# Graphene nanoflakes - structural and electronic properties

**ABSTRACT** The structures, cohesive energies and HOMO-LUMO gaps of graphene nanoflakes

and corresponding 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 Ns

is number of edge atoms). For the HOMO-LUMO gap the trends are more complex and

include also the topology of the edges.

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- Carbon 02/2014; 67:721-735. · 6.16 Impact Factor
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**ABSTRACT:**Graphene nanoflakes (GNFs) have more configurational degrees of freedom as compared to Graphene nanoribbons (GNRs) and are viable candidates for future nanodevices. GNFs can be devised with disparate geometries, and their electronic properties can be fine-tuned by genuine chemical functionalization. Hence, it is vital to know specific sites on GNFs where reaction is most feasible for chemical functionalization with donor− acceptor functional groups (nucleophiles/electrophiles). Here, we present spin-polarized and dispersion-corrected density functional theory based relative reactivity descriptor calculations to shed light on the reactivity pattern in small-sized GNFs. To have a clear understanding on the structure−property relationship, we consider GNFs with 24, 42, and 54 carbon atoms having various edges, namely, fully armchair, armchair/zigzag (arm-zig), and fully zigzag. All the edge atoms are saturated by hydrogen atoms. On the basis of the symmetry of the GNFs, susceptibility of assorted reactive sites pertinent to nucleophilic and electrophilic attacks is anticipated using relative reactivity descriptors. Further, we validate these relative reactivity descriptors for nucleophilic attack on armchair-C 24 H 14 and zigzag-C 24 H 12 by explicit adsorption of OH − , NH 2 − , and H 2 O molecules. Our study reveals that the reactivity pattern varies in small-sized GNFs as a function of shape. Importantly, few specific structural isomers have alternate Lewis acid−base pairs. It also manifests how the reactivity of peripheral and interior carbon atoms differ with shape and size of GNFs. With a discernment on site selectivity, GNFs can be functionalized by proper donor−acceptor groups at specific sites and hence can be used as potential candidates for molecular-and nanoelectronics.10/2014; - [Show abstract] [Hide abstract]

**ABSTRACT:**Boron and nitrogen substitutional impurities in graphene are analyzed using a self-consistent tight-binding approach. An analytical result for the impurity Green's function is derived taking broken electron-hole symmetry into account and validated by comparison to numerical diagonalization. The impurity potential depends sensitively on the impurity occupancy, leading to a self-consistency requirement. We solve this problem using the impurity Green's function and determine the self-consistent local density of states at the impurity site and, thereby, identify acceptor and donor energy resonances.Physical review. B, Condensed matter 04/2013; 87(15). · 3.66 Impact Factor

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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

G. Seifert

Physikalische Chemie, Technische Universit¨ at,

Bergstrasse 66b, 01062 Dresden, Germany

Abstract

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

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1.INTRODUCTION

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

nanoelectronics.

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.

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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

nanotechnology.

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

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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).

2.METHODS

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:

Ebind(N)

N

= ε∞+ γ(N)(1)

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8.48.68.89

DFT

7.7

7.8

7.9

8

8.1

8.2

DFTB

DZVP: R2 = 0.9054

TZVP: R2 = 0.8635

77.27.4

DFT

7.6 7.8

6.6

6.8

7

7.2

7.4

DFTB

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.

Limit structuresReorganization

(c)(b)

(e)

(f)

(g)

(h)

(a)

(d)

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)

C22.

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):

E(N)

N

= ε∞+ cNs

N,

(2)

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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

Ns

Nscales as

1

√Nand

energy of graphene nanoflakes can be written as:

E(N)

N

= ε∞+

c′

√N,

(3)

where c′is a constant. Thus, the correlation between the binding energy and the size of

nanoflakes should be proportional to

1

√Nas well as proportional toNs

N, if quantum effects of

the extended π-system are not important.

3.RESULTS AND DISCUSSION

3.1.Structural Properties

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

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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 1520 2530354045

N

1.2

1.25

1.3

1.35

1.4

1.45

Bond length [Å]

MC4n+2Dnh

MC4n+2D(n/2)h

MC4n

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.

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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.

0 20 40

N

60

0.7

0.8

0.9

1

[E(N)/N]/ε∞

CNTs

Ccage

MC4n+2

MC4n

GNFs

02040

N

60

Sz

Sa

Fz

Fa

Tz

Ta

PAHs

graphene

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

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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).

a

b

c

d

Kekule pattern

Clar’s sextet pattern

e

f

g

h

i

Kekule 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

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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

structures.17,41,42

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.

00.10.2 0.30.4

1/√N

0.7

0.8

0.9

1

[E(N)/]/ε∞

GNFs

00.10.20.3 0.4

1/√N

Tz

Ta

Sz

Sa

Fz

Fa

PAHs

C60

FIG. 7: (Online color) Calculated cohesive energies (in units of the graphene cohesive energy, ε∞) of GNFs

(left) and PAHs (right) versus

1

√N(N – total number of carbon atoms). For the abbreviations see Fig. 5.

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00.20.4 0.6 0.81

Ns/N

0.7

0.8

0.9

1

[E(N)/N]/ε∞

Fa

Fz

Sa

Sz

Ta

Tz

PAHs

UP GNFs

SP GNFs

IIII II

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

1

√Nif the

quantum effects of the π system have no major influence on the stability. Figure 7 shows

the cohesive energy plot as a function of

1

√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

from theNs

Nratio for GNFs and PAHs.

In the case of PAHs, the benzene molecule can be taken as the bottom limiting structure

with the ratio

NH

NC+NHequal to 0.5. The top limit stands, of course, for the infinite graphite

monolayer with the ratio

NH

NC+NH=0. One should keep in mind that this ratio should vary

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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

1

√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

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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.

50100150

0

1

2

3

4

5

6

HOMO-LUMO gap [eV]

GNFs

50100150

N

Sa

Sz

Fref

Ta

Tz

Fa

Fz

PAHs

204060 80

MC4n

MC4n+2

Ccage

CNTs

MCref

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.

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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

4.CONCLUSIONS

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

1

√N,

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

ratioNs

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

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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.

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- Available from Agnieszka Beata Kuc · May 27, 2014
- Available from Agnieszka Beata Kuc · May 27, 2014