Extremely flat band in bilayer graphene

Abstract

We propose a novel mechanism of flat band formation based on the relative biasing of only one sublattice against other sublattices in a honeycomb lattice bilayer. The mechanism allows modification of the band dispersion from parabolic to "Mexican hat"-like through the formation of a flattened band. The mechanism is well applicable for bilayer graphene-both doped and undoped. By angle-resolved photoemission from bilayer graphene on SiC, we demonstrate the possibility of realizing this extremely flattened band (< 2-meV dispersion), which extends two-dimensionally in a k-space area around the

K
¯

point and results in a disk-like constant energy cut. We argue that our two-dimensional flat band model and the experimental results have the potential to contribute to achieving superconductivity of graphene- or graphite-based systems at elevated temperatures.

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CONDENSED MATTER PHYSICS Copyright © 2018
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Extremely flat band in bilayer graphene
D. Marchenko
1
*, D. V. Evtushinsky
1†
, E. Golias
1‡
, A. Varykhalov
1
, Th. Seyller
2
, O. Rader
1
We propose a novel mechanism of flat band formation based on the relative biasing of only one sublattice
against other sublattices in a honeycomb lattice bilayer. The mechanism allows modification of the band dis-
persion from parabolic to “Mexican hat”–like through the formation of a flattened band. The mechanism is well ap-
plicable for bilayer graphene—both doped and undoped. By angle-resolved photoemission from bilayer graphene
on SiC, we demonstrate the possibility of realizing this extremely flattened band (< 2-meV dispersion), which extends
two-dimensionally in a k-space area around the

K point and results in a disk-like constant energy cut. We argue that
our two-dimensional flat band model and the experimental results have the potential to contribute to achieving
superconductivity of graphene- or graphite-based systems at elevated temperatures.
INTRODUCTION
Bilayer graphene (BLG) has, along with related two-dimensional (2D)
materials, extensively been studied by both transport and photo-
emission measurements. It is a material with an energy gap that opens
as soon as an asymmetry is imposed on the two graphene layers. This
tunable gap framed by van Hove singularities results from the “Mexican
hat”shape of the band structure (1) and is promising for low–power
consumption transistors for which on/off ratios of 3.5 × 10
4
are
expected (2). By using a substrate or even sandwiching the BLG with
other 2D systems, it is possible to achieve a wide range of physical
phenomena related to topological properties and control them by exter-
nal doping or gating: a valley Hall effect (3) with peaked Berry curvature
at the valley bottom, a gate-tunable topological valley transport (4), and
unconventional quantum Hall effects (5).
On the other hand, frequent reports of superconductivity in
graphite at elevated temperatures even above 300 K (6)raiseanumber
of questions. It is important to note that there is an established low-
and medium-temperature superconductivity in carbon known as a
phenomenon with strong doping dependence and connected to alkali
and alkaline earth metals. Examples are intercalated graphite such as
CaC
6
[T
C
= 11.6 K (7)], which can also be thinned down to an inter-
calated bilayer as a 2D superconductor [C
6
CaC
6
with T
C
=4K(8)],
as well as doped fullerides such as Cs
2
RbC
60
[T
C
=33K(9)]. Here,
the alkali and alkaline earth metals act to increase the carrier concen-
tration and density of states (DOS) at the Fermi energy. Most recently,
superconductivity was discovered in twisted BLG without any alkali
doping and a T
C
of 1.7 K (10).
The superconducting pairing mechanism is not fully established in
these materials, but for many of them, there is strong evidence for
phonon-mediated pairing and the validity of the Bardeen-Cooper-
Schrieffer (BCS) theory (11). The BCS theory predicts the relation
T
C
ºexp[−1/(Un(E
F
))] between the critical temperature T
C
of su-
perconductivity, the coupling constant Uof the effective attractive
interaction, and the DOS at the Fermi energy n(E
F
) such that, ac-
cording to the BCS theory, T
C
can be enhanced by increasing either
Uor n(E
F
).
The characteristic feature of graphite and graphene is, however,
their low or zero density of electronic states at the Fermi level, with
linear dependence on the energy. It has been argued that a flat band
system will enable superconductivity with strongly enhanced T
C
values (12). While Uremains difficult to assess, a flat band will lead
to maximal values of n(E
F
). Moreover, in this case of U≫W,theBCS
theory predicts T
C
ºUn(E
F
); i.e., the exponential suppression of T
C
with the interaction strength is removed (12). In this context, rhom-
bohedral (i.e., ABC) stacking of multilayered graphene has been
considered in theory (12,13). Consequently, there is much interest
to realize the rhombohedral stacking in experiment, and recently,
angle-resolved photoemission spectroscopy (ARPES) has shown a flat
band for five-layer graphene on 3C-SiC, which resembles the
calculated very flat dispersions (14). In the photoemission data, the flat
band is found to extend near K by about 0.8 Å
−1
(14). In this range,
the band disperses by only ~ 20 meV. Over the past years, theory has
consistently predicted flat band superconductivity accessible by doping
or gating and with enhanced critical temperature. These theoretical
approaches were inspired by cases such as the ABC-stacked graphite
or the van Hove singularity at the Mpointofmonolayergraphene
(MLG) and would substantially gain relevance if an extremely flat band
were found (15–17). Most recently, superconductivity has been observed
in twisted BLG (10,18) as the first purely carbon-based 2D supercon-
ductor. For “magic angles”between the two graphene layers, the moiré
pattern leads to flat band formation (19), and the observed super-
conductivity is directly assigned to the flat band effect.
In the present study, we investigate another way of band flattening
and DOS enhancement for the system with low DOS: BLG on SiC, which
does not require any twist. We use high-resolution ARPES measure-
ments to reveal the Mexican hat band structure of BLG on SiC in detail.
We show that there is a band portion that is much flatter, narrower,
and of higher photoemission intensity than expected, showing experi-
mentally no dispersion around the graphene K point (for ~ ±0.017 Å
−1
).
This means that a very high DOS is compressed here at a narrow energy
interval. This strong peak in the graphene DOS is, first of all, prom-
ising for the application of this system in high on/off ratio graphene-
based transistors (2). Because this extremely flattened band forms a
strong 2D-extended van Hove singularity and we find indications of
enhanced electron-phonon coupling, it could be used to achieve high-
temperature superconductivity in BLG.
On the basis of our theoretical analysis, we propose a novel
mechanism of band flattening in BLG by biasing of only one sub-
lattice relative to other sublattices, which is possible for both doped and
undoped BLG. We show that the band flattening effect is universal and
1
Helmholtz-Zentrum Berlin für Materialien und Energie, Elektronenspeicherring
BESSY II, Albert-Einstein-Straße 15, 12489 Berlin, Germany.
2
Institut für Physik,
Technische Universität Chemnitz, Reichenhainer Str. 70, 09126 Chemnitz, Germany.
*Corresponding author. Email: marchenko.dmitry@gmail.com
†Present address: Laboratory for Quantum Magnetism, École Polytechnique Fédérale
de Lausanne (EPFL), CH-1015 Lausanne, Switzerland.
‡Present address: Institut für Experimentalphysik, Freie Universität Berlin, Arnimallee
14, 14195 Berlin, Germany.
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achievable by different means of combining interlayer asymmetry, sub-
lattice asymmetry, and doping. The mechanism allows control of the
band dispersion all the way from parabolic through flat band forma-
tion to Mexican hat–like.
RESULTS
Experiment
Figure 1 shows ARPES data for a 6H-SiC sample with 1.2 monolayer
graphene (MLG) coverage. [As usual, the structural graphene mono-
layer at the interface, “zero-layer graphene”(ZLG), which is covalently
bonded and acts as buffer, is not counted.] For this coverage, the MLG
Dirac cone dispersion is expected to dominate with BLG contributing
just a faint intensity. However, there is an additional peculiar, very in-
tense, very sharp, and very flat band portion at 255-meV binding
energy that is not present in the case of MLG on SiC (20). The new
band is marked by white arrows in Fig. 1 (A to C). On the basis of
both calculations and experimental data, we attribute this band to the
bottom of one of the BLG bands. For this 1.2 monolayer coverage, the
photoemission intensities of the BLG bands are about four times lower
than those of the MLG bands, but at the same time, the flat band in-
tensity is about three times higher than that of the MLG bands.
There are examples in the literature where it is possible to see
this intense band in the data, but it has been ignored in discussions
so far (21–23), possibly because the resolution was not sufficient for
details of the band dispersion. We performed the measurements in
Fig. 1A with the electron wave vector perpendicular to the GKdirec-
tion and through the K point, at a temperature of 60 K and a photon
energy of 62 eV. We did not observe any difference between data atK
and K′. The features are much better visible as first derivative with
respect to energy in Fig. 1B: There is a faint indication that possibly
one more flat band at 150 meV (blue arrow) exists together with a
kink in the dispersion in the 150- to 160-meV energy range.
To judge the photoemission intensity distribution, we measured a
3DmaparoundtheK point. The cut along the GK direction in Fig. 1C
reveals that, in this experimental geometry,onlyhalfofthemonolayer
Dirac cone and only half of the bilayer dispersion are visible because
of a destructive interference from the two graphene sublattices (24,25).
We see the flat band on both sides from the Kpointwithsimilarinten-
sities. This is unusual for photoemission interference from graphene.
In Fig. 1D, two constant energy cuts are presented, taken from Fig. 1A
data at 235- and 255-meV binding energies. An energy difference of
20 meV, very small for ARPES otherwise, is enough to show the drastic
change in the constant energy cuts. At 235 meV, away from the flat band,
there is a nearly circular ring with intensity modulation due to the photo-
emission interference effect, but at 255 meV, one sees the shape of a
disk, without modulation by interference.
TherepresentationinFig.1Easastack of photoemission spectra
(only every 10th spectrum is shown) demonstrates the high photo-
emission intensity of the flat band. Figure 1F shows the spectrum that
exactly intersects the K point. The spectra were analyzed by simple
single-peak Gaussian fitting of the topmost portion of the photoemission
peaks. The resulting dispersion is shown as dots in Fig. 1G. We find
that the scattering of peak maxima energies is not more than 2 meV
0–0.04–0.08 0
–1
Wave vector k (A )
||
1.0
0.6
0.8
–1
Wave vector k (A )
||
–0.08 –0.04 0.04 0.080.04 0.08
BLG
BLG
MLG
K K
Binding energy (eV)
0.2
0.4
0.0 AB
150 meV
255 meV
–1
Wave vector k (A )
||
C
BLG
MLG
MLG + BLG
K
1.701.661.62 1.74 1.78
Binding energy (eV)
0.20.40.60.8 0.0
E
K
BLG
BLG
MLG
MLG + BLG
Flat band
E-cuts
MLG + BLG
k
||
k
||
k
||
Flat band
Binding energy (eV)
0.20.40.60.8 0.0
BLG MLG + BLG
Flat band
F
K
0.02 –1
A
D
255 meV
235 meV
E-cuts
k
||
k
|| k
||
Binding energy (eV)
G
K
0.2 0.10.3
–
1
Wave vector k (A )
||
0
–0.04
0.04
Fig. 1. Angle-resolved photoemission spectroscopy. (A) Data for the sample with 1.2 monolayer graphene (MLG) coverage around the 
K point of the graphene
Brillouin zone. The MLG Dirac cone dispersion, the faint BLG dispersion, and an intense nondispersing flattened band at 255-meV binding energy, marked by a white
arrow, can be seen. Measurements were done at hn= 62 eV and T=60K.(B) First derivative with respect to energy from the same data as in (A) where dispersions of all
bands are much more visible. A blue arrow shows the possible presence of one more flat band. (C) Measurements in the GK direction showing a destructive interference
effect for the monolayer and bilayer bands and its absence for the flat band. (D) Constant energy cuts taken at 235- and 255-meV binding energies. (E) Same data as in
(A) presented as a stack of spectra (only every 10th spectrum is shown). (F) Spectra at the 
K point showing the flattened band intensity and its narrow width. (G) Dispersion of
the maxima extracted from the spectra.
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for a k-space range of ±0.017 Å
−1
around the K point. In addition to
the flat band at 255 meV, there is a kink and a second flat band faintly
visible around 150-meV binding energy (Fig. 1, B and G, and fig. S1).
The sample with mostly BLG contribution is presented in fig. S1.
We measured the data at room temperature; hence, the energy broaden-
ing is slightly larger, but all the features under discussion are well vis-
ible. In summary, at the position of the bottom of one of the BLG
bands, we have a nondispersive and very sharp band with more than
an order of magnitude higher photoemission intensity than expected.
Density functional theory
We conducted density functional theory (DFT) calculations of MLG,
BLG, and trilayer graphene (TLG)/6H-SiC systems using the Vienna
Ab initio Simulation Package (VASP) package (see overview in fig. S2)
(26). The structure of the BLG/SiC used in the calculations is presented
in Fig. 2D. To compare with experimental data, the DFT dispersions
of both the MLG and BLG are presented in Fig. 2A. The experimen-
tally observed picture and the high DOS are well reproduced. In Fig. 2C,
we see a magnified view of the calculated flat band structure with a
5-meV dispersion around the K point. This value is small relative to
the overall band structure but significantly larger than the flat band
dispersion observed in our experiment.
Symbol sizes in Fig. 2A reflect the contribution of p
z
orbitals in the
topmost graphene layer to the calculated band structure. We see a
strong localization of the flat bandwavefunctionsonthetopmostgra-
phene layer. A closer examination shows that the flat band wave
function is localized not only on the top graphene layer but also on sole-
ly one graphene sublattice, B (see fig. S3).We observed a similar effect in
graphene/Ni(111), where upper and lower halves of the Dirac cone be-
long to different sublattices (27). We will return to this point below since
it helps us develop the low-energy Hamiltonian for the present problem.
Assuming rotational symmetry of the calculated bands around the 
K
point, as is supported by the experimental data in Fig. 1, we calculate the
DOS from the data in Fig. 2A, taking into account both layers of the
BLG. The result, presented in Fig. 2B, shows two very strong DOS sin-
gularities at the edges, separated by a gap. These DOS peaks represent
van Hove singularities, which, in the case of the flat band, have a very
strong DOS divergence because dE/dk,d
2
E/dk
2
,andd
3
E/dk
3
are zero at
the K point and the flat band is spread in k-space as a 2D disk-like area
in the Brillouin zone.
To investigate the role of the substrate for the flat band appearance,
we calculated the charge density redistribution in the BLG due to in-
teraction with the SiC substrate (ZLG/SiC) as a difference r
BLG/SiC
−
r
BLG
−r
SiC
. Details of the charge density redistribution provide us
information on the interplay between interlayer and sublattice asym-
metries in BLG on SiC, important for the formation of specific band
shapes different from the case of a free BLG. The result is presented in
Fig. 2 (E and F). Yellow-colored isosurfaces show gain of charge, and
DOS (arbitrary units) 0.0 0.02 0.04–0.02–0.04
–1
Wave vector k (A )
||
C
K
0.15
0.16
0.17
0.18
0.19
0.20
5 meV
1.0
0.2
0.4
0.6
0.8
1.2
Binding energy (eV)
0.0
–0.2
–0.4
Binding energy (eV)
B
–1
Wave vector k (A )
||
1.0
0.2
0.4
0.6
0.8
1.2
BLG
MLG
Binding energy (eV)
0.0
–0.2
–0.4
K
0.0 0.1 0.2–0.1–0.2
A
Flat band
BLG
SiC
ZLG
DE
A
B
C
BLG
Silicon carbide
H
F
Zero-layer graphene
Bilayer graphene Side view Tilted view
Gain of charge
Loss of charge
Fig. 2. DFT calculations. (A) Calculated band structures of MLG (blue) and BLG (red) on SiC around the 
K point with k
∥
perpendicular to the GK direction. Width of lines
corresponds to the contribution of p
z
orbitals to the top graphene layer. (B) DOS calculated from the data in (A), taking into account the wave function contribution to
both layers of BLG and assuming rotational symmetry of the calculated bands around the 
K point. (C) Magnified view of the calculated flat band dispersion around the 
K
point. (D) Slab structure of BLG on 6H-SiC(0001) used in DFT calculations. Between the BLG and the SiC substrate, there is a graphene buffer (zero) layer. The back side
of the slab is H passivated. The inset shows the model of the BLG/6H-SiC unit cell used in the calculations (only BLG is shown), with definition of sublattices A, B, and C
used in the present work. (E) Effect of BLG interaction with the ZLG/SiC system (ZLG/SiC). Yellow-colored isosurfaces show gain of charge, and light blue–colored
isosurfaces show loss of charge, indicating charge transfer from the ZLG/SiC system to the BLG (mainly bottom layer) and emergence of a strong sublattice asymmetry.
Letters A, B, and C in the top-left part of the figure indicate A, B, and C sublattices, correspondingly. (F) Same as (E) with tilted view for better clarity.
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light blue–colored isosurfaces show loss of charge, indicating both
charge transfer from the ZLG/SiC system into the BLG (mainly the
bottom layer) and emergence of a strong sublattice asymmetry at
the bottom layer. From this picture, we can also see that both sublattices
A and B of the top graphene layer and the sublattice A of the bottom
graphene layer are almost unaffected and show only small charge
asymmetry due to the interaction with the substrate. However, the C
sublattice, which is in the lower graphene layer, is strongly affected
by the interaction. We will further investigate this finding below.
Flat bands are also generally unstable against other types of order,
such as ferromagnetism and charge order, and much of the argumen-
tation in the present work holds for these cases as well. 2D ferromag-
netism has been investigated theoretically in gapped BLG (28).
Comparing the tendencies for ferromagnetism, charge order, and su-
perconductivity in flat band systems, it was recently concluded that
superconductivity will always prevail, provided the flat band is
brought sufficiently close to the Fermi level (15–17). This has been
concluded particularly for the flat band in ABC-stacked graphite
(15)andforpureMLGwithvanHovesingularity(15,17)andisalso
valid for the present case.
Model Hamiltonian
Intuitively, the simplest form of the BLG Hamiltonian around the
K point can be constructed by joining two MLG 2 × 2 matrices into
a larger 4 × 4 matrix and adding the required interaction terms.
This results in the Hamiltonian H
1
(Eq. 1), where tand bsubscripts
denote top and bottom graphene layers, respectively, and super-
script indices indicate BLG sublattices A, B, and C (see the inset in
Fig. 2D). Sublattice A atoms of the top layer are located above sublat-
tice A atoms of the bottom layer, and in the first approximation, we
add only the interaction between them (t
⊥
), neglecting interactions be-
tween other pairs of atoms located at larger distance from each other.
We get to Hamiltonian H
2
when we consider the top layer graphene
atoms to be at the same energy E
t
(i.e., without sublattice asymmetry)
and the bottom graphene atoms to experience sublattice asymmetry D
due to the interaction with the substrate. The substrate also causes a
difference between E
t
and E
b
, which represent interlayer asymmetry
between top and bottom layers, due to the interaction with the sub-
strate and the corresponding charge transfer. This Hamiltonian (Eq. 1)
can also be considered a simplified version of the Slonczewski-
Weiss-McClure model for bulk graphite (29) adapted to BLG.
H1¼
EA
tpt⊥0
p*EB
t00
t⊥0EA
bp
00p*EC
b
0
B
B
@
1
C
C
A
H2¼
Etpt⊥0
p*Et00
t⊥0EbD
2p
00p*EbþD
2
0
B
B
B
B
B
@
1
C
C
C
C
C
A
ð1Þ
where p¼ffiffi
3
p
2atðkxþikyÞ,withabeing the graphene lattice constant
and tbeing the nearest-neighbor hopping energy.
Fitting the Hamiltonian parameters to reproduce the experimental-
ly observed band structure yields values of E
t
=0.25eV,E
b
=0.35eV,
and D= 0.2 eV (modeling with 0.01-eV precision). From these values,
we immediately see the important property
Et¼EbD
2ð2Þ
Returning to the more general case of Hamiltonian H
1
shows that
there are several possible conditions that yield a flat band (several
orders of derivatives equal to zero at the Kpoint);theydiffermostly
in how charges are redistributed between sublattices and with which
sign. Only one case was found to reproduce a result reasonable from
the point of view of charge transfer signs and magnitudes for the case
of BLG on SiC
EB
t¼EA
bEA
b<EC
bð3Þ
In other words, the bottom-layer sublattice A is energetically equal
to the top-layer sublattice B. This condition is possible in the case of
both interlayer and sublattice asymmetries compensating each other at
the A sublattice of the bottom layer. Note that, here, the EA
tvalue is
unimportant for obtaining the flat band.
At the same time, any deviation from Eq. 3 by changing either the
sublattice or the interlayer asymmetry values produces a band with
finite curvature around the Kpoint.Figure3showsthissituationin
which a slight change of parameters modifies the flat band structure
either into a parabolic shape (positive effective mass) or into a Mex-
ican hat shape (negative effective mass), changing in particular the
number and positions of the band minima. In the intermediate stage,
we have a band with the first, second, and third derivatives equal to
zero at the K point. If we include more interaction parameters into the
Hamiltonians (Eq. 1), the overall picture will appear distorted, but the
possibility of achieving a flat band and the fine-tuning of the band
shape remain unaffected.
The condition (Eq. 3) for the graphene sublattice energies is very
close to the picture that we have in DFT calculations concerning the
charge density redistribution due to graphene-substrate interaction
(Fig.2,EandF).Aswehaveshownexperimentally, these parameters
are present in the BLG on the SiC system; however, they could be
reproduced in BLG or TLG on other substrates by the joint influence
of sublattice and interlayer asymmetries, e.g., by combination of chem-
ical doping and gating. Furthermore, the proposed mechanism is
promising beyond graphene because it is also applicable to other
2D layered systems such as multilayers of germanene and silicene.
DISCUSSION
The experimental high photoemission intensity of the flat band can be
partially explained by the 2D extent and broadening in k
x
and k
y
.This
broadening, however, is not the reason for the flatness and the ob-
served features of the band. If this broadening were to play a signifi-
cant role, we would see the narrowing and intensity enhancement effects
also at other BLG bands around the KpointwheretheirdE/dkbecomes
locally zero. The experiments do not show these effects.
In Fig. 1C, we see, unusual for graphene, the disappearance of the
interference pattern in the region of the flat band, resulting (Fig. 1D)
in a disk-like image of the constant energy cut at 255-meV binding
energy. The destructive interference arises because of localization of
thewavefunctionondifferent graphene sublattices (24,25); however,
for the flat band, the wave function is localized on one sublattice
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only, and preconditions for the destructive interference disappear
(fig. S3).
In the ARPES data, there are, in addition to the pronounced flat
band, a kink and a second flat band, faintly visible around 150-meV
binding energy (Fig. 1, B and G, and fig. S1). The nature of this kink is
not unambiguous as two different reasons could produce a similar re-
sult. First, it could be due to overlap of intensities from regions with
different numbers of graphene layers, particularly TLG.
Calculations of TLG on 6H-SiC in two possible stackings (ABA
and ABC) are presented in fig. S2. In fig. S4, they are shown taking
into account the contribution of thewavefunctiontothetopgraphene
layer and overlapped with the BLG for comparison. From these
figures, we see that the rhombohedral (ABC) TLG on SiC has its
own flat band structure with specific electron localization at a binding
energy lower than that of BLG. This TLG coverage may actually be
negligibly small, especially in the case of an extremely sharp and in-
tense photoemission feature. For undoped TLG, the band structure
was studied experimentally by Nanospot ARPES and shows cubic
band dispersion at the Fermi level (30) for rhombohedral stacking.
An example of MLG, BLG, and TLG on another substrate, Ir(111),
is presented in fig. S5. There are characteristic double- and triple-split
Dirac cones without flat bands. Because of the absence of n-doping,
only the bottom bands are visible.
Another possible explanation for the observed kinks is re-
normalization due to many-body effects as known from MLG
(20,31). The enhanced electron-phonon coupling in superconducting
CaC
6
produces in ARPES a very similar renormalization around
160 meV below the Fermi level (32). Thus, we want to address at
this point again the relevance for superconductivity. There are var-
ious possible pairing mechanisms for intrinsic superconductivity in
graphene. Besides conventional s-wave pairing (33), p + ip (33), d
(34), d + id (16), and f (16) have also been considered for graphene.
It should also be mentioned that the extra layer degree of freedom
in bilayer systems leads to more possibilities in pairing. In this way,
e.g., the possibility of interlayer pairing arises (35). Since the pairing
mechanism is not established despite the strong indication for electron-
phonon coupling, we want to briefly assess the role of strong electron
correlation (36). It is possible that electron correlation contributes to the
flatness of the band. For graphene, this has been predicted (31). We
have performed model calculations to simulate complete photoemission
spectra. As a result, the small broadening in the experiment at higher
energies is incompatible with a significant role of electron correlation
for the flat band dispersion.
Returning to the question of electron-phonon coupling, we note
that the disk-like constant energy surface around the KandK′points
of the graphene Brillouin zone favors enhanced intravalley and
K→K′intervalley scattering processes when the flat band is shifted
to the Fermi level. With small doping/gating of only a few milli–electron
volts, the Fermi surface can be changed between circle and disk shapes,
strongly affecting the number of possible scattering channels.
The measured band structure shows n-doping due to the substrate
influence; therefore, the Dirac cone and the flat band in discussion are
located significantly below the Fermi level. This means that it is nec-
essary to bring the flat band to the Fermi energy to examine possible
superconductivity. This is possible by doping (21)andgating(37). We
recall that the possibility of doping large amounts of charge carriers
to the graphene layer was realized by combined Ca intercalation and
K deposition, resulting in bringing a 1D extended van Hove singularity
along the MK direction from more than 1 eV above down to the Fermi
level (17). In the present case, four times less doping but of the p-type
must be accomplished. Fortunately, p-doping of BLG on SiC has been
demonstrated as well (21).ItwasshownthatF4-TCNQmoleculescom-
pensate n-doping of BLG on SiC and make it charge neutral (21).
Based on the proposed model, n-doping of only one graphene sub-
lattice (either B or C) of initially undoped BLG leads to gap opening
along with an instant flattening of the dispersion at K. With increased
doping, the flat band area increases, but the energy position remains
fixed (fig. S6). In a device, however, single-sublattice doping is difficult
to control. Thus, the main approach of modification of the band dis-
persion is supposed to be the gate biasing with corresponding change
of interlayer asymmetry until both interlayer and sublattice asym-
metries compensate each other at the A sublattice of the bottom layer.
By using a double-gate device configuration (37), it should become
possible to control both the doping and the interlayer asymmetries
independently and in operando.
CONCLUSIONS
Our high-resolution ARPES study of BLG on 6H-SiC shows that the
band structure around the K point has neither the predicted parabolic
nor the Mexican hat shape but stays in an intermediate stage causing
flat band dispersion. The band has a number of unusual properties
–1
Wave vector k (A )
||
Binding energy (eV)
0.247
A
K
0.0
0.015
–0.015
–1
Wave vector k (A )
||
B
K
0.249
0.251
0.0 0.015
–0.015
Interlayer asymmetry change Sublattice asymmetry change
Eb = 0.38 = 0.2
Eb = 0.35 = 0.2
Eb = 0.32 = 0.2
Eb = 0.35 = 0.1
Eb = 0.35 = 0.2
E = 0.35 = 0.3
b
Fig. 3. Interlayer/sublattice asymmetry. Demonstration of the possibility of transforming the flat band electronic structure into a band with either positive or negative
effective mass by changing (A) the interlayer asymmetry and (B) the sublattice asymmetry of the bottom layer. The calculation is based on the Hamiltonian H
2
.
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such as very high photoemission intensity, very high DOS without de-
tectable dispersion and narrow width, contribution of only one graphene
sublattice, and the absence of photoemission interference effects. We
explain the mechanism of the flat band appearance and show that, by
influencing sublattice and interlayer asymmetries, one can radically
control the band shape and its properties. Indications of enhanced
electron-phonon coupling, together with the discussed possibility of
creating and controlling the flat band, are related to the question of
high-temperature superconductivity in graphene- and graphite-based
systems, while, on the other hand, the mechanism of the flattening of
the dispersion is more universal and could be used in transport appli-
cations beyond BLG.
METHODS
Experiment
Measurements were performed using linearly polarized undulator
radiation from the UE112 beamline and hemispherical analyzers of
three experimental stations at BESSY II: “ARPES 1
2
”station equipped
with a Scienta R8000 analyzer for initial high-resolution studies (T=
60 K), “ARPES 1
3
”station with a Scienta R4000 analyzer for high-
resolution low-temperature studies (T=1K),andPHOENEXSstation
for room temperature core-level studies. Graphene on nitrogen-doped
6H-SiC(0001) was produced as described in (38). The substrate was
etched in molecular hydrogen at a pressure of 1 bar and a temperature
of 1425°C to remove polishing damage. Then, graphene was grown by
annealing the sample in 1 bar of argon at a temperature of 1675°C.
The sample was transferred in air into the photoemission station and
subsequently annealed. Low-energy electron diffraction (LEED), x-ray
photoelectron spectroscopy (XPS), photoemission electron microsco-
py, and ARPES from this sample showed that it is mostly one MLG
with 20% contribution of BLG. After the initial study, the sample was
stored in air, annealed at 250°C for 20 min, and studied again. The sam-
ple quality was the same without any significant changes in LEED, XPS,
and ARPES. Subsequently, the sample was annealed several times in
ultrahigh vacuum with increased temperatures in the range of 1300°
to 1400°C, until parts of the surface appeared with mostly BLG. At all
stages, ARPES and LEED measurements were conducted for control
of the sample quality and the graphene layer thickness. All results shown
here were from the same sample, and the same results were obtained
from other samples grown at other times, which only differ in the bilayer/
monolayer ratio.
Theory
DFT calculations were performed using VASP (26)withtheprojector
augmented-wave method in the generalized gradient approximation.
Van der Waals forces were taken into account using the DFT-D2
method. The calculated system is a slab of six Si-C layers (ABCACB
stacking), zero-layer graphene (buffer layer), and graphene layers on
top (one for MLG, two for BLG, and three for TLG) (see Fig. 2D). The
vacuum region between the slabs was about 27 Å. The back side was
hydrogen passivated.
SUPPLEMENTARY MATERIALS
Supplementary material for this article is available at http://advances.sciencemag.org/cgi/
content/full/4/11/eaau0059/DC1
Fig. S1. BLG ARPES.
Fig. S2. MLG, BLG, and TLG/SiC.
Fig. S3. BLG sublattice contributions.
Fig. S4. TLG/SiC.
Fig. S5. Graphene/Ir thickness dependence.
Fig. S6. Unsupported BLG.
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Acknowledgments: We thank P. Wehrfritz and F. Fromm for help with sample preparation.
Funding: This work was supported by SPP 1459 of the Deutsche Forschungsgemeinschaft and
by “Impuls- und Vernetzungsfonds der Helmholtz-Gemeinschaft”through a Helmholtz-Russia
Joint Research Group (grant no. HRJRG-408). Author contributions: D.M. and O.R. initiated the
project. T.S. prepared the samples. D.M., D.V.E., E.G., and A.V. conducted ARPES measurements.
D.M. and E.G. conducted DFT calculations. D.M. and O.R. wrote the manuscript with input from all
coauthors. Competing interests: The authors declare that they have no competing interests.
Data and materials availability: All data needed to evaluate the conclusions in the paper are
present in the paper and/or the Supplementary Materials. Additional data related to this paper
may be requested from the authors.
Submitted 8 May 2018
Accepted 4 October 2018
Published 9 November 2018
10.1126/sciadv.aau0059
Citation: D. Marchenko, D. V. Evtushinsky, E. Golias, A. Varykhalov, Th. Seyller, O. Rader,
Extremely flat band in bilayer graphene. Sci. Adv. 4, eaau0059 (2018).
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Extremely flat band in bilayer graphene
D. Marchenko, D. V. Evtushinsky, E. Golias, A. Varykhalov, Th. Seyller and O. Rader
DOI: 10.1126/sciadv.aau0059
(11), eaau0059.4Sci Adv
ARTICLE TOOLS http://advances.sciencemag.org/content/4/11/eaau0059
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SUPPLEMENTARY http://advances.sciencemag.org/content/suppl/2018/11/05/4.11.eaau0059.DC1
REFERENCES http://advances.sciencemag.org/content/4/11/eaau0059#BIBL
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