modelling the role of size, edge structure and

Modelling the Role of Size, Edge Structure and

Terminations on the Electronic Properties of

Graphene Nano-flakes

Amanda S Barnard

CSIRO Materials Science & Engineering, Clayton, Victoria, 3168, AUST.

E-mail: [email protected]

Ian K Snook

Applied Physics, RMIT University, Melbourne, Victoria, 3000, AUST.

E-mail: [email protected]

Abstract.

The addition of graphene nano-flakes to the suite of materials for graphene-based

nanotechnology requires a complete understanding of the relationship between shape,

structure, properties, and property dispersion. Due to the large number of configurational

degrees of freedom, this is a very challenging undertaking, particularly if morphological

ensembles contain a reasonable array of sizes, shapes (edges and corners) and edge/corner

terminations. We report results of density functional tight-binding simulations of zigzag

and armchair hexagonal graphene nano-flakes, with unterminated, monoyhydride or

dihydride terminated edges and corners. We find that hexagonal nano-flakes with armchair

edges are most likely to achievable experimentally, providing that sufficient H is present

during synthesis (or processing) to facilitate dihydride edge and corner passivation,

forming a circumference of sp3 hybridized C atoms. This is significant, since the energy

of the Fermi level, and electronic density of states in the vicinity of the Fermi level,

are sensitive to the structural and chemical characteristics of the atoms around the

circumference, which can be modified post-synthesis.

PACS numbers: 02.70.-c,31.15.ae,68.65.Pq,73.20.At,73.22.Pr,

Submitted to: Modelling Simulation Mater. Sci. Eng.

Hexagonal Graphene Nano-flakes 2

1. Introduction

Since this experimental discovery [1] there has been an enormous amount of theoretical

and experimental interest in graphene (a two dimensional sheet or membrane of sp2

bonded carbon atoms), because graphene has been found to exhibit an array of exceptional

properties, including very fast electron transport, room temperature quantum hall effect,

the highest mechanical strength and greatest thermal conductivity yet measured. In

particular its fascinating electrical properties have lead to the speculation that graphene

may one day replace silicon as the material of choice for most electronic applications

[2, 3]. However, despite having many exceptional properties graphene has one very severe

limitation from the point of view of electronics applications; it has no band-gap and a

vanishingly small density of states at the Fermi level, making it a semi-metal [4, 5, 6].

Several methods have been suggested to induce a band gap in graphene and thus

overcome this fundamental limitation, but by far the most popular is to cut a graphene

into one dimensional (1-D) nanoribbons [7, 8, 9]. In such cases the gap width depends

sensitively on the physical width of the ribbon [5, 6, 7, 8, 9]. This is due to the participation

of edge states in the electronic structure, which begin to play a significant role when there

are a large number of atoms residing on or near the newly created edges. Furthermore,

there are two major types of idealized edges, zigzag (ZZ) and armchair (AC), although

others also exist [10], and graphene nanoribbons may be metals, semiconductors, half-

metals, feromomagnets and antiferomagnets depending on the choice of edge structure,

ribbon width and chemical termination [5, 6, 11]. While this is a very attractive prospect,

the intrinsic relationship between properties and edge states also leads to new limitations

in terms of fabrication (for use in electronic devices) because it is difficult to consistently

and reliably produce graphene nanoribbons with edge states of a particular type, and the

very act of cutting graphene into ribbons leads to electron mobility degradation and loss

of performance in devices. Ideally, one would like to be able to exercise a greater degree

of control, or develop structures that can preserve their properties under a greater degree

of morphological dispersion.

The zero-dimensional (0-D) form of graphene, known as graphene nano-flakes or

graphene nano-dots, has also been produced, but has been much less extensively studied

than 2-D graphene or 1-D ribbons. This is surprising given that preliminary studies

indicated they have a range of properties which differ from those of their higher dimensional


counterparts, and offer great potential for a variety of electronic and magnetic applications.

Like graphene the basic structure of graphene nano-flakes is that of fused hexagonal rings

of aromatic benzene, with a nearest neighbour distance of 1.42 A and bond angles of 120◦,

akin to a giant aromatic molecule [12]. If the edges are hydrogenated, these are often

termed polycyclic aromatic hydrocarbons. Their desirable properties arise due to fact that

graphene nano-flakes have a much larger number of configurational degrees freedom (may

be cut into a much larger variety of different shapes), and possess corner states in addition

to edge states. Furthermore, graphene nano-flakes can potentially range in size from

molecular to semi-infinite 2-D structures [13], and consequently their electronic structures

will vary from having discrete molecular levels to being band-like as their dimensions are

made larger. This leads to the potential of spanning the range of electronic and magnetic

properties from molecular to 2-D by using graphene nano-flakes of different dimensions.

However, as indicated above, the widespread use of graphene nano-flakes in modern

nanotechnology will be dependent upon development of reliable methods of producing these

structure, and/or the development of a complete understanding of the consequences the

deliberate or incidental shapes and reconstructions have upon properties, and property

dispersion. Due to the large number of configurational degrees of freedom, the latter

represents a very challenging undertaking, particularly if one is to consider a reasonable

array of sizes, shapes (edges and corners) and edge/corner terminations; as would be

present in the majority of industrially relevant samples. It has already been shown that the

stable structure of graphene nano-flakes depends on the size [14], shape [14], temperature

[15], space charge [16, 17] and interactions with other chemical groups [18]. Clearly the

way forward is to create monodispersed samples and systematically study the change in

properties as different morphological features are changed incrementally; a task that is not

possible experimentally at this time.

Therefore, to initiate a more detailed understanding of the relationship between

properties and nanomorphology, we have begun by undertaking a series of density

functional tight-binding simulations of an incremental set of graphene nano-flakes with the

same shape (hexagonal), but different edge structures. Our structures range in size from

a simple molecular ring (beginning with the benzene radical, benzene and cyclohexane) to

∼6.7 nm in diameter, and are terminated exclusively with ZZ edges or exclusively with AC

edges. As suggested above, these edges either remain unterminated (radical) or possess


Figure 1. Examples of the types of hexagonal graphene nano-flakes included in the

structure set. Nano-flakes in the top row have zigzag (ZZ) edges, and nano-flakes in the

bottom row have armchair (AC) edges. The left column are radicals (with unterminated

edges), the central column have monohydride terminated edges (akin to benzene), and

the right column have dihydride terminated edges (akin to cyclohexane).

benzene-style monohydride terminations, or cyclohexane-style dihydride terminations, as

shown in the examples in figure 1. As we will show, the type of edge structure and

termination impacts the thermodynamic stability, and contributes to property dispersion,

particularly at small sizes. In addition to this, we find the structure of the corners can be

a deciding factor for both the stability and the electronic properties in the vicinity of the

Fermi level.

2. Computational Method

In this study, we chose to use the density functional based tight-binding method with

self-consistent charges (SCC-DFTB) [19, 20], which is a two-center approach to density

functional theory (DFT), as the tight-binding methodology has been shown to be idea

for studying the electronic properties of graphene [5]. Formally, the Kohn-Sham density

functional is expanded to second order around a reference electron density, which is

obtained from self-consistent density functional calculations of weakly confined neutral

atoms within the generalized gradient approximation (GGA). The confinement potential is


optimized to anticipate the charge density and effective potential in molecules and solids. A

minimal valence basis is established and one- and two-center tight-binding matrix elements

are calculated (rather than fitted) within DFT, and a universal short-range repulsive

potential accounts for double counting terms in the Coulomb and exchange-correlation

contributions as well as the internuclear repulsion. We used the pbc set of parameters

for C–C, C–H and H–H interactions, as developed by Kohler and Frauenheim [21]. Self-

consistency is included at the level of Mulliken charges, as described in reference [20]. In

this study we have only considered non-magnetic states. The convergence criterion for a

stationary point was 10−4 a.u. ≈ 5 meV/A for forces, and all structures were fully relaxed

prior to the calculations of their average binding energy and electronic band structure.

This approach has already proven highly successful in exploring the electronic properties of

graphene nano-flakes in the past [16, 17, 18], as well as those of other carbon nanomaterials

[22, 23, 24, 25].

3. Discussion of Results

3.1. Mechanical and Thermodynamic Stability

The first factor one must consider when attempting to identify relationships between

functional properties and the structure of graphene nano-flakes, is that if the edges and

corners are un-terminated, then reconstructions will occur, breaking the aromaticity and

lowering the total energy of the flake. Depending on the shape, the effects can be minimal,

or quite extreme, even at low temperature [15]. Reconstructions are generally more severe

at highly under-coordinated feature such as corners, and differ on zigzag and armchair

edges.

In the case of our computational samples, there are two dominant shapes (as shown

in figure 1), with either ZZ edges, which intersect at an AC-type corner, or with AC

edges which intersect at a ZZ-type corner. We have analysed the types of reconstructions

occurring at the edges and corners of these shapes, and find a number of consistent trends.

Firstly, when the edges and corners are unterminated, there is a net in-plane contraction

of the nano-flakes, giving rise to a shorter characteristic C–C bond length, that converges

rapidly to the bulk value at a (corner to corner) diameter of ∼3.7 nm. The smallest C24

ZZ flake has an average C–C bond length of 1.39±0.076 A, and the C–C bond length in an


(a) (b) (c)

Figure 2. Examples of the types of edge and corner reconstructions occurring in the

vicinity of (a) an AC-type corner at the intersection of two ZZ-edges, (b) one type of

ZZ-corner at the intersection of two AC-edges, and (c) another type of ZZ-corner at the

intersection of two AC-edges. The initial “bulk-like” atomic configurations are shown in

grey, and the relaxed configurations are shown in blue. The red arrows highlight the

in-plane direction of the reconstruction.

infinite graphene membrane was found to be 1.42 A. Of course, these contractions are not

homogeneous, and we found that edge and corner atoms would either expand or contract

from their bulk-like positions, depending upon their location with respect to these features.

To highlight the types of edge and corner reconstructions that were consistently

observed, we show three examples in figure 2. In these images, the initial (bulk-like)

configuration is shown in grey, and the relaxed configurations are superimposed in blue.

In each case, to aid in visualisation, the relaxed structure has been scaled to remove

the net contraction, and allow for a relative atom-specific in-plane reconstructions to be

easily compared. The red arrows have been added to indicated the direction the atoms

moved as a result of the reconstruction. In figure 2(a) we see that the AC-type corners

undergo a different type of reconstruction to the adjacent ZZ edges, with the two corner

atoms shortening the bond between them (to that of a sp1 bond, at 1.25 A) in addition

to contracting toward the centre of the nano-flake. The atoms residing on the ZZ edges

directly adjacent to these corners expand away from the centre of the nano-flake and

away from the corner, whereas those further away from the corner relax inward (when

under coordinated) or outward (when fully coordinated), which gives rise to an overall

straightening of the chain of atoms along the ZZ edge.

In figure 2(b), we can see a type of ZZ-corner with two C atoms relaxing inward and

away from the vertex. Directly adjacent to them two more under coordinated C atoms relax

inward, toward the vertex. This, along with the AC edge reconstruction that are consistent


with those of the AC corners in fig 2(a), has the overall effect of smoothing the corners. In

contrast, figure 2(c) shows there is another type of ZZ-corners that undergo a much more

dramatic reconstruction, which is quite dissimilar to that of the adjacent AC edges, and

those observed on the AC-corner. While the reconstruction pattern of along the AC edges

is consistent with the reconstruction on the AC-corner, the ZZ-corner is not consistent with

the pattern along a ZZ edge. Although there is a very severe inward contraction if the C

atom residing on the vertex, the adjacent corner atoms relax away from the vertex and

toward the vacuum simultaneously. This significantly shortens the C–C bond lengths of

the AC-corner atoms (∼1.35 A), but does not have a net smoothing effect. The protrusion

of the AC-corners becomes more pronounced, even though the corner has been effectively

flattened.

These complicated corner-dependent reconstruction patterns, along with the net

contraction, is eliminated when the edge and corner atoms are terminated with

hydrogen, and no discernable differences could be found between the configurations of

the monohydride or the dihydride terminal C atoms.

The reason for these reconstructions is ultimately to lower the energy of the nano-flake.

However, the energy differences due to the edge and corner reconstructions represent a small

fraction of the total energy difference, which is most significantly affected by size and the

density of H atoms residing on the circumference. Figure 3(a) shows the average excess

binding energy per atom (〈∆Eb〉) for each structure in the sample sets, with respect to an

infinite graphene sheet, calculated using:

〈∆Eb〉 =1

NC +NH

(E(NC , NH)−NCµC −NHµH). (1)

In this simple expression, E(NC , NH) is the total energy of the nano-flake extracted from

the simulations, µC is the chemical potential of a carbon atom in graphene, µH is the

chemical potential of a hydrogen atom in H2, and NC and NH are the number if carbon

and hydrogen atoms, respectively.

We are first struck by the obviously higher energy of the radical (unterminated) nano-

flakes, which show a smooth convergence toward planar graphene with increasing size.

The higher energy is, of course, due to the large number of two-fold coordinated atoms

around the edges and corners, but the results are surprisingly robust against variations

in the fraction of edge/corner atoms, irrespective of size. This is because, while the AC

nano-flakes have a larger number of two-fold coordinated atoms around the circumference,


(a) (b)

Figure 3. (a) The thermodynamic stability of the different sets of hexagonal graphene

nano-flakes, and (b) a closer view of the stability of the monohydride and dihydride

terminated structures. Results for the zigzag nano-flakes are shown by the closed symbol

and solid lines (to guide the eyes), and the armchair nano-flakes are shown by the open

symbols and dashed lines.

these atoms break aromaticity upon reconstruction, forming a circumferential chain of sp1

bonds, with a larger net (compensating) reconstruction energy.

In the case of the monohydride and dihydride terminated nano-flakes, the binding

energies are much closer to that of graphene (see figures 3(a) and (b)), but we can see

that the type of edges/corners and the density of H terminations are more important than

size. In general, the AC hexagonal nano-flakes are thermodynamically stable (by ∼0.5

eV) when the edges/corners are saturated with two H atoms, resulting in a circumference

of sp3 hybridized C atom, but not with single H atoms (which results in a circumference

of sp2 hybridized C atoms). In the former case the average binding energy per C atom

is exothermic, and in the later case it is endothermic. The ZZ nano-flakes are far less

sensitive this is nuance, and both are unstable with respect to an infinite graphene sheet.

This suggests that, contrary to common intuition, AC-type hexagonal nano-flakes are most

likely to achievable experimentally, providing that sufficient H is present during synthesis

(or processing) to facilitate dihydride edge and corner passivation.


3.2. Electronic Properties

Following the structural relaxation and investigation of the thermodynamic stability of

the graphene nano-flakes, the electronic properties were calculated, including the complete

electronic density of states (DOS), the energy of the Fermi level (Ef ), and the electronic

band gap (Eg). To begin with, the DOS is presented for all of the structures in figure 4.

In each case, we see a size-dependent transition from the discrete molecular levels of C24

(figure 4(a)), C42 (figure 4(b)), C24H12 (figure 4(c)), C42H18 (figure 4(d)), C24H24 (figure

4(e)), and C42H38 (figure 4(f)) — to the smooth band structure of C1176 (figure 4(a)), C1200

(figure 4(b)), C1176H84 (figure 4(c)), C1200H96 (figure 4(d)), C1176H168 (figure 4(e)), and

C1200H192 (figure 4(f)). In general, our results revealed that the DOS in the vicinity of the

Fermi level is sensitive to both the types of edges and corners, and the types of termination.

As a direct comparison of the types of edges, corners and terminated considered in

this study, we can exploit the fact that both the ZZ and AC sets of nano-flakes contain

a structure with exactly 1014 C atoms. Both the ZZ and AC radicals therefore have the

same molecular weight, but the AC nano-flake has a higher number of under-coordinated

atoms at the edges and corners. For the monohydride nano-flakes, the ZZ structure has 78

terminal H atoms, whereas the AC structure has 90 terminal H atoms. The ZZ and AC

dihydride structures therefore have 156 and 180 H atoms around the edges and corners,

respectively.

The DOS for these structures are provided in figure 5, off-set and corrected with

respect to the energy of the Fermi level (Ef , which is set to zero) for the purposes of

clarity. The ZZ radical in figure 5(a) and the AC radical in figure 5(b) show a finite DOS

at the Fermi level, and show minor differences in the occupation of states within Ef ± 2eV

due to the different edges and corners (see figure 2(a) and 2(c)). There is also a gap at

higher energies in the conduction band. The monohydride ZZ (figure 5(c)) and the AC

(figure 5(d)) structures are also very similar, but differ in that the ZZ structure has a finite

DOS at the Fermi level, whereas the DOS of the AC structure is zero at the Fermi level.

The higher energy gap in the conduction band is narrower. Finally, both the dihydride ZZ

(figure 5(e)) and AC (figure 5(f)) nano-flakes have a zero density of states at the Fermi

level, and the higher energy gap in the conduction band no longer exists. The closing

of the band gap is due to additional edge and corner states occupying the gap, and so

only small hexagonal graphene nano-flakes are likely to exhibit semiconducting behaviour,


Figure 4. Electronic density of states of (a) the ZZ radicals, (b) the AC radicals, (c) the

monohydride ZZ nano-flakes, (d) the monohydride AC nano-flakes, (e) the dihydride ZZ

nano-flakes, and (d) the dihydride AC nano-flakes.

unless passivation of the edges and corners can be prevented, as shown in figure 6(a).

Beyond the presence or lack of a band gap, the differences in the DOS at the Fermi

level are an important feature of these graphene nano-flakes, as is the energy of the Fermi

level itself, since this influences the compatibility of these structures with other materials

and molecules. In general, we find that Ef is sensitive to size, converging to that of planar

graphene at large sizes, but also to the type of terminations (see figure 6 (b)). The Ef of


Figure 5. Electronic density of states of (a) the C1014 ZZ radical, (b) the C1014 AC

radical, (c) the monohydride C1014H78 ZZ nano-flake, (d) the monohydride C1014H90 AC

nano-flake, (e) the dihydride C1014H156 ZZ nano-flake, and (d) the dihydride C1014H180

AC nano-flake. The results are off-set for clarity, and the energies (E) are corrected with

respect to the energy of the Fermi level (Ef ).

the unpassivated nano-flakes is lower than bulk graphene, the dihydride nano-flakes have

a slightly higher Fermi energy than bulk graphene, and the results for the monohydride

terminated nano-flakes are approximately equivalent to the bulk value (exhibiting little in

the way of size dependence). When the edges and corners are terminated the trends in the

ZZ and AC nano-flakes are remarkably similar; particularly at larger diameters.

However, the results for the radical nano-flakes show significant differences between

the ZZ and AC results. Specifically, the results for the unterminated AC nano-flakes

exhibit some very abrupt variations, which are not present in any other sample set. These

differences are corner effects. The higher Ef values belong to structures (see C84, C180,

C924 and C1200) with corners of the type shown in figure 2(b), and the lower Ef values

have corners of the type shown in figure 2(c). These effects are also evident in the DOS


(a) (b)

Figure 6. The convergence of (a) the band gaps, and (b) the energy of the Fermi levels.

Figure 7. Electronic density of states of the C216 ZZ radical the C180 and C222 AC

radicals, to highlight the role of corner states. The energies (E) are corrected with respect

to the energy of the Fermi level (Ef ).

in the vicinity of the Fermi level, but not specifically the presence or width of the band

gap (see figure 6(b)). To illustrate the impact of corners states, the DOS in the vicinity of

the Fermi levels of the C216 ZZ radical the C180 and C222 AC radicals is shown in figure 7.

Remember, the C180 and C222 nano-flakes are both AC structures, that differ in the type

of corners, but are effectively equivalent in terms of average diameter.


These results demonstrate that a greater variety of different electronic structures can

be developed via deliberate modification of structural parameters, even in these simple

hexagonal graphene nano-flakes samples. At this time, the controlled engineering of specific

shapes (and/or the control of polydispersivity) remains challenging to experimentalists,

but chemical functionalization and edge/corner passivation is significantly more mature

and can already provide a means of structure/property engineering [3, 12, 26, 27, 28].

As mentioned above, this study ws restricted to non-magnetic states, although ZZ edges

have been shown to introduce localized states in graphene nanoribbons, which can have

different spin ordering when opposite edges belong to different sub-lattices. Zigzag edges

can create ferromagnetic (FM) or antiferromagnetic (AFM) phases, and large magnetic

moments occur at ZZ edges, while there are no localized states and magnetic phases at AC

edges. The next stage in this investigation is to examine other spin states, and correlate the

physical structure, chemical character and size to magnetic properties [26]. In the future,

it will also e important to study the electronic properties of other graphene nano-flakes,

with different relative combinations of ZZ and AC edges, and different obtuse and acute ZZ

and AC corners. After a complete morphological exploration has been undertaken, it will

then be possible to build a statistical description of the structural and electron properties of

graphene nano-flakes, that more accurately represents the morphological ensembles present

in real specimens.

4. Conclusions

Presented here are results of density functional tight-binding simulations of sets of fully

reconstructed hexagonal graphene nano-flakes, with exclusively zigzag or armchair edges

around the circumference. We have shown how relative thermodynamic stability imporves

with increasing size, but is quite sensitive to the structure and chemistry character of

the edges and corners. We find that hexagonal nano-flakes with armchair edges are most

likely to achievable experimentally, provided that sufficient H is present during synthesis

(or processing) to facilitate dihydride edge and corner passivation. This means that stable

graphene nano-flakes will prefer to be surrounded by circumference of sp3 hybridized C

atoms, in contrast to the majority of sp2 hybridised atoms residing in the interior of the

flakes.

Looking beyond the role of zigzag and armchair edges, we have shown that corner


reconstructions are unique, are highly dependent on the structure of the vertex, and may

be remarkably dissimilar to the reconstructions of edges. Although the types of corners

have little impact on the thermodynamic stability, they are very important in determining

the electronic structure, particularly when the nano-flakes are small. The electronic density

of states in the vicinity of the Fermi level, and well as the energy of the Fermi level itself, is

strongly and predictably influenced by structural and chemical characteristics of the atoms

around the circumference (including the corners). Based on these findings we predict that,

just as the introduction of edges can open the band gap of graphene nanoribbons [7, 8, 9],

so too the introduction of specific corners and control over the degree of edge/corner

passivation can provide a means of engineering the Fermi level. Tuning the Fermi level is

an important factor in interfacing graphene with existing and future device components,

making graphene nano-flakes a unique material that will be invaluable in the development

of graphene-based nanotechnology.

Acknowledgments

Computational resources for this project have been supplied by the National Computing

Infrastructure (NCI) national facility under MAS Grant e74.

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