Frédéric Pourraz

Thierry Vallée

Céline Lorentz

Alain Duclos

SYNTHESIS: THE ONLINE TOOL PROVIDING EASY ACCESS AND SYSTEMATIC CROSS-CHECKING OF ALL USEFUL INFORMATION FOR AVALANCHE DANGER ASSESSMENT

Keywords: Risk-communication, avalanche forecasting, cross-checking information, data retrieving

Having a clear vision of current and future snow and avalanche situation depends on a limited number of criteria widely accepted in our community. According whether it is carried out by a forecaster or a practitioner, this task nevertheless requires a significant amount of information to be cross-checked. Due to either lack of time or negligence, such a cross-checking is not systematically carried out. Here again, human factors sometimes prevail over common sense. The originality of our approach lies in the stated wish to make rational use of digital tools to automatize this task, or at least simplify it. For this reason, we have developed the SYNTHESIS tool, accessible via the www.data-avalanche.org platform. On the one hand, it provides an innovative graphical representation that overlays recent avalanche observations, snow tests results, the positions of automatic measuring stations, the danger level, and the critical slope aspects from official bulletins. On the other hand, SYNTHESIS, as suggested by the name, brings together in a single place all useful information available online: avalanche danger bulletins, slope gradients map, descriptions of recent avalanche observations as well as the map of avalanche areas' history, data from automatic measuring stations, and finally snow tests graphical representations. Such information corresponds to the six criteria of the CRISTAL approach. The objective of SYNTHESIS is not to produce an automatic avalanche danger assessment, but to simplify as much as possible data retrieving and exploitation tasks.

INTRODUCTION

Forecasting avalanche risk without being on the field faces challenges encountered by forecasters and especially by practitioners in identifying and accessing the most useful information. Often overlooked or simply not carried out due to its tedious nature, this task is nonetheless necessary to have a clear understanding of avalanche activity and current snow conditions in a specific area.

The importance of systematically evaluating a limited number of tangible and observable criteria was established as early as the 2000s, especially to counter the influence of human factors on decision-making [McCammon, 2000]. Following the Obvious Clues Method [McCammon, 2006], complemented by the Bounded Vigilance Model [McCammon, 2009], it has now been more *Corresponding author address:

Frédéric Pourraz LISTIC -University Savoie Mont Blanc 5 chemin de bellevue 74944 Annecy-le-Vieux FRANCE tel: +33 450 666 002, email: frederic.pourraz@univ-smb.fr than ten years since [Duclos, 2012] adopted six criteria that form the structure of what we consider as "useful information". This approach is known as CRISTAL and is the specific subject of another article in this ISSW issue. As the corresponding digital data is scattered across the internet, SYNTHESIS was designed to gather, process, and present the data in the most readable manner (see Figure 1-1). Accessible both on PCs and smartphones, the objective is to not only cross-check specific information but also consolidate it in a single location.

In addition to data from automatic measuring stations, official bulletins, and other conventional sources, SYNTHESIS leverages data recorded by the French association data-avalanche (www.data-avalanche.org). This data consists of observations shared by the mountain community, covering both avalanche activity and stratigraphy. As of now, between 300 and 800 avalanches are recorded each winter (totaling 4635), often in near-real-time, described, geolocated, and illustrated by text and photos. Stratigraphy is primarily depicted through snow testing, often supplemented by complete stratigraphic profiles, conducted following the protocol described in another article in this ISSW issue (nearly 200 per winter in the French Alps currently, with a total Figure 1: SYNTHESIS online tool of 543 conducted since the winter of testing in 2020-2021). All of this data is now automatically processed and cross-checked, aiming to save mountaineers as much energy as possible for the crucial phases of analysis and decisionmaking that require deep thought. Currently, SYNTHESIS is primarily intended for use during the planning stages of a winter outings (Phase 1 of the 3x3 method [Munter, 1997]).

In Section 2, we will provide a more detailed presentation of the various cross-checked information. In Section 3, we will explain how to associate the relevant data with the evaluation of the six criteria of the CRISTAL approach. Section 4 will offer a critical perspective on the tool and its limitations, before concluding in Section 5.

CROSS-CHECKING INFORMATION

The task, previously entirely reliant on human effort, of cross-checking recent avalanche observations, snow tests, and certain information from the danger assessment bulletin, is now greatly facilitated by the graphical representation shown in Figure 1-1. This representation depicts a truncated cone, viewed from the top, with each face corresponding to a cardinal direction. The outer edges of the cone represent an altitude of 1000 meters. Its apex is truncated at the maximum altitude of the considered mountain massif, limited to 4000 meters in altitude. Level curves are also depicted every 200 meters of elevation change. This cone is currently available for each of the French Alpine massifs. The first step is to select a massif on the left-hand map (see Figure 12). The divisions used correspond to those employed by Météo-France, sometimes modified to correct certain inconsistencies. We will address the limitations of this division in Section 4. The color of each massif corresponds to the highest danger level for the day, extracted from the currently valid official bulletin (see legend in the upper left corner). It's important to note that all the data is automatically updated hourly. The white "i" icon, sometimes overlaying a massif, signifies that in addition to the danger level, one or more observations (avalanche(s) and/or snow test(s)) and/or one or more alerts (which will be discussed later) are also available. When viewed on a computer, the name of the massif is displayed when hovering the mouse over it. Below this map, there is a link for reporting a new avalanche observation (a historical practice of the data-avalanche association) as well as a link to a video presenting all the tool's functionalities (currently available only in French).

When selecting a massif, all the information displayed on the right panel is updated accordingly, starting with the name of the massif, date, and time of the last update. Regarding the cone, the danger level is more detailed than on the massif map, as differences can occur based on altitude thresholds occasionally mentioned in the bulletin. This later sometimes highlights more critical aspects than others. This is depicted by graying out the corresponding orientation sectors and the truncated cone summit to provide a clearer visualization of this information. Several icons (refer to legend in Figure 1-3) may overlay the cone, depending on the massifs and available observations. Light blue triangles correspond to recent avalanche observations, hexagons denote snow tests, and blue circles indicate the presence of an autonomous snow-meteorological station, such as the FlowCapt system [START_REF] Bourgeois | Double measurements of snow depth: Innovation for better management of avalanche risk in real time on the roads of Savoy (France)[END_REF], installed on the field. The color of the hexagons depends on the result of the respective test. Four different colors are used to represent the criticality of the test, indicating the propensity of local crack to propagate in an identified weak layer (as described in another article within this ISSW issue, as mentioned earlier). All these icons are positioned at a specific altitude and aspect, determined by the geolocation of the information they represent. They are also clickable, providing access to information details, including avalanche descriptions, propagation test outcomes, stratigraphic and temperature profiles, autonomous station measurements (windward and leeward snow depths, snow transport, air and surface snow temperatures, wind speed, etc.). This cone allows for the automatic cross-checking of various pieces of information, providing a better perception of the current snow situation. To complement this, users have the option to view all this information over a 10-day history by simply adjusting a slider (see figure 1234). Indeed, depending on the weather conditions of the preceding days, it is valuable to accumulate information over a shorter or longer time span. Figure 2 illustrates an example of this accumulation, starting from March 19, 2022, and going back up to 5 days prior. We can observe that even though the situation seems to be improving (the two snow tests conducted on the 19th are not in purple, unlike the test conducted 4 days earlier), avalanche observations have been reported in the preceding days, sometimes even outside the critical aspects indicated in the official bulletin. These avalanche observations are the subject of a current alert (see figure 12345). Indeed, any practitioner can sign up to receive email alerts from the data-avalanche association, and these laters are also visually accessible through SYNTHESIS. These alerts cover three types of information: the ongoing avalanche situation, precipitation (snow/rain), and snow transport by the wind. Each type is categorized into three levels: situation to monitor (yellow alerts), concerning situation (orange alerts), and unusual situation (red alerts).

Lastly, all this information (observations, measurement stations, and alerts) can also be viewed on a map (see tab in figure 123456). This second, more conventional visualization offers a more precise geolocation of data but loses the advantage of cross-checking. Therefore, these two types of visualization are complementary.

RETRIEVING USEFUL DATA

To facilitate the implementation of the CRISTAL approach introduced in section 1, SYNTHESIS also gathers all the relevant information for evaluating the six criteria, in order to systematize their consideration during each outing. Each criterion is represented by a clickable icon (see figure 1234567).

The first icon (danger level) provides access to the complete current snow bulletin, which is posted daily by the services of Météo-France. This bulletin naturally corresponds to the massif previously selected on the left-hand map (see figure 12).

The second icon (slopes steeper than 30°) provides access to the geographical portal of the French government. This website offers a 1:25,000 scale map, along with an additional layer that colors and identifies slopes steeper than 30°(a key factor in avalanche triggering), categorized in 5°increments (see figure 3). The third icon (recent avalanches observed) opens a dialog box that provides access to (i) the database of all avalanches recorded by the data-avalanche association: www.data-avalanche.org/now; (ii) the same geographical portal as mentioned earlier but with an other additional layer depicting the historical record of avalanches in the French mountains: CLPA (Map of Avalanche Phenomena Localization -see figure 4). Although not exhaustive and sometimes subject to uncertainties, this map can still provide valuable information about the potential extent of an avalanche. The fourth icon (temperature inncrease, snow thawing) and the fifth icon (new overloading: wind-driven accumulation, fresh snow, rain) both open a dialog box that provides access to the list of all autonomous snow and weather stations in the selected massif (FlowCapt stations and NIVOSE stations by Météo-France). Selecting a station provides access to all the measured data. This is particularly the case for FlowCapt stations, which offer a large number of metrics and unlimited access to the entire historical data (see figure 5). The data from NIVOSE stations, on the other hand, is much more limited. The sixth icon (a possible buried weak layer) opens a dialog box that provides access to (i) the database of all snow tests recorded by the dataavalanche association, which can be visualized graphically (see figure 6); (ii) the website https://www.regobs.no, which lists a few tests in France (mainly column tests). Finally, the CRISTAL logo (see figure 1234567) provides access to our tool that supports the approach (see figure 7). As we have just seen, SYNTHESIS facilitates the collection and cross-checking of useful information, both for practitioners and for forecasting purposes. After more than one season of successful operation, one particular point emerges as a certain limitation. The division by massif implied by the consideration of the danger assessment bulletin sometimes masks certain situations. Indeed, this delimitation is not always consistent with the snow and weather conditions, and a set of observations (recent avalanches and/or snow tests) indicative of a concerning avalanche situation can be at the boundary of 2 or more massifs. If the user restricts themselves to the information from a single massif, they may underestimate this situation. It is therefore important to have a broader view of the concept of massif by also exploring peripheral data. For this reason, all the information is also accessible on a map (see tab figure 123456), which provides a wider view of the data location but loses the advantage of data cross-checking through the cone representation.

All of this data is also used daily by three French "departments" (Savoie, Isère, Hautes-Alpes) for ensuring road safety and reopening major Alpine mountain passes at the end of the season. In order to facilitate this forecasting task and address the aforementioned limitation, we have also implemented a version organized by "departments" (see figure 8). This division provides an overview of multiple massifs, which is why information related to snow bulletins by individual massifs is absent from this representation. Nevertheless, all observations within the "departments" are crosschecked through the cone representation to offer a comprehensive view of the past and ongoing situation. However, situations that straddle the borders of two or more "departments" can still be obscured. For the upcoming winter, we are working on a new evolution of the tool that allows users to select sectors to be grouped into a single cone, aiming to address this issue.

CONCLUSION

The objective analysis of the current snow and avalanche situation is often complicated by the need to cross-check numerous pieces of information from various sources, accessible but scattered across the Internet. This situation leaves too much room for hasty interpretations often driven by human factors. SYNTHESIS has emerged from this premise and our belief that a rational use of digital tools would provide valu-able assistance. Developed within an academic and association framework, this application is now fully operational. It is available for free and unrestricted access.

Following an initial effort of data collection and cross-checking, SYNTHESIS offers an innovative graphical representation (see Figure 1-1) along with essential information corresponding to the six criteria of the CRISTAL approach (see Figure 1234567).

All within a single virtual cone (altitude and aspect bands), one can find recent avalanche observations, results of snow tests, the locations of automatic measuring stations, the danger level, and the critical aspects from official bulletins. All of these pieces of information are updated on an hourly basis and are presented in the context of the French Alpine massifs (see Figure 1-3). Moreover, users have the option to select a historical range of up to 10 days (see Figure 1234). Alerts are also included, providing additional interpreted information based on predetermined thresholds or expert insights: precipitation, snow transport by wind, or specific avalanche conditions (see Figure 12345).

Each of the 6 icons in the CRISTAL approach includes links to relevant information for the specific massif under consideration: current danger bulletin, automatic measuring stations, slope map, observed recent avalanches, envelope of recorded notable avalanches (CLPA), and snow tests.

The objective of SYNTHESIS is not to automatically produce an estimation of avalanche danger, but rather to simplify the tasks of data collection and analysis as much as possible. Since its public release in March 2022, it has already received a number of visits that exceeded our expectations: more than 35,000 in just over a year, with an average consultation time of around 4 minutes.

By consolidating all the relevant and available information online in one place, SYNTHESIS likely contributes to avalanche risk prevention and certainly to the education of practitioners.

Figure 2 :

 2 Figure 2: Information's history

Figure 3 :

 3 Figure 3: 2D map with slope steepness layer see: https://www.geoportail.gouv.fr

Figure 4 :

 4 Figure 4: 2D map with historical avalanches layer see: https://www.geoportail.gouv.fr

Figure 5 :

 5 Figure 5: FlowCapt example of mesured data: wind transport of snow, air temperature, snow surface temperature, snowpack height windward and leeward

Figure 6 :

 6 Figure 6: Graphical test's results see: https://www.data-avalanche.org/romansns

Figure 7 :

 7 Figure 7: CRISTAL's tool supporting the framework

Figure 8 :

 8 Figure 8: SYNTHESIS "departments" view


Attention : le but de votre recherche n'est pas d'emblée d'être dans le parti-pris, mais de conduire une reflexion qui vous amène à prendre parti, c'est-à-dire à trancher de manière justifiée et rationnelle. De même, le but n'est pas ici de défendre telle ou telle approche mais de clarifier ce qui est confus et qui pose donc nécessairement un problème.

  Etude d'un cas concret ICES -2023/2024 -Semestre 1 -M Chaigneau Genèse et généalogie du wokisme. Critique du dit « marxisme culturel ». AVERTISSEMENT ! Il s'agit d'un cours de méthodologie. L'étude du cas concret n'est ici qu'un exemple, parmi des millions d'autres possibles. Il ne s'agit donc pas d'un cours exhaustif sur la question traitée.

Francis Bacon, 2 - 2 -

 22 Novum organum, Livre I, 95, 1620. ICES -2023/2024 -Semestre 1 -M Chaigneau Éléments de problématisation. Source : wikipedia Source : https://www.toupie.org/Dictionnaire/Wokisme.htm ICES -2023/2024 -Semestre 1 -M Chaigneau Questionnement / Étonnement (au sens philosophique) "Le marxisme culturel est-il une réalité ? Le wokisme est-il le produit du marxisme ? Le 'wokisme' et les théories qui l'accompagnent ne sont-ils pas plutôt le fruit de la logique libérale ?" Questionnement / Premières hypothèses. (a) Le wokisme n'est pas de facto un neomarxisme ( Domination / exploitation -Postmarxisme / Marxisme) Ni un neoexistentialisme (retour aux essences) Ni un simple anti-racisme (Essentialisation des races -Bouteldja / Camus) Ni écologiste (adepte d'une forme de transhumanisme) Le wokisme n'est pas un progressisme (Chollet / Rousseau / Bouteldja / lagasnerie)Le wokisme, en terme marxiste justement, serait peut-être plutôt la superstructure idéologique du libéralisme.I -Étude comparativeApproche épistémologique du postmodernisme (wokisme) Fin des dits "Grands récits" et lutte contre le totalitarisme : Point de vue de la totalité 3 -Identité partielle sujet/objet / Monisme dialectique 4 -Détermination de classe 5 -Donation de sens par la praxis / l'histoire.

  Introduction : thème et problématique Développement : Ce qui est déjà connu / limites / Propositions. Conclusion : apport / ouverture (lacunes déjà ciblées).

	3. Remarques et premiers obstacles épistémologiques.
	3.1 De la difficulté de l'omniscience.
	3.1.1	Survoler l'ensemble du savoir ?
	3.1.2	Importance de la délimitation et des axes de recherches : ne pas voir trop large.
	3.2 Savoir où chercher.
	3.2.1	Ressources académiques / professionnelles (Cairn / Google Scholar)
	3.2.2	Constituer corpus / Bibliographie.
	3.3 Transdisciplinarité et axes de compétences.

3.3.1 Délimiter le déjà-vu et le déjà-fait 3.3.2 Enrichir la problématique de ses compétences extérieures. 3.3.3 Établir une bibliographie : quelles références incontournables ? spécifiques ? 4.. Plan typique d'une revue de littérature critique.

'étroite alliance de ces deux facultés: l'expérimentale et la rationnelle…."

	"Les philosophes qui se sont mêlés de traiter des sciences se
	partageaient en deux classes, à savoir: les empiriques et les
	dogmatiques. L'empirique, semblable à la fourmi, se contente
	d'amasser et de consommer ensuite ses provisions. Le
	dogmatique, tel que l'araignée, ourdit des toiles dont la matière
	est extraite de sa propre substance. L'abeille garde le milieu;
	elle tire la matière première des fleurs des champs et des
	jardins; puis, par un art qui lui est propre, elle la travaille et la
	digère. La vraie philosophie fait quelque chose de semblable;
	elle ne se repose pas uniquement ni même principalement sur
	les forces naturelles de l'esprit humain, et cette matière qu'elle
	tire de l'histoire naturelle, elle ne la jette pas dans la mémoire
	telle qu'elle l'a puisée dans ces deux sources, mais après
	l'avoir aussi travaillée et digérée, elle la met en magasin. Ainsi
	notre plus grande ressource et celle dont nous devons tout
	espérer, c'est l

-Présentation et défense du choix du sujet. Pourquoi ce sujet est intéressant ? -Quels sont les documents connus sur lesquels nous pourrions nous appuyer ? -Une synthèse problématisée. Quel problème pose ce sujet ? -Démarche critique : lacunes ciblées et/ou délimitation claire.

  1. L'approche généalogiste / réductionnisme discursifDepuis le XVIIIe siècle, le sexe n'a pas cessé de provoquer une sorte d'éréthisme discursif généralisé. Et ces discours sur le sexe ne se sont pas multipliés hors du pouvoir ou contre lui; mais là même où il s'exerçait et comme moyen de son exercice; partout ont été aménagées des incitations à parler, partout des dispositifs à entendre et à enregistrer, nous convainquons par un étrange scrupule que nous n'en disons jamais assez, que nous sommes trop timides et peureux, que nous nous cachons l'aveuglante évidence par inertie et par soumission, et que l'essentiel nous échappe toujours, qu'il faut encore partir à sa recherche. Sur le sexe, la plus intarissable, la plus impatiente des sociétés, il se pourrait que ce soit la nôtre.Le titre de cet exposé reprend une formule de Gilles Deleuze : « l'infâme dialectique ». Faisant écho à la maxime de Voltaire, « écraser l'infâme », qui visait le fanatisme religieux, elle donne une idée de la violence de ce rejet de la dialectique en France au cours du 20ème siècle La cause est aisée àidentifier : composante du marxisme, la dialectique et sa critique fonctionnent ici comme indicateur d'un rapport à Marx et plus largement d'un rapport au Parti communiste, force politique alors centrale et dont le relatif affaiblissement va se jouer au cours de cette période. Mais elle indique aussi un déplacement du débat politique sur le terrain philosophique, qui va produire des effets complexes. En effet, chez les philosophes du moment qui manifestent leur détestation de la dialectique, on peut relever un certain nombre de thématiques communes qui toutes ont une portée politique, mais une portée politique indirecte et qui va se déployer sur le terrain philosophique : critique de l'humanisme et du sujet, tournant linguistique, critique de la rationalité et de la représentation,dénonciation de la dialectique et anti-hégélianisme virulent, Alors que le modèle de Crenshaw décrit habilement le fonctionnement de ce que l'écrivain féministe afro-américaine Patricia Hill Collins a qualifiée de « matrice d'oppressions », la bidimensionnalité spatiale du modèle suggère son insuffisance comme une explication de la raison pour laquelle cette « matrice » existe(Collins, 1990). Qui a créé ces avenues? Pourquoi certaines personnes les empruntent-elles? Où ont-elles été construites et quand? Le modèle spatial décourage les questions comme celles-ci. Le fait que les femmes noires en question soient des travailleuses qui gagnent au mieux un salaire modeste, mais rendent les patrons de General Motors (GM) très riches, est simplement pris pour acquis. Autrement dit, pour revenir à la métaphore des routes qui se croisent, le sol sur lequel les routes ont été construites est un fait, même pas remis en question. Si wokisme = continuité logique libérale => cela pose une question fondamentale qui touche tout le spectre politique.5. Le corpus philosophico-politique est-il suffisant ou seulement un préalable ?

	Wokisme V.S marxisme (b)
	Hannah Arendt, le système totalitaire, 1951
	Jean-François Lyotard, la condition postmoderne,1979
	La French Theory / Théorie de la déconstruction : Gilles
	Deleuze, Jacques Derrida, Michel Foucault, Félix
	Guattari, Luce Irigaray, Julia Kristeva, Jacques Lacan,
	Jean-François Lyotard, Jacques Rancière et Monique
	Wittig
	théorisations du désir et de la sexualité,promotion de l'autonomie, montée des
	thématiques autogestionnaires et de la critique de l'État,redéfinition des
	exploités en tant qu'exclus, promotion d'une analyse moléculaire et de la
	micropolitique, critique généralisée de l'engagement traditionnel ainsi que des
	organisations politiques et syndicales, esthétisation et sophistication croissante
	du discours philosophique

partout des procédures pour observer, interroger et formuler. On le débusque et on le contraint à une existence discursive. De l'impératif singulier qui impose à chacun de faire de sa sexualité un discours permanent, jusqu'aux mécanismes multiples qui, dans l'ordre de l'économie, de la pédagogie, de la médecine, de la justice, incitent, extraient, aménagent, institutionnalisent le discours du sexe, c'est une immense prolixité que notre civilisation a requise et organisée. Peut-être aucun autre type de société n'a jamais accumulé, et dans une histoire relativement si courte, une telle quantité de discours sur le sexe. De lui, il se pourrait bien que nous parlions plus que de toute autre chose; nous nous acharnons à cette tâche; nous Michel Foucault, Histoire de la sexualité, I -l'incitation aux discours, Tel Gallimard, 1994, p. 26

2. La fin des grands récits et le totalitarisme

Source : Isabelle Garo, colloque, intervention : « L'infame dialectique » Le rejet de la dialectique dans la philosophie française de la seconde moitié du 20e siècle .Jeudi 24 mai 2012. "" Barbara Foley, Intersectionality: A Marxist Critique -How do gender, race and class correlate?, Septembre 2018 Conclusion de notre étude de cas. 1. Gauche # Marxisme 2. Wokisme # Marxisme 3. Wokisme = néolibéralisme culturel ? 4.


Srilakshmi Krishnamoorthy

Abinash Sarma
email: sarmaabinash15@iisertvm.ac.in

SOME NEW CONGRUENCES FOR ℓ-REGULAR MULTIPARTITIONS

Keywords: 2010 Mathematics Subject Classification. 05A17, 11P83, 11P84 Integer Partitions, Multipartitions, Regular Partitions, Congruences

For a positive integer n, let B ℓ 1 ,...,ℓr (n) denote the number of (ℓ

In this paper, we prove several infinite families of congruences satisfied by B (r) ℓ (n) for different values of ℓ and r.

≡ 0 (mod 5), where p i = 5 is a prime number such that p i ≡ 2 (mod 3) for i ∈ {1, • • • , t + 1} and p t+1 ∤ j.

Introduction

Given a positive integer n, a partition of n is said to be ℓ-regular, for a fixed positive integer ℓ, if none of its parts is divisible by ℓ. Let n = a 1 + a 2 + • • • + a r be a partition of n. Suppose, in turn, λ (i) is a partition of a i for each i ∈ {1, 2, . . . , r}. We call the ordered tuple (λ (1) , λ (2) , . . . , λ (r) ) as a r-multipartition of n; if the value of r is obvious from the context, we simply call it a multipartition of n. In particular, a 2-multipartition of n is called a bipartition of n. Let us fix r positive integers, namely, ℓ 1 , ℓ 2 , . . . , ℓ r such that λ (i) is a ℓ i -regular partition. Then (λ (1) , λ (2) , . . . , λ (r) ) is said to be a (ℓ 1 , ℓ 2 , . . . , ℓ r )-regular multipartition of n.

Let B ℓ1,...,ℓr (n) denote the number of (ℓ 1 , ℓ 2 , . . . , ℓ r )-regular multipartitions of n. By convention, we assume that B ℓ1,...,ℓr (0) = 1 for any r and any (ℓ 1 , ℓ 2 , . . . , ℓ r ). Note that the generating function for B ℓ1,...,ℓr (n) is given by

∞ n=0 B ℓ1,...,ℓr (n)q n = E ℓ1 E ℓ2 • • • E ℓr E r 1 , (1)
where E k := ∞ m=1 (1 -q km ) for k ∈ N. Note that, for any prime number p and any k ∈ N, we have

E kp ≡ E p k (mod p). (2)
Suppose ℓ 1 = ℓ 2 = • • • = ℓ r = ℓ. Then we say that (λ (1) , λ (2) , . . . , λ (r) ) is a ℓ-regular r-multipartition of n. In this case, we simplify the notation for B ℓ1,...,ℓr (n) as B (r) ℓ (n). Now from (1), we have

∞ n=0 B (r) ℓ (n)q n = E r ℓ E r 1 . (3)
Recently, for different values of ℓ, many properties of ℓ-regular partition function have been discussed by several mathematicians in [START_REF] Adiga | Congruences for 7 and 49-regular partitions modulo powers of 7[END_REF][START_REF] Ahmed | New congruences for ℓ-regular partitions for ℓ ∈ {5, 6, 7, 49[END_REF][START_REF] Carlson | Infinite families of infinite families of congruences for k-regular partitions[END_REF][START_REF] Cui | Arithmetic properties of ℓ-regular partitions[END_REF][START_REF] Dai | On the distribution of odd values of 2 a -regular partition functions[END_REF][START_REF] Furcy | Congruences for ℓ-regular partition functions modulo 3[END_REF][START_REF] Hou | Quadratic forms and congruences for ℓ-regular partitions modulo 3, 5 and 7[END_REF][START_REF] Ranganatha | Ramanujan-type congruences modulo powers of 5 and 7[END_REF][START_REF] Webb | Arithmetic of the 13-regular partition function modulo 3[END_REF][START_REF] Xia | Congruences for some l-regular partitions modulo l[END_REF]. For recent results for (ℓ 1 , ℓ 2 )-regular bipartition function, the reader may refer to [START_REF] Adiga | A simple proof of a conjecture of Dou on (3,7)-regular bipartitions modulo 3[END_REF][START_REF] Dou | Congruences for (3, 11)-regular bipartitions modulo 11[END_REF][START_REF] Kathiravan | Ramanujan-type congruences modulo m for (l, m)-regular bipartitions[END_REF][START_REF] Kathiravan | On ℓ-regular bipartitions modulo ℓ[END_REF][START_REF] Lin | Arithmetic of the 7-regular bipartition function modulo 3[END_REF][START_REF] Lin | An infinite family of congruences modulo 3 for 13-regular bipartitions[END_REF][START_REF] Wang | Arithmetic properties of (k, ℓ)-regular bipartitions[END_REF]. Saikia and Boruah [START_REF] Saikia | Congruences of ℓ-regular partition triples for ℓ ∈ {2, 3, 4, 5}[END_REF] proved certain congruences for B

(3) ℓ (n) taking ℓ ∈ {2, 3, 4, 5} and Gireesh and Naika [START_REF] Gireesh | On 3-regular partitions in 3-colors[END_REF] proved for B

(3) 3 (n). Chern, Tang, and Xia [START_REF] Chern | Arithmetic properties for 7-regular partition triples[END_REF] established infinite families of congruences for B

(3) 7 (n) modulo powers of 7. Baruah and Das [START_REF] Baruah | Generating functions and congruences for 9-regular and 27-regular partitions in 3 colours[END_REF] established the generating functions for B

(3) 9 (n) and B

(3) 27 (n). Very recently, Murugan and Fathima [START_REF] Murugan | Arithmetic properties of 3-regular 6-tuple partitions[END_REF] studied congruences for B [START_REF] Chern | Arithmetic properties for 7-regular partition triples[END_REF] 3 (n). The following are the main results of this paper.

Theorem 1.1. For any n ≥ 0 and t ≥ 0, we have

(i) B (12) 3 3p 2 1 • • • p 2 t+1 n + p 2 1 • • • p 2 t p t+1 (p t+1 + 3j) -1 ≡ 0 (mod 3)
, where p i is a prime number such that p i ≡ 2 (mod 3)

for i ∈ {1, • • • , t + 1} and p t+1 ∤ j. (ii) B (15) 3 3p 2 1 • • • p 2 t+1 n + p 2 1 •••p 2 t pt+1(5pt+1+12j)-5 4
≡ 0 (mod 3), where p i = 3 is a prime number such that p i ≡ 3 (mod 4) for i ∈ {1, • • • , t + 1} and p t+1 ∤ j.

(iii) B

(iv) B (7) 7

7p 2 1 • • • p 2 t+1 n + 7 p 2 1 •••p 2 t pt+1(pt+1+4j)-1 4
≡ 0 (mod 7), where p i = 7 is a prime number such that p i ≡ 3 (mod 4) for i ∈ {1,

3p 2t+2 n + p 2t+1 (p + 3j) -1 ≡ 0 (mod 3)
, where p is a prime number such that p ≡ 2 (mod 3) and p ∤ j.

(ii) B (15) 3 3p 2t+2 n + p 2t+1 (5p+12j)-5 4 ≡ 0 (mod 3), where p = 3 is a prime number such that p ≡ 3 (mod 4) and p ∤ j.

(iii) B ≡ 0 (mod 5), where p = 5 is a prime number such that p ≡ 2 (mod 3) and p ∤ j.

(iv) B (7) 7

7p 2t+2 n + 7 p 2t+1 (p+4j)-1 4
≡ 0 (mod 7), where p = 7 is a prime number such that p ≡ 3 (mod 4) and p ∤ j. ≡ 0 (mod 11), where p ∈ {3, 11} is a prime number such that p ≡ 3 (mod 4) and p ∤ j.

Theorem 1.3. For n ≥ 0 and t ≥ 0, we have

(i) B (6) 5 p 4t n + (p 4t -1) ≡ p 2t B (6) 5 (n) (mod 5), if B (6)
5 (p -1) ≡ 0 (mod 5), where p = 5 is a prime number.

(ii) B (6) 7 7p 4t n + 7p 4t -3 2 ≡ p 4t B (6) 7 (7n + 2) (mod 7), if B (6) 7 7p-3 2
≡ 0 (mod 7), where p = 7 is an odd prime number.

Theorem 1.4. For n ≥ 0 and t ≥ 0, we have

(i) B (12) 3 6p 2t+2 n + p 2t+1 (4p + 3j) -1 ≡ 0 (mod 6)
, where p is a prime number such that p ≡ 2 (mod 3) and j is an even number such that p ∤ j.

(ii) B ≡ 0 (mod 10), where p = 5 is a prime number such that p ≡ 2 (mod 3) and j is an even number such that p ∤ j.

(iii) B (15) 3

15p 2t+2 n + p 2t+1 ((12α+5)p+12j)-5 4
≡ 0 (mod 15), where α ∈ {1, 2, 3, 4}, p = 3 is a prime number such that p ≡ 3 (mod 4) and j is such that 5 | j but p ∤ j.

Theorem 1.5. For any n ≥ 0 and t ≥ 0, we have

(i) B (20t+1) 5
(20n + 14) ≡ 0 (mod 10 Note that Corollary 1.2 (i), (ii), (iii), (iv), (v) can be proved by taking p 1 = • • • = p t+1 = p in Theorem 1.1 (i), (ii), (iii), (iv), (v), respectively.

Preliminary

In this section, we list down a few results which will be used while proving our main theorems. Before stating the first lemma, we write, for any r ≥ 1, [START_REF] Newman | The coefficients of certain infinite products[END_REF], Theorem 1). Suppose that r is even, 0 < r ≤ 24. Let p be a prime such that r(p-1) ≡ 0 (mod 24). Set δ = r(p -1)/24. Then for any n ≥ 0

E r 1 = ∞ n=0 a r (n)q n . (4) Lemma 2.1 ([
a r (pn + δ) = a r (δ) • a r (n) -p r 2 -1 • a r n -δ p ,
with the convention that a r (x) = 0, if x is not an integer.

Let us list down the following 2-dissection, 5-dissection, 7-dissection, and 11-dissection results.

Lemma 2.2 ([13], Theorem 2.1). We have,

E 5 E 1 = E 8 E 2 20 E 2 2 E 40 + q E 3 4 E 10 E 40 E 3 2 E 8 E 20 . ( 5
)
Lemma 2.3 ([21] and [START_REF] Watson | Ramanujans Vermutung über Zerfällungszahlen[END_REF]). For R(q) := ∞ m=1 1-q 5m-4 1-q 5m-1 1-q 5m-3 1-q 5m-2 , we have

E 1 = E 25 1 R(q 5 ) -q -q 2 R q 5 . ( 6
)
Lemma 2.4 ([12], (10.5.1)).

For A i (q) := ∞ m=1 (1 -q 7m-i )(1 -q 7m-7+i ), i ∈ {1, 2, 3}, we have E 1 = E 49 A 2 (q 7 ) A 1 (q 7 ) -q A 3 (q 7 ) A 2 (q 7 ) -q 2 + q 5 A 1 (q 7 ) A 3 (q 7 ) . (7)
Lemma 2.5 ([12], (10.6.1)).

For B i (q) := ∞ m=1 (1 -q 11m-i )(1 -q 11m-11+i ), i ∈ {1, 2, 3, 4, 5}, we have E 1 = E 121
B 4 (q 11 ) B 2 (q 11 ) -q B 2 (q 11 ) B 1 (q 11 ) -q 2 B 5 (q 11 ) B 3 (q 11 ) + q 5 + q 7 B 3 (q 11 ) B 4 (q 11 ) -q 15 B 1 (q 11 ) B 5 (q 11 ) .

We conclude this section by writing down Dedekind's eta-function defined by

η(z) := q 1/24 ∞ m=1 (1 -q m ) . (9)

Congruences Modulo a Prime Number

In this section, we will prove Theorem 1.1, and Theorem 1.3.

Proof of Theorem 1.1. Part (i): Let us take ℓ = 3 and r = 12 in (3) to get ∞ n=0 B (12) 3 (n)q n = E 12 3 E 12 1 . Now, by (2), we get ∞ n=0 B (12) 3 (n)q n ≡ E 12 3 E 4 3 ≡ E 8 3 (mod 3).
In view of (9), we get

∞ n=0 B (12) 3
(3n)q 3n+1 ≡ η 8 (3z) (mod 3).

Writing

η 8 (3z) = ∞ n=0 a(n)q n , we get B (12) 3
(3n) ≡ a(3n + 1) (mod 3). [START_REF] Furcy | Congruences for ℓ-regular partition functions modulo 3[END_REF] Moreover, one can verify that a(n) = 0, if n ≡ 1 (mod 3).

Serre [START_REF] Serre | Sur la lacunarité des puissances de η[END_REF] showed that η 8 (3z) is a normalized Hecke eigenform of weight 4, level 9. Therefore, for any prime number p, operating the Hecke operator T p on η 8 (3z), we get

η 8 (3z) | T p = a(p) • η 8 (3z). (11) Since η 8 (3z) | T p = ∞ n=1 a(pn) + p 4-1 • a n p q n = ∞ n=1 a(pn) + p 3 • a n p q n ,
therefore, by comparing coefficients on both sides of ( 11), we get

a(pn) + p 3 • a n p = a(p) • a(n) ∀ n ≥ 1.
Let p be a prime number such that p ≡ 2 (mod 3); then we have

a(pn) + p 3 • a n p = 0. ( 12
)
Substituting n by pn + r with p ∤ r in the above equation, we get a(p 2 n + pr) = 0. Now, substituting n by 3n + 1 -pr, we get

a(3p 2 n + p 2 -p 3 r + pr) = 0. But, 3p 2 n + p 2 -p 3 r + pr = 3 p 2 n + p 2 -1 3 + pr(1-p 2 )
3 + 1. Thus, by [START_REF] Furcy | Congruences for ℓ-regular partition functions modulo 3[END_REF], we get

B (12) 3 3 p 2 n + p 2 -1 3 + pr(1 -p 2 ) 3 ≡ 0 (mod 3).
As gcd 1-p 2 3 , p = 1, so we have p ∤ r if and only if p ∤ j, where j := r(1-p 2 )

3

. Hence, for j ≡ 0 (mod p), we get

B (12) 3 3p 2 n + p 2 -1 + 3pj ≡ 0 (mod 3). ( 13
)
Substituting n by pn in [START_REF] Hirschhorn | The Power of q: A Personal Journey[END_REF], we get [START_REF] Furcy | Congruences for ℓ-regular partition functions modulo 3[END_REF], we get

a(p 2 n) = -p 3 • a(n) ≡ a(n) (mod 3). Now, substituting n by 3n + 1, we get a(3p 2 n + p 2 ) ≡ a(3n + 1) (mod 3). But, 3p 2 n + p 2 = 3 p 2 n + p 2 -1 3 + 1. Thus, by
B (12) 3 3 p 2 n + p 2 -1 3 ≡ B (12) 3 (3n) (mod 3). ( 14
)
Let t ≥ 0 and p i be a prime number such that

p i ≡ 2 (mod 3) for i ∈ {1, • • • , t + 1}. Since p 2 1 • • • p 2 t n + p 2 1 • • • p 2 t -1 3 = p 2 1 p 2 2 • • • p 2 t n + p 2 2 • • • p 2 t -1 3 + p 2 1 -1 3 ,
therefore by repeated use of ( 14), we get

B (12) 3 3p 2 1 • • • p 2 t n + (p 2 1 • • • p 2 t -1) ≡ B (12) 3
(3n) (mod 3).

Substituting n by p 2 t+1 n +

p 2 t+1 -1 3
+ p t+1 j with p t+1 ∤ j, we get the desired result in view of [START_REF] Hirschhorn | Elementary proofs of parity results for 5-regular partitions[END_REF]. Part (ii): Let us take ℓ = 3 and r = 15 in (3) to get

∞ n=0 B (15) 3 (n)q n = E 15 3 E 15 1 . Now, by (2), we get ∞ n=0 B (15) 3 (n)q n ≡ E 15 3 E 5 3 ≡ E 10 3 (mod 3).
In view of (9), we get

∞ n=0 B (15) 3
(3n)q 12n+5 ≡ η 10 (12z) (mod 3).

Writing η 10 (12z) = ∞ n=0 a(n)q n , we get B

(3n) ≡ a(12n + 5) (mod 3). ( 15)

Serre [START_REF] Serre | Sur la lacunarité des puissances de η[END_REF] showed that η 10 (12z) = 1 96 (ϕ + (z) -ϕ -(z)), where ϕ + (z) and ϕ -(z) are normalized Hecke eigenforms of weight 5, level 2 4 3 2 and character χ defined as

χ(n) = (-1) n-1 2 if gcd(n, 2 4 3 2 ) = 1, 0 otherwise. If we write ϕ ± (z) = ∞ n=0 b ± (n)q n ,
then for any prime number p, operating the Hecke operator T p on η 10 (12z), we get

η 10 (12z) | T p = 1 96 (ϕ + (z) | T p -ϕ -(z) | T p ) = 1 96 (b + (p) • ϕ + (z) -b -(p) • ϕ -(z)) . (16) Since η 10 (12z) | T p = ∞ n=1 a(pn) + χ(p) • p 5-1 • a n p q n = ∞ n=1 a(pn) + (-1) p-1 2 p 4 • a n p q n ,
therefore, by comparing coefficients on both sides of ( 16), we get a(pn) + (-1)

p-1 2 p 4 • a n p = 1 96 (b + (p) • b + (n) -b -(p) • b -(n)) ∀ n ≥ 1.
Moreover, Serre [START_REF] Serre | Sur la lacunarité des puissances de η[END_REF] further established that b ± (n) = 0, if n ≡ 1 (mod 4). Let p = 3 be a prime number such that p ≡ 3 (mod 4); then we have

a(pn) -p 4 • a n p = 0. ( 17
)
Substituting n by pn + r with p ∤ r in the above equation, we get a(p 2 n + pr) = 0. Now, substituting n by 12n + 5 -pr, we get

a(12p 2 n + 5p 2 -p 3 r + pr) = 0. But, 12p 2 n + 5p 2 -p 3 r + pr = 12 p 2 n + 5(p 2 -1)

12

+ pr(1-p 2 ) 12 + 5. Thus, by [START_REF] Kathiravan | Ramanujan-type congruences modulo m for (l, m)-regular bipartitions[END_REF], we get

B (15) 3 3 p 2 n + 5(p 2 -1) 12 + pr(1 -p 2 ) 12 ≡ 0 (mod 3).
As gcd 1-p 2 12 , p = 1, so we have p ∤ r if and only if p ∤ j, where j := r(1-p 2 )

12

. Hence, for j ≡ 0 (mod p), we get

B (15) 3 3p 2 n + 5(p 2 -1) 4 + 3pj ≡ 0 (mod 3). ( 18
)
Substituting n by pn in [START_REF] Lin | Arithmetic of the 7-regular bipartition function modulo 3[END_REF], we get

a(p 2 n) = p 4 • a(n) ≡ a(n) (mod 3).
Now, substituting n by 12n + 5, we get a(12p 2 n + 5p 2 ) ≡ a(12n + 5) (mod 3).

But, 12p 2 n + 5p 2 = 12 p 2 n + 5(p 2 -1) 12 + 5. Thus, by [START_REF] Kathiravan | Ramanujan-type congruences modulo m for (l, m)-regular bipartitions[END_REF], we get

B (15) 3 3 p 2 n + 5(p 2 -1) 12 ≡ B (15) 3 (3n) (mod 3). ( 19
)
Let t ≥ 0 and p i = 3 be a prime number such that p i ≡ 3 (mod 4) for i ∈ {1, • • • , t + 1}. Since

p 2 1 • • • p 2 t n + 5(p 2 1 • • • p 2 t -1) 12 = p 2 1 p 2 2 • • • p 2 t n + 5(p 2 2 • • • p 2 t -1) 12 + 5(p 2 1 -1) 12 ,
therefore by repeated use of ( 19), we get

B (15) 3 3p 2 1 • • • p 2 t n + 5(p 2 1 • • • p 2 t -1) 4 ≡ B (15) 3
(3n) (mod 3).

Substituting n by p 2 t+1 n +

5(p 2 t+1 -1) 12
+ p t+1 j with p t+1 ∤ j, we get the desired result in view of [START_REF] Lin | An infinite family of congruences modulo 3 for 13-regular bipartitions[END_REF]. In view of ( 9), we get

∞ n=0 B (10) 5
(5n)q 3n+1 ≡ η 8 (3z) (mod 5).

Rest of the proof follows similarly as Part (i). Part (iv): Let us take ℓ = 7 and r = 7 in (3) to get

∞ n=0 B (7) 7 (n)q n = E 7 7 E 7 1 .
Now, by (2), we get

∞ n=0 B (7) 7 (n)q n ≡ E 7 7 E 7 ≡ E 6 7 (mod 7).
In view of (9), we get

∞ n=0 B (7)
7 (7n)q 4n+1 ≡ η 6 (4z) (mod 7).

Writing

η 6 (4z) = ∞ n=0 a(n)q n , we get B ( 7
)
7 (7n) ≡ a(4n + 1) (mod 7). [START_REF] Newman | The coefficients of certain infinite products[END_REF] Moreover, one can verify that a(n) = 0, if n ≡ 1 (mod 4).

Serre [START_REF] Serre | Sur la lacunarité des puissances de η[END_REF] showed that η 6 (4z) is a normalized Hecke eigenform of weight 3, level 16 and character χ defined as

χ(n) = (-1) n-1 2 if n is odd, 0 if n is even.
Therefore, for any prime number p, operating the Hecke operator T p on η 6 (4z), we get

η 6 (4z) | T p = a(p) • η 6 (4z). ( 21
) Since η 6 (4z) | T p = ∞ n=1 a(pn) + χ(p) • p 3-1 • a n p q n = ∞ n=1 a(pn) + (-1) p-1 2 p 2 • a n p q n ,
therefore, by comparing coefficients on both sides of ( 21), we get

a(pn) + (-1) p-1 2 p 2 • a n p = a(p) • a(n) ∀ n ≥ 1.
Let p = 7 be a prime number such that p ≡ 3 (mod 4); then we have

a(pn) -p 2 • a n p = 0. ( 22
)
Substituting n by pn + r with p ∤ r in the above equation, we get a(p 2 n + pr) = 0. Now, substituting n by 4n + 1 -pr, we get

a(4p 2 n + p 2 -p 3 r + pr) = 0. But, 4p 2 n + p 2 -p 3 r + pr = 4 p 2 n + p 2 -1 4 + pr(1-p 2 )
4 + 1. Thus, by [START_REF] Newman | The coefficients of certain infinite products[END_REF], we get

B (7) 7 7 p 2 n + p 2 -1 4 + pr(1 -p 2 ) 4 ≡ 0 (mod 7).
As gcd 1-p 2 4 , p = 1, so we have p ∤ r if and only if p ∤ j, where j := r(1-p 2 )

4

. Hence, for j ≡ 0 (mod p), we get

B (7) 7 7p 2 n + 7(p 2 -1) 4 + 7pj ≡ 0 (mod 7). ( 23
)
Substituting n by pn in [START_REF] Ranganatha | Ramanujan-type congruences modulo powers of 5 and 7[END_REF], we get But, 4p 2 n + p 2 = 4 p 2 n + p 2 -1 4 + 1. Thus, by [START_REF] Newman | The coefficients of certain infinite products[END_REF], we get

a(p 2 n) = p 2 • a(n).
B (7) 7 7 p 2 n + p 2 -1 4 ≡ p 2 • B (7)
7 (7n) (mod 7). [START_REF] Serre | Sur la lacunarité des puissances de η[END_REF] Let t ≥ 0 and p i = 7 be a prime number such that p i ≡ 3 (mod 4) for i ∈ {1, • • • , t + 1}. Since

p 2 1 • • • p 2 t n + p 2 1 • • • p 2 t -1 4 = p 2 1 p 2 2 • • • p 2 t n + p 2 2 • • • p 2 t -1 4 + p 2 1 -1 4 ,
therefore by repeated use of ( 24), we get

B (7) 7 7p 2 1 • • • p 2 t n + 7(p 2 1 • • • p 2 t -1) 4 ≡ (p 2 1 • • • p 2 t+1 ) • B (7)
7 (7n) (mod 7).

Substituting n by p 2 t+1 n +

p 2 t+1 -1 4
+ p t+1 j with p t+1 ∤ j, we get the desired result in view of [START_REF] Saikia | Congruences of ℓ-regular partition triples for ℓ ∈ {2, 3, 4, 5}[END_REF]. In view of ( 9), we get

∞ n=0 B (11)
11 (11n)q 12n+5 ≡ η 10 (12z) (mod 11).

Rest of the proof follows similarly as Part (ii).

Proof of Theorem 1.3. Part (i): Let us take ℓ = 5 and r = 6 in (3) to get

∞ n=0 B (6) 5 (n)q n = E 6 5 E 6 1 .
Now, by (2), we get

∞ n=0 B (6) 5 (n)q n ≡ E 30 1 E 6 1 ≡ E 24 1 (mod 5).
In view of (4), we get

B (6) 5 (n) ≡ a 24 (n) (mod 5) ∀ n ≥ 0. ( 25
)
Let us apply Lemma 2.1 taking r = 24 to get that for a prime p,

a 24 (pn + (p -1)) = a 24 (p -1) • a 24 (n) -p 11 • a 24 n -(p -1) p .
Substituting n by pn + (p -1) repeatedly, we get

a 24 p 2 n + (p 2 -1) = a 24 (p -1) • a 24 (pn + (p -1)) -p 11 • a 24 (n) , (26)
a 24 p 3 n + (p 3 -1) = a 24 (p -1) • a 24 p 2 n + (p 2 -1) -p 11 • a 24 (pn + (p -1)) , (27)
a 24 p 4 n + (p 4 -1) = a 24 (p -1) • a 24 p 3 n + (p 3 -1) -p 11 • a 24 p 2 n + (p 2 -1) . (28)
In [START_REF] Xia | Congruences for some l-regular partitions modulo l[END_REF], replacing a 24 p 3 n + (p 3 -1) and a 24 p 2 n + (p 2 -1) according to ( 26) and ( 27), respectively, we get after simplifying Let us assume that p = 5 and that B .

The above equation can be rewritten as

∞ n=0 B (6) 7 (n)q n = E 6 7 E 7 1 • E 1 .
Now, by [START_REF] Cui | Arithmetic properties of ℓ-regular partitions[END_REF], we get

∞ n=0 B (6) 7 (n)q n = E 6 7 E 7
1

• E 49 B q 7 C(q 7 ) -q A q 7 B(q 7 ) -q 2 + q 5 C q 7 A(q 7 ) .

In view of (2), we get

∞ n=0 B (6) 7 (n)q n ≡ E 12 7 B q 7
C(q 7 ) -q A q 7 B(q 7 ) -q 2 + q 5 C q 7 A(q 7 ) (mod 7).

Collecting the terms involving q 7n+2 , we get

∞ n=0 B (6)
7 (7n + 2)q 7n+2 ≡ 6q 2 E 12 7 (mod 7). Now, dividing both sides by q 2 and substituting q by q 1/7 , we get

∞ n=0 B (6)
7 (7n + 2)q n ≡ 6E 12 1 (mod 7).

In view of ( 4), we get

B (6) 7 (7n + 2) ≡ 6a 12 (n) (mod 7) ∀ n ≥ 0. ( 29
)
Let us apply Lemma 2.1 taking r = 12 to get that for an odd prime p,

a 12 pn + p -1 2 = a 12 p -1 2 • a 12 (n) -p 5 • a 12 n -p-1 2 p .
Substituting n by pn + p-1 2 repeatedly, we get

a 12 p 2 n + p 2 -1 2 = a 12 p -1 2 • a 12 pn + p -1 2 -p 5 • a 12 (n) , (30)
a 12 p 3 n + p 3 -1 2 = a 12 p -1 2 • a 12 p 2 n + p 2 -1 2 -p 5 • a 12 pn + p -1 2 , ( 31
)
a 12 p 4 n + p 4 -1 2 = a 12 p -1 2 • a 12 p 3 n + p 3 -1 2 -p 5 • a 12 p 2 n + p 2 -1 2 . ( 32
)
In (32), replacing a 12 p 3 n + p 3 -1 2 and a 12 p 2 n + p 2 -1 2 according to (30) and (31), respectively, we get after simplifying

a 12 p 4 n + p 4 -1 2 = a 12 p -1 2 • a 12 p -1 2 2 -2p 5 • a 12 pn + p -1 2 -p 5 • a 12 p -1 2 2 -p 5 • a 12 (n).
Now, by (29), we get 6B (6) 7

7p 4 n + 7p 4 -3 2 ≡ 6B (6) 7 7p -3 2 • 6B (6) 7 7p -3 2 2 -2p 5 • 6B (6) 7 7pn + 7p -3 2 -p 5 • 6B (6) 7 7p -3 2 2 -p 5 • 6B (6)
7 (7n + 2) (mod 7).

Let us assume that p = 7 and that B (6) 7 7p-3) 2

≡ 0 (mod 7). Then we get

B (6) 7 7p 4 n + 7p 4 -3 2 ≡ p 4 B (6)
7 (7n + 2) (mod 7).

We get the desired result by inductively substituting n by p 4 n + p 4 -1 2 in the above congruence.

Congruences Modulo a Composite Number

In this section, we will prove Theorem 1.4, Theorem 1.5, Theorem 1.6, and Theorem 1.7.

Proof of Theorem 1.4. Part (i): Let us take ℓ = 3 and r = 12 in (3) to get

∞ n=0 B (12) 3 (n)q n = E 12 3 E 12 1
. Now, by (2), we get

∞ n=0 B (12) 3 (n)q n ≡ E 6 6 E 6 2
(mod 2).

Comparing coefficients of q 2n+1 on both sides, we get

B (12) 3 (2n + 1) ≡ 0 (mod 2) ∀ n ≥ 0. ( 33
)
On the other hand, let p be a prime number such that p ≡ 2 (mod 3) and j be an even number such that p ∤ j. By Corollary 1.2 (i), we get that for any n ≥ 0 and t ≥ 0,

B (12) 3 3p 2t+2 n + p 2t+1 (p + 3j) -1 ≡ 0 (mod 3).
Substituting n by 2n + 1, we get

B (12) 3 6p 2t+2 n + p 2t+1 (4p + 3j) -1 ≡ 0 (mod 3). Note that 6p 2t+2 n + p 2t+1 (4p + 3j) -1 = 2 3p 2t+2 n + p 2t+1 (4p+3j)-2 2 + 1.
Thus, in view of (33), we get

B (12) 3 6p 2t+2 n + p 2t+1 (4p + 3j) -1 ≡ 0 (mod 2),
and hence, the desired result.

Part (ii): Let us take ℓ = 5 and r = 10 in (3) to get Comparing coefficients of q 2n+1 on both sides, we get B

∞ n=0 B (10) 5 (n)
(2n + 1) ≡ 0 (mod 2) ∀ n ≥ 0. (34) On the other hand, let p = 5 be a prime number such that p ≡ 2 (mod 3) and j be an even number such that p ∤ j. By Corollary 1.2 (iii), we get that for any n ≥ 0 and t ≥ 0,

B (10) 5 5p 2t+2 n + 5 p 2t+1 (p + 3j) -1 3 ≡ 0 (mod 5).
Substituting n by 2n + 1, we get

B (10) 5 10p 2t+2 n + 5 p 2t+1 (4p + 3j) -1 3 ≡ 0 (mod 5).
Note that 10p 2t+2 n + 5 p 2t+1 (4p+3j)-1 3

= 2 5p 2t+2 n + 5p 2t+1 (4p+3j)-8 6 + 1. Thus, in view of (34), we get

B (10) 5 10p 2t+2 n + 5 p 2t+1 (4p + 3j) -1 3 ≡ 0 (mod 2),
and hence, the desired result. Part (iii): Let us take ℓ = 3 and r = 15 in (3) to get

∞ n=0 B (15) 3 (n
)q n = E 15 3 E 15 1
. Now, by (2), we get

∞ n=0 B (15) 3 (n)q n ≡ E 3 15 E 3 5 (mod 5).
Comparing coefficients of q 5n+α on both sides, where α ∈ {1, 2, 3, 4}, we get

B (15) 3 (5n + α) ≡ 0 (mod 5) ∀ n ≥ 0. (35)
On the other hand, let p = 3 be a prime number such that p ≡ 2 (mod 3) and j be a positive integer such that 5 | j but p ∤ j. By Corollary 1.2 (ii), we get that for n ≥ 0 and t ≥ 0,

B (15) 3 3p 2t+2 n + p 2t+1 (5p + 12j) -5 4 ≡ 0 (mod 3).
Substituting n by 5n + α, we get

B (15) 3 15p 2t+2 n + p 2t+1 ((12α + 5)p + 12j) -5 4 ≡ 0 (mod 3). Note that 15p 2t+2 n + p 2t+1 ((12α+5)p+12j)-5 4 = 5 3p 2t+2 n + 3p 2t+2 α-α ′ +3p 2t+1 j 5 + p 2t+2 -1 4 + α ′
, where α ′ ∈ {1, 2, 3, 4} is such that 3p 2t+2 α ≡ α ′ (mod 5). Thus, in view of (35), we get (n

)q n = E 20t+1 5 E 20t+1 1 . (36)
The above equation can be rewritten as

∞ n=0 B (20t+1) 5 (n)q n = E 20t 5 E 20t 1 • E 5 E 1 .
Now, by [START_REF] Carlson | Infinite families of infinite families of congruences for k-regular partitions[END_REF], we get

∞ n=0 B (20t+1) 5 (n)q n = E 20t 5 E 20t 1 E 8 E 2 20 E 2 2 E 40 + q E 3 4 E 10 E 40 E 3 2 E 8 E 20 .
In view of (2), we get

∞ n=0 B (20t+1) 5 (n)q n ≡ E 5t 20 E 5t 4 E 8 E 2 20 E 4 E 40 + q E 3 4 E 10 E 40 E 3 2 E 8 E 20
(mod 2).

Collecting the terms involving q 2n , we get

∞ n=0 B (20t+1) 5 (2n)q 2n ≡ E 5t 20 E 5t 4 • E 8 E 2 20 E 4 E 40 (mod 2).
Now, substituting q by q 1/2 , we get

∞ n=0 B (20t+1) 5 (2n)q n ≡ E 5t 10 E 5t 2 • E 4 E 2 10 E 2 E 20 (mod 2).
Comparing coefficients of q 2n+1 on both sides, we get

B (20t+1) 5 (4n + 2) ≡ 0 (mod 2) ∀ n ≥ 0. ( 37
)
On the other hand, (36) can be also rewritten as

∞ n=0 B (20t+1) 5 (n)q n = E 20t+1 5 E 20t+5 1 • E 4 1 .
Now, by [START_REF] Chern | Arithmetic properties for 7-regular partition triples[END_REF], we get

∞ n=0 B (20t+1) 5 (n)q n = E 20t+1 5 E 20t+5 1 • E 4 25 1 R(q 5 ) -q -q 2 R q 5 4 .
In view of (2), we get

∞ n=0 B (20t+1) 5 (n)q n ≡ E 20t+1 5 E 4 25 E 4t+1 5 1 R (q 5 ) 4 + q R (q 5 ) 3 + 2q 2
R (q 5 ) 2 + 3q 3 R (q 5 ) + 2q 5 R q 5 + 2q 6 R q 5 2 + 4q 7 R q 5 3 + q 8 R q 5 4 (mod 5).

Comparing coefficients of q 5n+4 on both sides, we get (n

)q n = E 20t+9 5 E 20t+9 1 .
The above equation can be rewritten as

∞ n=0 B (20t+9) 5 (n)q n = E 20t+8 5 E 20t+8 1 • E 5 E 1
as well as

∞ n=0 B (20t+9) 5 (n)q n = E 20t+9 5 E 20t+10 1 • E 1 .
Following similar arguments as in Part (i), we get that for any n ≥ 0, B

(4n + 2) ≡ 0 (mod 2), B

(5n + α ′ ) ≡ 0 (mod 5), where α ′ ∈ {3, 4}. Combining these, we get the desired result. Part (iii): Let us take ℓ = 5 and r = 20t + 17, for t ≥ 0, in (3) to get

∞ n=0 B (20t+17) 5 (n)q n = E 20t+17 5 E 20t+17 1 .
The above equation can be rewritten as

∞ n=0 B (20t+17) 5 (n)q n = E 20t+16 5 E 20t+16 1 • E 5 E 1 as well as ∞ n=0 B (20t+17) 5 (n)q n = E 20t+17 5 E 20t+20 1 • E 3 1 .
Following similar arguments as in Part (i), we get that for any n ≥ 0, B

(4n + 2) ≡ 0 (mod 2), B

(5n + α ′ ) ≡ 0 (mod 5), where α ′ ∈ {2, 3, 4}. Combining these, we get the desired result.

Proof of Theorem 1.6. Part (i): Let us take ℓ = 35 and r = 35t + 4, for t ≥ 0, in (3) to get

∞ n=0 B (35t+4) 35 (n)q n = E 35t+4 35 E 35t+4 1 . (39)
The above equation can be rewritten as

∞ n=0 B (35t+4) 35 (n)q n = E 35t+4 35 E 35t+5 1 • E 1 . Now, by (6), we get ∞ n=0 B (35t+4) 35 (n)q n = E 35t+4 35 E 35t+5 1 • E 25 1 R(q 5 ) -q -q 2 R q 5 .
In view of (2), we get

∞ n=0 B (35t+4) 35 (n)q n ≡ E 35t+4 35 E 25 E 7t+1 5 1 R(q 5 )
-q -q 2 R q 5 (mod 5).

Comparing coefficients of q 5n+α1 on both sides, where α 1 ∈ {3, 4}, we get

B (35t+4) 35 (5n + α 1 ) ≡ 0 (mod 5) ∀ n ≥ 0. ( 40
)
On the other hand, (39) can be also rewritten as

∞ n=0 B (35t+4) 35 (n)q n = E 35t+4 35 E 35t+7 1 • E 3 1 .
Now, by [START_REF] Cui | Arithmetic properties of ℓ-regular partitions[END_REF], we get

∞ n=0 B (35t+4) 35 (n)q n = E 35t+4 35 E 35t+7 1 • E 3 49 A 2 (q 7 ) A 1 (q 7 ) -q A 3 (q 7 ) A 2 (q 7 ) -q 2 + q 5 A 1 (q 7 ) A 3 (q 7 ) 3 .
In view of (2), we get

∞ n=0 B (35t+4) 35 (n)q n ≡ E 35t+4 35 E 3 49 E 5t+1 7 A 3 2 A 3 1 + 4q A 2 A 3 A 2 1 + 4q 2 A 2 2 A 2 1 + 3q 2 A 2 3 A 1 A 2 + 6q 3 A 3 A 1 + 6q 3 A 3 3 A 3 2 + 3q 4 A 2 A 1 + 4q 4 A 2 3 A 2 2 + 3q 5 A 2 2 A 1 A 3 + 4q 5 A 3 A 2 + q 7 A 2 A 3 + 3q 7 A 1 A 3 A 2 2 + 6q 8 A 1 A 2 + 3q 9 A 1 A 3 + 3q 10 A 1 A 2 A 2 3 + 4q 11 A 2 1 A 2 A 3 + 4q 12 A 2 1 A 2 3 + q 15 A 3 1 A 3 3 (mod 7),
where A i := A i (q 7 ) for i ∈ {1, 2, 3}. Comparing coefficients of q 7n+6 on both sides, we get

11 11p

 11 2t+2 n + 11 p 2t+1 (5p+12j)-5 12

  Part (iii): Let us take ℓ = 5 and r = 10 in (3) to get

  Now, substituting n by 4n + 1, we get a(4p 2 n + p 2 ) = p 2 • a(4n + 1).

  Part (v): Let us take ℓ = 11 and r = 11 in (3) to get

a 24 p

 24 4 n + (p 4 -1) = a 24 (p -1) • a 24 (p -1) 2 -2p 11 • a 24 (pn + (p -1)) -p 11 • a 24 (p -1) 2 -p 11 • a 24 (n). Now, by (25), we get B

p 4 n

 4 + (p 4 -1) ≡ p 2 B (6) 5 (n) (mod 5). We get the desired result by inductively substituting n by p 4 n + (p 4 -1) in the above congruence. Part (ii): Let us take ℓ = 7 and r = 6 in (3) to get

B ( 15 ) 3 15p

 153 2t+2 n + p 2t+1 ((12α + 5)p + 12j) -5 4 ≡ 0 (mod 5), and hence, the desired result. Proof of Theorem 1.5. Part (i): Let us take ℓ = 5 and r = 20t + 1, for t ≥ 0, in (3) to get

B (20t+1) 5 (5n + 4 )

 54 ≡ 0 (mod 5) ∀ n ≥ 0. (38) Combining (37) and (38), we get the desired result. Part (ii): Let us take ℓ = 5 and r = 20t + 9, for t ≥ 0, in (3) to get

B (35t+4) 35 (7n + 6 ) 1 • E 1 . 1 . 6 ) 1 • E 1 .α 2 ) 1 • E 3 1α 1 ∈ 1 .B 1 • E 1

 3561116112131111 ≡ 0 (mod 7) ∀ n ≥ 0. (41) Combining (40) and (41), we get the desired result. Part (ii): Let us take ℓ = 35 and r = 35t + 6, for t ≥ 0, in (3) to get Following similar arguments as in Part (i), we get that for any n ≥ 0, B (35t+6) 35 (5n + 4) ≡ 0 (mod 5), B (35t+6) 35 (7n + α 2 ) ≡ 0 (mod 7), where α 2 ∈ {3, 4, 6}. Combining these, we get the desired result. Part (iii): Let us take ℓ = 35 and r = 35t + 11, for t ≥ 0, in (3) to get Following similar arguments as in Part (i), we get that for any n ≥ 0, ≡ 0 (mod 7). Combining these, we get the desired result. Part (iv): Let us take ℓ = 35 and r = 35t + 27, for t ≥ 0, in (3) to get Following similar arguments as in Part (i), we get that for any n ≥ 0, ≡ 0 (mod 7), where α 1 ∈ {2, 3, 4} and α 2 ∈ {3, 4, 6}. Combining these, we get the desired result. Part (v): Let us take ℓ = 35 and r = 35t + 32, for t ≥ 0, in (3) to get Following similar arguments as in Part (i), we get that for any n ≥ 0, {2, 3, 4}. Combining these, we get the desired result. Part (vi): Let us take ℓ = 35 and r = 35t + 34, for t ≥ 0, in (3) to get The above equation can be rewritten as ∞ n=0

  • • • , t + 1} and p t+1 ∤ j.

	(v) B	(11) 11	11p 2 1 • • • p 2 t+1 n +	11 p 2 1 •••p 2 t pt+1(5pt+1+12j)-5 12
		(12)
		3

≡ 0 (mod 11), where p i ∈ {3, 11} is a prime number such that p i ≡ 3 (mod 4) for i ∈ {1, • • • , t + 1} and p t+1 ∤ j.

Corollary 1.2. For any n ≥ 0 and t ≥ 0, we have (i) B

Following similar arguments as in Part (i), we get that for any n ≥ 0, B (35t+34) 35 (5n + α 1 ) ≡ 0 (mod 5),

B (35t+34) 35

(7n + α 2 ) ≡ 0 (mod 7), where α 1 ∈ {3, 4} and α 2 ∈ {3, 4, 6}. Combining these, we get the desired result.

Proof of Theorem 1.7. Part (i): Let us take ℓ = 55 and r = 55t + 21, for t ≥ 0, in (3) to get

The above equation can be rewritten as

Now, by [START_REF] Chern | Arithmetic properties for 7-regular partition triples[END_REF], we get

In view of (2), we get

+ 2q 6 R q 5 2 + 4q 7 R q 5 3 + q 8 R q 5 4 (mod 5).

Comparing coefficients of q 5n+4 on both sides, we get

On the other hand, (42) can be also rewritten as

Now, by [START_REF] Dai | On the distribution of odd values of 2 a -regular partition functions[END_REF], we get

) -q 2 B 5 (q 11 ) B 3 (q 11 ) + q 5 + q 7 B 3 (q 11 ) B 4 (q 11 ) -q 15 B 1 (q 11 ) B 5 (q 11 ) .

In view of (2), we get

-q B 2 (q 11 ) B 1 (q 11 ) -q 2 B 5 (q 11 ) B 3 (q 11 ) + q 5 + q 7 B 3 (q 11 ) B 4 (q 11 ) -q 15 B 1 (q 11 ) B 5 (q 11 ) (mod 11).

Comparing coefficients of q 11n+α2 on both sides, where α 2 ∈ {3, 6, 8, 9, 10}, we get

Combining ( 43) and (44), we get the desired result.

Part (ii): Let us take ℓ = 55 and r = 55t + 32, for t ≥ 0, in (3) to get

The above equation can be rewritten as

Following similar arguments as in Part (i), we get that for any n ≥ 0, B

The above equation can be rewritten as

Following similar arguments as in Part (i), we get that for any n ≥ 0, B

(5n + α 1 ) ≡ 0 (mod 5), B

(11n + α 2 ) ≡ 0 (mod 11),

where α 1 ∈ {3, 4} and α 2 ∈ {3, 6, 8, 9, 10}. Combining these, we get the desired result.


Andrea Quintero

Patrice Gergaud

Tra Nguyen-Thanh

Jean-Michel Hartmann

Vincent Reboud

Eric Cassan

Philippe Rodriguez
email: philippe.rodriguez@cea.fr

Texture of NiGe(Sn) on Ge(100) and its evolution with Sn content

The texture of the Ni monostanogermanide phase on a Ge(100) substrate was evaluated during a solid-state reaction, with a focus on the impact of Sn addition. Complementary X-ray diffraction analyses involving in situ X-ray diffraction, in-plane reciprocal space maps (RSMs) and pole figures were used to that end. A sequential growth of the phases for the Ni/Ge(Sn) system was found. An Ni-rich phase formed first, followed by the NiGe(Sn) phase. The NiGe and NiGe(Sn) layers were polycrystalline with different out-of-plane orientations. The number of out-of-plane diffraction peaks decreased with the Sn content, while the preferred orientation changed. In-plane RSM analyses confirmed these results. Sn addition modified the out-of-plane and in-plane orientations. Pole figure analysis revealed that numerous epitaxial texture components were present for the Ni/Ge system, while Sn addition reduced the number of epitaxial texture components. On the other hand, segregated Sn crystallized with an epitaxial alignment with the Ge substrate underneath.

Introduction

NiGeSn intermetallics are promising for use as contacts in GeSn-based electronic and optoelectronic devices [START_REF] Gupta | 2011 International Electron Devices Meeting (IEDM)[END_REF][START_REF] Yang | IEEE International Electron Devices Meeting (IEDM)[END_REF]Wirths, Geiger et al., 2015 ;Tseng et al., 2013 ;Oehme et al., 2014 ). In particular, GeSn layers are envisioned as high-mobility channels in p-type metal oxide semiconductor field effect transistors (pMOSFETs) [START_REF] Gupta | 2011 International Electron Devices Meeting (IEDM)[END_REF] and p-type tunnel field effect transistors [START_REF] Yang | IEEE International Electron Devices Meeting (IEDM)[END_REF], source and drain stressors in Ge MOSFETs (Wirths, Troitsch et al., 2015 ) and p-i-n photodetectors (Tseng et al., 2013 ), light emitting diodes (Oehme et al., 2014 ;[START_REF] Casiez | IEEE Photonics Conference (IPC)[END_REF], and optically pumped lasers (Wirths, Geiger et al., 2015 ;[START_REF] Chrétien | [END_REF]. The possibility of obtaining low resistive values when using nickel makes the contacts a key module to achieve high performance (Vincent et al., 2011 ;Nishimura et al., 2011 ;Demeulemeester et al., 2011 ;[START_REF] Han | 2011 International Electron Devices Meeting (IEDM)[END_REF]. Ni/Ge or Ni/GeSn contacts are formed by a solid-state reaction during which Ni1-xGex or Ni1-x(GeSn)x polycrystalline layers are obtained (Balakrisnan et al., 2005 ;Gaudet et al., 2006 ;De Schutter, Van Stiphout et al., 2016 ;van Stiphout et al., 2017 ;Nishimura et al., 2011 ;Wirths et al., 2014 ;Wirths, Geiger et al., 2015 ;Liu et al., 2015 ;Schulte-Braucks, Glass et al., 2017 ;Quintero et al., 2018c ). In order to understand the properties of these polycrystalline films, it is essential to have a good understanding of their microstructure. One important aspect is the texture, which describes the orientations of the various grains and the frequency of occurrence of different orientations within the film (De Schutter, Van Stiphout et al., 2016 ;Detavernier et al., 2003 ). Balakrisnan et al. (2005 ) studied the texture of NiGe films formed on Ge(001) substrates and their evolution with the annealing temperature. Detailed analysis of (111) pole figures revealed the presence of six types of epitaxial grains, in which the NiGe (111)[0 1] ∥ Ge (001) [110] preferred orientation increased with the annealing temperature. Gaudet et al. (2006 ) showed that Ni germanides present several epitaxial and axiotaxial texture components on Ge(001) and Ge(111) substrates. The texture on Ge(001) samples corresponded to a combination of five different epitaxial components and three very weak axiotaxial components, for which either the NiGe (121), ( 220) or (211) plane was aligned with a Ge (110) plane. Finally, studies of van Stiphout et al. (2017 ) on Ni germanide thin films showed mostly epitaxial components together with weaker axiotaxial lines of the NiGe (211) and NiGe (202) planes aligned with the Ge (220) planes. On the other hand, annealing the film at temperatures higher than 673 K caused the NiGe structure to relax. This was also reflected in its texture axiotaxial components, which became more intense after relaxation owing to growth of axiotaxial grains.

The literature concerning the Ni/GeSn system texture is rather scarce. Different studies have evaluated the crystalline nature of the Ni-rich phase either on GeSn or SiGeSn substrates. The results showed that the Nirich phase corresponded to the hexagonal ɛ-Ni5Ge3 metastable phase (De Schutter, Van Stiphout et al., 2016 ;Quintero et al., 2018c ). Meanwhile, in-depth studies concerning the texture of the monostanogermanide phase are not available. Wirths and co-workers showed that the Ni(SiGeSn) layer was not in an epitaxial relationship with the SiGeSn layer underneath (Wirths et al., 2014 ;Wirths, Troitsch et al., 2015 ). Liu et al. (2015 ) reported variations of intensity for some diffraction peaks between reference and carbonimplanted samples. Additionally, the formation of certain preferred orientations like NiGeSn (112) was favored in the presence of carbon. Schulte-Braucks, Hofmann et al. (2017 ) analyzed Niy(Ge1-xSnx)z layers fabricated on several GeSn alloys with various Sn contents ranging from 0 to 12.5 at.%. Their θ-2θ X-ray diffraction (XRD) structural analysis revealed the coexistence of several Niy(Ge1-xSnx)z phases, with different crystalline orientations, in which the Sn content modified the phases that were present.

We therefore decided to study the impact of Sn addition on the monostanogermanide phase texture. NiGe and NiGeSn films (with various at.% of Sn) formed by rapid thermal annealing (RTA) of sputter-deposited Ni on Ge(100) or GeSn(100) substrates were evaluated with a focus on the impact of Sn addition on texture evolution. The Ni/Ge system was used as a reference.

Experimental procedure

A pure Ge layer and GeSn layers with 6, 10 and 15 at.% of Sn were epitaxially grown at 1023, 623, 598 and 573 K, respectively, on Ge-buffered Si(100) substrates in a 200 mm reduced pressure chemical vapor deposition tool (Aubin et al., 2017 ). The thickness of the GeSn layers (60 nm) was low enough for them to be fully compressively strained on the Ge strain-relaxed buffers underneath. In order to remove particles and native oxides from the surface, samples were dipped for 1 min in diluted HF (1% in water) and then rinsed in deionized water and dried with an N2 gun. Then, 10 nm of Ni were deposited at room temperature, followed by 7 nm of TiN capping layer deposited at 373 K by physical vapor deposition (magnetron sputtering). Given that no in situ pre-clean was used, the air break between the surface preparation and the sample transfer under vacuum for metallization was reduced to the bare minimum (less than 10 min). No air break occurred between the Ni and TiN deposition: samples were maintained at pressures as low as 1.33 × 10 -5 Pa during the transfer between the two deposition chambers.

The NiGe1-xSnx phase formation sequence and crystalline evolution were monitored by in situ XRD using an Empyrean PANalytical X-ray diffractometer equipped with a copper (Cu Kα) source, a linear detector (PIXcel1D) and an HTK 1200 Anton Paar furnace under secondary vacuum (pressure was maintained at about 0.1 Pa to avoid atmospheric contamination). θ-2θ scans on a large range of diffraction angles (20-90°) were performed with a 3° offset in ω in order to attenuate the symmetric (400) substrate reflection. This in situ measurement, from 323 to 873 K, reduced the risk of overlooking transient but crucial stages during the solidstate reaction. After cooling, complementary ex situ XRD θ-2θ scans using a Cu source were performed with an X'Pert Pro PANalytical X-ray diffractometer.

Additionally, in-plane reciprocal space map (RSM) characterizations were carried out on these samples with a Rigaku SmartLab X-ray diffractometer. The in-plane RSM configuration was as follows: A copper source with in-line focus was used. The primary optics configuration comprised a mirror, a parallel slit collimator with an in-plane angular divergence of 0.5°, a mask of 10 mm and an incident slit of 0.5 mm. The secondary optics configuration comprised a parallel slit analyzer with an angular aperture of 0.5°; the secondary slits in front of the proportional detector were fully open (20 × 20 mm). An in-plane continuous scan of the detector was performed with steps of 0.1°, while the in-plane orientation of the sample, φ, was changed by steps of 0.5°. The RSM maps were acquired over an in-plane 2θ range of 20-85° and a φ range of 95°. After integration over the full φ range, a 2θ phase diagram was finally obtained. Phase analysis was performed with the HighScore Plus software from PANalytical, using the International Centre for Diffraction Data (ICDD) PDF-4 database.

Pole figures were also recorded using the SmartLab setup (Nagao & Kagami, 2011 ). They were measured using the in-plane pole figure configuration developed by Rigaku [START_REF] Inaba | [END_REF] with an angular aperture of 0.5° in front of the detector. Pole figures were analyzed using in-house software developed in the Python language and using a Jupyter notebook to provide a graphic user-friendly interface. This analysis was completed by simulations with the CaRIne 3.1 software (CaRIne Crystallography, Senlis, France). The lattice parameters of NiGe(Sn) were refined with the HighScore software and were used as input for the Python software. The lattice parameters used for NiGe(Sn) in this study were a = 5.53, b = 3.43 and c = 5.87 Å.

The Python software provides an option to simulate several pole figures simultaneously. The simulation was done by creating a number of crystalline grains of NiGe or NiGe(Sn) phases with different normal orientation with respect to the sample surface, which were eventually rotated in 3D space. The stereographic projections of their poles were then calculated and superposed on experimental data. Each grain would simultaneously show their corresponding poles in the experimental pole figures. Visual checking of the position matching was sufficient and no optimization of the grain orientation was necessary because of the polycrystalline nature of the grains. Their experimental pole position was therefore much larger than the calculated position. This means that diffraction features can appear diffuse on experimental pole figures, whereas they appeared as discrete diffraction spots in the Python software. The final calculated orientation of each grain was then considered as the average orientation. Crystallographic data of phases described in this study and corresponding PDF file references are given in Table 1 .

Table 1

Summary of the crystallographic data of phases described in the article

Phase

Results

Out-of-plane XRD analyses

The Ni/Ge phase sequence followed by in situ XRD is shown in Fig. 1 as a contour map with temperature on the y axis and diffraction angle on the x axis. A similar phase sequence occurred in samples with 6, 10 and 15 at.% of Sn (from a straightforward comparison of in situ XRD contour maps). This has been detailed in previous studies (Quintero et al., 2018c(Quintero et al., , 2020a ) ). There are some differences, however, concerning the threshold temperatures above which mono(stano)germanide phases were formed (see Table 2 ). This trend was due to the impact of Sn segregation on Ni diffusion (Quintero et al., 2020a ). In this context, the term `Sn segregation' refers to the long-range diffusion of Sn towards the surface.

Table 2

Critical As far as the Ni/Ge system is concerned, a broad diffraction peak due to TiN 200 was observed at 42.7°. In addition, diffraction peaks related to polycrystalline NiGe were observed at 30.2, 34.6, 34.9, 44.1, 45.5, 53.3, 53.9 and 54.3°, which were linked to the 011, 111, 102, 112, 211, 301, 020 and 013 out-of-plane diffraction lines, respectively. The out-of-plane preferred diffraction line, i.e. the out-of-plane peak which presented the highest intensity, was the peak at 34.6°, which was due to the 111 reflection.

As far as the Ni/Ge1-xSnx systems are concerned, additional diffraction peaks due to Sn segregation appeared at 30.7, 32.0, 43.6 and 44.9°. They were related to β-Sn diffraction and were due to the 200, 101, 220 and 211 diffraction lines. The intensity of the peaks related to Sn diffraction lines increased with Sn content. As the data were acquired under similar conditions, this intensity increase can be related to the quantity of Sn matter and should be more significant for samples with higher Sn content. The broad diffraction peak due to TiN 200 progressively dissipated with increasing Sn content. This phenomenon was due to Sn segregation. The TiN capping layer follows the shape of the layer underneath (Quintero et al., 2020a ). Consequently, when Sn diffused towards the surface, the TiN layer was impacted and the TiN planes were no longer parallel to the substrate surface. This led to greater mosaicity of the TiN layer. The direct effect was that the diffraction intensity was more scattered.

Diffraction peaks related to the NiGe(Sn) phase were observed similarly to the Ni/Ge system. A variation of the intensity and the number of out-of-plane peaks was, however, observed. For the Ni/Ge0.94Sn0.06 system, the same diffraction lines as for the Ni/Ge system were observed. The intensity of the 111 diffraction line strongly decreased compared with that of the Ni/Ge system, while an increase of the 112 and 013 diffraction peak intensities occurred. For the Ni/Ge0.90Sn0.10 and Ni/Ge0.85Sn0.15 systems, a further reduction of the intensity of the out-of-plane peaks was noticed concerning the monostanogermanide phase. The 013 diffraction peak intensity still increased and became the diffraction line with the highest intensity, while other diffraction lines nearly dissipated. For the sample with 15 at.% of Sn, the 301 diffraction line also exhibited a strong intensity.

θ-2θ ex situ XRD showed that the out-of-plane orientations for the NiGe(Sn) phase were strongly impacted by Sn content. A reduction of the number of diffraction lines occurred and a transition from a 111 preferred orientation towards a 013 orientation was apparent when increasing the Sn content.

In-plane XRD analyses

In order to verify that the impact on out-of-plane orientations observed with the Sn content increase was due to a texture variation, in-plane RSMs (IPRSMs) were obtained for all samples. Fig. 3 shows the corresponding integrated patterns in φ. For the Ni/Ge system, numerous peaks due to NiGe in-plane diffraction lines were observed. These are indexed and identified with gray squares in Fig. 3 . Stereographic projections were performed for each outof-plane diffraction line found in Section 3.1 . An example of the stereographic projection of NiGe (111) ∥ Ge ( 00) is given in Fig. 4 . From this example, one can see that the (111) out-of plane orientation could lead to (112), ( 211) and ( 013) orientations in in-plane mode. Thanks to these simulations, for each out-ofplane diffraction line, potential corresponding in-plane orientations were found. They are listed in Table 3 .

Table 3

Potential in-plane orientations for the NiGe phase induced by out-of-plane orientations observed in Fig. 2 Note that one out-of-plane orientation can induce a family of in-plane orientations.

Out-of-plane orientations

Potential in-plane orientations (011) ( 200), ( 013), ( 401), (412) (111) ( 112), ( 013), (211) (102) ( 211), (020) (112) ( 111), ( 321), (122) (211) ( 102), ( 111), ( 211), ( 013), (123) (301) (020) (020) ( 104), ( 103), ( 102), ( 203), ( 101), ( 201), ( 301), ( 401), (100) (013) ( 200), ( 211), ( 111), ( 122), ( 011)

Figure 4
Stereographic projection of NiGe (111) ∥ Ge ( 00) obtained with the CaRIne Crystallography software.

The mosaicity of out-of-plane orientations can induce some in-plane orientations even if the theoretical angle between the out-of-plane orientation and the in-plane orientation is not strictly equal to 90°. In this case, the theoretical angles between out-of-plane and in-plane orientations are close enough to 90° to allow the observation of corresponding in-plane orientations during in-plane RSM analyses.

Apart from one or two exceptions, all in-plane diffraction lines indexed in Fig. 3 can be explained by the out-of-plane diffraction peaks shown in Fig. 2 . Note that the 2θ range for Fig. 2 was limited to 25-60°. Out-of-plane peaks that fall outside of this range could explain the in-plane diffraction lines that are not otherwise accounted for. In addition, out-of-plane orientations within the studied 2θ range with a structure factor equal to 0 can be present. On the other hand, a few potential in-plane diffraction lines were not visible in Fig. 3 . This could be because (i) the in-plane diffraction lines are expected to diffract beyond the 2θ range of Fig. 3 , (ii) the corresponding theoretical intensities are too low to be seen in Fig. 3 (which is the case for the 200 diffraction line) and (iii) the resolution of Fig. 3 is not high enough to distinguish two very close peaks [this is the case for the 102 diffraction line, which is not discernible owing to the presence of the intense 111 diffraction line, and for the 122 and 104 diffraction lines, which are not visible owing to the intense peak of Si/Ge(Sn)].

As for the out-of-plane configuration, the in-plane orientations varied with the Sn content. A strong reduction of the number of peaks related to the monostanogermanide phase occurred with increasing Sn at.%. Sn diffraction peaks were additionally observed for the Ni/GeSn samples, in which the peak intensities increased with Sn content as observed for the out-of-plane θ-2θ scans in Fig. 2 . In-plane diffraction peaks due to the presence of Sn are indexed and identified with blue stars in Fig. 3 . For NiGe, stereographic projections were performed for each Sn out-of-plane diffraction line shown in Fig. 2 to assess the corresponding inplane orientations.

In order to confirm the differences between the Ni/Ge and Ni/GeSn systems, full in-plane RSMs were analyzed, comparing Ni/Ge and Ni/Ge0.90Sn0.10 samples annealed in situ up to 873 K. Analyses were performed after the samples had completely cooled to room temperature. Fig. 5 shows the full IPRSM scan obtained for the Ni/Ge system. As for Fig. 3 , diffraction spots or lines corresponding to Si/Ge were indexed using triangle symbols, circle symbols were used for TiN and square symbols for the NiGe phase.

Figure 5

Full in-plane RSM scan for the Ni/Ge system annealed up to 873 K. The pattern was acquired once the sample had been cooled to room temperature.

The in-plane diffraction spots showing a high intensity (i) at 2θ = 42.5° and 2θ = 45.3° and at φ = 0 and 90° were due to TiN 200 and Ge 220, respectively, and (ii) at 2θ = 62.1° and 2θ = 66.0° around φ = 45° were due to TiN 220 and Ge 400, respectively. The TiN and Ge spots were well aligned, suggesting an epitaxial relationship between these phases.

Most of the features associated with the NiGe phase appeared as lines across the full range of φ values, which could be due to a fiber texture. Diffraction spots with a strong intensity at 2θ = 53.3, 54.2 and 65.7° and φ = 0 and 90° were due to NiGe 020, 013 and 104 diffraction lines. They show an epitaxial in-plane alignment with Ge 220. Additionally, the double spots at 2θ = 34.7° and 2θ = 72.9°, related to the NiGe 111 and 222 diffraction lines and separated by about 10° in φ, were observed to be slightly misaligned with Ge 400. Finally, the diffraction spots at 2θ = 45.5° and related to NiGe 211 were separated by about 20° in φ and were misaligned with Ge 400. All diffraction features shown in Fig. 5 were indexed, and a list is proposed in Table 4 .

Table 4

Indexing of diffraction features shown in All peaks indexed in Fig. 3 for the Ni/Ge system are observed and indexed in Fig. 5 . Compared with Fig. 3 , a few additional diffraction features are evident in Fig. 5 . This is because the full IPRSM exhibited a better resolution and a clear distinction between all diffraction features (i.e. lines, spots).

Fig. 6 shows the full IPRSM scan obtained for the Ni/Ge0.90Sn0.10 system. As for Fig. 3 , diffraction spots or lines corresponding to Si/Ge were indexed using triangle symbols, circle symbols were used for TiN, square symbols for the NiGe(Sn) phase and star symbols for Sn.

Figure 6

Full in-plane RSM scan for the Ni/Ge0.90Sn0.10 system annealed up to 873 K. The pattern was acquired once the sample had been cooled to room temperature.

A first observation is that the IPRSM obtained for the Ni/Ge0.90Sn0.10 system is even richer than that obtained for Ni/Ge. Numerous well defined spots were observed while the fiber texture tended to disappear. As for the Ni/Ge system, in-plane diffraction spots showing a high intensity (i) at 2θ = 42.5° and 2θ = 45.3° and at φ = 0 and 90° were due to TiN 200 and Ge(Sn) 220, respectively, and (ii) at 2θ = 62.1° and 2θ = 66.0° around φ = 45° were due to TiN 220 and Ge(Sn) 400, respectively. The TiN and Ge(Sn) spots were well aligned, suggesting an epitaxial relationship between these phases.

For the NiGe(Sn) phase, the main diffraction features were indexed as follows. The diffraction spots with strong intensities at 2θ = 34.7 and 72.9° related to NiGe(Sn) 111 and 222 diffraction lines, respectively, were separated by about 30-40° in φ and were misaligned with Ge(Sn) 400. The diffraction spots with a strong intensity at 2θ = 45.5° and related to NiGe(Sn) 211 were separated by about 10° in φ and were misaligned with Ge(Sn) 400. With a weaker intensity, the diffraction spots at 2θ = 30.2, 53.3, 53.7 and 68.9° and due to the NiGe(Sn) 011, 020, 013 and 400 diffraction lines, respectively, showed an epitaxial in-plane alignment with Ge(Sn) 220. The NiGe(Sn) 200 diffraction line is too weak to be seen in Fig. 6 , its theoretical intensity being very low. Additional diffraction features corresponding to the NiGe(Sn) phase were indexed in Fig. 6 . A full list is proposed in Table 5 .

Table 5

Indexing of diffraction features shown in Numerous additional diffraction features corresponding to the Sn phase were indexed in Fig. 6 . A full list is proposed in Table 5 .

Pole figure analyses

In order to fully understand the Ni germanide texture and the impact of Sn addition, pole figures (PFs) were acquired for the Ni/Ge and Ni/Ge0.90Sn0.10 systems for the Bragg peaks at 2θ = 30.6, 34.9, 45.5 and 54.1°. Samples were in situ annealed up to 873 K and analyses were performed after cooling to room temperature. The results are shown in Fig. 7 . A straightforward comparison reveals that, for a given Bragg angle, the features of the PF change significantly from Ni/Ge to Ni/Ge(Sn).

Figure 7

Pole figures for Ni/Ge and Ni/Ge0.90Sn0.10 samples annealed up to 873 K. The pole figures were measured after the samples had been cooled to room temperature. The symbols used for each component can be found in Tables 6 , 7 and 8 .

From the out-of-plane (Fig. 2 ) and in-plane (Fig. 3 ) XRD analyses, it was observed that the NiGe phase was polycrystalline. The PF results in Fig. 7 show that the grains were not randomly oriented. Epitaxial alignments of NiGe grains are evident and defined spots are present on the pole figures with a fourfold symmetry characteristic of the cubic Ge(100) substrate, as observed in previous studies (Balakrisnan et al., 2005 ;Gaudet et al., 2006 ;De Schutter, Van Stiphout et al., 2016 ). The orientation of grains of the intermetallic film can then be directly determined from the orientation of the Ge substrate. In addition to spots corresponding to the NiGe phase, signal from the Ge substrate (identified by black open circles in Fig. 7 ) was also indicated in the pole figures at 2θ = 30.6, 45.5 and 54.1°. These spots correspond to the signals of the Ge (111), Ge (220) and Ge (311) planes, respectively.

The complex and rich patterns obtained suggest that several texture components are present in the film.

Epitaxial components lead to high-intensity spots in pole figures, while fiber texture results in circular lines centered around the middle of the pole figure. Epitaxial alignments were determined by using the in-house Python software mentioned in Section 2 and by simulating stereographic projections with the CaRIne Crystallography software.

For the NiGe phase, numerous epitaxial components were identified. They were displayed in Table 6 . The symbols in Table 6 were used to mark the locations of the diffraction spots for that component in Fig. 7 .

Table 6

Summary of the orientations observed for the NiGe phase

For each component, the symbol used to mark the locations of diffraction spots for that component in Fig. 4 , many of the diffraction features indexed in the IPRSM of the Ni/Ge system correspond to fiber-like texture. Weak circular lines centered around the middle of the pole figures can be distinguished at the periphery of the PFs recorded at 30.6, 34.9 and 45.5°. Their presence confirms that some of the in-plane orientations are fiber textured. Second, during the analysis of Fig. 5 , it was reported that some diffraction spots were misaligned with Ge 400. This was confirmed by the analysis of the stereographic projections. For instance, the stereographic projection of NiGe ( ) ∥ Ge (

  a = 5.45, b = 3.427, c = 5.885 Pnam (62) ɛ-Ni5Ge3 04-003-4346 Hexagonal a = b = 3.913, c = 5a = b = 5.831, c = 3.182 I41/amd (141)

Figure 1 θ

 1 Figure

  threshold temperatures for the Ni mono(stano)germanide phase formation System Ni(Ge1-xSnx) growth temperature (K) Ni/Ge 453 Ni/Ge0.94Sn0.06 508 Ni/Ge0.90Sn0.10 518 Ni/Ge0.85Sn0.15 503 A variation of out-of-plane orientations of the monostanogermanide phase with the Sn content was also seen. Fig. 2 shows patterns of samples studied by θ-2θ in situ XRD up to 873 K and analyzed by θ-2θ ex situ XRD after cooling to room temperature.

Figure 2 θ

 2 Figure 2

Figure 3

 3 Figure 3

  ) 210 Two diffraction spots misaligned by 10° with Ge(Sn) 400 42.5 TiN 200 Epitaxial relationship with Ge(Sn) 220 43.9Sn 220 Epitaxial relationship with Ge(Sn) 220 44spots due to the presence of the β-Sn phase were indexed in Fig.6. For a given in-plane orientation, various relationships with the Ge(Sn) substrate are evident. This implies the presence of numerous diffraction spots for a given 2θ value. The main diffraction spots were observed at 2θ = 30.5° and at φ = 0, 45 and 90°. This suggests that Sn 200 showed an epitaxial in-plane alignment with Ge(Sn) 220 and Ge(Sn) 400, corresponding to two different texture components (see Section 3.3 ). Among other diffraction features linked to Sn, we note the diffraction spots at 2θ = 32.0° and at φ = 0 and 90°. They are due to Sn 101 and indicate an epitaxial in-plane alignment with Ge(Sn) 220. The diffraction spots at 2θ = 43.9° and at ϕ = 0 and 90° were due to Sn 220 and also show an epitaxial in-plane alignment with Ge(Sn) 220.

  always possible to link an in-plane orientation with each out-of-plane orientation. This was due to two phenomena. First, as mentioned in Section 3.2 and Table

Acknowledgements

The authors thank Virginie Loup (CEA-Leti) for performing the HF surface treatment and Nicolas Bernier and David Cooper (CEA-Leti) for the TEM/EDS analyses.

Funding information

This work was supported by the French National Research Agency (ANR) under the `Investissements d'avenir' programs ANR 10-AIRT-0005 (IRT NANOELEC) and ANR 10-EQPX-0030 (EQUIPEX FDSOI 11) and by the CEA DSM-DRT Phare project `Photonics'.

) planes were slightly misaligned by about 5° with Ge ( 00). This kind of misalignment cannot be referenced as a full epitaxial relationship.

On the other hand, the preferential in-plane alignment of NiGe ( 010), ( 013) and ( 104) with Ge (011) evidenced by the PF analysis perfectly matches observations made during the analysis of the IPRSM of the Ni/Ge system in Section 3.2 . Most of these alignments were explained by the similar d spacings of the following planes: dNiGe(030) = 1.142 Å, dNiGe(026) = 0.851 Å and dNiGe(033) = 0.987 Å compared with dGe(440) = 1.000 Å. However, and especially for NiGe (104), the similar d spacings and a complete coherency of lattice parameters are not the only possible explanation for the preferential alignment that we observe. It could also be due to a minimization of interfacial energy favoring an oriented growth, as reported for instance during the analysis of elastic strains in Au/Ni multilayers (Labat et al., 2000 ). Finally, it could be explained by the fact that we did not identify the hkl planes that have a real epitaxial relationship with Ge.

For the Ni/Ge0.90Sn0.10 system, completely different pole figures were obtained (see Fig. 7 ). A of the monogermanide texture occurred with Sn addition. Rich patterns with a fourfold symmetry, originated from the Ge(Sn) substrate, were obtained as for the Ni/Ge system. In addition, spots related to Sn segregation were observed in the pole figures recorded at 2θ = 30.6 and 45.5°, together with the substrate contribution at 2θ = 30.6, 45.5 and 54.1°.

For Ni/Ge0.90Sn0.10 samples, five texture components (numbered 1 to 5) relating to the intermetallic and substrate alignments were common with the Ni/Ge system, whereas two additional texture components (numbered 6 and 7) were specific to the Ni/Ge0.90Sn0.10 system. They are listed in Table 7 .

Table 7

Summary of the orientations observed for the NiGe(Sn) phase

For each component, the symbol used to mark the locations of diffraction spots for that component in Fig. 7 is given. The texture components in common with the Ni/Ge system presented a very weak intensity for the Ni/Ge0.90Sn0.10 system, except for the out-of-plane orientation corresponding to the (301) planes. Some of them were nearly indistinguishable. On the other hand, the texture component resulting from the (013) outof-plane orientation presented the most intense diffraction features. As mentioned above, some misalignments evidenced by the IPRSM analysis in Section 3.2 and confirmed by stereographic projection analysis cannot be referenced as a full epitaxial relationship. For instance, the stereographic projection of NiGe (00 ) ∥ Ge(Sn) ( 00) showed that the in-plane (2 0) planes were slightly misaligned with Ge(Sn) (00

Symbol

).

An example of the stereographic projection of NiGe(Sn) (013) ∥ Ge(Sn) ( 00) is given in Fig. 8 .

Figure 8

Stereographic projection of NiGe(Sn) (013) ∥ Ge(Sn) ( 00) obtained with the CaRIne Crystallography software.

The preferential in-plane alignment of ( 011), ( 010), ( 013) and (100) with Ge(Sn) (011) evidenced by the PF analysis perfectly matches observations made during the analysis of the IPRSM of the Ni/Ge0.90Sn0.10 system in Fig. 6 . The epitaxial relationship of NiGe(Sn) 011, 020, 013 and 400 with Ge 220 is also corroborated. These alignments were possible thanks to the similar d spacings of the following planes: dNiGe(Sn)(033) = 0.987 Å, dNiGe(Sn)(030) = 1.143 Å, dNiGe(Sn)(026) = 0.85 Å and dNiGe(Sn)(500) = 1.106 Å compared with dGeSn(440) = 1.000 Å.

In addition, numerous components could be related to Sn/Ge(Sn) alignment. They are listed in Table 8 and are identified in Fig. 7 .

Table 8

Summary of the orientations observed for the β-Sn phase

For each component, the symbol used to mark the locations of diffraction spots for that component in Fig. 7 is given.

Symbol Number Out-of-plane orientation

In-plane orientation 1 ( 00)Sn ∥ ( 00)Ge(Sn) (0 0)Sn ∥ (00

The preferential alignments of Sn with Ge(Sn) obtained from pole figure analyses fully match with the observations made for IPRSM analysis in Section 3.2 . Alignment of Sn (010) with Ge(Sn) (001) (orientations 1, 4 and 8) matches with the epitaxial relationship of Sn 200 and 400 with Ge(Sn) 400. Then, the alignment of Sn (010) with Ge(Sn) (011) (orientations 2 and 3) matches with the epitaxial relationship of Sn 200 with Ge(Sn) 220. Finally, the alignment of Sn ( 10) with Ge(Sn) (01 ) (orientation 7) matches with the epitaxial relationship of Sn 101 with Ge(Sn) 220.

For out-of-plane orientations 5 and 6 it was not possible to link, without any doubt, an in-plane orientation having an epitaxial relationship with Ge(Sn). Nevertheless, for texture component 5, the stereographic projection of Sn ( 0) ∥ Ge(Sn) ( 00) showed that the Sn in-plane (112) planes were slightly misaligned by about 5° with Ge(Sn) ( 01). For texture component 6, the stereographic projection of Sn (

) ∥ Ge(Sn) ( 00) can be linked to the Sn in-plane ( 211) and ( 312) orientations without any preferential alignment with Ge(Sn). This is in good agreement with the IPRSM analysis of the Ni/Ge0.90Sn0.10 system summarized in Table 5 . These kinds of misalignments cannot be referenced as a full epitaxial relationship.

It was previously shown that Sn segregation occurs first at grain boundaries. Then, Sn is distributed at the surface/interface (Quintero et al., 2020a ). Islands or dots appeared on the surface of Ni/GeSn samples when annealing temperatures were high (beyond 623 K). This phenomenon was due to Sn accumulation at the sample surface (Quintero et al., 2018a ,b ). Multiple XRD results have also shown that Sn solidified in a crystalline phase (Fig. 2 , 3 and 6 ). It is, however, remarkable that, even under these conditions, segregated Sn crystallized following epitaxial relationships with the Ge(Sn) substrate instead of being present as randomly oriented grains. This behavior can be related to Sn accumulation at the NiGe(Sn)/Ge(Sn) interface, evidenced at 623 K by ex situ transmission electron microscopy (TEM)/energy dispersive spectroscopy (EDS) analyses and at 543 K by in situ TEM/EDS analyses (see Fig. 9 ), which can allow Sn to follow an epitaxial alignment with the Ge(Sn) substrate. A similar phenomenon was observed when Sn segregated from Si/SiSn/Si multilayers. Spots of the β-Sn phase revealing the orientation relations (220)Si ∥ (200)β-Sn and (200)Si ∥ (101)β-Sn were found from studies by Gaiduk et al. (2014 ).

Figure 9

Cross-section TEM images coupled with EDS: ex situ analyses (upper row) and in situ analyses (bottom row).

Discussion

For intermetallic thin films, a preferred orientation of grains with respect to the single-crystal substrate underneath will be driven by a minimization of the interfacial energy. A low interfacial energy can be achieved by the optimization of bonding between the thin film and the substrate. The bonding can be optimized if the interface structure is periodic. In fact, both the intermetallic film and the substrate will exhibit a periodic crystal structure (assuming that both are crystalline layers). Therefore, if a matching of the intermetallic lattice and the substrate lattice along a direction within the plane of the interface is possible, periodicity can occur.

For the cases presented in Section 3.3 , preferential NiGe/Ge and NiGe(Sn)/Ge(Sn) alignments were found.

A preferential alignment of NiGe ( 010), ( 013) and ( 104) with Ge (011) was observed for Ni/Ge samples while a preferential alignment of NiGe(Sn) ( 011), ( 010), ( 013) and (100) with Ge(Sn) (011) was demonstrated for the Ni/Ge0.90Sn0.10 system.

The studied phases have different crystal structures (Table 1 ). The monogermanide or monostanogermanide phases are orthorhombic, while the Ge(Sn) substrates are cubic. For a case like this, interface matching is unlikely to occur. However, if the low-index planes (from both the intermetallic and the substrate) share similar d spacing, matching can take place. This corresponds to the principle of axiotaxy alignment (Detavernier et al., 2003 ).

It was verified that, for NiGe and NiGe(Sn) phases, the alignment was possible thanks to similar d-spacing values between the following planes: (i) dNiGe( 030 00). This result is in good agreement with previous work on NiGe texture analysis (Balakrisnan et al., 2005 ). However, for the Ni/Ge0.90Sn0.10 system, the main out-of-plane orientation was NiGe ( 0) ∥ Ge(Sn) ( 00). Although the number of texture components was similar for the two systems, the degree of texture for the Ni/Ge(Sn) system was greater. For the Ni/Ge system, a higher incidence of different orientations with respect to the substrate was observed. This was evidenced by out-of-plane XRD analyses and confirmed with IPRSM and pole figure analyses. Meanwhile, the NiGe(Sn) layers consisted of polycrystalline grains almost all of which shared the same orientation. In addition, looking at the main texture component, the difference of d spacing between the intermetallic and the substrate was smaller for the case of NiGe(Sn)/Ge(Sn) alignment. The difference between dNiGe(Sn)(033) and dGeSn( 440) is 1.3%, whereas it reaches nearly 15% for dNiGe(026) and dGeSn( 440). Thus, the change in texture induced by Sn addition might reduce the interface energy between the film and the substrate.

On the other hand, when axiotaxy occurs, the grains oriented with this texture are left with one rotational degree of freedom around the normal to the axiotaxy axis. This results in circular patterns aligned off-center on the pole figure. Our results did not reveal these kinds of circular patterns (see Fig. 7 ). Nonetheless, the well defined spots obtained were arranged in a way that might project semicircular lines off-center (see Fig. 7 for the PF recorded for the Ni/Ge system). Owing to the axiotaxy-related plane alignment, periodicity can be achieved in one direction. However, if a periodicity is well defined in a second direction, a two-dimensional periodic interface structure can be obtained. This alignment will then generate well defined epitaxial spots on the pole figures at certain positions along the axiotaxial lines (De Schutter, De Keys et al., 2016 ). When comparing Ni/Ge and Ni/GeSn PFs, it can be observed that the occurrence of well defined spots along axiotaxial lines is low or nonexistent for Ni/GeSn samples in contrast to the Ni/Ge system. Therefore, Sn addition might promote a better established epitaxial alignment between NiGe(Sn) films and the Ge(Sn) substrate. Meanwhile, for the Ni/Ge case, a complex combination of axiotaxial and epitaxial alignments occurred.

The thin-film texture can influence parameters such as phase formation, phase stability, morphological and electrical stability (De Schutter, De Keys et al., 2016 ). For example, the threshold temperatures for Ni/Ge phase growth changed when the substrate orientation varied from Ge (001) to Ge (111) (De Schutter, Van Stiphout et al., 2016 ). Meanwhile, Deduytsche et al. (2005 ) showed that agglomeration of an Ni/Si film can occur at lower temperatures on a single-crystalline (001)-oriented silicon-on-insulator substrate than on a polycrystalline Si substrate. Finally, this morphological degradation can reduce electrical performance. These parameters are critical for contact elaboration. Therefore, knowing the impact of texture and other critical properties of the intermetallic can be a great insight for integration purposes.

For the current study, linking the texture to the evolution of other properties was not straightforward. Even though a variation of threshold temperature for the growth of the monostanogermanide phase with the Sn content was evident (see Table 2 ), this phenomenon was mainly due to the Sn behavior during the solidstate reaction (Quintero et al., 2020a ). Meanwhile, agglomeration processes and Sn segregation can degrade morphological and electrical properties (Quintero et al., 2018a(Quintero et al., ,b , 2020b ) ), which makes a direct correlation between texture and morphological or electrical evolution difficult.

Conclusions

In summary, the impact of Sn addition on the Ni mono(stano)germanide phase texture was studied. A variation of the out-of-plane and in-plane orientations with Sn addition was found. A reduction of the number of peaks related to the mono(stano)germanide phase when increasing the Sn content was thus evidenced thanks to θ-2θ XRD and integrated in-plane RSM analysis, while the preferred orientation changed.

The analysis of pole figures revealed the presence of numerous epitaxial components for the Ni/Ge system, while the number of meaningful epitaxial components was reduced for the Ni/GeSn case, in which the texture varied. A preferred out-of-plane orientation of NiGe (111) ∥ Ge ( 00) was observed for the Ni/Ge system, while for the Ni/Ge0.90Sn0.10 system, the main out-of-plane orientation was NiGe(Sn) (0 ) ∥ Ge(Sn) ( 00). A preferential alignment of NiGe (010), NiGe (013) and NiGe (104) with Ge (011) was observed for the Ni/Ge samples while a preferential alignment of NiGe(Sn) (011), NiGe(Sn) (010), NiGe(Sn) (013) and NiGe(Sn) (100) with Ge(Sn) (011) was demonstrated for the Ni/Ge0.90Sn0.10 system. These alignments were possible thanks to similar plane spacing values, minimizing the lattice mismatch and interfacial energies.

Additionally, segregated and melted Sn was observed to solidify in a crystalline phase which exhibited an epitaxial alignment with the Ge/GeSn substrate underneath.