"We develop a selfconsistent solution of the Schroedinger and Poisson equations in semiconductor heterostructures with arbitrary doping profiles and layer geometries. An algorithm for this nonlinear problem is presented in a multiband k.P framework for the electronic band structure using the finite element method. The discretized functional integrals associated with the Schroedinger and Poisson equations are used in a variational approach. The finite element formulation allows us to evaluate functional derivatives needed to linearize Poisson’s equation in a natural manner. Illustrative examples are presented using a number of heterostructures including single quantum wells, an asymmetric double quantum well, p-i-n-i superlattices and trilayer superlattices."

Submitted by Maria Aparecida (cidazen@gmail.com) on 2017-11-16T12:37:19Z No. of bitstreams: 1 ANTONIO DE JESUS DOS SANTOS FERNANDES JÚNIOR.pdf: 2676930 bytes, checksum: 6ea43214cce79a1ebadd7cae50663205 (MD5) / Made available in DSpace on 2017-11-16T12:37:19Z (GMT). No. of bitstreams: 1 ANTONIO DE JESUS DOS SANTOS FERNANDES JÚNIOR.pdf: 2676930 bytes, checksum: 6ea43214cce79a1ebadd7cae50663205 (MD5) Previous issue date: 2017-09-05 / In this work,a nevel material heterostructured based on the assembly of double layered double hydroxide (LDH) suported in magadiite, was synthesized. For this purpose, layered double hydroxides of MgAl and ZnAl,both in the proportion 2:1 was synthesized by the method of coprecipitation in situ in the presence of magadiite.Diverse physicochemical techniques (DRX,FTIR,DTA,ICP-OES,MEV E TEM ) were employed to caractherize the materials syntehesized with the aim of verifying the existence of interactions in the interface of both components. Results gotten by DRX show plans of reflection typical of magadiite,confirming the formation o HDL in the surface of silicate. Nanocrystalline interactions among the groups of hydroxyl of each component. Metalic Oxide nanoparticles were supported in magadiite employing the thermic decomposition (calcination) of the component HDL present in the hetero-structure MAG/HDL,with the aim of getting catalyst formed by metalic oxides highly dispersed supported in magadiite. Calcined materials and non-calcined were evaluated as photocatalysts, using blue dye of methylene as a standard molecule. Photocatalytic analyses were made with “heterostructure” MAG/HDL and MAG/ODL showed a catalytic activity improved in order to degradate the dye,,where the concentration decreased considerablly in a short interlude of exposition to irradiation of uv light in camparison to the photlysis test.These results were gotten probably thanks to the resulting synergistic effect of textured properties interesting,charactheristics of synthesized heterostructured ,turning them in photocatalytic promising in the degradation of organic compounds. / Neste trabalho, um novo material heteroestruturado baseado na síntese de hidróxidos duplos lamelares suportados em magadiíta, foi sintetizado. Para este fim, hidróxidos duplos lamelares (HDL) de MgAl e ZnAl, ambos nas proporções 2:1, foram sintetizados pelo método de coprecipitação in situ em presença do silicato alcalino lamelar. Diversas técnicas físico-químicas (DRX, FTIR, DTA, ICP-OES, MEV e TEM) foram empregados para caracterizar os materiais sintetizados com o objetivo de verificar a existência a existência de interações na interface dos dois componentes. Resultados obtidos por DRX mostram planos de reflexão típicos de magadiíta, confirmando a formação de HDL na superfície do silicato. Interações nanocristalinas entre as porções foram evidenciadas por estudos de espectroscopia onde ocorrem interações entre os grupos hidroxilas de cada componente. Nanopartículas de óxidos metálicos foram suportados em magadiíta empregando a decomposição térmica (calcinação) do componente HDL presente na heteroestrutura MAG/HDL, com o objetivo de obter-se materiais catalisadores formados por óxidos metálicos altamente dispersos suportados em magadiíta. Os materiais calcinados e não calcinados foram avaliados como fotocatalisadores, usando o corante azul de metileno como molécula modelo. Testes fotocatalíticos realizados com as heteroestruturas MAG/HDL e MAG/ODL mostraram uma atividade catalítica melhorada para a degradação do corante, onde a concentração diminuiu consideravelmente em um curto intervalo de tempo de exposição à irradiação de luz UV em comparação ao teste de fotólise. Estes resultados foram obtidos provavelmente devido ao efeito sinérgico resultante das propriedades texturais interessantes, características das heteroestruturas sintetizadas, tornando-os fotocatalisadores promissores na degradação de compostos orgânincos.

Nanowire heterostructures play a crucial role in nanoscale electronics, i.e., one-dimensional electronics derives benefits from the growth of heterostructures along the nanowire axis. We use first-principles plane-wave calculations within density functional theory with the localized density approximation (LDA) to get information about the structural and electronic properties of bare and hydrogen passivated GaAs/AlAs nanowire heterostructures. We also take into account the reconstruction of the nanowire surfaces. Modeled nanowire heterostructures are constructed using bulk atomic positions along [001] and [111] direction of zinc-blende structures and cutting out wires from this GaAs/AlAs heterostructure crystal with a diameter of 1 nm. We study for the effects of the surface passivation on the band gap and the band offsets for the planar GaAs/AlAs bulk heterostructure system and GaAs/AlAs nanowire heterostructure system. It is possible to control the potential that carriers feel in semiconductor heterostructures. For the planar lattice-matched heterostructures, the macroscopic average of potential of the two materials is constant far from the interface and there is a discontinuity at the interface depending on the composition of the heterostructure. In order to obtain the valence band offset in the heterostructure system, the shift in the macroscopic potential at the interface and the difference between the valence band maximum values of the two constituents must be added. In nanoscale heterostructures, the potential profile presents a more complex picture. The results indicate that while the discontinuity remains close to the planar limit right at the interface, there are fluctuations on the average potential profile beyond the interface developed by the inhomogeneous surface termination, that is, there are variations of the band edges beyond the interface. We report a first-principles study of the electronic properties of surface dangling-bond (SDB) states in hydrogen passivated GaAs/AlAs nanowire heterostructures with a diameter of 1 nm, where the SDB is defined as the defect due to an incomplete passivation of a surface atom. The charge transition levels of SDB states serve as a common energy reference level, such that charge transition level value for group III and V atoms is a constant value and a periodic table atomic property. We have carried out first-principles electronic structure and total energy calculations of aluminum nanowires for a series of different diameters ranging from 3 Angtrom-10 Angstrom, which is cut out from a slab of ideal bulk structure along the [001] direction. First-principles calculations of aluminum nanowires have been carried out within the density-functional theory. We use the norm-conserving pseudopotentials that are shown to yield successful results for ultrathin nanowire regime. Our results show that the number of bands crossing the Fermi level decreases with decreasing wire diameter and all wires studied are metallic.

<p>Quantum Dot Infrared Photodetectors (QDIPs) are important alternatives to conventional infrared photodetectors with high potential to provide required detector performance, such as higher temperature operation and multispectral response, due to the 3-D quantum confinement of electrons, discrete energy levels, and intrinsic response to perpendicular incident light due to selection rules. However, excessive dark current density, which causes QDIPs to underperform theoretical predictions, is a limiting factor for the advancement of QDIP technologies. The purpose of this dissertation research is to achieve a better understanding of dopant incorporation into the active region of QDIPs, which is directly related to dark current control and spectral response. From this dissertation research, doping related dipole fields are found to be responsible for excessive dark current in QDIPs. </p><p>InAs/GaAs QDIPs were grown using solid source molecular beam epitaxy (MBE) with different doping conditions. The QDIPs were optically characterized using photoluminescence and Fourier transform infrared (FT-IR) spectroscopy. Devices were fabricated using standard cleanroom fabrication procedures. Dark current and capacitance measurements were performed under different temperature to reveal electronic properties of the materials and devices. A novel scanning capacitance microscopy (SCM) technique was used to study the band structure and carrier concentration on the cross section of a quantum dot (QD) heterostructure. In addition, dark current modeling and bandstructure calculations were performed to verify and better understand experimental results.</p><p>Two widely used QDIP doping methods with different doping concentrations have been studied in this dissertation research, namely direct doping in InAs QD layer, and modulation doping in the GaAs barrier above InAs QD layer. In the SCM experiment, electron redistribution has been observed due to band-bending in the modulation-doping region, while there is no band-bending observed in directly doped samples. A good agreement between the calculated bandstructure and experimental results leads to better understanding of doping in QD structures. The charge filling process in QDs has been observed by an innovative polarization-dependent FT-IR spectroscopy. The red-shift of QD absorbance peaks with increasing electron occupation supports a miniband electronic configuration for high-density QD ensembles. In addition, the FT-IR measurement indicates the existence of donor-complex (DX) defect centers in Si-doped QDIPs. The existence of DX centers and related dipole fields have been confirmed by dark current measurements to extract activation energies and by photocapacitance quenching measurements. </p><p>With the understanding achieved from experimental results, a further improved dark current model has been developed based on the previous model originally established by Ryzhii and improved by Stiff-Roberts. In the model described in this dissertation, two new factors have been considered. The inclusion of background drift current originating from Si shallow donors in the low bias region results in excellent agreement between calculated and measured dark currents at different temperatures, which has not been achieved by previous models. A very significant effect has been observed in that dark current leakage occurs due to the dipole field caused by doping induced charge distribution and impact-ionized DX centers. </p><p>Last but not least, QDIPs featuring the dipole interface doping (DID) method have been designed to reduce the dark current density without changing the activation energy (thus detection wavelength) of QDIPs. The DID samples involve an InAs QD layer directly-doped by Si, as well as Be doping in the GaAs barrier on both sides of the QD layer. The experimental result shows the dark current density has been significantly reduced by 104 times without any significant change to the corresponding activation energy. However, the high p-type doping in the GaAs barrier poses a challenge in that the Fermi level is reduced to be well below the QD energy states. High p-type doping is reported to reduce the dark current, photocurrent and the responsivity of the devices. </p><p>To conclude, it is significant to identify to effect of Si-induced defect centers on QDIP dark currents. The subsequent study reveals doping induced dipole fields can have significant effects on QDIP device performance, for example, causing charge leakage from QDs and reducing activation energy, thereby increasing dark current density. The DID approach developed in this work is a promising approach that could help address these issues by using controlled dipole fields to reduce dark current density without changing the minimum detectable energy of QDIPs.</p> / Dissertation

Ternary chalcopyrite compound semiconductors have received much attention as the absorbing layers in the polycrystalline thin film solar cell structures. Most widely used one is CuInSe2 and CuInGaSe2 structures, but there are some diffusion problems with copper atoms in the structure. On the other hand, AgInSe2 is promising material with several advantages over the CuInSe2. The aim of this study was to investigate and optimize the production and post-production methods of the Ag-In-Se thin film based heterostructure devices. In this study Ag-In-Se thin films were deposited on glass, ITO coated glass and Si wafer substrates by thermal evaporation and RF/DC sputtering methods. The structural, electrical and optical properties of the devices were investigated. The XRD measurements of the thermally evaporated films showed that as-grown films in amorphous nature. By annealing the films under nitrogen atmosphere, the AIS films turn to polycrystalline structure which including AgIn5Se8 and &amp / #948 / &amp / #8722 / In2Se3 multi-phases with n-type conductivity. p-Si/n-AIS heterojunctions showed very good diode behavior with 4 order rectification factor. Annealing under nitrogen atmosphere decreased the series resistance of the devices and calculated solar cell conversion efficiency and fill factor of devices increased up to n=2.6% and FF=63, respectively. The XRD measurements of layer-by-layer sputtered AIS films were showed that as-grown films amorphous in nature. The AIS thin films were annealed at 300oC temperature under selenium ambient and mono-phase AgInSe2 with desired p-type conductivity were obtained. n-Si/p-AIS heterojunctions showed very good diode behaviors with 6 order rectification factor. The results of the study showed that AIS thin film has a photoresponce maximum which is exactly matching with solar photon energy maxima. High series resistance of the devices increases the recombination in the junction and this results in the lower solar conversion efficiency. The adequate electrical, optical and structural properties of the AIS thin films reveals that p-AIS thin films could be used as a solar cell absorber layer with an appropriate window layer, such as CdS.

Heterostructures of the materials system GaN/AlGaN/GaN were grown by molecular beam epitaxy on 6H-SiC substrates and high electron mobility transistors (HEMTs) were fabricated. For devices with large gate periphery an air bridge technology was developed for the drain contacts of the finger structure. The devices showed DC drain currents of more than 1 A/mm and values of the transconductance between 120 and 140 mS/mm. A power added efficiency of 41 % was measured on devices with a gate length of 1 µm at 2 GHz and 45 V drain bias. Power values of 8 W/mm were obtained. Devices with submicron gates exhibited power values of 6.1 W/mm (7 GHz) and 3.16 W/mm (25 GHz) respectively. The rf dispersion of the drain current is very low, although the devices were not passivated. / Heterostrukturen im Materialsystem GaN/AlGaN/GaN wurden mittels Molekularstrahlepitaxie auf 6H-SiC-Substraten gewachsen und High-Electron-Mobility-Transistoren (HEMTs) daraus hergestellt. Für Bauelemente mit großer Gateperipherie wurde eine Air-Bridge-Technik entwickelt, um die Drainkontakte der Fingerstruktur zu verbinden. Die Bauelemente zeigten Drainströme von mehr als 1 A/mm und Steilheiten zwischen 120 und 140 mS/mm. An Transistoren mit Gatelängen von 1 µm konnten Leistungswirkungsgrade (Power Added Efficiency) von 41 % (bei 2 GHz und 45 V Drain-Source-Spannung) sowie eine Leistung von 8 W/mm erzielt werden. Bauelemente mit Gatelängen im Submikrometerbereich zeigten Leistungswerte von 6,1 W/mm (7 GHz) bzw. 3,16 W/mm (25 GHz). Die Drainstromdispersion ist sehr gering, obwohl die Bauelemente nicht passiviert wurden.

Ce travail est consacré à la réalisation de diodes électroluminescentes visibles à base de fils de GaN crûs sur Si(111) par épitaxie en phase vapeur de précurseurs organo-métalliques. Nous cherchons en particulier à comprendre les mécanismes de croissance des fils de GaN et les propriétés structurales et optiques de puits quantiques InGaN/GaN cœur/coquille déposés autour de ceux-ci. La croissance de fils orientés le long de l'axe -c sur saphir est dans un premier temps détaillée et expliquée. Nous montrons que l'injection de silane pendant la croissance des fils permet de former une couche de passivation de SiNx autour de ceux-ci. L'arrêt de l'injection de silane après quelques dizaines de secondes ne modifie pas la géométrie fil et ce procédé peut donc être utilisé pour contrôler le positionnement le long du fil de la zone de dépôts des puits quantiques InGaN/GaN. Ce procédé est ensuite transféré sur substrat Si(111) grâce au dépôt préalable d'une fine couche tampon d'AlN sur le substrat. Le dépôt de puits quantiques InGaN/GaN sur les facettes non-polaires m de ces fils et l'influence de différents paramètres de croissance sur leur émission de lumière sont étudiés. Nous montrons notamment l'existence de plusieurs familles des puits quantiques dans les fils dont les longueurs d'onde d'émission ont pu être indexées à l'aide de cartographies de cathodoluminescence. La concentration en indium des puits quantiques déposés a été estimée en comparant les énergies d'émissions des puits à des simulations utilisant la théorie k.p dans l'approximations 8 bandes pour les électrons et les trous et est comprise entre 8 et 24%. Enfin, des structures LED complètes ont été déposées sur les fils de GaN par MOVPE et une électroluminescence bleue à 450 nm à température ambiante est mesurée sur des fils uniques et sur des assemblées de fils sur silicium. / This work reports on the realization by metal organic vapor phase epitaxy of visible light emitting diodes based on GaN wires grown on Si(111) with a focus on understanding the wires growth mechanisms and the properties of InGaN/GaN core/shell heterostructures grown around them. First we report the MOVPE growth of –c oriented GaN wires on sapphire. We demonstrate that the injection of silane during the growth induces the formation of a SiNx passivation layer around the GaN wires, preventing the lateral expansion. The silane flow can be stopped after a certain time without modifying the wire geometry. This phenomenon is used to control the position of the InGaN/GaN multiple quantum well shells along the wires. The wire growth on sapphire has then been transferred to silicon substrate thanks to the deposition of a thin AlN buffer layer prior to the wire growth. The deposition of InGaN/GaN core/shell heterostructures on the non-polar m-plane side facets of the wires and the influence of different growth parameters on the light emission properties of the quantum wells are then studied. Several types of quantum wells grown on different facets of the wire surface are observed. These different families emit light at different wavelengths that have been indexed thanks to cathodoluminescence mapping. The indium concentration in the quantum wells deposited is estimated between 8 and 24 %, depending on the growth conditions. This estimation has been made by comparing the emission wavelength of the quantum well to the recombination energy of electrons and wells simulated using the 8x8 band k.p theory for electron and hole masses. Finally, complete LED structures have been deposited on GaN wires by MOVPE and blue electroluminescence at 450 nm has been measured on single wires and assemblies of wires on Si(111).

abstract: The research described in this dissertation has involved the use of transmission electron microcopy (TEM) to characterize the structural properties of II-VI and III-V compound semiconductor heterostructures and superlattices. The microstructure of thick ZnTe epilayers (~2.4 µm) grown by molecular beam epitaxy (MBE) under virtually identical conditions on GaSb, InAs, InP and GaAs (100) substrates were compared using TEM. High-resolution electron micrographs revealed a highly coherent interface for the ZnTe/GaSb sample, and showed extensive areas with well-separated interfacial misfit dislocations for the ZnTe/InAs sample. Lomer edge dislocations and 60o dislocations were commonly observed at the interfaces of the ZnTe/InP and ZnTe/GaAs samples. The amount of residual strain at the interfaces was estimated to be 0.01% for the ZnTe/InP sample and -0.09% for the ZnTe/GaAs sample. Strong PL spectra for all ZnTe samples were observed from 80 to 300 K. High quality GaSb grown by MBE on ZnTe/GaSb (001) virtual substrates with a temperature ramp at the beginning of the GaSb growth has been demonstrated. High-resolution X-ray diffraction (XRD) showed clear Pendellösung thickness fringes from both GaSb and ZnTe epilayers. Cross-section TEM images showed excellent crystallinity and smooth morphology for both ZnTe/GaSb and GaSb/ZnTe interfaces. Plan-view TEM image revealed the presence of Lomer dislocations at the interfaces and threading dislocations in the top GaSb layer. The defect density was estimated to be ~1 x107/cm2. The PL spectra showed improved optical properties when using the GaSb transition layer grown on ZnTe with a temperature ramp. The structural properties of strain-balanced InAs/InAs1-xSbx SLs grown on GaSb (001) substrates by metalorganic chemical vapor deposition (MOCVD) and MBE, have been studied using XRD and TEM. Excellent structural quality of the InAs/InAs1-xSbx SLs grown by MOCVD has been demonstrated. Well-defined ordered-alloy structures within individual InAs1-xSbx layers were observed for samples grown by modulated MBE. However, the ordering disappeared when defects propagating through the SL layers appeared during growth. For samples grown by conventional MBE, high-resolution images revealed that interfaces for InAs1-xSbx grown on InAs layers were sharper than for InAs grown on InAs1-xSbx layers, most likely due to a Sb surfactant segregation effect. / Dissertation/Thesis / Ph.D. Physics 2012

abstract: Polycrystalline CdS/CdTe solar cells continue to dominate the thin-film photovoltaics industry with an achieved record efficiency of over 22% demonstrated by First Solar, yet monocrystalline CdTe devices have received considerably less attention over the years. Monocrystalline CdTe double-heterostructure solar cells show great promise with respect to addressing the problem of low Voc with the passing of the 1 V benchmark. Rapid progress has been made in driving the efficiency in these devices ever closer to the record presently held by polycrystalline thin-films. This achievement is primarily due to the utilization of a remote p-n heterojunction in which the heavily doped contact materials, which are so problematic in terms of increasing non-radiative recombination inside the absorber, are moved outside of the CdTe double heterostructure with two MgyCd1-yTe barrier layers to provide confinement and passivation at the CdTe surfaces. Using this design, the pursuit and demonstration of efficiencies beyond 20% in CdTe solar cells is reported through the study and optimization of the structure barriers, contacts layers, and optical design. Further development of a wider bandgap MgxCd1-xTe solar cell based on the same design is included with the intention of applying this knowledge to the development of a tandem solar cell constructed on a silicon subcell. The exploration of different hole-contact materials—ZnTe, CuZnS, and a-Si:H—and their optimization is presented throughout the work. Devices utilizing a-Si:H hole contacts exhibit open-circuit voltages of up to 1.11 V, a maximum total-area efficiency of 18.5% measured under AM1.5G, and an active-area efficiency of 20.3% for CdTe absorber based devices. The achievement of voltages beyond 1.1V while still maintaining relatively high fill factors with no rollover, either before or after open-circuit, is a promising indicator that this approach can result in devices surpassing the 22% record set by polycrystalline designs. MgxCd1-xTe absorber based devices have been demonstrated with open-circuit voltages of up to 1.176 V and a maximum active-area efficiency of 11.2%. A discussion of the various loss mechanisms present within these devices, both optical and electrical, concludes with the presentation of a series of potential design changes meant to address these issues. / Dissertation/Thesis / Doctoral Dissertation Electrical Engineering 2017

Made available in DSpace on 2015-05-14T12:14:10Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 1315751 bytes, checksum: ac1c804a66de6eb3a5e3c4a1cb7d3e04 (MD5) Previous issue date: 2014-02-19 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / In this thesis we study the electronic properties of several systems of condensed matter physics using two different continuum models, the effective mass theory and the effective Dirac Hamiltonian. In several systems, there is an effective mass depending on position. Some models for the kinetic energy operator were proposed to describe these systems, but there is no definition of which one is the most appropriate. It is one of the oldest open questions in solid state physics. We propose a new model, where we consider all permutations among the operators and show that it satisfies the fundamental requirements of quantum mechanics. We use this model to obtain the minibands structure of a heterostructure composed by two different materials and compare our model with other models previously proposed. We also get the Schrödinger equation for a particle constrained to a curved surface with position dependent mass. We follow the da Costa approach, where there is a geometric potential. We show that the position dependent mass does not affect the geometric potential, contributing only to the kinetic part. We use this equation to study the electronic transport in a junction of two cylinders with different radii, with the effective mass varying with the cylinder radius. Using the effective Dirac Hamiltonian, we consider a graphene sheet on a periodic substrate heterostructure composed by two different materials. Each material induces a specific energy gap and Fermi velocity in the graphene, so the Dirac Hamiltonian has a gap (mass) term and a Fermi velocity depending on position. We write this operator taking into account that it has to be Hermitian and we obtain the minigaps induced by the substrate in the electronic structure of graphene. Motivated by experimental results, we study the effects of rotation on the electronic structure of carbon nanotubes, fullerene C60 and topological insulators, using an effective Dirac operator. In the carbon nanotube and C60 cases, the rotation adds a shift in the energy levels and a break in spin degeneracy. In the topological insulator case, the rotation adds only a shift in the energy. / Nesta tese estudamos as propriedades eletrônicas de diversos sistemas físicos da matéria condensada utilizando dois modelos contínuos distintos, o modelo de massa efetiva e o hamiltoniano de Dirac efetivo. Em vários sistemas existe o aparecimento de uma massa efetiva dependente da posição. Diversos modelos para o hamiltoniano cinético com massa efetiva variável foram propostos, mas não existe uma definição de qual seja o mais adequado. Essa é uma das questões mais antigas em aberto na física do estado sólido. Nós propomos um novo modelo, levando em conta todas as permutações possíveis entre os operadores e mostramos que ele satisfaz os requisitos fundamentais da mecânica quântica. Nós usamos esse modelo para obter a estrutura de minibandas de uma heteroestrutura formada por dois materiais diferentes e comparamos o nosso modelo com outros modelos propostos anteriormente. Também obtemos a equação de Schrödinger para uma partícula confinada a uma superfície curva com massa efetiva dependendo da posição. Seguimos a abordagem de da Costa, onde surge um potencial geométrico. Mostramos que a massa variável não altera a potencial geométrico, contribuindo apenas para a parte cinética. Nós usamos a equação obtida para estudar as propriedades de transporte eletrônico em uma junção de dois cilindros de raios diferentes, com a massa efetiva variando com o raio do cilindro. Utilizando um hamiltoniano de Dirac efetivo nós consideramos o grafeno sobre um substrato formado por uma heteroestrutura com dois materiais diferentes. Cada material induz um gap de energia e uma velocidade de Fermi específica no grafeno, fazendo com que tenhamos um hamiltoniano de Dirac com termo de gap (massa) e velocidade de Fermi dependendo da posição. Nós escrevemos esse operador levando em conta que ele tem que ser hermitiano e obtemos a estrutura de minibandas induzidas na estrutura eletrônica do grafeno pelo substrato. Motivados por resultados experimentais, nós também estudamos os efeitos da rotação na estrutura eletrônica do nanotubo de carbono, fulereno C60 e isolante topológico, utilizando um hamiltoniano de Dirac efetivo. Para o nanotubo de carbono e o C60 a rotação adiciona um deslocamento nos níveis de energia e há uma quebra na degenerescência de spin. No isolante topológico a rotação induz apenas um deslocamento na energia.