Graphene Models

The quantity x is a dimensionless quantity which is conventionally restricted to a range of —-ir < x < tt, a central Brillouin zone. For the case yj = 0 (i.e., S a pure translation), x corresponds to a normalized quasimomentum for a system with one-dimensional translational periodicity (i.e., x s kh, where k is the traditional wavevector from Bloch s theorem in solid-state band-structure theory). In the previous analysis of helical symmetry, with H the lattice vector in the graphene sheet defining the helical symmetry generator, X in the graphene model corresponds similarly to the product x = k-H where k is the two-dimensional quasimomentum vector of graphene. 

For the nanotubes, then, the appropriate symmetries for an allowed band crossing are only present for the serpentine ([ , ]) and the sawtooth ([ ,0]) conformations, which will both have C point group symmetries that will allow band crossings, and with rotation groups generated by the operations equivalent by conformal mapping to the lattice translations Rj -t- R2 and Ri, respectively. However, examination of the graphene model shows that only the serpentine nanotubes will have states of the correct symmetry (i.e., different parities under the reflection operation) at the K point where the bands can cross. Consider the K point at (K — K2)/3. The serpentine case always sat-... 

Within the Slater-Koster appro.ximation, we can easily test the validity of the approximations made in eqn (7) based on the graphene model. In Fig. 5 we depict the band gaps using the empirical tight-binding method for nanotube radii less than 1.5 nm. The non-metallic nanotubes n m) are shown in the... 

Two types of graphene models are often employed in computations. The simpler one is a cluster model, where a molecule of a relatively large polyaromatic hydrocarbon, for example, circumcoronene, is used as the model of a pristine graphene sheet. To mimic defective or doped graphene, some atoms may be cut off from the large polyaromatic hydrocarbon model molecule or substituted with a dopant atom. 

The energy of Co atom binding to graphene sheet predicted with GGA-PW91 is more than five times lower than that to the coronene molecule predicted by CASSCF (complete active space self-consistent field) with a complex basis set of triple- quality (Table 11.3, rows 8 and 9). Such a severe discrepancy is surprising even considering the different graphene models used. This also may be caused by the incompleteness of active space in the CASSCF calculations as only it-orbitals of coronene and d-orbitals of Co were included. 

Metal atom (adsorption site) Graphene model (kcal mol ) d(A) Computational protocol References... 

Metal n Graphene model Computational protocol (kcalmol h References... 

Fig. 2. By rolling up a graphene sheet (a single layer of ear-bon atoms from a 3D graphite erystal) as a cylinder and capping each end of the eyiinder with half of a fullerene molecule, a fullerene-derived tubule, one layer in thickness, is formed. Shown here is a schematic theoretical model for a single-wall carbon tubule with the tubule axis OB (see Fig. 1) normal to (a) the 6 = 30° direction (an armchair tubule), (b) the 6 = 0° direction (a zigzag tubule), and (c) a general direction B with 0 < 6 < 30° (a chiral tubule). The actual tubules shown in the figure correspond to (n,m) values of (a) (5,5), (b) (9,0), and (c) (10,5).

Fig. 6. Band gap as a function of nanotube radius using first-principles LDF method. Solid line shows estimates using graphene sheet model with...

Abstract—Experimental and theoretical studies of the vibrational modes of carbon nanotubes are reviewed. The closing of a 2D graphene sheet into a tubule is found to lead to several new infrared (IR)- and Raman-active modes. The number of these modes is found to depend on the tubule symmetry and not on the diameter. Their diameter-dependent frequencies are calculated using a zone-folding model. Results of Raman scattering studies on arc-derived carbons containing nested or single-wall nanotubes are discussed. They are compared to theory and to that observed for other sp carbons also present in the sample. 

An important question relating to the structure of nanotubes is Are nanotubes made of embedded closed tubes, like "Russian dolls," or are they composed of a single graphene layer which is spirally wound, like a roll of paper Ijima et al. [2] espouse the "Russian doll" model based on TEM work which shows that the same number of sheets appear on each side of the central channel. Dravid et al. [4], however, support a "paper roll" structural model for nanotubes. 

The optimised interlayer distance of a concentric bilayered CNT by density-functional theory treatment was calculated to be 3.39 A [23] compared with the experimental value of 3.4 A [24]. Modification of the electronic structure (especially metallic state) due to the inner tube has been examined for two kinds of models of concentric bilayered CNT, (5, 5)-(10, 10) and (9, 0)-(18, 0), in the framework of the Huckel-type treatment [25]. The stacked layer patterns considered are illustrated in Fig. 8. It has been predicted that metallic property would not change within this stacking mode due to symmetry reason, which is almost similar to the case in the interlayer interaction of two graphene sheets [26]. Moreover, in the three-dimensional graphite, the interlayer distance of which is 3.35 A [27], there is only a slight overlapping (0.03-0.04 eV) of the HO and the LU bands at the Fermi level of a sheet of graphite plane [28,29],... 

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