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Edge stability, reconstruction, zero-energy states and magnetism in triangular
graphene quantum dots with zigzag edges
O. Voznyy,1 A. D. Gu¨c¸lu¨,1 P. Potasz,1, 2 and P. Hawrylak1
1Institute for Microstructural Sciences, National Research Council of Canada , Ottawa, Canada
2Institute of Physics, Wroclaw University of Technology, Wroclaw, Poland
(Dated: November 2, 2010)
We present the results of ab-initio density functional theory based calculations of the stability
and reconstruction of zigzag edges in triangular graphene quantum dots. We show that, while the
reconstructed pentagon-heptagon zigzag edge structure is more stable in the absence of hydrogen,
ideal zigzag edges are energetically favored by hydrogen passivation. Zero-energy band exists in
both structures when passivated by hydrogen, however in case of pentagon-heptagon zigzag, this
band is found to have stronger dispersion, leading to the loss of net magnetization.
Graphene, a single layer honeycomb lattice of carbon
atoms, exhibits fascinating properties due to relativistic-
like nature of quasiparticle dispersion close to the
Fermi level[1–5]. Graphene’s potential for nanoelectron-
ics applications motivated considerable amount of re-
search in graphene nanoribbons[6–10] and, more recently,
graphene quantum dots[11–24]. In such low-dimensional
structures, the character of the edge drastically affects
the electronic properties near the Fermi level[24–27].
In particular, assuming stable zigzag edge in triangu-
lar graphene quantum dots (TGQDs), tight-binding and
density functional theory based methods predicted col-
lapse of the energy spectrum near the Fermi level to a
shell of degenerate states, isolated from the rest of the
spectrum by a well defined gap[16–21, 23, 24]. It was
shown that in this band of degenerate states, strong
electron-electron interactions lead to ferromagnetism
with peculiar magnetic[19, 21] and optical[22] properties.
Recently, the potential of nanoscale graphene flakes for
use in photovoltaics was demonstrated[28]. Demonstra-
tion of multiexciton generation in carbon nanotubes[29,
30] suggests its possibility in other graphene-related ma-
terials. Combined with the possibility to control the
bandgap and with the presence of intermediate band in
the gap, it makes TGQDs an attractive material for third
generation solar cells[31].
The stability[8, 32–35], control over[10, 36, 37], and
physical effects[24–27] of edges in graphene structures
were studied experimentally and theoretically. It was
predicted[8, 32, 34] that in nanoribbons the zigzag
configuration (ZZ) is not necessarly the most stable,
but a transition to reconstructed edge, terminated by
pentagon-heptagon pairs (ZZ57), can occur. The ZZ57
reconstruction was also observed experimentally[33, 34,
38] in graphene boundaries. In general, confining Dirac
fermions in two-dimensions requires the understanding of
the role of the edges in finite area graphene based nanos-
tructures.
In this work, using ab-initio methods we investigate the
reconstruction of the edges and corners of TGQDs and
their effect on zero-energy states and magnetism. We
(a) (b)
(c) (d)
FIG. 1: (Color online) Triangular graphene quantum dot
edge configurations considered in this work: (a) Ideal zigzag
edges, ZZ, (b) ZZ57 reconstruction with pentagon-heptagon-
pentagon corner configuration, (c) ZZ57 reconstruction with
heptagon-hexagon-pentagon corner, and (d) ZZ57 reconstruc-
tion with hexagon-hexagon-pentagon corner.
focus on the competition between ZZ and ZZ57 config-
urations which were shown to be the most stable in pre-
vious works[8, 32, 34]. For hydrogen passivated edges,
we find ZZ structure to be the most stable. We show
that, in TGQDs, the reconstruction can occur in various
ways due to the presence of corners, and the most stable
reconstructions cause breaking of the reflection symme-
try of the triangular structure. Moreover, in hydrogen
passivated ZZ57 structures, zero energy states survive,
although the ferromagnetism is destroyed due to stronger
dispersion.
Calculations have been performed within the density
functional theory (DFT) approach as implemented in the
SIESTA code[39]. We have used the generalized gradient
approximation (GGA) with the Perdew-Burke-Ernzerhof
exchange-correlation functional (PBE)[40], double-ζ plus
polarization orbital (DZP) bases for all atoms (i.e. 2s, 2p
and 2d orbitals for carbon, thus, both σ- and pi-bonds are
included on equal footing) and Troullier-Martins norm-

32 4 6 8 10 12 14 16 18 20 22 24
-1
0
1
2
3
AFM


E
FM
-E
A
FM
(e
V
)
n
ZZ
ZZ
57
FM
FIG. 4: (Color online) Total energy difference between ferro-
magnetic and antiferromagnetic states as a function of the size
of the triangle for hydrogen-passivated ZZ (blue squares) and
ZZ57 (red circles). For ZZ the ground state is ferromagnetic
for all sizes studied, while for ZZ57 it is antiferromagnetic for
n > 4
Zero-energy states are localized exclusively on the sub-
lattice to which the ZZ edges belong and are exactly
degenerate within the nearest neighbor tight-binding
model[18, 19, 23]. Fig.3 compares the DFT electronic
spectra near the Fermi level for the ground states of
hydrogen-passivated unreconstructed (ZZ) and recon-
structed (ZZ57) TGQDs with n = 12. Introduction of
the ZZ57 edge reconstruction smears the distinction be-
tween sublattices. Nevertheless, the zero-energy band
survives in a reconstructed (ZZ57) TGQD. As can be
seen from the electronic density of occupied states of the
shell, they are still predominantly localized on the edges.
Moreover, the number of zero-energy states remains the
same. However, the dispersion of this band increases
almost three-fold due to reduction of the structure sym-
metry. Lifting of the band degeneracy becomes observed
even in the nearest neighbor tight-binding model with
equal hoppings, and is more pronounced for the struc-
tures in Fig.1(c) and (d) which additionally lift the re-
flection symmetry present in Fig.1(b).
Magnetization of the ZZ configuration was inves-
tigated in detail through meanfield[18, 19] and exact
diagonalization[21] calculations. It was shown that the
electrons in the zero-energy band are spin-polarized. The
up- and down-spin edge states are split around the Fermi
level such that only up-spin states are filled. Our calcu-
lated dispersion of the up-spin states is ∼0.03 eV/state
(Fig.3(a)). On the other hand, the ground state of ZZ57
configuration is antiferromagnetic, i.e. there is no split-
ting between the up- and down-spin states (Fig.3(b)).
Nevertheless, calculations for the ferromagnetic ZZ57 can
still be performed by adjusting the Fermi level for up and
down spins independently. The energy sectrum obtained
in such a way is similar to the one for ZZ but with neg-
ative ∆min. The interplay of the ∆max and the spin-up
band dispersion in such FM calculation can be monitored
to predict whether the ground state will be ferromagnetic
(∆min > 0) or antiferromagnetic (∆min < 0). Apart
from the significant increase of the band dispersion, we
note that the spin up-down splitting ∆max reduces by a
factor of two in ZZ57 structure. One can see from charge
density plot in Fig.3(b) that zero-energy states can now
populate both A and B sublattices even close to the cen-
ter of the dot (see outlined regions). We speculate that
the resulting reduction in the the peak charge density on
each site is responsible for the reduced on-site repulsion
between spin-up and spin-down electrons. Stronger dis-
persion and reduced up-down spin splitting favor kinetic
energy minimization versus exchange energy and destroy
the ferromagnetism in ZZ57. It should be noted that
partial polarization can still be possible in ZZ57. Par-
ticularly, we observed it for structures with symmetric
corners (Fig.1(b)) which exhibit smaller dispersion.
Our conclusions based on the analysis of the energy
spectra are supported by the total energy calculations
depicted in Fig.4 . For ZZ structure the gap ∆min is
always positive and the total energy of the FM config-
uration is lower than that of AFM (blue squares). For
ZZ57 configuration, on the opposite, the ground state
clearly remains AFM for all sizes with the exception of
the case with n=4. Here the band consists of only 3
states and their dispersion cannot overcome the splitting
between spin-up and spin-down states, resulting in FM
configuration beeing more stable. The total energy dif-
ference between the FM and AFM configurations for ZZ,
remains almost constant (in the range 0.3-0.5 eV) for the
triangle sizes studied here, and reduces with size if di-
vided by the number of edge atoms. Such a small value,
comparable to the numerical accuracy of the method,
makes it difficult to make reliable predictions regarding
magnetization of larger dots.
In order to investigate whether the magnetization of
the edges would be preserved on mesoscale we plot in
Fig.5 the evolution of the energy spectra with the TGQD
size. For this plot we performed an additional calculation
for the case of n = 40 (1761 carbon atoms total). We did
not perform the geometry optimization for this case due
to high computational cost, however, based on the results
for smaller structures we expect that this would have mi-
nor effect on the spectrum. This alows us to notice the
reduction of the splitting ∆max between the spin-up and
spin-down states with the growing size, not appreciated
previously[19]. Our GGA gap between zero-energy bands
(∆min) and that between the valence and conduction
bands are larger than LDA gaps reported previously[19],
as also observed for graphene nanoribbons[41]. Both
gaps show sublinear behavior, complicating the extrapo-
lation to triangles of infinite size. This behavior, however,
should change to linear for larger structures where the ef-
fect of edges reduces[22], converging both gaps to zero,
as expected for Dirac fermions. An important difference

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