Introduction

In the beginning, science was mostly about distinguishing and classifying objects and notions in order to establish boundaries and binary contrasts between pairs of notions such as earth and sky, land and sea, domestic and wild, useful and useless, allowed and forbidden, vowels and consonants. Research has, however, consistently shown that such boundaries and distinctions are fuzzy and blurred, and at best function as prototypes or targets. The more we know, the more old, reassuring certainties collapse, the more scatter plots and probabilities replace the clear dividing lines.

The realm of linguistics is no exception to this. Seducing and detailed as they may seem, the neat rules of 'standard' or prescriptive grammars fail to fully describe and encompass the complexity of human language as it is attested in the daily speech of billions of people around the world.

And with respect to the specific field of Creole studies, much remains to be done in order to take language variation into account in a more comprehensive manner. If we look at APICS (Michaelis & al. 2013), the widest ever published cross linguistic study of Pidgin and Creole languages, many of the linguistic varieties described therein appear as relatively homogeneous systems. This seemingly uniform structure is linked to traditional descriptive approaches, whereby the grammar-writer is supposed to choose one (or a few) form(s) or paradigm(s) considered to be most representative of the language, in this case the Creole. Similarly noteworthy is the scarcity in Creole studies of linguistic atlases (exceptions seem to be found mostly in the domain of French-based Creoles, e.g. Carayol & Chaudenson 1984-1995(Reunion), Fattier 2000 (Haiti) and Le Dû and Brun-trigaud 2011 (Lesser Antilles) 1 ), revealing a frequent lack of attention paid to diatopic variation in Creole and Pidgin languages.

However, research on Creoles has played a significant role in critically examining the very notions of language and speech community and in enhancing our understanding of the nature of 1 See also Quint (forthcoming) for a work devoted to Portuguese-based Creoles.

Léglise, I., Migge,B. Quint,N.,Approaches to variation in Creole studies,Journal for Pidgin and Creole Languages,[1][2][3][4][5][6][7][8][9][10][11] variation and how it relates to society. At the end of the 19 th century, a first generation of scholars (e.g. Schuchardt) chose to study Creole and Pidgin languages at a time when most people, including researchers, considered them as 'corrupt', 'degraded' or 'incomplete' forms of their respective lexifiers and therefore unworthy of academic attention. This early generation of scholars stood out precisely because they took linguistic variation into consideration rather than dismiss it. They opened up the possibility to systematically investigate variation in language.

In the second half of the 20 th century, two important concepts emerged in relation to Creoles and variation: diglossia and the post-creole continuum. Following [START_REF] Ferguson | Diglossia[END_REF] example of Haitian and French in contact in Haiti, the French-speaking research tradition on Creoles adopted the notion of diglossia which argued that French and French-based creoles are in complementary social distribution and have different levels of overt prestige. The concept was very productive for a long time probably because the Haitian situation appeared to closely resemble that of other regions in which French-based Creoles, the low-prestige languages, are in contact with French, the high-prestige language, such as the French overseas regions of the Lesser Antilles and La Réunion in the Indian Ocean. The notion was useful for capturing the status of the languages in a territory [START_REF] Hazaël-Massieux | Approche socio-linguistique de la situation de diglossie françaiscréole en Guadeloupe[END_REF] for Guadeloupe; Carayol and Chaudenson 1978 for Réunion;Prudent 1982 for the Lesser Antilles), for accounting for language use according to interlocutors and social settings and for representing the discourses of Creole speakers on minoritization [START_REF] March | Diglossie et créolité : un point de vue sociolinguistique[END_REF][START_REF] Bavoux | Représentations et attitudes dans les aires créolophones[END_REF]. By focusing on the relationship between a high and a low variety, the framework echoed the very hierarchy between the acrolect and basilect varieties observed by researchers working on English-based creoles. The approach was not accepted by all though. [START_REF] Carayol | Diglossie et continuum linguistique à la Réunion[END_REF], for example, promoted the notion of a continuum and [START_REF] Prudent | Diglossie et interlecte[END_REF]Prudent ( , 1993)), in an attempt to show the limits of the continuum, proposed the notion of an interlectal zone, an area which borrows from the two languages that make up the linguistic repertoires of bilinguals. More recent work within the French tradition shows, from a linguistic perspective, how those interlectal zones can be operationalized for analytical purposes via floating zones (Ledegen 2012)2 or multiple transcripts Léglise,I.,Migge,B. Quint,N.,Approaches to variation in Creole studies,Journal for Pidgin and Creole Languages,[1][2][3][4][5][6][7][8][9][10][11] using annotation tools that were developed at the crossroad between corpus and contact linguistics [START_REF] Vaillant | A la croisée des langues : Annotation et fouille de corpus plurilingues[END_REF][START_REF] Léglise | Plurilingual Corpora and Polylanguaging, Where Corpus Linguistics Meets Contact Linguistics[END_REF].

The post-creole continuum model was coined by [START_REF] Decamp | Towards a generative analysis of a post-Creole speech continuum[END_REF] and popularized by [START_REF] Bickerton | Dynamics of a creole system[END_REF], and has played a dominant role in conceptualizing variation in research on English-based Creoles. It argues that in communities where an English-based creole, the socalled basilect, co-exists with its original lexifier, the acrolect, a 'set of varieties' linguistically intermediate between them, also referred to as the mesolects, will emerge when speakers of the basilect start acquiring the acrolect. Mesolects combine English-like forms with basilectal functional and semantic content [START_REF] Craig | Creole and standard: Partial learning, base grammar and the mesolect[END_REF]. In communities such as Jamaica and Guyana, speakers are assumed to occupy different positions on the linguistic continuum based on their degree of acquisition of (features of) the acrolect or their relative use of basilectal features. As speakers' level of knowledge of English rises across society, basilectal features are assumed to contract, giving rise, over time, to the dominance of English-like varieties, or so-called intermediate creoles. The process of decreolization is allegedly at the heart of this change.

Although originally conceptualized as a qualitative process of language change, whereby basilectal features are gradually replaced by acrolectal ones, Rickford (1983) successfully argued in favor of a quantitative conceptualization. He demonstrates that language change in creole continua is due to a decline in the number of people who use basilectal varieties, rather than mostly a replacement of linguistic features (Rickford 1983). An important tenet of the post-creole continuum is that the lects of the continuum cannot be separated from each other due to continuous variation between them. In [START_REF] Bickerton | Dynamics of a creole system[END_REF] view, this continuous set of lects constitutes a linguistic system. This conceptualization of language practices in creole speaking communities was challenged in subsequent years by quantitative sociolinguistic research. Focusing predominantly on morphosyntactic patterns of variation such as the copular domain and pronouns, the research aimed to establish the social and linguistic factors that condition variation between English and Creole features. It revealed that creole communities are made up of two or more co-existing socially and linguistically distinct language systems [START_REF] Devonish | On the existence of autonomous language varieties in "Creole Continuum Situations[END_REF], namely a Creole and a European (or another dominant) language. Traditionally, each variety is associated with a particular type of context -Creole: informal contexts; European language: formal contextsand with particular social groupings -Creole: rural and lower-class populations; European language: urban and middle class populations. But Creole communities do not conform to [START_REF] Ferguson | Diglossia[END_REF] notion of diglossia (Winford 1985) because most contexts and populations do not just make use of one language. Code-switching is productively used by community members to negotiate social relationships and social identities (Escure 1982;[START_REF] Fenigsen | Meaningful routines: Meaning-making and the face value of Barbardian greetings[END_REF]Migge 2005aMigge , 2007) ) as in any bilingual or multilingual community. In countries such as Guyana and Jamaica intermediate varieties also constitute focused systems in their own right [START_REF] Rickford | The Insights of the Mesolect[END_REF][START_REF] Patrick | Urban Jamaican Creole: Variation in the Mesolect[END_REF]. They are linked to urbanized working class populations and code-switching takes place between them, the 'deep' Creole and Standard Englishes. Creole continua then constitute a repertoire of varieties that speakers manipulate in various ways (Winford 1997).

Although this research has helped to demystify the notion of post-creole continuum and shed some light on the nature of language variation in Creole communities, it continues to uphold the tacit assumption that the two linguistic poles of the continuum, 'the' Standard and 'the' Creole are the main resources for variable language choices. Creoles have different varieties [START_REF] Patrick | Style and register in Jamaican Patwa[END_REF][START_REF] Migge | On the emergence of new language varieties: The case of the Eastern Maroon Creole in French Guiana[END_REF][START_REF] Dupré | X MSG'ing among French Reunion 18to 25-year olds: A pilot study of mobile-mediated communication in a diglossic context[END_REF][START_REF] Jourdan | Pijin and shifting language ideologies in Solomon Islands[END_REF], Quint 2014) and variation may also take place between these varieties. Early quantitative sociolinguistic studies using macro social categories showed that Caribbean communities are stratified according to social class (e.g. [START_REF] Winford | A sociolinguistic description of two communities in Trinidad[END_REF]Rickford 1991;[START_REF] Patrick | Urban Jamaican Creole: Variation in the Mesolect[END_REF]) and ethnicity [START_REF] Escure | Linguistic variation and ethnic interaction in Belize: Creole/Carib. In Language and Ethnic Relations[END_REF](Escure , 1982) ) while more recent, anthropologically oriented research demonstrates that variation is not only constrained by factors such as gender [START_REF] Sidnell | Gender and pronominal variation in an Indo-Guyanese Creolespeaking community[END_REF], context, audience and aims of the interaction but is also actively deployed to manage social relationships and identities [START_REF] Fenigsen | Meaningful routines: Meaning-making and the face value of Barbardian greetings[END_REF]. In their research on the emergence of new newer creole varieties, [START_REF] Migge | On the emergence of new language varieties: The case of the Eastern Maroon Creole in French Guiana[END_REF] also showed that processes of identity formation give rise to processes associated with koineization involving mixing of features from different related regional varieties of the Creole, levelling of such features, formal reduction, and finally focusing of a new mixed variety [START_REF] Siegel | Koines and koineization[END_REF]. Moreover, while earlier research investigating linguistic prestige and language attitudes [START_REF] Ferguson | Diglossia[END_REF], Rickford 1985;Winford 1985;Mühleisen 2001;[START_REF] Wassink | Historic low prestige and seeds of change : attitude toward Jamaican Creole[END_REF] focused on the distinction between covert and overt prestige and how they relate to the two main varieties, 'the' Standard and 'the' Creole, Migge & Léglise (2013) demonstrate how language ideologies play a significant role in the different ways in which both linguists and speakers conceptualize language practices as either languages or varieties that can neither be easily characterized as monostylistic nor as monolingual. [START_REF] Léglise | Language and Identity Construction on the French Guiana-Suriname Border[END_REF] research on the Surinamese and French Guianese border zone also demonstrates that different sets of ideologies, such as traditional emic and modern state-based language ideologies as well as emerging discourses in the urban context may co-exist and productively function as an important force field for shaping Maroon's social and language practices and identities. Finally, [START_REF] Migge | The speech event kuutu in the Eastern Maroon community[END_REF]Migge's ( , 2005aMigge's ( , 2005b[START_REF] Migge | On the emergence of new language varieties: The case of the Eastern Maroon Creole in French Guiana[END_REF] research among urbanizing Maroons suggests that changing notions of linguistic politeness are an important motivation for changes in language use, leading to both the emergence of new language varieties or practices and changes in the social values of existing social varieties.

here is also research on variation in Creoles from the perspective of mediated language use such as the use of Creoles in literature otherwise written in a dominant language (e.g. M hleisen 2002; Lalla 2005; Pollard 2014), the use of Creoles in broadcast media (e.g. Garrett 2000; Managan 2011), music performances (e.g. [START_REF] Herzfeld | The Limonese calypso as an identity marker[END_REF][START_REF] Farquharson | Linguistic Ideologies and the historical development of language use patterns in Jamaican music[END_REF]) and in electronic media or computer-mediated communication (CMC) [START_REF] Deuber | Dynamics of orthographic standardization in Jamaican Creole and Nigerian Pidgin[END_REF][START_REF] Dupré | X MSG'ing among French Reunion 18to 25-year olds: A pilot study of mobile-mediated communication in a diglossic context[END_REF][START_REF] Moll | Jamaican Creole Goes Web: Sociolinguistic Styling and Authenticity in a Digital 'Yaad[END_REF]Migge 2020). This research investigates variation in language use and discursive practices focusing on both the identification of linguistic, including orthographic, and language use patterns and their social meanings and functions.

Recent qualitative studies on variation among multilingual speakers, influenced both by linguistic anthropology and 3rd wave sociolinguistic research [START_REF] Eckert | Meaning and Linguistic Variation: The Third Wave in Sociolinguistics[END_REF] show that variation is often idiosyncratic. However, it is productively employed to index interactional relationships, stances and new transnational and cosmopolitan social identities. Speakers, as language users, deploy their entire linguistic repertoire, made up of social styles, dialectal variables and a variety of heterogeneous language forms, as resources to position themselves in their everyday interactions (Léglise & Sanchez Moreano 2017). Linguistically heterogeneous practices [START_REF] Léglise | Multilingualism and Heterogeneous Language Practices: New Research Areas and Issues in The Global South (translation)[END_REF]) perform various discursive and social functions and constitute important resources for conveying various types of social and interactional meanings.

To sum up, research employing early quantitative sociolinguistic methods was instrumental in shedding light on the makeup of linguistic varieties in creole communities and the linguistic aspects of variable patterns. However, it did not provide a comprehensive analysis Léglise,I.,Migge,B. Quint,N.,Approaches to variation in Creole studies,Journal for Pidgin and Creole Languages,[1][2][3][4][5][6][7][8][9][10][11] of the micro-social aspects of variation because it focused on macro-level relationships and inferred social meanings from statistical correlations of linguistic and macro-social categories. It was the greater use of qualitative methods, including ethnographic investigation of the social context and attention to language ideologies through discourse-based analysis of monolingual and multilingual variable practices across different types of social domains and interactional contexts that opened up more detailed insights into the linkages between language and microsocial practices. As in non-creole contexts, language and social ideologies and changes in both play a crucial role in motivating language variation and change, both its occurrence and directionality, but changes are rarely uni-directional and linguistic practices are seldom associated to just one social or interactional meaning. Variable linguistic practices and contact patterns are important indices of ideological processes, but an understanding of their indexical work requires close attention to both people's practices and discourses.

The papers in this special issue explore variation in a range of Creoles and represent different approaches to researching variation. Two of the papers follow a typical quantitative sociolinguistic approach. They focus on one linguistic feature, or variable, and correlate its distribution with independent linguistic factors (Riccelli's paper) and with linguistic and social factors (Cardoso and Costa's paper) to uncover the constraints that govern the distribution of that variable. Evans' paper differs from these studies in that it investigates variation in legal translations from English to Kwéyòl. It attempts to establish the degree of pragmatic and legal equivalence in the interpretations of on sight translations of an important English legal phrase, the pre-trial right to silence or police caution, and the social factors that condition this variation.

The other two papers in the special issue follow a linguistic anthropological approach to variation in that they investigate types of variable linguistic practices, rather than single variables, and speaker's discourses about them, in order to understand how speakers conceptualize these practices (Schneider) and how these conceptualisations feed into identity formation (Jourdan & Angeli). Britta Schneider's extensive work on a Belizean village draws on the fundamental question of linguistic ideologies to understand patterns of linguistic prestige. As in many other places, several languages coexist in the village with Kriol (the local English-based Creole language) being the main spoken vernacular. Nevertheless, the way Kriol speakers feel towards Léglise,I.,Migge,B. Quint,N.,Approaches to variation in Creole studies,Journal for Pidgin and Creole Languages,[1][2][3][4][5][6][7][8][9][10][11] their language is not easy to define. Kriol is at the same time an indexical marker of local and Belizean identity and a variety that is not always consciously viewed as really different from English, the only official language of Belize. She shows that prestige relations are not necessarily binary but that different language ideologies exist side-by-side and that linguistic choices have several indexical meanings.

Jourdan and Angeli's paper, which is based on long-term ethnographic research in the Solomon Islands, demonstrates how the development of weak normativity in Solomon Islands Pijin in the capital Honiara closely parallels the emergence of a new socio-cultural group, the Pijin-speaking urbanites. The linguistic practices that index their social identity draw on the existing linguistic resources, the indigeneous Micronesian languages, rural and more conservative L2 varieties of Pijin and English, the sole official language in the country. This Honiaran urbanite identity is far from linguistically fixed as several varieties of Pijin coexist and influence each other, in addition to English which is also present (with diverse types of insertion) in most of the spoken productions of Honiaran Pijin speakers.

Evans' paper investigates variation in legal language focusing on how the pre-trial right to silence or police caution is translated on the spot from English to Kwéyòl by police personnel in St. Lucia. Her analysis reveals considerable variation in illocutionary force but fairly strong form-based similarity between translations. Evans suggests that this discrepancy is due to social factors such as the language competence and professional experience of the translator. She argues that only the existence of a standardized and legally binding translation of the police caution can guarantee the equality of access to justice for speakers of Kwéyòl.

Cardoso and Costa show how in Sri Lanka Portuguese Creole, the paradigm of personal pronouns displays a much wider variation than had hitherto been recorded by previous descriptive studies. These results seem to be due to several factors, such as linguistic obsolescence (differences based on age) and diatopic variationwhich Cardoso and Costa are probably among the first to really exploreand presumably also metatypy due to close contact with the main local substrate/adstrate, namely Tamil, a Dravidian language, and English, one of the main media of education in Sri Lanka, which both exert a growing influence on language use of the tiny Creole community whose direct contact with its lexifier (Portuguese) was severed Léglise,I.,Migge,B. Quint,N.,Approaches to variation in Creole studies,Journal for Pidgin and Creole Languages,[1][2][3][4][5][6][7][8][9][10][11] several centuries ago. Thus, in spite (or because?) of their reduced number, the speakers of Sri Lanka Portuguese Creole seem to display a significant degree of variation in their language practices.

Riccelli's analysis of the subject domain in Santiaguense Capeverdean introduces us to yet another dimension of variation. Although the debate about the existence and the status of null subjects in this variety has been ongoing for more than two decades, Riccelli is probably the first researcher to study the expression of the subject argument in Santiaguense based on spoken language corpora. His analyses reveal that semantic notions such as ANIMACY, SPECIFICITY, and DEFINITENESS condition the (non)-occurrence of null subjects in Santiago Capeverdean. In addition, Riccelli notices that the way these factors influence the realization of the subject follows a gradient rather than a categorical or discrete distribution.

The papers in this special issue explore linguistic variation at various levels, whether semantic, diatopic or social and their link to language ideologies, that resist simplistic approaches and concepts. Such explorations undoubtedly pave the way for a better understanding of Creole languages in their complex sociocultural environments.

  Isabelle Leglise, Bettina Migge, Nicolas Quint To cite this version: Introduction Isabelle Léglise (CNRS, SeDyL), Bettina Migge (UCD, SeDyL) & Nicolas Quint (CNRS, LLACAN)

  Léglise, I.,Migge, B. Quint, N., Approaches to variation in Creole studies, Journal for Pidgin and Creole Languages,[1][2][3][4][5][6][7][8][9][10][11]

  Léglise, I.,Migge, B. Quint, N., Approaches to variation in Creole studies, Journal for Pidgin and Creole Languages,[1][2][3][4][5][6][7][8][9][10][11]

Ledegen shows that such floating predicates can account for as much as 16% of the total corpus in the speech of people from la Réunion Island.


Kazem Zhour
email: kazem.zhour@gmail.com

Andrei Postnikov
email: andrei.postnikov@univ-lorraine.fr

Theoretical mapping of interaction between potassium atoms adsorbed on boron carbide monolayer

Keywords: semiconductor, monolayer, alkali metal, adsorption, diffusion

First-principles calculations using density functional theory and two methods in comparison, Quantum ESPRESSO and Siesta, are done on large supercells which describe different placements of two adsorbed potassium atoms on the monolayer of boron carbide BC 3 . The energy of single K adsorption over the center of C 6 ring, of the C 4 B 2 hexagon and over a boron atom are estimated to be 1.66, 1.63 and 1.58 eV, respectively; the comparison of these results with some available data is discussed. Exact theoretical estimations may be somehow sensitive to the lattice spacing practically used in the calculation. The interaction of two potassium atoms adsorbed at close enough distances (less than ≃10 Å) is negligible if the adsorption occurs at the opposite sides of the BC 3 layer, but creates a steep repulsive potential at distances less than ≃8 Å if the K atoms are adsorbed on the same side of the monolayer. Relaxation patterns resulting from the two K atoms being trapped at adjacent adsorption sites in the lattice are explained. The results suggest that the density of adsorbed K atoms on BC 3 can be interestingly high.

Introduction

Boron carbide (BC 3 ) recently emerged [1] as an interesting two-dimensional (2D) material, in its structure and some properties close to graphene, with the difference that its single layer is semiconducting and not semi-metallic. Certain expectations are related with using this material functionalized with metals, for so different purposes as alkali metal storage in batteries [2], enhanced sensibilisation for gas sensoring [3], and hydrogen storage [4]. These lines of research brought about recent works of first-principles simulation. Zhao et al. [5] probed different placements of Li, Na and K atoms over different sites of BC 3 , compared the corresponding adsorption energies, and traced the energy profiles across the barriers between adjacent adsorption sites. They used the VASP package on 3×3 supercells (of primitive cells with 2 formula units) and generalized gradient approximation for the exchange-correlation after the prescription of Perdew-Burke-Erznerhof (GGA-PBE) [6,7]. Naqvi et al. [8] used the same calculation method (on a smaller, 2×2 supercell) to probe the docking of the same alkali metals and moreover of the alkaline earth metals, with subsequent adsorption of CH 4 , CO 2 and CO molecules at the metal sites. Bafekry et al. [9] [11] 1.422 1.565 5.174 0.62 VASP GGA [5] 1.42 1.57 5.17 0.66 Siesta GGA [9] 1.422 1.562 5.17 0.7 explored different placements and estimated corresponding adsorption energies for a long list of adatoms and small molecules, in a series of calculations using OpenMX and Siesta methods with GGA-PBE, for 2×2 supercell. Whereas these studies permit to provide some consistent idea of how would the potassium atoms place themselves onto, and diffuse over, the BC 3 layer, we find it interesting to address a hitherto unexplored issue of interaction of K atoms at the surface. This may help to parametrise this interaction to adequately model the diffusion, and/or to estimate maximal density of K atoms to be placed on the surface, which may be helpful for the above discussed issues, i.e., battery electrodes or gas sensors. In view to probe this interaction, we considered supercells of larger size that in the mentioned previous studies, and decorated with two potassium atoms. Two calculation methods have been used in comparison; -Siesta for principal large-size calculations and Quantum ESPRESSO for smaller-size benchmarks. The exchange-correlation potential used was GGA-PBE as in most of the other recent works; other calculation details can be found in Section 5. We start however with a short discussion of pristine BC 3 and the K adsorption on it in Section 2, checking ourselves against the previous results, and come to novel results concerning K-K interaction in Section 3. The conclusions are summarized in Sec. 4.

Pristine BC 3 and adsorption of K on it

Planar Boron carbide, described in Ref. [1], is characterized by a honeycomb planar lattice like that of graphene, in which one carbon atom out of four (equivalently, two atoms within a 2×2 graphene supercell) are substituted by boron. In principle, one can imagine different relative placements of B atoms on the honeycomb lattice, corresponding to para-, orthoand meta-isomers (referred to as types A, B and C in Ref. [10]). However, the para-isomer is singled out by its more regular and symmetric arrangement in which every carbon atom has exactly one boron neighbour, and there are no B-B bonds. This isomer retains the hexagonal (super)structure, which makes a pattern of C 6 (perfect) hexagons with the C-C bond lengths d C-C ≈1.4 Å and C 4 B 2 (distorted) hexagons around the perfect ones, with the C-B bond lengths d C-B ≈1.6 Å. The lattice parameter of this hexagonal unit cell which hosts two formula units is a = √ 3(d C-C + d C-B ). Differently from perfect infinite graphene which has a zero gap, BC 3 has the band gap of 0.65 eV (in a single sheet). A summary of previous data concerning pristine BC 3 , along with our present results, is given in Table 1.

The works aimed at the study of K adsorption on BC 3 from first principles are not numerous. Understandably, the possible adsorption sites probed in previous works (notably Zhao et al. [5]) are over the center of one or another hexagon, over the C-C or C-B bonds, or over the one or the other atom species. Zhao et al. argued (with respect to Li, Na and K on BC 3 , which we confirm in our study of K), that only three of these sites, over the both hexagons and atop a boron atom, emerge as local energy minima for adsorption.

There seems to be a consensus among the calculations done that the hollow site over the center of the C 6 hexagon, referred to as HC in the following, is the preferential one, closely followed by the hollow site over the center of the C 4 B 2 hexagon, referred to as HB. With respect to the value of the adsorption energy, however, there seems to be a controversy we show in Table 2. We performed this calculation using sufficiently large supercell (in terms of B 2 C 6 unit cells) per adsorption atom and inspected several possible sources of controversy. Without much details given in the papers cited, good "suspects" could be the basis set superposition error (BSSE, Ref. [12]) (potentially dangerous in methods with atom-centered basis functions, e.g., Ref. [9]), and the neglection of the magnetic state for an isolated potassium atom, both factors tending to overestimate the adsorption energy. Our result shown in Table 2 is obtained from comparing the energies of two sufficiently thick supercells of identical size, possessing the same 73 atoms: the relaxed system upon K adsorption, and the pristine BC 3 layer with the K atom removed away from the layer at a distance that would preclude the overlap of their basis functions. This does not fully removes the BSSE (that could have been done by introducing ghost atoms), but substantially cancels systematic errors.

A smooth variation of the total energy and magnetic moment on approaching the K atom to the BC 3 sheet (without performing any structure relaxation and only converging the electronic structure) is depicted in Fig. 1. For the record, the lattice parameter in these trial calculations done with Siesta was fixed at a = 5.143 Å, somehow smaller than the value we preferred to use is our supercell calculations. However, the comparison with our "approved" estimations for the K adsorption energy (obtained in more accurate and fully relaxed calculations, to be discussed below) with the present "scan" helps to estimate the stability of results against small variations of the lattice parameter and to give a certain confidence in the values reported. The depth of the total energy profile gives a fair estimate of the adsorption energy. The magnetic moment exhibits instability throughout the range of distances 3 to 7 Å, as the 4s state of potassium, which carries an unpaired spin in a free atom, starts to interact with the wavefunctions of the CB 3 layer. Upon adsorption and after atomic and electronic relaxation, the magnetic moment is fully dissolved (differently from the situation with graphene, on which the adsorbed K atom retains its magnetic moment). 1The results obtained with the fixed lattice parameter of the supercell as optimized in QE calculation and following the unconstrained relaxation, comparing the total energies from "adsorbed K" and "distant K" situations within the same calculation setup, are shown in Table 2. For comparison, the calculation done with summing up the energies of BC 3 -only supercell and "single-K-only" supercell yields the adsorption energies of ≃2.1 eV.

Independently, we calculated the adsorption energy at HC and HB sites by the QE method, also for the 3×3×1 supercell, referring in this case to the total energy differences of the kind (Supercell with adsorbed K) -[(Supercell without K) + (Single K atom in the box)], which approach is less problematic when using a planewave-basis code not prone to BSSE. The results are 2.31 eV for the HC site and 2.25 eV for the HB site. Notwithstanding a slight variance with the Siesta results, we can anyway conclude that the adsorption energy of K on BC 3 is likely to be of the order of 2 eV, close to the result by Zhao et al., rather than the other two reported values, apparently strongly under-or correspondingly overestimated. The magnitude of the adsorption energy being thus settled, the differences between the adsorption at different sites seem mild and less controversial. In comparison with the adsorption energies of 1.66 eV at HC position, we arrive at 1.63 eV at HB and 1.58 eV in AB (on top of the B atom). This span is in good agreement with the energy profiles ("diffusion paths") shown in Fig. 5(d) of Ref. [5], according to which the HB position is ≃0.05 eV higher than HC, and the barrier between two adjacent hexagons is about 0.12 eV high. Our QE estimation of the energy difference between HC and HB sites confirms the energy difference between HC and HB to be ≃0.05 eV. The effect of disperse interactions, included in Ref. [5] within the simplified Grimme scheme [13] but neglected in our work, may be in part responsible for the remaining numerical difference. Apart from Ref. [5], we are not aware of other works dealing with the "diffusion profiles" of K on BC 3 . Our simulation confirms a peculiar property of potassium to combine appreciable adsorption energy (hence well sticking to the surface of BC 3 ) with very flat energy landscape (hence presumably good mobility over this surface). In the next section, we come to the analysis of interaction between K atoms.

Interaction of K atoms adsorbed on BC 3

Spatial map of K-K interaction energies on the BC 3 surface

In order to gain a substantial and reliable information concerning the interaction between K atoms adsorbed on BC 3 , we staged a series of simulations on large supercells containing two potassium atoms. One of them, the "reference atom", was adsorbed at a (lowestenergy) HC position over the center of a C 6 ring, and the other , "trial" atom -at different local-minima positions at different distances from the reference atom. Within the 5×5 supercell (200 atoms), all non-equivalent adsorption positions have been explored within the "irreducible wedge" (shown in inset of Fig. 2) closest to a given HC site. This amounted to four HC positions, 9 HB positions (over the center of C 4 B 2 hexagons) and 7 AB positions (atop of a boron atom), hence the positions confirmed (see previous Section) to correspond to stable (local energy minima) adsorption sites. Moreover, the configurations in which the second (trial) atom was adsorbed at the opposite side of the CB 3 monolayer has been tested -on all the HC, HB and AB sites, mentioned above, plus directly at the reference site, in a mirror position from the reference atom. In all cases, an unconstrained relaxation has been performed (fixing however the supercell parameters), which yielded the total energy and the equilibrium geometry.

For K-K distances starting from ∼7Å onwards, the relaxation patterns are barely visible: the adsorbed atoms reside in their nominal positions, so that only tiny modifications of interatomic distances come about. The CB 3 layer in all cases looks extremely rigid and planar, without a marked tendency towards bending or warping. At shorter distances, the repulsion between the adsorbed atoms pushes them out of symmetry positions; the examples of this will be discussed below. In fact, the stabilisation of the trial atom in the nearest HB or AB position to the reference atom becomes impossible, because the atoms drift away from such configuration. This refers to the atoms adsorbed at the same side of the CB 3 plane. The atoms adsorbed "across the plane" remain practically insensitive to the closeness of the reference atom, exhibiting no perturbations in their relaxation pattern, and the energies which are largely independent on the in-plane distance to the reference atom.

The "interaction energies" plotted in Fig. 2 are the free energies from the calculations, expressed relative to the lowest one over all configurations corresponding to the given placement (i.e., either above, or below the BC 3 layer) of the trial atom. In fact, this lowest energy stems from the most distant HC configuration, which effectively sets the "infinity" limit for the two K atoms being in maximally distant lowest-energy (i.e., HC) positions.

We note that these "infinity limit" energies are not exactly equal for the trial atom being on the same side of the BC 3 plane as the reference atom, or on the opposite side. In fact, the second energy is lower by 0.05 eV. This disparity (which would disappear be the atoms at really sufficiently far distance) can be considered as a measure of the effect of our limited supercell size.

Choosing such "distant K-K" limit of two adsorbed atoms as the reference level leaves the adsorption energy at a HC site out of our grasp; however, this issue has been separately addressed in the previous Section. On the contrary, the differences between adsorption energies at HC, HB and AB sites, or otherwise the barrier heights, are easily readable from Fig. 2, and are instructive to discuss.

We note first that for "inverted" adsorption of a trial atom, the energy of its interaction The upper panel maps the interactions between K atoms adsorbed at the same side of the CB 3 plane, the bottom panel -between atoms adsorbed at the opposite sides, in both cases -as function of the in-plane distances between the adsorption sites. The inset shows the projections of (ideal) adsorption sites of the three types onto the "irreducible part" (drawn as a triangle) of a 5×5 supercell. Positions included in the mapping are marked by symbols, as in the main plot. A primitive cell of BC 3 is outlined by a dashed contour.

with the reference one is practically independent on interatomic distance, and follows the universal trend that HB positions are, on the average, higher in energy by ≃0.03 eV than the HC ones, whereas the AB positions are higher than the HC by ≃0.08 eV. This simply passes the information about barrier heights (differences between adsorption energies at different sites), which was already addressed in the previous Section. However, here it appears not as a "random" number but as a "statistically credible" result. We can again refer to the similarity of the present estimation of barrier heights with those given by Zhao et al. [5], notably in Fig. 5(d) of this publication. The bottom panel of Fig. 2 suggests that the K atoms "do not see each other" across the BC 3 plane till the K-K distance becomes really short, within the closest adsorption sites. Still, this does not bring about neither an "asymmetric" relaxation nor noticeable augmentation of the z-coordinate; the adsorbed atoms remain well centered at their respective sites. The "double occupation" of a HC adsorption site from both sides of the BC 3 plane, whereby the two K atoms remain stable at a distance 5.4 Å (hence at a regular height above the layer) is by far not so energetically unfavourable as it could have been a priori anticipated; it costs only 0.1 eV to bring a K atom from any distant HC site onto the reference site.

The established hierarchy of barrier heights, revealed via interaction energies being independent on the K-K distance, is also valid for adsorptions at the same side of the BC 3 plane (see upper panel of Fig. 2), provided the K atoms are placed not closer than ≃10 Å. At shorter distances, the energy barrier rapidly increases, forming a nearly rigid core of the radius of ≃6.1 Å, irrespectively of the adsorption site concerned.

Unusual relaxation patterns of closely placed adsorbed atoms

The placement of trial atoms at adsorption sites close (within ≃8 Å) of the reference atom resulted in a noticeable repulsion of both atoms in their relaxed configurations, as shown in Fig. 3. The reference atom displaces in its "nest" over the C 6 ring without however leaving the latter, whereas the trial atom could have been pushed from its initial "on top of B" position onto that above the C-B bond; see the case AB in Fig. 3. It was not possible to stabilise the trial atoms in AB or HB positions closest to the reference atom, since the interacting atoms were pushed apart into "second-neighbors" configurations, shown in Fig. 3. This strengthens an idea of "hard core" repulsion potential which effectively prevents the potassium atoms to come closer than about 6 Å. This minimal distance corresponds to both atoms residing over the second-neighbour C 6 -C 6 or C 6 -C 4 B 2 hexagons, slightly pushed apart from the respective centers, as shown in Fig. 3. We note that the z-coordinates of potassium atoms are not much affected by their relaxation "situation"; also the warping of the BC 3 layer, although noticeable around such "strong" relaxations, remains small and difficult to make sense of.

Conclusion

Summarizing, we found that potassium atoms adsorbed over the BC 3 monolayer interact strongly repulsively at distances smaller than ≃10 Å, and otherwise remain relatively insensitive to the mutual presence, feeling the "usual" landscape of potential barriers of 0.1 eV in their movement over the BC 3 plane. A combination of relatively high adsorption energy (≃1.7 eV) with respectively low barriers, known already from early works, presumed already an interesting interplay of sticking and mobility. Now we conclude that even relatively high concentration of adsorbed K atoms might not prevent them from acting as a highly mobile "gas" on the BC 3 surface. This may open perspectives for further more realistic simulations and interesting applications.

Calculation methods and technical details

We used two first-principles DFT methods in comparison, the plane-wave pseudopotential Quantum ESPRESSO (QE) [14,15] and numerical orbitals pseudopotential Siesta [16,17]. Exchange-correlation was treated within the GGA, using the PBE parametrisation [6,7]. Siesta is generally expected to be more efficient than planewave methods in treating large open low-coordinated systems, but, at the same time, potentially sensitive to the "quality" of fixed basis functions adopted. After some tests, we found it sufficient to use double-zeta basis functions for all atoms, including notably 3p as a valence state for potassium. The pseudopotentials used were norm-conserving, generated according to the Trouiller -Martins scheme [18] for the following free-atom configurations (the cutoff radii in Bohr being indicated in parentheses for each l-channel) : K 4s 1 (3.14) 3p 6 (1.83) 3d 0 (3.14) 4f 0 (2.54), B 2s 2 (1.74) 2p 1 (1.74) 3d 0 (1.74) 4f 0 (1.74), C 2s 2 (1.54) 2p 2 (1.54) 3d 0 (1.54) 4f 0 (1.54). The PAO.EnergyShift parameter in Siesta was set to 0.015 Ry, that resulted in the maximal extension of basis functions of 5 Å.

In QE calculations, the kinetic energy and the charge density cutoffs were set to 50 Ry and 400 Ry, respectively. The convergence threshold for self-consistency in energy was set to 10 -7 Ry. A 8×8×1 undisplaced k mesh was employed in electronic structure calculations for pristine BC 3 , with appropriate reduction in larger supercells. Ultrasoft pseudopotentials have been used.

Supporting Information

which includes detailed tables with adsorption geometry and energies for all the adsorption sites probed is available from the Wiley Online Library or from the author.

Figure 1 :

 1 Figure 1: Total energy (upper panel, blue line) and magnetic moment (lower panel, red line) profiles as functions of the height of the K atom over the BC 3 layer. Calculations are done with lattice constant fixed at a = 5.143 Å without structure relaxation.

Figure 2 :

 2 Figure2: Map of interaction energies between K atoms adsorbed at different sites of the BC 3 lattice and the reference K atom in the HC position (the bottom blue point in the inset). HC positions (over C 6 hexagons), HB positions (over C 4 B 2 hexagons) and AB positions (over B atoms) are indicated by different symbols. The upper panel maps the interactions between K atoms adsorbed at the same side of the CB 3 plane, the bottom panel -between atoms adsorbed at the opposite sides, in both cases -as function of the in-plane distances between the adsorption sites. The inset shows the projections of (ideal) adsorption sites of the three types onto the "irreducible part" (drawn as a triangle) of a 5×5 supercell. Positions included in the mapping are marked by symbols, as in the main plot. A primitive cell of BC 3 is outlined by a dashed contour.

Figure 3 :

 3 Figure 3: Relaxed structures (in side and top views) of trial K atoms initially placed at HC, HB and AB positions close to the reference K atom (that in the left of each plot). The arrows indicate the movement of the both K atoms apart.

Table 1 :

 1 Calculated properties of single-layer BC 3 .

	Method	Ref.	d C-C (Å) d C-B (Å) a (Å) E gap (eV)
	Siesta GGA present work 1.428	1.564	5.183 0.67
	QE GGA	present work 1.421	1.564	5.170 0.51
	VASP GGA

Table 2 :

 2 Calculated parameters of K adsorption over the C 6 hexagon on BC 3 . E a : adsorption energy (corresponding to energy gain on adsorption, in all cases); h: the height of the adsorbed atom over the BC 3 plane. See text for detail.

	Method	Ref.	Supercell	E a (eV)	h (Å)
	Siesta GGA [9]	2×2×1	0.82 a	2.569
	VASP GGA	[8]	2×2×1	4.72	2.53
	VASP GGA	[5]	3×3×1	1.75	2.57
	Siesta GGA present work 3×3×1 *	1.66 a , 1.63 b , 1.58 c 2.68 a , 2.67 b
	QE GGA	present work 3×3×1 *	2.31 a , 2.25 b	2.63

a , 2.62 b * a=5.17 Å fixed; a HC position; b HB position; c over C-B bond.

The asymptotic value of the magnetic moment corresponding to large distances in Fig.1does not reach 1 µ B -probably due to a purely technical drawback that very fine k and energy mesh is needed to cope with very narrow K4s energy levels.

Acknowledgements

8

The authors acknowledge the support of the Erasmus mobility program which made possible for K.Zh. his research stay in Metz, and the mesocenter of calculation EXPLOR at the Université de Lorraine (project 2019CPMXX0918) for access to computation facilities.