#
Dirac Cones in Graphene, Interlayer Interaction in Layered Materials, and the Band Gap in MoS_{2}

## Abstract

**:**

_{2}monolayers; and (iii) the issue of a 2D screening in estimates of the band gap for MoS

_{2}monolayers.

## 1. Introduction

_{2}, a typical representative of layered transition metal dichalcogenides, has been roused not only by its well-known lubrication properties but also by its very suitable band gap width (about 1 eV) for applications in nanoelectronics [10].

_{2}monolayer exhibits a dramatic increase in luminescence quantum efficiency compared to the bulk material [11,12]. The explanation for these phenomena was found with the help of the band structure calculations [12,13,14,15,16], which have revealed the transformation of the band gap from an indirect one for the bulk MoS

_{2}to a direct one for the monolayer and an increase in the width of the band gap.

_{2}monolayers.

_{F}band crossing at the K point were obtained from band structure calculations performed with one or another form of Schrödinger’s equation (for example, within DFT or tight-binding approximation), which afterwards were suggested to be invalid.

_{2}monolayers (Section 3); (iii) screening and estimates of the band gap for MoS

_{2}monolayer (Section 3). The band structures of model layered systems, distributions of electronic densities, and interlayer interactions, presented in Figure 1, Figure 2, Figure 3 and Figure 4, were calculated with the ABINIT program package, using the supercell model, norm-conserving pseudopotentials, and LDA approximation for exchange-correlation; see [15] for details. Main conclusions and final remarks are presented in Section 4.

## 2. Dirac Cones and Buckling in Graphene and Similar Layers

_{F}. Similar shifts of the bands were reported as a result of the interaction of graphene monolayers with the substrate surface [7,19,29,30,31,32]. For example, it was found, by angle-resolved photoelectron spectroscopy, that graphene on Ir(111) displays a Dirac cone with the Dirac point shifted only slightly above the Fermi level. The moiré resulting from the overlaid graphene and Ir(111) surface lattices imposes a superperiodic potential giving rise to Dirac cone replicas and the opening of minigaps in the band structure [33].

_{6v}to C

_{3v}, by breaking the equivalency of the graphene sites, leads to the opening of a band gap in the otherwise gapless semiconductor graphene [23]. It should be noted that there is an interplay between the energy cost or strain energy for graphene structural reconstructions and reduction in energy opening up a band gap, but when a reduction of the symmetry is allowed, graphene can lower the total free energy of the system and a band gap will open at the Dirac point. The opening of the gap due to the layer distortion can be illustrated by means of calculations for a model distorted graphene layer, in which every second atom is shifted by ~0.05 Å in the direction towards one of the 3 nearest neighbors (as depicted in the insert in Figure 1b). This type of distortion indeed results in the opening of the band gap (Figure 1b), as anticipated.

_{F}is small. For graphene layers, symmetric with respect to the z = 0 plane (AA), in contrast, the bilayer becomes a semiconductor with the direct gap of 0.45 eV (Figure 2b). It should be noted that the drastic change of the electronic structure from semimetal to semiconductor caused by the relative shift of graphene monolayers indicates an importance of the interlayer interaction in graphite and raises the question of whether this interaction is indeed vdW in nature.

_{F}, that is, to a semiconducting state of the layer.

_{F}at K point, which, nonetheless, can be calculated without involving the Dirac equations, for a free perfect infinite graphene monolayer. In reality, however, a free perfect monolayer never can be obtained, and therefore the perfect cone bands at K point of BZ for graphene, as follows also from all performed to-date photoemission experiments, can exist only in theory, while in reality the cone will unavoidably be distorted because of the related change of symmetry. In other words, the cone-like bands observed in photoemission experiments are not true Dirac cones, as they, in fact, do not correspond to zero-mass Fermions. This is good news, because it allows for ordinary non-relativistic DFT calculations for graphene-containing layered structures.

## 3. The Interlayer Interaction in Layered Crystals

_{2}) is usually explained by van der Waals (vdW) forces. It should be clarified in this regard that physical chemistry attributes all interactions beyond Coulomb and exchange interactions to vdW forces, which therefore include electrostatic forces between permanent dipoles (Keesom forces), permanent dipoles and a corresponding induced dipoles (Debye forces), and London dispersion forces (which are sometimes explained in terms of the interaction between instantaneously induced dipoles). The modern understanding of dispersion forces does not involve the concept of temporal virtual dipoles and explains the attraction between neutral molecules in terms of electrostatic forces. This type of interatomic interaction is of the same origin as the exchange interaction between electrons which can be explained as Coulomb interaction of fermions having antisymmetric wave functions. In solid state physics, it is usually just the dispersion forces that are called van der Waals forces.

_{2}bilayer [15] (Figure 3). The most important result of the calculations is the revealed significant overlap of the wave functions of adjacent MoS

_{2}layers (the distribution of the plane-averaged electronic density of the bilayer along the normal to the surface is shown in Figure 3b). In contrast, the GGA gives a substantially different distribution of electronic density (shown by dashed line in Figure 3b). In particular, the GGA electronic density at the middle point between the layers is found to be approximately 10 times less than in LDA calculations. Accordingly, the energy of the interlayer interaction, estimated with GGA, decreases to 0.008 eV/cell from the LDA value of 0.12 eV/cell.

_{2}can be significantly increased by means of H or alkali metal intercalation. In particular, for hydrogen-intercalated MoS

_{2}bilayers, due to forming S–H–S bonds, the interaction energy increases from 0.12 eV to 0.60 eV [62]. Very similar results were also reported for Li [63] and Na [64] intercalated MoS

_{2}bilayers. However, in contrast to intercalated hydrogen, Li and Na do not reconstruct the MoS

_{2}bilayer, which retains the central symmetry pertinent to the bulk. In all cases, the intercalation leads to metallization, which is evident from the appearance of the bands crossing Fermi level and significant density of states at E

_{F}.

## 4. Band Gap and Screening in MoS_{2} Layers

_{2}and free MoS

_{2}monolayer, calculated within LDA [15], are shown in Figure 4. The reduced symmetry of the monolayer (because of the absence of the inversion symmetry) with respect to the bulk MoS

_{2}reveals itself in the k-dependent spin-orbit splitting of the bands [65,66,67,68,69] (Rashba effect). At K point of BZ the splitting of the topmost valence band is quite pronounced (0.15 eV).

_{↑}(

**k**) = E

_{↓}(

**k**)], originated from the combination of time-reversal [E

_{↑}(

**k**) = E

_{↓}(−

**k**)] and inversion symmetry [E

_{↑}(

**k**) = E

_{↑}(−

**k**)] [65,66], similar relativistic calculations have not indicated any spin-orbit splitting of the bands.

_{2}, the LDA-estimated gap of 0.76 eV is significantly less than the experimental value 1.2–1.3 eV.

_{2}, GGA, due to increased (by 10/7 [73]) exchange term in the PBE [74] exchange-correlation functional [72] usually gives somewhat better estimates of the gaps [75]. (It should be mentioned in this regard that GGA has severe intrinsic problems, in particular, the sum rule for the exchange-correlation hole cannot be satisfied, which leads to an incorrect behavior of the wave function, so that, for example, Friedel oscillations do not appear with the gradient corrections to LDA [61]; furthermore, the gradient expansion does not converge (that is, invalid) for any realistic systems, where the density gradients always exceed the convergence criterion for an order of magnitude [61,76,77]).

_{2}, estimated in [14] using a sophisticated quasiparticle self-consistent GW approximation (QSGW), is found to be 1.297 eV, which is in a perfect agreement with the generally accepted experimental value. For a MoS

_{2}monolayer, however, the gap calculated either with GW or QSGW approximation (2.759 eV [14]), exceeds the 1.90 eV energy of a prominent photoluminescence (PL) band by ~0.9 eV. To explain this dramatic difference, it was suggested [14] that the PL band originates from the recombination of a 2D Wannier–Mott exciton with enormously high binding energy because of inherently 2D screening of the Coulomb interaction in the monolayer.

_{2}, MoSe

_{2}, MoTe

_{2}, WS

_{2}, and WSe

_{2}monolayers were studied also using the GW approximation in conjunction with the Bethe–Salpeter equation [78]. The transition energies for monolayer MoS

_{2}are shown to be in excellent agreement with available absorption and photoluminescence measurements. Similar conclusions were derived also from the study of many-body effects and diversity of exciton states and their role in the formation of the optical spectrum of MoS

_{2}by Qiu et al. [79] and in recently performed STS and PL studies of MoSe

_{2}layers on graphene substrate, supported by GW and Bethe–Salpeter calculations of exciton binding energies [80,81]).

_{2}monolayer, Mak et al. [11] reported an abrupt rise of photoconductivity by 3 orders of magnitude at ~1.8 eV, which was attributed to a direct gap photoexcitation. (It should be noted, however, that this value is lower than the energy of the PL peak, reported in this study, which is somewhat confusing and might be attributed to some influence of the substrate and conditions of the experiment).

_{2}and ReS

_{2}by high resolution electron energy loss spectroscopy (HREELS) [87] have reported an indirect band gap of 1.27 eV obtained from the multilayer regions (i.e., essentially bulk MoS

_{2}). For the monolayer, the band gap becomes direct (with the valence band maximum and conduction band minimum at the K point of the Brillouin zone) and increases to 1.98 eV. For monolayer MoS

_{2}, the twin excitons (1.8 and 1.95 eV) originating at the K point are observed (note the 0.15 eV exchange splitting due to Rashba effect, c.f. with the theoretically estimated splitting, Figure 4b). It should be noted that the energies of the exciton peaks determined by HREELS [87] well agree with the energies of PL lines for the MoS

_{2}monolayers [12]. Then, the difference between the width of the band gap for the monolayer (1.98 eV) and the energies of PL lines (1.8 and 1.95 eV) is about 0.03–0.18 eV, which is in reasonable agreement with anticipated (i.e., usual) values of Wannier-Mott exciton binding energies [82]. The energy of trion, which also was reported to be detected for the monolayer, is also in this range.

_{2}on graphene [88,89,90,91,92,93]. It was found that the electronic structure of two-dimensional (2D) semiconductors can be significantly altered by screening effects. The results obtained using time- and angle-resolved photoemission (ARPES) reveal a significant (~400 meV) reduction of the band gap of the MoS

_{2}layer induced by optical excitation [89]. The band gap and photoluminescence shift were reported to depend on the orientation of the graphene and MoS

_{2}monolayers [90,91]. The changes in electronic structure of graphene caused by interaction with MoS

_{2}monolayer were suggested to be less pronounced. From ARPES study of this heterostructure Diaz et al. [92] concluded that the Dirac cone of graphene remains intact and no significant charge transfer doping was detected. Later, Pierucci et al. [93] confirmed that, close to the Fermi level, graphene exhibits a robust, almost perfect, gapless Dirac cone, but suggested the graphene to be n-doped.

_{2}monolayer is in the range of 1.8–2.15 eV. Then, the 2.9 eV value obtained in GW calculations either should be attributed to the difference between “electronic” and “optical” gaps, produced by excitons with enormously large binding energies, or to apparent problems with evaluations of 2D dielectric function. In my view, the latter explanation is more consistent since it also explains the values obtained for the gaps in photoconductivity and HREELS measurements.

## 5. Conclusions

_{F}at the K point (which, nonetheless, can be calculated without involving the Dirac equations), for a free perfect infinite graphene monolayer. It is well known that a free perfect monolayer can never exist in reality because of an inherent instability, and therefore the perfect cone bands at the K point of BZ for graphene, as also follows from all performed to-date photoemission experiments, can exist only in theory, while in reality the cone will unavoidably become distorted because of the related change of symmetry. In other words, the cone-like bands observed in photoemission experiments are not true Dirac cones, as they, in fact, do not correspond to zero-mass Fermions. This is a good news, because allows for ordinary non-relativistic DFT calculations for graphene-containing layered structures.

## Conflicts of Interest

## Abbreviations

DFT | Density Functional Theory |

LDA | Local Density Approximation |

GGA | Generalized Gradient Approximation |

GW | GW quasi-particle Hedin's approach |

QSGW | Quasiparticle Self-consistent GW approximation |

BZ | Brillouin Zone |

vdW | van der Waals |

HREELS | High Resolution Electron Energy Loss Spectroscopy |

2D | 2-Dimensional |

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**Figure 1.**Band structures in vicinity of Fermi level for (

**a**) an ideal free graphene; (

**b**) distorted graphene monolayer; and (

**c**) rippled Sn monolayer (stanene).

**Figure 2.**Band structures for free graphene bilayers with the AB layer configuration, pertinent to a bulk graphite (

**a**); and the AA configuration, in which carbon atoms of the second layer are positioned atop the atoms of the first layer (

**b**).

**Figure 3.**The dependence of the energy of interlayer interaction (per unit cell) for the MoS

_{2}bilayer on the distance between the layers (

**a**) and the plane-averaged electronic density of the bilayer along the normal to the surface (

**b**). LDA and GGA results are shown by solid and dashed lines, respectively.

**Figure 4.**The band structures of the (

**a**) bulk MoS

_{2}and (

**b**) MoS

_{2}monolayer calculated within LDA in a fully relativistic (i.e., with account for spin-orbit coupling) approximation. Shaded areas denote the band gaps. Valence band maxima and conduction band minima are marked by circles.

© 2016 by the author; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Yakovkin, I.N.
Dirac Cones in Graphene, Interlayer Interaction in Layered Materials, and the Band Gap in MoS_{2}. *Crystals* **2016**, *6*, 143.
https://doi.org/10.3390/cryst6110143

**AMA Style**

Yakovkin IN.
Dirac Cones in Graphene, Interlayer Interaction in Layered Materials, and the Band Gap in MoS_{2}. *Crystals*. 2016; 6(11):143.
https://doi.org/10.3390/cryst6110143

**Chicago/Turabian Style**

Yakovkin, Ivan N.
2016. "Dirac Cones in Graphene, Interlayer Interaction in Layered Materials, and the Band Gap in MoS_{2}" *Crystals* 6, no. 11: 143.
https://doi.org/10.3390/cryst6110143