Cone-like graphene nanostructures: electronic and optical properties
© Ulloa et al.; licensee Springer. 2013
Received: 15 July 2013
Accepted: 1 September 2013
Published: 12 September 2013
A theoretical study of electronic and optical properties of graphene nanodisks and nanocones is presented within the framework of a tight-binding scheme. The electronic densities of states and absorption coefficients are calculated for such structures with different sizes and topologies. A discrete position approximation is used to describe the electronic states taking into account the effect of the overlap integral to first order. For small finite systems, both total and local densities of states depend sensitively on the number of atoms and characteristic geometry of the structures. Results for the local densities of charge reveal a finite charge distribution around some atoms at the apices and borders of the cone structures. For structures with more than 5,000 atoms, the contribution to the total density of states near the Fermi level essentially comes from states localized at the edges. For other energies, the average density of states exhibits similar features to the case of a graphene lattice. Results for the absorption spectra of nanocones show a peculiar dependence on the photon polarization in the infrared range for all investigated structures.
Since the first observation  of carbon nanocones (CNCs), large progress has been made on synthesis, characterization, and manipulation of CNCs and carbon nanodisks (CNDs) [2–6]. Differently from a planar graphene, the CNCs show a mixing of geometric, topological, and symmetry aspects that are exhibited in a non-homogeneous distribution of the electronic states through the structure. Particular effects of such feature are the charge accumulation at the cone apix and the selective polarized light absorption that may be used in technological applications.
There are different theoretical schemes to describe the electronic properties of cone-like structures. Models based on the Dirac equation [7, 8] give a convenient insight of properties in the long wavelength limit. However, for finite-size graphenes, the longest stationary wavelength occurs in the border, and a correct description of the states near the Fermi level is given in terms of edge states [9, 10]. The boundary conditions appearing when the nanosystems exhibit edges, such as the cases of nanoribbons, nanodisks, and nanorings, are quite well defined within a tight-binding formalism. Contrarily, in the continuum model, different approaches are followed to incorporate boundary conditions including the case of infinite mass  that have been critically examined and compared to tight-binding results. Ab initio models [12, 13] are able to predict detailed features, but they are restricted to structures composed of a few hundred atoms due to their considerable computational costs. Calculations based on a single π orbital are able to describe the relevant electronic properties [14–16]. In that spirit, we calculate the electronic structure and optical spectra of CNDs and CNCs within a tight-binding approach. CNC-structured systems generated by pentagonal and heptagonal defects were previously studied using a Green function recursive method [14, 17]. An interesting point to raise about the advantages of the tight-binding model is the fact that differently from the Dirac model, it is not essential to define two sublattices (A and B). For nanocones, this is a relevant point since for odd number of pentagons it is not possible to define the A/B sublattices.
In this work, finite-size systems (from 200 up to 5,000 atoms) are studied by performing direct diagonalizations of the stationary wave equation in the framework of a first-neighbor tight-binding approach. Each carbon atom has three nearest neighbors, except the border atoms for which dangling bonds are present. The overlap integral s is considered different from zero. As we will show later, this has important effects on the cone energy spectrum.
It is important to mention that relaxation mechanisms of the nanocone lattice are not explicitly included in the theoretical calculation. However, some stability criteria were adopted: (1) adjacent pentagonal defects are forbidden; (2) carbon atoms at the edges must have two next neighbors at least; (3) once the number of defects is chosen, the structures should exhibit the higher allowed symmetry (D6h group for the disk, D5 for the one-pentagon nanocone, and D2 for the nanocone with two pentagon defects). On the other hand, a statistical model to examine the feasibility and stability of nanocones has recently been reported . Combined with classical molecular dynamics simulations and ab initio calculations, the results show that different nanocones can be obtained. An important result is that a small cone (consisting of only 70 atoms) is found to be quite stable at room temperature. One should remark that the nanosystems studied in the present work are composed with more than 5,000 atoms and an analysis based on ab initio methods of molecular dynamics should be prohibited.
Although some of the graphene electronic properties are present in the CNCs, deviations are always manifested as a consequence of the different atomic arrangements, the finite-size of the nanocones, and also the possible point symmetry of the distinct cones. In the absence of external fields, the calculated density of states (DOS) shows a peak at the Fermi energy, and the local density of states (LDOS) shows that electron states are localized at the cone base. On the other hand, the symmetries observed in the LDOS at different energies allow a systematic description of the electronic structure and selection rules of optical transitions driven by polarized radiation. Unlike the nanodisk, the presence of topological disorder in nanocones involves a deviation from the electrical neutrality at the apex and at the edges.
where the |π j 〉 denotes the atomic orbitals 2p at site . Note that the overlapping between neighboring orbitals prohibits the set |π j 〉 to be an orthogonal basis. Only in the ideal case of zero overlap s=0, the coefficients in might be considered equal to the discrete amplitude probability to find an electron at the j-th atom (described by the one electron state |Ψ〉). We use the s≠0 basis, |π j 〉, to construct the eigenvalue equation and the base to calculate the properties related to discrete positions. Of course, to relate both bases, it is required to know the projection.
It is important to mention that, in ab initio calculations of carbon systems with edges, the atomic edges are passivated by hydrogen atoms. For graphene nanoribbons, the hydrogen passivation effects are better described when hybridized sigma-orbitals are considered . However, for a single pi-orbital model, position-dependent hopping amplitude is usually adopted. In the case of armchair ribbons, a single correction at the carbon atoms layering at the dimmer positions of the edges sites is enough to obtain similar results to the density functional theory (DFT) calculations, while for zigzag nanoribbons, the agreement between electronic structures obtained from tight-binding models with no passivation and DFT models including H-passivation are remarkably good for energies next to the Fermi energy. Making a parallel to nanocone systems, we believe that passivation effects may be neglect in a first approximation and that the main characteristics of the electronic properties are preserved within this simple model.
as it is shown in the subsection ‘Discrete position approach.’
with γ=0 and 1, for N C even and odd, respectively.
with εi,j corresponding to the energies of occupied and unoccupied states, respectively.
Discrete position approach
with the S−1≈Δ(0)−s Δ(1)+O(s2) matrix being different from the N C ×N C identity matrix Δ(0).
as the matrix elements of a position-dependent function in the π-base.
Results and discussion
Electronic density of states
As expected, for small finite systems, the DOS, LDOS, and the position of the Fermi energy depend on the number of atoms considered in the numerical calculation and on their characteristic geometries [21–23] and topology [24, 25]. The experimental results by Ritter and Lyding  give actually a true conclusion about the influence of edge structure on the electronic structures of graphene quantum dots and nanoribbons. A remarkable difference between CND and CNCs structures is the existence of a finite DOS above the Fermi level for nanocones. This clear metallic character of the DOS for nanocones is more robust for the two-pentagon CNC [22, 26]. This feature is a consequence of a symmetry break induced by the presence of topological defects in the CNC lattices, which generates new states above the Fermi energy not present in the CND structure. The contributions to the DOS coming from the apex atoms states are apparent in the LDOS of Figure 3e,f. Also notice that for the two-pentagon case, in which there is a large topological disorder, the LDOS spectra exhibit significant differences depending on the point symmetry of the considered atom (cf. Figure 2).
Electric charge distribution
LEC (fundamental charge units) at some relevant atoms in the cone apices shown in Figure 2 b,c
Concerning the different polarization directions, one should notice that, as occurs in C6v symmetric systems, α z =0 and α x =α y for the nanodisk. On the other hand, the absorption coefficients for the different cones studied (single and two pentagons) are finite for parallel polarization, and it depends on the structure details: as α z increases for a two-pentagon CNC structure, αx,y decreases. Due to the lack of π/2-rotation symmetry, one should expect, in principle, different results for x- and y-polarizations for any nanocone. However, such difference is observable just for the absorption coefficient of the two-pentagon CNC system, mainly in the range of low photon energies. The fact that α x =α y , for the case of one-pentagon CNC structure, may be explained using similar symmetry arguments applied to C6v symmetry dots , extended to the C5v symmetric cones. In the case of a two-pentagon CNC, the apex exhibits a C2v symmetry, preventing the cone to be a C4v symmetric system. As the apex plays a minor role, α x and α y will be slightly different. A large difference between the α z and the αx,y CNC absorption spectra occurs in the limit of low radiation energy. The α z coefficient goes to zero as whereas αx,y shows oscillatory features. The behavior of the absorption for parallel polarization is due to the localization of the electronic states at the atomic sites around the cone border. As the spatial distribution of those states are restricted to a narrow extension along the z coordinate, the z degree of freedom is frozen for low excitation energies.
Here, we have presented a theoretical study on the electronic properties of nanodisks and nanocones in the framework of a tight-binding approach. We have proposed a discrete position approximation to describe the electronic states which takes into account the effect of the overlap integral to first order. While the |π〉 base keeps the phenomenology of the overlap between neighboring atomic orbitals, the |π0〉 base allows the construction of diagonal matrices of position-dependent operators. A transformation rule was set up to take advantage of these two bases scenarios. Although the theoretical framework adopted does not explicitly include relaxation mechanisms, some stability criteria were adopted, and our analysis may be considered as a good first approximation to describe the main electronic structure and optical properties of such sizeable nanocones.
We have investigated the effects on the DOS and LDOS of the size and topology of CND and CNC structures. We have found that both total and local density of states sensitively depend on the number of atoms and characteristic geometry of the structures. One important aspect is the fact that cone and disk edges play a relevant role on the LDOS at the Fermi energy. For small finite systems, the presence of states localized in the cone apices determines the form of the DOS close to the Fermi energy. The observed features indicate that small nanocones could present good field-emission properties. This is corroborated by the calculation of the LEC that indicates the existence of finite charges at the apex region of the nanocones. For large systems, the contribution to the DOS near the Fermi level is mainly due to states localized in the edges of the structures whereas for other energies, the DOS exhibits similar features to the case of a graphene lattice.
The absorption coefficient for different CNC structures shows a peculiar dependence on the photon polarization in the infrared range for the investigated systems. The symmetry reduction of the two-pentagon nanocones causes the formation of very rich absorption spectra, with comparable weights for distinct polarizations. Although we have not found experimental data concerning to one-layer nanocones, we do believe that absorption measurements may be used as a natural route to distinguish between different nanocone geometries. The breaking of the degeneracy for different polarizations is found to be more pronounced for small nanocones. Absorption experiments may be used as natural measurements to distinguish between different nanocone geometries.
This work was supported by Fondecyt grant 1100672 and USM internal grant 11.13.31. AL thanks Brazilian agencies FAPERJ (under grant E-26/101522/2010), CNPq, and the Instituto Nacional de Ciência e Tecnologia em Nanomateriais de Carbono. LEO thanks the Brazilian agencies CNPq and FAPESP (Proc. 2012/51691-0) for partial financial support. PU thanks DGIP and Mecesup PhD scholarships.
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