La mise en évidence des nanotubes de carbone mono-feuillets (SWCNTs) en 1993 par S. Iijima a été une des découvertes qui a marqué la recherche dans le domaine des nanos. L'étude de leurs propriétés a révélé qu'elles étaient supérieures à celles de matériaux déjà connus et a laissé entrevoir un champ très large d'applications potentielles. Les méthodes de synthèse actuelles permettent de produire des SWCNTs en grande quantité. Cependant, les échantillons de SWCNTs peuvent être hétérogènes tant au niveau de leur composition (impuretés carbonées et métaux catalytiques) qu'en ce qui concerne les caractéristiques des SWCNTs eux-mêmes. Des traitements post-synthèses sont donc indispensables pour améliorer les caractéristiques des échantillons. Dans ce travail, nous avons dans un premier temps mis en oeuvre une procédure de purification (multi-étape) couramment utilisée. En accord avec la littérature, l'optimisation des paramètres expérimentaux notamment en ce qui concerne le traitement d'oxydation n'a pas permis d'obtenir de bons rendements de purification et une sélectivité de réaction performante. Une méthode de purification alternative a été développée. Elle consiste essentiellement à un traitement thermique sous flux de dichlore. Elle a permis de diminuer significativement la proportion d'impuretés métalliques tout en préservant les nanotubes. Elle s'est également révélée efficace avec des échantillons produits selon différentes méthodes. L'autre difficulté majeure dans l'utilisation des nanotubes est leur processabilité. La modification de l'état de surface des nanotubes passe par la fonctionnalisation chimique. Pour cela, nous avons étudié le greffage de fonctions sondes / The discovery of single-walled carbon nanotubes (SWCNTs) in 1993 by S. Iijima has been one of the milestones of scientific research in the domain of nanos. The study of their properties has revealed that they were superior to those of materials already known and has let to foresee a field of numerous potential applications. The actual methods of synthesis enable to produce SWCNTs in large quantities. However, the SWCNT samples can be heterogeneous regarding both their composition (carbonaceous and metal impurities) and the features of the as-produced SWCNTs. Post-synthesis treatments are therefore indispensable to improve the characteristics of the samples. In this work, firstly we have carried out a standard multi-step purification procedure. In agreement with literature, the optimization of experimental parameters notably those used for the oxidation treatment did not permit us to obtain neither good yields of purification nor efficient selectivity of reaction. An alternative purification method has thus been developed. It consists mainly in a thermal treatment under a dichlore stream. It has enabled to significantly decrease the proportion of metallic impurities in the samples while preserving the nanotubes. It has also shown that this new purification process was efficient with samples produced by different methods. The other main difficulty in using nanotubes is their processability. Chemical functionalization is then needed to modify nanotube surface properties. In that case, we have studied the grafting of probe functions

Le dopage non covalent de nanostructures carbonées par transfert de charge depuis/vers des molécules donneuses ou acceptrices (EDA) ou bien par des molécules d’acide sulfurique H2SO4, est considéré comme potentiellement intéressant pour de nombreuses applications. Parmi celles-ci on peut citer: capteur chimique, transistor à effet de champ, et d’autre l'électronique. Cependant, d'un point de vue théorique, on en sait peu au sujet de ces processus de transfert de charge par électrons ou par trous.Dans un premier temps, nous nous sommes intéressés à l’interaction entre des molécules d’acide sulfurique et des nanostructures modèles, car elles sont capables de doper des nanotubes, de s’intercaler dans le graphite et même d’aligner les tubes dans une phase nématique, ce qui pourrait mener à la création de matériaux composites à forte valeur ajoutée.Bien que certaines études théoriques DFT ont été menées récemment, leurs résultats restent source de confusion. Par exemple, même s’il est rapporté un transfert de charge entre une molécule de H2SO4 et un plan de graphène, tous nos efforts pour reproduire ces calculs ont été infructueux. Nous proposons dans ce travail de thèse, un mécanisme de réaction qui expliquent la "protonation" des parois du tube, tel que proposé dans la littérature. Enfin nous proposons un scénario possible pour une meilleure compréhension de la structuration à grande échelle des molécules d'acide autour de points d'ancrage, telles que des défauts, de la structure carbonée / Non-covalent doping of carbon nanostructures by charge transfer from/to donor/acceptor molecules (EDA) or by H2SO4 molecules, be it with holes or electrons, is usually thought as potentially interesting for many applications of carbon based nano-devices. However, from a theoretical point of view, little is known about such “charge transfer” processes.Employing first-principles method based on Density Functional Theory (DFT), we have studied in details, and proposed a model to rationalize, the interaction between a prototypical donor molecule the tetrathiafulvalene (TTF), a standard acceptor organic molecule, tetracyanoethylene (TCNE) and carbon nanostructures: graphene layer and SWNTs with various chiral indices. Main results concern structural and thermodynamic aspects including dispersion forces effects, and evidently electronic structure modifications of the nanostructures. Various adsorption modes and concentration effects have been investigated. At very low coverage values, we have reported a charge transfer between graphene and TCNE or TTF. Moreover, we have shown that the charge transfer can be enhanced by increasing the concentration of those two EDA molecules, as it has been demonstrated experimentally. Those results are beneficial for comprehending the nonchemical doping mechanism in graphene structure by means of charge transfers. Considering the interaction between these prototypical molecules and carbon nanotubes, we have found that charge transfers tend to decrease while the curvature of nanotube is increasing. Besides, a strong influence of the metallic/semi-conductor character of the SWNTs can be observed and be explained by the change of polarisability of the curved carboneous substrates. Additionally, we have studied the adsorption properties of sulfuric acid molecules, in its non-hydrated form, on carboneous nanostructures. Against the common believe, no charge transfer is observed in the H2SO4@graphene or H2SO4@CNTs cases, even at very high concentrations. Instead, in order to elucidate the origin of p-doping observed experimentally, we have proposed that molecule is responsible of the reversible doping. Besides we have shown that a proton transfer could cause the experimental phenomenon of crystallization of H2SO4 molecules on SWNT’s surface. Finally in such process, defects like vacancy are of first importance, since they could provide anchorage points for hydrogen atoms. The results of the present work will certainly help to understand the charge transfer and doping mechanism of carbon nanostructures by means of non-covalent functionalization, which is a promising method for their future applications

Multi-walled carbon nanotube aggregates can be suspended in the aqueous phase by natural organic matter. These aggregates are ingested by filter feeding zooplankton. Ingested aggregates result in decreased growth and decreased reproduction. These effects may be caused by reduction in energy input from normal feeding behavior. pH alters natural organic matter structure through changes in electrostatic repulsion. Altered natural organic matter structure changes multi-walled carbon nanotube aggregate size. This size variation with variation in pH is significant, but not large enough a change in size to alter toxicity, as the aggregate size range remains well within the particle size selection of the organisms.

The technological interest in a material depends very much on its electrical, magnetic, optical and/or mechanical properties. In carbon nanotubes the atoms form a cylindrical structure with a diameter of the order 1 nm, but the nanotubes can be up to several hundred micrometers in length. This makes carbon nanotubes a remarkable model for one-dimensional systems. A lot of efforts have been dedicated to manufacturing carbon nanotubes, which is expected to be the material for the next generation of devices. Despite all the attention that carbon nanotubes have received from the scientific community, only rather limited progress has been made in the theoretical understanding of their physical properties. In this work, we attempt to provide an understanding of the electron-phonon and electron-defect interactions in semiconducting zigzag carbon nanotubes using a tight-binding approach. The electronic energy dispersion relations are calculated by applying the zone-folding technique to the dispersion relations of graphene. A fourth-nearest-neighbour force constant model has been applied to study the vibrational modes in the carbon nanotubes. Both the electron-phonon interaction and the electron-defect interaction are formulated within the tight-binding approximation, and analyzed in terms of their quantum mechanical scattering rates. Apart from the scattering rates, their components in terms of phonon absorption, phonon emission, backscattering and forward scattering have been determined and analyzed. The scattering rates for (5,0), (7,0), (10,0), (13,0) and (25,0) carbon nanotubes at room temperature and at 10K are presented and discussed. The phonon scattering rate is dependent on the lattice temperature in the interval 0-0.17 eV. We find that backscattering and phonon emission are dominant over forward scattering and phonon absorption in most of the energy interval. However, forward scattering and phonon absorption can be comparable to backscattering and phonon emission in limited energy intervals. The phonon modes associated with each peak in the electron-phonon scattering rates have been identified, and the similarities in the phonon scattering rates between different nanotubes are discussed. The dependence of the defect scattering rate on the tube diameter is similar to that of the phonon scattering rate. Both the phonon and the defect scattering rates show strong dependence on the tube diameter (i.e., the scattering rate decreases as a function of the index of the nanotube). It is observed that the backscattering and forward scattering for electrons interacting with defects occur with same frequency at all energies, in sharp contrast to the situation for phonon scattering. It is demonstrated that the differences in the scattering rate between different tubes are mainly due to the differences in their band structures.

[pt] Neste trabalho estudamos a síntese e caracterização de nanotubos de carbono de paredes simples com incorporação de fósforo. Os nanotubos foram produzidos em duas diferentes temperaturas (800 e 850 Graus Celsius), usando um precursor em pó (fase sólida) sem diluição em nenhum liquido. O sistema utilizado na síntese foi o CVD em alto vácuo. Para a caracterização comparamos as amostras incorporadas com SWNTs crescidos com etanol. Usamos a espectroscopia Raman, espectroscopia de fotoelétrons induzida por raios x (XPS) e microscopia eletrônica de transmissão (TEM) para caracterizar as amostras. A espectroscopia Raman e a microscopia eletrônica de transmissão foram usadas para confirmar a presença de SWNTs na amostra. Encontramos evidencia de que o fósforo foi incorporado nas amostras ao compararmos os espectros Raman dos SWNTs com amostra de referência. A partir dos resultados de XPS, observamos a presença de fósforo ligado a átomos de carbono que nós faz concluir que houve incorporação de fósforo nos nanotubos. / [en] In this work was studied the synthesis and characterization of single wall carbon nanotubes (SWCN) with phosphorus incorporation. The nanotubes were produced in two different temperatures (800 and 850 Degrees Celsius), using a powder precursor (solid phase) without liquid dilution. The system used for the synthesis was High Vacuum Chemical Vapor Deposition (HVCVD). The samples were characterized comparing the samples with phosphorus with SWNT synthetized with ethanol. Raman spectroscopy, x-ray photoelectron spectroscopy (XPS) on transmission electron microscopy (TEM) were employed to characterize the samples. The Raman spectroscopy and TEM were employed to confirm the presence of SWNT in the samples. We found evidences that phosphorus were incorporated comparing the Raman spectra with the reference sample. The XPS results showed the presence of phosphorus atoms bonded to carbon atoms which make us conclude that the phosphorus were incorporated to the nanotubes.

Ces travaux de thèse consistent à formuler des nanocomposites électriquement conducteurs pour des applications micro-ondes. L’objectif principal est la mise en œuvre de matériaux absorbant les ondes radar, plus particulièrement sur la bande X (8-12 GHz). La polyaniline et les nanotubes de carbone, dispersés dans une matrice époxyde, ont été sélectionnés pour apporter les propriétés d’absorption aux fréquences visées. Différentes morphologies de polyaniline ont été synthétisées afin d’étudier leur influence sur les propriétés d’absorption des composites. L’utilisation d’une polyaniline à morphologie feuillet, présentant une forte anisotropie et un facteur de forme élevé, permet d’augmenter la conductivité et les pertes diélectriques des composites. De plus, leur association avec des nanotubes de carbone améliore significativement les propriétés d’absorption aux fréquences micro-ondes. Des écrans absorbants radar performants qui présentent des coefficients de réflexion inférieurs à -20 dB ont pu être modélisés et mis en œuvre, confirmant le potentiel de ces matériaux pour des applications de furtivité radar. / This thesis deals with the formulation of electrically conductive nanocomposites for microwave applications. The main purpose is to process radar-absorbent materials, more particularly at the X band. (8-12 GHz). Polyaniline and carbon nanotubes, dispersed in an epoxyde matrix, have been selected. Different morphologies of polyaniline have been synthesized to study its impact on the absorption properties of composites. Using flake-like polyaniline showing high anisotropy and aspect ratio increases conductivity and dielectric losses of composites. Moreover, its association with carbon nanotubes significantly improves the absorption properties at microwaves frequencies. Efficient radar absorbing screens, showing reflection losses lower than -20 dB, have been calculated and processed confirming the potential of these materials for stealth applications.

A dissertation submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg, in fulfilment of requirements for the degree of Master of Science. Johannesburg, 2016. / Metal dusting (MD) is a severe type of corrosion that occurs mainly in petrochemical industries. The occurrence of MD is mainly due to syngas attacking Fe-, Ni- and Co-based alloys at elevated temperatures. More recently, literature has shown that apart from syngas, liquid hydrocarbon sources have been causing problems on platformer units in refineries. In the first part of this study a highly corrosion resistant Ni-based alloy (Hastelloy C276), in its polished form, was subjected to MD conditions at 800 °C using a liquid hydrocarbon (toluene) and helium (carrier gas) for 1 h. Exposure to these conditions revealed the formation of carbon nanofilaments and graphite layers which were confirmed by laser Raman spectroscopy, scanning electron microscopy (SEM) and electron probe microanalysis (EPMA). Burning off the carbon nanofilaments and the graphite layers in laboratory air for 1 h at 800 °C revealed that pits were formed on the Hastelloy C276. These same pits were not evident when Hastelloy C276 was exposed to either the carrier gas (helium) or laboratory air alone. Besides MD being a continuous problem in industry, this mechanism has been shown to be beneficial in the synthesis of carbon nanofilaments viz., carbon nanofibers (CNTs) and nanotubes (CNFs). In the second part of this study, unpolished Hastelloy C276 blocks (as opposed to polished blocks) were used to synthesize carbon nanofilaments. This was done as prior studies had shown that carbon nanofilaments were produced with better quality and greater yields this way. Here the flow rate (80, 160 and 240 mL/min) and reaction duration (10, 15, 30, 45, 60, 120 and 240 min) were studied using toluene (a liquid hydrocarbon). Scanning electron microscopy (SEM) and transmission electron microscopy (TEM) were used to assess the quality and quantity of the carbon nanofilaments synthesized. Besides the formation of carbon nanofilaments, a less important material known as graphite particle structures (GPSs) were also synthesized. These studies collectively showed that MD had taken place on the surface of Hastelloy C276 when exposed to toluene at 800 °C. / TG2016

Du fait de leurs exceptionnelles propriétés, un très vaste champ d'applications s'ouvre aux nanotubes de carbone. La connaissance de leur toxicité est un enjeu primordial afin de connaître les risques et les dangers que ces objets représentent.Le travail présenté dans cette thèse vise à la synthèse de dérivés ayant pour but de favoriser les interactions entre les nanotubes et le milieu vivant.La synthèse d'une série de dérivés du cholestérol utilisables pour la fonctionnalisation des nanotubes de carbone a été mise au point. L'utilisation de tels dérivés a pour but de favoriser les interactions entre les objets greffés et les membranes cellulaires.Par ailleurs, de nombreux intermédiaires de synthèse présentent des propriétés mésomorphes. Celles-ci ont été étudiées au moyen de différentes méthodes telles que la microscopie optique, la calorimétrie différentielle à balayage et la diffusion des rayons X.Des nanotubes HiPco ont pu être purifiés et fonctionnalisés. L'analyse de leur cytotoxicité a été abordée, celle-ci a permis de mettre en avant une diminution de la toxicité des nanotubes greffés comparée à celles de nanotubes purifiés. / Synthesis and study of mesomorphic properties of grafts for carbon nanotubes functionalizationIn order to study the toxicity of carbon nanotubes, cholesterol based derivatives were synthesized. Due to their exceptional properties, a wide field of applications is opened for carbon nanotubes. Regarding this field, knowing their toxicity becomes a key issue.The work presented here aims towards the synthesis of chemical derivatives designed to increase interactions between carbon nanotubes and a biological environment. The synthesis of a series of cholesterol derivatives aimed for the functionalization of carbon nanotubes has been developed. Such derivatives are intended to interact with cell membranes.Moreover, many synthetic intermediates have mesomorphic properties. These properties have been studied using different methods such as polarized optical microscopy (POM), differential scanning calorimetry (DSC) and X-ray scattering (MWAXS).HiPco nanotubes have been purified, functionalized and characterized with techniques such as Raman spectroscopy and X-ray diffraction (XRD).Finally, a study of carbon nanotubes toxicity is presented.

Au cours de ce travail de thèse, je présente les résultats de travaux de recherche visant à la fabrication et au développement d'une nouvelle génération d'implants cérébraux permettant l'enregistrement stable dans le temps de signaux électrophysiologiques. L'objectif est d'offrir une solution technologique pour le développement de nouvelles applications chroniques utiles pour l'investigation du fonctionnement du cerveau (ex : dynamique des réseaux de neurones, traitement de l'information) mais aussi nécessaires pour la mise au point de nouvelles technologies, de type Brain Computer Interface (BCI), qui devraient, à terme, permettre d'augmenter l'autonomie des patients atteints de troubles moteurs ou victimes d'accident ayant conduit à un handicap physique.Nous rapportons, à travers diverses études expérimentales, des arguments soutenant l'utilisation de matériaux nanostructurés et en particulier les nanotubes de carbone, pour permettre d'améliorer les performances électriques et pour diminuer leur impact sur la réponse immunitaire du cerveau induite par l'implantation d'un corps étranger. Ceci représente la principale limitation à l'utilisation d'implants de mesure sur de longues périodes.Notre étude couvre ainsi l'ensemble des problématiques relatives aux implants cérébraux, en abordant les techniques de fabrication, le protocole de caractérisation électrique et enfinl'évaluation biologique in vitro et in vivo.Dans un premier temps, nous avons développé une matrice de microélectrodes nanostructurées en utilisant les techniques de microfabrication. Ceci a permis de mettre en avant la diminution de l'impédance interfaciale des électrodes liée à l'augmentation de leurs surfaces développées et, par conséquent, l'amélioration de la qualité du signal due à la réduction du niveau de bruit de la mesure. Une analyse électrochimique complémentaire, nous a dans le même temps, permis d'établir un modèle qualitatif représentant les différents phénomènes mis en jeu et soulignant l'importance des phénomènes diffusifs lors de la mesure et leurs éventuelles conséquences in vivo.Cette première génération d'électrodes nous a également permis de mener une campagne de tests in vitro pour évaluer la cytotoxicité des nanotubes à partir de différents modèles cellulaires: cultures de lignées neuronales (NG108-15) et cellules primaires (cellules gliales7et neurones du cortex ou de l'hippocampe). Suite à ces expériences, aucune forme de toxicité n'a été décelée. Bien au contraire, l'évolution des cellules gliales pendant la culture tend à défendre l'idée que les nanotubes de carbone ont le potentiel de réduire la réaction immunitaire du cerveau. Nous avons testé ce phénomène in vivo grâce à l'analyse histologique de cerveaux de rats après une implantation sous-durale d'un mois avec des échantillons en nitrure de titane (TiN) avec ou sans nanotubes de carbone. Nos résultats confirment une réduction significative de la cicatrice gliale lors d'implantations avec des échantillons nanostructurés.Ces résultats suggèrent un possible impact des nanotubes de carbone sur la stabilité de la mesure dans le temps, autrement dit sur sa « biostabilité ». Nous avons donc implanté un macaque Fascicularis avec un système comportant 16 électrodes en TiN et 16 électrodes nanostructurées de 1mm², amincies à 25μm et montées sur un substrat en polyimide flexible.L'étude qui se situe dans le cadre du projet Neurolink (financé par l'ANR/PNANO),représente la première validation expérimentale sur une longue période et démontre les avantages tant électriques que biologiques de l'utilisation des nanotubes de carbone comme interface entre l'électrode et les neurones. / During this work of thesis, I present the results of research works aiming at the manufacturing and to the development of a new generation of intellectual implants allowing the stable recording in the time of signals électrophysiologiques. The objective is to offer a technological solution for the development of new useful chronic applications for the investigation of the functioning of the brain (ex: dynamics of the networks of neurones, data processing) but also necessities for the development of new technologies, Brain Computer Interface ( BCI), which should, eventually, allow to increase the autonomy of the patients reached(affected) by driving disorders(confusions) or the victims of accident having led(driven) to a physical handicap. [résumé incomplet]

O principal objetivo dessa dissertação de mestrado é o estudo das propriedades estruturais e eletrônicas de umdosmateriaismais promissores para a nanoeletrônica, os nanotubos de carbono, na presença de defeitos do tipo vacâncias. Os defeitos estudados neste trabalho são as monovacâncias e multivacâncias em nanotubos de carbono armchair (5, 5). Para isso, realizamos cálculos quânticos de primeiros princípios baseados na teoria do funcional da densidade. Primeiramente, foi realizado um estudo sistemático das monovacâncias variando a sua concentração, para que possamos entender qual é a influência da distância entre esses defeitos no sistema. A partir de 14.784Å o efeito do potencial inserido pela vacância praticamente é desprezível. Seguindo essa sistemática, observamos também que a partir de 19.712Å a interação elástica entre defeitos é desprezível. Em todas as estruturas de bandas ocorre o surgimento de bandas de impureza, caracterizadas por uma localização de carga no defeito, principalmente no \"dangling bond\". Utilizando o procedimento NEB (nudge elastic band) obtivemos um valor de 0.55eV para a barreira de migração da vacância. Esse valor fornece um tempo aproximado de 0.156ms a 300K, para que a vacância salte de um sítio para outro na rede do nanotubo. Tendo isso, determinamos que a monovacância é um defeito que apresenta uma grande mobilidade no sistema, o que pode eventualmente ocorrer de uma encontrar-se com outras, formando sistemas maiores, como clusters de monovacâncias ou multivacâncias. Seguindo o nosso objetivo, realizamos um estudo sistemático das multivacâncias. Determinamos todas as reconstruções desses defeitos, nas quais observamos que as multivacâncias de índice par, apresentavam somente pentágonos e uma região central, e em contrapartida as de índice ímpar apresentavam pentágonos, uma região central e um dangling bond na sua estrutura final. Definidas as estruturas e as energias das multivacâncias, calculamos suas propriedades eletrônicas, onde é notório que a influência desses defeitos é muito mais drástica que as monovacâncias, devido a sua grande extensão no sistema. Realizamos cálculos de transporte eletrônico para todos os sistemas, utilizando o método de funções de Green fora do equilíbrio (NEGF-DFT), desenvolvido no nosso grupo. A partir desses cálculos, observamos que todos os defeitos modificam a estrutura eletrônica do sistema, uns mais outros menos, mas até para as multivacâncias maiores como a hexavacância, o sistema mantém o seu caráter metálico. Através das técnicas de funções de Green, mostramos que quando estamos utilizando condições periódicas de contorno há uma surgimento de minigap\'s espúrios. Estes não aparecem quando o vínculo de simetria translacional é quebrado via técnicas de funções de Green. / The main purpose of this dissertation is the study of the electronic and structural properties in the presence of different types of vacancies, in one of the most promising materials for nanoelectronics, carbon nanotubes. The defects studied in this work are monovacancy and multivacancies in (5, 5) armchair carbon nanotubes. For that purpose perform quantum ab initio calculations based on density functional theory (DFT). Firstly, was made a systematic study of monovacancies varying its concentration, so we can understand what is the influence of the distance between these defects in the system. Undergoes reconstruction pentagon nonagon from14.784Å on words the effect of the potential created by the vacancy is practically negligible. Following this systematic approach, we also found that from 19.712Å the structural interaction between the defects tends to zero. In all band structures one observes impurity bands, characterized by charge localization in the defect, mainly in the dangling bond. Using the NEB (nudge elastic band) procedure we obtained a value of 0.55eV for the migration barrier of the monovacancy. This value provides an approximate time of 0.156ms at 300K for the vacancy to jump from one site to another on the nanotube network. Hence, we determined that the monovacancy presents a great mobility in the system, which may possibly inccur in two ore more vacancies merging with others to form a larger defect, such as clusters of monovacancies or a hole multivacancy. Following our goal, we conducted a systematic study of multivacancies. We determined all reconstructions of these defects. We observed that multivacancies with and even number of carbon atoms removed had only pentagons and a central region. In contrast the CNT with and odd numbers of carbon atoms removed have pentagons, a central region and a dangling bond in its final structure. Once we have defined the structures and the energies of the multivacancies, we calculated their electronic properties. It is clear that the influence of these defects is much more drastic than the monovacancy, due to the size of the defect in the system. We calculated the electronic transport for all systems, using a non-equilibrium Green\'s functions method (NEGF-DFT), developed in our group. From these calculations, we observed that all the defects alter the electronic structure of the system, but even for the largest multivacancy, the system maintains its metallic character. Using the Green\'s functions techniques, we show that the use of periodic boundary conditions leads to the emergence of spurious mini-gap\'s. These do not appear when the translational symmetry is broken using Green\'s function approach and a true open system is considered.