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Graphene on incommensurate substrates:
trigonal warping and emerging Dirac cone replicas with halved group velocity
Carmine Ortix,1Liping Yang,1and Jeroen van den Brink1
1Institute for Theoretical Solid State Physics, IFW Dresden, D01171 Dresden, Germany
(Dated: November 3, 2011)
The adhesion of graphene on slightly lattice-mismatched surfaces, for instance of hexagonal boron
nitride (hBN) or Ir(111), gives rise to a complex landscape of sublattice symmetry-breaking poten-
tials for the Dirac fermions. Whereas a gap at the Dirac point opens for perfectly lattice-matched
graphene on hBN, we show that the small lattice incommensurability prevents the opening of this
gap and rather leads to a renormalized Dirac dispersion with a trigonal warping. This warping
breaks the effective time reversal symmetry in a single valley. On top of this a new set of massless
Dirac fermions is generated, which are characterized by a group velocity that is half the one of
pristine graphene.
PACS numbers: 73.22.Pr, 73.21.Cd, 72.80.Vp
Introduction – One of the main experimental challenges
towards the realization of next-generation graphene elec-
tronics technology is the possibility to access the low
energy Dirac point physics.
strates, for instance, are not ideal for graphene because
of the trapped charges in the oxide.
induced charge traps limit the device performances and
make the low energy physics inaccessible [1]. It has been
recently shown that placing graphene on hexagonal boron
nitride (hBN) yields improved device performances [2] –
graphene on hBN can have mobilities and charge inho-
mogeneities almost an order of magnitude better than
graphene devices on SiO2.
hBN is interesting because it has the same honeycomb
lattice structure of graphene, but only with two atoms in
the unit cell, B and N, that are chemically inequivalent.
Precisely this causes hBN to be a wide bandgap insula-
tor. When graphene is placed on top of a hBN surface,
the lowest energy stacking configuration has one set of C
atoms on top of B and the other C sublattice in the mid-
dle of the BN hexagons [3, 4] – assuming perfect lattice
matching between graphene and hBN. Consequently the
substrate-induced potential breaks the graphene sublat-
tice symmetry. This leads to a gap at the Dirac point
and hence a robust mass for the Dirac fermions. First
principles band structure calculations [3] put this gap at
∼ 50 meV – an energy roughly twice as large as kBT
at room temperature. However, recent scanning tunnel-
ing microscopy experiments [5, 6] do not detect a sizable
bandgap.
Within an effective continuum approach, here we show
that this discrepancy originates from the 1.8 % lattice
mismatch [7] between graphene and hBN which leads to
a Moir´ e superstructure with periodicity much larger than
the graphene lattice constant. In this Moir´ e lattice, car-
bon atoms are embedded in a local environment of boron
and nitrogen atoms that is varying continuously and pe-
riodically. This leads to a complex landscape of local
sublattice symmetry-breaking terms which prevent the
Silicon oxide (SiO2) sub-
These impurity-
opening of a band gap at the Dirac point. Due to the in-
commensurability, the Dirac cones are instead preserved
in renormalized form, with a threefold global symmetry
due to a substrate-induced trigonal warping, which is in
excellent agreement with experimental observations [8].
In addition we also show that a new set of massless Dirac
fermions is generated at the corners of the supercell Bril-
louin zone. These quasiparticles are characterized by a
collinear group velocity vF which in the relevant weak
coupling regime equals one half of v0
ity in pristine graphene. As a set of these newly gener-
ated massless Dirac fermions does not overlap in energy
with any other states, gated or doped graphene triangu-
lar Moir´ e superlattices provide a clear way to probe these
Dirac fermions.
Before presenting the calculations that explicate these
results, we wish to point out that very similar physics
arises for graphene on incommensurate substrates other
than hBN, in particular for the experimentally relevant
Moir´ e superlattices formed by graphene epitaxially grown
on Ir(111) surfaces [8, 9]. As the (111) surface iridium
atoms form a triangular lattice, there are two distinct lo-
cal Ir-C configurations with a high symmetry [10]. The
first one occurs when a C atom is on top of Ir, situ-
ating its three neighbors in throughs between Ir sites –
the natural equivalence of the two triangular graphene
sublattices is therefore broken. In the other high sym-
metry Ir-C configuration, the honeycomb carbon ring is
centered above an iridium atom.
fect of the Ir charges does not break the sublattice sym-
metry and therefore, no gap should open at the Dirac
point. While H decorated graphene/Ir(111) superstruc-
tures have been reported to give rise to absolute band
gap openings [9, 10], recent angle-resolved photoemission
spectroscopy data have rather shown an anisotropic be-
havior of the massless Dirac fermions close to the Dirac
points due to an enhanced trigonal warping [8].
A number of interesting theoretical predictions exist
on graphene superlattices.
F, the Fermi veloc-
In this case the ef-
It is known that external
arXiv:1111.0399v1 [cond-mat.mtrl-sci] 2 Nov 2011
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2
one-dimensional periodic potentials can lead to a huge
anisotropic renormalization of the electronic spectrum
[11, 12], emerging zero modes [13] and even to a Landau-
like level spectrum as a result of the presence of extra
Dirac points [14]. Dirac cone replicas at different k point
in the Brillouin zone (BZ) have been reported also in
triangular graphene superlattices [15] as well as in bi-
layer graphene superlattices [16, 17].
however, rely on a description that disregards local sub-
lattice symmetry-breaking terms, which are crucial when
investigating the opening/closing of gaps in graphene on
slightly incommensurate hBN or Ir(111). In this Letter,
we investigate the modification of the electronic spec-
trum of graphene Moir´ e superlattices taking explicitly
into account these essential, slowly varying, sublattice
symmetry-breaking terms in an effective continuum ap-
proach.
These findings,
Effective-Hamiltonian – We start out by taking into
account the interaction induced by the substrate charges
as an external electrostatic potential for graphene’s Dirac
electrons. The potential has a triangular periodicity that
coincides with the arrangements of the centers of the
BN hexagons (Ir atoms at the 111 surface): V(r) =
?
nitudes depend on the modulus of G alone.
following, we restrict the sum to the six wavevectors
with equal magnitude G = 4π/(3aS), G/G = (±1,0),
(±cosπ/3,±sinπ/3) , where
hexagons (Ir-Ir) distance. As the mismatch between aS
and the graphene carbon-carbon distance a is small, we
can evaluate the effect of the substrate-induced electro-
static potential on the two triangular graphene sublat-
tices A/B as the sum of products of rapidly varying
parts exp(iGSR· rA/B
positions and GSRrescaled wavevectors with magnitude
GSR= 4π/(3a), times slowly varying parts exp(i?G · r)
the rescaled ”coarse-grained” wavevectors with magni-
tude?G = 4π|δa|/3a2where δa = aS− a indicates the
been assumed positive.As a result, the effect of the
substrate charges leads to an average external potential
acting equally on each carbon atom and a mass term
breaking the graphene sublattice symmetry given by
V±(r) = [VA(r) ± VB(r)]/2 = V0/2?
parts of the potential are now encoded in the non-trivial
phase factors φ? G=?G/?G · δ/a where δ = −a(1,0) is the
Since the large periodicity of the Moir´ e super-
structure prevents intervalley scattering, we can de-
scribe the low-energy quasiparticles near the corners
K± =
onal Brillouin zone as 4-dimensional spinors Ψ
GVGeiG·rwhere the G’s are the reciprocal lattice vec-
tors and VG the corresponding amplitudes whose mag-
In the
√3aS indicates the BN
j
), rA/B
j
being the actual atomic
which we treat in the continuum limit.The?G’s are
lattice mismatch which without loss of generality has
? G
?1 ± expiφ? G
?×
exp(i?G · r).
graphene nearest-neighbor vector.
The contribution of the rapidly varying
?2π/(3a),±2π/(3√3a)?
of the graphene hexag-
=
-1.0-0.50
V0
0.51.0
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
εD
0 0.10.2
V0
0.30.4
0.5
0.6
0.7
0.8
0.9
1.0
vF / vF
0
(a)
(b)
-0.1
-0.05
0
0.05
0.1
-0.1-0.05 0
kx
0.05 0.1
V0=0.1
V0=0.3
V0=0.5
ky
(d)
-0.1
-0.05
0
0.05
0.1
-0.1-0.05 0
kx
0.05 0.1
ky
V0=0.1
V0=0.3
V0=0.5
εF-εD=0.05
(c)
εF-εD=0.05
FIG. 1: (Color online) (a) Renormalization of the Fermi veloc-
ity at the Dirac point as a function of V0. The points are the
results of the exact diagonalization of the low-energy Hamilto-
nian whereas the continuous line is the analytical result from
second-order perturbation theory. (b) Energy of the Dirac
point as a function of the external potential amplitude V0.
(c),(d) Fermi lines for different values of the potential ampli-
tude V0 close to the K+ (c) and the K−(d) valleys. Energies
and wave vectors are in units of ?vF? G and? G respectively.
?ψK+,A,ψK+,B,ψK−,B,ψK−,A
an effective Hamiltonian
?, characterizing the elec-
tronic amplitudes on the two crystalline sublattices, with
ˆ H = v0
Here we use direct products of Pauli matrices σx,y,z,
σ0 ≡ˆ1 acting in the sublattice space and τx,y,z,τ0 ≡
ˆ1 acting on the valley degree of freedom (K±).
V±(r) ≡ 0,ˆ H has a chiral symmetry which can be ex-
pressed as
ˆ H,τ0⊗ σz
relation implies that in each valley any eigenstate Ψ?
with energy ? has a particle-hole partner τ0⊗ σzΨ with
energy −?. This property implies the doubly degener-
acy of the zero energy states in each valley.
presence of substrate-induced interactions of the form
as in Eq. 1, the system still possesses a chiral symme-
try provided the external superlattice potentials satisfy
V±(r + T) = −V±(r). In this case it is possible to de-
fine a new chiral operator [14] τ0⊗ σzS where S is a
shift operator SΨ(r) = Ψ(r + T). For the electrostatic
potentials defined above, a translation vector for which
the triangular potential V+(r + T) ≡ −V+(r) is absent
thereby implying particle-hole symmetry breaking and a
consequent lifting of the zero energy states degeneracy.
Fτ0⊗ p · σ + V+(r)τ0⊗ σ0+ V−(r)τz⊗ σz. (1)
For
??
≡ 0.This anticommutation
In the
This, however, does not lead to the opening of any ab-
solute band gap since the?G ≡ 0 component of the local
Therefore, the Dirac cones are preserved with the effect
sublattice symmetry breaking terms identically vanishes.
Page 3
3
FIG. 2:
the two graphene valleys K+ (a) and K− (b). The arrows indicate the corners?K± of the SBZ where the new Dirac cones are
(Color online) Energy dispersion relations for a graphene triangual Moir´ e superlattice with V0 = 0.1?v0
F? G close to
generated . (c) Topology of the Fermi lines close to the emergent Dirac cones. Units are the same as in Fig.1.
of particle-hole asymmetry eventually leading to a shift
of the conical points [shown in Fig.1(b)] of the two val-
leys reminiscent of the graphene doping caused by ad-
sorption of metal substrates [18]. We also find the Dirac
cones to be renormalized in triangular Moir´ e superlat-
tices. In Fig.1(a) we show the behavior of the collinear
Fermi velocity at the conical points for different values of
the interaction strength V0. The substrate-induced inter-
action leads to a decrease of the Fermi velocity as can be
found in the weak potential limit by treating the effect
of the electrostatic potentials in second-order perturba-
tion theory [c.f. continuous line in Fig.1(a)] according to
which
?
vF= v0
F
1 −
6V2
F?G2
0
?2v02
?
.
The local sublattice symmetry-breaking term breaks
the effective time-reversal symmetry on a single valley
[19],?T = i(τ0⊗ σy)ˆC, withˆC the operator of complex
[ψ?
ible in Fig.1(c),(d) where we show the topology of
the Fermi lines close to the Dirac points in the two
graphene valleys.There is a trigonal warping which
breaks the k → −k symmetry of the Fermi lines, i.e.
?(K±,k) ?= ?(K±,−k), consistent with the threefold
symmetry of the bandstructure experimentally detected
in Ir(111) superlattices [8].
an opposite effect on the two valleys since the exter-
nal electrostatic potentials do not break the true time-
reversal symmetry interchanging the valleys [19] T =
conjugation and
K+,B,−ψ?
?T
?ψK+,A,ψK+,B,ψK−,B,ψK−,A
?
=
K+,A,ψ?
K−,A,−ψ?
K−,B].This is clearly vis-
The trigonal warping has
−(τy⊗ σy)ˆC with T
[ψ?
lines fulfill ?(K±,k) ≡ ?(K∓,−k) as can been shown in
the weak potential limit where the energy dispersion of
the low-energy quasiparticles reads
?ψK+,A,ψK+,B,ψK−,B,ψK−,A
?
=
K−,A,−ψ?
K−,B,−ψ?
K+,B,ψ?
K+,A].Hence, the Fermi
?(K±,k) ? ?vFs|k| −
3s
?v0
F
|k|2
?G3V2
0
?
1 ±
√3sin3θk
?
. (2)
Here θkis the angle of the vector k with respect to theˆkx
direction and s = ±1 for quasi-electrons and quasi-holes
respectively. By increasing the strength of the poten-
tial V0[c.f. Fig.1(c),(d)], the warping effect is enhanced
and results in an anisotropy much larger than the one
expected in freestanding graphene [8].
Emergence of Dirac cone replicas – We have also ob-
tained the energy dispersion in the full supercell Brillouin
zone (SBZ) by exact diagonalization of the Hamiltonian
Eq. 1. The effect of the substrate-induced external po-
tential V±(r) has been incorporated into our calculations
through the scattering matrix elements between the chi-
ral eigenstates of the graphene quasiparticles
?ψK±,A
ψs(K±,k) =
ψK±,B
?
=
1
√2
?
1
se±iθk
?
eik·r.
Fig.2 shows the ensuing energy dispersion of the first and
second bands above and below the original Dirac points
in each valley. Contrary to Ref. 15 we do not find a gen-
eration of Dirac cones at the six?
second band above and below the original Dirac points
M points of the SBZ.
Indeed, the energy separation between the first and the
Page 4
4
goes to zero respectively at the (K±,?K±), (K±,?K∓) cor-
the sublattice symmetry-breaking term V−(r) which does
not allow for a sixfold symmetry of the bandstructure.
The topology of the Fermi lines close to the SBZ corners
clearly shows the emergence of Dirac cone replicas. How-
ever, while above the original Dirac points [c.f. Fig.2(c)]
these new massless quasiparticles are obscured by other
states, below the original Dirac points there is an energy
window where there are no other states than the new
massless Dirac fermions. Therefore there is one energy
value – apart from the original Dirac point – where the
density of states (DOS) vanishes linearly.
noticing that this asymmetry of the DOS reflects the
particle-hole symmetry breaking discussed above.
Further insight into the properties of the Dirac cone
replicas is obtained by introducing an effective Hamil-
tonian close to the three equivalent corners of the SBZ
[20].In the following we will restrict to consider the
behavior close to the (K±,?K∓) points, relevant for the
points. In the absence of external electrostatic poten-
tials V±(r) ≡ 0, there are three degenerate hole excita-
tions with energy ?(K±,?K∓) = −?vF? K. This degener-
tials and, as a result, one finds a singlet excitation with
energy ?S(K±,?K∓) = −?vF? K −V0and a doubly degen-
It can be easily shown that the effective Hamiltonian in
the vicinity of this doubly degenerate state corresponds
precisely to a massless two-dimensional Dirac equation
with Fermi velocity vR
F/2 and an isotropic disper-
sion relation
ners of the SBZ. This qualitative difference is caused by
It is worth
Dirac cone replicas generated below the original Dirac
acy is lifted by the substrate-induced electrostatic poten-
erate state with energy ?D(K±,?K∓) = −?vF? K + V0/2.
F= v0
?D(K±,δk±) = ?D(K±,?K∓) + ?v0
where we introduced δk±= k −?K∓and s?= ±1 is the
perfect agreement with the numerical results obtained by
exact diagonalization of the Hamiltonian. This is shown
in Fig.3(a),(b) where we plot the behavior of the Dirac
point energy ?D(K±,?K∓) and the Fermi velocity vR
close to the Dirac points, we find a trigonal warping re-
specting the threefold symmetry of the band structure as
it is shown in Fig.3(c),(d) where we plot the Fermi lines
for different values of V0.
Conclusions – By setting up an effective continuum
approach, we have derived the electronic properties
of graphene Moir´ e superlattices generated by adhesion
of graphene sheets onto lattice mismatched substrates.
While the complex landscape of sublattice symmetry-
breaking terms prevents the opening of a bandgap at
the Dirac point, we have demonstrated that with mis-
matched substrates one can tailor the low-energy band
F
2s?δk±
(3)
band index. The foregoing weak coupling analysis is in
Fas
a function of the potential strength V0. Away from but
-0.1
-0.05
0
0.05
0.1
-0.1-0.05 0
kx
0.05 0.1
V0=0.1
V0=0.3
V0=0.5
ky
εF-εD=0.05
00.10.20.30.40.5
-0.60
-0.55
-0.50
-0.45
-0.40
V0
εD
0 0.10.2 0.30.40.5
0.50
0.55
0.60
0.65
0.70
0.75
0.80
V0
vF
R / vF
0
-0.1
-0.05
0
0.05
0.1
-0.1-0.05 0
kx
0.05 0.1
V0=0.1
V0=0.3
V0=0.5
ky
εF-εD=0.05
(a)(b)
(d)(c)
FIG. 3:
?D(K±,?K∓) (a) as a function of the external potential am-
perturbation theory whereas the points are the result of the
exact diagonalization. (b) Group velocity at the Dirac point
replicas as function of the strength of the potential V0. (c),(d)
Fermi lines close to the emergent Dirac points ?D(K+,?K−)
(Color online) Behavior of the Dirac point energy
plitude V0. The continuous line is the result of the degenerate
(c) and ?D(K−,?K+) for different values of the amplitude V0.
dispersion. In agreement with recent experiments [8], we
have found a threefold-symmetry of the band structure
associated to a substrate-induced trigonal warping of the
Dirac cones. In addition a new set of Dirac fermions is
generated in graphene Moir´ e superlattices. By properly
and explicitly accounting for local sub-lattice symmetry
terms, we have shown that these new quasiparticles are
generated at the corners of the supercell Brillouin zone
and are characterized by a collinear group velocity at the
conical points of ∼ v0
L.Y. thanks F. Guinea for valuable discussions. This
research was supported by the Dutch Science Foundation
NWO/FOM.
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