Graphene nanoflakes - structural and electronic properties
A. Kuc and T. Heine
School of Engineering and Science, Jacobs University Bremen,
Campus Ring 1, 28759 Bremen, Germany
Physikalische Chemie, Technische Universit¨ at,
Bergstrasse 66b, 01062 Dresden, Germany
The structures, cohesive energies and HOMO-LUMO gaps of graphene nanoflakes and corre-
sponding polycyclic aromatic hydrocarbons for a large variety of size and topology are investigated
at the density functional based tight-binding level. Polyacene-like and honeycomb-like graphene
nanoflakes were chosen as the topological limit structures. The influence of unsaturated edge atoms
and dangling bonds on the stability is discussed. Our survey shows a linear trend for the cohesive
energy as function of Ns/N (N − total number of atoms and Nsis number of edge atoms). For the
HOMO-LUMO gap the trends are more complex and include also the topology of the edges.
Keywords: Graphene Nanoflakes, Polycyclic Aromatic Hydrocarbons, Topology Trends, Energetic Stability,
Electronic Properties, DFTB
Graphene, as a single layer of carbon atoms arranged packed densely in a two-dimensional
honeycomb crystal lattice, has attracted an enormous interest in the area of solid state elec-
tronics and composite materials, due to its high mechanical, thermal and chemical stability
and excellent electronic properties.
Graphene nanoribbons (GNRs) and nanoflakes (GNFs), that are finite in both dimen-
sions, can be considered as fragments or molecular subunits of graphene. Since their initial
successful fabrication,1the dimensions of GNRs and GNFs have rapidly reduced from the
microscale down to nanometer sizes either by top-down2or bottom-up3,4approaches. This
gave a possibility to explore low-dimensional transport and perspective for carbon-based
Depending on the size and shape, GNFs possess the ability to form ordered columnar me-
sophases.5Since the basic functional components of future electronics and spiroelectronics
devices are required to be on the nanometer scale it is important to understand the properties
of GNFs and their saturated counterparts, polycyclic aromatic hydrocarbons (PAHs).
Isolated GNFs and GNRs can be presently produced using different experimental ap-
proaches. The bottom-up approach by thermal annealing nanographene molecules results in
a conductive graphene film.4Alternatively, nanographene can be produced by soft-landing
of ions generated by solvent-free matrix assisted laser desorption/ionisation.6The product
is transfered to the gas phase, purified and adsorbed at surfaces. As top-down-techniques,
GNRs with widths varying from several tens of nanometers down to 2 nm have been fab-
ricated either by etching7–9or by means of chemical treatment of graphene or graphite.2It
has been reported that GNRs with certain edge chirality would open the band gap.10
GNFs with controlled thicknesses have been isolated in solution using density gradient
ultracentrifugation.11Cong et al12. have fabricated arrays of graphene nanodiscs (GNDs)
using nanosphere lithography (GNDs are GNFs with smooth edges and spherical shape).
Fabrication of GNRs with smooth edges is essential for many applications, however, it
is difficult to produce such edges by conventional physico-chemical methods. Jia et al.13
have shown that an efficient edge-reconstruction process, at the atomic scale, can be ob-
tained for graphitic nanoribbons by Joule heating using an integrated transmission electron
microscope-scanning tunnelling microscope (TEM-STM) system.
Very recently, few groups have independently reported very elegant methods for GNR
production.14–16The methods are based on the longitudinal unzipping of multi-walled CNTs
and involve an Ar plasma etching of the nanotubes. These procedures are simple and
inexpensive, and lead to GNRs with well-defined widths and edge structures.
It is widely known that stable GNFs with sizes C3–C10form linear conformations, C11–
C20form annular structures and Cns with n > 20 form fullerenes.17–19However, it appears
that so far little is known on the stability, structure and properties of larger (planar) carbon
nanostructures, and a computational study is timely.
Graphite and graphene are zero-gap semiconductors. If carbon particles are reduced in
size to a level where quantum effects are significant, large energy gaps may appear.20Indeed,
as large as 8.5 eV energy gaps have been predicted for clusters with few carbon atoms,21
while gaps below 2 eV were found for carbon cages composed of up to 80 carbon atoms.22–24
The knowledge and understanding of the size-tuned properties, e.g. binding energy and
HOMO-LUMO gap, would make carbon particles interesting candidates for applications in
FIG. 1: Exemplary graphene nanoflakes isomers considered for C26.
In this paper, we focus on small and medium neutral GNFs and corresponding PAHs.
We have restricted our calculations to even number of atoms, as well as to planar sixfold
ring systems (polyhexe structures; see Fig. 1). The determination of topologically distinct
structures (isomers) is difficult, as the number of isomers is strictly related to the number of
atoms (n) and increases rapidly with increasing n. As an example, for the molecular formula
C6H6, about 330 isomers can be written, considering the geometrical- and stereo-isomers.25
The complete set of isomers for any graphene flage can be created using graph theory.26,27
In this work, we do not aim to investigate all possible graphene flake structures, instead we
want to derive trends in stability and electronic structure on the basis of an extensive survey
of flake structures inhibiting different molecular weight and topology. We have created a
number of possible isomers for structures with N=6–34, 38, 42, 50, 54, 60, and 74 (N is the
number of carbon atoms). For some specific topologies, e.g. triangles, nanoribbons (stripes)
or circular flakes, the number of atoms was up to 220. Since the number of isomers rapidly
increases with increasing N the completeness in selection was related only to the smallest
structures (N <28).
All structures and corresponding cohesive energies were calculated using DFTB (Density
Functional based Tight-Binding).28,29For a recent review on the method see Oliveira et al.30
To proof the reliability of the this method we have performed DFT (Density Functional The-
ory) calculations as well. The VWN31(Vosko, Wilk, Nusair) exchange-correlation potential
was used for full DFT optimization with DZVP and TZVP basis sets.32Both DFTB and
DFT results are compared in Fig. 2, and we find good agreement between the two methods.
For a given number of carbon atoms the limiting geometries were chosen: polyacene
and polyphenanthrene chains, zig-zag and armchair honey-comb-like flakes. The exemplary
structures for C42are shown in Fig. 3.
In order to examine the the size-dependent trends of properties for various GNF and
PAH isomers, we have calculated the cohesive energies. The cohesive energies were further
related to the energy of the reference system – a graphene layer. One may write the binding
energy Ebind(per atom) of nanoflakes as:
= ε∞+ γ(N)(1)
DZVP: R2 = 0.9054
TZVP: R2 = 0.8635
DZVP: R2 = 0.9993
TZVP: R2 = 0.9993
FIG. 2: Comparison of the DFTB and the DFT cohesive energies for carbon flakes: (left) GNFs and (right)
PAHs. The energies are given in units of the graphene cohesive energy.
FIG. 3: Considered limiting geometries for C42GNFs: (a) – polyphenanthrenes (armchair nanoribbons)
and (d) – polyacenes (zig-zag nanoribbons), (b) – armchair and (c) – zig-zag circular flakes. Exemplary
GNFs with unstable polyhexe forms before and after full geometry optimization: (e) C14, (f) C18, (g, h)
where ε∞is the binding energy per atom in the infinite graphene layer and γ(N) is a kind
of a surface or ’edge’ energy.
For planar nanoflakes this surface energy can be expressed as a ratio of the number of
edge atoms (Ns) to the total number of atoms (N):
= ε∞+ cNs
where c is a constant. In case of GNFs, the Nsstands for the number of unsaturated carbon
atoms, the atoms that have only two neighbors, while for PAHs it is directly related to the
number of hydrogen atoms.
Since the number of atoms N in a planar flake is proportional to R2(R can be a radius
for circular flakes or one half of the diagonal for other shapes), then
energy of graphene nanoflakes can be written as:
where c′is a constant. Thus, the correlation between the binding energy and the size of
nanoflakes should be proportional to
√Nas well as proportional toNs
N, if quantum effects of
the extended π-system are not important.
3. RESULTS AND DISCUSSION
In this work, we have studied structural, energetic and electronic properties graphene
nanoflakes (GNFs) and their saturated counterparts – polycyclic aromatic hydrocarbons
(PAHs). The family of GNFs has dangling bonds at the circumference of the flakes which are
saturated with hydrogen atoms in the case of PAHs. Therefore, some of the GNF structures
suffered reorganization of the atoms arrangement during the optimization, formation of
monocycles (MC), ’holes’ or 5-fold rings (see Fig. 3(e-h)). This effect is not observed in
the PAH systems, what suggests the possibility to stabilize small graphene flakes in their
polyhexe forms by hydrogenation.
Several specific topologies can be mentioned, among them circular flakes, triangles,
nanoribbons. The latter ones are characterized by the number of hexagonal units that
determine the flake’s width. The nanoribbons, with the width of one hexagonal unit, are
polyacenes and polyphenanthrenes, depending on the type of edges (zig-zag or armchair).
All GNFs in the polyacene form are unstable in their polyhexe forms and they transform
into monocyclic rings or partial rings (’holes’) after full optimization. Therefore, we have
included also monocyclic rings in our studies for comparison and validation of our method,
as these systems have been investigated intensively - see e.g.33–36
We have divided carbon MCs into two families: 4n+2 (with symmetry DNhand D(N/2)h,
N – number of carbon atoms) and 4n (n – natural number), following the H¨ uckel rule of
aromaticity. The 4n+2 MCs with symmetry DNhare called cumulenic and have all bond
lengths and bond angles equal. MCs with D(N/2)hsymmetry are those with alternating bond
lengths (or bond angles). The bond lengths of all studied carbon MCs are shown in Fig. 4.
10 15202530 3540 45
Bond length [Å]
FIG. 4: Calculated bond lengths in the 4n+2 (with symmetry DNhand D(N/2)h) and 4n monocycles.
Our results and the previous theoretical studies show that the energetically most stable
4n MCs have always bond-lengths alternant structures.33–36This results from the first-
order Jahn-Teller distortion but the alternation decreases with increasing the ring size.
Furthermore, we have found that the 4n+2 ground state structures prevent the bond-length
alternation up to N = 10, while larger rings show a bond length alternation (D(N/2)h) at
all sizes and the cumulenic isomer (DNh) is a structural transition state. This results are in
good agreement with other works at DFT and quantum Monte Carlo level of calculations,33,35
however, the energy difference between cumulenic and alternant structure of C10is extremely
small of ≃1 kcal mol−1.
3.2. Energetic Stability
The calculated cohesive energies of all studied GNF and PAH isomers as a function of the
number of carbon atoms are given in Fig. 5. In addition, cohesive energies of other carbon
allotropes (fullerenes and carbon nanotubes, CNTs) are shown. Some of the high-symmetry
topologies (triangles, circles, nanoribbons) are marked as well.
FIG. 5: (Online color) Calculated cohesive energies (in units of the graphene cohesive energy, ε∞) of GNFs
and PAHs versus the number of carbon atoms (N). Some of the topologies together with the type of edges
(z – zig-zag or a – armchair) are highlighted: (squares) S – stripes (nanoribbons), (circles) F – circular
flakes, (triangles) T – triangles, MC – monocycles. For comparison finite carbon nanotubes (CNTs; stars)
and fullerenes (carbon cages; pluses) are given.
The most stable topologies,for a given N, are circular flakes, while the least stable are the
very narrow nanoribbons, namely polyacenes and polyphenanthrenes. In case of GNFs, the
polyacene form transforms into monocycles or ’holes’, therefore the least stable structures
will be those with partial polyacene forms. Nanoribbons become more stable with increasing
their width approaching the stability of graphene for very wide systems. In fact, the most and
least stable topologies correspond to our suggested limiting structures. Other energetically
favorable topologies are those of triangular flakes. Recently, it was shown by Ci et al9. that
graphene fragments can be shape-controlled by multistage cutting and that the two main
shapes obtained in such a process are triangles and few-nanometer wide nanoribbons.
Generally, the zig-zag type of edges is more stable than the armchair one. However, in
the case of nanoribbons, polyphenanthrenes are more stable than polyacenes, what can be
seen in the case of PAHs. This result is in qualitative agreement with other theoretical
studies.37The reason for the difference in the stabilities can be attributed to the differences
in the geometric properties, namely to the number of the Kekul´ e numbers, which is larger
for armchair edges. Considering C14H10one can draw 5 Kekul´ e patterns for phenanthrene
but only 4 patterns for anthracene (see Fig 6).
Clar’s sextet pattern
Clar’s sextet pattern
FIG. 6: Kekul´ e and corresponding Clar’s sextet patterns of anthracene (a-d) and phenanthrene (e-i).
The set of Kekul´ e patterns can be corresponded to the set of aromatic sextet patterns
according to Clar’s notation of the π-sextets.38,39According to Clar, not all of the Kekul´ e
valence structures in PAHs are equally important, and benzene may be the only system
that is exceptional. The dominant ones are those Kekul´ e valence structures which, after
superimposing, will give the largest number of isolated π-sextet rings.40Therefore, PAHs
larger than benzene tend to form with maximum number of aromatic sextets. Fig 6 shows
that for polyacenes we can draw at maximum one Clar sextet, while for polyphenanthrenes
this number increases with the length.
All the GNFs are less stable than graphene layer and the isolated C60molecule, but are
as stable as smaller fullerenes and (5,5) CNTs with finite length. In fact, it was previously
shown that for N<30 the graphene-like clusters are more stable than the respective fullerene
Considering carbon MCs, we have found that they are more stable than carbon flakes up
to N = 20 and this is in agreement with other theoretical works.17,41Polyacene GNFs are
not stable in their polyhexe forms for smaller sizes and they reorganize into MCs, what is
connected with the overlapping of free p-orbitals at the edges of GNF planes due to dangling
bonds. This effect causes a gain in the binding energy.
0 0.1 0.20.30.4
00.1 0.20.3 0.4
FIG. 7: (Online color) Calculated cohesive energies (in units of the graphene cohesive energy, ε∞) of GNFs
(left) and PAHs (right) versus
√N(N – total number of carbon atoms). For the abbreviations see Fig. 5.
0 0.20.40.6 0.81
FIG. 8: (Online color) Calculated cohesive energies (in units of the graphene monolayer cohesive energy,
ε∞) of GNFs and PAHs versusNs
N(Ns– number of edge atoms, N – total number of carbon atoms). For
the abbreviations see Fig. 5.
According to Eq. 3 (see Sec. 2), the cohesive energy should increase with
quantum effects of the π system have no major influence on the stability. Figure 7 shows
the cohesive energy plot as a function of
√N. For a given topology, e.g. triangles, stripes,
circular flakes, and flake sizes N >18 there are linear trends of cohesive energies. However,
the overall behavior is far from linearity. The energies of circular and triangular flakes
converge to the energy of graphene layer, while the narrow nanoribbons converge to the
energies of infinite polyacene and polyphenanthrene. The same trend is observed for both,
GNFs and PAHs (see Fig. 7).
As one can see from Fig. 8 there is a much better linear correlation of the binding energies
Nratio for GNFs and PAHs.
In the case of PAHs, the benzene molecule can be taken as the bottom limiting structure
with the ratio
NC+NHequal to 0.5. The top limit stands, of course, for the infinite graphite
monolayer with the ratio
NC+NH=0. One should keep in mind that this ratio should vary
for other isomers between 0 and 0.5.
In Fig. 8 the studied GNFs can be divided into two groups: (a) those, that keep the
polyhexe form after geometry optimization and (b)those, where the polyhexe form of a
flake is not stable. Furthermore, one can distinguish also between three size regions in the
cohesive energy plot. In the region I only GNFs stable in the hexagonal forms are cumulated,
while the systems that suffer strong reorganization in π-electron system are present in the
region III. The intermediate structures are marked in the region II. The higher cohesive
energies in the region III are due to the formation of monocyclic and 5-membered rings,
where stronger overlap of p-orbitals appears.
Taking into consideration only stable forms of GNFs, the energy scales linearly with the
size of a flake for all the topologies studied. All the energies converge to that of graphene
single layer, unlike for the
√Nratio. It can be concluded that the number of the edge atoms
is very important for the measure of surface energy, γ(N). This can be understood better
by the fact that the deviations in the energy of carbon flakes from that of graphene layer
are mainly due to the edge atoms. The points in the region I can be described by the trend
line, whose slope gives γ(N).
As one can also see from Fig. 8, a perfect linear trend of the cohesive energy versus
Ns/N is obtained for PAHs. In this case, the surface energy, γ(N), is a sum of the surface
energy of carbon atoms (εs) and the surface energy of hydrogen atoms (εH). While, for
the GNFs (homonuclear molecules) it can be calculated in a clear way, there is no unique
procedure to determine the binding energy in heteronuclear molecules. Still one can make an
approximation that εsis the correction of ε∞and should be equal zero in the case of PAHs,
and then the εHcan be calculated. In conclusion, small graphene flakes can be stabilized in
their polyhexe forms by saturation of dangling bonds by hydrogen atoms.
3.3. Electronic Properties
We have also studied the electronic properties of carbon flakes in terms of their size and
topology. The HOMO-LUMO gaps (∆) were calculated for all flakes studied in this work.
In addition, we have calculated much larger flakes (up to N = 220) to compare the results
with the experimental work of M¨ uller et al.5
Increasing the size leads to decrease of the HOMO-LUMO gap. All nanoflakes studied
in this survey are semiconductors or insulators (see Fig. 9). However, large clusters tend to
a gap closing, similar to graphene. The size and shape dependent trends are divided into
several groups of nanoflakes: polyacenes, polyphenanthrenes, monocycles, and graphene
flakes of armchair and zig-zag edges. Generally, for each specific topology, there is a size-
dependent trend and the ∆s decrease slowly or rapidly with the number of aromatic sextets,
depending on the type of edges and the shape of flakes.
HOMO-LUMO gap [eV]
FIG. 9: (Online color) Calculated HOMO-LUMO gap versus N. S – stripes (polyacenes and polyphenan-
threnes), F – circular flakes, Fref– flakes as studied in Ref. 5, T – triangles, MC – monocycles, MCref– data
from Ref., 36 z – zig-zag, a – armchair.
The ∆ of polyphenantherene GNFs are lower than the corresponding PAH forms and they
converge to the values of 1.4 eV and 2.5 eV, respectively. When increasing the width of
armchair-edged nanoribbons, however, the band gap approaches zero.43,44Generally, the zig-
zag type of peripheries lower the HOMO-LUMO gap in comparison to the armchair edges,
as predicted by Stein et al.45Therefore, the gaps of polyphenanthrenes are larger than those
of polyacenes. The ∆ of the PAH polyacenes decreases very rapidly with increasing the
length and the width of nanoribbons, although a very small gap always occur. This is due
to a higher-order Peierls distortion effect.46In the zig-zag nanoribbons, the frontier orbitals
are localized at the edges, while in the case of armchair-edge structures they are distributed
evenly over the carbon structure.
We have found that the metallic character, for the range of sizes studied here, is found
for the zig-zag-edge PAH structures with triangular topology. The ∆ decreases rapidly to
zero already for N=20. This is in a very good agreement with the recent work of Ezawa et
al47. who have found that the band gap decreases inversely to the length, and zero-energy
states emerge as the length goes to infinity. Infinite-length nanoribbons have the flat band
made of degenerated zero-energy states.47The circular PAH flakes converge ∆ to zero very
slowly. For example, the armchair flake with 222 carbon atoms has still ∆ of around 1.1
eV. The corresponding GNF structures do not show smooth trends and both types of edges,
zig-zag and armchair, give similar values of ∆.
Moreover, all carbon monocycles are semiconducting and the calculated ∆ converge
quickly to around 1.8 eV for both, 4n+2 and 4n, groups of MCs. The variation of ∆
is, however, much smaller for the 4n MCs and faster convergence is obtained. This result is
in qualitative (for the 4n MCs also quantitative) agreement with the work of Xu et al.36
In this work, we have presented results of an extensive DFTB study on the structural
and energetic properties of graphene nanoflakes and the corresponding polycyclic aromatic
hydrocarbons. A wide range of isomers (topologies) for a given number of carbon atoms, N,
was considered (N =6-34 (even numbers), 38, 42, 50, 54, 60, 67 and 74). The results show
clear trends in the energetic stabilities of different topological types with changing the flake
size. We have focused on the energy trends of neutral flakes considering the total number of
carbon atoms, as well as, the number of edge atoms. Only six-fold ring connections in the
planar clusters were considered in this study.
The established model consideration describes the cohesive energy trends in terms of the
number of atoms qualitatively well. For both, GNFs and PAHs, the energy scales as
however, different topologies converge to the energies of the corresponding infinite structures.
Only the triangular and circular flakes approach the energy of graphene reference structure.
Our simple model consideration predicts a linear behavior of cohesive energy versus the
N(where N is the total number of carbon atoms and Nsdenotes the number of edge
atoms). Good linear scaling is obtained for the all the PAHs and these GNFs, which are
stable in their six-fold connections. Small GNFs undergo a reorganization of π-electron
system forming monocycles and ’holes’, what causes a gain in energy and the deviations
from the linearity. At this point we must, however, emphasize that Eq. 3 has been tested
for few topologies. It is not obvious if this model can be generalized to hold for the rich
manifold of other graphene nanoflake topologies.
We have also found interesting size- and topology-dependent trends in the electronic
properties. Calculations of HOMO-LUMO gaps shows a variety of properties in electronic
conduction, from metals to typical semiconductors and insulators. Generally, the zig-zag
type of edges lower the energy gap and almost all zig-zag triangular PAHs are metallic.
Armchair triangular and all circular PAHs approach the electronic properties of graphene
very slowly. GNFs do not show smooth trends of ∆ but the ∆ decreases with increasing
number of stable aromatic sextets.
Comparison between the DFTB and DFT calculation shows a good agreement, validating
our method for its usage for systems built of sp2carbon atoms.
1K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V.
Grigorieva, and A. A. Firsov. Electric field effect in atomically thin carbon films. Science,
306(5696):666–669, October 2004.
2L. J. Li, Y. Y. Xia, M. W. Zhao, C. Song, J. L. Li, and X. D. Liu. The electronic structure of
a single-walled aluminosilicate nanotube. Nanotechnology, 19:1, 2008.
3X. Y. Yang, X. Dou, A. Rouhanipour, L. J. Zhi, H. J. Rader, and K. Mullen. Two-dimensional
graphene nanoribbons. J. Am. Chem. Soc., 130(13):4216–+, April 2008.
4X. R. Wang, Y. J. Ouyang, X. L. Li, H. L. Wang, J. Guo, and H. J. Dai. Room-temperature
all-semiconducting sub-10-nm graphene nanoribbon field-effect transistors. Phys. Rev. Lett.,
100(20):206803, May 2008.
5S. Mueller and K. Muellen. Phil. Trans. R. Soc. A, 365:1453–1472, 2007.
6H. J. Rader, A. Rouhanipour, A. M. Talarico, V. Palermo, P. Samori, and K. Mullen. Processing
of giant graphene molecules by soft-landing mass spectrometry. Nat. Mater., 5:276–280, 2006.
7M. C. Lemme, T. J. Echtermeyer, M. Baus, and H. Kurz. A graphene field-effect device. Ieee
Electron Device Letters, 28(4):282–284, 2007.
8Leonardo C. Campos, Vitor R. Manfrinato, Javier D. Sanchez-Yamagishi, Jing Kong, and Pablo
Jarillo-Herrero. Anisotropic etching and nanoribbon formation in single-layer graphene. Nano
Lett., 9:2600–2604, 2009.
9Lijie Ci, Li Song, Deep Jariwala, Ana Laura Elias, Wei Gao, Mauricio Terrones, and Pulickel M.
Ajayan. Graphene shape control by multistage cutting and transfer. Adv. Mater., 9999, 2009.
10X. L. Li, X. R. Wang, L. Zhang, S. W. Lee, and H. J. Dai. Chemically derived, ultrasmooth
graphene nanoribbon semiconductors. Science, 319(5867):1229–1232, February 2008.
11Alexander A. Green and Mark C. Hersam. Solution phase production of graphene with con-
trolled thickness via density differentiation. Nano Lett., 2009.
12C. X. Cong, T. Yu, Z. H. Ni, L. Liu, Z. X. Shen, and W. Huang. Fabrication of graphene
nanodisk arrays using nanosphere lithography. J. Phys. Chem. C, 113(16):6529–6532, April
13X. T. Jia, M. Hofmann, V. Meunier, B. G. Sumpter, J. Campos-Delgado, J. M. Romo-Herrera,
H. B. Son, Y. P. Hsieh, A. Reina, J. Kong, M. Terrones, and M. S. Dresselhaus. Controlled for-
mation of sharp zigzag and armchair edges in graphitic nanoribbons. Science, 323(5922):1701–
1705, March 2009.
14L. Tapaszto, G. Dobrik, P. Nemes-Incze, G. Vertesy, P. Lambin, and L. P. Biro. Tuning the
electronic structure of graphene by ion irradiation. Phys. Rev. B, 78(23), 2008.
15A. Hirsch. Unzipping carbon nanotubes: A peeling method for the formation of graphene
nanoribbons. Angew. Chem. Int. Ed., 48(36):6594–6596, 2009.
16L. Y. Jiao, L. Zhang, X. R. Wang, G. Diankov, and H. J. Dai. Narrow graphene nanoribbons
from carbon nanotubes. Nature, 458(7240):877–880, April 2009.
17R. O. Jones and G. Seifert. Phys. Rev. Lett., 79:443–446, 1997.
18F. Zerbetto.Carbon rings snapping.
Journal Of The American Chemical Society,
19T. Heine and F. Zerbetto. Dynamics of carbon clusters: chemical equilibration of rings and
bi-cyclic rings. Chemical Physics Letters, 358(5-6):359–367, 2002.
20M. R. C. Hunt and R. E. Palmer. The development of metallic behaviour in clusters on surfaces.
Philosophical Transactions Of The Royal Society Of London Series A-Mathematical Physical
And Engineering Sciences, 356(1735):231–247, 1998.
21C. X. Liang and H. F. Schaefer. Electronic-structures of linear-c4, linear-c6, linear-c8, and
linear-c10 carbon clusters and a symmetry-breaking phenomenon. Chemical Physics Letters,
22S. Saito and A. Oshiyama. Electronic and geometric structures of c70. Physical Review B,
23S. J. Woo, E. Kim, and Y. H. Lee. Geometric, electronic, and vibrational-structures of c-50,
c-60, c-70, and c-80. Physical Review B, 47(11):6721–6727, 1993.
24B. L. Zhang, C. Z. Wang, K. M. Ho, C. H. Xu, and C. T. Chan. The geometry of small fullerene
cages - c-20 to c-70. Journal Of Chemical Physics, 97(7):5007–5011, 1992.
25T. C. Dinadayalane, U. D. Priyakumar, and G. N. Sastry. J. Phys. Chem. A, 108:11433–11448,
26M. Randic. Aromaticity of polycyclic conjugated hydrocarbons. Chemical Reviews, 103(9):3449–
27E. Estrada and G. Patlewicz. Croat. Chem. Acta, 77:203–211, 2004.
28D. Porezag, T. Frauenheim, T. Kohler, G. Seifert, and R. Kaschner. Construction of tight-
binding-like potentials on the basis of density-functional theory: Application to carbon. Phys.
Rev. B, 51:12947–12957, 1995.
29G. Seifert, D. Porezag, and T. Frauenheim. Int. J. Quant. Chem., 58:185–192, 1996.
30A. F. Oliveira, G. Seifert, T. Heine, and H. A. Duarte. Density-functional based tight-binding:
an approximate dft method. J. Br. Chem. Soc., 20(7):1193–1205, 2009.
31S. H. Vosko, L. Wilk, and M. Nusair. Accurate spin-dependent electron liquid correlation
energies for local spin-density calculations: A critical analysis. Canad. J. Phys., 58:1200–1211,
32N. Godbout, D. R. Salahub, J. Andzelm, and E. Wimmer. Optimization of gaussian-type basis-
sets for local spin-density functional calculations .1. boron through neon, optimization technique
and validation. Canadian Journal Of Chemistry-Revue Canadienne De Chimie, 70(2):560–571,
33E. J. Bylaska, J. H. Weare, and R. Kawai. Development of bond-length alternation in very large
carbon rings: Lda pseudopotential results. Physical Review B-Condensed Matter, 58(12):R7488–
34M. Saito and Y. Okamoto. Second-order jahn-teller effect on carbon 4n+2 member ring clusters.
Physical Review B, 60(12):8939–8942, 1999.
35T. Torelli and L. Mitas. Electron correlation in c4n+2 carbon rings: Aromatic versus dimerized
structures. Physical Review Letters, 85(8):1702–1705, 2000. Download full-text
36S. H. Xu, M. Y. Zhang, Y. Y. Zhao, B. G. Chen, J. A. Zhang, and C. C. Sun. Chem. Phys.
Lett., 421:444–447, 2006.
37J. Wu and Y. S. Jiang. The valence bond calculations for conjugated hydrocarbons having 24-28
pi-electrons. Journal Of Computational Chemistry, 21(10):856–869, 2000.
38N. Ohkami, A. Motoyama, T. Yamaguchi, and H. Hosoya. Graph-theoretical analysis of the
clars aromatic sextet - mathematical properties of the set of the kekule patterns and the sextet
polynomial for polycyclic aromatic-hydrocarbons. Tetrahedron, 37(6):1113–1122, 1981.
39E. Clar. The Aromatic Sextet. Wiley, London, 1972.
40M. Randic, S. El-Basil, S. Nikolic, and N. Trinajstic. Clar polynomials of large benzenoid
systems. Journal Of Chemical Information And Computer Sciences, 38(4):563–574, 1998.
41R. O. Jones. Europ. Phys. J. D, 9:81–84, 1999.
42G. Seifert, K. Vietze, and P. W. Fowler. CP442, Electronic Properties of Nevel Materials: XII
Int. Winterschool, 1998.
43K. Yoshizawa, K. Yahara, K. Tanaka, and T. Yamabe. Bandgap oscillation in polyphenan-
threnes. Journal Of Physical Chemistry B, 102(3):498–506, 1998.
44R. Sharma, N. Nair, and M. S. Strano. Structure-reactivity relationships for graphene nanorib-
bons. J. Phys. Chem. C, 113:14771–14777, 2009.
45S. E. Stein and R. L. Brown. Pi-electron properties of large condensed polyaromatic hydrocar-
bons. Journal Of The American Chemical Society, 109(12):3721–3729, 1987.
46M. Kertesz and R. Hoffmann. Higher-order peierls distortion of one-dimensional carbon skele-
tons. Solid State Communications, 47(2):97–102, 1983.
47M. Ezawa. Metallic graphene nanodisks: Electronic and magnetic properties. Physical Review
B, 76:245415–1–6, 2007.