Maram Caesar
email: maram.caesar@edu.mnhn.fr

Roger Roy

Frédéric Legendre

Philippe Grandcolas

Roseli Pellens

Frederic Legendre

Catalogue of Dictyoptera from Syria and neighbouring countries (Lebanon, Turkey, Iraq and Jordan)

Keywords: Mantodea, Isoptera, Blattaria, species list, types, Mediterranean region, biodiversity assessment

niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

The Mediterranean Region is widely recognised as one of the world's Biodiversity Hotspots [START_REF] Myers | Biodiversity hotspots for conservation priorities[END_REF]. Its wonderful dry climate with diverse but harsh soil conditions gave rise to an enormous diversification of plants, the group with the highest rates of endemicity, but also amongst vertebrates, insects, and many other organisms. However, the Mediterranean basin biomes (Mediterranean Grasslands and the Mediterranean Forests) are distributed in many countries and comprises several biodiversity hotspots (e.g. [START_REF] Medail | Hot-spots analysis for conservation of plant biodiversity in the Mediterranean basin[END_REF][START_REF] Medail | Biodiversity hotspots in the Mediterranean basin: Setting global conservation priorities[END_REF], and the present knowledge of the existing biodiversity and its present status is quite dissimilar from one place to another.

Blattaria, Mantodea and Isoptera are large groups of worldwide-distributed insects with an important diversity in the Mediterranean region. However, at present time it is hard to have an idea of the number of species and of their areas of distribution because the existing taxonomic data is spread in hundreds of documents demanding a huge effort each time we need information about species' occurrences, or distribution area. So, as a first step to systematize the present knowledge on the distribution of these three insect orders along the Mediterranean biomes, we prepared a catalogue of Dictyoptera (Mantodea, Isoptera and Blattaria) in five Eastern Mediterranean countries (Syria, Lebanon, Turkey, Iraq and Jordan), and we analysed the evolution of the knowledge about these groups along time.

Results and discussion

There are 75 species of Dictyoptera known from these five countries. These species belong to 15 families (eight for Mantodea, four for Isoptera and three for Blattaria). Mantodea is by far the dictyopteran group with the highest richness with 43 species and subspecies occurring in this region, followed by Blattaria, with 21, and Isoptera with 11. This is at odds with general trends of diversity within these groups, as globally Blattaria is the more specious, with more than 4000 species described (Princis, 1962(Princis, -1971) ) and with a high number of new species at each new place sampled (Grandcolas, 1994a,c;1997;[START_REF] Pellens | Fragmentação florestal em Mata Atlântica de Tabuleiros: os efeitos da heterogeneidade da paisagem sobre a diversidade de artrópodos edáficos[END_REF][START_REF] Pellens | Catalogue of Blattaria (Insecta) from Brazil[END_REF][START_REF] Grandcolas | Capitulo 27. Blattaria[END_REF]; Isoptera has often low local species richness and includes a total of 3106 living and fossil species [START_REF] Krishna | Treatise on the Isoptera of the World[END_REF]; and Mantodea has the lowest total number of known species (more than 2400 species) [START_REF] Svenson | Phylogeny of Mantodea based on molecular data: evolution of a charismatic predator[END_REF]. These discrepancies suggest that, concerning the ratio between local and global number of species, Mantodea diversified more in these countries than the two others. More species of Dictyoptera were recorded in Turkey (34%), followed by Iraq (23%) then Syria (22%), Jordan (15%) and Lebanon (7%). Roughly these results seem to be strongly associated with the size of the countries since Turkey is the largest one; Lebanon the smallest; and the three others have intermediate surfaces, and this diversity might reflect environmental diversity within each of them. However, other factors such as facility of access to natural sites, and the existence of local scientists interested in these groups might also play an important role in the knowledge of the existing fauna.

The species of Dictyoptera occurring in this region started to be studied in 1773, when Bolivaria brachyptera (Pallas, 1773) (Mantodea) was described, and the number of species increased at different paces for these three insect orders along time. Most of species of Blattodea and Isoptera are known from before 1950. For Mantodea, however, an important number of species (28%) were described more recently, and the accumulation curve (Figure 1A) suggests that the number of species in the region will still increase. For Isoptera all known species, except one, were described at the very beginning of the 20 th century. For Blattaria, some species were described between 1937 and 1957, and never after this date.

Twenty-nine of these species (i.e., 38%) have so far been recorded only in these five countries (22 of Mantodea, one of Isoptera and six of Blattaria) and, as shown in Figure 1B, the greatest part of these species was described during the 20 th century. It also shows that the total number of species is much smaller for (Figure 1 B).

In taxonomic studies, the number of times a species is cited reveals the dynamics of their study, as for example changes in nomenclature, reviews of the genera and resolution of synonymies. If we consider the number of times the species of Dictyoptera were cited along the time period comprised in this study, we can see that, as for many other taxonomic groups and in diverse scientific domains, it is above all from the beginning of the twentieth century they started to be more studied. Globally Mantodea is by far the dictyopteran group the most studied, with a total of 91 citations, and a significantly higher number of citations at each time period. But since 1958, Isoptera became more studied, being more cited than Mantodea in some decades (Figure 2). This information shows that despite more studies of Isoptera from this region in the second half of the 20th century, only one new species was described (see Figure 1 B). This suggests that the present number of known species might be close to the real number of Isoptera species occurring in these countries. The 75 species with occurrence in these five countries are represented by 107 types or type series distributed in 29 institutions. Four of them (NHM, NHMW, MHNG, ZMHB) house 52% of the types, and each of them has between 13 and 15 types of these species. Seven other institutions house 35% of the types (MNHN, MLUH, UZIU, AMNH, MRSN, ZIN, IPUC), with between 4 and 7 types in each of them. The remaining 22 types (20%) are housed in 18 other institutions. Catalogue of Mantodea, Isoptera and Blattaria from Syria, Lebanon, Turkey, Iraq and Jordan ORDER MANTODEA Family: Amorphoscelidae Amorphoscelis pantherina Roy, 1966 Amorphoscelis pantherina Roy, 1966: 268-270;Ehrmann, 2002: 62;2011: 6;Otte & Spearman, 2005: 25. Geographical records: Iraq. General distribution: Iraq, Turkey-SE. Type series: Holotype male ANSP. Type locality: Iraq: Arbil Liwa.

Family: Rivetinidae

Bolivaria brachyptera (Pallas, 1773) Mantis brachyptera [START_REF] Pallas | Reise durch verschiedene Provinzen des Russischen Reiches in den Jahren 1768-1774[END_REF]: 728. Bolivaria brachyptera Giglio-Tos, 1927: 482;Ramme, 1951: 327;Çiplak & Demirsoy, 1997: 109;Ehrmann, 2002:

FIGURE 1 .

 1 FIGURE 1. A) Accumulated number of species of Mantodea, Isoptera and Blattaria occurring in five Middle East countries (Syria, Lebanon, Turkey, Iraq and Jordan) described along time. B) Accumulated number of species of Mantodea, Isoptera and Blattaria known to occur only in these countries.

FIGURE 2 .

 2 FIGURE 2. Number of citations of the species of Mantodea, Isoptera and Blattaria occurring in five Middle East countries (Syria, Lebanon, Turkey, Iraq and Jordan). The time period goes from the first species described (Bolivaria brachyptera (Pallas, 1773)-Mantodea) to 2007 when the last study concerning species from these groups was published.

Acknowledgements

This paper was made in the framework of the PhD thesis of Maram Caesar at the ED 227 du Muséum National d'Histoire naturelle and ISYEB. MC has a PhD grant from Syrian government (Grant from Syrian government -BGF).

General distribution: Also recorded in Afghanistan, Iran, Turkmenistan. Type: Syntypes AMNH. Ref: Krishna et al., 2013: 289-290. Anacanthotermes ubachi [START_REF] Navás | Algunos Ortopteros y Neuropteros de Palestina[END_REF] Geographical records: Turkey, Syria, Iraq, Jordan. General distribution: Also recorded in Saudi Arabia. Type: Syntypes (depository unknown). Ref: Krishna et al., 2013: 291. Anacanthotermes vagans [START_REF] Hagen | Monographie der Termiten[END_REF] Geographical records: Iraq. General distribution: Also recorded in Afghanistan, Iran, Kuwait, Pakistan, Saudi Arabia. Type: Syntype NHMW. Holotype NHMW. Ref: Krishna et al., 2013: 292. Family: Kalotermitidae Kalotermes flavicollis [START_REF] Fabricius | Entomologia systematica emendata et aucta. Secundum: classes, ordines, genera, species. adjectis: synonimis, locis, observationibus, descriptionibus[END_REF] Geographical records: Syria, Turkey. General distribution: Also recorded from Algeria to Egypt, Levant, NW Caucasus, France, Italy, Libya, Morocco, Portugal, Spain, Tunisia, Yugoslavia. Type: Lectotype ZMUC. Ref: Krishna et al., 2013: 509-515.

Family: Rhinotermitidae

Reticulitermes clypeatus [START_REF] Lash | A new species of Reticulitermes (Isoptera) from Jerusalem, Palestine[END_REF] Geographical records: Iraq. General distribution: Also recorded in Iran, Romania. Type: Holotype AMNH. Paratypes AMNH, SANC, FRI. Ref: Krishna et al., 2013: 794. Reticulitermes lucifugus lucifugus [START_REF] Rossi | Mantissa insectorum, exhibens species nuper in Etruria collectas: adjectis faunae etruscae illustrationibus, ac emendationibus, Fauna Etrusca[END_REF] Geographical records: Iraq, Turkey. General distribution: Also recorded in Algeria, Austria, Azerbaijan, Cyprus, Bosnia, Herzegovina, Croatia and Yugoslavia, Egypt, France, Germany, Greece, Iran, Italy, Madeira Islands, Morocco, Portugal, Romania, Russia, Spain, Switzerland, Ukraine. Type: Syntypes ZMHB. Ref: Krishna et al., 2013: 838-842.

Family: Termitidae

Amitermes corpulentus Al-Alawy, [START_REF] Al-Alawy | Description of new species of termites (Insecta, Isoptera) from Iraq[END_REF] Geographical records: Iraq. General distribution: Also recorded in Iran. Type: Holotype INHM. Paratype INHM, BMNH, DZCU. Ref: [START_REF] Krishna | Treatise on the Isoptera of the World[END_REF]Krishna et al., : 2011. . Amitermes vilis [START_REF] Hagen | Monographie der Termiten[END_REF] Termes vilis Hagen, 1858: 185-186, 240. Geographical records: Iraq, Jordan. General distribution: Also recorded in Yemen, Afghanistan, Iran, Oman, Saudi Arabia, Turkmenistan. Type: Syntypes NHMW, AMNH, ZIHS. Ref: [START_REF] Krishna | Treatise on the Isoptera of the World[END_REF]Krishna et al., : 2058Krishna et al., -2059. [START_REF] Silvestri | A new termite from Mesopotamia[END_REF] Geographical records: Iraq. General distribution: Also recorded in Yemen, North Yemen, South Yemen, Iran, Kuwait, Oman, Saudi Arabia. Type: Syntypes IEA, AMNH. Ref: [START_REF] Krishna | Treatise on the Isoptera of the World[END_REF]Krishna et al., : 2196Krishna et al., -2197. (Chopard, 1921) Polyphaga persica Chopard, 1921: 763. Arenivaga (Heterogamisca) persica (Chopard, 1921); Princis, 1962: 68. Heterogamisca persica (Chopard, 1921); Grandcolas, 1994b: 157. Geographical records: Iraq. General distribution: Iran, Iraq, Afghanistan. Type: Holotype NHM. Type locality: Iraq: (Perse, Quazvin, Mésopotamie). Ref: Princis, 1961Princis, -1971: 68. : 68. (http://blattodea.speciesfile.org/Common/basic/Taxa.aspx?TaxonNameID=1178078).

Microcerotermes diversus

Microcerotermes gabrielis

Heterogamisca simillima [START_REF] Chopard | Orthoptera palaearctica critica. VII. Les Polyphagiens de la faune paléarctique[END_REF] Heterogamodes simillima Chopard, 1929: 313. Arenivaga (Heterogamisca) simillima (Chopard, 1921); Princis, 1962: 68. Geographical records: Jordan. General distribution: From Jordan to the Mediterranean Sea. Type: Holotype NHM. Type locality: Jordan: Transjordanie. Ref: Princis, 1961Princis, -1971: 68. : 68. (http://blattodea.speciesfile.org/common/basic/taxa.aspx/Taxa.aspx?TaxonNameID=1178079)

Heterogamodes ursina [START_REF] Burmeister | Handbuch der Entomologie[END_REF] Heterogamia ursina Burmeister, 1838: 489. Heterogamia africana Brunner de Wattenwyl, 1865: 357. Heterogamia syriaca Krauss, 1890: 241. Heterogamia livida Giglio-Tos, 1893: 3. Heterogamodes ursina [START_REF] Burmeister | Handbuch der Entomologie[END_REF]; Princis, 1962: 56;Grandcolas, 1994b: 157. Geographical records: Syria. General distribution: Syria; Egypt; Libya; Tunisia; Algeria; Morocco; Africa (Sahara). Type: MLUH. Type locality: Syria, Egypt. Ref: Princis, 1962: 56. (http://blattodea.speciesfile.org/Common/basic/Taxa.aspx?TaxonNameID=1178086) There are specimens identified as ursina in the MHNG Collection, but they are not indicated as types or as part of a type series.

Polyphaga aegyptiaca [START_REF] Linnaeus | Systema naturae per regna tria naturae, secundum classes, ordines, genera, species, cum characteribus, differentiis, synonymis, locis. Tomus 1, editio decima, reformata. Holmaie[END_REF] Blatta aegyptiaca Linnaeus, 1758: 424. Polyphaga aegyptiaca [START_REF] Linnaeus | Systema naturae per regna tria naturae, secundum classes, ordines, genera, species, cum characteribus, differentiis, synonymis, locis. Tomus 1, editio decima, reformata. Holmaie[END_REF]; Princis, 1962: 48;Grandcolas, 1994b: 157 [START_REF] Linnaeus | Systema naturae per regna tria naturae, secundum classes, ordines, genera, species, cum characteribus, differentiis, synonymis, locis. Tomus 1, editio decima, reformata. Holmaie[END_REF]; Grandcolas, 1994b: 157. Geographical records: Syria. General distribution: From Egypt to Syria, probably more abundant in Syria and surrounding countries (Chopard, 1929: 304). Type: UZIU. Type locality: Africa. Specimens of africana in Syria and Lebanon in MLUH, MNHN, NHMW, NHM, ZMUZ. of ursine and syriaca in MHNG. Ref : Princis, 1961: Princis, -1971: 66;: 66;Chopard, 1929: 301-305. (http://blattodea.speciesfile.org/Common/basic/Taxa.aspx?TaxonNameID=1177349) There are specimens identified as africana in MHNG, but they are not indicated as types or as part of a type series.

Hemelytroblatta ebneri [START_REF] Chopard | Orthoptera palaearctica critica. VII. Les Polyphagiens de la faune paléarctique[END_REF] Heterogamodes ebneri Chopard, 1929: 305-306. Arenivaga (Psammoblatta) ebneri [START_REF] Chopard | Orthoptera palaearctica critica. VII. Les Polyphagiens de la faune paléarctique[END_REF]; Princis, 1962: 67. Hemelytroblatta africana [START_REF] Linnaeus | Systema naturae per regna tria naturae, secundum classes, ordines, genera, species, cum characteribus, differentiis, synonymis, locis. Tomus 1, editio decima, reformata. Holmaie[END_REF]; Grandcolas, 1994b: 157. Geographical records: Turkey, Syria. General distribution: Turkey, Syria, Rhodos. Type: NHMW. Type locality: Turkey: Taurus: Makri. Ref: Princis, 1961Princis, -1971: 67. : 67. There are specimens identified as ebneri in MHNG, but they are not indicated as types or as part of a type series.

Hemelytroblatta latifrons [START_REF] Chopard | Orthoptera palaearctica critica. VII. Les Polyphagiens de la faune paléarctique[END_REF] Heterogamodes latifrons Chopard, 1929: 319. Psammoblatta persica Chopard, 1929: 316. Arenivaga (Psammoblatta) latifrons [START_REF] Chopard | Orthoptera palaearctica critica. VII. Les Polyphagiens de la faune paléarctique[END_REF]; Princis, 1962: 66. Geographical record: Iraq. General distribution: Iraq, Iran, Pakistan. Type: MNHN. Type locality : Perse: Bender-Bouchir. Ref: Princis, 1961Princis, -1971: 66. : 66. (http://blattodea.speciesfile.org/Common/basic/Taxa.aspx?TaxonNameID=1178047) (http://isyeb.mnhn.fr/IMG/pdf/Types_Blattaria_MNHN_modif.pdf)

Hemelytroblatta livida (Brunner de Wattenwyl, 1865) Heterogamia livida Brunner de Wattenwyl, 1865: 359. Arenivaga (Psammoblatta) livida (Brunner de Wattenwyl, 1865); Princis, 1962: 62. Geographical records: Iraq, Turkey, Syria, Jordan. General distribution: Cyprus, Greece, Asia Minor. Type: Paratype NHMW. Type locality: Chypre. Ref: Princis, 1961-1971: 62, Giglio-tos, 1893: 3. A recent list of types in NHMW confirms its presence in this institution.

Hemelytroblatta marismortui [START_REF] Janson | Some Account of the fauna and flora of Sinai. Petra and Wady 'Arabah[END_REF] Heterogamia (Polyphaga) maris-mortui Janson, 1891: 184. Arenivaga (Psammoblatta) maris-mortui (Janson, 1891); Princis, 1962: 65. Hemelytroblatta africana [START_REF] Linnaeus | Systema naturae per regna tria naturae, secundum classes, ordines, genera, species, cum characteribus, differentiis, synonymis, locis. Tomus 1, editio decima, reformata. Holmaie[END_REF]; Grandcolas, 1994b: 157. Geographical records: Iraq, Syria. General distribution: Syria, Iraq, Sinaï (Egypt). Type: NHM. Type locality: Egypt: Sinaï. Ref: Princis, 1961Princis, -1971: 65. : 65. (http://blattodea.speciesfile.org/Common/basic/Taxa.aspx?TaxonNameID=1178051) Although the original publications mention NHM as the depository institution, recent institutional lists do not confirm its presence.

Family: Ectobiidae

Subfamily: Ectobiinae

Arbiblatta syriaca (Bey-Bienko, 1938) Phyllodromica (Arbiblatta) syriaca [START_REF] Bey-Bienko | On some new or interesting Asiatic Blattodea[END_REF]: 232. Hololampra brevipennis Ramme, 1951: 415. Geographical records: Syria. General distribution: Syria. Type: NHMW in the Brunner von Wattenwyl's collection; Paratype in the former Zoological Institute, Leningard, now ZIN. Type locality: Syria Ref: Princis, 1961Princis, -1971Princis, : 1108. . (http://blattodea.speciesfile.org/Common/basic/Taxa.aspx?TaxonNameID=1175944) Ectobius vittiventris [START_REF] Costa | Pecie nuove o rare d'Insetti delle Montagne del Matese[END_REF] Blatta vittiventris [START_REF] Costa | Pecie nuove o rare d'Insetti delle Montagne del Matese[END_REF]: 111. Ectobius neolividus Fruhstorfer, 1921: 13, 18, 56, 58, 63, 78. Geographical records: Turkey. General distribution: Switzerland; Sicily, Italy; Yugoslavia; Romania; Bulgaria; Turkey; Caucasus mountains. Type: Syntype ZMHB. Type locality: Switzerland. Ref: Princis, 1961Princis, -1971Princis, : 1050Princis, -1052;;Baur & Corey, 2004: 615. (http://nmbe-xen25.unibe.ch/sites/default/files/uploads/pubinv/2120.pdf) There are specimens identified as vittiventris in MHNG, but they are not indicated as types or as part of a type series.

Luridiblatta beybienkoi [START_REF] Maran | Wissenschaftliche Ergebnisse der zoologischen Expedition des Nationalmuseums in Prag nach der Turkei. 21. Blattoidea[END_REF] Phyllodromica beybienkoi Maran, 1957: 165. Geographical records: Turkey. General distribution: Turkey. Type: Holotype NMP. Type locality: Turkey: Karapinar, Toros. Ref: Maran, 1957: 165. Phyllodromica brevipennis [START_REF] Fischer | Orthoptera Europaea. Mantodea[END_REF] Blatta brevipennis Fischer, 1853: 102. Alphebia carpetana (nec Bolívar, 1873) Giglio-tos, 1893: 3. Geographical records: Syria, Turkey, Lebanon. General distribution: Austria; Italy; Yugoslavia; Hungary; Albania; Bulgaria; Turkey; Syria. Type: NHMW in the Brunner von Wattenwyl's collection. Type locality: Italy: Istria. Ref: Princis, 1971Princis, : 1103;;Giglio-tos, 1893: 3. (http://blattodea.speciesfile.org/Common/basic/Taxa.aspx?TaxonNameID=1176174) There are specimens identified as brevipennis in MHNG, but they are not indicated as types or as part of a type series.

Phyllodromica marginata [START_REF] Schreber | Beschreibungen merkwürdiger Insekten. Erstes Stück[END_REF] Blatta marginata Schreber, 1781: 88. Geographical records: Turkey. General distribution: (Sardinia Island), Italy; (Sicily), Italy; Austria; Yugoslavia; Albania; Greece; Bulgaria; Hungary; Romania; Ukraine; Crimea; Turkey. Type: MLUH. Type locality: Italy. Ref: Princis, 1971: 1091-1093. (http://blattodea.speciesfile.org/Common/basic/Taxa.aspx?TaxonNameID=1176252) There are specimens identified as marginata in MHNG, but they are not indicated as types or as part of a type series.

Phyllodromica megerlei Fieber, 1853 Blatta megerlei Fieber, 1853: 94. Hololampra f. erythronota Ramme, 1951: 324. Phyllodromica megerlei ssp. asiatica Bey-Bienko, 1950: 232. Geographical records: Syria, Turkey. General distribution: Germany; Switzerland; Slovakia; Czech Republic; Austria; Italy; Slovenia; Croatia; Serbia; Bosnia and Herzegovina; Hungary; Romania; Moldova; Republic of Ukraine. Type: According to [START_REF] Vidlicka | Revision of the megerlei-group of the cockroach genus Phyllodromica Fieber (Blattaria: Blattelidae, Ectobiinae)[END_REF], the type is missing. Ref: Princis, 1961Princis, -1971Princis, : 1094Princis, -1096;;Vidlicka & Majzlan, 1997: 165. (http://blattodea.speciesfile.org/Common/basic/Taxa.aspx?TaxonNameID=1176209) There are specimens identified as megerlei in MHNG, but they are not indicated as types or as part of a type series. There are specimens of asiatica in Syria and Turkey in ZMHB.

Phyllodromica pallida (Brunner von Wattenwyl, 1882) Aphlebia pallida Brunner von Wattenwyl, 1882: 42. Geographical records: Turkey. General distribution: Greece; Yugoslavia; Turkey. Type: NHMW. Type locality: Greece. Ref : Princis 1971: Princis : 1104. . (http://blattodea.speciesfile.org/Common/basic/Taxa.aspx?TaxonNameID=1176247) There are specimens identified as pallida in MHNG, but they are not indicated as types or as part of a type series. Harz, 1977: 33-34, probably not valid). Type locality: unknown Ref: Princis, 1961-1971: 850, Giglio-tos, 1893: 3. (http://blattodea.speciesfile.org/Common/basic/Taxa.aspx?TaxonNameID=1175303) There are specimens identified as decipiens in NHM, but they are not indicated as types or as part of a type series.

Phyllodromica znojkoi

Family: Blattidae

Subfamily: Blattinae Blatta orientalis Linnaeus, 1758 Blatta orientalis Linnaeus, 1758: 724. Geographical records: Ankara-Baraj, Anatolia (Turkey), Syria. General distribution: cosmopolitan in temperate regions [southern Russian origin]. Type: Lectotype, Paralectotype NHM in the collection of the Linnean Society; Neotype MHNG, [START_REF] Marshall | The orthopteroid insects described by Linnaeus, with notes on the Linnaean collection[END_REF]: 390. Ref: Princis, 1962-1971: 475-507;Marshall, 1983: 390;Giglio-tos 1893: 3. Remarks: Domestic species. Cosmopolitan There are specimens identified as orientalis in MLUH, but they are not indicated as types or as part of a type series. The congeneric Blatta furcata (Karny, 1908) could be probably found in the region as well [START_REF] Bohn | Blatta furcata (Karny), the nearest relative of the oriental cockroach Blatta orientalis L. (Insecta: Blattodea: Blattidae)[END_REF].


Keywords: plasmonics, single emitter, defects, nanoantennas, graphene

In this work we theoretically investigate the impact of the atomic scale lattice imperfections of graphene nanoakes on their nonlinear response enhanced by the resonance between an incident electromagnetic eld and localized plasmon. As a case study we address the second harmonic generation from graphene plasmonic nanoantennas of different symmetry with missing carbon atom vacancy defects in the honeycomb lattice.

Using the many-body time-dependent density matrix approach we nd that one defect in the nanoake comprising over ve thousand carbon atoms can strongly impact the nonlinear hyperpolarizability and override the symmetry constraints. The eect reported here cannot be captured using relaxation time approximation within quatum or classical framework. Results obtained in this work have thus important implications for design of nonlinear graphene devices.

Owing to an electron motion conned to the 2D plane of atoms, and its band-structure characterized by the Dirac cone at the K-point, graphene shows remarkable properties in electron transport, light-matter interaction, and spintronics. [START_REF] Cox | Plasmon-assisted high-harmonic generation in graphene[END_REF] In particular, multiple experimental and theoretical studies demonstrated that electrical doping allows one to control the linear and nonlinear optical response of graphene nanostructures, which is to large extent enabled by tunability of their collective electronic excitations, 2D graphene plasmons. [START_REF] Grigorenko | Graphene plasmonics[END_REF]913 The possibility of an active control along with strong nonlinearity makes graphene nanostructures a versatile platforms for linear [START_REF] Grigorenko | Graphene plasmonics[END_REF][START_REF] García De Abajo | Graphene plasmonics: challenges and opportunities[END_REF][START_REF] Low | Graphene plasmonics for terahertz to mid-infrared applications[END_REF] and nonlinear [START_REF] Vermeulen | Perspectives on nonlinear optics of graphene: Opportunities and challenges[END_REF]1422 applications in information technologies, wavelength conversion, and sensing among others. This vast range of possible applications challenges the nanofabrication of graphene nanodevices optimised for the desired linear or nonlinear eects. In this regard, the possible guidance from the theory cannot be overestimated. Since the fabricated nanodevices are unavoidably unperfect on atomic scale (lattice imperfections, substrate eects, adatoms), one naturally questions to which extent the optical response of the realistic nanostructure diers from that of an ideal one. So far, most of the theoretical studies of the linear and nonlinear optical properties of graphene and graphene nanostructures either neglected the lattice imperfections, or accounted for their eect introducing empirical or calculated relaxation time associated with defect scattering, [START_REF] Das Sarma | Electronic transport in twodimensional graphene[END_REF]2325 while considering an ideal graphene lattice (quantum treatments), 12,[START_REF] Jablan | Multiplasmon absorption in graphene[END_REF][START_REF] Cox | Plasmon-assisted high-harmonic generation in graphene[END_REF]2632 or homogeneous 2D material (classical treatments). [START_REF] García De Abajo | Graphene plasmonics: challenges and opportunities[END_REF][START_REF] Low | Graphene plasmonics for terahertz to mid-infrared applications[END_REF][START_REF] You | Four-wave mixing of topological edge plasmons in graphene metasurfaces[END_REF]3339 In this regard, e.g. the Drude dielectric functions [START_REF] Low | Graphene plasmonics for terahertz to mid-infrared applications[END_REF][START_REF] Jablan | Multiplasmon absorption in graphene[END_REF]3942 contain the relaxation time as a parameter simplifying such an approach.

In this work, using a case study of the second harmonic generation from graphene nanoakes (GNF's) with lattice defects represented by a carbon atom vacancy, we theoretically demonstrate the "local" eect of the defect scattering which can not be captured by relaxation time approximations described above (see also Ref. [START_REF] Karimi | Nonlinear optical response in graphene nanoribbons: The critical role of electron scattering[END_REF] ). We consider the situation where the incident electromagnetic radiation is at resonance with localized plasmon of the doped GNF. The plasmon excitation is associated with strong enhancement of the near eld and thus of the nonlinear response of the nanostructure. [START_REF] Cox | Plasmon-assisted high-harmonic generation in graphene[END_REF][START_REF] Mikhailov | Theory of the giant plasmon-enhanced second-harmonic generation in graphene and semiconductor two-dimensional electron systems[END_REF][START_REF] Cox | Quantum eects in the nonlinear response of graphene plasmons[END_REF] We compare the nonlinear response of GNF's with geometries chosen such that for an ideal lattice with no defects the second harmonic generation is either (1) ecient, or (2) it is inhibited by the centrosymmetry of the object. Performing quantum-many body calculations we show that single atom vacancy defect concentrations as low as one per several thousands of lattice atoms lift the symmetry constraint and lead to an ecient second harmonic generation.

The eective nonlinear second-order susceptibility of the defective GNF is larger than that of such strongly non-linear low-dimensional materials as transition metal dichalogenides. [START_REF] Ullah | Harmonic generation in low-dimensional materials[END_REF] We further demonstrate that irrespective of the nonlinear response for an ideal geometry, nanofaricated GNF's might have similar second harmonic generation eciency because of the unavoidable defect contribution. Our results thus provide a quantitative insights into the nonlinear optical properties of small graphene nanostructures with lattice defects. The ndings reported here are of importance for setting realistic objectives when engineering nonlinear graphene devices.

As representative examples, we study the nonlinear second-order response of the circular (C-type, N c = 5520 carbon atoms), hexagonal (H-type, N c = 5514 carbon atoms), and triangular (T-type, N c = 5556 carbon atoms) GNF's often considered in the context of graphene plasmonics [START_REF] Koppens | In 2D Materials: Properties and Devices[END_REF] as prototypical nanoantennas allowing enhancement of the incident electromagnetic eld and thus of the nonlinear eects. 12,[START_REF] Cox | Plasmon-assisted high-harmonic generation in graphene[END_REF][START_REF] Cox | Quantum eects in the nonlinear response of graphene plasmons[END_REF]4547 The geometry of the nanostructures with carbon atom layer located in the (x, y)-plane is sketched in Figure 1.

The H-type and T-type GNF's have armchair edges reducing the localized plasmon decay via adge scattering. [START_REF] Koppens | In 2D Materials: Properties and Devices[END_REF][START_REF] Thongrattanasiri | Quantum nite-size eects in graphene plasmons[END_REF] The charge doping with 100 (C-type), 20 (H-type), and 20 (T-type) electrons added to the neutral nanostructure, and large enough size of the nanostructures (∼ 10 nm), allow to obtain well-dened dipolar plasmon resonances in the optical response of the defect-free GNF's [START_REF] Thongrattanasiri | Quantum nite-size eects in graphene plasmons[END_REF][START_REF] Aguillon | Plasmons in graphene nanostructures with point defects and impurities[END_REF] as shown in Figure 1).

Inline with widely used approaches, the single atom vacancy defect is modeled by removing the carbon atom located at the defect site of the honeycomb lattice of GNF's. While being an approximation to more complex realistic situations 5054 this model of the vacancy defects captures the main physics behind the electronic and optical properties of defective graphene. [START_REF] Das Sarma | Electronic transport in twodimensional graphene[END_REF][START_REF] Stauber | Electronic transport in graphene: A semiclassical approach including midgap states[END_REF][START_REF] Yuan | Modeling electronic structure and transport properties of graphene with resonant scattering centers[END_REF]5557 The dynamics of the electron density of GNF's in response to an incident electromagnetic wave is described within the framework of the time-dependent density matrix approach using the tight-binding description of the electronic structure of graphene (TDDM-TB).

The TDDM-TB approach is based on earlier developments 12,[START_REF] Cox | Plasmon-assisted high-harmonic generation in graphene[END_REF][START_REF] Cox | Quantum eects in the nonlinear response of graphene plasmons[END_REF][START_REF] Thongrattanasiri | Quantum nite-size eects in graphene plasmons[END_REF][START_REF] Aguillon | Plasmons in graphene nanostructures with point defects and impurities[END_REF]58,[START_REF] Kosik | Revising quantum optical phenomena in adatoms coupled to graphene nanoantennas[END_REF] and it allows to address large GNFs with well-established collective plasmon modes. The details on the actual implementation of the method are given in Supporting Information (SI). In brief, using the split-operator time-propagation, [START_REF] Leforestier | A comparison of dierent propagation schemes for the time dependent Schrödinger equation[END_REF] we solve the Gorini-Kossakowski-Lindblad-Sudarshan (GKLS) quantum master equation for the time-evolution of the density matrix ρ6164 (unless otherwise stated atomic units are used throughout the paper)

∂ ρ ∂t = - i ℏ Ĥ, ρ + L D (ρ) . (1)
The tight-binding hamiltonian of the system Ĥ depends on the occupation ρjj of the lattice cites (j = 1, ..., N c ) via the non-retarded Coulomb interactions between them. It also comprises the potential owing to the incident x-polarized electromagnetic plane wave with an electric eld given by E(t) = e x E 0 e -iωt , where e x is the unit-length vector along x-axis (see Figure 1). The dissipative part of the Lindblad super-operator, is represented in a simple form L D (ρ) = -γ ρ -ρ(0) describing relaxation to the ground-state density matrix ρ(0)

with a rate γ. Here we use γ = 15 meV to account for the plasmon decay with excitation of the optical phonons in graphene. [START_REF] Jablan | Plasmonics in graphene at infrared frequencies[END_REF][START_REF] Low | Graphene plasmonics for terahertz to mid-infrared applications[END_REF][START_REF] Karimi | Plasmons in graphene nanoribbons[END_REF][START_REF] Novko | Dopant induced plasmon decay in graphene[END_REF] The frequency-resolved quantities are obtained from the time-dependent quantities by time-to-frequency Fourier transform.

The plasmon modes of GNF's appear in the calculated linear absorption spectra as absorption resonances allowing analysis of their energies and widths. The second-order hyperpolarizability tensor α(2) (2ω, ω, ω) of the 2D nanostructure is obtained from TDDM-TB calculations using the dipole induced at the second-harmonic (R j stands for the position vector of the lattice site j)

P(2ω) = - Nc j=1 R j ρjj (2ω), (2)
To this end, considering the reection symmetry with respect to (x, y)-plane, the P(2ω) is represented in the form

P(2ω) = m,n,k e m α (2) mnk (2ω, ω, ω) E n (ω) E k (ω). (3)
The indexes m, n, k take the values x, y, and E n (ω) is the projection of the incident electric eld vector E(ω) on the corresponding coordinate axis

E n (ω) = e m • E(ω).
The eciency of the second-order nonlinear process is given by the total power P radiated into the far eld at the second harmonic frequency. Considering the x-polarized incident eld, it can be expressed as

P = 2 3 ck 4 |P(2ω)| 2 = 2 3 ck 4 α (2) xxx (2ω, ω, ω) 2 + α (2) yxx (2ω, ω, ω) 2 E 4 0 , ( 4
)
where c is the speed of light in vacuum, and the wave vector k is obtained from 2ω = kc.

Therefore, we will characterize the systems by the parameter

α (2) (2ω) ≡ α (2) xxx (2ω, ω, ω) 2 + α (2) yxx (2ω, ω, ω) 2 , (5)
dubbed as second-order hyperpolarizability in the rest of the text.

Prior to the discussion of the second harmonic generation, we describe the major features When present, carbon atom vacancies in graphene lattice produce very ecient scattering centers increasing decay and dephasing of the localized plasmons. [START_REF] Low | Graphene plasmonics for terahertz to mid-infrared applications[END_REF][START_REF] Principi | Impact of disorder on Dirac plasmon losses[END_REF][START_REF] Aguillon | Plasmons in graphene nanostructures with point defects and impurities[END_REF]6670 Indeed, even a single vacancy (≈ 1/5500 vacancy concentration) notably broadens the plasmon resonance in absorption spectra, while 3 vacancies (≈ 1/2800 vacancy concentration) double the plasmon resonance width.

Because of the large incident eld enhancement, the nonlinear response of GNF's is strongest when the incident radiation excites dipolar plasmon of GNF's, [START_REF] Cox | Plasmon-assisted high-harmonic generation in graphene[END_REF][START_REF] Cox | Quantum eects in the nonlinear response of graphene plasmons[END_REF] as we further discuss in SI with an example of the H-type nanoake. Therefore, we focus our work on the resonant situation ω = ω DP , and we will use the term "resonant" for the corresponding nonlinear response and second-order hyperpolarizability α (2) (2ω). It is noteworthy that from results shown in Figure 1 one would expect that broadening of the localized plasmons by the vacancy defects leads to smaller 2 nd harmonic generation. As we discuss below, while this eect indeed exists, it is not always the leading consequence of the presence of the lattice imperfections.

We now turn to the second harmonic generation from GNF's with and without carbon the defect-free C-type and H-type GNF's i.e. no second harmonic generation is possible in the far eld because of the centrosymmetric geometry of these nanoparticles. As soon as the defect breaks the symmetry of the C-type and H-type GNF's, a nonlinear dipole can be induced at the second harmonic frequency with dipolar second harmonic emission into the far eld. It is worth noting that the second harmonic is present in the near eld of the pristine centrosymmetric nanostructures (see SI), where it results from the high order multipoles.

In Figure 2 we show the resonant hyperpolarizability of the defective C-type and H-type GNF's calculated with TDDM-TB as function of the position of the carbon atom vacancy.

The defect is following the armchair sequence along the x-axis (y = 0). It follows from our results that even a single carbon atom vacancy present in the nanostructure comprising over 5500 carbon atoms, and displaced from the geometrical center by only one C-C bond length is very ecient in allowing frequency conversion. The resulting hyperpolarizability α (2) (2ω) is of the same order of magnitude as that of an ideal T-type GNF (7×10 9 a.u. or 6×10 -23 esu)

which is a prototypical nanoantenna for the second-order nonlinear applications. It is also essentially larger than the nonlinear response of the metal nanostructures of comparable size as analysed in details in Ref. 12 (see also Refs. [START_REF] Hendry | Coherent nonlinear optical response of graphene[END_REF][START_REF] Karimi | Nonlinear optical response in graphene nanoribbons: The critical role of electron scattering[END_REF] ).

The role of the vacancy defect in allowing the resonant second harmonic generation from centrosymmetric GNF's is further analysed in Figure 3. We compare the charge density induced at ω (linear response) and 2ω (nonlinear response) in an ideal and defective H- Finally, in Figure 4 we analyse the second harmonic generation from defective GNF's with dierent geometries (see Figure 1). Results are shown as function of the number of carbon atom vacancies N V for the x-polarised incident plane wave with fundamental frequency ω at resonance with dipolar plasmon of the nanostructure ω = ω DP .

Two distinct behaviours of resonant hyperpolarizability α [START_REF] Geim | The rise of graphene[END_REF] (2ω) on N V are obtained when comparing (i) the C-type, H-type GNF's with geometry silencing the second harmonic generation for an ideal nanoparticle because of the symmetry constraints, and (ii) the Ttype GNF allowing an ecient frequency conversion for an ideal nanoparticle. Upon adding defects to an ideal lattice of geometry (i) the second-order hyperpolarizability GNF's rst increases, reaches an optimum value at N V = 1-3, and then decreases. This nding indicates a local eect that can not be captured by an eective relaxation time approximation. For the T-type GNF, increasing number of defects continuously reduces the nonlinear response.

The dependence of resonant α (2) (2ω) on vacancy concentration can be understood as follows. For the C-type and H-type GNF's two eects compete. First, the vacancies produce the symmetry break necessary for the second harmonic generation. Second, the broadening of the localized plasmon by the vacancy scattering (see Figure 1), reduces the near elds, and thus the nonlinearity. The interplay between these two eects leads to the existence of the optimal vacancy concentration maximising the second-order nonlinearity. This is in contrast with the T-type GNF which is an ecient frequency convertor for an ideal lattice. In this situation the defects degrade the resonant nonlinear response via the plasmon broadening mechanism and α [START_REF] Geim | The rise of graphene[END_REF] (2ω) monotonously decreases with increasing N V . Noteworthy, we obtain that when ∼ 10 vacancies are present in the lattice (0.2% vacancy concentration), GNF's studied here feature similar α [START_REF] Geim | The rise of graphene[END_REF] (2ω) irrespective of their geometry. I.e. the vacancy eect determines the second harmonic generation in the system.

C -type H -type

Before closing the discussion of the results obtained in this study several comments are in order regarding model representation of the vacancy defect used here, and neglect of the role of the substrate that hosts plasmonic GNF's in practical situations, and can be a source of nonlinear eects. [START_REF] Dean | Second harmonic generation from graphene and graphitic lms[END_REF][START_REF] Dean | Graphene and few-layer graphite probed by secondharmonic generation: Theory and experiment[END_REF] To test the robust character of our results with respect to the model representation of the vacancy defect we calculated resonant α [START_REF] Geim | The rise of graphene[END_REF] (2ω) of the defective H-type GNF (i) considering the actual V 1 (5-9) mono-vacancy defects [START_REF] Banhart | Structural defects in graphene[END_REF][START_REF] Bhatt | Various defects in graphene: a review[END_REF][START_REF] Kotakoski | Stone-Wales-type transformations in carbon nanostructures driven by electron irradiation[END_REF] with atomic arrangement determined from ab initio calculations in Refs.; [START_REF] Banhart | Structural defects in graphene[END_REF][START_REF] Zobelli | A comparative study of density functional and density functional tight binding calculations of defects in graphene[END_REF] (ii) considering the defects formed by the substitutional boron and nitrogen atoms. [START_REF] Araujo | Defects and impurities in graphene-like materials[END_REF][START_REF] Panchakarla | structure, and properties of Boron-and Nitrogen-doped graphene[END_REF] These calculations (see details in SI) fully support our ndings on the role of the lattice defects in the second harmonic generation from GNF's. It is also worth mentioning that the charged adatom or dipolar admolecule considered earlier in the context of sensing using centrosymmetric GNF's [START_REF] Yu | Nonlinear plasmonic sensing with nanographene[END_REF] and representing another type of the point defect, lead to comparable nonlinear hyperpolarizability of GNF's.

Furthermore, from the calculated second-order hyperpolarizabilities of GNF's one can obtain their eective surface χ

s and bulk χ

b resonant nonlinear susceptibilities. Thus, the single vacancy per 5500 carbon atoms leads to χ which can be compared with nonlinear second-order response of the 2D layered materials reported in literature. Present results are nearly 10 3 times larger than that of hexagonal boron nitride, [START_REF] Ullah | Harmonic generation in low-dimensional materials[END_REF][START_REF] Li | Probing symmetry properties of few-layer MoS2 and h-BN by optical second-harmonic generation[END_REF] or of graphene or multilayer graphene triggered by the dc current, [START_REF] An | Optical secondharmonic generation induced by electric current in graphene on Si and SiC substrates[END_REF][START_REF] Brun | Intense and tunable second-harmonic generation in biased bilayer graphene[END_REF] electrical doping, [START_REF] Zhang | Doping-induced second-harmonic generation in centrosymmetric graphene from quadrupole response[END_REF] or symmetry break [START_REF] Zhang | Doping-induced second-harmonic generation in centrosymmetric graphene from quadrupole response[END_REF][START_REF] Lobet | Probing Graphene χ (2) ( Using a Gold Photon Sieve[END_REF] (see also the review by R. Zhou et al [START_REF] Zhou | Engineering the harmonic generation in graphene[END_REF] ). Moreover, the eective χ

s and χ

b found here are larger than the second-order susceptibility of such strongly nonlinear materials as transition metal dichalogenides. [START_REF] Ullah | Harmonic generation in low-dimensional materials[END_REF][START_REF] Li | Probing symmetry properties of few-layer MoS2 and h-BN by optical second-harmonic generation[END_REF]8284 (See further details in SI.) Therefore, despite the fabrication of plasmonic GNF's implies their deposition on the substrate, such as glass, SiC, or h-BN, the eect of the lattice defects reported here would override [START_REF] Janisch | Extraordinary second harmonic generation in tungsten disulde monolayers[END_REF] the nonlinearity linked with presence of the substrate.

In conclusion, using the many-body time-dependent density matrix approach we addressed the inuence of the lattice defects on the second-order nonlinear response of the doped graphene nanostructures. The incident electromagnetic wave is at resonance with localized dipolar plasmon of the nanostructure which increases the nonlinearity. As a case study we considered the single carbon atom vacancy defects within nanoakes of the geometries prototypical for graphene plasmonics.

As main ndings of our work we would like to emphasize the following For graphene nanostructures with geometries inhibiting the second harmonic generation in the case of an ideal lattice, a single carbon atom vacancy defect in GNF comprising 5500 carbon atoms lifts this constraint and renders the frequency conversion very ecient. In such a situation, when additionally the incident eld is at resonance with dipolar plasmon of the doped GNF, the second harmonic emission into the far eld is comparable to that of a triangular graphene nanoantenna considered in the literature as a prototypical system for non-linear applications.

The resonant second-order hyperpolarizability α (2) (2ω) calculated for the defective GNF's is essentially larger than that of e.g. metal nanostructures of comparable size.

Moreover, the eective second-order nonlinear susceptibility obtained from α [START_REF] Geim | The rise of graphene[END_REF] (2ω) for the T-type or defected C-type and H-type GNF's are larger than the data reported for such strongly nonlinear 2D materials as transition metal dichalogenides.

For the vacancy concentrations reaching 0.2%, GNF's of dierent geometries feature similar large α (2) (2ω). Thus, irrespective of their geometry, defective GNF's are equally ecient for nonlinear applications.

We believe that the quantitative character of our analysis and our qualitative ndings are of importance for design and fabrication of graphene nanodevices for nonlinear applications.

Figure 1 :

 1 Figure 1: Upper panels: sketch of the circular (C-type), hexagonal (H-type), and triangular (T-type) GNF's studied in this work. Note that for the sake of the visibility of the edges and vacancies, we show GNF's that have spatial dimensions twice smaller than actually studied structures. Two randomly generated missing atom vacancies of the graphene lattice are identied with cyan circles. The blue double arrow indicates the polarization of the incident electromagnetic eld. Lower panels: Linear optical response of the doped GNF's. Absorption spectra calculated with TDDM-TB approach are presented as function of the frequency of the incident electromagnetic plane wave. Black line: absorption cross-section, σ abs for the defect-free nanoparticles. The colored full lines: absorption cross-sections of GNF's with 1 (blue lines), 3 (red lines), and 10 (green lines) carbon atom vacancy defects at random positions within nanostructure. The results are obtained by an average over 20 random samples. The hatched areas of the corresponding color show the averages crosssection ± standard deviation.

  of the linear response of GNF's considered in this work. The charge doping of GNF's results in well-dened resonances in the absorption spectra shown in Figure 1. Absorption resonances are associated with excitation of localised dipolar plasmon modes 10 (DP) at frequencies ω DP = 0.53 eV (C-type), ω DP = 0.37 eV (H-type), and ω DP = 0.32 eV (T-type).

  atom vacancy defects. An ideal defect-free T-type GNF has been thoroughly studied in the literature.12,[START_REF] Cox | Plasmon-assisted high-harmonic generation in graphene[END_REF][START_REF] Cox | Quantum eects in the nonlinear response of graphene plasmons[END_REF][START_REF] Deng | Quantum mechanical analysis of nonlinear optical response of interacting graphene nanoakes[END_REF] It is characterised by high resonant hyperpolarizability, and permits an ecient second harmonic generation in the far eld. In sheer contrast, α[START_REF] Geim | The rise of graphene[END_REF] (2ω) = 0 forC -type H -typeVacancy distance from the center of the nanostructure (a.u.)

Figure 2 :

 2 Figure 2: The second-order hyperpolarizability α (2) (2ω) for the x-polarized electromagnetic plane wave incident at the doped C-type and H-type GNF's (see Figure 1) with single atom vacancy. The incident radiation (fundamental frequency ω) is at resonance with dipolar plasmon of the nanostructure ω = ω DP . Results are shown in atomic units (10 9 a.u. = 0.864 × 10 -23 esu) as function of the vacancy distance from the center of GNF. The vacancy is displaced along the symmetry x-axis (y = 0, armchair sequence of the lattice sites).

Figure 3 :

 3 Figure 3: Maps of the charge densities induced at rst and second harmonic in the doped H-type GNF. The incident x-polarized electromagnetic plane wave with the eld amplitude E 0 = 5.14 × 10 -3 V/nm and frequency ω = 0.37 eV is at resonnace with dipolar plasmon of the nanostructure ω = ω DP . Results are shown as function of x-and y-coordinates dened in Figure 1. Red (blue) color stands for the positive (negative) value of the charge density. The range of the variation of the linear (nonlinear) induced density is ±4 × 10 -4 a.u. (±2 × 10 -5 a.u.). Results for the ideal (no defects) and defective (1 carbon atom vacancy) nanostructures are shown. The blue circle indicates the position of the vacancy defect.

Figure 4 :

 4 Figure 4: The nonlinear response of the doped defective C-type, H-type, and T-type GNF's calculated at resonance between the incident x-polarized electromagnetic plane wave and dipolar plasmon of the nanoparticle ω = ω DP . Resonant hyper-polarisability α (2) (2ω) is shown in atomic units (1 × 10 9 a.u. = 0.864 × 10 -23 esu) as a function of the number of randomly located carbon atom vacancies, N V . Each data point is obtained by an average over 20 random samples. Each sample corresponds to the N V vacancies that are randomly generated within GNF. The error bars show the standard deviation. Notice that for the T-type GNF the scaling factor ×1/3 is applied.

( 2 )

 2 s ≈ 2.5 × 10 7 pm 2 V and χ (2) b ≈ 7.6 × 10 4 pm V ,

Acknowledgement

The authors gratefully acknowledge Dr. Arkady Krasheninnikov and Dr Alberto Zobelli for sharing ab initio geometry of the monovacancy V 1 (5-9) defect in graphene.

Supporting Information Available

The following les are available free of charge. Supporting information providing details on the Time-dependent density matrix method based on the tight-binding description of the electronic structure of graphene.

The eect of the resonance between the incident electromagnetic radiation and localized dipolar plasmon of graphene nanoakes on the second harmonic generation.

The near eld induced at the second harmonic.

The nonlinear response of graphene nanoakes with V 1 (5 -9) single atom vacancies and substitutional Nitrogen and Boron atoms.

The eective surface and volume nonlinear susceptibility of defected graphene nanoakes and its comparison with measured nonlinear susceptibility of 2D materials.