Picq M, Véricel E, Hachem M Bernoud-Hubac N
email: michel.lagarde@insa-lyon.fr

Biological properties of a DHA-containing structured phospholipid (AceDoPC) to target the brain

Keywords: Metabolism, Blood-Brain Barrier, Stroke

1-acetyl,2-docosahexaenoyl-glycerophosphocholine (AceDoPC) has been made to prevent docosahexaenoyl (DHA) to move to the sn-1 position as it rapidly does when present in 1lyso,2-docosahexaenoyl-GPC (lysoPC-DHA), an efficient DHA transporter to the brain. When incubated with human blood, AceDoPC behaves closer to lysoPC-DHA than PC-DHA in terms of binding to plasma albumin and lipoproteins, and DHA incorporation into platelets and red cells. In addition, AceDoPC prevents more efficiently the deleterious effects of the experimental stroke in rats than does unesterified DHA. Also, AceDoPC inhibits plateletactivating factor-induced human blood platelet aggregation. Overall, AceDoPC might act as an efficient DHA transporter to the brain, and as a neuro-protective agent by itself.

Introduction

Docosahexaenoic acid (DHA) is the main polyunsaturated fatty acid (PUFA) esterified in the mammal brain phospholipids, and almost the only omega-3 PUFA in this tissue [START_REF] Crawford | Evidence for the unique function of docosahexaenoic acid during the evolution of the modern hominid brain[END_REF]. It is recognized as an important PUFA for the brain development and learning abilities, as well as in the retina for visual acuity [START_REF] Gordon | Nutrition and cognitive function[END_REF]. In addition to be an important endogenous component in these organs, DHA is a relevant nutrient from marine food, together with eicosapentaenoic acid (EPA), both being active to prevent cardiovascular events [START_REF] Ws | Omega-3 fatty acids and cardiovascular disease: a case for omega-3 index as a new risk factor[END_REF]. One important issue in the propensity of DHA to accumulate into the brain is the circulating form for its efficient crossing of the blood-brain barrier (BBB). Early work from our group suggested that PUFA esterified at the sn-2 position of 1-lyso,2-acyl-glycerophosphocholine were more efficiently taken up by rat brain than the corresponding non-esterified fatty acids (4). This observation is especially valid for DHA that shows the highest difference between the two forms, with a ten-fold preference for 1-lyso,2-docosahexaenoylglycerophosphocholine (lysoPC-DHA) than non-esterified DHA, when they were both bound to albumin [START_REF] Thies | Preferential incorporation of sn-2 lysoPC DHA over unesterified DHA in the young rat brain[END_REF]. This preference was confirmed by using an in vitro re-constituted BBB with a co-culture of brain capillary endothelial cells and astrocytes, whereas no preference was shown when those brain endothelial cells where replaced by aortic ones [START_REF] Bernoud | Preferential transfer of 2-docosahexaenoyl-1-lysophosphatidylcholine through an in vitro blood-brain barrier over unesterified docosahexaenoic acid[END_REF]. In the latter case, lysoPC-DHA was esterified and integrated in astrocyte PC (more than 50% of the initial lysoPC-DHA) with virtually no remaining of the lyso form. This suggests that lysoPC-DHA was acylated into PC. Interestingly, this preference for lysoPC was significantly lower when 1-lyso,2-palmitoyl-PC was used instead [START_REF] Bernoud | Preferential transfer of 2-docosahexaenoyl-1-lysophosphatidylcholine through an in vitro blood-brain barrier over unesterified docosahexaenoic acid[END_REF]. All these results are in favor of a specific passage of the BBB by lysoPC-DHA. One question then arose: are 1-lyso,2-acyl-glycerophospholipids, such as lysoPC-DHA, stable enough to cross the BBB without isomerization into 1-acyl,2-lyso counterparts, a more stable form of lysophospholipids (lysoPL) [START_REF] Plückthun | Acyl and phosphoryl migration in lysophospholipids: Importance in phospholipid synthesis and phospholipase specificity[END_REF]. Indeed, the primary alcohol, present in 1-lyso,2-acyl-PL is more reactive for acylation than the secondary alcohol in 1-acyl,2-lyso-PL, which should facilitate the position isomerization. The measurement of human blood lysoPC, the main lysoPL species, showed us a concentration of 150 nmol/ml in plasma, with one quarter being composed of PUFA (essentially linoleic acid (20%), arachidonic acid (4%), DHA (1%) [START_REF] Croset | Characterization of plasma unsaturated lysophosphatidylcholines in human and rat[END_REF]. However, studies on the migration of linoleoyl residue from the sn-2 position of lysoPC to the sn-1 position showed that three quarters migrated after just 20 min incubation in plasma kept at 37°C and physiological pH [START_REF] Croset | Characterization of plasma unsaturated lysophosphatidylcholines in human and rat[END_REF]. This prompted us to make a structured PC trying to mimic as far as possible 1-lyso,2-DHA-PC. Some results mentioned in this short review are published in patents but not yet in full articles.

Preparation of 1-acetyl,2-docosahexaenoyl-glycerophosphocholine (AceDoPC)

Figure 1 shows the chemical structure of AceDoPC. In order to keep the structured phospholipid closest to lysoPC-DHA, especially for its polarity, we have acetylated 1-lyso,2-DHA-PC via an organic process [START_REF] Polette | Synthesis of acetyl,docosahexaenoyl-glycerophosphocholine and its characterization using nuclear magnetic resonance[END_REF]. Although the lyso species was purified by HPLC immediately prior to acetylation, around 20% of 1-docosahexaenoyl,2-acetyl-PC was obtained, presumably due to DHA migration to the sn-1 position during the chemical process. We then decided to proceed through an enzymatic transacylation from a purified 1-acyl,2-DHA-PC (PC-DHA), issued from a marine biomass. AceDoPC was produced by using an immobilized triacylglycerol lipase acting upon pure PC-DHA in absence of water but presence of ethyl acetate/vinyl acetate as acetylation agents to replace the acyl moiety by acetyl at the sn-1 position. This transacylation process minimizes the migration of DHA, making AceDoPC an almost pure 1-acetyl,2-DHA-PC [START_REF] Lagarde | Method for preparing acetyl,docosahexaenoylglycerophosphocholine and use thereof for the delivery of polyunsaturated fatty acids[END_REF][START_REF] Picq | DHA metabolism: targeting the brain and lipoxygenation[END_REF]. Figure 2 shows the purity of AceDoPC, either by radiochromatography when using the radiolabelled phospholipid (Fig. 2A), or by HPLC analysis of the unlabelled molecule (Fig. 2B). Of note, AceDoPC exhibits polar properties closer to lysoPC-DHA than to DHA-containing PC as evaluated by chromatographic migration.

Behavior of AceDoPC in human blood

To assess that AceDoPC could be a privileged carrier of DHA to the brain, we first studied its distribution compared to PC-DHA within different human blood compartments. To this goal, we synthesized 14 C-labeled-DHA in PC-DHA from the marine microalgae Cryptecodinium cohnii incubated with 14 C-labeled acetate, and prepared 14 C-labeled-DHA in AceDoPC using the transacylation process reported above [START_REF] Lagarde | Method for preparing acetyl,docosahexaenoylglycerophosphocholine and use thereof for the delivery of polyunsaturated fatty acids[END_REF]. The labeled phospholipids were then incubated with human blood and the radioactivity distribution was measured in blood platelets and red cells, the latter compartment being accepted as an index of the brain DHA accretion. We found a greater incorporation of labelled DHA into red cells after incubation of whole blood with AceDoPC compared to PC-DHA. In contrast, this preference for AceDoPC was not observed with platelets (unpublished results). These results are in agreement with in vivo ones in which intake of DHA led to a slow but sustained rise in red cells, preferentially from lysoPC-DHA, while a faster but time-limited accumulation occurred in platelets from nonesterified DHA [START_REF] Brossard | Human plasma albumin transports [13C]docosahexaenoic acid in two lipid forms to blood cells[END_REF]13). This means that AceDoPC may mimic lysoPC-DHA more closely than PC-DHA. The analysis of the radioactivity distribution within human plasma indicated that AceDoPC was mainly bound to HDL and albumin (around 60 and 40%, respectively), whereas PC-DHA was mainly distributed within lipoproteins (around 85% within HDL, LDL and VLDL) and less than 15% within albumin (unpublished results). This suggests that AceDoPC behaves like lysoPC rather than PC. Overall, these results fit with the hypothesis that AceDoPC could be a privileged carrier of DHA to red cells and in turn to the brain.

Effect of AceDoPC on the ischemic stroke in rats

As DHA has been shown to induce neuroprotection in rats that underwent a transient cerebral ischemia [START_REF] Belayev | Robust docosahexaenoic acidmediated neuroprotection in a rat model of transient, focal cerebral ischemia[END_REF], we used this model for comparing DHA with AceDoPC. The external carotid artery of rats was inserted with a coated monofilament for a one-hour transient occlusion, then non-esterified DHA or AceDoPC, solubilized in rat plasma to allow their binding to albumin and lipoproteins, was intravenously injected with 1ml of plasma containing 1 µmol of each DHA form. Magnetic Resonance Imaging of the brain was performed as well as behavioral tests 24h after the injection. The lesion sizes due to the initial stroke were stable in rats receiving the plasma alone as a control, while they were decreased in rats receiving nonesterified DHA, and further decreased (twice as much) in rats receiving AceDoPC. Neuroscores tended to be improved (p=0.07) in the AceDoPC group [START_REF] Chauveau | Brain-targeting form of docosahexaenoic acid for experimental stroke treatment: MRI evaluation and antioxidant impact[END_REF]. It is interesting to note that the dose used was 2 to 4 times lower than those used in the Belayev's study [START_REF] Belayev | Robust docosahexaenoic acidmediated neuroprotection in a rat model of transient, focal cerebral ischemia[END_REF], who stated that very high dose of DHA were inactive. Overall, our study makes promising the use of moderate injected dose of AceDoPC following stroke. The mechanism of neuroprotection observed remains to be clarified. The measurement of isoprostanes (evaluated with 8-epi-prostaglandin F 2 ) in the brain 24h after the injection of either DHA or AceDoPC showed that, although non-significantly decreased, a lower value was obtained after AceDoPC compared to DHA. Interestingly, 8-epi-prostaglandin F 2 was significantly lower when both treatments were pooled in the statistical analysis, suggesting a possible reduction of oxidative stress. Whether the influx of DHA was higher in the AceDoPC group was not tested in this study, but an ongoing investigation using radioactively labeled molecules shows higher DHA accretion into normal rodent brains from AceDoPC compared to non-esterified DHA (Hachem et al., unpublished results).

Anti-PAF effect of AceDoPC

Platelet-activating factor (PAF) has been characterized as 1-O-alkyl,2-acetylglycerophosphocholine [START_REF] Vargaftig | Background and present status of research on platelet-activating factor (PAF-acether)[END_REF]. In addition to aggregate blood platelets, PAF is recognized as a pro-inflammatory molecule [START_REF] Larsen | Mediators of inflammation[END_REF] as well as a broncho-constricting agent [START_REF] Mcmanus | Platelet activating factor in pulmonary pathobiology[END_REF]. Apart from the O-alkyl moiety that differs from the docosahexaenoyl one, AceDoPC could be considered as a position isomer of PAF. The aggregation of human blood platelets isolated from their plasma, induced by PAF (hexadecyl,acetyl-glycerophosphocholine), was then evaluated in presence or absence of AceDoPC. An IC 50 was observed with 10µM AceDoPC while a 15% inhibition could be observed with 1µM AceDoPC [START_REF] Lagarde | Phosphatidylcholines en tant que modulateurs de l'activité du facteur d'activation plaquettaire[END_REF]. Although this inhibition may be considered as modest with regard to the active concentrations, this biological property allows some extrapolation on the anti-inflammatory potential of the molecule in addition to its capacity to be beneficial to the brain.

Conclusions

AceDoPC is a proposed DHA-containing structured phospholipid that mimics in a way lysoPC-DHA in terms of polarity and distribution within blood cells and albumin/lipoproteins. The acetylation of the sn-1 position keeps docosahexaenoyl at the sn-2 position that is the physiological one. Whether AceDoPC facilitates incorporation of DHA into the brain through its lysoPC-DHA moiety is under investigation. On the other hand, AceDoPC proved to be able to prevent more efficiently deleterious effects of the experimental ischemic stroke than did non-esterified DHA. Also, it may exert antiinflammatory activity by inhibiting PAF effects, putatively resulting from some antagonism due to the inverted position of the acetyl moiety but keeping the same polar head. Altogether, AceDoPC might be considered as a dual structured phospholipid to provide DHA to the brain and exert anti-atherothombotic effects. 2B: Analysis of unlabelled AceDoPC by reverse phase high performance liquid chromatography (X Bridge C 18 column) using a linear solvent gradient: solvent A and B were a mixture of methanol/water/acetonitrile (95/35/2.5 and 100/4/2.5, v/v, respectively) from 5 to 35 min (1ml/min). AceDoPC was detected at 205 nm after 27 min. 2B: Analysis of unlabelled AceDoPC by reverse phase high performance liquid chromatography (X Bridge C 18 column) using a linear solvent gradient: solvent A and B were a mixture of methanol/water/acetonitrile (95/35/2.5 and 100/4/2.5, v/v, respectively) from 5 to 35 min (1ml/min). AceDoPC was detected at 205 nm after 27 min.

Figure 1 :

 1 Figure 1: Structure of 1-acetyl, 2-docosahexaenoyl-glycerophosphocholine (AceDoPC).

Figure 2 :

 2 Figure 2: Typical tracings showing the purity of labelled and unlabelled AceDoPC. 2A: Radiochromatogram of radiolabeled 14 C-(DHA)-labelled AceDoPC (Thin-layer chromatography with chloroform/methanol/water (65/25/4) as eluent).2B: Analysis of unlabelled AceDoPC by reverse phase high performance liquid chromatography (X Bridge C 18 column) using a linear solvent gradient: solvent A and B were a mixture of methanol/water/acetonitrile (95/35/2.5 and 100/4/2.5, v/v, respectively) from 5 to 35 min (1ml/min). AceDoPC was detected at 205 nm after 27 min.

Figure 1 :

 1 Figure 1: Structure of 1-acetyl, 2-docosahexaenoyl-glycerophosphocholine (AceDoPC).

Figure 2 :

 2 Figure 2: Typical tracings showing the purity of labelled and unlabelled AceDoPC.

2A:

  Radiochromatogram of radiolabelled14 C-(DHA)-labelled AceDoPC (Thin-layer chromatography with chloroform/methanol/water (65/25/4) as eluent).

Acknowledgements

The authors thank Inserm, INSA-Lyon, and ANR "Neuroprotect" 2008, for their financial support, and Dr Valérie Pruneta-Deloche for human blood lipoprotein separation by FPLC.


Antoine Mischler Dooapp

Martin Monperrus

An Approach for Discovering Traceability Links between Regulatory Documents and Source Code Through User-Interface Labels

In application domains that are regulated, software vendors must maintain traceability links between the regulatory items and the code base implementing them. In this paper, we present a traceability approach based on the intuition that the regulatory documents and the user-interface of the corresponding software applications are very close. First, they use the same terminology. Second, most important regulatory pieces of information appear in the graphical user-interface because the end-users in those application domains care about the regulation (by construction). We evaluate our approach in the domain of green building. The evaluation involves a domain expert, lead architect of a commercial product within this area. The evaluation shows that the recovered traceability links are accurate.

Introduction

Some companies develop software applications in domains that are regulated. For instance, in the USA, healthcare related software products must comply with the Health Insurance Portability and Accountability Act (HIPAA).

Regulation, as software, evolves and it is essential to have traceability links between the regulation artifacts and the code artifacts [START_REF] Sannier | INCREMENT une approche hybride pour modéliser et analyser dans le large les exigences réglementaires de sûreté[END_REF]. Then, to modify a software application in respond to regulatory changes, software development teams follow those traceability links and are quickly pointed to the software modules to change. The whole point of research on traceability is on how to set up and maintain those links accurately.

The contribution of this paper is a novel fully-automated approach for recovering traceability links between regulatory documents and source code. This approach is based on the intuition that the labels of graphical user interfaces (GUIs) can act as proxy to the domain-specific terminology. If a user-interface label uses specific terms of requirements of regulatory documents, it is very likely that the code elements interacting with the label are indeed related to the requirements under consideration.

Our approach takes as input the code base of a software application that implements a specific regulation. This application must contain a human-machine interface. An indexing phase stores all user interface labels. When a developer wants to locate code elements that relate to a piece of regulation (whether a header, a sentence or a paragraph), she/he gives it as query to the system, which retrieves back a list of code elements (e.g. a list of classes in object-oriented software). The matching is done between the regulation terminology and the user-interface terminology.

The added value of our approach is the productivity gain when developers want to modify or add new features in a code base implementing a regulation that is unknown to them. Our traceability system directly points them to relevant pieces of code. To this extent, the goal is not to absolutely retrieve all related code elements but to retrieve quickly good ones. Once one element is found, the collaborating classes are probably also of interest.

Logically, we evaluate our approach as a recommendation system. First, we extracted 53 requirements from the official documents regulating the construction of green buildings in Europe. Then, we run our traceability system on those 53 requirements and the code base of a proprietary product complying with the regulation. The first author, expert of the code base, assessed the relevance of the top 25 recovered traceability links. Overall, for 44/53 queries, our prototype system found at least one correct traceability link. For 19/44 answered queries, the first recommended code item was correct.

To sum up, our contributions are:

• a fully-automated approach for recovering traceability links between regulatory documents and source code based on user interface labels;

• an evaluation on a real work regulation and a complying commercial product, done by a domain expert.

The remainder of this paper is as follows. Section 1 describes our approach, section 3 presents its evaluation, section 4 discusses the related work and section suggests future work.

Approach

Our traceability approach is based on the intuition that most regulation items will result, at least partially, in a label in the user-interface of the application. The whole process has two steps: first, we populate a traceability database by analyzing user-interface labels. Then, we automatically extract the traceability links by querying this database to retrieve the related code items. This process is summarized in Figure 1.

Our approach is both generic and specific. It is generic in the sense that it can be applied to any application domain. It is specific in the sense that the tool implementing the norm must have a user-interface. Even if the user-interface only spans a subset of features or requirements of the regulation documents, the traceability provided in this scope it is valuable.

Application codebase

UI labels indexing

Populating the traceability database offline

UI labels extraction The first step consists in extracting all user interface labels from the application source. In large internationalized applications, UI labels are generally stored in resource files. A resource file is a map of key-value pairs for a given language. Each entry in the resource bundle represents the translation of an UI label for this language. The value is the translation of the key and the key is used in the code to reference this label. At the end of this step, we obtain a database with all labels used in the UI.

Code items extraction

The second step consists in retrieving the code items that use the labels collected in the previous step. In our database, each label is paired with a key that is used in the code to reference that particular label. In most cases, a literal search of the key in the codebase allows us to retrieve the code item related to the UI label.

In some cases, no result matches the key of the UI label. There two reasons for this. First, this happens when the label is not used in the UI, it is either deprecated or unused due to an accidental label duplication after a refactoring. Deprecation of labels happens during the lifecycle of an application, when developers update the UI and forget to clean the deprecated resources.

The second reason is that the key is sometimes dynamically built at runtime and can not be retrieved using a literal search in the codebase. In this case, we propose an approach based on dynamic traces. This involves generating a database of runtime traces of the application. The application is used according to a usage scenario and a monitor logs all accesses to the UI label resources. This enables us to recover links between UI labels and code elements that can not be observed statically. This approach has a limitation: we cannot guarantee that all dynamic keys have been used during the runtime scenario.

Synonym Normalization Synonym normalization [START_REF] Leuser | Tackling Semi-automatic Trace Recovery for Large Specifications[END_REF] consists in replacing synonymous terms for a single concept by the same word. In the domain of air tightness, the regulation defines a list of terms and symbols that might be used as synonyms for frequently used values or concepts. For example, "n50" can be used to represent the "taux de renouvellement d'air sous 50 Pa". Our prototype system implements synonym normalization as preprocessing to indexing. Hence, searching for "n50" of its long explicit version yields the same results.

Online retrieval of traceability links

The database of traceability links populated offline is then used to retrieve traceability links on demand in an online manner.

Search To find code elements related to a regulatory item, a developer types a query in natural language. For instance, in the domains of cash machines, the developer might type "incorrect password". The traceability system then answers that class "ScreenA" and "ScreenB" uses an UI label that matches the query "incorrect password".

Given the developer query (e.g. a regulatory item), we select relevant UI labels from the database using an information retrieval library. A tf-idf (term frequency-inverse document frequency1 ) statistic is used to find the labels related that are most related to the query.

For each matched label, the source code classes that use this label are retrieved using the literal or trace search described in Section 2.1. If a matching label is not used according to both techniques, the developer is told that a matching label has been found but no code elements can be identified. It is her responsibility to tell whether the label is a dead one or not. From the traceability viewpoint, one can not be state beforehand that the {key,label} is actually not used in the application.

To sum up, a search results in a collection of traceability links matching the user query.

Integrated display in IDE An IDE integration of the search process is important for an easy use of the traceability system during daily development. The IDE integration should allow the user to enter its query and displays the corresponding traceability links. Our prototype is presented in section 2.3.

Prototype implementation

We have developed a prototype that implements our approach. Our prototype consists of two artifacts. First, a Java library performs the traceability population (see 2.1) and the online retrieval of traceability links (see 2.2). Second, the integration of the approach into an IDE is demonstrated with a plugin for the Intellij IDEA development environment.

Library The traceability database if first populated by scanning the applications resources and retrieving all resource bundles. Our library indexes key-value pairs using Apache Lucene [START_REF]Lucene homepage[END_REF], a well known and mature indexing open-source application written in Java. The indexing uses the language specific indexing components of Lucene (e.g. FrenchAnalyzer if the regulation and user-interface are in French). Those components perform tokenization, stop word removal and stemming in a language specific manner.

Then, we analyze all source files of the application to extract the resource bundles keys used in the graphical user-interface. Our implementation uses Spoon [START_REF]Spoon homepage[END_REF] to analyze the Java source code and extract all literals. These literals are then stored in memory to keep track of their usage in source code.

Intellij IDEA plugin

The traceability UI is implemented as an Intellij IDEA plugin. This plugin provides a menu entry that allows the user to enter its search query. The library is then used to perform this online query and returns the collection of traceability links. These links are displayed in the search result window of the IDE. For each link, the plugin displays the corresponding resource bundle key-value and the weight of the result. By clicking on a link, the IDE opens the Java class where the key usage has been identified.

The plugin is available in the JetBains Plugin Repository and can be downloaded and tested [START_REF] Monperrus | Natural language code search plugin for intellijidea[END_REF].

Evaluation

We now evaluate our approach to recover traceability links between source code and regulatory documents. We set up an evaluation process in the context of the first author's company: green building.

Background

Application Domain DooApp develops and sells a product for measuring and reporting on the thermal performance of buildings with respect to air tightness. The product, called Infiltrea, drives a blower door and generates a report that is part of the certification of green buildings.

The first author of this paper is the co-founder of a company called DooApp, that develops and sells a product for measuring and reporting on the thermal performance of buildings with respect to air tightness. The product, called Infiltrea, drives a blower door and generates a report that is part of the official certification of green buildings.

Regulations The measure of air permeability is regulated by a set of regulatory documents at the European and French level, in particular EN-13829 [START_REF] Afnor | Thermal performance of buildings -determination of air permeability of buildings -fan pressurization method[END_REF].

The regulation allows one to compare test results from different test operators and countries. The reports generated by Dooapp's product Infiltrea, complies with this regulation.

Evaluation Process

The evaluation is as follows. First, we have collected 53 regulation fragments in EN-13829 [START_REF] Afnor | Thermal performance of buildings -determination of air permeability of buildings -fan pressurization method[END_REF]. Each of those regulation fragments is an atomic feature of the thermal measurement product under study. Second, for each of them, we have used our prototype implementation to extract the matching pieces of code in Infiltrea's codebase. Third, the first author (the domain and codebase expert) has assessed each of the top-20 matched source code elements for each of the 53 regulatory items to see whether they were correct or not. In this context, there are two potential issues: false positive and false negative answers. False positive answers are code elements that our system points as related to the query while they actually have nothing to do with the regulation item under study. False negative answers are related code elements that are not identified as such during traceability recovery. Since our approach is meant to be used as a recommendation system to gain productivity, having some false negatives is not an issue2 . Each false positive is wasted time for the developer. Consequently, we try to minimize the false positives and to some extent the false negatives. To capture those two facets, we compute performance metrics such as precision and recall.

Dataset Setup

Within EN-13829, we have identified 53 key regulatory items to be used as evaluation dataset. For instance, a regulation item states that the measurement process involves "Computing the air flow rate at reference pressure". In this expression, the key words are "reference pressure" and "air flow rate": a software application in the domain of EN-13829 must handle the notion of reference pressure. So, we can except these key words to be used in the graphical userinterface.

Let us now consider a developer who wants to find the code implementing this regulatory item. One natural solution is to use the regulatory sentence as "input query" of a traceability system. This is the reason for which the 53 regulatory items form the evaluation dataset of our system.

Each regulatory item is saved in two forms: a short form representing the key description sentence such as "Computing the air flow rate at reference pressure", and a long form composed of the full paragraph extracted from the regulation.

Since the first author is an expert of the application domain, we think that the 53 regulatory items are valid both in terms of importance and scope. By importance, we mean that those regulation fragments indeed matters from a business value viewpoint. By scope, we mean that the identified fragments cover one and only one domain feature or constraint.

The short regulation items have an average length of 9 words and the long form an average length of 44 words.

Traceability Link Collection

For each of the 53 regulation items them, we have used our prototype implementation described in Section 2 to extract the related pieces of code in the code base of Infiltrea (the software product developed by the company of the first author).

Figure 3 gives the distribution of the number of traceability results for the 53 considered regulatory items (number of user interface key-value pairs). A traceability link consists of a pair {user-interface label, code element}. The median number of traceability links is 65; showing that many UI labels match the regulation items (more or less closely). This is a first piece of evidence that our intuition holds: the UI labels indeed capture the terminology and spirit of the regulation.

Traceability Link Assessment

For each regulatory item of the dataset, the first author manually assessed up to 20 traceability links (in the top-20) as given by our prototype implementation. He assessed the results for both the short and long version of the regulatory items. The assessment was supported by a specific web-application tool shown in Figure 4. In this evaluation tool, the traceability links are presented with four components: the UI label (for instance "DeltaP02 est égal . . . "), the Java file that uses this UI label (in blue), the key used to refer to this label (the key name usually has some semantic meaning), and the resource file name defining the pair {key,label}. The two first components are the essential part of the traceability link (the label and the corresponding Java class).

The last two components are specific to the UI technology we are considering. Still, it is important to present them to the domain and code expert because it helps him to quickly grasp which part of the code is involved in the traceability link, better than with only the class name.

For each traceability link, the domain and codebase expert rates on a Likert scale whether the traceability link is appropriate: correct, related ("pas loin" in French), and wrong ("faux" in French). "Correct" means that the assessed traceability link refers to a class that actually implements the regulatory item. "Related" means that it refers to a class that does not directly implement the regulatory item, but closely collaborate with a class that implements it. "Wrong" means that it refers to a class that does not implement the regulatory item under consideration.

In total, we have collected 1473 manual assessment of retrieved traceability links. This evaluation data is then used as input to compute performance metrics.

Figure 4: Screenshot of the web-based assessment tool (in French). The domain and codebase expert rates on a Likert scale whether the traceability link is appropriate: correct ("correct"), related ("pas loin"), wrong ("faux").

Metrics

Our traceability system is a recommendation system. Developers query it with a regulation related query (a sentence or a paragraph) and in response, the system recommends a list of code elements that are very probably related to the query (a list classes of object-oriented systems in our prototype system) To evaluate it, we use precision and recall. The golden truth is the manual data presented in the previous section 3.5. Precision and recall are computed at the top 5 and top 10 positions.

Table 1 gives the results of this evaluation. The traceability system is queried with both the short version of the regulatory items (one sentence) and the long version (one paragraph).

Let us consider the performance for the short queries (middle column). Over the 53 real world queries, the system finds a correct traceability link for 44 of them. For 19/44 answered queries, the first recommended code item is correct, it actually implements the requirement that is given as input (this may not be the only related code elements: in this context, "correct" does not mean complete). Overall, the median rank of the first correct traceability link is 2. This means that for most queries, the developer finds a correct traceability link in the top 2 results. Over the top 5 returned results, the average precision is 0.22, which means that in average one code elements out of the top five recommended ones is relevant to the query. This number means that there is almost a good traceability link in the top 5 results (given that there are often a single code element to find, in this case the precision is 1/5 = 0.2 by construction).

For the long version (second column of Table 1), the precision and recall decrease for the top 5 and top 25 results. This corresponds to the fact that the The evaluation metric of our traceability system. Over 53 real world queries, the system is capable of correctly answering most of them. When it answers, the first correct traceability link is very often in the top results.

Metric

long version contains a lot of words that interfere with the core of the query. This noise decreases the quality of the results. However, one query that was unanswered with the short version becomes answered with the long version. This is expected, the long version contains more word that can match UI labels. This evaluation based on real regulations and manual expert assessment shows that our UI-based approach to traceability actually works in practice. For 44/53 queries, the system finds at least one correct traceability link. For 19/44 answered queries, the first recommended code item is correct. There are good reasons to trust the results, since the first author is a domain and codebase expert: 1) the 53 evaluation query extracted from the regulatory documents are realistic and important in the considered application domain (domain expertise); 2) the 1473 codebase elements pointed by the traceability links have been evaluated by a codebase expert to rate the links (code expertise).

User Study

Finally, we performed a small user study of the tool. Within a weekly 1-hour technical seminar, we presented the traceability tool to the 3 developers of Infiltrea. We then asked them to use it in an informal manner, related to their current development duties. Those 3 developers are the exact type of users targeted by our traceability approach: they work in a regulated domain and are daily performing maintenance tasks on a large code base implementing the corresponding regulation. They do not master the whole code and regularly need help to locate relevant code elements.

After 1 hour of free usage, the feedback was positive (note there is a clear bias due to the fact that the new IDE feature was proposed by a colleague). A good anecdote is the following: the only case when a developer complained, he realized that he looked for a feature he was about to develop. Since the feature was not yet implemented hence no traceability was found. The query was meaningful, it was related to the regulation, but no traceability link could ever be found by construction.

This small user study gives us confidence that the tool could be valuable for the developers. However, beyond the bias towards positive feedback, the developers were novice neither in the domain nor in the code base. A more comprehensive and expensive user study would simulate this situation. Today, the tool is still in use in the development team of the company.

Limitations

The main limitation of our approach is that it only works with graphical software applications (whether desktop or web-based). For other types of software, we need another proxy to the domain terminology, such as the code element names (as usually done, see our discussion on related work e.g. 4).

Our approach is based on the terminological matching between the regulation in natural language and the user-interface labels (also in natural language). It is not tied to a particular language. If both the regulation and the user-interface are written in Tamil for instance, it would work. However, there is one case where our approach is not applicable: when the regulation is in one language (say English) and the software application implementing it in another one (say French).

Within graphical software applications, we have studied only one application domain (air permeability measurement). It may be the case that within this application domain, the terminology and concepts of the regulatory documents is more present in the user-interface than in other domains. If true, this is a case to the external validity of our empirical results.

Related Work

Winkler and Pilgrim [START_REF] Winkler | A survey of traceability in requirements engineering and model-driven development[END_REF] and Spanoudakis and Zisman [START_REF] Spanoudakis | Software traceability: a roadmap[END_REF] provide an introduction and overview to software requirements traceability. They describe the different types of traceability relations and present the existing approaches to traceability mining including manual, semi-automatic and automatic approaches. On the one side, manual approaches require the developer or analyst to track each link by hand. On the other side, automatic approaches involve generating links between the requirement and the code artifacts dynamically.

Antoniol et al. [START_REF] Antoniol | Recovering traceability links between code and documentation[END_REF] introduced automatic traceability based on Information Retrieval (IR) methods. They compared a Vector Space Model and a probabilistic approach [START_REF] Antoniol | Recovering traceability links between code and documentation[END_REF]. Both methods produced promising results. Marcus and Maletic [START_REF] Marcus | Recovering documentation-tosource-code traceability links using latent semantic indexing[END_REF] explored Latent Semantic Indexing methods obtaining results as good as Antoniol et al. with less preprocessing. The effectiveness of IR techniques in traceability have been furthered examined by De Lucia [START_REF] De | Recovering traceability links in software artifact management systems using information retrieval methods[END_REF] and Hayes et al. [START_REF] Huffman | Advancing candidate link generation for requirements tracing: The study of methods[END_REF].

One issue encountered by the previous methods relates to terminology mismatch. Code artifacts do not always use the same terminology as in the re-quirements. Cleland-Huang et al. [START_REF] Cleland-Huang | A machine learning approach for tracing regulatory codes to product specific requirements[END_REF] focused on machine learning approaches including term mining and training using manually extracted tracing matrices to cope with this issue and improve the precision of automated trace-retrieval methods. Our work proposes another approach to address this terminology issue by introducing a novel process based on user interface labels. Some traceability tools propose traceability links management such as TOOR developed by Pinheiro and Goguen [START_REF] Pinheiro | An object-oriented tool for tracing requirements[END_REF]. Those tools do not include traceability link recovery features. Asuncion et al. [START_REF] Hazeline Asuncion | An end-to-end software traceability tool in an industrial context[END_REF] provide a software application intended to support traceability link recovery. However, this software application is based on automated prospective traceability and does not use informationretrieval methods.

De Lucia et al. [START_REF] De | Recovering traceability links in software artifact management systems using information retrieval methods[END_REF] developed ADAMS, a traceability link recovery tool based on latent semantic indexing to retrieve traceability links. Leuser and Ott use a tool called TraceTool [START_REF] Leuser | Tackling Semi-automatic Trace Recovery for Large Specifications[END_REF] implementing two IR algorithms (tf/idf and LSI) to evaluate various optimizations on large specifications written in German. Klock et al. generalized this approach with Traceclipse, an Eclipse plug-in providing a generic platform for integrating traceability link recovery into an IDE [START_REF] Klock | Traceclipse: an eclipse plug-in for traceability link recovery and management[END_REF]. Those authors do not explore the terminological richness of userinterface labels.

Conclusion

In this paper, we have introduced a new approach for traceability link recovery using the user interface texts. This approach leverages the terminological richness of UI labels in order to recommend relevant code elements. Our prototype system uses a variety of optimizations related to indexing, relations between UI labels and code, etc. We have evaluated our approach using a commercial system that implements a European regulation in the domain of green building. A domain expert has assessed 1000+ recovered traceability links. The evaluation concludes that our system is very often able to provide a relevant code item in the top-5 results. An area of future work would be to compare our approach based on user-interfaces against one based on code symbols. We are also trying to get an industrial impact with a free plugin implementing the traceability approach in the IntelliJ integrated development environment.

Figure 1 :

 1 Figure 1: Traceability link extraction overview.

Figure 2 :

 2 Figure2: Our Evaluation Process. The collection of regulation fragments and the manual assessment of results is done by an expert who masters both the application domain (green building) and code base of the application under study.

Figure 3 :

 3 Figure 3: Distribution of the number of traceability results for the 53 considered regulatory items. The median number of referred Java classes is 65; showing that many UI labels match more or less closely the regulation items.

Table 1 :

 1

	Short	Long query
	query

see https://en.wikipedia.org/wiki/Tf-idf, last accessedMarch 11

The verification process later in the process (out of scope of our approach) is responsible for checking that nothing has been forgotten during the implementation of the change.


L Leotoing

D Guines

E Ragneau

An in-plane tensile test for rheological and formability identification : comparison between experimental and numerical FLC

Keywords: Biaxial testing, cruciform specimens, forming limit diagram, rheological behavior PACS: 81.70.-q

HAL is

INTRODUCTION

Metal forming operations are carried out under multi-axial states of stress, then limiting the mechanical behavior identification of materials to uniaxial tensile tests can lead to a misrepresentation of this behavior. The use of a more realistic loading during identification tests such as biaxial loading conditions must lead to a more accurate representation of the expected behavior of the material. For the evaluation of formability, the well-known method consists in plotting a Forming Limit Diagram (FLD) for a given material. In order to overcome the drawbacks of the classical tests, like Marciniak test or Nakazima test, a cruciform specimen can be an interesting alternative if the strain path at the onset of necking is directly controlled by the testing speed of each actuator of the biaxial machine, independently from the specimen shape and without any friction effects. Moreover, this device is well adapted to make tests under dynamic conditions and then evaluate the effect of strain rate on formability of materials. Indeed, strain rate sensitivity has been identified as an important factor for determining formability and can alter significantly the level and shape of forming limit curves. However, to date, very few studies have been done to characterize the forming behavior of sheet metals in a large range of strain rates, mainly due to a lack of reliable experimental data.

In this study, a servo-hydraulic testing machine provided with four independent actuators is used to realize inplane biaxial tensile tests on sheet specimens. For this biaxial test, a dedicated cruciform specimen shape has been designed and optimized. Then, an aluminium alloy (AA5086) has been tested to identify from the same specimen both formability limits and whole elasto-plastic behavior, including hardening law and anisotropic yield criterion [1]. In order to validate this material identification, a comparison is proposed with the classical method which consists in testing three uniaxial specimens, in rolling, transverse and 45 • directions. The accuracy of the two approaches is evaluated through the determination of numerical FLCs which are compared with the experimental one.

BIAXIAL EXPERIMENTAL DEVICE

Biaxial testing machine and performance

In this work, a servo-hydraulic testing machine provided with four independent dynamic actuators is used. The center point of the specimen is always maintained stationary throughout the test by an efficient servo-hydraulic control. For each actuator, the loading capacity is 50KN and the maximum velocity can reach up 2m/s. The load on each axis is measured by two specific gage sensors placed between the grip and actuator rod.

Cruciform specimen shape

In order to determine the forming limits under complex loading paths, a specimen shape has been defined. The main difficulty to design an appropriate cruciform specimen shape consists in forcing the onset of necking in the central zone and not in the arms of the specimen. Although cruciform specimens have been investigated quite extensively, no standard geometry exists to this day [2]. Moreover, for dynamic tests, the maximum stiffness of the specimen, the initial one, has to be adjusted according to the experimental setup capacities in order to control tensile test velocities. Other considerations like manufacturing constraints must also be taken into account.

To ensure that necking always appears in the central zone, the design of the cruciform specimens must induce the greatest deformations in this zone and no strain localizations in the other areas (grooves, fillets, ...). From finite element simulations, different specimen geometries have been investigated. The more effective and the more promising specimen shape has been optimized in order to make efficient its use for a whole forming limit curve identification [1]. The cruciform specimen shape obtained at the end of the optimization stage is presented Figure 1. For this geometry, the strain path value at the central point of the specimen is directly linked to the velocity ratio of actuators. At the end of the test, at the onset of necking, the strain path remains constant for all speed ratios. The linearity of the strain paths, obtained from the optimized geometry of the specimen, is illustrated Figure 2 for three different strain path values.

Strain measurement

The Digital Image Correlation technique (DIC) is used to evaluate the strain field during the experiments. To capture the consecutive images during the test, a Fastcam ultima APX-RS digital CMOS camera associated with a macro lens is used. The maximum acquisition rate is 3000i/s for a 1024 × 1024 pixels full resolution range. For the cruciform specimen geometry defined above, the dimensions of the filmed central zone are about 30 × 30mm. A resolution of

RHEOLOGICAL BEHAVIOR IDENTIFICATION

The elastic part of the aluminium alloy behavior is described by Hooke's model with Young modulus, E = 67293MPa, and Poisson ratio, ν = 0.3. For the plastic part, an anisotropic Hill48 yield criterion is considered and the hardening behavior of the material is described by a Voces's law :

σ = σ 0 + Q √ 1 -exp(-Bε) (1)
where σ and ε are the equivalent stress and the equivalent plastic strain respectively, σ 0 is the yield stress.

From experimental data and a FE model of equi-biaxial tensile tests, an inverse procedure of identification is carried out to determine the material parameters (constants of both hardening law and anisotropic yield criterion). The experimental data are the global force measurements on the two specimen axis and local displacements of two points (points 1 and 2) chosen in the central zone of the cruciform specimen (Figure 3). The best set of parameters is obtained by imposing experimental forces on the FE model and minimizing the difference in a least-square sense between the experimental displacements and the numerical ones. The commercial software packages ABAQUS and ModeFrontier are used respectively to simulate the equi-biaxial tensile test and for the optimization procedure. Table 1 shows the results of this identification. In order to validate the material identification based on biaxial experiments, material parameters obtained from the classical method which consists in testing three uniaxial specimens, in rolling, transverse and 45 • directions are also presented Table 1.

FLC IDENTIFICATION

For the numerical identification of forming limit curves, the main difficulty lies in the choice of an appropriate criterion to detect the onset of necking. Different failure criteria have been discussed by Zhang et al [3]. If the necking occurs in a zone, a sharp change of strain can be observed, corresponding to the onset of a plastic instability. Moreover, after the onset of this phenomenon, the level of strain remains constant in the other adjacent zones. This behavior can be observed by following the evolution of the equivalent plastic strain for different nodes in the specimen. To illustrate this phenomenon, the evolution of the equivalent plastic strain for four nodes (1 to 4) of the central zone of the cruciform specimen have been plotted Figure 4 for equi-biaxial conditions. An homogeneous evolution for these four curves is observed up to a moment of about t=3.5 s, after which the equivalent plastic strain for node 1 diverges rapidly compared to that of nodes 3 and 4. As expected, for these two nodes, the equivalent plastic strain remains constant after the onset of necking in the zone of node 1.

In this work, the criterion widely used in the M-K model studies is chosen to predict the onset of localized necking. When the equivalent plastic strain increment ratio between a point located in the necking zone (node 1) and a point in a adjacent zone (nodes 3 or 4) attains a critical value, the onset of localized necking is assumed to occur and the corresponding major and minor strains calculated at node 1 are retained as a point on the FLC. In the literature, this critical value generally varies from 7 [4] to 10 [5] and it is necessary to rigorously fix it for our specific geometry. From Figure 4, the plastic instability is clearly located with the evolution of the equivalent plastic strain at node 1 thanks to the onset of a bifurcation point at t Bi f urcation = 3.4s. We propose to plot the evolution of the equivalent plastic strain increment ratio between nodes 1 and 3 (∆ε 1 /∆ε 3 ) and between nodes 1 and 4 (∆ε 1 /∆ε 4 ). In order to apply this method for experimental identification of forming limit curves, the increment must not be calculated for a short time, otherwise many fluctuations associated with the accuracy of the experimental strain measurement could appear. The increment is calculated for a time of 1s and the results are plotted in Figure 5. For the time corresponding to the onset of necking (t Bi f urcation ), the critical values are identified for the two increment ratios. Afterwards, the higher and more sensible ratio ∆ε 1 /∆ε 4 will be used and a critical value of 8 seems to be appropriate to detect the onset of the bifurcation point and then the phenomenon of necking (Figure 5). For the time corresponding to the occurrence of necking, the numerical limit strain, i.e., the major and minor principal strains are calculated. Moreover, it is found that this critical value is well appropriate for the different speed ratios imposed on the two axes. To cover the whole forming limit diagram, the following speed ratios are tested : 1, 0.75, 0.5, 0.4, 0.25, 0.1, 0, -0.02, -0.1 and free for one axis. Then, a whole forming limit curve identified by use of this criterion can be plotted.

EXPERIMENTAL AND NUMERICAL FORMING LIMIT CURVE

The procedure described above is applied for the detection of both numerical and experimental forming limit strains. For the numerical ones, the Voce hardening law and the anisotropic Hill48 yield criterion identified either from biaxial experiments or uniaxial ones have been implemented in the FE model of the biaxial tensile test of the cruciform specimen. Figure 6 presents a comparison between the experimental and numerical formability of AA5086. One can see that the correlation, between experimental and numerical forming limit curves, is rather good, especially for the values of minor strain close to zero which is the zone of major interest for a FLC. The minimum of the FLC is not exactly reached for a zero value of the minor strain, this well known effect [5] is due to the non-linearity of the strain path at the beginning of the test. Figure 6 shows that the level of the numerical forming limit curve is sensitive to the identified material parameter values especially for the right hand side of the FLC. The numerical FLC from biaxial identification seems to be more precise that the one obtained from the three uniaxial tests. These results validate the identification procedure proposed in this work.

CONCLUSIONS AND PERSPECTIVES

The presented biaxial experimental device associated with a dedicated cruciform specimen is successfully used for a complete elasto-plastic characterization of the metallic sheet behavior of an aluminium alloy. The strain path controlled by displacements of actuators permits to cover the whole forming limit diagram. A good correlation between experimental and numerical results is shown, so the identified elasto-plastic behavior seems to be adapted for the numerical prediction of the forming limit curve, especially for large strains. The procedure, validated in quasi-static conditions, is now successfully adapted in dynamic ones in order to evaluate the influence of strain rate on formability.

FIGURE 1 .

 1 FIGURE 1. Specimen shape.

FIGURE 2 .

 2 FIGURE 2. Measured strain paths.

2 FIGURE 3 .

 23 FIGURE 3. Experimental and simulated displacements of two points in the central zone of the biaxial specimen.

FIGURE 4 .

 4 FIGURE 4. Evolution of the equivalent plastic strain for nodes 1 to 4 in the central zone of the specimen.

4 FIGURE 5 .

 45 FIGURE 5. Evolution of the equivalent plastic strain increment ratios (∆ε 1 /∆ε 3 ) and (∆ε 1 /∆ε 4 ).

FIGURE 6 .

 6 FIGURE 6. Experimental and numerical formability of AA5086.

TABLE 1 .

 1 Identified material parameters of the Voce law.

	Law	σ 0 (MPa) Q(MPa)	B	F	G	H	N
	Biaxial	134.3	342.6	2.7 0.723 0.662 0.338 1.545
	Uniaxial	130.2	300.4	3.9	0.7	0.637 0.363 1.494


Michel Fruchart

David Carpentier

Krzysztof Gawędzki

Parallel Transport and Band Theory in Crystals

Keywords: numbers: 71, 20, -b 03, 65, Vf 73, 43, Cd

We show that different conventions for Bloch Hamiltonians on non-Bravais lattices correspond to different natural definitions of parallel transport of Bloch eigenstates. Generically the Berry curvatures associated with these parallel transports differ, while physical quantities are naturally related to a canonical choice of the parallel transport.

An increasing effort has been recently devoted to the characterization of geometrical and topological properties of electronic bands in crystals. Several physical concepts, including the semi-classical evolution and the polarizability, were related through the geometrical notions of Berry connection, curvature, and phase to the parallel transport within electronic bands [1][2][3][4]. It was recently proposed to experimentally access such Berry properties in cold atoms lattices [5,6]. Blount already noted [START_REF] Blount | Solid State Physics[END_REF] that a quantity analogous to an electromagnetic vector potential characterizes isolated bands in a crystal. With the advent of the integer quantum Hall effect, the first Chern class of electronic bands was described via a connection defining parallel transport of eigenstates over the Brillouin zone [START_REF] Thouless | [END_REF]. Independently, Berry [1] introduced a similar notion of parallel transport to describe the adiabatic evolution of eigenstates of a time dependent Hamiltonian. Both notions of parallel transport were related soon after [9], and the properties of parallel transport of eigenstates in crystals are now associated with a Berry connection, although they are not necessarily related to a Hamiltonian adiabatic evolution.

Electronic eigenstates in a crystal can be deduced from a Bloch Hamiltonian depending on a quasi-momentum in the Brillouin zone [START_REF] Blount | Solid State Physics[END_REF]. However, as noted by Zak, this Hamiltonian does not have a unique matrix form for a given model on a crystal [10]. Several conventions exist for bases of Bloch states over the Brillouin zone, as recently discussed in the context of graphene-like systems in [11,12]. Such different conventions define different ways of transporting eigenstates parallel to the bases. The different parallel transports do not necessarily define unique "Berry geometrical properties" when restricted to a (few) band(s). It is the purpose of this paper to show that different "Berry curvatures" are obtained out of commonly used Bloch conventions for models defined on non-Bravais lattices such as graphene. Moreover, we point to a canonical Bloch convention that allows to define parallel transport in an (almost) unambiguous way. When projected on a (few) band(s), the latter defines the physically relevant "Berry connection" characterizing measurable properties of the crystal. The above situation differs from the Hamiltonian adiabatic evolution initially considered by Berry where the parallel transport in the whole Hilbert space is unambiguously defined from the start.

Let us begin by considering a crystal C, a discrete subset of a d-dimensional Euclidean space E d (the locations of atoms) acted upon by a Bravais lattice Γ ⊂ R d of discrete translations, Γ ≃ Z d . For non-Bravais crystals, several classes of translationally equivalent points of C called sublattices exist: the cardinal N of C/Γ is larger than 1. For example, the hexagonal lattice of graphene possesses N = 2 sublattices A and B marked, respectively, as full and empty circles in the insets of Fig. 1. The Bravais lattice is composed of vectors connecting sites of the same sublattice. The set C/Γ may be represented by a fundamental domain F ⊂ C, also called a unit cell, which has one point in each sublattice. A translation of F by a Bravais vector is also a possible choice for a unit cell but when N > 1 then there are choices of F that are not related in this way. Two such choices for graphene are illustrated in the insets of Fig. 1a andb.

Along with the translational degrees of freedom, one may consider a finite number of internal degrees of freedom of atoms, e.g. the spin of electrons, represented as a finitedimensional complex vector space V equipped with a scalar product •, • V . The Hilbert space of crystalline states is then the space H = ℓ 2 (C, V ) of V -valued squaresummable functions on C with the scalar product ψ|χ = x∈C ψ(x)|χ(x) V . The translation of states ψ ∈ H by a vector γ of the Bravais lattice is defined by the unitary operator T γ such that T γ ψ(x) = ψ(x -γ) for x ∈ C. Below, we shall assume for simplicity that V = C but all considerations generalize in a straightforward way to the case with internal degrees of freedom.

Operators T γ define a representation of the translation group Γ in H which may be decomposed into irreducible 1-dimensional components. Irreducible representations of Γ are its characters γ → e ik•γ , where k ∈ R d . As

e ik•γ = e i(k+G)•γ for G in the reciprocal lattice Γ ⋆ (com- posed of vectors G with G • γ ∈ 2πZ), the characters of Γ form a d-dimensional Brillouin torus BZ = R d /Γ ⋆ .
The decomposition of H into irreducible components is realized by the Fourier transform ψ → ψ, where

ψ k (x) = γ∈Γ e -ik•γ ψ(x -γ). (1)
Note that ψ k = ψ k+G and

ψ k (x -γ) = e ik•γ ψ k (x) (2)
so that T γ ψ k (x) = e ik•γ ψ k (x). Property (2) defines the Bloch functions on C with quasi-momentum k. For fixed k, such functions ϕ k (x) form a finite-dimensional space H k = H k+G that may be equipped with the scalar product

ϕ k |χ k k = C/Γ ϕ k (x)χ k (x).
As a Bloch function is determined by its values on a unit cell F ⊂ C, dim(H k ) = N . Note that spaces H k are defined canonically for each k, without any further choices.

Geometrically, the collection of vector spaces H k forms a complex N -dimensional vector bundle H over the Brillouin torus BZ. We shall call H the Bloch bundle. Spaces H k are the fibers of H and their scalar product equips H with a Hermitian structure. Sections of H are maps

k → ϕ k ∈ H k . They are smooth if functions k → ϕ k (x)
are smooth for all x. The Fourier transform ( 1) is an isomorphism between H and the space of square-integrable sections k → ψ k of the Bloch bundle H defined over BZ. It preserves the norm, as stated by the Plancherel formula. Its inverse is given by the normalized integral of ψ k (x) over the Brillouin torus.

To compare vectors in different fibers along a curve in the base space of a bundle one needs a parallel transport, as in the Bloch bundle where fibers H k are distinct vector spaces for different k. Such a transport is usually not given a priori and requires a prescription, provided by a so-called connection. Equivalently, a covariant derivative of sections encodes infinitesimal parallel transport along coordinate axes in the base space. Connections always exist, in particular in the Bloch bundle, but are nonunique and may be non-flat in the sense that the parallel transport of a state along a closed loop does not end in the same state even if the loop may be continuously deformed to a point. Only special bundles carry flat connections. The Bloch bundle does because it may be trivialized (but not in a canonical way). A trivialization of H is a family of smooth sections k → e i k , i = 1, . . . , N , defined over BZ (i.e. with e i k = e i k+G ), which for each k form an orthonormal basis (a frame) of H k . An example of a trivialization of H is provided by the Fourier transforms of functions δ x,xi concentrated at points x i , i = 1, . . . , N , of a fixed unit cell F ⊂ C. We shall denote by e Ii k the corresponding vectors in H k . Bloch functions decompose as

ϕ k = i ϕ k (x i )e Ii k .
The trivialization of H defined this way depends on the choice of F. If F ′ is another unit cell then x ′ i = x i +γ i with γ i ∈ Γ for an appropriate numbering of its points so that δ x,x ′ i = T γi δ x,xi and, consequently, e ′Ii k = e ik•γi e Ii k . Each trivialization permits to identify the Bloch bundle H with the trivial bundle BZ × C N and to equip H with a flat connection that we identify with the covariant derivative ∇ = dk µ ∇ µ acting on smooth sections k → ϕ k of H by the formula

e i k |∇ϕ k k = d e i k |ϕ k k (3)
where d = dk µ ∂ µ is the exterior derivative of functions on BZ. Flatness of the connection means that ∇ 2 = 0. It follows from the property d 2 = 0 of the exterior derivative. We shall denote by ∇ I the flat connection associated to the trivialization k → e Ii k of H defined above. ∇ I depends on the choice of unit cell F but is independent of the numbering of its points. For another unit cell F ′ related to F as discussed above, we have

∇ ′I = ∇ I - i i |e Ii k e Ii k | dk • γ i .
The difference is a differential 1-form with values in linear transformations of the fibers of H.

There exists a more canonical and physically more relevant way to equip the Bloch bundle with a flat connection. Upon a choice of the origin x 0 in the Euclidean space E d , we may identify Bloch functions with Γ-periodic functions on C by writing for any

ϕ k ∈ H k , ϕ k (x) = e -ik•(x-x0) u k (x). (4)
Clearly

u k (x + γ) = u k (x) and u k+G (x) = e iG•(x-x0) u k (x)
for γ ∈ Γ and G ∈ Γ ⋆ . This allows to identify the Bloch bundle with the quotient of the trivial bundle R d ×ℓ 2 (C/Γ) over R d , with the fiber composed of Γ-periodic functions u(x) on C, by the action

(k, u(x)) -→ (k + G, e iG•(x-x0) u(x)) (5)
of the reciprocal lattice Γ ⋆ . Note that an element G ∈ Γ ⋆ acts on u(x) by multiplying it by the Γ-periodic function e iG•(x-x0) , preserving the ℓ 2 (C/Γ) scalar product. The trivial bundle R d × ℓ 2 (C/Γ) has a natural flat connection given by the exterior derivative of its sections. As the functions e iG•(x-x0) do not depend on k, this connection commutes with the action of Γ ⋆ and, consequently, it induces a connection ∇ II on the Bloch bundle H. The induced connection is still flat: (∇ II ) 2 = 0, but it has a non-trivial holonomy along the non-contractible loops of the Brillouin torus BZ. This holonomy may be identified via relation (4) with the above action of Γ ⋆ on ℓ 2 (C/Γ). Physical importance of connection ∇ II resides in the fact that -i∇ II µ corresponds under the Fourier transform to the position operator multiplying wave functions ψ(x) by (x -x 0 ) µ . We can still use eq. ( 3) to define ∇ II but now with Bloch functions e i k (x) = e -ik•(x-x0) u i (x), where u i , i = 1, . . . , N , form any orthonormal basis of the space ℓ 2 (C/Γ) of Γ-periodic functions on C (note that e i k+G = e i k in this case). For the particular choice with u i (x) = γ∈Γ T γ δ x,xi , where x i are the points of a unit cell F, we shall denote the corresponding Bloch functions by e IIi k (x). Although connection ∇ II does not depend on the choice of F, it does depend on the choice of the origin x 0 of the Euclidean space, but in a very simple way. If we choose another origin x ′ 0 then e ′IIi k = e ik•(x ′ 0 -x0) e IIi so that ∇ ′II = ∇ II -i dk • (x ′ 0 -x 0 ), i.e. the two connections differ by a closed scalar 1-form. To compare connections ∇ I and ∇ II , one notes that e IIi k = e -ik•(xi-x0) e Ii k for the same choice of F. Hence ∇ II = ∇ I -i i |e Ii k e Ii k | dk • (x i -x 0 ). Often we restrict our attention to electronic states in a Γ-invariant subspace of H, such as valence bands in an insulator. This amounts to consider a subbundle E of the Bloch bundle, i.e. a collection of M -dimensional vector subspaces E k ⊂ H k smoothly dependent on k and such that E k+G = E k . Any connection ∇ on the bundle H projects to a connection E ∇ on E for which the covariant derivative of sections k → ϕ k ∈ E k is given by

E ∇ϕ k = P k ∇ϕ k (6)
where P k is the orthogonal projector from H k to E k . In general, flat connections project to connections with curvature. In particular, if ∇ is a flat connection obtained by formula ( 3) then

e i k | E ∇ϕ k k = j P ij k d e j k |ϕ k k where P ij k = e i k |P k |e j k k . In this case e i k | E ∇ 2 ϕ k k = j F ij k e j k |ϕ k k ( 7
)
where F ij = mn P im dP mn ∧ dP nj is the matrix curvature 2-form and

F = i F ii = tr (P dP ∧ dP ) is its scalar version.
The integrals divided by -2πi of F over 2-dimensional (sub-tori in) BZ give the 1 st Chern number(s) of the vector bundle E that are independent of the choice of connection. We may work, in particular, with the connections E ∇ I or E ∇ II obtained by projection of ∇ I and ∇ II to E. Scalar curvature F I corresponding to connection E ∇ I depends in general on the unit cell F ⊂ C whereas scalar curvature F II corresponding to connection E ∇ II does not depend on x 0 and is canonically defined for each subbundle E ⊂ H. To see this note that the difference between different frames of sections k → e i k and k → e ′i k involved in the definitions (3) has the form e i k = j U ji k e ′j k for unitary matrices U k . The corresponding projectors P ′ k and P k given by matrices P ′ij k and P ij k are related by the equality

P ′ k = U k P k U -1
k leading to the relation

F ′ = F + tr (dP ∧ U -1 dU -P U -1 dU ∧ U -1 dU ). ( 8
)
In general, operators U do not commute with P and we obtain different scalar curvatures. An exception is the relation between the sections k → e IIi k corresponding to different x 0 where operators U are scalar and P ′ = P resulting in the same scalar curvatures.

We shall use the above geometric setup for the case of the subbundle E ⊂ H of the valence-band states of an insulator. To this end, let us consider a tight-binding Hamiltonian

H = x,y∈C h x,y |x y| (9)
in the Hilbert space H, where |y denotes the state with localized wave function x → δ x,y . We shall assume that h x,y = h y,x vanishes for |x-y| outside some fixed range. If h x,y = h x+γ,y+γ for γ ∈ Γ then H commutes with Bravais lattice translations T γ and maps Bloch functions into Bloch functions, defining Bloch Hamiltonians H k = H k+G acting in the finite-dimensional spaces H k . Under the Fourier transform (1),

Hψ k = H k ψ k . (10)
Given a frame of sections k → e i k of the Bloch bundle H, Hamiltonians H k may be represented by N × N Bloch matrices e i k |H k |e j k k ≡ H ij k = H ji k related by unitary transformations for different choices of the frame but describing the same physics. In particular, we may obtain Bloch matrices H I k = H I k+G corresponding to the choice of frames e Ii k or H II k = H II k+G corresponding to the frames e IIi k , a situation considered in [13]. These matrices are the usual standard forms of Bloch Hamiltonians. In the context of graphene, they correspond to the two conventions discussed in [12]. The spectrum E k1 ≤ • • • ≤ E kN of the Bloch matrices is independent of the frame and coincides with the spectrum of operators H k . For insulators, the Fermi energy ǫ F has a value that lies in the spectral gap, i.e. it differs from all E kn . The subspaces E k ⊂ H k corresponding to E kn < ǫ F form the valence-band subbundle E of the Bloch bundle H. The space of sections of E is mapped by the inverse Fourier transform to the subspace E ⊂ H of the electronic states with energy < ǫ F that are filled at zero-temperature. The geometrical properties of bundle E have a bearing on the low temperature physics of the insulator. The previous discussion applies directly to the valence-band subbundle. In particular, we may equip E with different connections E ∇ I or E ∇ II whose scalar curvatures differ but give rise to the same Chern numbers.

We can express the projected connections E ∇ in terms of a local frame k → ϕ a k = i ϕ a i e i k , a = 1, . . . , M , of E, e.g. composed of the eigenstates of Hamiltonians H k with energies E ka < ǫ F over regions in BZ where no such energy levels cross. For ∇ given by (3), one has:

ϕ a k | E ∇ϕ b k k = i ϕ a ki dϕ b ki ≡ A ab k ( 11
)
where A ab = -A ba is the local connection 1-form, and F = a dA aa . The same relation defines the Berry connection extracted from the change of eigenstates under adiabatic changes of the Hamiltonian. Although there is usually no underlying physical adiabatic process involved in the definition of connections projected on the valence-band subbundle, those are often dubbed "Berry connections". We may just talk of Berry connections E ∇ I and E ∇ II on E. In particular, the (almost) canonical connection E ∇ II is related to the position operator projected on the subspace E = P -H of states with energy < ǫ F : the operator (x-x 0 ) - µ ≡ P -(x-x 0 ) µ P -corresponds under the Fourier transform to the covariant derivative -i E ∇ II µ and, for a single valence band, the commutator [(x -x 0 ) - µ , (xx 0 ) - ν ] measuring the non-commutativity of the projected position operators corresponds to the multiplication by the component F II νµ of the canonical scalar curvature F II . One may trace back the occurrence of Berry connection ∇ II in physical properties [3,4] to the above relations.

We now illustrate the different choices of Berry connections on the gapped graphene model with an alternate potential on different sublattices. The model is described by the Hamiltonian

H = t x,y |x y| + x∈C v x |x x| (12)
where x, y run through the nearest neighbor pairs on the hexagonal crystal C and v x = +v (resp. -v) on its sublattice A (resp. B). In convention I, for the unit cell F = {x A , x B } shown in Fig. 1a, the Bloch matrix Hamiltonians are

H I k = +v g I k g I k -v (13)
with

g I k = t[1 + exp(ik • b 1 ) + exp(-ik • b 2 )]
, where b i = ε ijk (a j -a k ) are the Bravais vectors between secondnearest neighbors and a i , i = 1, 2, 3, the ones between nearest neighbors shown on the inset in Fig. 1c. A second choice of unit cell F ′ shown in This ends our discussion of the two main conventions used to define Berry connections in subbundles of Bloch states, the main subject of the present paper. We showed how in one of those conventions the scalar curvature depends on additional choices and we identified another, more physical, convention in which the scalar curvature is unambiguously defined and relates to the position operator.

3 FIG. 1 .

 31 FIG. 1. The (normalized) Berry curvature F k of the valence band in a gapped graphene model (12) with v/t = 1 is plotted on the Brillouin zone (dashed hexagone) for conventions I (a and b) and II (c). The corresponding choices of unit cells for convention I are shown in the insets. Vectors ai connecting nearest neighbors in sublattices A and B of the graphene lattice are shown in inset (c). In all three cases, curvature F k is concentrated around the Dirac points of graphene. It depends strongly on the unit cell for convention I and is uniquely defined and respects the symmetries of the crystal for convention II.

  Fig. 1b leads to g ′I k = t[1 + exp(ik • b 2 ) + exp(-ik • b 3 )]. These Hamiltonians are periodic in k but they explicitly depend on the choice of unit cell. The Hamiltonians H II k in canonical convention II can be deduced from H I k using the change of basis matrix U k = e ik•x0 diag(e -ik•x A , e -ik•x B ) with an arbitrary x 0 . It takes the form (13) but with g I k replaced by gII k = e ik•a3 g I k = t[exp(ik • a 1 ) + exp(ik • a 2 ) + exp(ik • a 3 )]. In all cases the spectrum is E k± = ±(v 2 + |g k | 2 ) 1/2and, for ǫ F = 0, the valence band corresponds to the minus sign. Curvature F of the valence-band subbundle has only one component F 12 . It is represented in Fig.1for the three above conventions for the gapped graphene with unit v/t. The dependence of the Berry curvature on the Bloch conventions is clearly illustrated. In particular, for convention I, a rotation of the unit cell rotates the curvature plot while a translation of the unit cell amounts to a U (1) gauge transformation and does not change the curvature.

Acknowledgments. D.C. and M.F. thank B. Douçot, J.-N. Fuchs, G. Montambaux and F. Piéchon for insightful discussions. This work was supported by a grant from the Agence Nationale de la Recherche (ANR Blanc-2010 IsoTop).


Gilles Nogues

Thomas Auzelle

Martien Den Hertog

Bruno Gayral

Bruno Daudin

Cathodoluminescence of stacking fault bound excitons for local probing of the exciton diffusion length in single GaN nanowires

teaching and research institutions in France or abroad, or from public or private research centers.

Cathodoluminescence of stacking fault bound excitons for local probing of the exciton diffusion length in single GaN nanowires Gilles Nogues, 1, 2, a) Thomas Auzelle, 3 Martien Den Hertog, 1, 2 Bruno Gayral, 3 and Bruno Daudin 3 1) Univ. Grenoble Alpes, Inst. NEEL, F-38042 Grenoble, France 2) CNRS, Inst. NEEL, F-38042 Grenoble, France We perform correlated studies of individual GaN nanowires in scanning electron microscopy combined to low temperature cathodoluminescence, microphotoluminescence and scanning transmission electron microscopy. We show that some nanowires exhibit well localized regions emitting light at the energy of a stacking fault bound exciton (3.42 eV) and are able to observe the presence of a single stacking fault in these regions. Precise measurements of the cathodoluminescence signal in the vicinity of the stacking fault gives access to the exciton diffusion length near this location.

Carrier diffusion length is a key quantity in optoelectronics, as it notably plays a significant role in the competition between radiative and non-radiative processes, especially in materials with large densities of defects. Concerning nitride semiconductors, various studies have established that carrier diffusion length in InGaN and GaN bidimensional (2D) layers are rather small, in the range of 50-250 nm [1][2][3][4][5][6][7] . Such lengths are smaller than the typical distance between dislocations 8 , one among other possible reasons for the surprising efficiency of radiative recombination in InGaN quantum wells in spite of the high density of defects in current heterostructures. With the aim of further efficiency improvement, a current trend in nitride optoelectronics research is to explore the potential of nanowires (NWs) as building-blocks for light emission or absorption devices. This approach has already led to the realization of NW heterostructure-based lightemitting diodes (LEDs) [9][10][11][12] . In the case of these pioneering works, the small diameter of NWs raises the issue of the influence of size effects on carrier diffusion. More generally, the issue of carrier diffusion length in GaN NWs is poorly documented to date. Recent works [13][14][15] suggest that recombinations are faster in InGaN nanorods than in 2D layers, which is correlated to a smaller carrier diffusion length and has been tentatively assigned to possible surface damage. In this letter we address the issue of exciton diffusion length in single GaN NWs with a diameter in the range of 100 nm. For this purpose, low-temperature cathodoluminescence (CL) experiments have been performed on individual NWs in correlation with microphotoluminescence (µPL) and scanning transmission electron microscopy (STEM) studies that reveal the presence of I1 basal stacking faults (SFs). It is well known from 2D-layer studies that such SFs are radiative recombination centers [16][17][18] . Previous TEM/CL studies on GaN epilayers have already evidenced the correlation between the presence of different types of SFs with well identified emission peaks 16,17 . The density of structural a) Electronic mail: gilles.nogues@neel.cnrs.fr defects was relatively large in those samples. In the case of our NWs, we show that emission peaks are linked to the presence of a single defect, acting as a quasi-punctual recombination center. By varying the distance between the SF and the the CL excitation spot we investigate the diffusion length along the NW axis.

The nanowires are grown by Plasma-assisted Molecular Beam Epitaxy (MBE) on a 2-inch Si(111) substrate. Desoxydation of the silicon is done in situ by annealing at 950 • C until the clear appearance of the 7×7 surface reconstruction at 820 • C. The growth temperature, set at 820 • C, is determined by the measurement of the corresponding Ga desorption time 19,20 . A thin AlN buffer layer (2-3 nm thick) is grown directly onto the silicon. Used as a seed layer for the GaN nanowires, it decreases the tilt of the NWs relative to the normal to the surface 21 . This helps to obtain well separated nanowires, even for extensive growth time. NWs were grown for 18 h in Nrich conditions (Ga/N ratio of 0.3) 22 . They are L =3 µm long and their diameters range from 50 to 100 nm. As grown NWs are mechanically dispersed onto 2 kinds of substrates. Substrate S1 is used for CL and µPL studies. It is made of a Si wafer. Localization marks are patterned by standard deep UV (DUV) optical lithography followed by reactive ion etching (RIE), using the resist as a mask. It is then sputtered with a 100 nmthick Al layer and a 20 nm-thick SiO 2 layer. For combined TEM and CL studies, substrate S2 consists of a home-made 35 nm thick Si 3 N 4 membrane with a window size of 90 µm. Arrays of nitride membranes are fabricated starting from a 200 µm thick Si (100) wafer with a layer of low stress Si 3 N 4 of 35 nm on top of a 240 nm thick SiO 2 layer on each side. The fabrication procedure is described in Ref. 23 , with the difference that for these membranes the SiO 2 layer is removed below the Si 3 N 4 layer by RIE and a sequential KOH etch. Optical DUV lithography and electron beam metallization are used to pattern Ti-Au markers on the membranes to locate the same NW in different experiments.

CL images are taken at 5 K in a FEI scanning electron microscope (SEM) and analyzed through a 45 cm spectrometer and UV-optimized grating with 600 grooves/mm. Figures 1(a-b) present simultaneously recorded SEM and CL images of 3 NWs on substrate S1. The color scale of the CL image is set in order to show that light comes from the whole length of the NWs. Nevertheless, a strong emission arises from 3 bright spots which saturate the image. Figure 1(c) presents three spectra with a fixed electron beam excitation at three locations (#1 to #3) shown on image 1(b). The spectrum #3 is representative of the emission from the whole length and is similar to the ensemble PL measurements on the as-grown sample. One observes a peak at 3.48 eV that corresponds to the emission from neutral donor bound exciton (D 0 X) and a peak at 3.28 eV with its phonon replica at lower energy that are attibuted to donor-acceptor pairs (DAP). Spectra #2 and #3 show that the bright spots are associated to an extra emission peak at 3.42 eV attributed to the emission from excitons trapped by stacking faults (SFX) 16,24,25 . CL observations allow us to locate well isolated NWs with SFX emission that are further studied by microphotoluminescence. µPL spectra are similar to Fig. 1(c) with a magnitude of the 3.42 eV peak slightly smaller or comparable to the D 0 X one. This is well understood by considering that the laser excitation spot is larger than the NW and that the µPL spectrum integrates light emitted over its whole volume. The linewidth of the D 0 X peak varies significantly from one NW to the other between 2 and 20 meV, whereas the 3.42 eV peak linewidth is always in the 10-20 meV range. The larger linewidth of the 3.42 eV peak has been reported before 26 and is still of unknown origin.

Further evidence of the correlation between the emission at 3.42 eV and the presence of a SF is given by joint CL and STEM studies of the same NW on a S2 substrate. A well isolated NW is first identified in lowtemperature CL [Figs. 2(b-d is more homogeneous, except for two areas where it is quenched. One of them corresponds to the place where the 3.42 eV emission occurs, the other one is in the middle of the NW. In Figure 2 The profile of the CL signal along the NW longitudinal axis at 3.42 eV is therefore related to the probability for an exciton to be trapped by a single SF acting as a radiative recombination center. We define n FX (x, t) (resp. n D 0 X and n SFX ) as the linear density of free excitons (FX) (resp. neutral donor and SF bound excitons) between x and x + dx. Extending the model of Corfdir et al. 18 , and using the same assumptions (absence of non-radiative decay channels and detrapping processes of bound excitons), one writes:

∂n FX ∂t = D ∂ 2 n FX ∂x 2 + p(x) - 1 τ r,FX + 4s d n FX (1)
-

1 τ FX→D 0 X + 1 τ FX→DAP + γ FX→SFX (x) n FX ,
where D is the diffusion constant of the FX in the material. p(x) represents the excitation term. It is assumed constant over the wire length in µPL. On the contrary it is point like in the case of CL. The FX population can decay either radiatively, with time τ r,FX , or through surface effect, s being the surface recombination velocity and d the NW diameter [START_REF] Shockley | Electrons and Holes in Semiconductors[END_REF] . The last line of Eq. 1 corresponds to the trapping of free excitons by neutral donors (characteristic time τ FX→D 0 X ), DAPs (τ FX→DAP ) or the SF. In the latter case, γ FX→SFX (x) has a non zero value 1/τ 0 F X→SF X only close to the SF, with a characteristic extension w SF . In the same manner one has:

∂n D 0 X ∂t = n FX τ FX→D 0 X - n D 0 X τ r,D 0 X , (2)
∂n SFX ∂t = γ FX→SFX (x)n FX - n SFX τ r,SFX , (3)
where τ r,D 0 X (resp. τ r,SFX ) is the radiative decay time of the neutral donor (resp. stacking fault) bound exciton. The total SFX (resp. D 0 X) fluorescence signal is L/2 -L/2 n SFX /τ r,SFX dx (resp. n D 0 X /τ r,D 0 X ). µPL observations show that their stationnary values have the same amplitude. Hence the SF capture rate is dramatically larger than the one of the neutral donors, in agreement with the results of Ref. 18 . From Eqs. 2 and 3, one infers τ 0 FX→SFX ≃ τ FX→D 0 X * w SF /L. Using a SF capture range 24 w SF =3 nm, 1/τ 0 F X→SF X is 300 times larger than the capture rate by neutral donors.

We now assume a single NW extending from -L/2 to L/2 and containing a single SF at x = 0. It is excited at position x p by the electron beam, p(x) = p 0 δ(x -x p ). Outside the SF capture range, the stationnary solution to Eq. 1 is n FX (x) = n 0 exp(-|x -x p |/L NW ), where L NW = √ Dτ eff is the diffusion length in the NW, with:

1 τ eff = 1 τ r,FX + 4s d + 1 τ FX→D 0 X + 1 τ FX→DAP . (4)
Time-resolved photoluminescence experiments in ensemble of GaN nanorods reported τ r,FX ≃100 ps [START_REF] Harris | [END_REF]29 and s =2.7 10 4 cm/s 13,14 . In the set of NWs we have studied d =110±10 nm. Hence the two first terms of Eq. 4 contribute with the same magnitude to the effective decay rate of the FX. The capture rate by neutral donors or DAPs depends on their density. For pure undoped thick layers of GaN, CL experiments report exciton diffusion lengths in the vicinity of threading dislocations ranging from L 2D =81 30 to 201 nm 31 . In our NW sample, the absence of a peak at the FX energy and the presence of D 0 X and DAP peaks in the spectra let us infer that the capture rate 1/τ FX→D 0 X + 1/τ FX→DAP is larger than the radiative or surface decay terms.

Close to a SF, the effective decay rate 1/τ eff is dominated by the SF capture rate 1/τ 0 FX→SFX , which is L/w SF larger than the D 0 X capture rate. The corresponding diffusion length is therefore divided by a factor L/w SF . It is in the 10 nm range, of the same order of magnitude as w SF . This means that a large fraction of incoming excitons is trapped by the SF and later converted into a photon. The SF fluorescence signal is therefore proportional to n FX (x = 0) = n 0 exp(-|x p |/L NW ).

Figures 3(b) and (d) plot the CL signal in logscale as a function of the longitudinal wire coordinate x for two different NWs on a S1 substrate. The peaks are well fitted by convoluting an exponential decay with a Gaussian function of width w p =30±10 nm for electron acceleration voltages V ≥10 kV. This is larger than w SF and rather reflects the size of the excitation region by the electron beam. For V =5 kV, w p is degraded to 70 nm. We attribute this effect to backscattered electrons in the NW at lower energy. At distances larger than w p from the peak center, the tails on each side are well fitted by exponential functions with characteristic length L NW (dashed lines). Average of L NW measured for an ensemble of 12 NWs gives L NW =68±16 nm< L 2D . Based on the previous discussion, it confirms that the capture by of neutral donors or DAP dominates the FX effective decay rate. We also observe a large dispersion on the values of L NW , with sometimes abrupt changes like around x =800 nm on Fig. 3(d). This is evidence of long range variations of 1/τ eff . The latter can arise from changes of the density of neutral donors or DAPs or the presence of an extra non radiative decay term -n FX /τ nr,FX (x) in Eq. 1 due to other kind of impurities.

Further evidence of long range variations of the NW parameters is given by the CL signal at the D 0 X emission energy [Figures 3(a) and(c)]. Its amplitude variations can only be accounted for if one assumes that τ FX→D 0 X varies along the wire length or if one adds an extra non radiative decay term -n D 0 X /τ nr,D 0 X (x) to Eq. 2 due to the trapping of the D 0 X by other defects. For example, a SF can capture excitons from the neighboring neutral donors 18 resulting in a localized quenching area [arrows on Figs. 3(a,c)]. However, we also observe extended NW areas where a partial quenching of the signal occurs. This can only happen if the previously introduced capture or decay rates experience long range variations and are not just point-like as for the SF defects. We observe no clear correlation between the D 0 X signal magnitude and L N W . This means that 1/τ eff is determined by other parameters like DAP capture rate or non radiative decay.

In conclusion, we are able to observe single SFs present in individual NWs with different optical and structural techniques. The CL profile in the vicinity of a SF is well understood by a simple diffusion model of the exciton.

FIG. 1 .

 1 FIG. 1. (a) SEM image of 3 dispersed NWs. (b) Corresponding image of the CL signal between 3.40 eV and 3.50 eV. (c) CL spectra at 5 K with a spot excitation at points #1 to #3 shown on (b).

  FIG. 2. Same NW observed in BF STEM (a), SEM (b), CL at 3.48 eV (c) and 3.42 eV (d). (e) HAADF STEM and (f) BF STEM images of the region marked by a red rectangle in (a). (g) BF STEM zoom on the blue rectangle in (a).

  (a) a bright field scanning TEM (BF STEM) image of the same NW is shown oriented along the [2-1-10] direction. We note that between the two observations, the NW remained at the same location but rotated onto itself. It explains why the small piece of GaN attached to the NW is at different positions between Figs. 2(a) and (b). The region marked by the red rectangle in Fig. 2(a) is shown at higher magnification in BF STEM [Fig. 2(f)] and high angular dark field (HAADF) STEM images [Fig. 2(e)] which reveal the presence of a stacking fault. For sake of clarity it is indicated by an arrow in both images, as well as a line following the stacking of the Ga columns in the 2H hexagonal wurtzite structure (ABABA.. stacking), that is disrupted by the insertion of one C plane (..ABABC AC.. stacking) characteristic of a cubic zinc blende phase. Comparison of the CL and STEM data clearly shows that the 3.42 eV emission exhibits an excellent spatial correlation with the location of the SF. On the other hand Fig. 2(g) shows a higher magnification BF STEM image of the region indicated with the blue rectangle in Fig 2(a), that is correlated with a quenching of D 0 X CL emission. In this region we observe no feature indicating the presence of crystal defects. A continuous smooth contrast over this region was confirmed by high resolution STEM, as for the rest of the NW (not shown here).

FIG. 3 .

 3 FIG. 3. CL intensity profiles along the longitudinal coordinate x for two nanowires NW9 (a-b) and NW10 (c-d). (a) and (c): D 0 X signal in linear scale. Arrows mark localized areas where it is unambiguously quenched by the presence of a SF. (b) and (d): SFX signal in log scale. Dashed lines are exponential fits with characterisctic length LNW indicated next to each line.

  3) CEA, INAC, F-38054 Grenoble, France

	(Dated: 12 March 2014)

ACKNOWLEDGMENTS

This work was performed in the CEA/CNRS joint team "Nanophysique and semiconductors" of Institut Néel and INAC, and in the team "Structure and properties of materials. Extreme conditions" of Institut Néel. We acknowledge help from the technical support teams of Institut Néel: "Optics and microscopies" (Fabrice Donatini) and "Nanofab" (Bruno Fernandez) and benefited from the access to the technological platform NanoCarac of CEA-Minatech. We acknowledge financial support from ANR programs JCJC (project COSMOS, ANR-12-JS10-0002) and P2N (project FIDEL, ANR-11-NANO-0029).