Substrate-induced bandgap in graphene on hexagonal boron nitride
Substrate-induced bandgap in graphene on hexagonal boron nitride
Gianluca Giovannetti1,2, Petr A. Khomyakov2, Geert Brocks2, Paul J. Kelly2 and Jeroen van den Brink1
Instituut-Lorentz for Theoretical Physics, Universiteit Leiden,
P.O. Box 9506, 2300 RA Leiden, The Netherlands
Faculty of Science and Technology and MESA+ Institute for Nanotechnology,
University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands.
(Dated: October 22, 2018)
We determine the electronic structure of a graphene sheet on top of a lattice-matched hexagonal
boron nitride (h-BN) substrate using ab initio density functional calculations. The most stable
configuration has one carbon atom on top of a boron atom, the other centered above a BN ring.
The resulting inequivalence of the two carbon sites leads to the opening of a gap of 53 meV at the
Dirac points of graphene and to finite masses for the Dirac fermions. Alternative orientations of the
graphene sheet on the BN substrate generate similar band gaps and masses. The band gap induced
by the BN surface can greatly improve room temperature pinch-off characteristics of graphene-based
field effect transistors.
PACS numbers: 71.20.-b, 73.22.-f, 73.20.-r
Introduction Less than 3 years ago it was discovered
that graphene – a one-atom-thick carbon sheet – can be
deposited on a silicon-oxide surface by micromechnical
cleavage of high quality graphite [1]. The graphene flakes
are micrometers in size, sufficiently large to have con-
tacts attached so as to construct field effect transistors
(FETs). Electrical transport measurements made clear
that at room temperature graphene has an electron mo-
bility of at least 10 000 cm2/Vs, a value ten times higher
than the mobility of silicon wafers used in microproces-
sors [1, 2, 3]. The high mobility is not much affected by
a field-induced excess of electrons or holes.
A graphene sheet has a honeycomb structure with two
crystallographically equivalent atoms in its primitive unit
cell. Two bands with pz character belonging to different
irreducible representations cross precisely at the Fermi
energy at the K and K ′ points in momentum space. As
a result undoped graphene is a zero-gap semiconductor.
The linear dispersion of the bands results in quasiparti-
cles with zero mass, so-called Dirac fermions. At energies
close to the degeneracy point the electronic states form
perfect Dirac cones. The absence of a gap, preventing
the Dirac fermions from attaining a finite mass and com-
plicating the use of graphene in electronic devices [4], is
related to the equivalence of the two carbon sublattices
of graphene.
The relativistic nature of the Dirac fermions gives
rise to counterintuitive phenomena. One, known as the
Klein paradox, is that relativistic electrons exhibit per-
fect transmission through arbitrarily high and wide po-
tential barriers. This effect is related to an unwanted
characteristic of graphene FETs, namely that pinch-off
is far from complete [5]. If one applies a gate voltage
so that either holes or electrons are injected into the
graphene sheet, the FET is open and its conductivity
high. One can then try to block the current by tun-
ing the gate voltage to move the graphene layer towards
the charge neutrality point where the Fermi energy co-
incides with the Dirac points; at this energy the density
of states vanishes and nominally there are no carriers
present. However, it turns out that in spite of the lack of
electronic states the conductivity does not vanish in this
case. Rather, it assumes the minimal value σmin = 4e
where h is Planck’s constant and e the unit of charge.
Thus even when pinched-off to its maximum the FET
still supports an appreciable electrical current, which is
intrinsic to graphene and related to the fact that the
Dirac fermions are massless [2, 3, 4, 5, 6, 7].
Inducing a gap The poor pinch-off can only be reme-
died by generating a mass for the Dirac fermions. A
number of possibilities exist to do so. One is to use
bi-layer graphene which will have a gap if the top and
bottom layers are made inequivalent, for instance by ap-
plying a bias potential [5]. Another is the use of graphene
nanoribbons, where gaps arise from the lateral constric-
tion of the electrons in the ribbon. The size of the gap
then depends on the detailed structure of the ribbon
edges [8, 9, 10]. We investigate an alternative possibility
and consider graphene on a substrate that makes the two
carbon sublattices inequivalent. This breaks the sublat-
tice symmetry directly, generating an intrinsic and robust
mass for the Dirac fermions.
As a substrate, hexagonal boron nitride (h-BN) is a
suitable choice [11]. This wide gap insulator has a layered
structure very similar to that of graphene but the two
atoms in the unit cell are chemically inequivalent. Placed
on top of h-BN the two carbon sublattices of graphene
become inequivalent as a result of the interaction with
the substrate. Our band structure calculations in the
local-density approximation show that a gap of at least
53 meV – an energy roughly twice as large as kBT at
room temperature – is induced. This can be compared
to graphene on a copper (111) metallic surface where the
gap is found to be much smaller and can even vanish,
depending on the orientation of the graphene sheet.
Stable structure The lattice mismatch of graphene with
http://arxiv.org/abs/0704.1994v2
carbonnitrogen boron
FIG. 1: (Color online) The three inequivalent orientations
of single-layer graphene on a h-BN surface. Left: sideview,
right: topview.
hexagonal boron nitride is less than 2%. Just as in
graphite, the interaction between adjacent BN layers is
weak. The h-BN layers have an AA′ stacking: the boron
atoms in layer A are directly above the nitrogen atoms
in layer A′. Within the local density approximation
(LDA) the minimum energy separation of adjacent layers
is found to be 3.24 Å, which is reasonably close to the
experimental value of 3.33 Å. Because generalized gra-
dient approximation (GGA) calculations give essentially
no bonding between BN planes and lead to excessively
large values of c [12], we opt for electronic structure cal-
culations within the LDA. Electronically, h-BN is a wide
gap insulator, with experimentally a gap of 5.97 eV [13].
This gap is underestimated by about 33% in LDA. A
quasi-particle GW correction on top of the LDA brings
it into very close agreement with experiment [14, 15] and
reinterprets experiment in terms of an indirect gap. For
the composite graphene layer on top of h-BN system, we
use the LDA lattice parameter for graphene, a = 2.445
On the basis of this structural information we con-
struct a unit cell with 4 layers of h-BN and a graphene
 0.25
 0.15
 0.05
−0.05
 3.9 3.7 3.5 3.3 3.1 2.9 2.7 2.5
d ( °A)
FIG. 2: (Color online) Total energy E of graphene on h-BN
surface for the three configurations (a), (b), and (c) as a func-
tion of the distance between the graphene sheet and the top
h-BN layer.
top layer. We represent the vacuum above graphene with
an empty space of 12 to 15 Å. The results to be presented
below converge quickly as a function of the number of h-
BN layers and the width of the vacuum space, consistent
with weak interlayer interactions. No significant differ-
ence in the final results were found when 6 layers of h-BN
were used. The in-plane periodicity is that of a single
graphene sheet with a hexagonal unit cell containing two
carbon atoms. We consider three inequivalent orienta-
tions of the graphene sheet with respect to the h-BN, see
Fig. 1:
- the (a) configuration with one carbon over B, the other
carbon over N.
- the (b) configuration with one carbon over N, the other
carbon centered above a h-BN hexagon.
- the (c) configuration with one carbon over B, the other
carbon centered above a h-BN hexagon.
The self-consistent calculations were performed with
the Vienna Ab-initio Simulation Package (VASP) [16, 17]
using a plane wave basis and a kinetic energy cutoff of
600 eV. The Brillouin Zone (BZ) summations were car-
ried out with the tetrahedron method and a 36× 36× 1
grid which included the Γ, K and M points. A dipole
correction avoids interactions between periodic images of
the slab along the z-direction [18].
The total energies of the three configurations are shown
as a function of the distance between the h-BN surface
and the graphene sheet in Fig. 2. For all distances, the
lowest-energy configuration is (c) with one carbon on top
of a boron atom and the other above a h-BN ring. The
equilibrium separation of 3.22 Å for configuration (c) is
smaller than 3.50 Å for configuration (a) and 3.40 Å for
configuration (b). For all three configurations the energy
landscape is seen to be very flat around the energy min-
imum. Though symmetry does not require inequivalent
DOS C B N Bands
Γ K M 
FIG. 3: (Color online) Band structure along the ΓK and KM
directions in reciprocal space, total and projected densities of
states (DOS) for the relaxed structure (c) of graphene on h-
BN. Carbon, Boron and Nitrogen projected DOS are shown,
with a projection on the p-states in-plane (red/thick grey
lines) and out-of-plane (blue/thin grey lines). The inset is
a magnification of the bands around the K point, where the
gap opens.
carbon atoms to be equidistant from the BN layer, in
practice the stiffness of the graphene sheet prevents any
significant buckling.
Band structure With the stable structures in hand,
we compute the corresponding electronic band structures
and projected densities of states which are shown in Fig. 3
for configuration (c). For the h-BN derived bands a gap
of 4.7 eV at theK-point is found, which is nearly identical
to the LDA gap value at this particular point in the Bril-
louin zone found for bulk h-BN [15]. Within this boron
nitride gap, the bands have entirely carbon character as
expected on the basis of the weak interlayer interactions
in both bulk h-BN and graphite. On the electron-volt
scale of Fig. 3 the Dirac cone around the K-point appears
to be preserved. However, zooming in on that point in
the BZ (see inset) reveals that a gap of 53 meV is opened
and the dispersion around the Dirac points is quadratic.
The band gaps for the three different configurations
are shown in Fig. 4 as a function of the distance between
the graphene sheet and the h-BN surface. Decreasing
this distance increases the gap, as expected for a phys-
ical picture based upon a symmetry-breaking substrate
potential. The band gaps that are opened at the equi-
librium geometries of the (a) and (b) configurations are
56 meV and 46 meV, respectively, which are compara-
ble to the band gap obtained for configuration (c). The
largest gap is found for the (a) configuration with one
carbon atom above a boron atom and the other above
a nitrogen atom. Again, this is expected for gap open-
ing induced by breaking the symmetry of the two carbon
FIG. 4: (Color online) The values of the gaps for the three
configurations (a), (b), and (c) as a function of the distance
between the graphene sheet and the top h-BN layer. The
calculated equilibrium separations are indicated by vertical
arrows.
atoms. Since LDA generally underestimates the gap, the
values that we obtain put a lower bound on the induced
band gaps, which we thus find to be significantly larger
than kBT at room temperature.
Although the lattice mismatch between graphene and
h-BN constants is less than 2% and can be neglected in
a first approximation, in a real system incommensurabil-
ity will occur and we expect the strong in-plane bond-
ing of both graphene and h-BN to prevail over the weak
inter-plane bonding. For graphene on Ir(111) where the
lattice mismatch is ∼ 10%, Moiré patterns have been ob-
served in STM images [19]. There, first-principles calcu-
lations showed that regions could be identified where the
graphene was in registry with the underlying substrate in
high symmetry configurations analogous to the (a), (b)
and (c) configurations discussed above, and transition
regions with little or no symmetry [19]. The graphene
separation from the substrate varied across the surface
leading to bending of the graphene sheet. If we could
take the lattice mismatch into account in a large super-
cell in a similar fashion, some areas of graphene would be
forced into the higher energy (a) and (b) configurations
with larger separations to the BN substrate. However,
the corresponding band gaps are all of the order of the
50 meV we find for the lowest energy (c) configuration, or
higher. It seems reasonable to conclude that the broad-
ening resulting from lattice mismatch will not reduce the
gap substantially.
Cu(111) substrate The situation changes markedly for
graphene on a Cu(111) surface. The copper surface layer
forms a triangular lattice, matching that of graphene
to better than 4%. We consider two configurations of
graphene on Cu(111). Either the center of each car-
bon hexagon is on top of a Cu atom, which we call the
symmetric configuration in the following, or every sec-
ond carbon atom is on top of a Cu atom, which we call
the asymmetric configuration. For the asymmetric and
symmetric configurations, LDA calculations yield equi-
librium separations of 3.3 Å and 3.4 Å which are compa-
rable to those of graphene on h-BN (Fig. 2). The total
energy difference between the two configurations is only
about 9 meV. In the asymmetric configuration a small
gap of 11 meV is opened in the graphene band structure,
whereas in the symmetric configuration the gap remains
very close to zero. In both cases we find very little mix-
ing between copper and carbon states. The difference
between the gaps can be explained by the fact that the
symmetric configuration preserves the graphene symme-
try in the top Cu surface layer, whereas the symmetry
is broken in the asymmetric configuration. The effect of
this symmetry breaking is small, however, and the re-
sulting band gap is much smaller than that induced by
h-BN and comparable to the typical thermal broadening
reported in experiments [2, 4]. Taking into account the
graphene-Cu lattice mismatch in for instance a super cell
calculation [19] will not to change this conclusion.
For both configurations of graphene on Cu, a charge
rearrangement at the interface is found which moves the
Fermi level away from the induced gap [20] by much
more than the magnitude of the gap itself. This is in
contrast to a h-BN substrate, where the Fermi level re-
mains in the induced gap. Around the Fermi level of
graphene on Cu the band dispersion is still linear. Con-
sequently, the properties characteristic of graphene which
result from the linear dispersion should be preserved. In
for instance tunneling experiments that require adsorp-
tion of graphene on a metallic (Cu) substrate [21] one
should still be able to observe the intrinsic linear elec-
tronic structure of graphene near the Fermi energy, but
no longer at the Dirac points.
Conclusions Our density functional calculations show
that the carbon atoms of a graphene sheet preferentially
orient themselves directly above the boron atoms of a
h-BN substrate, with one carbon sublattice above the
boron sublattice and the other carbon centered above a h-
BN ring. Although graphene interacts only weakly with
the h-BN substrate, even when a few angstroms away
the presence of h-BN induces a band gap of 53 meV,
generating an effective mass for the Dirac fermions of
4.7·10−3 me, whereme is the electron mass. The gap that
opens at the Dirac points is considerably larger than the
one for graphene on Cu(111). Additional quasi-particle
interactions, for instance taken into account within aGW
scheme, will increase the value of the gap. The opening
of a band gap in graphene on h-BN offers the potential
to improve the characteristics of graphene-based FETs,
decreasing the minimum conductance by orders of magni-
tude. Other interesting features such as the valley degree
of freedom, which is related to the degeneracy of the K
and K ′ points in the Brillouin zone, remain intact and
can still be used to control an electronic device [22]. Also
the half-integer quantum Hall effect – a peculiar charac-
teristic of graphene – remains unchanged [2, 23, 24].
This work was financially supported by “NanoNed”,
a nanotechnology programme of the Dutch Ministry of
Economic Affairs and by the “Nederlandse Organisatie
voor Wetenschappelijk Onderzoek (NWO)” via the re-
search programs of “Chemische Wetenschappen (CW)”
and the “Stichting voor Fundamenteel Onderzoek der
Materie (FOM)”. Part of the calculations were performed
with a grant of computer time from the “Stichting Na-
tionale Computerfaciliteiten (NCF)”.
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