acquired along zx and zy plane of SiN devices programmed in the RESET state are reported in Fig. 6. The EDX maps along the zy plane (Fig. 6a) shows a not uniform active region on the top of the heater, well evident in the Sb map, compatible with what observed in simulations. On the contrary, along the zx plane (Fig. 6b), the elemental distribution results to be more homogeneous due to the signal integration along the whole lamella thickness, which confirms a higher elemental profile change mainly only along the y-axis. A slight evidence of the shape of the active region is however present in the Sb and Te maps.

SiC encapsulation

Same analyses performed for SiN are reported here for SiC in Fig. 7 (simulations) and Fig. 8 (TEM/EDX). The melting of the PCL is reached for both center and border at 0.5 a.u.. with a h 930K evolution at increasing current almost identical at both y positions. This confirms that the thickness of the melted region (i.e. that becomes amorphous volume after the RESET pulse application) is uniform over the heater/PCL interface along the y-direction, as expected from Fig. 4b. The position of T max (Fig. 7b) at increasing current is also confirming a good homogeneity of the temperature distribution along the y-axis in the cell (i.e. well matched behavior between center and border). As observed for SiN, at low current T max is localized in the PCL, while increasing the current T max shifts inside the heater. The TEM/EDX of a RESET cell along zy plane (Fig. 8a) shows a uniform elemental distribution over the heater/PCL interface in all the Ge, Sb and Te maps. Moreover, the active volume is clearly evidenced along zy plane (Fig. 8b) since uniformly achieved on the whole heater/PCL surface (y-direction) and along the whole lamella thickness (of about one hundred nm), with an elemental distribution compatible with previous works [START_REF] Zuliani | Overcoming temperature limitations in Phase Change Memories with optimized Ge x Sb y Te z[END_REF][START_REF] Sousa | Operation fundamentals in 12Mb Phase Change Memory based on innovative Ge-rich GST materials featuring high reliability performance[END_REF]. Such homogeneity, achieved thanks to SiC encapsulation, is in agreement with simulations results.

Electro-thermal theoretical analysis and discussions

The double slope behavior evidenced in the RI characteristic of both devices in Fig. 2, can be correlated to the different programming scenario that is achieved in the cell for different current amplitudes. As observed in simulations, the transition of the temperature profile from the PCL to the heater, is the origin of the double slope. In first approximation we can describe the electro-thermal behavior of the cell in 2D along the zy plane, as previously reported [START_REF] Boniardi | Optimization metrics for Phase Change Memory (PCM) cell architectures[END_REF]. We can consider that the resistance value R achieved in the PCM device after the pulse application can be expressed as a function of the amorphous thickness obtained in the cell:

R ≃ h am 0 ρ am dz πz w ∝ ln(h am ) (1)
where h am is the amorphous thickness, ρ am the resistivity of the amorphous phase and w the width of the heater along the y direction. We considered a uniform semi-cylindrical amorphous shape over the heater. In such system, the heat distribution is driven by the stationary heat transfer equation, that can be simplified as:

∂ 2 T ∂z 2 ≃ - ρ ON I 2 k th,e f f (2)
where ρ ON represents the device resistivity when the PCL is melted during the pulse application, T is the temperature along the z-axis, k th,e f f is the effective thermal conductivity associated to the overall system, while I is the current applied to the device and which has the role to induce a specific temperature profile inside the cell thanks to the achieved current density J at the PCL/heater interface (I = Jwt with t the heater thickness along the x-axis). Considering the height of the melted region equal to the height of the amorphous final region (h 930K = h am ) we can write from equation eq. ( 2):

∆T i ∝ I 2 k th,e f f h 2 am ( 3
)
where ∆T i represents the temperature increase achieved at the heater/PCL interface (with respect to room temperature), and giving rise to the following correlation between h am and I:

dh am dI ∝ ∆T i k th,e f f I 2 (4)
Introducing the known correlation between power (P) and temperature we have:

∆T i = R T H P ∝ R T H I 2 (5)
where R T H is the global thermal resistance of the device. We can finally simplify eq. ( 4), considering the linear proportionality between R T H and the reciprocal of the effective thermal conductivity of the system k th,e f f :

dh am dI ∝ R T H k th,e f f √ ∆T i ∝ 1 k th,e f f ( 6
)
This correlation is in agreement with expectations, namely that the thickness of the amorphous region achieved in the cell when a specific current is applied increases when the thermal conductivity of the surrounding materials is reduced. Therefore, since the resistance R depends on h am , we can finally correlate the slope of the RI curve of Fig. 2 (i.e. s 1 or s 2 ) to the thermal conductivity of the system:

s = d ln(R) dI = d ln(R) dh am dh am dI ∝ 1 k th,e f f (7)
The electrical results obtained from 4kb were performed on totally equivalent devices, except for the different encapsulation layer. This allows us to validate the relation in eq. ( 7), by calculating the ratios between the slopes in Table 1 and correlating them with the ratios between the measured thermal conductivities. Therefore, the ratio between the two s 1 is SiC and SiN, and the two s 2 should depend only on the ratio between the thermal conductivities of the two materials and this is confirmed by the following calculation:

s 1,S iC s 1,S iN = s 2,S iC s 2,S iN = k th,S iN k th,S iC ∼ 1.8 (8)
This evidences that the difference between the evolution of SiC and SiN RI curves is strongly related to the value of the thermal conductivity of the encapsulation layer which is in direct contact with the PCL and the heater. Therefore, the choice of the encapsulation layer is fundamental in the tuning of the zy plane temperature profile of a Wall based PCM and further optimization can allow an even steeper slope in the RI curve, leading to a current reduction for the RESET operation.

The lower uniformity along y-axis observed in SiN devices can explain the lower I m found in Fig. 2 with respect to SiC. Indeed, some amorphous residuals featuring low thermal conductivity could be generated by such non-homogeneous temperature profile, reducing the current needed to melt the active volume in the PCL. The amorphous residuals are compatible also with the increased SET resistance in SiN devices with respect to SiC.

An other validation of eq. ( 7) can be obtained by comparing the transition from s 1 to s 2 observed in both encapsulation layers. Our hypothesis, confirmed in simulations, supports a transition of T max from the PCL at low currents towards the heater at higher currents. This is an effect which depends on both the phase-change material electrical resistivity evolution in temperature and the amount of PCL that triggers the melting during the programming operations. Based on such hypothesis, we can consider that the main change between the two regimes, in terms of materials surrounding T max , would be along the x-axis, since along y-axis the encapsulation dielectric remains unchanged. Indeed, when T max is inside the PCL, it is surrounded by the phase-change material itself along x-axis as it is visible in the zx scheme of Fig. 1. In such case the thermal conductivity of the PCL influences s 1 . On the contrary, when T max is localized inside the heater, it is surrounded by a SiN based dielectric (referred as "DIEL." in the scheme of Fig. 1) along the x-axis, with an effect on s 2 . Such SiN is of the same nature as the one used for the encapsulation (i.e. PECVD) with equivalent thermal conductivity. The ratio between s 1 and s 2 is equivalent in both SiC and SiN devices:

s 1,S iC s 2,S iC ∼ s 1,S iN s 2,S iN ∼ 2 (9)
This confirms the hypothesis of the equivalent origin of the transition between the two slopes in both encapsulation layers, namely the change of position of T max at increasing current. Following our reasoning it would mean that:

s 1 s 2 = k th,S iN k th,PCL ∼ 2 → k th,PCL ∼ 0.37 W/mK (10)
This result is in agreement with previous works reporting the thermal conductivity of fcc GeSbTe alloys [START_REF] Lyeo | Thermal conductivity of phase-change material Ge 2 Sb 2 Te 5[END_REF][START_REF] Siegert | Impact of vacancy ordering on thermal transport in crystalline phase-change materials[END_REF]. Considering the heterogeneity of GGST layer after BEOL process, facing phases segregation (i.e. formation of fcc GeSbTe phase and Ge phase) as reported in [START_REF] Sousa | Operation fundamentals in 12Mb Phase Change Memory based on innovative Ge-rich GST materials featuring high reliability performance[END_REF][START_REF] Zuliani | Overcoming temperature limitations in Phase Change Memories with optimized Ge x Sb y Te z[END_REF], the thermal conductivity value found in our calculation is in agreement with the presence of grain boundaries and residual amorphous regions in the PCL, outside the active volume. Moreover, being the thermal conductivity of crystalline Ge higher than fcc GeSbTe, the thermal transport in the GGST system should be driven by the GeSbTe phase segregated. This last result highlights the high importance of the engineering of the dielectrics surrounding the heater element in a Wall structure, in order to make the programming operation more efficient. For a given PCL, the encapsulation layer thermal conductivity influences the steepness of s 1 while the thermal conductivity of the dielectrics surrounding the heater influences s 2 . Therefore, in a Wall PCM structure featuring a highly resistive heater element, with a T max localized inside the heater at each programming current, the slope s 2 dominates the RI characteristic and the engineering of the dielectrics surrounding the heater becomes fundamental for drastic current reduction.

Conclusion

In this work we report the improved electrical performances in Wall based state-of-the-art 4kb PCM arrays thanks to the engineering of the device encapsulation introducing an optimized SiC layer featuring low thermal conductivity. Thanks to a more homogeneous amorphous volume obtained in the PCL, we demonstrate an improved data retention in GGST alloys of more than one hour at 250 • C. Moreover, the programming operation becomes more efficient being the SET to RESET transition steeper due to the improved thermal confinement. The results are supported by 3D electro-thermal simulations and TEM/EDX analyses. Finally, a theoretical model is proposed revealing the correlation between the slopes of the RI characteristics and the thermal conductivities of the materials constituting the device. The model is validated by using the measured thermal conductivities of the dielectrics surrounding the PCL and the heater, and the PCL thermal conductivity available in literature. Finally, we provide a demonstration of the strong impact of the dielectrics thermal conductivities and of the thermal boundaries surrounding the heater element, providing a path towards ultra low power next generation PCM devices.

PCL

Fig. 1 .

 1 Fig. 1. Simplified description of our heater based PCM device. TEM images performed on the zx (a) and zy (b) planes on a PCM device with a nominal heater width of 300 nm are reported along with the structure description. The encapsulation layer (SiN or SiC), the heater, the phase-change layer (PCL) and the and dielectrics (DIEL.) are evidenced.

Fig. 2 .

 2 Fig.2. Resistance as a function of the programming current (RI) for 4kb PCM arrays based on GGST alloy with respectively SiN and SiC as encapsulation layers. The slopes s 1 and s 2 are calculated as l . n(R)/dI for both devices and reported in Table1.

  Fig.2. Resistance as a function of the programming current (RI) for 4kb PCM arrays based on GGST alloy with respectively SiN and SiC as encapsulation layers. The slopes s 1 and s 2 are calculated as l . n(R)/dI for both devices and reported in Table1.

1

 1 Fig.2. Resistance as a function of the programming current (RI) for 4kb PCM arrays based on GGST alloy with respectively SiN and SiC as encapsulation layers. The slopes s 1 and s 2 are calculated as l . n(R)/dI for both devices and reported in Table1.

  Fig.2. Resistance as a function of the programming current (RI) for 4kb PCM arrays based on GGST alloy with respectively SiN and SiC as encapsulation layers. The slopes s 1 and s 2 are calculated as l . n(R)/dI for both devices and reported in Table1.

Fig. 3 .Fig. 4 .Fig. 5 .Fig. 6 .Fig. 7 .Fig. 8 .

 345678 Fig. 3. SET and RESET data retention performed by isochronal annealing of one hour at increasing temperatures for both SiN (a) and SiC (b) where the minimum resistance value in the box represents the first quantile, while the bar covers the whole range of data measured. About the 25% of SiN based devices face a RESET failure at 250 • C while SiC ones retain RESET up to more than one hour at the same temperature. The SET resistance faces a resistance drift for both SiC and SiN up to 150 • C, compatibly with previous works [15], followed by a crystallization.

Table 1 .

 1 Slopes (s 1 , s 2 ) and variance for both SiC and SiN devices, obtained from the RI characteristics reported in Fig.2.up to 250 • C, while SiN devices programmed in RESET state face a 25% of failures after such annealing. As demonstrated in the following, the different behavior is likely related to a different in the amorphous region morphology between SiN and SiC which is achieved in the device active volume (i.e. programmed volume) after programming it in the RESET state.

	• C before to trigger the crystallization mechanism at higher

⋆ This work was partially funded by European commission, French State and Auvergne-Rhone Alpes region through ECSEL project WAKEMEUP and French Nano2022 program.