Modelling the Role of Size, Edge Structure and
Terminations on the Electronic Properties of Trigonal
Graphene Nano-flakes
H. Q. Shi a∗, A.S. Barnard b , and I. K. Snook
a
b
a
Applied Physics, RMIT University, Melbourne, Victoria, 3000, AUST
CSIRO Materials Science and Engineering, Clayton, Victoria, 3168, AUST
September 22, 2011
Abstract
Graphene nano-flakes provide a range of opportunities for engineering graphene
for future applications, due to the large number of configurational degrees of freedom associated with the addition of different types of corners and edge stages in the
structure. Since these materials can, in principle, span the molecular to macroscale
dimensions, the electronic properties may also be discrete or continuous, depending
on the application in mind. However, since the widespread use of graphene nanoflakes will require them to be predictable, stable and robust against variations associated with some degree of structural polydispersivity, the development of a complete
understanding the relationship between structure, properties and property dispersion is essential. In this paper we used electronic structure computer simulations to
model the thermodynamic, mechanical and electronic properties of trigonal graphene
∗
[email protected]
1
nano-flakes with acute (highly reactive) corners. We find that these acute corners
introduce new features different to the obtuse corners characteristic of hexagonal
graphene nano-flakes, and introduce different electronic states in the vicinity of the
Fermi level. The structure and properties are sensitive to size and functionalization,
and may provide new insights into the engineering of graphene nano-flake components.
Keywords: graphene, stability, band gap, ionization potential, electron affinity,
nanoparticle
1
Introduction
Since graphene, a two dimensional sheet or membrane of sp2 -bonded carbon atoms, was
successfully isolated 1 in 2004, it has been found to exhibit an array of exceptional properties
that are not characteristic of graphite or small aromatic molecules. These include very fast
electron transport, room temperature quantum hall effect, the highest mechanical strength
and greatest thermal conductivity yet measured . In particular its fascinating electronic
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 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 .
Several methods have been suggested to introduce a band gap in graphene and thus
overcome this fundamental limitation, but by far the most popular approach is to cut
graphene into nanoribbons 6–8 . Restricting the structure in another dimension introduces
new states into the electronic structure, but adds another degree of complexity. There
are two major types of idealized edges, zigzag (ZZ) and armchair (AC), although others
also exist 9 , and the gap width depends sensitively on the physical width of the ribbon 4–8 .
The participation of edge states in the electronic structure is significant when there is a
large fraction of the atoms residing on or near the newly created edges, but this diminishes
as the ribbons become wider, and eventually the electronic structure converges to that of
2
graphene.
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 investigated than
two dimensional (2-D) graphene or one dimensional (1-D) ribbons. Graphene nano-flakes
have a much larger number of configurational degrees freedom as they may be cut into a
much larger variety of different shapes, and possess corner states in addition to edge states.
In addition to this, graphene nano-flakes can potentially range in size from molecular to
semi-infinite 2-D structures 10 , 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 size and shape, particularly given that (while there are a limited number of edge structures) there are a large
number of different types of corners. The corners of graphene nano-flakes may be obtuse,
right or acute, and may be decorated by one, two, three, or four under-coordinated atoms.
In addition to the different electronic states of these corners, their reactivity and affinity
for functionalization will be entirely different. In a recent study it was shown that simple
hexagonal graphene nano-flakes can have three different types of obtuse 120◦ corners, each
with characteristic reconstructions that affect the electronic structure 15 .
However, the widespread use of graphene nano-flakes in modern nanotechnology will
be dependent upon development of reliable methods of producing these structures, and the
degree of control we can exercise during manufacturing. To become reliable components
in modern nanotechnology, graphene nano-flakes must be reasonably predictable, reproducible and stable. Moreover, given the economic constraints that will inevitably be applied
during production, these materials must also be robust against variations associated with
some degree of structural polydispersivity. The development of a complete understanding
of the consequences the deliberate or incidental shapes (and reconstructions) have upon
properties and property dispersion is essential, and can provide insight for the application
of graphene nano-flakes in future devices.
Due to the large number of configurational degrees of freedom, this represents a very
3
challenging undertaking, particularly if one is to consider a reasonable array of sizes, shapes
(edges and corners), edge/corner terminations, and a reasonable array of sizes; 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, shape, space charge 11,12 ,
interactions with other chemical groups 13 and temperature 14 . The most definitive way forward is to create monodispersed samples and systematically study the change in properties
as different morphological features are changed incrementally; a task that would require
an unrealistic amount of time and effort.
As mentioned above, a study of hexagonal graphene nano-flakes with unterminated,
monoyhydride or dihydride terminated ZZ and AC edges and corners has recently been
reported, based on density functional tight-binding simulations 15 . While this study had the
advantage of using sets of structures with exclusively ZZ or AC edges (effectively decoupling
the size/shape/structure dependencies), it was restricted to obtuse corners and could not
probe the properties one might expect if the corners were acute. Therefore, in this work we
have undertaken a complementary series of the comparable simulations on incremental sets
of trigonal graphene nano-flakes (from 22 or 897 atoms) that are also terminated exclusively
with AC or ZZ edges. This gives rise to different reconstructions, thermodynamic stability
and electronic states in the vicinity of the Fermi level, and the effect of benzene-style
monohydride terminations, or cyclohexane-style dihydride terminations on the edges and
corners has also been investigated. In particular, we find that AC edges and corner play
an important role in property dispersion, shifting the Fermi energy, the band gap, the
ionization potential and the electron affinity.
2
Computational Method
In this study we have used the density functional based tight-binding method with selfconsistent charges (SCC-DFTB) 16,17 which is a two-center approach to density functional
theory (DFT), that has been shown to be ideal for studying the electronic properties of
graphene 4 . In this approach, the Kohn-Sham density functional is expanded to second
4
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 explicitly calculated
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, and self-consistency is included at the level of Mulliken charges, as described in
reference 17. Although not strictly an observable quantity, the Mulliken charges in SCCDFTB are not extracted ex post facto, and form an integral part of the energy functional
which expresses local density fluctuations around a given atom. Mulliken charge fluctuations are calculated from the eigenvalue coefficients, and are algorithmically independent
from bonding considerations and spatial partitioning schemes. Therefore, in this method
they are very useful in illustrating bonding trends in (charged and neutral) heteronuclear
systems, and near the surface of molecules and clusters 18 .
We used the pbc set of parameters for C–C, C–H and H–H interactions, as developed
by Köhler and Frauenheim 19 . In this study we have only considered non-magnetic states.
The convergence criterion for a stationary point was 10−5 a.u. ≈ 0.5 meV/Å 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 11–13 , and means
that the present work is directly comparable with our previous studies.
3
3.1
Discussion of Results
Mechanical and Thermodynamic Stability
As mentioned above, there is one shape but there are two dominant structures in computational samples (as shown in figure 1), with either AC or ZZ edges. These intersect at
5
ZZ-type corners (when there are ZZ edges), or at AC-type corners (when there are AC
edges). This is different to the obtuse corners of hexagonal graphene nano-flakes, where
ZZ edges intersect at AC corners, and vice versa. Our AC structures range in size from
36 to 918 carbon atoms (∼1 nm to ∼6.5 nm in average diameter), and our ZZ structures
range from 22 to 897 carbon atoms (∼1 nm to ∼6.5 nm in average diameter). When the
edges and corners are unterminated, all of our virtual samples contain two different types
of carbon atoms hybridization, depending on their coordination, i.e. two-fold (sp1 ) and
three-fold (sp2 ) coordinated atoms. The atoms at the edges and corners are sp1 hybridized,
whereas the interior atoms are sp2 hybridized, and hence there are three kinds of bonds:
sp1 –sp1 , sp1 –sp2 , and sp2 –sp2 bonds (between all interior atoms). For AC nano-flakes,
three sp1 –sp1 bonds form a corner, and the carbon chains at the edge consist of alternating
sp1 –sp1 and sp1 –sp2 bonds. In contrast, the ZZ nano-flakes have sp1 –sp1 bonds at the
corners, but only sp1 –sp2 bonds along the edges. The sp2 -sp2 bonds locate in the inner
area for both shapes. Sample structures are provided in figure 1.
Figure 2 shows the average binding energy per atom (h∆Eb i) for each structure in the
sample sets, with respect to an infinite graphene sheet, calculated as:
h∆Eb i =
1
E(NC ) − NC µC .
NC
(1)
In this simple expression, E(NC ) is the total energy of the nano-flake extracted from the
simulations, µC is the chemical potential of a carbon atom in graphene, and NC is the
number of carbon atoms. If the edges and corners are un-terminated, as they would be
prior to deliberate functionalization, reconstructions in the vicinity of the sp1 hybridized
atoms lower the total energy of the nano-flake. This is captured in E(NC ).
Results of h∆Eb i for each of the trigonal nano-flakes, and the comparable hexagonal
nano-flakes 15 (for the purposes of comparison) are shown in figure 2. Here we can see the
higher energy of the trigonal nano-flakes with respect to hexagonal nano-flakes (of comparable size), and the smooth convergence of all sets toward planar graphene with increasing
size. This reduced stability with respect to hexagonal nano-flakes is due to the larger
fraction of two-fold coordinated atoms (sp1 atoms) around the edges and corners. Like
6
(a)
(b)
Figure 1: (a) The AC shape and (b) the ZZ shape.
7
Figure 2: The thermodynamic stability of the different sets of trigonal graphene nanoflakes, where results for the radical ZZ (AC) nano-flakes are shown by the closed blue
(red) symbols. Corresponding results for ZZ and AC hexagonal graphene nano-flakes are
provided for the purposes of comparison, with open symbols. The solid lines are to guide
to the eye. The thermodynamic stability of hexagonal nano-flakes is studied in more detail
in reference 15.
8
the hexagonal nano-flakes, the trigonal structures with AC edges/corners are marginally
more stable than the ZZ counterparts. This is because, while the AC nano-flakes have a
larger fraction of two-fold coordinated atoms around the circumference, these atoms break
aromaticity upon reconstruction, which offers some energetic compensation.
We have analysed the types of reconstructions occurring at the edges and corners of
these shapes, and find a number of consistent trends. Firstly, we found dilation in the
structures of AC and ZZ shapes. Overall, the edge-atoms contract toward the centre
of the nano-flake and the whole flake is dilated. The dilations of this type represent a
first-order (stretching) reconstruction. 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. A sp1 –sp1 bond is
significantly shortened from 1.42 Å (C–C bond length in an infinite graphene membrane)
to ∼1.3 Å or less in all cases. For example, in the AC-C630 (AC nano-flake consisting of
630 carbon atoms and 274 rings, as shown in figure 3(a)), the average sp1 –sp1 bond-length
is 1.26 Å. A sp1 –sp2 bond however, exhibited little in the way of reconstruction. In AC
(ZZ) sets, the average length of sp1 –sp2 bonds is 1.42 (∼1.41) Å, which is consistent with
sp2 –sp2 bond lengths. The reconstruction of sp1 –sp1 and sp1 –sp2 bonds unavoidably leads
to second-order reconstructions (bending), as the C–sp1 –C angle is enlarged with respect
to the ideal 120◦ . The average C–sp1 –C angle in the AC (ZZ) sets is found to be 124.6◦
(123.4◦ ), respectively. The shortened sp1 –sp1 bonds and enlarged C–sp1 –C angles have
the overall effect of smoothing the corners and straightening the chain of atoms along the
edges. Both first and second order reconstructions remain “in-plane”.
In addition to the first-order and second-order reconstructions described above, there
also exists third-order reconstructions, involving torsions. Torsions are “out-of-plane” reconstructions, which present as ripples, in order to disperse the forces and lower the total
energy. Out of the fifteen AC nano-flakes we investigated, eleven exhibited rippling upon
relaxation; but only one of the ZZ nano-flakes underwent the same reconstruction. We can
9
Figure 3: Top (upper) and side (lower) views of the various reconstructed structures. (a)
The flat reconstructed structure with W (C) = 0 for AC-C630 , (b) the rippled reconstructed
structure W (C) = 1.73 for AC-C630 , and (c) the slightly rippled reconstructed structure
with W (C) = 0.003 for ZZ-C781 . The initial “bulk-like” atomic configurations are shown
in grey, and the relaxed configurations are shown in yellow.
10
quantify the degree of rippling using the interfacial width W (C), which is defined as:
W (C) = [hz 2 (C)i − hz(C)i2 ]1/2 .
(2)
where z(C) is the magnitude of z coordinate of a carbon atom, perpendicular to the plane
of the flake. For the rippled reconstructed structure of the AC-C630 nano-flake, W (C) is
1.73, while for the rippled reconstructed of ZZ structure with 781 carbon atoms and 351
rings W (C) is only 0.003 (see figure 3(b) and (c)).
More information on the structure and distribution of different types of C–C bonds
in the 30 trigonal nano-flakes included in this study may be found in the Supplementary
Information.
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 fundamental
band gap (Eg ), which is calculated via the ionization potential (I) and electron affinity
(A):
Eg = E(N + 1) + E(N − 1) − 2E(N )
(3)
Eg = [E(N − 1) − E(N )] + [E(N + 1) − E(N )]
(4)
and,
=I −A
where E(M ) is the energy of the M electron system, I is the ionization potential and A is
the electron affinity 20 .
At this most fundamental level there is no mention of quantities related to band structure, but this does become important when we consider the mode of excitation and the
creation of electron hole pairs. If we describe the electronic structure of a solid in terms
of the band structure model an elementary excitation may be described as involving the
11
promotion of an electron from the top of the valence band to the bottom of the conduction
band. In this “one electron” or quasi-particle description of this process, the band gap is
effectively the difference in energy between the minimum energy state in the conduction
band and the maximum energy state in the valence band. Since the band gap of a semiconductor may be one of two types (direct, or indirect), the minimal-energy state in the
conduction band, and the maximal-energy state in the valence band are characterized by
a k-vector in the Brillouin zone. If the k-vectors are the same the gap is direct, requiring
only a photon to excite the electron into the conduction band; whereas if the k-vectors are
different the gap is indirect and a photon is required for the energy change and a phonon
is required for the momentum change (i.e. the change in k-vector).
The fundamental band gap for all of the trigonal nano-flakes is presented in figure 4(a),
along with the results for the sets of hexagonal nano-flakes reported in reference 15. Here
we can see that, unlike the hexagonal nano-flakes, the band gap of small trigonal nanoflakes is very small, and quickly closes as the size is increased (irrespective of chirality).
The closing of the band gap is due to additional edge and corner states occupying the gap,
and so only small trigonal nano-flakes are likely to exhibit semiconducting behavior. The
ionization potential and electron affinity for the trigonal nano-flakes are shown in 4(b) and
(c). Both exhibit convergence with increasing size, and the ionization potential is found to
be independent of the type of edges/corners. However, for the small trigonal nano-flakes,
the presence of AC or ZZ edges does have an influence on the electron affinity, which leads
to the difference at the fundamental band gap at the same sizes.
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 is also sensitive to the type of terminations (see figure
4 (d)). The Ef of the trigonal nano-flakes is lower than bulk graphene, but higher than
the comparable hexagonal nano-flakes. In previous work it was found that the Ef of the
hexagonal nano-flakes was largely independent of the type of edges, but was sensitive to
12
(a)
(b)
(c)
(d)
Figure 4: The convergence of (a) the fundamental band gaps, (b) the ionization potential,
(c) the electron affinity, and (d) the energy of the Fermi levels.
13
the type of corner reconstructions 15 (which gives rise to the fluctuations in figure 4(d)).
In the case of the trigonal nano-flakes, the types of corners are consistent at all sizes, but
differ between sets, so the size-dependent trends are smooth but displaced with respect
to one another. Once again, the structure of the nano-flakes provides the opportunity to
engineer the energy of the Fermi level.
To capture more information on the electronic structure of the trigonal nano-flakes, the
DOS is presented for the AC and ZZ structures in figure 5(a) and 5(b), corrected with
respect to the energy of the Fermi level (Ef , which is set to zero) for the purpose of clarity.
In each case, we see a size-dependent transition from the discrete molecular levels of C36
(figure 5(a)) and C22 (figure 5(b)), to the smooth band structure of C918 (figure 5(a)) and
C897 (figure 5(b)). However, there exists some significant differences in the DOS at the
Fermi level between the AC and ZZ nano-flakes. The DOS of the AC radicals in figure
5(a) is zero at the Fermi level, while the ZZ structures in figure 5(b) have a finite DOS
at the Fermi level (with the exception of the the smallest ZZ C22 nano-flake, which has a
discrete molecule-like structure). In the case of the hexagonal nano-flakes it was reported
that both AC and ZZ nano-flakes have a finite DOS at the Fermi level 15 , so it appears that
the trigonal AC radicals are exceptional.
Of course, electronic properties are not the only features that are influenced by shape
and the presence of acute or obtuse corners. A recent study of the vibrational and optical
properties of trigonal and hexagonal graphene nano-flakes with a small number of atoms
(<250), using classical molecular dynamics and time-dependent density functional theory
calculations 21 revealed that the band gap could be tuned to spans the full range of the
visible spectrum. The calculated UV-vis absorption spectra of trigonal AC nano-flakes
showed how the UV-vis maximum absorption energies and intensities scaled linearly with
edge length, and a nonzero net spin in the ZZ trigonal nano-flakes was found to increased
linearly with the length of the ZZ edges. In addition to this, the authors reported the
identification of shape- and border-related peaks in the infrared spectra that could be used
as “fingerprints” in the identification and isolation of these types of structures.
14
(a)
(b)
Figure 5: Electronic density of states of (a) the AC radicals and (b) the ZZ radicals.
15
Table 1: The calculated binding energy, Fermi energy, fundamental band gap, ionization
potential, and electron affinity (in eV) for trigonal graphene nano-flakes with different edge
structures (AC and ZZ) and different types of edge/corner terminations.
Nano-flake
C546
Radical
C546
C546
Monohydride Dihydride
C526
Radical
C526
C526
Monohydride Dihydride
hEb i
0.33
0.02
-0.03
0.36
0.04
0.05
Ef
-5.04
-4.45
-4.08
-5.38
-4.53
-3.77
Eg
0.003
0.003
0.002
0.002
0.002
0.001
I
0.01
0.008
0.007
0.01
0.008
0.006
A
-0.008
-0.006
-0.005
-0.009
-0.007
-0.005
3.3
Effect of Circumferential Passivation
Since the electronic properties of graphene nano-flakes are so dependent on the presence
and structure of the edges and corners, it is also important to consider the influence of
edge passivation. To test the impact of edge passivation we have terminated the edges of
two exemplary structures, the AC-C546 structure and the ZZ-C526 structure, with either
benzene-style monohydride terminations, or cyclohexane-style dihydride terminations on
the edges and corners. The structures were then re-optimized, and the results are listed
along with the unterminated radical structures in table 1.
We can see from these comparisons that the AC and ZZ trigonal nano-flakes are thermodynamically stabilized (by ∼0.3 eV) when the edges/corners are saturated with single
(or double) H atoms. Both monohydride and dihydride terminations result in a circumference of sp2 – (or sp3 –) hybridized C atoms, and average binding energies that are almost
the same as determined for “infinite” bulk graphene when in a bath of H2 . Alternative
reservoirs will give a different picture of thermodynamic stability, but the relative stability
will be consistent with these results.
Although hydrogen adsorption has an important influence on the thermodynamic and
mechanical properties, our results predict that it has little effect on the fundamental band
16
gap, ionization potential, and electron affinity, for both AC and ZZ converged structures.
When the nano-flakes are small, hydrogen adsorption has a minor effect on the ionization
potential and electron affinity; reducing the values for both. The effect on the fundamental
band gap is very small, but the change in the energy of the Fermi level is more significant.
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-graphene value.
Based on the shifts in the Fermi level, it is expected that the band structure in the
vicinity of the Fermi level will be altered. Therefore, the DOS for flat radical, monohydride, and di-hydride ZZ-C526 structures are directly compared in figure 6(a). In all
structures, the occupied bonding and unoccupied antibonding states are associated with
the s orbitals located at an energy of −2.2 and 1.4 eV, respectively. Here we can easily
see that the addition of hydrogen weakens both the bonding and antibonding states of the
s orbitals, and decreases the DOS in the vicinity of Fermi level significantly. Adsorption
of hydrogen atoms also strengthens the bonding and antibonding states of the sp hybrid
orbitals.
In addition to this, the DOS for radical, mono-hydride, and di-hydride AC-C546 structures are directly compared with corresponding hexagonal nano-flakes in figures 6(b), 6(c)
and 6(d), respectively. In the radical case we can see that there is a difference in the DOS in
the vicinity of the Fermi level, in both the first order bonding and anti-bonding states. The
radical trigonal structures has zero DOS, while the hexagonal structure has a finite DOS
in the vicinity of Fermi level. When the edges and corners are passivated, this difference
is eliminated, regardless of whether monohydride or dihydride terminations are applied.
Since both the trigonal and hexagonal nano-flakes have AC edges and the same number of
atoms, the differences in the DOS can be reliably attributed to the different shape and the
acute or obtuse corners. The shapes and corners affects the first order bonding (∼ −2 eV)
and anti-bonding states (1 eV to 2 eV), but the second order bonding (∼ −5.2 eV) and
anti-bonding (∼ −5.6 eV) remain largely unchanged. This is due to the main contribution
coming from the sp2 carbon atoms in the central area of the nano-flakes.
17
(a)
(b)
(c)
(d)
Figure 6: Electronic density of states graphene nano-flakes, comparing (a) the C526 trigonal
ZZ nano-flakes with each type of edge/corner terminations, (b) the C546 trigonal and hexagonal AC radial (bare) nano-flakes, (c) the C546 trigonal and hexagonal AC monohyrideterminated nano-flakes, and (d) the C546 trigonal and hexagonal AC dihyride-terminated
nano-flakes. In all cases the energy of the Fermi level has been set to zero. The electronic
density of states of hexagonal nano-flakes is studied in more detail in reference 15.
18
4
Conclusions
Presented here are results of density functional tight-binding simulations of sets of fully
reconstructed trigonal graphene nano-flakes, with exclusively zigzag or armchair edges and
corners around the circumference. The results show how the thermodynamic stability improves with increasing size, but is quite sensitive to the structure and chemical character
of the edges and corners. We find that trigonal nano-flakes with armchair edges are more
likely to be achievable experimentally than trigonal nano-flakes with zigzag edges, but this
shape is less stable than the hexagonal counterpart. 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 provided a means of structure/property engineering 3,22–25 .
If they were to be synthesized, the density of electronic states in the vicinity of the
Fermi level exhibit differences that may be technologically exploitable. Although the overall shape and type of corners can play an important role for the electronic properties,
stabilization with terminal groups passivating the edges and corners will likely eliminate
any shape-dependent advantage. This issue requires further experimental and theoretical
investigation, as the results do predict that the introduction of specific edges and corners,
and the adsorption of terminal groups 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
This project has been supported by the Australian Research Council under Discovery
Grant DP110101362. Computational resources for this project have been supplied by the
National Computing Infrastructure (NCI) national facility under MAS Grant e74.
19
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