INTRODUCTION
Bilayer graphene (BLG) has, along with related two-dimensional (2D) materials, extensively been studied by both transport and photoemission 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 external 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) raise a number 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 [
TC = 11.6 K (
7)], which can also be thinned down to an intercalated bilayer as a 2D superconductor [C
6CaC
6 with
TC = 4 K (
8)], as well as doped fullerides such as Cs
2RbC
60 [
TC = 33 K (
9)]. Here, the alkali and alkaline earth metals act to increase the carrier concentration and density of states (DOS) at the Fermi energy. Most recently, superconductivity was discovered in twisted BLG without any alkali doping and a
TC 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
TC ∝ exp[− 1/(
Un(
EF))] between the critical temperature
TC of superconductivity, the coupling constant
U of the effective attractive interaction, and the DOS at the Fermi energy
n(
EF) such that, according to the BCS theory,
TC can be enhanced by increasing either
U or
n(
EF).
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
TC values (
12). While
U remains difficult to assess, a flat band will lead to maximal values of
n(
EF). Moreover, in this case of
U ≫
W, the BCS theory predicts
TC ∝
Un(
EF); i.e., the exponential suppression of
TC with the interaction strength is removed (
12). In this context, rhombohedral (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
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
point of monolayer graphene (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 superconductor. For “magic angles” between the two graphene layers, the moiré pattern leads to flat band formation (
19), and the observed superconductivity 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 measurements 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 experimentally no dispersion around the graphene
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, promising 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 sublattice relative to other sublattices, which is possible for both doped and undoped BLG. We show that the band flattening effect is universal and achievable by different means of combining interlayer asymmetry, sublattice asymmetry, and doping. The mechanism allows control of the band dispersion all the way from parabolic through flat band formation to Mexican hat–like.
DISCUSSION
The experimental high photoemission intensity of the flat band can be partially explained by the 2D extent and broadening in
kx and
ky. This broadening, however, is not the reason for the flatness and the observed features of the band. If this broadening were to play a significant role, we would see the narrowing and intensity enhancement effects also at other BLG bands around the
point where their d
E/d
k becomes 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 the wave function on different graphene sublattices (
24,
25); however, for the flat band, the wave function is localized on one sublattice 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 result. 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 the wave function to the top graphene 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 intense 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 renormalization 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 various 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
and
points of the graphene Brillouin zone favors enhanced intravalley and
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 necessary to bring the flat band to the Fermi energy to examine possible superconductivity. This is possible by doping (
21) and gating (
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
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). It was shown that F4-TCNQ molecules compensate n-doping of BLG on SiC and make it charge neutral (
21).
Based on the proposed model, n-doping of only one graphene sublattice (either B or C) of initially undoped BLG leads to gap opening along with an instant flattening of the dispersion at
. 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 dispersion is supposed to be the gate biasing with corresponding change of interlayer asymmetry until both interlayer and sublattice asymmetries 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.