| Carbon widely exists in nature. Carbon materials are found in variety forms such as graphite, diamond, fullerenes, and carbon nanotubes. The reason why carbon assumes many structural forms is that a carbon atom can form several distinct types of valence bonds, where the chemical bonds refer to the hybridization of orbitals by physicists. In sp 3 hybridization, fourσbonds defining a regular tetrahedron are sufficient to form a three-dimensional structure known as the diamond structure. It is interesting that hybridization which forms a planar structure in two-dimensiona1 graphite also forms a planar local structure in the closed polyhedra (0-dimensional) of the fullerene family and in the cylinders (1-dimensional) called carbon nanotubes. sp2A carbon nanotube is a honeycomb lattice rolled into a cylinder. The diameter of a carbon nanotube is of nanometer size and the length of the nanotube can be more than lpm. The nanotube diameter is much smaller in size than the most advanced semiconductor devices obtained so far. Thus the availability of carbon nanotubes may have a large impact on semiconductor physics because of its very small size and the special electronic properties that are unique to carbon nanotubes.Graphene is a zero-band gap semi-metal/semiconductor materials, its carrier mobility is much higher than silicon. Graphene has obvious two-dimensional electronic characteristics. In 2004, the single-layer graphene with the ideal two-dimensional structure and unique electronic properties have been successfully prepared, which triggered the research boom of a new wave of carbon-based materials. Recently, a significant Quantum Hall Effect and fractional Quantum Hall Effect observed in graphene have confirmed that it is very promising material of nano-electronic devices. In 2006-2008 years, graphene has been made of ballistic transport transistor, FET plane, which is attracted a large number of interests of scientists. It is succeeded in creating a planar field-effect transistor and in observing to the quantum interference effect, and in developing graphene-based circuits based on this study. Therefore, there is an important value of theoretical research in graphene, and it has quickly become the materials science and condensed matter physics research hotspot in recent years.In the semiconductor surface, due to the existence of its own defects, adsorption material, oxides and other reasons, quantum state of the surface electron is able to form discrete energy levels or very narrow energy band, which is called the surface state. It can capture or release carriers, or form recombination centers, so that the semiconductor has often a surface charge, which affects its electrical properties. The existence of surface states makes the local density of states near the surface different from that of interior. Especially when the energy of these surface states is very close to Fermi energy, that is, near the Fermi surface, the surface state will be very significant impact on the properties of solid surface. Therefore, it is of great significance to research the structure of the surface state.In this paper, we have regarded the semi-infinite armchair nanoribbon and semi-infinite zigzag carbon nanotube as infinite armchair ribbon or nanotube cutting edge along a zigzag edge, and study their surface densities of states. Using the transfer matrix method and iterative method, we calculated the surface electronic density of states in the ribbon and nanotube under the nearest-neighbor approximation and the next nearest-neighbor approximation respectively. we can see that the surface electronic density of states in the semi-infinite ribbon or carbon nanotube give a sharp peak at Fermi level ,and are much greater by the impact of the width N of the device area than by the impact of length L. The density of states difference between ribbon and carbon nanotube is that the electronic local density of states of the ribbon in each point of each layer is not the same, however, that of carbon nanotube is the same. Because the various atoms in each layer is in the same physical environment in carbon nanotube, the interaction between atom and atom or electron and electron is the same, but that of carbon nanotube is not the same . In the next nearest-neighbor approximation, the surface density of state peaks in the semi-infinite ribbon and carbon nanotube not only deviated from the Dirac point, but also occurred regularly splits with the increased of N. The overall law of the ribbon is from the beginning of N = 4, N for each additional 3, on the increase in a peak of the density of state. The splitting law of the density of state peak in the carbon nanotube is different from that of the ribbon, which may be due to the unique structure of carbon nanotube. Before the ribbon is rolled into the carbon nanotube, both upper and lower armchair edges in the distance for the N atoms in the ribbon, because the atomic distance on both sides is very far, the interaction between them is very weak, it can approximately be regarded as no interaction between them. However, after the ribbon is rolled into carbon nanotube, due to the requirements of the periodic boundary conditions, the (NN +1)th atom in each layer are regarded as overlap with the first atom in the same layer, and the (NN +2 )th atom and the second layer atom overlap, thus the two atoms that there was no interaction between the upper and lower armchair edges cause some the nearest-neighbor and the next nearest-neighbor interactions, which make carbon nanotube and ribbon nature not exactly the same, there is bound to be some differences.In this section, we also considered the changes of the surface density of state by adding layer by layer Linear confined potential in the semi-infinite Armchair ribbon device area under the nearest-neighbor and the next nearest-neighbor approximation. It is found that the changes of the Linear confined potential cause transfer of the surface levels, this makes the density of state peak deviate from the Dirac point, and occur to split with the increase of N. Under the next nearest-neighbor approximation, the peak splitting number of the density of state reduces with the increasing Linear confined potential.Secondly, we study on the electric transport properties of the infinite Armchair graphene ribbon and the infinite Zigzag-type carbon nanotube, respectively. Through the use of transfer matrix method or iterative method in the nearest neighbor and the next nearest-neighbor approximation, We work out the conductivity of the ribbon or carbon nanotube, respectively. We find that with the changes of width N , Armchair-type ribbon and zigzag-type carbon nanotube can have metallic or semiconducting properties, which also reflects in the conductance spectra. When they are the metal, there are e-channels in the vicinity of Dirac point. The eyeable ribbon in the conductance spectrum corresponding to the conductance G = 1, whereas carbon nanotube corresponds to conductance G = 2. On the other hand, when they are the semiconductor, that is, there does not exist electronic access near the Dirac point, the conductivities spectra in the ribbon and carbon nanotube correspond to G = 0. When we only consider the nearest neighbor approximation, the conductivity spectras of the ribbon and carbon nanotube are stage-like symmetrical at Dirac point. After taking the nearest neighbor approximation as amended, a complete spectrum of the conductance the ribbon or carbon nanotube is still stage-like, but no longer a completely axisymmetric distribution, only half of its cycle of conductance spectra shows a symmetric distribution, but the symmetry center has deviated from the zero energy.Finally, we studied the vertical magnetic field on the impact of electron transport. We find that the infinite Armchair-type ribbon and the Zigzag-type carbon nanotube exist an obvious magnetoresistance in the magnetic field. In order to obtain greater magnetoresistance, we get some conclusions by comparing the two: First, It should select the corresponding energy of the most dramatic changes in the conductivity jump as electronic incident energy in their respective conductivity-energy spectra of the ribbon and carbon nanotube. Second, it is necessary to increase the whole the size of the carbon nanotube or ribbon, to the extent possible, the magnetic field zone size is relatively large. Third, while considering the impact of the nearest neighbor approximation, we take the impact of the nearest neighbor approximation into account. These can significantly enhance the magnetoresistive effect. Overall, under the same conditions, magnetoresistive effect of carbon nanotube is much stronger than that of the ribbon. The some qualitative results studied in this paper will provide theoretical guidance for the application of graphene and carbon nanotube with nano-scale devices on magnetoresistance.
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