OPEN
SUBJECT AREAS:
COMPUTATIONAL
SCIENCE
Bandgap Opening by Patterning
Graphene
Marc Dvorak, William Oswald & Zhigang Wu
TWO-DIMENSIONAL MATERIALS
ELECTRONIC PROPERTIES AND
DEVICES
ELECTRONIC AND SPINTRONIC
DEVICES
Received
18 March 2013
Accepted
10 July 2013
Published
26 July 2013
Correspondence and
requests for materials
should be addressed to
Z.W. (zhiwu@mines.
edu)
Department of Physics, Colorado School of Mines, Golden, CO 80401, USA.
Owing to its remarkable electronic and transport properties, graphene has great potential of replacing
silicon for next-generation electronics and optoelectronics; but its zero bandgap associated with Dirac
fermions prevents such applications. Among numerous attempts to create semiconducting graphene,
periodic patterning using defects, passivation, doping, nanoscale perforation, etc., is particularly promising
and has been realized experimentally. However, despite extensive theoretical investigations, the precise role
of periodic modulations on electronic structures of graphene remains elusive. Here we employ both the
tight-binding modeling and first-principles electronic structure calculations to show that the appearance of
bandgap in patterned graphene has a geometric symmetry origin. Thus the analytic rule of gap-opening by
patterning graphene is derived, which indicates that if a modified graphene is a semiconductor, its two
corresponding carbon nanotubes, whose chiral vectors equal graphene’s supercell lattice vectors, are both
semimetals.
G
raphene has risen as a fascinating system in condensed matter physics for not only fundamental science
but also technological applications1–4 because of its many unique and amazing properties originating
from its massless fermions2,5 due to the linear dispersion of energy bands near the Dirac points (K). For
example, under gate voltage charge carriers in graphene can be tuned continuously between electrons and holes1,
and their mobilities exceed 2.5 3 105 cm2V21s21 under ambient conditions6. This certainly makes graphene
extremely attractive for electronic and optoelectronic applications3,4; however, graphene is a semimetal with
bandgap (Eg) closing at K, which prohibits switching off the graphene channels in field-effect transistors and
building functional junctions in graphene optoelectronics.
Extraordinary efforts7–16 have been spent to create semiconducting graphene materials that maintain the
exceptional transport property. One scheme is obtaining bandgap through quantum confinement in graphene
nanoribbons7 (GNRs). But it is difficult to control the magnitude of bandgap in GNRs, since Eg is very sensitive to
ribbon edge and width; moreover, a narrow GNR, which is necessary for a sizable gap, cannot carry sufficiently
large currents. Other methods include substrate-induced8,9 and strain-induced10 bandgaps, but the former is
essentially not tunable, while the latter is practically limited by the ability of tuning Eg to the desired value. An
alternative approach is to periodically modify graphene17, which is crucial to preserve its anomalously high charge
carriers mobilities, by normal or inverse Stone-Thrower-Wales type of defects11,12, hydrogen passivation13, boron
and nitrogen doping14, and nanoscale holes creating graphene nanomeshes15–17 (GNMs), etc. In particular, Balog
et al.13 demonstrated the existence of bandgap in graphene by patterned hydrogen adsorption, and Bai et al.16
made functional GNM field-effect transistors capable of supporting currents 100 times of those in GNR-based
devices.
However, it is still a mystery that a periodically modulated graphene can either open up a substantial gap or
remain gapless (or a tiny gap) if its supercell lattice changes only slightly18–28, which complicates the creation of
bandgap by patterning and is apparently at odds with the proposal that bandgap in these graphene structures is
due to quantum confinement between neighboring modulated sites16. The quantitative description of quantum
confinement that results in energy gap in a patterned graphene is missing; instead, previous theoretical investigations18–28 mainly focused on some special patterns whose supercell lattices form rectangular or hexagonal
structures. A number of interesting empirical rules have been unraveled, but a fundamental and precise understanding on how electronic structures are affected by regularly patterning graphene has not yet been revealed.
Results
Analytical theory. In this work we adopt and extend the tight-binding model2, which describes the sp2-bonding in
graphene extremely well, to include periodic structural alterations modeled by an external potential U(r) applied
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on graphene29. We denote the primitive lattice vectors of the pristine
graphene by a1 and a2, and the supercell lattice vectors of the
patterned graphene can be written as
R1 ~n1 a1 zm1 a2
R2 ~n2 a1 zm2 a2
ð1Þ
with (n1, m1, n2, m2) four integers. If the perturbed sites form twodimensional Bravais lattices, then the external potential U(r) has the
same translational symmetry
U ðrÞ~U ðrzl1 R1 zl2 R2 Þ,
ð2Þ
where l1 and l2 are two integers.
At Dirac points K, the intrinsic tight-binding off-diagonal matrix
elements l0f(k) of the pristine graphene vanishes, giving rise to a zero
bandgap; therefore, a non-vanishing U(K), the Fourier transformation of U(r) at Dirac points will result in an energy gap. This leads to
the following bandgap opening rule:
n1 {m1 ~3p and n2 {m2 ~3q, ðGap openingÞ
ð3Þ
Otherwise,
ðGap closureÞ
where p and q are two integers. Our modeling is briefly summarized
in the Methods section, and the physical reasoning and mathematical
derivations are detailed in Supplementary Information.
Equation (3) indicates that only one ninth of arbitrarily patterned
graphenes are semiconducting while the rest remain semi-metallic;
thus the patterning periodicity must be carefully chosen to create a
gap. Furthermore, Equation (1) suggests that a patterned graphene
can be mapped to a pair of carbon nanotubes30 (CNTs) with chiral
vectors Ch equal to R1 and R2, respectively; then bandgap opening in
a patterned graphene has an exactly opposite correlation to that in
CNTs: it possesses a sizable bandgap only if both of its corresponding
CNTs are gapless at the tight-binding level.
We illustrate our analytical theory by discussing a crucial and
representative class of patterned graphene, namely GNMs. Fig. 1
plots various unit cells of GNMs, which are parallelograms with
different angles, side lengths, and orientations with respect to the
pristine graphene lattice. Because of the honeycomb structure of
graphene, 60u and 90u (rectangle) parallelograms are especially
important and have also been realized experimentally15,16. Like other
patterned graphenes, a GNM can be mapped to two CNTs. Since
unfolding a CNT generates a GNR, a GNM can also be mapped to a
pair of GNRs, as illustrated in Figs. 1g and 1h. Experimentally, GNRs
have been created by unzipping CNTs longitudinally31, whereas
CNTs could also be produced by wrapping GNRs around32. A
GNR and its corresponding CNT have similar dependency of electronic band structure on their transverse vectors, and the well-known
three-fold alternation in electronic structures is not surprising, given
the hexagonal symmetry of graphene2.
However, a GNM (or other types of patterned graphene)’s
dependence of bandgap opening on supercell lattice vectors is oppositely to that of its corresponding CNTs (or GNRs), because along
each lattice vector of a GNM the matching condition on Dirac points
is exactly the same as the circular confinement condition of a CNT. A
CNT is characterized by its chiral vector defined by a pair of integers
(n, m): Ch 5 na1 1 ma2. The tight-binding model asserts that a CNT
is semi-metallic if n 2 m ; 0 mod 3, or semiconducting otherwise30,
Figure 1 | Unit cells of GNMs. (a), (b) and (c) 60u parallelograms, with two lattice vectors along the zigzag (blue), armchair (gray) and chiral (green)
directions, respectively. (d) Oblique parallelogram (red). (e) and (f) Rectangles whose lattice vectors are along the zigzag and armchair directions,
respectively (yellow), or along two chiral directions (purple). Dangling bonds in GNMs and GNRs are passivated by hydrogen atoms (small gray dots),
while carbon atoms are denoted by brown dots. (g) and (h) The relationship among GNMs, GNRs and CNTs, as explained in the text.
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since its allowed k points satisfy k ? Ch 5 2pl with l integers.
Figs. 2d–f illustrate that only when these lines of k points pass
through graphene’s Dirac points K, a CNT remains semimetallic.
On the other hand, as shown in Figs. 2g–i, the reciprocal lattice
vectors of a GNM are the intersections of two sets of parallel lines
representing the allowed k points of its two matching CNTs. To
create a bandgap, the reciprocal lattice vectors of a GNM must
overlap the Dirac points, requiring both sets of parallel lines pass
through K; consequently, a semiconducting GNM corresponds to
two semimetallic CNTs.
First-principles calculations. In the following we demonstrate that
the above conclusion drawn from analytical modeling is in excellent
agreement with first-principles results. We present electronic structures obtained using the density functional theory (DFT33,34) for
GNMs; in addition, our DFT calculations on periodically H-passivated and boron-nitrogen doped graphene show similar results,
which will be published elsewhere. Although it is well known that
DFT can severely underestimate bandgaps in semiconductors and
insulators, DFT makes good predictions on the trends of electronic
band structures and bandgap35. In particular, DFT correctly predicts
a zero bandgap (closing at the Dirac points) for pristine graphene2,
and the predicted trends for bandgaps and band structures in
graphene-based nanostructures, such as CNTs and GNRs, are
mostly correct, compared with the many-body perturbation theory
within the GW approximation and experimental data7,30,36. If these
materials are semimetallic (or nearly semimetallic), DFT predicts
them to have a zero (or nearly zero) bandgap; on the other hand, if
they are semiconductors, DFT predicts sizable Eg, which tend to be
underestimated.
Figs. 2j–l compare DFT electronic band structures of three oblique
GNMs with (n1, m1, n2, m2) 5 (9, 23, 24, 9), (7, 4, 24, 8) and (9, 22,
24, 9), whose crystal structures are shown in Figs. 2g–i, respectively.
Although these three GNMs have same holes and similar unit cells,
only the middle one has a substantial gap because its two matching
CNTs are semimetallic. The left GNM corresponding to one metallic
and one semiconducting CNT, has a tiny gap at M2 and it is nearly
semimetallic; the GNM on the right remains gapless, with both of its
matching CNTs semiconducting. These first-principles results agree
well with our analytic theory, clearly demonstrating the sensitivity of
Figure 2 | Band gap opening/closure in GNMs and their corresponding CNTs. (a), (b) and (c) Crystal structures of a pair of CNTs in each panel,
whose chiral vectors are equal to the lattice vectors of GNMs plotted on the left side of (g), (h) and (i), and whose allowed k-points (parallel lines) are
plotted in (d), (e) and (f), respectively. The right side of (g), (h) and (i) show the reciprocal lattices of GNMs plotted in (a), (b) and (c), respectively. In
(d)–(i) the hexagons indicate the Brillouin zone of the pristine graphene, with 6 corners the Dirac points. (j), (k) and (l) Electronic band structures of
GNMs plotted in (g), (h) and (i), respectively. Brillouin zones of all GNMs in the text are plotted in Supplementary Fig. 2.
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band structure to patterning periodicity, whose small variation could
modify electronic structures significantly. They also suggest that in a
GNM the top valence bands and bottom conduction bands derived
from the p orbitals are nearly symmetric with respect to the Fermi
energy (or the middle of energy gap), consistent with the tight-binding model for graphene in which only the nearest-neighbor interactions are taken into account.
Next we discuss rectangular and hexagonal GNMs, which have a
one-to-one correlation to CNTs, because their lattice vectors R1 and
R2 are not independent. R1 and R2 of a rectangular GNM form the
chiral and translational (T) vectors of a CNT subject to an integer
multiplier j: R1 5 Ch and R2 5 jT, or R1 5 jT and R2 5 Ch. Here T 5
t1a1 1 t2a2, with t1 5 (2m 1 n)/t, t2 5 2(2n 1 m)/t, and t the greatest
common divisor of 2m 1 n and 2n 1 m. If n 2 m is not a multiple of
3, then t1 2 t2 ; 0 mod 3; thus at least n1 2 m1 or n2 2 m2 is divisible
by 3. We set n2 2 m2 ; 0 mod 3, so that the existence of a bandgap in
a rectangular GNM depends only on R1 by the difference between n1
and m1: it is semiconducting if n1 2 m1 ; 0 mod 3 or semi-metallic
otherwise, exactly in contrast to a CNT with Ch 5 R1, as summarized
in Fig. 3i. The DFT band structures of two rectangular GNMs
(Figs. 3a and 3b) with (n1, m1, n2, m2) 5 (6, 3, 24, 5) and (6, 4,
27, 8) are plotted in Figs. 3e and 3f. In both cases n2 2 m2 are
divisible by 3; the former is a semiconductor with a direct gap at
the C-point, while the latter is semimetallic with gap closing at the
C2M2 segment. In fact, all semiconducting GNMs open up a direct
bandgaps at the C point.
In a rectangular GNM, R1 and R2 are perpendicular to each other
with conjugate chiralities so that at least one of its two matching
CNTs are metallic, and then one third of rectangular GNMs are
semiconducting. On the other hand, a 60u parallelogram GNM’s
lattice vectors R1 and R2 have the same chirality subject to integer
multipliers. In general we cannot assign a pair of integers to characterize a 60u parallelogram GNM, because R1 and R2 might be
different in length. For R1 and R2 with the same length, a 60u parallelogram becomes a 60u rhombus and such GNMs have the hexagonal symmetry, in which n1 2 m1 and n2 2 m2 are either both
divisible or neither divisible by 3. Therefore the existence of a bandgap in a hexagonal GNM also depends only on R1 by n1 2 m1, and
one third of them possess bandgaps, compared with only one ninth
of 60u parallelogram GNMs semiconducting. Figs. 3g and 3h plot
DFT band structures of two hexagonal GNMs (Figs. 3c and 3d) with
(n1, m1, n2, m2) 5 (5, 4, 9, 25) and (7, 1, 8, 27), respectively. As
predicted by Eq. (3), the former is semimetallic, while the latter is a
semiconductor.
Previous theory and simulations18–28 studied many special GNMs,
namely the zigzag (Fig. 1a), armchair (Fig. 1b), and zigzag-armchair
(Fig. 1e) GNMs, in which R1 and R2 are both along the zigzag or
armchair direction, or along zigzag and armchair directions respectively. A zigzag GNM is characterized by a pair of integers (Z1, Z2)
with (n1, m1, n2, m2) 5 (Z1, 0, 0, Z2), an armchair by (A1, A2) with (n1,
m1, n2, m2) 5 (2A1, 2A1, A2, A2) and a zigzag-armchair by (Z, A)
with (n1, m1, n2, m2) 5 (Z, 2Z, A, A). The bandgap opening rules for
these special cases as functions of structural parameters A and Z are
presented in Figs. 4e, 4f and 4j, respectively. One third of zigzagarmchair GNMs are semiconducting, which is independent of the
value of A and depends only on whether Z ; 0 mod 3. A zigzag GNM
acquires a bandgap if both Z1 ; 0 mod 3 and Z2 ; 0 mod 3, so only
one ninth of them are semiconducting, whereas all armchair GNMs
Figure 3 | Rectangular and hexagonal GNMs. (a) and (b) Crystal structures of two rectangular GNMs; (e) and (f) the corresponding electronic band
structures. (c) and (d) Crystal structures of two hexagonal GNMs; (g) and (h) the corresponding electronic band structures. (i) Bandgap opening (red)
and closure (blue) for rectangular and hexagonal GNMs (hexagons) and CNTs (circles) characterized by (n1, m1) for vector R1 of GNMs and Ch of CNTs,
respectively, i.e., Ch 5 R1 5 n1a1 1 m1a2. Here the solid arrows in (i) denote the zigzag direction, while the dashed line indicates the armchair vector.
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are semiconductors regardless of the values of A1 and A2. These
predictions have been verified by previous18–28 and present DFT calculations whose results are plotted in Figs. 4a–d, 4g and 4h.
Comparing band structures plotted in Figs. 2–4, one finds that for
the same symmetry of patterning, semiconducting graphene alternates with semimetallic ones, except for the armchair case where all
of them are semiconductors. Apparently, the armchair structure is
the preferred choice to open bandgap by patterning graphene.
We have carried out additional DFT calculations for electronic
structures of other patterned graphenes with a variety of unit cells
and defect sizes other than what reported in Figs. 2–4. All these
results (in Supplementary Information) agree with Equation (3) on
bandgap existence. Furthermore, the magnitude of bandgap of a
semiconducting patterned graphene depends on the onsite perturbation strength and inter-site distances; in particular, Eg of GNMs as
functions of hole size and unit-cell area can be well fitted to a simple
scaling rule proposed by Pedersen et al.18, as demonstrated in Fig. 4i.
Hence once a semiconducting patterning is chosen, Eg can be tuned
to the ideal values for target applications by adjusting the types and
sizes of and distances between structural modulations.
Discussion
Using a perturbative tight-binding model we have derived a universal
bandgap opening rule for graphene with arbitrary Bravais periodic
patterning. A simple reverse relation on existence of bandgap
between a modified graphene and its two analogous CNTs is found,
which is confirmed by first-principles electronic structure calculations. Current modeling can be easily generalized from Bravais to
non-Bravais supercells, such as the honeycomb structure, and thus
the DFT results for the honeycomb GNMs26 can also be well
explained. We conclude that bandgap opening in patterned graphene
is not directly caused by quantum confinement as in CNTs or GNRs,
though confinement still plays a role in determining the magnitude
of Eg once a gap is present; instead, it originates from geometric
symmetry breaking at the Dirac points, leading to the exactly opposite bandgap opening conditions between modified graphene and
CNTs.
Thus our present work has solved an outstanding problem on how
to properly pattern graphene to induce bandgaps, pointing to practical routes of geometrically designing graphene with controllable
and desirable electronic behaviors. Previous experiments13–17 have
demonstrated that patterning graphene could open up a substantial
bandgap suitable for transistors, but none of them have yet reported
the predicted sensitivity of electronic behavior to patterning periodicity. Their patterned graphene structures are rather rough, which
might greatly hamper transport of charge carriers, while a high-quality accurate periodic patterning is expected to largely maintain the
ultrahigh electron mobility. This requires the ability of tailoring
Figure 4 | Zigzag, armchair, and zigzag-armchair GNMs. Electronic band structures of two zigzag (a, b), armchair (c, d), and zigzag-armchair
(g, h), GNMs. Their crystal structures are plotted in Supplementary Fig. 3. (e), (f) and (j). Band gap opening (red) and closure (blue) for zigzag (e),
armchair (f) and zigzag-armchair (j) GNMs characterized by (Z1, Z2), (A1, A2) and (A, Z), respectively. (i) Bandgap (eV) as a function of hole size
indicated by the number of removed carbon atoms (Nremoved) per unit cell for a zigzag (red), an armchair (green) or a zigzag-armchair (black) GNM.
Computed data (closed circles) are fitted to smooth curves, as described in Supplementary Information.
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graphene structures precisely at the atomic scale37 – an exciting and
rapidly advancing frontier in nanoscience and nanotechnologies for
realizing electronics and optoelectronics towards ever smaller length
scales.
Finally, the restriction of our modeling, as well as the key to its
success, is that the valence and conduction bands in graphene touch
only at the Dirac points. In other materials with vanishingly small
density of states near the Fermi energy or, equivalently, bandgap
closing only at a small discrete set of k points, it is possible to perturb
all these touching k points by patterning. Moreover, it is appealing to
investigate the effects of periodic perturbation and structural symmetry breaking on strongly-correlated materials showing intriguing
quantum/topological orders, such as topological insulators38, whose
gapless surface states are protected by time-reversal symmetry and
the topology of the material.
Methods
First-principles calculations. Our first-principles electronic structure calculations
based on DFT were performed employing the SIESTA package39 using a triple-f
polarized atomic basis set for carbon atoms, whose numerical accuracy has been
rigorously examined against the planewave-based VASP program40. The twodimensional Brillouin zones of GNMs were sampled on a 2 3 2 Monkhorst-Pack kgrid with good convergence due to very large unit cells. The generalized gradient
approximation41 was used for the exchange-correlation functional. All calculations
are spin-polarized. GNM structures are optimized until the maximum atomic force is
less than 0.02 eV/Å. Dangling bonds in GNMs are passivated by hydrogen atoms.
Tight-binding model. Graphene has two sublattices (A and B) whose Hamiltonian is
written as
E0
l0 f ð k Þ
HK ~
,
ð4Þ
l0 f ð k Þ E 0
where E0 and l0 are two parameters. The eigenvalues are
E+ ðk Þ~E0 +l0 jf ðk Þj,
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
pffiffiffi
pffiffiffi 3
3
ky a cos kx a ,
jf ðkÞj~ 3z2 cos 3ky a z4 cos
2
2
ð5Þ
ð6Þ
with a < 1.42 Å, the neighboring carbon-carbon distance. The structural
modifications are modeled by an external periodic potential U(r). If the scattering
matrix element between two degenerate pseudo-spin states at the Dirac points
l1 ðKÞ~N hwA jU ðKÞjwB i,
ð7Þ
is non-zero, where wA and wB are orbitals for a pair of the nearest neighbors, N is the
effective average number of the nearest neighbors per site, and U(k) is the Fourier
transformation of U(r), or the scattering matrix element between two degenerate
states at Dirac points K and K942,43
jhKjUr jK’ij2 ~jU ðK’{KÞj2 cos2 ðhK,K’ =2Þ,
ð8Þ
is non-zero, where hK,K9 is the angle between K and K9, then there is a sizable bandgap
in the defected graphene. Eqs. 7 and 8 lead to the same bandgap opening condition of
U(K) ? 0.
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(1996).
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made simple. Phys. Rev. Lett. 77, 3865 (1996).
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back scattering. J. Phys. Soc. Japan 67, 1704–1713 (1998).
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Acknowledgements
Z.W. gratefully acknowledges financial support from USA Department of Energy (DOE)
Early Career Award (No. DE-SC0006433). Computations were carried out at the Golden
Energy Computing Organization (GECO) at Colorado School of Mines and at National
Energy Research Scientific Computing Center (NERSC).
6
www.nature.com/scientificreports
Author contributions
Competing financial interests: The authors declare no competing financial interests.
Z.W. conceived and designed this project and derived the theoretical modeling. M.D. was
involved in analytical derivations and performed first-principles calculations. W.O. carried
out some initial calculations. Z.W. and M.D. wrote the paper.
How to cite this article: Dvorak, M., Oswald, W. & Wu, Z. Bandgap Opening by Patterning
Graphene. Sci. Rep. 3, 2289; DOI:10.1038/srep02289 (2013).
Additional information
This work is licensed under a Creative Commons AttributionNonCommercial-ShareAlike 3.0 Unported license. To view a copy of this license,
visit http://creativecommons.org/licenses/by-nc-sa/3.0
Supplementary information accompanies this paper at http://www.nature.com/
scientificreports
SCIENTIFIC REPORTS | 3 : 2289 | DOI: 10.1038/srep02289
7
Bandgap Opening by Patterning Graphene:
Supplementary Information
Marc Dvorak, William Oswald & Zhigang Wu
Department of Physics, Colorado School of Mines, Golden, CO 80401
I. Tight-Binding Model for Pristine Graphene
As shown in Fig. 1, graphene has two sublattices (A and B) with the tight-binding model Hamiltonian1
written as
H
(0)
k
⎛ E
⎜
= ⎜⎜⎜ 0 *
⎜⎝ λ0 f (k)
λ0 f (k) ⎞⎟⎟
⎟⎟ ,
⎟⎟
E0
⎠
(1)
where E 0 and λ0 are two parameters, defined by
E 0 ≡ φA H (0) φA = φB H (0) φB ,
(2)
where φA and φB are orbitals on the A-site and B-site, and
λ0 ≡ φA H (0) φB ,
(3)
with φA and φB a pair of the nearest-neighbor orbitals. The eigenvalues are
E ±(0)(k) = E 0 ± λ0 f (k) ,
(4)
Supplementary Figure 1 | The lattice structure (Left) and the Brillouin zone (Right) of graphene.
Here a1 and a 2 are the lattice vectors in real space, while b1 and b2 are reciprocal lattice vectors. The
Dirac points, K and K′ , are located on the six corners of the Brillouin zone. Here the black and grey
dots on the left panel represent the A- and B-sublattices, respectively.
1
and
⎛ 3
⎞⎟ ⎛ 3 ⎞
⎜
f (k) = 3 + 2cos( 3kya) + 4 cos ⎜⎜ kya ⎟⎟⎟ cos ⎜⎜⎜ kxa ⎟⎟⎟ ,
⎜⎜⎝ 2
⎟⎠ ⎝ 2 ⎟⎠
(5)
with a ≈ 1.42 Å, the neighboring carbon-carbon distance. At the Dirac points K and K′ , f (K) =
f (K′) = 0, leading to bandgap closure.
II. Perturbative Model for Graphene under Periodic Potential
We model the effects of periodic structural modifications on graphene by a periodic potential U(r) :
U(r) =U(r +  1R1 +  2R 2 ) ,
(6)
where  1 and  2 are two integers, while R1 and R 2 are lattice vectors defining the periodicity of the
applied periodic potential (structural modifications):
R1 = n1a1 + m1a 2,
R 2 = n2a1 + m2a 2 .
(7)
Here n1 , m1 , n2 , and m2 are integers, and a1 and a 2 are the primitive lattice vectors of the pristine
graphene:
⎛ 3 1 ⎞⎟
⎜
a1 = 3a ⎜⎜ , ⎟⎟⎟,
⎜⎜⎝ 2 2 ⎟⎠
⎛ 3
1 ⎞⎟
⎜
a 2 = 3a ⎜⎜ , − ⎟⎟⎟.
⎜⎜⎝ 2
2 ⎟⎠
(8)
We denote the periodic potential in reciprocal space by U(k) :
+∞ +∞
1
−ik⋅r
U(k) =
∫ ∫ U(r)e dx dy .
2π −∞
−∞
(9)
Treating the periodic potential as a perturbation to the pristine graphene, the changes in the diagonal
matrix elements can be approximated by
E1(k) ≡ φA U(k) φA = φB U(k) φB ,
(10)
while the changes in the off-diagonal matrix elements are approximated by
λ1(k) ≡ N φA U(k) φB .
(11)
Here φA and φB are orbitals for a nearest-neighbor pair, and N is the effective average number of the
nearest neighbors per site. The structural modifications, such as nanoscale holes and partial passivation,
destroy the local graphene structure, so that only the long-range periodicity induced by such modulations
remains in real space. As a result, E1(k) and λ1(k) possess the translational symmetry of U(k) in recip2
rocal space, not the translational symmetry of the pristine graphene. In our model, both the diagonal and
the off-diagonal matrix elements are perturbed by the external potential. This is because the periodic
structural modifications not only change the onsite energy but also the hopping parameter. For example,
in a graphene nanomesh, near hole edges the hopping between the nearest A and B sites are affected: both
the hopping strength and the number of the nearest neighbors are modified.
Obviously, at Dirac points if λ1(K) ≠ 0 [ ⇒U(K) ≠ 0 ], i.e., the scattering between the degenerate
pseudo-spin (sublattices A and B) portion of the wave function is non-zero, then the degeneracy is lifted at
Dirac points, leading to bandgap opening. In addition, the scattering matrix element between two states
with different wave vectors is2,3
k U(r) k′
2
= U(k′ − k) cos2 (θk,k′ / 2) ,
2
(12)
where θk,k′ is the angle between k and k′ . If the scattering between degenerate states at Dirac
points K and K′ is non-zero, i.e., U(K′ − K) ≠ 0 , then the degeneracy at the Dirac points can be lifted as well. Interestingly, because K′ is equivalent to 2K , the second condition of bandgap opening is
equivalent to the first condition of U(K) ≠ 0 .
III. Bandgap Opening Rule
In order to find the analytical relation between the existence of bandgap and the translational symmetry of
structural modifications, we employ a periodic potential U(r) using the δ-function,
U(r) =
+∞
∑
α,β=−∞
gδ(r − αR1 − βR 2 ) ,
(13)
where parameter g indicates the strength of the external potential, with α and β integers. This is truly a
simplest model describing only the translational symmetry of the structural modifications on graphene,
but it is sufficient for investigating whether the gap remains closed at the Dirac points. Within this model,
we have derived that bandgap opening requires U(K) ≠ 0 , i.e.,
⎪⎧⎪U(K) = 0,
⎨
⎪⎪⎩U(K) ≠ 0,
Bandgap closure
Bandgap opening
(14)
Next, we evaluate U(k) , the Fourier transform of U(r) :
+∞ +∞
+∞ +∞
1
g +∞
−ik⋅r
U(k) =
U(r)e
dx
dy
=
δ(r − αR1 − βR 2 )e −ik⋅r dx dy .
∑
∫∫
∫
∫
2π −∞
2π
α,β=−∞ −∞ −∞
−∞
Here we can express R1 and R 2 in terms of lattice constant a :
3
(15)
⎡
⎤
⎡
⎤
3
3
3
3
r − αR1 − βR 2 = ⎢x − α (n1 + m1 )a − β (n2 + m2 )a ⎥ x̂ + ⎢⎢y − α
(n1 −m1 )a − β
(n2 −m2 )a ⎥⎥ ŷ
⎢
⎥
2
2
2
2
⎢⎣
⎥⎦
⎣
⎦
= (x − αx1 − βx 2 ) x̂ + (y − αy1 − βy 2 )ŷ,
where x1 ≡
3
3
3
3
(n1 + m1 )a, x 2 ≡ (n2 + m2 )a, y1 ≡
(n1 −m1 )a, and y 2 ≡
(n −m2 )a.
2
2
2
2 2
δ (r − αR1 − βR 2 ) = δ (x − αx1 − βx 2 ) δ (y − αy1 − βy 2 ).
Therefore,
(16)
Then,
+∞
⎤ ⎡ +∞
⎤
g +∞ ⎡⎢
−ikx x
⎥ ⎢ δ y − αy − βy e −ikyy dy ⎥
δ
x
−
αx
−
βx
e
dx
∑ ∫ (
⎥ ⎢∫ (
⎥
1
2)
1
2)
2π α,β=−∞ ⎢⎢−∞
⎥⎦ ⎢⎣−∞
⎥⎦
⎣
g +∞ −ikx (αx1 +βx 2 ) −iky (αy1 +βy2 )
=
e
∑ e
2π α,β=−∞
g ⎡⎢ +∞ −iα(kx x1 +kyy1 ) ⎤⎥ ⎡⎢ +∞ −iβ(kx x 2 +kyy2 ) ⎤⎥
=
∑e
∑e
⎥
⎥ ⎢ β=−∞
2π ⎢⎣ α=−∞
⎦⎣
⎦
+∞
+∞
⎤
⎤⎡
g ⎡⎢
=
2π ∑ δ(kx x1 + kyy1 − 2πα)⎥ ⎢2π ∑ δ(kx x 2 + kyy 2 − 2πβ)⎥
⎥
⎥ ⎢ β=−∞
2π ⎢⎣ α=−∞
⎦⎣
⎦
U(k) =
= 2πg
+∞
∑
α,β=−∞
δ(kx x1 + kyy1 − 2πα)δ(kx x 2 + kyy 2 − 2πβ).
For U(k) ≠ 0 , k points have to satisfy the following equations:
⎧
⎪
kx x1 + kyy1 = 2πα,
⎪
⎪
⎨
⎪
k x + kyy 2 = 2πβ.
⎪
⎪
⎩ x 2
(17)
Eq. 17 defines a grid of perturbed k points at which U(k) ≠ 0 , as plotted in Fig. 2 in the paper. When
these k points include the Dirac points K and K′ , we expect band gap opening. The Dirac points in the
Brillouin zone are
⎛ 2π
2π ⎞⎟
⎟⎟ ,
K = ⎜⎜⎜ , ±
⎜⎝ 3a
3 3a ⎟⎠
(18)
and we insert these into Eq. 17:
2π
2π
x1 ±
y1 = 2πα;
3a
3 3a
2π
2π
x2 ±
y 2 = 2πβ .
3a
3 3a
Plugging in the expressions for x1, y1, x 2, and y 2 ,
4
(19)
⎡
⎢
⎢
⎢⎣
⎤
2π ⎡⎢ 3
2π ⎡⎢
(n2 + m2 )a ⎥ ±
⎥ 3 3a ⎢⎢
3a ⎢⎣ 2
⎦
⎣
⎤
2π ⎡⎢ 3
2π
(n1 + m1 )a ⎥ ±
⎢
⎥
3a ⎣ 2
⎦ 3 3a
⎤
3
(n1 −m1 )a ⎥⎥ = 2πα,
2
⎥⎦
⎤
3
(n2 −m2 )a ⎥⎥ = 2πβ,
2
⎥⎦
(20)
which reduce to
1
(n1 + m1 ) ± (n1 −m1 ) = 2α,
3
1
(n2 + m2 ) ± (n2 −m2 ) = 2β.
3
(21)
Eq. 21 is satisfied for integers α and β only if
n1 −m1 = 3p
and n2 −m2 = 3q ,
(22)
where p and q are integers. Thus we have derived the crucial bandgap opening rule (Eq. 3) in the paper.
IV. Dependence of Bandgap in GNMs on Hole Size
Our derivation is general enough that it likely holds for several different perturbative potentials that are
reasonably localized. We have studied a series of holes in graphene nanomeshes (GNMs), and our results
suggest that as hole size is increased, the bandgap in semiconducting GNMs increases. If a bandgap exists, magnitude of Eg is primarily determined by hole area and the supercell area. We used the following
scaling rule proposed by Pedersen et al.4,
Eg = g
α
N removed
N total
,
(23)
to fit our bandgap data. Originally the parameter α is set to be 0.5, and we found that α ≈ 0.3 gave the
best fitting to GNM with fixing unit cells and varying hole sizes (the number of missing C atoms in a
supercell), as plotted in Fig. 4i in the paper and Fig. 4b in Oswald et al.’s work5, respectively.
5
V. Brillouin Zones
Supplementary Figure 2 | Brillouin Zones for GNMs. Left to right, top to bottom, the (n1,m1;n2,m2 )
coordinates for these GNM structures presented in the paper are (6, 3; −4, 5), (6, 4; −7, 8), (8, −4; 4, 4),
(10, −5; 6, 6), (5, 4; 9, −5), (11, 0; 0, 9), (12, 0; 0, 12), (9, −2; −4, 9), (9, −3; −4, 9), (7, 1; 8, −7), (6, −6; 4,
4), (7, −7; 4, 4), and (7, 4; −4, 8).
6
VI. Zigzag, armchair and zigzag-armchair GNMs
Supplementary Figure 3 | Zigzag, armchair and zigzag-armchair GNMs. Crystal structures of two
zigzag (a, b), armchair (c, d) and zigzag-armchair (e, f) GNMs, whose electronic band structures are plotted in Figure 4 in the paper.
7
VII. Summary of Bandgap Results
n1
m1
n2
m2
Hole Size
TB Gap?
DFT Gap (eV)
10
9
10
6
11
12
3
4
4
5
6
6
7
8
9
−10
−9
−10
−6
−11
−12
2
1
3
1
3
4
2
2
6
4
7
7
6
8
8
−7
−6
−10
−7
−4
−7
−11
−6
−7
4
7
7
6
8
8
8
9
11
11
5
8
16
9
8
24
24
24
24
24
24
6
6
6
6
6
6
6
6
6
N
Y
N
Y
N
Y
N
Y
N
N
Y
N
N
Y
Y
0
0.41
0
0.698
0
0.273
0
0.62
0
0
0.61
0
0
0.35
0.26
Zigzag
10
11
12
12
12
0
0
0
0
0
0
0
0
0
0
10
12
12
12
12
24
24
6
24
54
N
N
Y
Y
Y
0
0
0.21
0.38
0.5
Armchair
8
8
10
10
14
14
14
14
−4
−4
−5
−5
−7
−7
−7
−7
6
5
5
6
8
8
8
8
6
5
5
6
8
8
8
8
24
24
24
24
6
24
54
96
Y
Y
Y
Y
Y
Y
Y
Y
1.05
0.78
0.68
0.55
0.18
0.32
0.41
0.41
60º Parallelogram
3
5
7
5
4
1
8
9
8
−3
−5
−7
24
24
24
N
N
Y
0
0
0.98
9
9
9
9
9
9
10
10
7
−1
−2
−3
−1
−2
−3
1
2
4
−4
−4
−4
−4
−4
−4
2
2
−4
9
9
9
9
9
9
9
9
8
6
6
6
24
24
24
24
24
24
N
N
N
N
N
N
N
N
Y
0
0
< 0.05
0
0
< 0.1
0
0
0.62
Zigzag-Armchair
Rectangular
Chiral Rectangular
Oblique
Parallelogram
8
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9