Abdesamed Réda

Amina Meherhera
email: fameherhera@esi.dz

Mekideche Imane
email: fimekideche@esi.dz

Dr Zemmouchi Ghomari
email: aghomari@esi.dz

Pr Ghomari

A Survey of Current Approaches for Transforming Open Data to Linked Data

Keywords: Linked Data, Open Data, Transforming Approaches

come   L'archive ouverte pluridisciplinaire

Introduction

The world wide web counts today billions of web pages and documents that are full of data; the classic web shows significant limitations in terms of organising, extracting, and processing the available data on documents. Thus, the necessity of moving to another version of the web, the semantic web, introduced by Tim-Berners-Lee in 2009, which enables data providers to publish their data efficiently so that it can be machine-readable and facilitate its reuse. One of these techniques is the linked data paradigm, which aims to enable uniform practices for publishing interconnected data on the web using semantic web technologies [START_REF] Heath | Linked data: Evolving the web into a global data space[END_REF].

This transforms the classical web from a network of documents to a network of data allowing software agents and machines to interpret the data.

The adoption of the interconnected data principles by a large number of companies, research centers, and institutions around the world has led to the creation of a global data space with interconnected data from different domains, such as people, companies, books, films, statistical and scientific data. [START_REF] Heath | Linked data: Evolving the web into a global data space[END_REF]. This data network is called the "Linked Open Data (LOD) Cloud", a massive network of published datasets available via the existing web infrastructure and interconnected according to the linked data principles. The linked data principles provide general guidance for making a given dataset available on the web in linked data format. However, there are several different tools, methods, and techniques for generating and publishing these datasets. In this paper, we present a bibliographical study of linked data in its context, the semantic web, as well as the set of existing approaches and current efforts to perform the transformation from open data to linked data. For this aim, we conduct a systematic survey on the literature related to transforming web open data to linked data. Our aim is to provide new interpretations of existing literature. Our survey is composed of four sections. Section 2 covers basic concepts of the paper domain. Section 3, describes the process of systematic survey. Section 4 presents the selection and the study of existing approaches and initiatives to transform open web data to linked data. We classified them according to some criteria, such as automation degree, supported files format and application domain. We assessed the methodological quality of studies and discussed the different initiatives, where we will identify the significant limitations and challenges they face. Section 5 concludes the paper and presents its perspectives.

Background

In this chapter, we introduce the basic concepts of the semantic web and linked data.

Definitions:

The semantic web is an extension of the World Wide Web that aims to publish data in a format that is machine-readable and thus allows computers to intelligently search, combine and process web content based on its semantics. [START_REF] Hitzler | Foundations of semantic web technologies[END_REF] It is based on a set of best practices for publishing and interlinking structured data of multiple sources on the web. Theses best practices are known as the linked data principles mentioned in section 2.2. [START_REF] Bizer | Linked data: The story so far[END_REF] Linked data makes the World Wide Web into a global database that we call the web of data. Developers can query linked data from multiple sources at once and combine it on the fly, something challenging to do with traditional data management technologies. [START_REF] Wood | Linked Data[END_REF] We can't talk about linked data without Open data which can be summed up in the statement: "Open means anyone can freely access, use, modify, and share for any purpose "3 . Publishing Data on the web as linked data requires some special technologies: RDF for representing data and SPARQL for querying data are the most important semantic web tehnologies.

RDF 4 is a language for representing information about resources in web. It is based on the idea of identifying things using web identifiers (called Uniform Resource Identifiers, or URIs) and describing resources in terms of simple properties and property values. This enables RDF to represent simple statements about resources as a graph of nodes and arcs representing the resources, their properties and values. 5 In RDF, the description of a resource is represented as a number of triples. The three parts of each triple are called: subject (the URI identifying the resource), object (can be either a literal value or a URI) and predicate (indicates the relation between the subject and the object). SPARQL 6 , is a set of specifications that provide language and protocol to query and manipulate RDF graph content on the web or in an RDF store 7

Linked data principles

Tim-Berners-Lee, the inventor of the web and the initiator of the Linked Open Data project, introduces the four linked data principles:

1. Use URIs as names for things. 2. Use HTTP URIs, so that people can look up those names. 3. When someone looks up a URI, provide useful information, using the standards (RDF, SPARQL). 4. Include links to other URIs, so that they can discover more things.

Linked Data applications

"Linked data successes are difficult to see because they are under the hood" [START_REF] Wood | Linked Data[END_REF]. Linked data has been widely used since its inception, we cite here some of the success stories of linked data: [START_REF] Zemmouchi-Ghomari | Linked Data, a manner to realize the web of data[END_REF] -Google Knowledge Graph: it was introduced by Google in 2012 to enhance search results and allow users to extend their knowledge by searching for related topics. [START_REF] Pelikánová | Google knowledge graph[END_REF]. The google snippet that is shown in the right side when searching for something on Google is one of its use cases. -Open Government Data (OGD): a collaboration between US, UK, France and Singapoure governments to publish machine readable datasets. Actually, the data.gov offers more than 5 billions RDF triples. -BBC website: One of the most known success stories of linked data is the BBC website that uses the linked data principles to publish their data. 3 Methodology

In our work, we followed a systematic literature approach: a synthesis of the scientific literature in response to a specific question. It uses explicit methods for restrictively searching, selecting, and analyzing data. Our approach is based on the guidelines proposed by [START_REF] Kitchenham | Procedures for performing systematic reviews[END_REF] and [START_REF] Dyba | Applying systematic reviews to diverse study types: An experience report[END_REF]. The procedure is:

1. Definition of the terms/questions of the research. 3.2 Selecting the sources (digital libraries) to be searched.

In order to cover the widest possible range of relevant publications, we have identified and used the most widely used electronic libraries, namely : ACM Digital Library, IEEE Xplore, ScienceDirect, Google Scholar, HAL, SemanticScholar.

4 Literature Review

Research and selection of studies :

In order to select the most relevant articles, we have proceeded to gathering, analysing and then filtering articles in four steps:

1. Searching for articles using the keywords mentioned in section 3.1.

2. Selecting articles with titles related to our study: the result of this step was: 42 articles.

3. Quick reading of the abstract, introduction and conclusion to select according to reliability (number of citations, variety and quality of references, reputation of the author, source of the article) and relevance (The article proposed an approach to transform open data to linked data or contains relevant information for our study): The result of this step was: 22 articles. 4. Finally, we have read the articles and applied an other filter on them to select those which:

proposed a new approach for transforming open data to linked data proposed an approach producing linked data of good quality (respecting the four linked data principles). performed tests on their approach and good results have been revealed.

explained the steps of their approach in a non ambiguous way. The result of this step was: 12 articles.

Figure 1 represents the process that we have followed to filter and select articles. In this section, we will talk about the approaches resulting from the three filters:

- [START_REF] Lebo | Converting governmental datasets into linked data[END_REF] : proposed an incremental approach of transforming governmental datasets available in data.gov to linked data by converting to RDF format and then using existing vocabularies for interlinking. - [START_REF] Hyland | The joy of data-a cookbook for publishing linked government data on the web[END_REF]: the authors proposed a step-by-step process for modelling data, converting and publishing new datasets that has enabled many governments to expose data for further visualisation, reuse and analysis -Linked data life cycles8 : the authors state that existing data management methods which assume control of data, schema and data generation cannot be used in the web environment because of its open and decentralised nature. This is an approach suggested by the W3C group as it covers all stages of the linked data life cycle, but unfortunately it focuses on government data. - [START_REF] Mulwad | Automatically generating government linked data from tables[END_REF] : proposed a technique to automatically produce good quality linked data by deducing the semantics of columns headers, cell values and relations between columns inferred by knowledge base from different data sources such as the LOD cloud . - [START_REF] Gupta | Integrating linked open data with unstructured text for intelligence gathering tasks[END_REF]: creates a RDF Triple Store populated with triples produced from text documents to extract relations between terrorists or terrorism organisms.

- [START_REF] Villazón-Terrazas | Methodological guidelines for publishing government linked data[END_REF] : covers all stages of government linked data life cycle.

- [START_REF] Auer | Managing the life-cycle of linked data with the lod2 stack[END_REF]: provides software tools for the integration of the very heterogeneous LOD2 data in a coherent framework. - [START_REF] Atemezing | Transforming meteorological data into linked data[END_REF]: worked on transforming data coming on streaming of AEMET 9 dataset to linked data by developing a specific ontology for the dataset. - [START_REF] Bischof | City data pipeline[END_REF]: it aims to integrate city data into a common data model using semantic web technologies in order to help city managers with their decisions by providing automated analytical support. - [START_REF] Sharma | Automatically converting tabular data to rdf: An ontological approach[END_REF]: an approach based on conceptual mapping between CSV components and ontology components and keeping relations between columns by proposing algorithms to detect the datatype or the object. - [START_REF] Rinciog | Govlod: Towards a linked open data portal[END_REF] : presented a platform that aims to change the paradigm of open government data portals in Romania by publishing data in LOD format providing an easy way for developers to create applications without having to process the initial files (by providing a SPARQL engine). - [START_REF] Norman | Exploring a linked data approach in accessing physical oceanography archive[END_REF]: the goal of the datalift project was to foster the emergence of the web data by industrialising and facilitating the structuring, publication, interlinking and use of critical mass of data, providing access to spatial datasets from oceanographic archives on the web. - [START_REF] Jaglan | Lod: Linking and querying shared data on web[END_REF] : proposed a framework to link and interrogate data published as csv files on the web (the data.gov plateform) .

After briefly describing the contribution of each article , we classified the approaches according to different criteria:

-Supported files formats -Application domain -Degree of automation

The result of the classification are represented in the figure 2 4.3 Evaluation of methodological quality of the studies : Some approaches analyzed in this article, like [START_REF] Hyland | The joy of data-a cookbook for publishing linked government data on the web[END_REF] and [START_REF] Villazón-Terrazas | Methodological guidelines for publishing government linked data[END_REF] included most steps of the generally accepted linked data life-cycle but missed guidelines on how to use the generated linked data. They do not provide detailed guidelines for publishing the generated linked dataset on the web. Another point to consider is that most of the studies used explicit data sources and published them on the web as linked data without proper semantic description and documentation. We noticed that most are focused on public data. After analysing and comparing existing approaches in literature and selected in this article, we present the most common limitations:

-Some approaches are specific to a domain or a dataset and are hard to be adapted to another domain or other data available in the web.

Fig. 2: Classification of selected articles -Some approaches accept restricted files format as entries (CSV files are the most supported). -Some approaches are not linking to other datasets and thus not complying with the fourth linked data principle. -Some approaches have some steps that need human intervention (are semiautomatic).

Synthesis of the results :

The linked data principles provide general guidance for making a given dataset available on the web in linked data format. However, there are several different tools, methods and techniques for generating and publishing these datasets, and their application depends on the type, the nature of data and a list of other factors. These methods and techniques are brought together in several linked data life-cycle methodologies, which offer different approaches for dealing with linked data in specific areas and for specific purposes.

In the section 4.2 we have explained only those which passed the three filters explained in the section 4.1, but there are more, in several application areas and with different architectures and other specifications.

[8], [START_REF] Ngomo | Introduction to linked data and its lifecycle on the web[END_REF], [START_REF] De Meester | Declarative data transformations for linked data generation: the case of dbpedia[END_REF], [START_REF] Corlosquet | Produce and consume linked data with drupal![END_REF] proposed eventually approaches to produce linked data but are generally specific to a use case and can hardly be adapted to other use-cases.

In [START_REF] Luis Redondo-García | Linked data methodologies for managing information about television content: Applying linked data principles in the ontotv system, in order to improve the collection processes and the way television information is accessed[END_REF], [START_REF] Zengenene | Towards a methodology for publishing library linked data[END_REF] and [START_REF] Hallo | Transforming library catalogs into linked data[END_REF], the authors presented methodologies for linking data in the media and library domains respectively. [START_REF] Seán O'riain | Xbrl and open data for global financial ecosystems: A linked data approach[END_REF] in finance, [START_REF] Rowe | Data. dcs: Converting legacy data into linked data[END_REF] in education for the Department of Computer Science at the University of Sheffield,UK, [START_REF] Tao | Semantator: Semantic annotator for converting biomedical text to linked data[END_REF] in healthcare...and many others.

As for the conversion step, we can notice that most of them share the same steps but use different techniques to achieve the task. The difference between approaches is the way the mapping is done: for example [START_REF] Lebo | Converting governmental datasets into linked data[END_REF] and [START_REF] Hallo | Transforming library catalogs into linked data[END_REF] create RDF resources then map them to resources from existing vocabularies while [START_REF] Mulwad | Automatically generating government linked data from tables[END_REF] and [START_REF] Jaglan | Lod: Linking and querying shared data on web[END_REF] maps directly to resources of existing vocabularies. A special case is [START_REF] Atemezing | Transforming meteorological data into linked data[END_REF] who used preset RDF templates and fill in the data from files since all incoming files have same format. Some approaches proposed additional steps to enhance the quality of data produced: [START_REF] Mulwad | Automatically generating government linked data from tables[END_REF] proposed to detect links between columns to create new RDF classes for a better precision while [START_REF] Sharma | Automatically converting tabular data to rdf: An ontological approach[END_REF] worked on algorithms to detect the data types of cells (literal or object URIs). [START_REF] Hallo | Transforming library catalogs into linked data[END_REF] proposed an approach that is totally different: it converts data to a relational database before transforming it to RDF format.

Conclusion

In this paper, we have analyzed several approaches to transform open data to linked data, then compared them and classified them according to different criteria and identified the most significant limitations they face.

The study of the existing linked data approaches provided us with a practical analysis of the methods and techniques that can be used in each step of the lifecycle for a linked data dataset. We drew experience on the different ways to get the data, different modeling approaches, the best ontology (re)use practices, the various methods for data transformation which, the different ways of publishing the dataset on the web, as well as the extensive ways the datasets can be used in real-world applications and services. We identified the specifics, advantages, and drawbacks of the existing linked data methodologies. Our future efforts will be guided by the perspective of the semantic web realization through linked data principles. We plan to propose a generic and fully automatic transformation approach that supports more than one entry file type and provides links to external datasets.

3. 1 -

 1 Defining the terms/questions of the research: The identification of research questions is essentially what distinguishes systematic research from traditional research. The goal of this survey is to analyse existing initiatives, tools and approaches to transform web open data into linked data. We therefore define the following as generic research questions: What are the existing approaches to transform open data to linked data? -How can they be classified ? -What technical aspects, features and functions are supported in the existing approaches ? -What are the current challenges and general trends related to the transformation of open web data to linked data? Keywords in search : Transform to linked data, Convert to linked data, Produce linked data , From open data to linked data.

Fig. 1 :

 1 Fig. 1: The followed process of articles selection

http://opendefinition.org/

https://www.slideshare.net/mediasemanticweb/linked-data-life-cycles


Zhijun Jiang

Charles Paillard

Hongjun Xiang

L Bellaiche

Linear versus nonlinear electro-optic effects in materials

Two schemes are proposed to compute the nonlinear electro-optic (EO) tensor for the first time. In the first scheme, we compute the linear EO tensor of the structure under a finite electric field, while we compute the refractive index of the structure under a finite electric field in the second scheme. Such schemes are applied to Pb(Zr,Ti)O 3 and BaTiO 3 ferroelectric oxides. It is found to reproduce a recently observed feature, namely why Pb(Zr 0.52 Ti 0.48 )O 3 adopts a mostly linear EO response while BaTiO 3 exhibits a strongly nonlinear conversion between electric and optical properties. Furthermore, the atomistic insight provided by the proposed ab-initio scheme reveals the origin of such qualitatively different responses, in terms of the field-induced behavior of the frequencies of some phonon modes and of some force constants.

Most materials exhibit a change in their refractive index when under applied static or lowfrequency electric fields, a phenomenon known as the electro-optic (EO) effect [START_REF] Weber | Handbook of Optical Materials[END_REF][START_REF] Lines | Principles and Applications of Ferroelectrics and Related Materials[END_REF] and which is promising for some technologies [START_REF] Zgonik | [END_REF][4][5][6]. In particular, having large nonlinear electro-optic coefficients would open the door for the design of novel devices [7][START_REF] Saleh | Fundamentals of Photonics[END_REF][START_REF] Yariv | Photonics: Optical Electronics in Modern Communications[END_REF][START_REF] Boyd | Nonlinear Optics[END_REF][START_REF] Didomenico | [END_REF][12][13][14][15][16]. For instance, it is important for EO modulation [13], high-speed optical shutters [14], electro-optical detection [15], and electrooptical switching [16]. Understanding at an atomistic level linear versus nonlinear EO effects should also be of large fundamental interest. For instance, it should resolve the current mystery of why a recent experiment [5] observed, in the THz regime, a linear electro-optic coefficient in Pb(Zr,Ti)O 3 while BaTiO 3 films rather exhibit significant nonlinear (second-order) electro-optic coefficients.

In view of such facts, having a first-principles-based technique allowing the computation of nonlinear conversion between electric and optical quantities but also providing a deep atomistic insight of such conversion is highly desired. However, such technique and ab-initio capabilities do not presently exist.

The aims of this paper are to demonstrate that such technique (1) can, in fact, be easily developed and applied to any material; (2) reproduces the experimental finding about the different nature (i.e., linear versus nonlinear) of the EO response of Pb(Zr,Ti)O 3 versus BaTiO 3 ; and [START_REF] Zgonik | [END_REF] explains such difference, via the field-induced behavior of some specific phonon frequencies and of the force constants of some chemical bonds.

Here, we employed the ABINIT package [17] with the local density approximation (LDA) to the density functional theory (DFT) and norm-conserving pseudopotentials [18], chosen in part to facilitate the computation of electro-optic coefficients [4,19]. The alchemical mixing approximation implemented in the ABINIT package [20], which is the pseudopotentials specific implementation of the virtual crystal approximation, is also adopted to investigate the Pb(Zr 0.52 Ti 0.48 )O 3 (PZT) solid solutions. We use a 8 × 8 × 8 grid of special k points and a plane-wave kinetic energy cutoff of 50 Hartree. The effects of dc electric fields applied along the [111] direction on structural properties of the rhombohedral R3m phase of both BaTiO 3 (BTO) and PZT are simulated by taking advantage of the method developed in Refs. [21][22][23][24] (note that the R3m phase is the well-known ground state of BTO, and that we chose to study a Ti composition of 48% in PZT in order to have a stable rhombohedral ferroelectric state as well). Technically, for each considered magnitude of the dc electric field, both the lattice parameters and the atomic positions were fully relaxed until the force acting on each atom is smaller than 5 × 10 -5 Hartree/Bohr. The resulting field-induced structures are then used as input for the ab-initio method of Refs. [4,19,25], that is based on the linear response of the optical dielectric tensor induced by a static (or low frequency) electric field E k and that has been proven to accurately compute EO coefficients in ferroelectric oxides (note that no electric field is incorporated when employing this latter method on the field-induced structures). Such coefficients obey the following equation

∆(ε -1 ) ij = 3 k=1 R η ijk E k , (1)
where (ε -1 ) ij is the inverse of the electronic dielectric tensor that depends on the electric field. It is important to realize that, in our case, R η ijk is a clamped (strain-free) EO tensor that can practically depend on E k since we used the crystal structure spanned by this electric field for its calculation.

In particular, plotting R η ijk versus E k will naturally determine if the materials under investigation only adopt a linear EO effect (in that case, R η ijk will be independent of E k ) or rather a nonlinear conversion between electric and optical quantities (which will make R η ijk dependent on E k ). As detailed in Refs. [4,19], R η ijk can be expressed as the sum of two contributions: a bare electronic part, R el ijk , which is proportional to the nonlinear optical dielectric susceptibility χ

ijk , and an ionic part, R ion ijk , which is caused by the relaxation of the atomic positions due to the applied electric field and which depends on the first-order change of the linear dielectric susceptibility. The origin of the ionic contribution is related to the Raman susceptibility α m ij of mode m, the transverse optic mode polarity p m k and phonon mode frequencies ω m . The clamped (strain-free) EO tensor is thus given by:

R η ijk = R el ijk + R ion ijk = -8π n 2 i n 2 j χ (2) ijk - 4π n 2 i n 2 j √ Ω 0 m α m ij p m k ω 2 m , (2)
where n i and n j are the principal refractive indices, and Ω 0 is the unit cell volume. As taken advantage in previous works [4,6,19,25], Eq. ( 2) can be used to provide a deep insight into EO coefficients. Examples include the determination of the modes m mostly responsible for the value of these coefficients as well as their enhancement via the softening of these modes (i.e., ω m approaching a zero value). Note also that there is an unclamped (stress-free that adds a contribution involving elasto-optic effects and piezoelectricity to the clamped one) EO tensor, that is given by R σ ijk = R η ijk + 3 α,β=1 p ijαβ d kαβ where p ijαβ are elasto-optic coefficients and d kαβ are piezoelectric strain coefficients [4,6,19,25], but that we numerically found (see Fig. S1 of the Supplemental Material (SM) [START_REF]See Supplemental Material at ... for more details about (i) clamped and unclamped electro-optic (EO) coefficients; (ii) another possible ab-initio way to compute EO coefficients for systems under electric field, and its comparison with the one described in the manuscript; (iii) all the zone-center phonon frequencies; and (iv) the electronic band gap of Pb([END_REF]) that R σ ijk and R η ijk behave in a similar qualitative and even quantitative way with the applied electric field, in both PZT and BTO. Consequently, we focus here on R η ijk . Note also that, as detailed in the SM [START_REF]See Supplemental Material at ... for more details about (i) clamped and unclamped electro-optic (EO) coefficients; (ii) another possible ab-initio way to compute EO coefficients for systems under electric field, and its comparison with the one described in the manuscript; (iii) all the zone-center phonon frequencies; and (iv) the electronic band gap of Pb([END_REF], we also computed the EO tensor associated with the aforementioned field-induced structures by using another (more brut force) method -since we are not aware that nonlinear electro-optic effects have ever been investigated using first-principles-based calculations. This latter method and the one explained above provide very similar results, which therefore attests of the validity of the approach adopted in this manuscript.

Let us now choose the Cartesian axes such as the z-axis is along the [111] polarization pseudocubic direction and the y-axis is perpendicular to the mirror plane of the R3m structure, for both PZT and BTO. With this choice of axes and when adopting the Voigt notation, the R η ijk EO tensor has four independent elements: R η 11 , R η 13 , R η 33 and R η 51 . Figures 1(a) and 1(b) show all these components as a function of the electric field applied along the [111] direction in PZT and BTO, respectively, as calculated from Eq. ( 2). Regarding PZT, Fig. 1(a) indicates that the clamped EO coefficients are predicted to be, at zero electric field and by order of increasing strength, R η 11 = -4.9 pm/V, R η 13 = 5.6 pm/V, R η 33 = 11.9 pm/V and R η 51 = 12.6 pm/V (note that we are not aware of any measurement or calculation of these coefficients in rhombohedral PZT). When the electric field is turned on and increases in PZT, R η 33 quickly becomes the largest element and basically only very slightly linearly decreases, therefore indicating the occurrence of a predominant linear EO coefficient accompanied by a weak second-order EO response. In fact, one can nicely fit R η 33 by r η 33 + s η 333 E 3 , which provides a linear (Pockels effect) EO coefficient of r η 33 = 11.6 pm/V and a quadratic (Kerr effect) EO parameter of s η 333 = -3.6× 10 -21 m 2 /V 2 . Our predicted r η 33 at 0 K is of the same order of magnitude and rather consistent with the measurement of Ref. [5] giving a value of 67.3 pm/V at room temperature for Pb(Zr 0.52 Ti 0.48 )O 3 at an applied ac frequency of 1 THz, when realizing that temperature decreases the soft-mode frequencies and therefore enhances EO coefficient -as clearly indicated by Eq. [START_REF] Lines | Principles and Applications of Ferroelectrics and Related Materials[END_REF].

For BTO at zero field, all the clamped EO coefficients are larger in magnitude that those of PZT, as shown in Fig. 1(b) that reports a value of R η 11 = 8.9 pm/V, R η 13 = 21.0 pm/V, R η 33 = 42.2 pm/V and R η 51 = 33.8 pm/V (note that these values are consistent with the previously reported ones of Ref. [19] but using the experimental lattice constants of BTO). Moreover and in sharp contrast with PZT, Fig. 1(b) also reveals that all the elements of R η ijk in BTO strongly depends on the magnitude of the electric field. Such numerical finding is fully in-line with a recent experiment [5] observing a predominantly linear EO response in PZT films versus a nonlinear electro-optic response of BTO thin films, in the THz frequency range. In fact, we numerically find that our computed R η 33 of BTO of Fig. 1(b) can be very well fitted by r η 33 + s η 333 E 3 + c η 3333 E 2 3 , with r η 33 = 39.6 pm/V, s η 333 = -6.4× 10 -20 m 2 /V 2 and c η 3333 = 5.1 × 10 -29 m 3 /V 3 . Note, however, that the magnitude of the predicted second-order EO coefficient of 6.4× 10 -20 m 2 /V 2 is about 200 times smaller in magnitude than that measured in Ref. [5] for an ac frequency of 1 THz (note that the other method described in the SM [START_REF]See Supplemental Material at ... for more details about (i) clamped and unclamped electro-optic (EO) coefficients; (ii) another possible ab-initio way to compute EO coefficients for systems under electric field, and its comparison with the one described in the manuscript; (iii) all the zone-center phonon frequencies; and (iv) the electronic band gap of Pb([END_REF] does not rely on linear response and provides similar result for the EO coefficients). Possible reasons behind such discrepancy is that we study here the R3m phase at 0 K while experiments are conducted on the tetragonal phase of BTO at room temperature, that is very close (namely by about 20 K [START_REF] Lines | Principles and Applications of Ferroelectrics and Related Materials[END_REF]31,32]) to the tetragonal-to-orthorhombic phase transition where a large enhancement of the EO responses is expected [33] due to the softening of some phonon frequencies -as evidenced by Eq. ( 2). Other possible reasons may be that ab-initio electric fields can typically provide an overestimation by one or two orders of magnitude with respect to experimental ones [34][35][36][37], or that the experiment in Ref. [5] is conducted on strained and unpoled samples while we study bulk polar materials. Such latter hypothesis is even more reasonable when assuming the formula indicated in Refs. [7,[START_REF] Didomenico | [END_REF]33], that are: R 0 13 = 2 3 (g 11 + 2g 12 -g 44 ) P s χ 33 , R 0 33 = 2 3 (g 11 + 2g 12 + 2g 44 ) P s χ 33 , and R 0 51 = 2 3 g 11 -g 12 + 1 2 g 44 P s χ 11 , where R 0 ij are linear EO coefficients, P s is the spontaneous polarization, χ 33 and χ 11 are the dielectric constants along the cand a-axes, respectively, and g ij are specific quadratic EO coefficients -all under infinitesimally small electric fields. As a matter of fact, plugging our numerical values for R 0 ij , P s , χ 33 and χ 11 into these formula gives for the R3m phase of BTO bulk: g 11 = 273, g 12 = -2 and g 44 = 68 in units of 10 -2 m 4 /C 2 , which is precisely the order of magnitude reported in Ref. [7] for the g ij coefficients of BTO bulk at room temperature (i.e., in the P 4mm phase).

In order to understand the origin of the linear electro-optic response in PZT versus the nonlinear EO behavior in BTO, we determined the contribution of each zone-center phonon mode for the R η 33 coefficient, as well as the density functional perturbation theory (DFPT)-predicted frequency of these modes (see Fig. S2 of the SM [START_REF]See Supplemental Material at ... for more details about (i) clamped and unclamped electro-optic (EO) coefficients; (ii) another possible ab-initio way to compute EO coefficients for systems under electric field, and its comparison with the one described in the manuscript; (iii) all the zone-center phonon frequencies; and (iv) the electronic band gap of Pb([END_REF] for all these zone-center phonons), for each of these two systems and for each investigated electric field. For PZT, Fig. 2(a) reveals that R η 33 mostly arises from the A 1 modes, with these two modes having frequencies [see Fig. 2(c)] behaving in such a manner that the ω -2 inverse of their square is only weakly (and nearly linearly dependent) on the applied electric field -as evidenced in Fig. 2(e). Incorporating such latter fact when looking again at Eq. ( 2) naturally explains why R η 33 is mostly independent on the electric field, that is why the EO response of PZT system is basically linear (with the slight change of ω -2 with field generating a weak second-order EO response). Note that the electronic part of the clamped EO tensor (first term of Eq. ( 2)) is found to be small, as revealed by Fig. 2(a) therefore indicating the predominant role of ionic contributions for R η 33 of PZT [the same finding holds for BTO, as demonstrated in Fig. 2(b)]. Note also that, as indicated in the inset of Fig. 2(c In contrast, for BTO, R η 33 takes most of its value from the A

(2)

1 mode for fields smaller than 3.1 × 10 8 V/m, with this mode having a frequency strongly increasing, and thus an inverse of the square of such frequency strongly decreasing, with the field. Consequently and according to Eq. ( 2) too, the electro-optic response of BTO is highly nonlinear, and is significantly reduced, for fields smaller than 3.1 × 10 8 V/m. Note also that the nonlinear behavior of ω -2 with field is the culprit behind the existence of third-order EO coefficient. The inset of Fig. 2(d

1 modes then occurs in BTO for a field equal to 3.1 × 10 8 V/m. Such anticrossing results in these modes repelling each other near this critical field, therefore yielding a gap between these two phonon frequencies and thus between the inverse of their square, as clearly seen in the inset of Fig. 2(f). This anticrossing also leads to the eigenvectors of these two modes inverting their atomic character before versus after this critical field [25]. It also makes the A

(3) 1 mode being the dominating one for R η 33 above this critical field, with the resulting EO response being still nonlinear -since the frequency of the A

1 mode also strongly depends on the magnitude of the field above 3.1 × 10 8 V/m. One can thus safely conclude that the linear character of the EO response in PZT versus the nonlinear electro-optic response of BTO mainly originate from the different behavior that the ω -2 of their corresponding predominant modes adopt in these two important perovskite oxides.

The next issue to address is therefore to understand why these ω -2 behave in a different manner in PZT and BTO. For that, we reported the electric-field dependence of the force constants Ti/Zr-Ti/Zr, O-O and Ti/Zr-O bonds of PZT in the left column of Fig. 3, and of the force constants of Ti-Ti, O-O and Ti-O of BTO in the the right column of Fig. 3. The choice to concentrate on these specific force constants (rather than those involving Pb or Ba ions) stems from the atomic character of the eigenvectors associated with the A 3 show nearly linear versus strongly nonlinear behaviors as a function of electric field in PZT and BTO, respectively, which therefore connects (and explains) the different nature of the macroscopic electro-optic response in these two systems to some specific atomistic bond characteristics.

In summary, a first-principle technique is developed to tackle nonlinear electro-optic response of materials at an ab-initio level for the first time, to the best of our knowledge. This method simply consists of first employing the development of Ref. [38] to determine the crystal and atomic structure induced by electric fields and then use such structure as input of the method of Refs. [4,19] to extract EO coefficients as a function of electric field (note that other atomistic methods, such as those of Refs. [39,40], can be used to extract the field-induced structure). This method is presently applied to the R3m phase of PZT and BTO ferroelectric perovskite oxides, and is also found to provide similar results than another, more brut-force technique further proposed and explained in the SM [START_REF]See Supplemental Material at ... for more details about (i) clamped and unclamped electro-optic (EO) coefficients; (ii) another possible ab-initio way to compute EO coefficients for systems under electric field, and its comparison with the one described in the manuscript; (iii) all the zone-center phonon frequencies; and (iv) the electronic band gap of Pb([END_REF]. Both of these methods reproduce a recent striking experimental finding, that is why the EO response of PZT and BTO is linear versus nonlinear, respectively [5] (note that the SM [START_REF]See Supplemental Material at ... for more details about (i) clamped and unclamped electro-optic (EO) coefficients; (ii) another possible ab-initio way to compute EO coefficients for systems under electric field, and its comparison with the one described in the manuscript; (iii) all the zone-center phonon frequencies; and (iv) the electronic band gap of Pb([END_REF] also shows that other optical properties can behave in a different qualitative way between these two important materials). The scheme indicated in this manuscript also naturally reveals that it is the field-induced behavior of the frequency of some specific phonon modes and of some force constants that are responsible for the difference in nature for the conversion between electric and optical properties in PZT and BTO. We thus hope that the present study enhances the knowledge of light-matter interactions and functional materials, and will also motivate the development of other techniques allowing the investigation of complex interplay between light and physical properties. A particular advantage of the proposed method is that it can be easily employed for the quest of materials with large nonlinear EO response.

  ), the Ti/Zr ions move along the spontaneous polarization [111] pseudo-cubic direction while the three oxygen ions move along the [ 11 0], [ 10 1] and [0 11 ] directions, respectively, in the A

  ) shows that Ti ions move along the [111] direction while the O ions are displaced along the [ 111 ] direction in the A (2) 1 mode. Furthermore and as evidenced in Figs. 2(d) and 2(f), a striking anticrossing between the A

( 3 ) 1

 31 mode in PZT and the A (2) 1 mode in BTO [see again the inset of Figs. 2(c) and 2(d), respectively]. The selected force constants of Fig.

FIG. 2 .FIG. 3 .

 23 FIG. 1. Clamped EO coefficients as a function of electric field applied along the [111] direction in (a) PZT and (b) BTO, respectively.


S Omid Sayedaghaee

Bin Xu

Sergey Prosandeev

Charles Paillard

L Bellaiche

Novel Dynamical Magnetoelectric Effects in Multiferroic BiFeO 3

An atomistic effective Hamiltonian scheme is employed within Molecular Dynamics simulations to investigate how the electrical polarization and magnetization of the multiferroic BiFeO 3 respond to time-dependent ac magnetic fields of various frequencies, as well as to reveal the frequency dependency of the dynamical (quadratic) magnetoelectric coefficient. We found the occurrence of vibrations having phonon frequencies in both the time-dependency of the electrical polarization and magnetization (for any applied ac frequency), therefore making such vibrations of electromagnonic nature, when the homogeneous strain of the system is frozen (case 1). Moreover, the quadratic magnetoelectric coupling constant is monotonous and almost dispersionless in the sub-THz range in this case 1. In contrast, when the homogeneous strain can fully relax (case 2), two additional low-frequency and strain-mediated oscillations emerge in the time-dependent behavior of the polarization and magnetization, which result in resonances in the quadratic magnetoelectric coefficient. Such additional oscillations consist of a mixing between acoustic phonons, optical phonons and magnons, and reflect the existence of a new quasiparticle that can be coined "electroacoustic-magnon". This latter finding can prompt experimentalists to shape their samples to take advantage of, and tune, the magnetostrictive-induced mechanical resonance frequency, in order to achieve large dynamical magnetoelectric couplings.

trical and magnetic moments. Such coupling is promising for designing novel devices by controlling magnetization with electric fields, or conversely, electrical polarization with magnetic fields [1][2][3][4][5][6]. Two paths can be taken to realize this magnetoelectric coupling: direct coupling of polarization with magnetic field vs mediated by strain. The latter is particularly investigated by mixing efficient magnetostrictive and piezoelectric materials, in composites [7] or heterostructures [8,9]. These two types of coupling have been mostly for static properties [1-4, 10, 11]. In other words, how strain affects dynamical properties of multiferroics is mostly an uncharted territory. In particular, it is yet unclear whether ME coefficients can be improved with mechanical resonances in a single phase materials, as in laminar composites [8,12]. It is also legitimate to investigate the effect of strain on electromagnons (mixing of phonons and magnons [13][14][15][16]), or even on the formation of novel type of (dynamical) objects.

To resolve such issues, we (1) conducted molecular dynamics (MD) simulations within an effective Hamiltonian scheme on BiFeO 3 (BFO), a prototypical multiferroic, subject to ac magnetic fields; and (2) monitored the resulting time-dependency of its electrical polarization and magnetization. It is found, that (i) electromagnons (of phonon frequencies) exist independently of allowing the homogeneous strain to relax; and (ii) relaxation of the homogeneous strain results in the emergence of a new type of quasiparticle consisting of acoustic vibrations coupled to phonons and magnons, and generating resonances in the quadratic ME coefficient.

Here, we use the effective Hamiltonian (H eff ) scheme of BiFeO 3 described in Ref. [17]. Its total energy E BFO ({u i }, {η H }, {η I }, {ω i }, {m i }) includes four types of degrees of freedom: 1) the local modes {u i }, proportional to the local electric dipoles [18,19]; 2) the homogeneous {η H } and inhomogeneous {η I } strain tensors [18,19]; 3) the pseudo-vectors {ω i } that characterize the oxygen octahedral tiltings [20] (also called antiferrodistortive (AFD) motions); and 4) the magnetic moments {m i } of the Fe ions (in all cases, the subscript i labels unit cells in our simulated supercells). The total energy of H eff for BFO is a sum of three main energies

E tot = E FE ({u i }, {η l })+E AFD ({u i }, {η l }, {ω i })+E MAG ({m i }, {u i }, {η l }, {ω i }), where {η l }
is the total strain tensor (i.e., that incorporates both the homogenous and inhomogeneous components). E FE is the energy involving the local modes and elastic deformations, while E AFD is the energy that gathers the AFD motions and their couplings with local modes and strains. Moreover, E MAG contains the magnetic degrees of freedom and their couplings with local modes, AFD tiltings and strains, and reads [21]:

E MAG = ijαγ Q ijαγ m iα m jγ + ijαγ S ijαγ m iα m jγ + ij,αγνδ E ij,αγνδ m iα m jγ u iν u iδ (1)
+ ij,αγνδ F ij,αγνδ m iα m jγ ω iν ω iδ + ijl,αγ G ijl,αγ η l (i)m iα m jγ + ij L ij (ω i -ω j ) • (m i × m j ) ,
where α, γ, ν, δ denote the Cartesian components, and the indices i and j run over sites.

The six terms of Eq. ( 1) are, respectively, the dipolar interactions between the magnetic moments, the short-range magnetic exchange coupling, the coupling between the magnetic moments with local modes, AFD motions and strain, and a particular Dzyaloshinskii-Moriya (DM) interaction involving the oxygen octahedral tiltings. Under a magnetic field, an additional termi m i •H is also incorporated into the total energy. This effective Hamiltonian is then adopted for MD simulations, by solving the equations of motion for local modes, oxygen octahedral tilting, strains and magnetic moments, as detailed in Refs. [22][23][24]. We adopt a 12×12×12 supercell in terms of the 5-atom perovskite unit cell, with periodic boundary conditions. MD simulations are carried out at 1 K under the N P T ensemble when the homogeneous strain can relax during the simulations versus the N V T ensemble when the total strain is frozen during the computations. More details about the MD computations and the effective Hamiltonian schemes for BFO are given in the Supplemental Material [21,[25][26][27][28][29][30][31][32][33][34][35][36].

We apply to our considered state of BFO a magnetic field with two components, both aligned along the [11 2] direction: a dc field of magnitude H dc =245 T and an ac field given by h ac sin(ωt) where h ac = 61.2 T and ν = ω/2π is the frequency of the applied ac magnetic field. These fields are chosen to have high magnitude to numerically observe the response of polarization since ME coefficients are known to be rather small in BiFeO 3 [21,32,37,38].

Let us first restrict ourselves to the case when the homogeneous strain is not allowed to relax (therefore adopting the homogeneous strain of the R3c state under a sole dc magnetic field of 245 T) during the MD simulations -while the inhomogeneous strain can still vary.

Figure 1a depicts the temporal behavior of the component of the polarization along the [111] direction in the R3c phase for a frequency (ν) of 160 GHz. Moreover, Fig. 1b displays the corresponding Fourier transform, and demonstrates that four main frequencies govern the temporal evolution of the electrical polarization: the applied ac frequency and its double (i.e., 160 and 320 GHz), which reveals the occurrence of dynamical magnetoelectric effects, and two higher frequencies that are of the order of 4300 and 7000 GHz, which are natural phonon frequencies (see Refs. [22,26,[39][40][41][42][43][44][45][46]).

Interestingly and as evidenced in Fig. 1d, these four frequencies also appear, in the Fourier transform of the curve representing the component of the magnetization along the [11 2] direction as a function of time (shown in Fig. 1c). The frequency of 160 GHz seen in Fig. 1d emerges from the energy coupling the magnetic moment with the ac magnetic field, while the frequency at 320 GHz is characteristic of non-linear magnetic couplings.

Remarkably, the high frequencies around 4300 GHz seen in Fig. 1d, and at lesser extent at 7000 GHz, reveal that natural phonons mix with magnons and affect the temporal evolution of the magnetization under a time-dependent magnetic field. Natural phonons thus become electromagnons [14][15][16], consistent with Ref. [47] for the R3c phase of BiFeO 3 -these electromagnons are presently numerically found to originate from the fact that polarization and oxygen octahedral tiltings affect the magnetic exchange parameters (third and fourth terms of Eq. ( 1)). Note that magnons having frequencies smaller than 160 GHz are not seen in Fig. 1d because the applied magnetic fields are too large and thus force the magnetic moments to mostly follow them, in addition to possessing smaller oscillations arising from the coupling of the magnetic moments with the aforementioned phonons (see Fig. 1c).

Let us now allow the homogeneous and inhomogeneous strains to fully relax during the MD simulations, and determine how it affects the temporal evolutions of the polarization and magnetization (see Figs 2a and2c, respectively), as well as their Fourier transforms (cf and η H,3 also possess another frequency of the order of 67 GHz that slightly appears in the Fourier transform of the magnetization as shown in Fig. 2d (note that, on the other hand, we did not find any frequency higher than 320 GHz in the Fourier transforms of all homogeneous strain components, including the phonon frequencies). Comparing the results between the cases of the fixed versus relaxed homogeneous strain therefore demonstrates that, in our simulations, the homogeneous strain tensor has its own natural frequencies of the order of 90 and 267 GHz that then couple with oscillations of both the polarization and magnetization (note that these two frequencies are indeed natural frequencies of the homogeneous strain because they are also numerically found (not shown here) in the Fourier transform of the homogeneous strain when only a dc magnetic field is applied or even when no magnetic field is imposed on BFO, with the homogeneous strain having the possibility to relax during all these additional simulations). In other words, one can create a new type of quasiparticle mixing acoustic phonons, optical phonons and magnons, when applying ac magnetic fields with specific frequencies (i.e., 90 and 267 GHz here). Such creation of this quasiparticle can be understood as follows: the magnetic field at these frequencies naturally activates magnons, via the direct interaction between magnetic field and magnetic moments, which in turn dynamically couple with the strain and its natural frequencies via the magnetostrictive effect. This dynamical strain then activates optical phonons at these frequencies, because of couplings between strain and electrical dipoles (via electrostrictive and piezoelectric effects), therefore resulting in the formation of this quasiparticle. We propose to name such quasiparticle as "electro-acoustic-magnons" to emphasize that, unlike "traditional" electromagnons, strain also plays a role in their creation. Note that pump-probe experiments revealed acoustic excitations having similar frequencies than our predicted "electro-acoustic-magnons", namely around 30 GHz and 50 GHz for transverse and longitudinal acoustic modes in BFO respectively [48,49] (note also that the presently calculated natural frequencies of the "electro-acoustic-magnons" naturally depend on the choice of the homogeneous strain mass adopted in our MD simulations). Moreover, a peak at about 300 GHz has been observed in the Raman spectrum of the spin-canted magnetic structure of epitaxial BFO films [36]. Furthermore, a phenomenon analogous to our proposed "electro-acoustic-magnons" has just been reported in Ref. [50], that is a dynamical coupling between nuclear spins and electromechanical phonons. Such phenomenon consists of applying an ac electric field at the natural frequency of a resonator, which leads to an electrically tunable phonon that imparts diagonal and shear strains oscillating with time and which then dynamically couple with spins of nuclei (via a quadruple interaction between strains and spins there). This resulting dynamical coupling between spins and electromechanical phonons was indicated to open up quantum state engineering, such as coherent coupling between sound and nuclei and mechanical cooling of solid-state nuclei [50]. Such interesting possibilities therefore hint that our presently discovered "electro-acoustic-magnons" may lead to novel and important devices.

Let us now reveal how these "electro-acoustic-magnons" affect the dynamical magnetoelectric coefficients. For that, let us first recall the following equation of any component i of the polarization under magnetic fields [51,52]:

P i = P s i + α ij H j + 1 2 β ijk H j H k , in which P s i
is the i component of the spontaneous polarization, while H j and H k are components of the magnetic field. Moreover, α ij and β ijk are linear and quadratic magnetoelectric coefficients, respectively. Assuming that the linear ME coefficient is negligible compared to the quadratic coefficient (as suggested in Refs. [21,53] for large fields) leads to the following equation for the polarization (along the [111] direction) when applying a magnetic field (along the [11 2] direction) H = H dc + h ac e i(ωt-π 2 ) :

P (t) = P 0 + β(0, ω)H dc h ac e i(ωt-π 2 ) + β(ω, ω) 1 2 h 2 ac e i(2ωt-π) (2)
The second and third terms on the right-hand side of Eq.( 2) characterize the magnetic-field induction of a polarization component with the same frequency of applied ac magnetic field and a second harmonic generation, respectively -which explains the occurrence of a strong Fourier transform at 160 GHz and a weaker one at 320 GHz in Figs. 1b and1d. Note that 2) is confirmed by our MD data.

Taking now into account that β(0, ω) is a complex number (especially close to resonant frequencies) we rewrite β(0, ω) = β (0, ω) + iβ (0, ω), whose separate contributions can be computed thanks to Eq. ( 2) via:

β (0, ω) = 2 L L 0 (P (t) -P 0 ) sin ωtdt H dc h ac and β (0, ω) = 2 L L 0 (P (t) -P 0 ) cos ωtdt H dc h ac ( 3
)
where L is the overall simulation time.

We now apply, in addition to the dc magnetic field of 245 T, ac magnetic fields of the same magnitude of 61.2 T than in Figs 1 and 2 but of different frequencies ranging between 20 GHz and 500 GHz (both fields applied along [11 2]). Using Eqs (3), β(0, ω) is determined in the frozen (Fig. 3a) and relaxed (Fig. 3b) homogeneous strain cases. When the homogeneous strain is fixed in the simulations, the imaginary part of β(0, ω) is basically null for any ac frequency while the real part β (0, ω) is nearly independent of the frequency taking a value of about 2.0 × 10 -8 C/m 2 T 2 = 0.32 × 10 -19 s/A in magnitude -that agrees very well with the β 311 coefficient of 0.3 × 10 -19 s/A measured in Ref. [38]. On the other hand, when the homogeneous strain fully relaxes, β(0, ω) exhibits two resonances at precisely the two frequencies of the electro-acoustic-magnons, as evidenced by strong peaks of the β (0, ω) imaginary part at 90 and 267 GHz that are accompanied by strong negative values immediately followed by strong positive values of the β(0, ω) real part in the near vicinity of these two frequencies. Relaxing the homogeneous strain has thus dramatic consequences on the dynamical quadratic ME coefficients near some resonant frequencies because such strain dynamically couples with both the polarization and magnetization. The divergences of β(0, ω) at these two resonances that are induced by indirect (i.e strain-mediated) ME coupling therefore differ in nature from the divergences of the linear dynamical ME coefficient predicted to occur at magnons or phonon frequencies in Ref. [54] since these latter originate from a direct coupling between polarization and magnetism. Note that electromechanically mediated resonance in magnetoelectric coefficients was already observed in laminar piezoelectric-magnetoelectric composite structures [8,12]. Here, it is of interest in the design of magneto-electric based sensors, for which the resonance frequency can be tuned by properly designing the shape and size of a single phase material (such as BiFeO 3 ).

Note also that we are unable to extract β(0, ω) for high frequencies, because the time dependency of the polarization becomes noisy due to interference between the phonon frequencies and the applied magnetic field frequency. This explains why we limited ourselves to frequencies up to 500 GHz in Fig. 3 and prevents us from checking if β(0, ω) has also resonances at the phonon/electromagnons frequencies of about 4300 and 7000 GHz.

In summary, molecular dynamics effective Hamiltonian simulations predicting the response of the polarization and magnetization to time-dependent magnetic fields of different ac frequency, allowed to extract the dispersion of the quadratic ME coefficient up to 500

GHz. In particular, electromagnons having phonon frequencies are found whether the homogeneous strain is frozen or relaxed during the simulations. Strain-mediated resonances in the magnetoelectric coupling are also reported, and are of large interest to design devices with proper shape of the sample for dynamical applications, since such shape can tune the resonant mechanical frequencies [55]. Those resonances can be described by a new type of quasiparticle that we coin "electric-acoustic-magnon", which arises when the frequency of the ac magnetic field resonantly excites homogeneous strain modes. This quasiparticle consists of a mixing of acoustic phonon, optical phonon and magnon. The calculations reported in this manuscript were done at very low temperature and high magnitude of the magnetic fields in order to have less fluctuation of the order parameters (e.g., polarization and magnetization) -yielding less numerical noise and thus better accuracy for the magnetoelectric response. However, as shown in the Supplementary Materials, our findings (e.g., resonances in the quadratic magnetoelectric coefficients originating from our discovered "electro-acousticmagnons") still qualitatively hold at 300K and also for smaller magnetic fields. We strongly believe that these results are not only relevant to BiFeO 3 but rather to many multiferroics due to couplings between polarization, magnetization and strains in such systems. We thus hope that the present results deepen the current knowledge of multiferroics, in general, and of dynamical magneto-electric effects, in particular.

Fig. 1a also

  Fig.1aalso shows the fit of P (t) by a function of the form A + B sin ωt by means of a solid line. Such fit nicely goes throughout the numerical MD data, therefore implying that (i) the linear ME coefficient and β(ω, ω) can be neglected in front of β(0, ω) for the ac frequency of 160 GHz (the deviation of the MD data with respect to the fit consists of rapid oscillations associated with the phonon frequencies of about 4300 and 7000 GHz); and (ii) the validity of Eq. (2) is confirmed by our MD data.

FIG. 1 :

 1 FIG. 1: (Color online) Temporal evolution of the polarization (panel a) and magnetization (panel c) in BiFeO 3 under a dc magnetic field of 245 T of magnitude coexisting with an ac magnetic field of 61.2 T of magnitude and of 160 GHz of frequency, along with their resulting Fourier transforms (panels b and d, respectively) as a function of frequency, in the case that the homogeneous strain is frozen in the MD simulations. The dc and ac magnetic fields are applied along the [11 2] direction. The displayed polarization is along the [111] direction while the magnetization is along the [11 2] direction. The solid line in Panel a represents the fit of the MD data by a function of the form A + B sin ωt where ω/2π=160 GHz.

FIG. 2 :FIG. 3 :

 23 FIG. 2: (Color online) Temporal evolution of the polarization (panel a), magnetization (panel c), diagonal elements of the homogeneous strain tensor (panel e) and shear elements of the homogeneous strain tensor (panel g) in BiFeO 3 under a dc magnetic field of 245 T of magnitude coexisting with an ac magnetic field of 61.2 T of magnitude and of 160 GHz of frequency, along with their resulting Fourier transforms (Panels b, d, f and h, respectively) as a function of frequency, when the homogeneous strain is allowed to relax during the MD simulations. The dc and ac magnetic fields are applied along the [11 2] direction. The displayed polarization is along the [111] direction while the magnetization is along the [11 2] direction.

  Figures 2f and 2h further indicate that the diagonal and shear elements of the homogeneous strain tensor adopt the frequency of the applied magnetic fields of 160 GHz too, and that η H,1 , η H,2

	Remarkably, allowing the homogeneous strain to relax generates two additional frequen-
	cies in the Fourier transform of the polarization-versus-time curve with respect to the case of
	fixed homogeneous strains. These two frequencies are about 90 and 267 GHz, respectively,
	and can also be seen in the Fourier transform of the magnetization-versus-time functions.
	Figures 2h reveals that the frequency of 90 GHz originates from the oscillations of the shear
	elements of η

Figs 2d and 2d, respectively). Figures

2e and 2f

further report the variation of the diagonal elements of the homogenous strain tensor (η H,1 , η H,2 and η H,3 ) as a function of time and of their Fourier transforms, respectively, while Figs 2g and 2h provide similar information but for the shear elements of the homogeneous strain tensor (η H,4 , η H,5 and η H,6 ). H,4 , η H,5 and η H,6 , while that of 267 GHz can be traced back to the vibrations of the diagonal (η H,1 , η H,2 and η H,3 ) elements of the homogeneous strain according to Fig. 2f.

  the mechanical and thus "electro-acoustic-magnon" frequencies). What is interesting is that mechanical frequencies are known to be practically dependent on the sample size. As a result, alternating the sample size should allow to control the frequencies of our discovered "electro-acoustic-magnons".

	S.O.S. and L.B. thank the DARPA Grant No. HR0011-15-2-0038 (under the MATRIX
	program). B.X. acknowledges funding from Air Force Office of Scientific Research under
	Grant No. FA9550-16-1-0065. S.P. thanks ONR Grant No. N00014-17-1-2818. C.P. ac-
	knowledges the ARO grant W911NF-16-1-0227. Some computations were also made possible
	owing to MRI Grant No. 0722625 from NSF, ONR Grant No.N00014-15-1-2881 (DURIP),
	and a Challenge grant from the Department of Defense. S.P. also appreciates support of
	RMES 3.1649.2017/4.6 and RFBR 18-52-00029 Bel a.

of