Diego Torres

Alicia Diaz

Hala Skaf-Molli

Pascal Molli
email: pascal.molli@univ-nantes.fr

Semdrops: A Social Semantic Tagging Approach for Emerging Semantic Data

Keywords: Social Tagging, Semantic Data, Semantic Social Tagging

This paper proposes a collective intelligence strategy for emerging semantic data. It presents a combination of social web practices with semantic web technologies to enrich existing web resources with semantic data. The paper introduces a social semantic tagging approach called Semdrops. Semdrops defines a conceptual model which is an extension of the Gruber's tag model where the tag concept is extended to semantic tag. Semdrops is implemented as a Firefox add-on tool that turns the web browser into a collaborative semantic data editor. To validate Semdrops's approach, we conducted an evaluation and usability studies and compared the results with automatic generation methods of semantic data such as DBpedia. The studies demonstrated that Semdrops is an effective and complementary approach to produce adequate semantic data on the Web.

I. INTRODUCTION

"The Semantic Web is an extension of the current web in which information is given well-defined meaning, better enabling computers and people to work in cooperation" [START_REF] Berners-Lee | The semantic web[END_REF]. Web resources of semantic web are semantically annotated with special meta-data that represent formally the meaning. The availability of this machine-readable meta-data enable intelligent access of Web resources by automated agents, adequate queries and combination of several sources of information [START_REF] Berners-Lee | The semantic web[END_REF].

Ontologies form the backbone of the Semantic Web systems and they are the means to semantically annotate Web resources. Due to the wide variety of topics presented on the web, existing ontologies partially cover a small part of the current web. For instance, in the fields of medicine, computers, society and business, a lot of efforts have been done to develop ontologies [START_REF] Hepp | Goodrelations: An ontology for describing products and services offers on the web[END_REF], [START_REF] Breslin | Towards semantically-interlinked online communities[END_REF], [START_REF] Miles | Skos: Simple knowledge organisation for the web skos: Simple knowledge organisation for the web[END_REF], but then quickly decreases in other domains. These ontologies are developed by domain experts resulting in sophisticated ontologies that are used in specific applications beyond the Web. Besides, developing and combing large ontologies involve sophisticated tasks.

Recently, there is a shift towards the development of lightweight ontologies for specific domains because the definition and combination of them are easier than large and sophisticated ones. Consequently, these lightweight ontologies are widely used in Semantic Web applications. Some popular examples are FOAF 1 , SKOS 2 and SIOC 3 .

Although existing lightweight ontologies are defined by domain experts, many studies have demonstrated the significance of taking advantage of the social force to contribute to the semantic web vision [START_REF] Siorpaes | Human intelligence in the process of semantic content creation[END_REF], [START_REF] Hope | Convergence of web 2.0 and semantic web: A semantic tagging and searching system for creating and searching blogs[END_REF]. In this sense, it is important to provide users with tools that support the emergence of semantic data. Semantic data are the seed to make emerge lightweight ontologies. Semantic wikis [START_REF] Krötzsch | Semantic wikipedia[END_REF], [START_REF] Auer | Ontowiki -a tool for social, semantic collaboration[END_REF], [START_REF] Buffa | Sweetwiki: A semantic wiki[END_REF] enable the emergence of lightweight domain ontologies by the community efforts. However, the emerged ontologies are limited to the Semantic Wiki scope and cannot be transparently extended to any Web resource.

Our objective is to provide a tool that allow the emergence of semantic data without being limited to semantic wikis. We propose an approach that allow the emergency of semantic data by means of identification and modelling of the tagging activity. This is a step to the semantification of the web based on social effort. The proposed approach has:

• to promote semantic data proliferation.

• to promote the semantic data quality • to be applicable to any web resources.

• to be used by a wide community • to provide an easy interaction mode. In order to achieve all of these requirements, the challenge now is to define a simple semantic social tagging model whose integration into the system provides an easy interaction modality that enables to attach semantic tags to a web resource without modifying the resource itself. In this work, we propose a social semantic tagging approach called Semdrops. Semdrops defines a conceptual model which is an extension of the Gruber tag model [START_REF] Gruber | Ontology of folksonomy: A mash-up of apples and oranges[END_REF], where the tag concept is extended to semantic tag. We implement Semdrops as a Firefox add-on tool that turns the web browser into a collaborative semantic data editor in situ for any web resource.

To validate Semdrops's approaches, we conducted an evaluation and usability studies and we compared the results with automatic generation methods of semantic data such as DBpedia. The studies demonstrated that Semdrops is an effective and complementary approach to produce adequate semantic data on the Web. This paper is organized as follow. Section II details the social semantic approach and defines the conceptual model of Semdrops. Section III details Semdrops the architecture and implementation. Section IV presents an evaluation and usability studies. Section V discusses the results of the evaluation. Section VI presents related works. Finally, section VII presents conclusions and future work.

II. SEMDROPS APPROACH

Semdrops uses the force of mass collaboration for facilitating the emergence of semantic data. This data can be used to emerge lightweight ontology. It is based on social semantic tagging strategy [START_REF] Kim | Social semantic cloud of tag: Semantic model for social tagging[END_REF] that adopts the characteristics of social tagging enhanced with semantic meta-data. In order to make easy the semantic annotation activity to the user, Semdrops proposes a simple semantic representational model to conceptualise the social semantic tagging activity, as it is shown in section II-A. This model captures the semantic annotations performed by the users. This means that the model allows to represent the attachment of social semantic tags to a web resource in an easy manner. The model also enables tagging a resource without modifying the resource itself. Section II-C presents a detailed example.

Although, Semdrops social semantic tag model was designed to make easy the tagging activity by regular users, it has to be mapped to existing semantic representation models like RDF, OWL in order to make it compatible with the semantic web technologies. In this work, we particularly use Semantic MediaWiki to make this mapping as is detailed in the section II-B.

A. Semantic Social Tagging Model

According to the literature, the most elementary building blocks of a tagging model is a tripartite model [START_REF] Newman | Tag ontology[END_REF] makes up of the taggers, the tags and the resources being tagged. Gruber in [START_REF] Gruber | Ontology of folksonomy: A mash-up of apples and oranges[END_REF] introduced to model one more element, the source of the tags, and named it the four-places relation model. Semdrops conceptual model is an extension of the four-places relation one, where the tag concept is extended to semantic tag. Semdrops defines four main concepts: Web resource, tagger, semantic tag and semantic support.

Web resource. A Web resource is any Resource in the web which could be identifiable by an URI.

Tagger. The tagger is the responsible of adding semantic tags to the web resource. A tagger can be an individual user or a community.

Semantic Tags. According to mc sharaefel et al. [START_REF] Schraefel | Semantic web meets web 2.0 (and vice versa): The value of the mundane for the semantic web[END_REF], a semantic tag is one where the tag itself is backed up by an RDF. Semantic tags are semantic annotations used to add a semantic description over a Web resource in an • Category Tag (CT).This semantic tag represents the belonging of a Web Resources to a Category. Tagging a resource by means of a CT defines the name of the tag as identifier of the category and the Web Resource as individuals which belongs to that category. Also, it is possible to organise categories specifying detailed levels by means of hierarchies using subsumption. A sub-category is defined as a subset of individuals of the super-category. • Property Tag (PT). This semantic tag describes a relationship between two Web Resources. In this case, the name of the property types the relationship between the two web resources, even there are not a navigational link between them. Having PT, the navigation is improved by a semantic relationship. • Attribute Tag (AT). This semantic tag describes a structural attribute for a web resource, for example the height of a mountain. It uses the label as name for the attribute and it is related with a simple data type as a number or a literal.

Semantic Support. We use this support to store semantic tags. The underlying semantic support has an impact on how Semdrops maps semantic tags to an existing semantic support data model. This is done by specifying a set of mapping rules. This semantic support makes Semdrops model compatible with other semantic web technologies. It can be any RDF repository e.g. Sesame [START_REF] Broekstra | Sesame: A generic architecture for storing and querying rdf and rdf schema[END_REF], Jena [START_REF] Mcbride | Jena: A semantic web toolkit[END_REF] or even Semantic Mediawiki [START_REF] Krötzsch | Semantic wikipedia[END_REF].

B. Using Semantic Mediawiki as Semantic Support

In this work, we particularly use Semantic MediaWiki (SMW) as a semantic support. We have chosen SMW because on one hand we have already used semantic wikis for knowledge personalisation [START_REF] Torres | Supporting personal semantic annotations in p2p? semantic wikis[END_REF] and on the other hand this allows using the powerful features of SMW e.g. query semantic tags and export semantic data.

Mapping Rules: We define a set of rules to transform Semdrops's semantic tags into suitable SMW semantic annotations in order to store them in the semantic support of SMW. In order to carry out this mapping, a web resource is mapped to a wiki article in terms of SMW.

• Category Semantic Tag. Let X be the URL identifying the Web Resource, and CatName the label for a category.

CategoryT ag(X, CatN ame) ≡ SemW iki(X) ← ([[Category : CatN ame]]).

Where SemW iki(page) ← (code) means that the wiki page titled page has the string code included in its content. • Property Semantic Tag. Let X, Y be the URLs identifying two different web resources and let PropName be the label of a property.

P ropertyT ag(X, P ropN ame, OW ebContent) ≡ SemW iki(X) ← ([[P ropN ame :: OW ebContent]])

• Attribute Semantic Tag. Let X be the URL identifying a web resource and let AtrName be the label of an attribute and AValue can be a literal or numerical value.

AttributeT ag(X, AtrN ame, AV alue) ≡ SemW iki(X) ← ([[AtrN ame := AV alue)]]

The table I summarizes the mapping rules. The left hand is the Semdrops semantic tags, in the middle the equivalent SMW and the right the equivalent in owl-like.

C. How to Use the Model ?

In this example, we consider a user who is reading the Paris city page 4 on Wikipedia. She would like to tag that Paris is a City and also it is the location of the movie Amelie.

Following the workflow in figure 1, first she has to decide which kind of semantic tags want to use. She uses a Category Tag to describe Paris as a City. For that, the literal City is used as the name of the Category Tag. Next, she decides to use Property Tag to describe the relationship between Paris and the film Amelie. A Property Tag requires a label to type the relationship and also a URI for two involved resources: the movie and the city. Therefore the string was movie location of types the relationships and the Amelie Wikipedia's page represents the movie. Figure 2 (a) shows the initial wikipedia page and the figure 2 (b) shows the tagged page. The left side of the screen shows the semantic tags for the current page. The semantic tags tree has three main labels: Category, Links and Attributes. The user can visualize the semantic tags grouped by these three types and can fold and unfold the list of each group of tags.

During the semantic tagging activity, Semdrops maps the semantic tags into the semantic support model of SMW by using the mapping rules detailed in the previous section. All the new meta-data is generated without modifying the original resource.

In this scenario, a new semantic relation is created between Paris city and the Amelie movie, there was no prior relationship between these two resources even a navigational link.

1) Semdrops as Mashups Tool: The previous scenario shows how semantic tags can be added easily to a web resource. For instance, the semantic tags describing the web page of Paris in Wikipedia could be associated with those defined in the Paris wiki page at SemanticWeb.org 5 . In order to combine different sources, Semdrops includes the ability to access linked semantic data in cross semantic repositories. It takes special attention in owl:sameAs tags in the RDF definition of the web resource. For example, if in the RDF description of a web resource appears the tag owl:sameAs:: targetResource, the semantic tags of the targetResource will be added as semantic tags of the original resource.

Figure 2 (b) shows the use of Semdrops with the semantic tags of the example in the section II-C with a cross reference to the Paris page on SemanticWeb.org. In this example, the semantic tags with the green icon are from the cross reference, and the category City has a special icon because it was produced in both semantic repositories: SMW repository and the SemanticWeb.org repository.

III. IMPLEMENTATION

Semdrops uses a Model-View-Controller architecture as it is detailed in the figure 3. It includes a semantic MediaWiki as a semantic support for storing and retrieving the semantic tags.

Semdrops is implemented as a Firefox sidebar add-on, it is available as an open source and can be downloaded from http://sourceforge.net/projects/semdrops/. It implements the semantic social tagging model. The semantic tags tree has three main labels: Category, Links and Attributes. The user can visualize the semantic tags grouped by these three types and can fold and unfold the list of each group of tags as shown in the figure 2. Semdrops implements drag & drop functionality, this improves user interaction and facilitates semantic tags generation. For instance, in the example of figure 2, a user can drag the France link and drop it on the Links label. This action opens the new link dialog box

IV. EVALUATION STUDY

To demonstrate the capability of Semdrops to produce adequate semantic data, we conducted an evaluation study. In this study, we have asked a group of participants to use Semdrops to semantically tags a set of predefined web resources; more precisely, a set of Wikipedia pages. We have conducted also an usability study to measure the satisfaction of participants regarding Semdrops.

In the following sections, we detail the evaluation methodology, the qualitative and quantitative analysis and the usability study.

A. Evaluation Methodology

We have asked 20 people to participate to the evaluation, 15 males and 5 females. They are between 23 and 35 years old and fans of the TV Lost serie 6 , most of them are in the domain of computer sciences. In order to familiarize the participants with Semdrops, we have provided them a textual user guide and a screen-cast showing a using example 7 .

Every participant was able to use Semdrops to add semantic tags to two specific web resources: the Wikipedia pages Jhon Locke (Lost) 8 and Richard Alpert (Lost) 9 , at any time and during four days. The participants were in different places and did not communicate with each other during the evaluation. Additionally, they only know that other people were involved in evaluation but they do not know who they were because we do not want the participants to communicate outside the system.

The participants were asked to read and analyse the wiki pages and after that add the semantic tags if they consider appropriate to semantically describe these wiki pages. They could also either delete or revise the semantic tags which they consider inappropriate. English language was used to name the tags in order to compare the results later with DBpedia.

B. Evaluation Results

1) Quantitative Analysis -Number of coincident tags with DBPedia:

We manually compare Semdrops semantic tags with semantic information coming from the DBpedia database for the same web resources. The figure 4 represents a comparison between the set of the semantic tags added by Semdrops and that semantic information coming from the DBpedia database. Numbers in the figure represents the amount of semantic annotations. To make the manual comparison, we preprocessed the semantic information from For both considered Wikipedia pages (Jhon Locke and Richard Alpert ), we observed that:

• In general, the number of semantic tags added by the social effort using Semdrops was the double of the number of semantic tags that the automatically generated by DBpedia project. • The most significant augmentation is the number of Property Tags. It was three times more in both Wikipedia pages. 2) Qualitative analysis: This analysis is in charge of finding out whether the participants have used the Semdrops conceptual model properly and made the right conceptualisation. This reveals the right understanding of the meaning of the category, property and attributes primitives by the participants. To do this, we manually inspected the Semdrops log file and observed that:

• All categories were well defined. This means that participants did not have any problem with the use of categories. They use the category tag according to the definition given in the section II. • Sometimes the participants confuse the use of property and attribute tag. For example, some participants use birthday with value on May 30, 1956 as a Property tag instead of an Attribute tag.

Proportion of Semantic Tags

• There were other cases where the right conceptualization might be controversial due to the information given in the wiki page was not enough (either because the abstraction level was not evident or the related concept was difficult to be expressed in a Wiki). Consequently, some attributes were not clear to be either an attribute or a property tag, we considered these cases also as not well defined. To summarize, the rate of wrong property tags is near to 15 %, while it increases to 30% in the case of attributes as shown in the figure 5.

3) Added Value Analysis: By analysing the kind of contribution of participants, we observed that:

• Most participants were worry about adding meaning to the navigational links, 67 % of the property tags were defined over existing navigational links. For example, the navigational link between the John Locke and the Anthony Cooper Wikipedias pages was typed with son of property, representing that John Locke is the son of Anthony Cooper. • In addition, new relationships among other resources were added. The 33 % of the added property tags establish new links with other Wikipedia pages. This result is illustrated in the figure 6.

It is important to notice that the enrichment of the existing content, in terms of both navigability and meaning, was possible to achieve by human interactions. • At the first glance, the number of Categories defined with Semdrops seems to be higher of those defined in the initial Wikipedia pages. However, we later discovered that the most Semdrops categories already exist in Wikipedia. This was due to in Wikipedia the categories are organized in a hierarchy structure and only the most specific ones are shown in the wiki pages. • Nearly 90 % of Categories defined with Semdrops were related each other and could be organized in a hierarchy. Although, the subsumption property is part of the proposed semantic tag model, the current early version of Semdrops prototype does not support it. However, this result encourage us to extend Semdrops implementation to incorporate it. To summarize, DBPedia has a good description of Category hierarchies and structural information of the page from Wikipedias infoboxes. On the other hand, Semdrops is better to make emerge semantic data that is not easy to obtain automatically. Both kind of semantic data are highly complementary.

C. Usability Evaluation

At the end of the evaluation, we made a set of questions to measure the satisfaction of the participants of the system. We use the System Usability Scale (SUS) of Brooke [START_REF] Brooke | SUS: A quick and dirty usability scale[END_REF]. SUS is a Likert scale simple questionnaire giving a subjective global value of usability. The obtained values indicate that the average of SUS usability was 71.5 over 100 with a standard deviation of 8.6. This results show that the participants had acceptable satisfaction of the system [START_REF] Bangor | An Empirical Evaluation of the System Usability Scale[END_REF].

In addition, we asked users the level of comfort for making category, property and attribute tags. Most of users had felt comfortable with the creation of categories and properties but less comfortable with the use of attributes.

V. DISCUSSION AND LEARNED LESSONS The conducted evaluation shows that semantic data obtained by using Semdrops are complementary with those of DBpedia. DBpedia has semantic data related to the structural content of the wikipages i.e. most properties reached in Wikipedia are related with navigational structure, for example is dbprop:redirect of or is dbprop:disambiguates of or infoboxes. While, Semdrops promotes semantic data proliferation related to the nature of the wiki article based on users background.

In addition, during the evaluation, we notice that users created properties following three main patterns :

• semantic enrichment of existing navigational links by adding semantic types. • new properties were defined by using information in wikipage that were not related to a navigational link. • emergence of new properties that relate pages that do not have any reference in the content of the wikipage. Other interesting point was the generation of attributes. Attributes tags had the highest rate of mistakes. In most cases this happened because for the tagger was not clear the difference between an attribute and a property tag.

Finally, by the manual examination of the log file, we notice that some semantic tags were rectified by users. For example, the spelling of the word bald to describe Jhon Locke character was rectified twice by different users each time. There is a community regulation similar to that present in wiki community practices. In this evaluation, we made manual analysis of the log file, an automatic one could help to consolidate the generated information.

VI. RELATED WORK

In the last years, many efforts focus on the semantification of the web. One of the most relevant is the Linking Open Data (LOD) 10 initiative. The objective of LOD is to connect 10 http://linkeddata.org related data that was not previously connected using URI and RDF. LOD generates a large amount of RDF statements, most of them are generated automatically, only less than 1 % of semantic contents are a user-generated [19]. DBPedia [START_REF] Bizer | Dbpedia -a crystallization point for the web of data[END_REF] is a major application of LOD. DBpedia is the Linked Data version of Wikipedia, it is based on an automatic approach to generate RDF statements, by converting structured information in Wikipedia pages into RDF statements. In Semdrops approach, the semantic data emerge by the social effort.

Other approaches to increase the semantic data on the web are based on the enrichment of social tags with semantic information:

• Annotea [START_REF] Kahan | Annotea: an open rdf infrastructure for shared web annotations[END_REF] defines a simple ontology to generate semantic bookmarks.

• In [START_REF] Mukherjee | Semantic bookmarking for non-visual web access[END_REF], semantic bookmarks are generated automatically to structure hypermedia documents. This approach is limited to specific web resources i.e. the bookmarks.

In Semdrops, semantic tags are not restricted to semantic bookmarks; they can be added to any web resource. • SOBOLEO 11 [START_REF] Zacharias | Soboleo -social bookmarking and lighweight engineering of ontologies[END_REF] uses SKOS taxonomy to annotate bookmarks. Although, SOBOLEO allows assigning tags to web resources and organize them into hierarchies, it does not support a rich semantic model. • In Limpenss Ph.D. thesis [START_REF] Limpens | Multi-points of view enrichment of folksonomies[END_REF] and Monin et al. [START_REF] Monnin | Speech acts meet tagging: Nicetag ontology[END_REF] is introduced the niceTag ontology which subclassifies the isRelatedTo tag into a set of sub-properties to conceptualize the different possible uses of tags presented by Golder & Huberman. Instead, Semdrops defines an open structure through which new properties definition can be added by the users. • Huynh-Kim-Ban et al. [START_REF] Huynh-Kim | Merging semantic and participative approaches for organising teachers' documents[END_REF] introduced an extension of the tag model presented in del.icio.us which includes both synonymy and inclusion relationships among tags, but they do not include the possibility to express, for example, attributes as Semdrops does. OntoGame [START_REF] Siorpaes | Ontogame: Weaving the semantic web by online games[END_REF] combines Web 2.0 and semantic definition to determine correctly class-instance correspondence and super class or instance-of relations for Wikipedia articles. This work was very useful to define the building block of Semdrops semantic social tagging model, however, the last one defines a more rich semantic model. Some other approaches propose semantical enrichment of social tagging [START_REF] Kim | The State of the Art in Tag Ontologies: A Semantic Model for Tagging and Folksonomies[END_REF], [START_REF] Gruber | Ontology of folksonomy: A mash-up of apples and oranges[END_REF] by defining a tag ontology for this purpose. For instance, TagOntology [START_REF] Newman | Tag ontology[END_REF] proposed a tripartite system (User, Resource, Tag); MOAT [START_REF] Passant | Meaning Of A Tag: A collaborative approach to bridge the gap between tagging and Linked Data[END_REF] allows the use of a URI to define the meaning of a tag, and SCOT [START_REF] Kim | Social semantic cloud of tag: Semantic model for social tagging[END_REF] enforces the social context involved in folksonomies. All of these approaches focus on giving semantic to the tags. These model can be reused to give semantic to the Semdrops tags.

Other approach proposes to use semantic annotations to enrich wikis systems; resulting in semantic wiki systems. Semantic wikis [START_REF] Krötzsch | Semantic wikipedia[END_REF], [START_REF] Auer | Ontowiki -a tool for social, semantic collaboration[END_REF], [START_REF] Buffa | Sweetwiki: A semantic wiki[END_REF] enable the emergence of lightweight domain ontologies by the community efforts. However, the emerged ontologies are limited to the Semantic Wiki scope and cannot be transparently extended to any Web resource. Semdrops is inspired in the semantic wikis approach but goes beyond allowing the semantic annotation of existing web content transparently.

VII. CONCLUSIONS AND FURTHER WORK

This paper introduced a social semantic tagging approach called Semdrops. Semdrops defines a simple and rich social semantic tagging model. This model is implemented as a Firefox add-on tool that turns the web browser into a collaborative semantic data editor in situ for any web resource. The evaluation and usability studies demonstrated that Semdrops :

• promotes semantic data proliferation thanks to its simple social semantic tagging model. • promotes the semantic data quality by community regulation.

• is applicable to any web resources thanks for using URI to identify them. • can be used by a wide community thanks to its simple social semantic tagging and its implementation as a Firefox add-on. • provides an easy interaction mode as demonstrated the usability study.

As future works, in order to consolidate results we will conduct a bigger evaluation with more resources and we will compare results with other work such as [START_REF] Meiyu | Image semantic description and automatic semantic annotation[END_REF]. Also, we will evaluate how lightweight ontologies can emerge from the semantic data. For this propose we will profit from the existence of the subsumption concept in our model. Further, we will study inconsistency and conflict problems.

For a more pragmatical point of view, Semdrops needs to be extended to make the captured semantic data more usable in the context of the semantic web. For instance, Semdrops should adopt the LOD philosophy to share semantic data, and even it could incorporates properties already existing in the semantic web technologies like the sameAs property.

Besides, we are interesting to apply Semdrops in a more general scenario where social tagging is combined with personal one. In this sense, it is needed to extend Semdrops approach to support personal tagging.

Figure 2 .Figure 3 .

 23 Figure 2. Semantic Tags for Wikipedia's Paris page: (a) On the top, the Wikipedia's Paris page without the Social Semantic Tags. (b) On the bottom, Semdrops in action for the same Wikipedia page.

Figure 4 .Figure 5 .

 45 Figure 4. Quantities of semantic Tags

Figure 6 .

 6 Figure 6. Proportion of Semantic Tags

http://www.foaf-project.org/

http://www.w3.org/2004/02/skos/

http://sioc-project.org/

http://en.wikipedia.org/wiki/Paris

SemanticWeb.org is a Semantic Mediawiki used mainly for content related with Semantic Web

http://abc.go.com/shows/lost

The screen cast is available at http://www.youtube.com/watch?v= N3Lr32iA9Vw

http://en.wikipedia.org/wiki/John Locke (Lost)

http://en.wikipedia.org/wiki/Richard Alpert (Lost)

ACKNOWLEDGMENT

This work was partially funded by: the Laccir R1210LAC007 project, and by the PAE 37279-PICT 02203 which is sponsored by the ANPCyT, Argentina.


Juliette Ben Arous
email: juliette.ben.arous@gmail.com

Jonas Binding
email: jonas.binding@espci.fr

Jean-François Léger
email: leger@biologie.ens.fr

Mariano Casado
email: casado@biologie.ens.fr

Piotr Topilko
email: piotr.topilko@ens.fr

Sylvain Gigan
email: sylvain.gigan@espci.fr

A Claude Boccara
email: claude.boccara@espci.fr

Laurent Bourdieu
email: laurent.bourdieu@ens.fr

Single myelin fiber imaging in living rodents without labeling by deep-OCM

Keywords: Myelin, full-field optical coherence tomography, endogenous contrast, optical aberration, live imaging

Myelin sheath disruption is responsible for multiple neuropathies in the central and peripheral nervous system. Myelin imaging has thus become an important diagnosis tool.

However, in vivo imaging has been limited to either low-resolution techniques unable to resolve individual fibers or to low-penetration imaging of single fibers, which can't provide quantitative information about large volumes of tissue, as required for diagnostic purposes.

Here, we perform myelin imaging without labeling and at micron-scale resolution with more than 300μm penetration depth on living rodents. This was achieved with a prototype (termed 'deep-OCM') of a high-numerical aperture infrared full-field optical coherence microscope, which includes aberration correction for the compensation of refractive index mismatch and high framerate interferometric measurements. We were able to measure the density of individual myelinated fibers in the rat cortex over a large volume of gray matter. In the peripheral nervous system, deep-OCM allows, after minor surgery, in-situ imaging of single myelinated fibers over a large fraction of the sciatic nerve. This allows quantitative comparison of normal and Krox20 mutant mice, in which myelination in the peripheral nervous system is impaired. This opens promising perspectives for myelin chronic imaging in demyelinating diseases and for minimally invasive medical diagnosis.

1) INTRODUCTION

The myelin sheath is an essential component of the vertebrate nervous system composed of glial membrane layers tightly wrapped around axonal fibers. Its high lipid content allows electrical insulation of the axonal membrane and enables saltatory conduction over long distances ensuring rapid, reliable and energetically efficient conduction of action potentials. Disruption of the myelin sheath is involved in central nervous system pathologies such as multiple sclerosis and leukodystrophies and peripheral nervous system (PNS) neuropathies such as Charcot-Marie-Tooth disease [START_REF] Scherer | Molecular mechanisms of inherited demyelinating neuropathies[END_REF][START_REF] Baumann | Biology of oligodendrocyte and myelin in the mammalian central nervous system[END_REF].

Quantitative myelin imaging is a necessary tool to follow demyelination or to evaluate eventual remyelinating treatments. Myelin in the white matter can be visualized in vitro and in vivo with magnetic resonance imaging, positron emission tomography and near-infrared imaging with a 0.1 to 1 millimeter resolution [START_REF] Laule | Magnetic resonance imaging of myelin[END_REF][START_REF] Mori | Principles of diffusion tensor imaging and its applications to basic neuroscience research[END_REF][START_REF] Stankoff | Imaging of CNS myelin by positron-emission tomography[END_REF][START_REF] Wang | Longitudinal near-infrared imaging of myelination[END_REF]. This resolution is insufficient to detect single myelinated axons, thus preventing the study of demyelination in gray matter, where myelinated fibers are sparsely organized. However demyelinating lesions in the gray matter are known to be an important hallmark of pathologies like multiple sclerosis [START_REF] Geurts | Consensus recommendations for MS cortical lesion scoring using double inversion recovery MRI[END_REF].

Different approaches have been recently designed to overcome this limitation.

Coherent Anti-stokes Raman Scattering microscopy allows the visualization of single fibers without exogenous labeling [START_REF] Fu | Ex vivo and in vivo imaging of myelin fibers in mouse brain by coherent anti-Stokes Raman scattering microscopy[END_REF][START_REF] Imitola | Multimodal coherent anti-Stokes Raman scattering microscopy reveals microglia-associated myelin and axonal dysfunction in multiple sclerosis-like lesions in mice[END_REF]. This technique has a penetration depth of only a few tens of micrometers and cannot provide quantitative information on large volumes of nervous tissue.

Third Harmonic Generation provides signals from myelin [START_REF] Farrar | In Vivo Imaging of Myelin in the Vertebrate Central Nervous System Using Third Harmonic Generation Microscopy[END_REF] with an improved penetration but is still a costly and complicated technique and its use for in vivo recording in the cortex

has not yet been demonstrated. A non invasive method allowing imaging of myelin with individual fiber resolution is therefore still lacking.

Optical coherence tomography (OCT) is a simple yet powerful technique based on interference that allows non-invasive imaging of biological samples and provides optical sectioning deep into scattering tissues [START_REF] Fujimoto | Optical coherence tomography for ultrahigh resolution in vivo imaging[END_REF][START_REF] Fujimoto | Optical biopsy and imaging using optical coherence tomography[END_REF]. Full-field OCT (ff-OCT) is a high spatial resolution version of OCT, the only providing micron resolution in 3D [START_REF] Vabre | Thermal-light full-field optical coherence tomography[END_REF]. OCT uses the backscattered light due to refractive index (n) variations in the sample as a source of endogenous contrast. The lipid-rich myelin sheath has a high refractive index (n~1.44) and lies in an aqueous background (n~1.33). Here we show that myelin produces a strong endogenous backscattering signal. To take advantage of this signal, we have designed a prototype of a high-numerical aperture infrared ff-OCT microscope with automatic correction of defocus aberration [START_REF] Labiau | Defocus test and defocus correction in full-field optical coherence tomography[END_REF][START_REF] Binding | Brain refractive index measured in vivo with high-NA defocus-corrected full-field OCT and consequences on two-photon microscopy[END_REF]. This prototype, designated as deep-OCM, allows high resolution myelin imaging deep in tissue without staining. We present two applications of myelin imaging in large volumes of tissue with single fiber resolution: the myelination of individual fibers during development in the rat cortex and the comparison of sciatic nerve structure in wild type and mutant mice in which myelination in the PNS is impaired.

2) MATERIALS AND METHODS

2.1) Animals.

All animal manipulation and surgical procedures were in accordance with the European Community guidelines on the care and use of animals (86/609/CEE, CE official journal L358, 18th December 1986), French legislation (décret no. 97/748, 19th October 1987, J. O. République française, 20th October 1987), and the recommendations of the CNRS. Male Wistar rats, female Swiss CNP-EGFP mice (16) (CNP: cyclic nucleotide phosphodiesterase, 4 months old), C57BL/6/DBA2 wild type and Krox20-/mice were used in this study [START_REF] Topilko | Krox-20 controls myelination in the peripheral nervous system[END_REF][START_REF] Coulpier | CNS/PNS boundary transgression by central glia in the absence of Schwann cells or Krox20/Egr2 function[END_REF].

2.2) Slice preparation.

For in vivo fixation, animals were deeply anesthetized with urethane and perfused transcardially with cold phosphate buffer solution (PBS) followed by 4% paraformaldehyde in PBS solution. In the somato-sensory cortex of rats (P19 and P85) and CNP-EGFP mice, 100μm thick slices were cut tangentially to the flattened cortical surface. In the cerebellum of a P19 rat, 300μm thick sagittal slices were cut. All slices were stored in PBS before imaging.

For myelin immuno-fluorescence labeling, after bleaching (10% H 2 O 2 , 13.3% DMSO, 53.3% methanol), cortical slices were incubated with rat monoclonal anti-MBP (Myelin Basic Protein) antibodies (1:100, Millipore Bioscience research) followed by goat anti rat Cy3 secondary antibodies (1:500, Jackson Immuno Research). All fluorescence images were acquired using a Leica SP2 confocal microscope with a 10x objective.

2.3) Sciatic nerve preparation.

For in vitro imaging, the sciatic nerves of two mice (wild type and Krox20-/mutant, P14) were extracted and laid on glass slides in PBS for immediate imaging. The nerves were not held by a coverslip to prevent them from being crushed. For in vivo imaging, a wild type mouse (2 months old) was anesthetized with ketamine (100mg/kg) and xylazine (10mg/kg). A small incision was performed in the left hind limb and the sciatic nerve was exposed. A plastic holder was slipped under the nerve to maintain it and prevent movements. Imaging was performed in PBS without coverslip.

2.4) Surgical procedure for in vivo cortex imaging.

Three 3 week old (P20-21; weight ~80g), three 6 week old (P45-46; weight ~250g) and two 12 week old (P86-87; weight ~400g) male Wistar rats were used for in vivo somatosensory cortex imaging. Rats were anesthetized by urethane injection (1.5 g/kg). Supplementary doses of urethane were applied when necessary. The body temperature of the animal was held at 38°C on a heating blanket during the whole experiment. Animals were maintained in a stereotaxic frame during the surgical procedure. A craniotomy (~3mm in diameter) centered on the somato-sensory cortex was performed on the rat's left hemisphere.

In some cases, the dura mater was removed, as indicated in the figure legends. The craniotomy was sealed with a cover slip with 5mm diameter which was glued onto the thinned skull to be in close contact with the brain, in order to prevent brain movements during the imaging procedure. For ff-OCT imaging, the animal was maintained with a metal holder glued on its skull to ensure optical stability. At the end of the experiments, rats were injected with a lethal dose of urethane.

2.5) Deep-OCM set-up.

The microscope (Fig. 1(a)) consists of a low spatial and temporal coherence light source (250 W halogen lamp) with the standard Köhler microscope illumination, a Linnik interferometer (i.e. a Michelson interferometer with an objective in each arm), a tube lens and a camera. The water immersion objectives used were either Zeiss IR-Achroplan 40x/0.8 W or Olympus UMPlanF 10x/0.3 W. For dispersion correction, a coverslip identical to the one used to cover the sample was placed in the reference arm. To avoid the back-reflection from the coverslip on the camera, the former was slightly tilted. A 50 cm doublet was used as a tube lens to image the sample and the reference arm mirror onto the camera. Two different cameras were used: an infrared InGaAs camera (Xeva-1.7-320, Xenics Infrared Solutions, Leuven, Belgium) and a silicon camera (Pantera 1M60, Dalsa, Waterloo, Canada). Myelin fibers were detected with both sensors. Deep-OCM images taken with the InGaAs camera are noted as "IR" in the text, whereas images from the silicon camera are described as "deep-red".

The InGaAs sensor chip had 320x256 pixels (pixel size: 30μm) and the silicon chip had 1024x1024 pixels (pixel size: 12 μm). They were operated at 66Hz and 57Hz respectively, triggered by a NI 6722 digital I/O board (National Instruments, Austin, Texas) controlled by the software Light-CT (LLTech, Paris). A synchronized rectangular 33 or 28.5 Hz signal drove the Piezo Linear Stage Actuator (PI P-753.11C, Physik Instrumente GmbH, Karlsruhe) onto which the reference arm mirror was mounted. The piezo actuator amplitude was adapted to produce a change in reference arm length of Ȝ/2, so that the relative phase between both arms of the interferometer would change by S. Taking the difference of subsequent camera images would therefore separate the interference signal from the background light, implementing a two-step phase stepping protocol. The software light-CT (LLTech Inc.) running on a standard PC allowed real-time display of the infrared images at 33 Hz, while the higher pixel number of the silicon camera limited display speed to about 15 Hz. Both the sample z position and the relative arm length could be changed with two motorized linear stages (M-VP-25XA, Newport).

With this setup the axial resolution (sectioning ability) is mainly due to the coherence length of the source whereas the lateral resolution is due to the numerical aperture of the microscope objectives. In the lowest resolution configuration, with 10x objectives and the InGaAs camera (peak of the effective spectrum: Ȝ max =1100nm), the lateral sampling of the system was 1.07μm, the lateral resolution 2.2μm (calculated using the Rayleigh formula 0.6*O max /NA, where NA is the objective numerical aperture), and the axial resolution 3.3μm (measured as the full width at half maximum of the interferogram). In the highest resolution configuration, with 40x objectives and the silicon camera (Ȝ max =770nm), the lateral sampling of the system was 0.1μm, the lateral resolution 0.5μm and the axial resolution 1.15μm.

2.6) Automatic compensation of defocus aberration due to refractive index mismatch

Refractive index mismatch between the immersion liquid in the sample arm and the sample causes depth-dependent defocus in ff-OCT [START_REF] Labiau | Defocus test and defocus correction in full-field optical coherence tomography[END_REF][START_REF] Binding | Brain refractive index measured in vivo with high-NA defocus-corrected full-field OCT and consequences on two-photon microscopy[END_REF]. Refractive index mismatch compensation was performed on-line as described previously [START_REF] Labiau | Defocus test and defocus correction in full-field optical coherence tomography[END_REF][START_REF] Binding | Brain refractive index measured in vivo with high-NA defocus-corrected full-field OCT and consequences on two-photon microscopy[END_REF]. Briefly, compensation is obtained by keeping the coherence volume at constant depth inside the sample and moving the actual focus into this same position (Fig. 1(b)). Due to the aqueous immersion (group index n g ), a movement l of the sample objective away from the sample increases the (paraxial) optical path in water by 2.n g .l, while the optical path in air only decreases by 2.l. To keep the coherence volume at constant depth in the sample, a displacement l of the sample objective should be therefore associated with a displacement

) 1 ( ) ( g n d " "
of the reference arm. The metric used to determine the optimal positions l and d(l) was the total image intensity (15).

2.7) Stitching.

For images with a large field of view, a prototype of a ff-OCT microscope from the

2.8) Quantification

For myelin quantification, vertical stacks of images were acquired in confocal immuno-fluorescence and in deep-OCM (both in vitro and in vivo). The stacks had a total depth of 60 to 90 μm and were acquired either just below the brain surface (in vivo) or in the first tangential slice from the brain surface (in vitro). The total length of myelin filaments per unitary surface was computed as follows. The length of myelin fibers contained within the depth of field in each independent frame was first measured manually using ImageJ. The average and standard error of the mean of fiber length density per unitary surface was then calculated for 23 to 37 frames chosen in at least 4 different stacks in each imaging condition.

Images from stacks obtained in rats of a given age were pooled together. Frame sizes were Fiber diameters, lengths, and interfringe spacings were determined manually using the ImageJ Plot Profile tool. Fiber inclination A was calculated from the fibers' apparent length L via tan(A)=L/l z , where l z is defined as the full width of the interferogram measured at 10% of the maximum (l z =2.3 μm for 40x deep-red deep-OCM).

Statistical comparisons were performed by standard Student's t-tests.

3) RESULTS

3.1) Deep-OCM allows in vitro and in vivo imaging of myelinated fibers

Deep-OCM has the following unique features (Fig. 1(a) and Materials and Methods).

It allows either deep-red imaging (wavelength range 700-850 nm, measured at half the peak height) for maximum lateral resolution or near-infrared imaging (wavelength range 1000-1170 nm) to increase further the depth penetration due to decreased scattering at longer wavelengths [START_REF] Sacchet | Simultaneous dual-band ultra-high resolution full-field optical coherence tomography[END_REF]. In all cases, fast (~30Hz) image acquisition is performed to prevent motion artifacts and loss of contrast due to sample motion. The objectives used are either a 10x objective for the observation of large fields of view or a 40x objective with a numerical aperture (NA) of 0.8 to maximize lateral resolution. Large NA makes the setup highly sensitive to defocus induced by refractive index mismatch between immersion medium and tissue [START_REF] Labiau | Defocus test and defocus correction in full-field optical coherence tomography[END_REF][START_REF] Binding | Brain refractive index measured in vivo with high-NA defocus-corrected full-field OCT and consequences on two-photon microscopy[END_REF]. Refraction at the brain surface indeed causes the actual focus z A of the objective to be shifted deeper into the sample with respect to the nominal focus z N (Fig. 1(b))

whereas the coherence volume penetrates the tissue slower than the nominal focus (Fig. 1(b)) [START_REF] Binding | Brain refractive index measured in vivo with high-NA defocus-corrected full-field OCT and consequences on two-photon microscopy[END_REF]. This spatial separation of coherence volume and actual focus results in a loss in OCT signal: even the small refractive index mismatch between immersion water and brain tissue of around 2% is sufficient for depth penetration not to be limited by scattering-induced exponential signal decrease, as expected from Beer-Lambert law, but by signal loss due to defocus [START_REF] Binding | Brain refractive index measured in vivo with high-NA defocus-corrected full-field OCT and consequences on two-photon microscopy[END_REF]. We therefore systematically applied automated defocus compensation, increasing the penetration depth by typically a factor of 2.5 [START_REF] Binding | Brain refractive index measured in vivo with high-NA defocus-corrected full-field OCT and consequences on two-photon microscopy[END_REF]. In these conditions, deep-OCM provides micrometer resolution in the 3 dimensions and in the depth of the tissue.

3.2) In vitro imaging of individual myelinated fibers.

Using deep-OCM, we were able to observe fibers in the rodent brain tissue. They show up with very high endogenous contrast due to backscattering of the incident light (Fig.

1(b) and 1(c)). Compensation of index-mismatch-induced defocus was mandatory to observe

the fibers with high contrast and resolution (Fig. 1(b)). In 40x deep-red images of rat cortical slices (Fig. 1(c)), the observed diameter of the fibers was 0.6 ± 0.1 μm and was therefore at the diffraction limit (microscope lateral resolution: 0.56 μm), in good agreement with the diameter of individual myelinated fibers in the rat brain [START_REF] Partadiredja | The number, size, and type of axons in rat subcortical white matter on left and right sides: a stereological, ultrastructural study[END_REF]. Note that these fibers appear discontinuous in the images, because of the presence of interference fringes on their reflective surface. The fringe spacing along the fibers allows the quantitative determination of the fiber inclination (Fig. 2).

To confirm that the observed fibers were myelinated axons, we compared deep-OCM and confocal fluorescence images of fiber bundles in the same slice of the corpus callosum of a mouse expressing the enhanced green fluorescent protein (EGFP) under the control of the 2ƍ-3ƍ-cyclic nucleotide 3ƍ-phosphodiesterase (CNP) promoter which causes specific labeling of oligodendrocytes. We observed the same myelin bundles in both images, whereas the cellular bodies of oligodendrocytes were only visible in the fluorescence image (Fig. 1(d)).

In order to quantify the proportion of myelinated fibers detected by deep-OCM imaging, 40x deep-red images of a fixed rat horizontal cortical slice (Fig. 3 Individual myelinated fibers could be readily observed in the cerebellar white matter (Fig.

4(d))

, beyond the depth sectioning ability of standard brightfield microscopy.

Myelin imaging was also performed in the Peripheral Nervous System. Deep-OCM 3D reconstruction was used to obtain the structure of an acutely dissected mouse sciatic nerve in vitro, in which individual myelinated fibers could be discriminated (Fig. 5(a), Movie 1).

Deep-OCM imaging allows optical reconstruction of a large fraction (about one half) of the whole nerve (Fig. 5(a)) with micrometer resolution. In order to check our ability to evaluate pathological demyelinating situations, we imaged the sciatic nerves of Krox20 mutant mice.

Krox20 is a transcription factor whose inactivation in Schwann cells results in impaired myelination of the PNS [START_REF] Topilko | Krox-20 controls myelination in the peripheral nervous system[END_REF][START_REF] Coulpier | CNS/PNS boundary transgression by central glia in the absence of Schwann cells or Krox20/Egr2 function[END_REF]. We observed a strong difference between nerves of Krox20 mutant and wild type mice (Fig. 5(b)). Indeed, individual fibers could not be discriminated in mutant animals. Moreover, in identical imaging conditions, the reflected light amplitude was ~40% stronger in wild type nerves than in mutant nerves. Thus, deep-OCM can be used to evaluate pathological myelin alteration in peripheral nerves.

3.3) In vivo imaging of individual myelinated fibers.

We tested whether sciatic nerve deep-OCM imaging could be performed in vivo in an anesthetized wild type mouse, using minimal surgery to expose the nerve and stabilize it (Fig.

6(a)

). This was necessary to prevent respiratory or heart beat movements. In vitro and in vivo images were of comparable quality (Fig. 6(b) vs. Fig. 5(a)).

Finally, we performed in vivo deep-OCM myelin imaging in the somato-sensory cortex of anesthetized rats. The animals were maintained in the imaging system by a tight head fixation to prevent movement during image acquisition (Fig. 7(a)). The use of IR imaging and of refractive index-mismatch correction are particularly important in deep in vivo imaging where they improve considerably the contrast and resolution of images. Two configurations were tested. In the first situation, the dura was removed, which potentially increases brain motion artifacts. In the second situation, the dura was preserved, despite its large backscattering. Detection of myelin fibers in the rat cortex was possible in both conditions (Fig. 7(b) and Fig. 7(c)).

Using deep-OCM, we followed the developmental profile of myelinated fibers from 3 to 12 weeks (Fig. 7(c) and Fig. 7(d)). Myelinated axons were observed in vivo up to 350μm deep into the somato-sensory cortex of 6 and 12 week old animals (Fig. 7(c), middle and right), whereas no fibers were observed in the cortex of 3 week old animals (Fig. 7(c), left).

The density of fibers in the first 150μm below the brain surface was consistent with the one measured in vitro (Fig. 7(d) and Fig. 3). It was similarly high for 6 and 12 week old rats (Student t-test, P=0.42, Fig. 7(d)), and was reduced in both cases at levels deeper than 150μm, in good agreement with previous histological studies ( [START_REF] Mikula | Internet-enabled high-resolution brain mapping and virtual microscopy[END_REF][START_REF] Trotts | Interactive visualization of multiresolution image stacks in 3D[END_REF] and http://brainmaps.org).

4) DISCUSSION

In this study we have shown that deep-OCM allows quantitative imaging of individual myelinated fibers in vitro and in vivo in the rodent nervous system, with micron-size resolution, without labeling and over large volumes of tissue, up to 350μm deep into the tissue. Deep-OCM combines several technical advances compared to standard ff-OCT microscopy, such as fast acquisition rate, IR imaging, large numerical aperture microscope objectives and real time defocus correction [START_REF] Labiau | Defocus test and defocus correction in full-field optical coherence tomography[END_REF][START_REF] Binding | Brain refractive index measured in vivo with high-NA defocus-corrected full-field OCT and consequences on two-photon microscopy[END_REF][START_REF] Sacchet | Simultaneous dual-band ultra-high resolution full-field optical coherence tomography[END_REF]. Deep-OCM imaging of myelinated fibers benefits from their large backscattered signal. This may arise from the high local refractive index of fibers due to their high lipid content. However, whereas imaging at moderate depth was possible in a standard configuration using deep-red light, imaging at greater depth was only possible by combining infrared detection with the correction of defocus aberration due to refractive index mismatch between immersion medium and brain tissue. Without defocus correction, both signal strength and resolution quickly degrade with depth. Finally, imaging at high temporal resolution with fast cameras in addition to a tight tissue mechanical fixation was required to avoid motion artifacts due to breathing or heart beat.

In the PNS, fast image acquisition of the myelin structure of sciatic nerves at high resolution was achieved both in vitro and in vivo, and defects of the myelin sheath observed in the raw images could be easily quantified from the reflected amplitude. As the access to the peripheral nervous system is quite simple and requires only minor surgery, this method may be used, in the future, in human patients with peripheral neuropathies. Deep-OCM-based myelin imaging of the sciatic nerve could replace nerve biopsy, which is a standard procedure for diagnosing certain neuropathies. It could thereby help avoid neuropathic side effects associated to nerve biopsies.

Deep-OCM also allowed us to record images of individual myelinated fibers in vivo within the somato-sensory cortex of anesthetized rats, without exogenous labeling, and either when the dura matter was removed or directly below the intact dura mater. Quantification of myelinated fiber density was obtained from the image analysis. Similar axonal densities have been measured in the superficial layers of the rat cortex [START_REF] Eayrs | Postnatal development of the cerebral cortex in the rat[END_REF]. By this approach, we could observe the developmental appearance of myelin structures in the adult rat brain. Compared to other techniques that work in this resolution and penetration depth regime, such as twophoton microscopy of genetically labeled oligodendrocytes, deep-OCM has the advantage that it images directly the local increase of refractive index caused by the high lipid content of the myelin sheath. This should allow extensive and even chronic imaging without the risk of signal deterioration due to bleaching, as would be the case in two-photon microscopy. Such chronic imaging of individual myelinated fibers in the upper layers of the rat cortex, which

  120 x 120 μm (immuno-fluorescence confocal), 105 x 105 μm (in vitro 40X deep-red deep-OCM) and 79 x 63 μm (in vivo 40X IR deep-OCM). The volume density of fiber length was obtained from the surface density by taking into account the thickness of each optical section (5.6μm in confocal microscopy, 1.15μm in in vitro deep-OCM, 3.3μm in in vivo deep-OCM).

Fig. 4 (

 4 Fig.4(a)). The stereotyped organization of the cerebellar cortex allows unequivocal

Figure 1 .

 1 Figure 1. Deep-OCM myelin imaging. (a) Deep-OCM setup incorporating IR and deep-red

Figure 2 .

 2 Figure 2. Interferometric measurement of the inclination of myelinated fibers. (a-b) In

Figure 3 .

 3 Figure 3. In vitro deep-OCM imaging and quantification in the rat somato-sensory

  are not visible in whole-brain imaging techniques such as MRI due to the limited resolution, could be of considerable interest in neuropathies progression studies. Our observations open perspectives of application to chronic experiments in rodents, notably for the longitudinal study of cortical demyelination and remyelination. To our knowledge, no other existing technique gives access to this information.In conclusion, deep-OCM makes a fine-scale analysis of myelination processes possible, both in the cerebral cortex and in the peripheral nervous system. Such analysis currently requires time-consuming and invasive histological methods. Our technique is relatively non-invasive, affordable and fast, and provides real time imaging without complex off-line image treatment. It opens promising possibilities for in vivo acute or chronic myelin imaging as well as for the diagnosis of myelin pathologies.

Acknowledgements

We would like to thank Marie-Stéphane Aigrot, Bruno Stankoff and Catherine Lubetzki for their helpful comments. We thank them also for transferring the transgenic CNP-GFP mice and Vittorio Gallo for authorizing us to use them. We would also like to thank Charles Brossollet, Benjamin Mathieu, and Gaspard Gerschenfeld for their help and Shawn Mikula for critical discussions. This work was supported by ANR RIB grants MICADO ANR-07-RIB-010-02 and ANR-07-RIB-010-04. J.B. was funded by a PhD fellowship from Fondation Pierre-Gilles de Gennes.

OCM and by confocal immuno-fluorescence, for 3 week old and 12 week old animals. Error bars are the standard error of the mean.


Julien Soret

Marie-Bernadette Lepetit

Electronic structure of the Ca 3 Co 4 O 9 compound from ab initio local interactions

We used fully correlated ab initio calculations to determine the effective parameters of Hubbard and t -J models for the thermoelectric misfit compound Ca3Co4O9. As for the NaxCoO2 family the Fermi level orbitals are the a1g orbitals of the cobalt atoms ; the e ′ g being always lower in energy by more than 240 meV. The electron correlation is found very large U/t ∼ 26 as well as the parameters fluctuations as a function of the structural modulation. The main consequences are a partial a1g electrons localization and a fluctuation of the in-plane magnetic exchange from AFM to FM. The behavior of the Seebeck coefficient as a function of temperature is discussed in view of the ab initio results, as well as the 496 K phase transition.

I. INTRODUCTION

First studied for their ionic conduction properties 1 , the layered cobalt oxides have been, in the last years, the object of a regained attention, due to the discovery of peculiar transport and magnetic properties. One can cite, for instance, the superconducting state of the Na 0.35 CoO 2 -1.3H 2 O, 2 the large thermoelectric power (TEP) in the Na 0.7 CoO 2 , 3 or the rich phases diagram of the Na x CoO 2 family. 4 In terms of applications the large thermoelectric power (TEP) found in some materials of this family is certainly the most interesting property. Indeed, in addition to Seebeck coefficients larger than 100µV/K at room temperature, these systems present the atypical association of a low resistivity and a low thermal conductivity. It results a large figure of merit ZT = S 2 T /ρκ at room temperature 5,6 (where Z is the factor of merit, S the Seebeck coefficient, T the temperature, ρ the electrical resistivity, κ the thermal conductivity).

The motivation was at the origin of the search for new compounds, based on the same CoO 2 layers (known to be responsible for the electronic and magnetic properties), but with better chemical stability than the Na x CoO 2 ones. Indeed, in air, these systems can both easily accept water or other small molecules in between the CoO 2 layers and loose part of their sodium content, both processes affecting the desired properties. The main idea was to replace the alkaline layer by a more complex oxide one. Many systems were thus synthesized such as Tl α [(Sr, Ca) ) occupies a special place. Indeed, this chemically "simple" system has been used as the reference system for the study of the thermoelectric properties in CoO 2 -based compounds. In addition to its interesting thermoelectric properties the Ca 3 Co 4 O 9 present puzzling magnetic transitions 11 with the inset of an incommensurate spin density wave (IC-SDW) at low temperature, an often called "ferrimagnetic" transition around 19 K and a transition inter-preted as a cobalt spin state transition around 380 K. The long-range IC-SDW sets in place at 27 K however from 100 K a short range IC-SDW order is observed by muon relaxation 11 . Indeed, the CoO 2 subsystem has been shown to be responsible for the low energy properties of the compound (transport, TEP, magnetism, etc.), since it supports the Fermi level electrons 6,10 while the CoCa 2 O 3 subsystem was proved to be gapped. In addition, in the related Na x CoO 2 family the Fermi level properties were proved to be strongly dependant both on the doping 4 and on the local structural parameters 13 .

The purpose of the present work will thus be to study the effect of the incommensurate modulations on the low energy degrees of freedom of the Ca 3 Co 4 O 9 system. At this point one should notice that, unfortunately, density functional (DFT) electronic structure calculations fail to properly describe the electronic structure of the CoO 2based systems. Indeed, LDA as well as LDA+U calculations 14 exhibit a conducting CoCa 2 O 3 subsystem dominating the density of states at the Fermi level. DFT is known to illy treat strong correlation effects, however, in many strongly correlated systems, it is nevertheless able to yield reasonable Fermi surfaces and the correct magnetic ground state. In the CoO 2 -based systems, this is not the case. In the Na x CoO 2 compounds DFT calculations yield a ferromagnetic ground state 15 while experimentally antiferromagnetic 16 and incorrect Fermi surfaces 17 . We will thus need to rely on alternating approaches, such as embedded cluster quantum chemical methods, explicitely treating the strong correlation at the Fermi level. Such methods also present the advantages to be able to treat the misfit character of the system. For this purpose we will use the same ab initio method that gave accurate and reliable results on the evaluation of such local effective parameters in other strongly correlated systems, that is the embedded fragment CAS+DDCI method 18 . This method yielded very good results in systems such as the copper or vanadium oxides 19 , the Na x CoO 2 13,20 , etc., in periodic systems as well as misfit ones 21 .

The modulated structures will be taken from reference 12 . We will compute the magnetic effective exchange between two nearest neighbors (NN) cobalt ions, the effective transfer, the correlation strength and the on-site parameters variations that could be responsible for electron localization.

II. METHOD AND COMPUTATIONAL DETAILS

The embedded fragment CAS+DDCI method is a spectroscopy method specifically designed to accurately treat the strongly correlated character of a set of orbitals (designed as active or magnetic orbitals), as well as the screening effects on all processes on this set of magnetic orbitals. Such a calculation implies large diagonalizations and can only be done on a formally finite system.

The embedded fragments are thus composed of a quantum part, treated within the CAS+DDCI quantum chemical method, and an embedding that reproduces the main effects of the rest of the crystal on the quantum part. The quantum part should include the cobalt ions involved in the sought interaction and their first coordination shell. Such a choice insures a proper treatment of the correlation effect on the cobalt 3d shell as well as the screening effects on the cobalt 3d electrons 18 .

The main effects of the rest of the crystal on the quantum part are double. First, there is the short range exclusion of the quantum electrons from the space occupied by the electrons of the rest of the crystal. This effect is modeled using total ions pseudo-potentials 22 (TIPs) that represent the electronic structure of the first shells of atoms surrounding the quantum part. The second important effect is the Madelung potential. For this purpose, we used a set of renormalized charges. As far as the two periodic directions are concerned, we used the real space method developed by A. Gellé et al. 23 that reproduces the Madelung potential with an exponential convergence. For the incommensurate direction, we used the same method as in reference 21, imposing the nullity of both charge and dipole moment.

The question is now "what formal charges should we use for the different atoms ?". The calcium and oxygen atoms will be taken as Ca 2+ and O 2-ions. The question of the cobalt valence is a bit more difficult. Indeed, there is no reason for the cobalts to present the same valency in the two subsystems. Two arguments can help us in finding a reasonable evaluation of the charge transfer between the two layers. The first argument is a simple comparison between the average first neighbor cobaltoxygen distances in the CoO 2 layers for Ca 3 Co 4 O 9 and for the Na x CoO 2 family. With an average Co-O distance of 1.904 Å the cobalt valence could be estimated from figure 5 of reference 24 to be similar to the Na 2/3 CoO 2 system, that is about Co 3.33+ in the CoO 2 layer, and thus Co 3.05+ in the rock-salt layer. The second argument comes from a recent evaluation of the cobalt valency using atomic-column resolved energy-loss spectroscopy (EELS) 25 . Indeed Klie et al found a single valency of Co 3+ in the rock-salt layers and a mixed valency in the CoO 2 layers. We thus used a Co 3+ formal charge for the rock-salt subsystem and, insuring electro-neutrality, a Co 3.38 formal charge for the CoO 2 subsystem.

In the quantum calculations, the core electrons where treated using effective core potentials 26 , while the valence and semi-valence electrons where treated using the associated Gaussian atomic basis set 26 .

III. RESULTS : THE COBALT 3d ORBITALS

In the CoO 2 subsystem of the Ca 3 Co 4 O 9 compound, the cobalt ions are in a mixed valence state Co 3+ (3d 6 ) and Co 4+ (3d 5 ). It is sometimes written in the literature 10,27 that the Co 3+ ions are in a low spin state while the Co 4+ ion are in a high spin state. This assumption is in contradiction with what was found for the Na x CoO 2 family where both cobalt ions are in a low spin state -Co 3+ : t 6 2g e 0 g and Co 4+ : t 5 2g e 0 g . Since these 3d orbitals are responsible for the physical properties, it is of prime importance to first understand the cobalt 3d orbital splitting and spin states.

We thus computed the different spin states of a CoO 6 embedded fragment (Co 3+ as well as Co 4+ ionic states) for all crystallographically independent cobalt sites in the periodic directions, and a representative set of them in the misfit direction. We found that in all cases both the Co 3+ and Co 4+ ions are in a low spin state. Indeed, even for the Co 4+ valence state the higher spin states are at least at 700 meV (∼ 8000 K) above the low spin state.

As in the Na x CoO 2 family, the CoO 6 octahedra are compressed along the c axis. One thus expects a splitting of the regular octahedron t 2g orbitals into a doublet e ′ g and a singlet a 1g . The energetic order between these orbitals is crucial since it determines the nature of the magnetic orbitals of the Co 4+ ions. A simple crystal field evaluation yields the e ′ g orbitals at the Fermi level. However, it was shown that due to the O h to S 6 symmetry reduction, the e ′ g orbitals can and do hybridize with the high energy e g ones. It results an energetic stabilization of the e ′ g orbitals compared to the a 1g one and thus a non degenerated atomic ground state for the Co 4+ ions 28 : (e ′ g ) 4 a 1g . In the present system the misfit character of the two subsystems tells us that further symmetry breaking occurs and that all cobalt 3d orbitals are non degenerated. 1 and e ′ g 3 a 1g 2 states of the CoO 6 embedded fragments. Indeed, these excitations energies can be associated with the effective orbital splitting of a 3-bands Hubbard model. One sees immediately that whatever the Co site, the a 1g orbital is always of higher energy than the e ′ g ones ; that is the hole is always located in the a 1g orbital. The excitation energy for having a hole located in a e ′ g orbital is always larger than 240 meV ≃ 2800 K. The average ε a1g -ε e ′ g splitting is 261 meV, while the difference between the two e ′ g orbitals ranges between 13 meV to 42 meV. Finally one should notice that the variations of the ε a1g -ε e ′ g splitting as a function of the different crystallographic sites remains small (< ±10 meV).

As mentioned above the relative position of the a 1g and e ′ g orbitals are related to the e ′ g -e g hybridization. As expected from the weak modulation of the orbital energy splitting, this hybridization remains stable, close for all sites to its average value of 10 degrees.

IV. RESULTS : a1g-BASED ONE BAND MODELS

The previous section clearly told us that the a 1g -e ′ g orbital splitting is large enough (2800 K) to justify a onband model for our system. In this section we will thus extract from our calculations both the effective exchange integrals and the parameters of an extended Hubbard model from our ab-initio calculations.

The NN effective exchange can be obtained as the energy difference between the singlet and triplet states of the Co 2 O 10 embedded fragments. The extended Hubbard model parameters necessitate in addition the states wave functions information. Indeed, the parameters are determined so that the model reproduce both the singlettriplet excitation energy and relative weight of the dominant configurations in the states wave functions. One should however notice that not all the parameters of the extended Hubbard model can be independently determined for the data. Indeed, only the NN hopping integral : t, the difference between the average on-site repulsion and the NN repulsion : Ū -V = (U 1 + U 2 ) /2 -V 12 and the difference between the orbital energies and the on-site repulsions : δε a1g -δU can be independently determined.

In this non-periodic system, the models parameters were determined for a representative set (sixteen dimers were computed) of incommensurate distortions values, then fitted as a function of the incommensurate dimension parameter τ .

One can define two types of non equivalent nearest neighbor dimers that propagate along the incommensurate b direction (see figure 3); the A dimers ( a -b direction) and the C dimers ( b direction). Indeed, the B and D dimers can be deduced from the preceeding ones, through the expressions:

τ A → τ B = 0.5 b CoO2 b RS -τ A (1)
τ C → τ D = -τ C (2)
In the following discussions, we will refer to " b direction dimers" for the C and D dimers and to " a ± b directions dimers" for the A and B ones.

A. Effective exchange and hopping Let us notice the following points.

J(τ ) = J 0 + J 1 cos (2π[τ + ϕ]) + J 2 cos (4π[τ + ϕ]) t(τ ) = t 0 + t 1 cos (2π[τ + ξ]) + t 2 cos (4π[τ + ξ]) (3) Dimer J0 J1 J2 ϕ t0 t1 t2 ξ A 2.
• The NN exchange J may be either antiferromagnetic (AFM) or ferromagnetic (FM) (resp. negative and positive) according to the specific dimer. • The NN exchange and hopping are strongly modulated for all dimer types. These modulations are essentially related to the Co-O distance modulations and not to the Co-Co ones. Indeed, despite the fact the Co-Co distances are nearly constant for the C and D dimers ( b direction dimers), and strongly fluctuating for the A and B ones ( a ± b directions dimers) 12 , the variation range of both exchanges and hoppings present similar amplitudes (∼7 meV and ∼13 meV respectively). In fact, one should notice that the amplitude of both the super-exchange AFM contribution to J and the hopping integral decrease when the Co-O distance increases.

• J absolute values are very small in amplitude, with average values around 2.56 meV for b direction dimers, and -1.54 meV for a ± b direction dimers (see Table I). The NN exchange is thus mostly ferromagnetic along the b direction and antiferro-magnetic along the a ± b directions. Comparing these exchange values with the ones obtained in the Na x CoO 2 family, one sees that while for the superconducting system 20 (x = 0.35), the x = 0.5 and the x = 0 systems 13 the exchange are always AFM, in the present system both AFM and FM interactions were found according to the Co-Co bond. In fact the antiferromagnetic character of J decreases with increasing x (see table II) for the Na x CoO 2 compounds. These theoretical findings are in agreement with experimental data. Indeed, for low x values the CoO 2 layers of the Na x CoO 2 systems were found to be antiferromagnetically coupled 16 , while for large x values, neutrons scattering exhibit A-type antiferromagnetism 29 , that is ferromagnetic correlations within the CoO 2 layers which are anti-ferromagnetically coupled. In fact, Lang et al NMR studies 4 allowed to propose a magnetic phases diagram with a temperature dependant boundary between the AFM and FM correlations within the CoO 2 layers. At low temperature the critical doping x * is in the 0.6 ≤ x * ≤ 0.7 range and decreases with increasing temperature (x * (300 K) ∼ 0.5). Comparing now our computed J values for the Ca 3 Co 4 O 9 compound with both theoretical and experimental results for the Na x CoO 2 family, one sees that the present compound -associated with x ∼ 0.62 -compares well with the Na x CoO 2 family. Indeed, it fluctuating FM/AFM effective exchange agrees well with its assumed doping of the CoO 2 layers close to the phase boundary.

B. Effective on-site repulsion

Figure 5 reports the difference between the average onsite repulsion on a dimer, Ū , and the NN repulsion

V : Ū -V = (U 1 + U 2 ) /2 -V 12 .
The structural analysis shows us that the Co-Co distance variation is very weak for the b direction dimers, while it is large for the a ± b dimers (See figure 5). Since the NN repulsion is essentially related to the Co-Co distances one can assume that V is nearly constant along the b direction, but strongly varies along the a ± b one. Let us analyze figure 5 in this light. The Ū -V curve exhibits only weak modulations for the C dimers. Since both V and U can be assumed as nearly constant for these dimers, the observed modulations may be related to either of them. In any case one can conclude that U remains large and nearly constant from site to site (fluctuations range smaller than 0.2 eV to be compared to an average Ū -V of 3.88 eV). The sites along the A, B and C, D dimers being the same, it results that the strong variation observed on the a ± b direction dimers is essentially due to NN repulsion variations V (τ ) associated with the strong fluctuations of the Co-Co distances. The amplitude of the variation is quite large, with about 1eV for an average value of 3.5eV (see table III). It is not possible using our data to quantify further the NN repulsion, however our results suggest that V is large (a few eV in amplitude). An accurate simple model should thus take into account the NN repulsion between the a 1g orbitals and its strong variations ; the on-site repulsion being taken as site-independent. The average values Ū -V values found in the present work are consistent with the values found for the Na x CoO 2 family. Indeed the super-conducting compound exhibits U -V = 3.6 eV while it was found for the Na 0.5 CoO 2 system 2.6 eV and 2.8 eV according to the crystallographic site. Table III. Parameters for Ū -V fits according to the equation

Ū (τ ) -V (τ ) = a0 + a1 cos (2π[τ + ζ]) + a2 cos (4π[τ + ζ]),
where τ is the fourth crystallographic dimension parameter.

C. Effective a1g orbital energy

The on-site repulsion being considered constant, the energy difference between NN a 1g orbitals can be extracted from our results. Figure 6 reports the effective a 1g orbital energy : ε a1g (τ ) (ε a1g (0) being taken as the energy reference). One can see that the amplitude of the ε a1g (τ ) modulations is quite large (> 200 meV). Compared to this modulation, the a 1g -e ′ g energy splitting is negligible with an amplitude 10 times weaker. It results that the ε a1g (τ ) modulation can be seen as an effective potential on the whole cobalt atom. Associated with the strong NN repulsion V 30 , one can thus expect a partial charge localization, with Co 3+ ions on the low energy sites. This result agrees with the observations on the Na x CoO 2 family for which a partial charge localization is observed for x ≥ 0.5 31 .

V. DISCUSSION

Let us now take a closer look on the Seebeck coefficient in the Ca 3 Co 4 O 9 system. The first point to be stressed is the fact that the TEP can be quite different according to the intercalation layer (S 300K (Na

0.7 CoO 2 ) ≃ 100µV/K 3 , S 300K ([Ca 2 CoO 3 ] 0.62 [CoO 2 ]) ≃ 120µV/K 6,10 , S 300K ([CaBiO 2 ] 2 [CoO 2 ] 1.69 ) ≃ 160µV/K 32 ).
The second point is the shape of the Seebeck coefficient as a function of temperature. At low temperature the Seebeck coefficient increases quasi-linearly up to ∼100 K and then saturates to a constant value up to about ∼400 K 33 . These two behaviors can be fitted on a correlated Fermi liquid behavior, for the low temperature part 34 (T < 100 K), and on a Heikes type of configurational entropy for the saturating part 35 (150 K≤ T ≤ 400 K).

The generalized Heikes formula 36 is a high temperature approximation. It supposes that the system is at very large temperature compared to the scale of the low energy degrees of freedom. In the present compound we showed (see section III) that these degrees of freedom are associated with the a 1g orbital of the CoO 2 layers cobalt atoms 3d-shell. Using this formula and the non-degenerated character of the magnetic a 1g orbital, one finds (using a cobalt valency of 3.38 as discussed previously in section II), a TEP value of 125 µV/K. This value is in good agreement with the experimental value of 120 µV/K found for the Ca3Co 4 O 9 saturation plateau 6,10 . One may conclude from this agreement that, at room temperature, the high temperature limit of the triangular one band model based on the a 1g orbitals is already reached. However, room temperature only corresponds to 25 meV, while the band width of a tight binding model on a triangular lattice is W = 9|t| ; using our present transfer estimations (|t| ≥ 100 meV) it comes a band width of the order of 10000 K. How can we account for such a large difference in order of magnitude? Why does the Heikes model fits so nicely the room temperature Seebeck coefficient for the CoO 2 -based compounds?

Different arguments can be advanced in order to explain this high temperature behavior of the TEP at 300 K. Let us first analyze the behavior of the Seebeck coefficient, as a function of temperature within a one-band tight binding model on a triangular lattice. The Boltzmann equation yields the following Seebeck expression

S(T ) = - e k B T (ε -µ) ∂f0 ∂ε dε ∂f0 ∂ε dε (4)
= -e k B T ε ∂f0 ∂ε dε ∂f0 ∂ε dε

+ e k B T µ (5)
= S tr (T ) + S µ (T )

where f 0 (T, ε) is the Fermi function, ε the orbitals energy, µ the electrochemical potential ; the collision time as well as the degeneracy is supposed energy independent. Figure 7 reports S(T ) and its two components S tr (T ) and S µ (T ). At low temperature S tr (T ) dominates, reaches a maximum for T /W ∼ 1/4, then slowly decreases toward zero in the high temperature limit. On the contrary the entropic part S µ (T ) increases for all temperatures and reaches its high temperature limit for temperatures of the order of a few band widths. The sum of the two terms results in a Seebeck coefficient reaching its saturation value at much lower temperature than its entropic part. Indeed for T /W = 0.5, S is already larger than 90% of its high temperature value. This simple analysis already allows us to reduce, from a few band widths to half the band width, the estimation of the temperature at which the TEP saturation should be observed. This is however not enough to account for the observed saturation temperature. Indeed W/2 = 9|t|/2 still accounts for about 5000 K.

A second argument is the extremely large value of the correlation strength in these compounds. Indeed, our calculations yield for the average U/t a value of 26. Despite this very large U/t, transport measurements 6,10 find the system metallic. One thus expect its Fermi level quasiparticle pic to be strongly renormalized due to the strong correlation effects 37 . Finally let us remember that our results suggest a partial electron localization due the large site energy mod- ulation (> 200 meV) and a large value of NN repulsion. It results that the lattice pattern seen by the Fermi level electrons can be expected to be quite different from the homogeneous triangular lattice assumed in these systems. This effect is also expected to strongly reduce the quasiparticle bandwidth at the Fermi level. All together, it seems quite possible that the bandwidth reduction is large enough (20 to 30 times) so that 0.5W eff ∼ 200 K, and thus that the high temperature Heikes model appears valid for the present (and related) systems.

Let us now take a look on the TEP behavior at temperatures larger than 400 K. Indeed, at T c = 396 K the Ca 3 Co 4 O 9 compound experiences a first-order metalsemiconductor phase transition associated with a linear increase of the Seebeck coefficient for T > T c 33 . Different hypotheses have been proposed in order to account for this phase transition and the high temperature phase properties. The first one associates the transition with a cobalt spin-state transition 10 . However our calculations show that the first atomic excited spin state of the cobalt atom is at least at 700 meV, i.e. 8100 K, above the ground state. Such an energy difference is clearly incompatible with a phase transition energy scale of 400 K. The second hypothesis associates the transition with a change in the cobalt t 2g orbitals occupancies. However this hypothesis does not resist to the energy scale of the a 1g -e ′ g energy splitting found in section III. Indeed, the 240 meV≃2800 K is still far too large for the 400 K energy scale of the transition. The last hypothesis is an electron localization of the e ′ g orbitals, associated with a structural phase transition 33 . Even if the assumption that the e ′ g orbitals are the Fermi level ones has been proven incorrect (see reference 13 or for the present system section III), the idea of an electron localization induced by an increase of the in-plane distances in the CoO 2 layers seem quite possible and would explain the metal-semi-conductor phase transition. Muguera and Grebille 38 solved the crystallographic structure in the high temperature phase. They showed that the transition is due to a rearrangement of the central CoO layer of the Ca 2 CoO 3 subsystem. This rearrangement acts on the CoO 2 layers through the O-Ca interaction between the two subsystems. It results in a modification of both the cobalts and oxygens modulations (but not of the average structure) along the a and c directions. While such a weak structural modification can account for the metal-semiconductor transition (due to the large correlation strength and associated closeness to the Mott transition), it cannot account for the associated increase of the Seebeck coefficient. Indeed, such an increase necessitate the opening of new degrees of freedom at the Fermi level and the weakness of the structural modifications within the CoO 2 layers cannot account the opening of either the higher cobalt spin states or the e ′ g orbitals. This study however show us that there is another degree of freedom up to now ignored that should be taken into account, that is the partial charge localization due to the action of the subsystem modulation over the Fermi level orbital energies. Indeed a modification of the modulation vector will be expected to modify the energy profile pictured in figure 6, thus the charge localization configuration and the lattice pattern seen by the Fermi electrons at the given temperature. Out of any other energetically compatible hypothesis, the latter seems the most plausible.

VI. CONCLUSION

We performed ab initio calculations of the effective local parameters of the thermoelectric Ca 3 Co 4 O 9 compound, taking into account the incommensurate distortions. As in the Na x CoO 2 family, cobalts a 1g -e ′ g orbitals splitting is large so that the low energy physics can be accounted for using the a 1g orbitals only. We showed that the incommensurate modulation of the CoO 2 layer cannot be neglected in order to understand the compound physical properties. Indeed, site energies, NN exchange, NN hopping as well as first neighbor repulsions are strongly modulated due to the misfit character of the system. It particular the site energies large modulation let us expect a partial Fermi electrons localization. The effective exchange remains weak in amplitude, however fluctuates between antiferromagnetic coupling and ferromagnetic one. This results can be compared with the on-plane AFM-FM transition line in the Na x CoO 2 family associated at T = 0 K with similar cobalt valency as found in the present compound.

We discussed the temperature behavior of the Seebeck coefficient and showed that the cobalts a 1g orbitals bandwidth is strongly renormalized, due to very strong electron correlation (U/t ∼ 26), and the modification of the effective lattice pattern seen by the itinerant electrons, associated with the partial charge localization. This strong bandwidth renormalization is most proba-bly responsible for the plateau due to the high temperature behavior observed in the Seebeck coefficient on the [150 K, 400 K] range. We further showed that he sudden increase of the TEP for T > T c = 396 K, cannot be attributed either by a cobalt spin state change or by a modification of the a 1g vs e ′ g orbitals occupation. In fact the 396 K metal-semiconductor transition is associated with a structural rearrangement within the Ca 2 CoO 3 subsystem inducing a modulation modification within the CoO 2 layer. This modification induces a site energy pattern change and thus a localization pattern change thought to be the best candidate for the change in the Seebeck behavior.

Figure 1 .

 1 Figure 1. (Color online) Structural representation of the Ca3Co4O9 compound. Green : CoO2 layers of edge-sharing octahedra, blue : cobalt octahedra in the rock-salt layers and light gray : calcium ions.

Figure 2

 2 Figure 2 displays the ε a1g -ε e ′ g orbital energy splitting as a function of the fourth crystallographic dimension parameter (τ = mod(n b CoO2 /b RS , 1), n ∈ N refers to the CoO 2 subsystem cells in the b direction -see figure 3 -b CoO2 and b RS are the lattice parameters of the two subsystems in the incommensurate direction) associated with the incommensurate modulations of the CoO 2 layers. Let us recall that this structural modulation takes place along the b direction. These effective orbital splittings where obtained as the energy difference of fully correlated e ′ g 4 a 1g1 and e ′

Figure 3 .

 3 Figure 3. (Color online) Schematic representation of the different dimers.

Figure 4 (

 4 Figure 4(a) reports the variations of the effective exchange integral J and figure 4(b) of the effective hopping integral t, as a function of the fourth crystallographic dimension parameter τ . Table I reports the parameters for the fit of the computed values according to the formula

Figure 4 .

 4 Figure 4. (Color online) (a) Effective exchange, J, and (b) effective transfer, t, as a function of the fourth crystallographic dimension parameter τ . Symbols represent the computed values, lines the fits.

Figure 5 .

 5 Figure 5. (Color online) Left scale : difference (meV) between the average on-site repulsion on a dimer U and the effective NN repulsion V as a function of the fourth crystallographic dimension parameter τ . Right scale : Co-Co NN distances ( Å) as a function of τ .

Figure 6 .

 6 Figure 6. a1g orbital energy (meV) as a function of τ . The τ = 0 value is taken as the energy reference.

Figure 7 .

 7 Figure 7. (Color online) Seebeck coefficient of a triangular lattice as a function of the temperature, normalized to the band width. Blue dashed curve : the entropic term : Sµ(T ), red dot-dashed curve : the Fermi liquid part : Str(T ).

  1-β ] 1+x CoO 2 , 7 Bi α [A 0.75 Bi 0.25 O] 3+3x/2 CoO 2 (A = Ca, Sr) 8 , Sr 2 O 2 CoO 2 , 9 etc. Among these layered cobalt oxides, the so-called Ca 3 Co 4 O 9 10 ([Ca 2 CoO 3 ] 0.62 [CoO 2 ]

  This value is very comparable to the values found on different Na x CoO 2 systems 13 . Figure 2. (Color online) 3d cobalt orbital energy splittings : εa 1g -ε e ′ g , as a function of the incommensurate modulation parameter τ . Blue curve with crosses : εa 1g -ε e ′ g1 splitting, red curve with plus signs : εa 1g -ε e ′ g2 splitting, black curve with stars : average εa 1g -ε e ′ g splitting.

				ε a 1g -ε e' g1 ε a 1g -ε e' g2 ε a 1g -(ε e' g1 + ε e' g2 )/2
				∆E = ε a 1g -ε e' g
		290
		280
	∆E (meV)	250 260 270
		240
		0 230	0.2	0.4	τ	0.6	0.8	1

Table I .

 I Parameters for effective exchange J and effective transfer t fits as a function of the incommensurate dimension parameter τ . The fitted values are associated with equations 3.

		56 -6.38 1.36 0.00 -135.81 -38.15 12.65 0.00
	C	-1.54 -0.02 3.49 0.60 -146.82 0.11 27.85 0.10

Table II .

 II (meV) -5213 -3620 -11/-19/-19/-2713 [-2.5,10.3] / [-5.0,2.0] Effective exchange values (in meV) as a function of the CoO2 filing parameter x, present system compared to different compounds of the NaxCoO2 family.

	System		NaxCoO2		Ca3Co4O9
	x	0	0.35	0.5	∼ 0.62
	J

ACKNOWLEDGMENTS

The authors thank Prof. Dominique Grebille and Dr. Olivier Perez for helpful discussion on the crystallographic aspects, as well as Dr. Daniel Maynau for providing us with the CASDI suite of programs. These calculations where done using the CNRS IDRIS computational facilities under project n • 1842 and the CRIHAN computational facilities under project n • 2007013.