Critical Sublattice Symmetry Breaking: A Universal Criterion for Dirac

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Critical Sublattice Symmetry Breaking: A Universal Criterion for Dirac Cone Splitting Ritesh Kumar, Deya Das, Enrique Munoz, and Abhishek K. Singh J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.9b07602 • Publication Date (Web): 02 Sep 2019 Downloaded from pubs.acs.org on September 2, 2019

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The Journal of Physical Chemistry

Critical Sublattice Symmetry Breaking: A Universal Criterion for Dirac Cone Splitting Ritesh Kumar,† Deya Das,† Enrique Muñoz,‡ and Abhishek K. Singh∗,† Materials Research Center, Indian Institute of Science, Bangalore, Karnataka 560012, India, and Facultad de Física, Pontificia Universidad Católica de Chile, Vicuña Mackenna 4860 Santiago, Chile E-mail: [email protected]

Abstract

iting its application in electronic industry. Intensive efforts on opening the bandgap in graphene led to development of a few promising approaches including synthesis of nanoribbons, 4–6 application of bias potential on bilayer, 7 doping, 8–10 and chemical functionalization. 11–13 Recent progress in monolayer transfer technologies has fuelled the synthesis of unprecedented number of van der Waals heterostructures of graphene 14 having varied electronic properties ranging from metal to semiconductors. So far, bandgap opening has been achieved in the heterostructures of graphene with hBN, 15,16 SiC, 17,18 h-C3 N4 , 19 CH, 20 g-GeC. 21 Often times, this bandgap opening has been attributed to the sublattice symmetry breaking. 16,20,22,23 Interestingly, in several heterostructures like graphene/MoS2 (gr/MoS2 ), 24 graphene/SnS2 25 (gr/SnS2 ), or graphene/phosphorene (gr/phos) 26 in spite of a sublattice symmetry breaking, the Dirac cone does not split. On the other hand, there are heterostructures, where even after Dirac cone splitting, the system remains metallic. This is due to (up) downward shift of either half of the Dirac cone inside the (conduction) valence band, leading to finite states at the Fermi level. 27,28 These results are quite puzzling as a consistent understanding of factors influencing the electronic structure of graphene heterostructure remains elusive. In this article, we report that a critical sublattice symmetry breaking is required to split the Dirac cone in graphene heterostructure. Moreover, we develop a simple and robust scheme to identify inequivalence in the nearest neighbour sublattice pairs. In order to split the Dirac cone, the number of such inequivalent pair must be in excess of

Sublattice symmetry breaking has been identified as the necessary condition for bandgap opening in monolayer graphene-on-substrate heterostructures. In many of them, however, in spite of sublattice symmetry breaking, the Dirac cone of graphene remains preserved. Here, we report using firstprinciples density functional theory (DFT) and a simple tight-binding (TB) model, that the presence of more than 50% symmetrically inequivalent carbon atoms are required to split the Dirac cone. Additionally, we find that the Dirac cone must also lie within the bandgap of the other 2D layer to get a semiconducting (non-metallic) heterostructure. The robustness of these two criteria have been validated in a series of heterostructures of graphene. The simplicity and robustness of the proposed model provides a useful design principle for materials scientists and engineers, thus potentially expanding the applicability of graphene bilayer heterostructures to a multitude of semiconductor devices.

Introduction Electronic structure of graphene has a unique linear dispersion and vanishing carrier mass at the Fermi level, which have opened the doors to an enormous number of possible applications. 1,2 However, the absence of bandgap results into a lower on-off ratio in pristine graphene-based devices, 3 thereby, lim∗

To whom correspondence should be addressed Materials Research Center, Indian Institute of Science, Bangalore, Karnataka 560012, India ‡ Facultad de Física, Pontificia Universidad Católica de Chile, Vicuña Mackenna 4860 Santiago, Chile †

ACS Paragon Plus Environment

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50%. The robustness of this criterion is validated on a series of graphene heterostructures including carbon nitrides, transition metal dichalcogenides (TMDCs), oxides, MXene, and phosphorene. Furthermore, to get a semiconducting graphene heterostructure, the split-Dirac cone must lie within the band gap of the other 2D material. In addition, a simple analytical model leading to an intuitive stoichiometric criteria required to split the Dirac cone, was developed to gain insight into this phenomenon. However, the proposed scheme cannot predict the bandgap of graphene-based heterostructures. In summary, we present a simple, robust and reliable criteria that can be applied by materials scientists and engineers in the design of 2D graphene-based heterostructures with tailored electronic properties. Very recently, bandgap opening of graphene due to proximity effect have been observed, which require strong topological insulators as substrates. 29 For this effect to be prominent, the heterostructure usually consist of three layers, 30,31 making it out of scope in the present study.

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Figure 1: Schematic for classifying any two given carbon atoms in the graphene layer into symmetrically equivalent or inequivalent. The carbon atoms in the graphene layer are shown by cyan-colored circles, and atoms in the layer below graphene are shown by bigger circles. The carbon atoms under consideration are represented as Ci ’s, and the nearest neighbour carbon atoms are represented as Nj ’s (j = 1, 2, 3).

Results and discussion

Methodology

Thermodynamic and dynamic stability

The first-principles calculations were performed using density functional theory (DFT) as implemented in the Vienna ab initio simulation package (VASP). 32 Electron-ion interactions were described by all electron projector augmented wave (PAW) pseudopotentials 33 and electronic exchange and correlation were approximated by a Perdew-BurkeErnzerhof (PBE) generalized gradient approximation (GGA). 34 The periodic images were separated by a 20 Å vacuum along z-direction to prevent spurious interactions. For structure optimization, the Brillouin zone was sampled by a 9x9x1 MonkhorstPack grid. All structures were fully relaxed using a conjugate gradient scheme until the energies and each component of forces were less than 10−5 eV and 0.005 eV Å−1 , respectively. The empirical atom-pairwise corrections proposed by Grimme in terms of the DFT-D2 scheme was used for describing the long-range van der Waals interactions. 35 The dynamical stability of heterostructures were investigated by calculating phonon bandstructures using the Parlinski–Li–Kawazoe method as implemented in the PHONOPY 36 package.

A total of 27 heterostructures were considered for this study, see Table S1 in the supporting information (SI). The lattice parameters for the heterostructures were chosen to minimize the lattice mismatch between graphene and the 2D substrates. To check the thermodynamic stability of heterostructures, we calculated interface binding energy according to the expression: 37,38 Eb =

Eheterostructure − Esubstrate − Egraphene . (1) A

where, Eheterostructure , Esubstrate , Egraphene , and A represent the total energies of the heterostructure, 2D substrate layer, and graphene layer, and area of the unit cell of heterostructure, repspectively. Eb for a heterostructure should be exothermic for it to be thermodynamically stable, and it was found that 18 out of the 27 studied heterostructures satisfy this criteria (Table S1). Most of these 18 heterostructures have also been synthesized experimentally, such as gr/h-BN, 15 gr/MoS2 , 24,39 gr/t-C3 N4 , 40 etc, thus confirming the applicability of the equation 1. We also checked the dynamical stability of some of the heterostructures using phonon band structure calculations (Fig. S1-S4). It was observed that while the two stackings of gr/PtSe2 (Fig. S1(a) and S1(b)) and gr/C3 N (Fig. S2(a) and S2(b)) are


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types of carbon atoms, we consider { U1 , R over type 1 V (R) = U2 , R over type 2

dynamically stable, the two stackings of gr/h-BN heterostructures exhibit very small imaginary frequencies (Fig. S3(a) and S3(b)), which could be removed by increasing the supercell size or changing the functional. Gr/BC3 heterostructure, which is thermodynamically unstable according to equation 1, is also found to be dynamically stable (Fig. S4(a) and S4(b)).

(3)

In the vicinity of the Dirac point, the predicted energy spectrum of graphene in presence of the substrate is given by (see SI for mathematical derivation): √ ( )2 ∆ 2 E± (q) = δ ± ¯hvF q + . (4) 2¯hvF

Scheme for assigning sublattice symmetry breaking Next, we develop a method to quantify the degree of sublattice symmetry breaking in graphene heterostructures by classifying any two nearest neighbour carbon atoms in the graphene layer to be either symmetrically equivalent or inequivalent, as shown in Fig. 1. If two carbon atoms in graphene along with their first nearest neighbours lie above similar atoms of the substrate layer, then they are assigned to be equivalent. For example, C1 (Fig. 1(ii)) and C2 (Fig. 1(iii)) and their nearest neighbours (N1 , N2 and N3 ) are present above atom 1 of the substrate layer, and hence are equivalent. If the arrangement of any of the atoms right below the pair of carbon atoms and their nearest neighbours change, then these are considered inequivalent. For example, atoms C1 (Fig. 1(ii)), C3 (Fig. 1(i)), C4 (Fig. 1(iv)) and C5 (Fig. 1(v)) according to our scheme are all inequivalent. Similarly, any such change in the arrangement of atoms lying below the pair of carbon atoms and their nearest neighbours are considered to be inequivalent. A count of equivalent and inequivalent atoms are then correlated to the Dirac cone splitting through a model.

Here, we have defined the Dirac cone splitting parameter 1 (1) (1) (2) (2) ∆ = (nA − nB )U1 + (nA − nB )U2 , (5) n and the Dirac cone shift by δ=

] 1 [ (1) (1) (2) (2) (nA + nB )U1 + (nA + nB )U2 , 2n

(6)

respectively. In these equations, U1 and U2 are the effective potentials acting on a carbon atom, when it is above type 1 and type 2 atom of the substrate, (1) respectively. On the other hand, the integers nA , (1) nB represent the number of A or B carbon atoms, which are on top of a type 1 atom of the substrate, (2) (2) with the analogous definitions for nA , nB , representing those on the top of a type 2 atom. The integer n represents the number of A and B sublattices inside the supercell due to the presence of the substrate. The Eq.(5) predicts a Dirac cone split(1) (1) (2) (2) ting if either nA − nB ̸= 0 or nA − nB ̸= 0, a condition which should be satisfied only when there is a critical breaking of the sublattice symmetry.

Tight-binding formalism Some examples

The model captures the effect of chiral symmetry breaking due to the presence of the substrate, by introducing an interaction term to the tight-binding Hamiltonian for a free-standing graphene: ∑ ˆ int = H V (RA )ˆ a†σ (RA )ˆ aσ (RA )

We next validate the counting scheme and the model on all of the 27 heterostructures. In the gr/h-BN with AA and AB stackings (Fig. 2(a) and 2(b)), the sublattice symmetry is broken in both of the cases. In the AA stacking, carbon atoms of A (B ) sublattice are directly above boron (nitrogen) atom, thereby breaking the inherent equivalence of the two sublattices. Taking boron atoms to be of type 1 and the nitrogen atoms to be of type 2, then (1) (2) from the notation of Eqs.(4–6): nA = 1, nA = 0, (1) (2) nB = 0, nB = 1. Therefore, for this case we have (1) (1) (2) (2) nA − nB = 1, while nA − nB = −1, and thus Eq.(5) predicts a Dirac cone splitting. On the other

σ,RA

+

∑

V (RB )ˆb†σ (RB )ˆbσ (RB ).

(2)

σ,RB

Here, V (R) represents the effective local potential that acts at sites RA , RB of the A and B graphene sublattices, respectively, due to the presence of the substrate (please see SI). In particular, for a graphene-on-substrate stacking defining two


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According to this convention, there will be a total of 27 (A,B ) sublattice pairs corresponding to 3 neighbours for each of the 9 A sublattices in gr/h-C3 N4 . In the AA stacked gr/h-C3 N4 , (A1,B 1) sublattice pair is arranged in an inequivalent configuration as shown in Fig. 1(iv) and 1(v). Similarly, (A9,B 9) pair is also arranged in an inequivalent configuration. On the other hand, in the AB stacked gr/hC3 N4 , (A1,B 1) sublattice pair is symmetrically inequivalent of the type shown in Fig. 1(i) and 1(iv), and (A9,B 9) pair is yet another inequivalent configuration. By taking the carbon and nitrogen atoms in h-C3 N4 layer to be of type 1 and 2 atoms, re(1) (2) (1) spectively, we find that nA = 6, nA = 0, nB = 0 (2) and nB = 8 from Fig. 2(c). Therefore, Eq.(5) predicts a Dirac cone splitting for AA stacking. On (1) (2) the other hand, for AB stacking we have nA , nA , (1) (2) nB , and nB , being equal to 0, 0, 6 and 0, respectively as shown in Fig. 2(d). Eq.(5) again predicts a Dirac cone splitting. The bandgap opening in the graphene layer is, therefore, observed in both the cases (Fig. 3(c) and 3(d)).

hand, in AB stacking, A and B carbon atoms are directly above the nitrogen atoms and the center of the hexagon of BN sheets, respectively. Therefore, (2) (2) (2) (2) nA − nB = 1 (as nA = 1, nB = 0) indicating the breaking of the sublattice symmetry. As a result, the bandgap is found to open in both the cases (Fig. 3(a) and 3(b)). The amount of splitting in Dirac cone (termed as ∆) in the AA stacking is 76 meV, while that in the AB stacking is 51 meV. (a)

B (b)

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(f) B1 B2 B1 B2 B1 A3 A4 A4 B3 B4 B3 B3 B4 B3 A1 A2 A1 A2 B1 B2 B1 B1 B2 B1 B1 A3

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Figure 2: Top view of (a) AA and (b) AB stacking of gr/h-BN, (c) AA and (d) AB stacking of gr/h-C3 N4 , (e) AA and (f) AB stacking of gr/C3 N heterostructures. Cyan, voilet, grey and orangecolored circles represent C (graphene), B, C (hC3 N4 and C3 N) and N atoms, respectively. The size of circles in the bottom layers have been made larger to show the kind of atoms that are present below C atoms in the graphene layer.

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We next extend this approach to analyze the sublattice symmetry breaking on other 2D sheets having lattices with more than two atom basis. We used a convention for systematic naming of these sublattice pairs, wherein half of the atomic positions in the graphene supercell are assigned Ai and the other half B i, where i is an integer. Such labelling convention ensures the two adjacent carbon atoms to be always of A and B type. For example, in the case of AA and AB stackings of gr/h-C3 N4 (Fig. 2(c) and 2(d)), A1 is surrounded by B 1, B 2 and B 4 and A2 is surrounded by B 2, B 3 and B 5. Using the scheme developed above, we search for asymmetry among the arrangements of the pairs formed by nearest neighbours of A or B sublattices, e.g., (A1,B 1), (A1,B 2), (A1,B 4), (A4,B 4), (A4,B 5), (A4,B 7), etc. (Fig. 2(c) and Fig. 2(d)).

-3 K (e) 3 E - Ef (eV)

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Figure 3: Electronic band structures of AA and AB stackings of (a,b) gr/h-BN, (c,d) gr/h-C3 N4 , and (e,f) gr/C3 N heterostructures, respectively. The magnified band structures near the Dirac cones of graphene are also shown. In the case of gr/C3 N, the tendency for Dirac cone splitting is dependent on the type of stack-


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ing. While the Dirac cone splits in the AB stacking, no such splitting is observed in the AA stacking (Fig. 3(e) and 3(f)). In the AA stacking (Fig. 2(e)), the six sublattice pairs (A1,B 3), (A3,B 3), (A2,B 1), (A2,B 2), (A2,B 4), and (A4,B 3) are inequivalent, while in the AB stacking (Fig. 2(f)), all 12 (A,B ) sublattice pairs are inequivalent. For (1) (1) (2) (2) AA stacking, nA − nB = 0 and nA − nB = 0 (section 2, case 5 in SI), and hence Eq.(5) predicts no splitting, in agreement with the DFT calculations. On the other hand, for AB stacking (1) (1) (2) (2) nA − nB = 3, nA − nB = 1 (section 2, case 6 in SI), and hence Dirac point splitting is predicted from Eq.(5), which is also observed from the DFT calculations. An identical trend is also observed in the two stackings of gr/BC3 heterostructure (Fig. S6(a) and (b)). Even though both BC3 and C3 N are isostructural, due to the larger lattice mismatch of graphene with BC3 (4.5%), C atoms in graphene layer do not lie above the atoms in BC3 layer, perfectly. Moreover, AB stacked gr/C3 N is metallic and the AB stacked gr/BC3 is semiconducting, even though there is splitting of Dirac cone in both the cases. The reason for this difference will be discussed in later part of the manuscript. Interestingly, the ∆ values for AA stackings are higher than those for AB stackings, where splitting of Dirac cone takes place irrespective of the stacking arrangement (Table S1). This is because in AA stackings, most of the carbon atoms belonging to A and B sublattices lie above different atoms of the 2D sheet as shown in Fig. 1(iv) and 1(v). On the other hand, in the case of AB stackings, the majority of the (A, B ) pairs are such that one of them lie on top of an atom of the substrate layer, while there is no atom below its nearest neighbour (as shown in Fig. 1(i)). For example, in the AA stacked gr/C3 N3 heterostructure, there are only 5 such (A, B ) pairs, while in the AB stacking, the number of such pairs increases to 15 (Fig. S6(c) and S6(d)). This indicates that the strength of symmetry breaking due to (C1 , C5 ) type of configurations (in Fig. 1), is much more than the (C1 , C3 ) type of configurations. Gr/h-C3 N4 heterostructure is the only case deviating from this behaviour. The difference may be attributed to the corrugated (or non-planar) structure of h-C3 N4 in the AA stacking, due to which the effect of sublattice symmetry breaking may not be as strong as that from the planar h-C3 N4 layer in the AB-stacked arrangement. We next shifted our study to the heterostructures of graphene with transition metal dichalcogenides

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Figure 4: Local density of states (LDOS) projected onto the combined-A (denoted by filled grey region) and B (denoted by solid red curve) sublattices of graphene present on (a) and (b) h-BN, (c) and (d) h-C3 N4 , and (e) and (f) C3 N substrates. The dashed blue and solid green curves represent combined-A and B sublattice states of isolated graphene layer (same unit cell size as that of 2D substrate), and dashed cyan line indicate position of Dirac cone of the graphene (ED ). The banddecomposed charge densities corresponding to π and π ∗ states are shown as yellow isosurfaces. (TMDCs). However, there are not many reports on bandgap opening of the graphene with TMDCs as substrates. 28,41 We also, find that most of the gr/TMDC heterostructures do not split the Dirac cone (Fig. 6), except with PtSe2 . Their honeycomb lattice do not superimpose on the graphene layer, therefore one cannot make perfect AA or AB stackings of gr/TMDC heterostructures. Hence, they are different from all the heterostructures discussed earlier and quantifying the amount of sublattice symmetry breaking for these cases is challenging. We can still consider the sublattice pairs that are partially present above different atoms to be inequivalent. The number of symmetrically inequivalent sublattice pairs in these heterostructures are usually less than 50% of the total sublattice pairs (Table S1). For example, in gr/MoS2 , there are 18 inequivalent out of 48 pairs (26.7%), and in gr/SnS2 , there are 3 inequivalent out of 27 pairs (11.1%). In one of the stackings of gr/PtSe2 (Fig. S9(b)), there is a very small opening of bandgap


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(Fig. 4 and Table S1). In these cases, one of the states of carbon belonging to A or B sublattices (a π state) is stabilized, while the other one (a π ∗ state) is not (Fig. 4). The lifting of the degeneracy of π and π ∗ states at the Dirac point of graphene has indeed been identified as one of the reasons for opening of Dirac cone. 42,43 Consistent with this observation, we find that whenever less than 50% of (A, B ) sublattice pairs are inequivalent, the LDOS for combined-A and combined-B sublattices are identical. As is evident from the band-decomposed charge densities shown in Fig. 4, the π state do not necessarily remain a bonding state after heterostructuring (all except 4(e)). This is due to the VBM charge densities being mostly localized on different atoms, representing the antibonding character (compared with the delocalized band-decomposed VBM charge density of isolated graphene in Fig. S10(a)).

of about 18 meV (Fig. 6) and the number of inequivalent (A,B ) sublattice pairs are 18 out of 27 (66.7%). While in the other stacking of gr/PtSe2 , there are 9 inequivalent pairs (33.3%, Fig. S9(a)), and hence no bandgap opening is observed (Fig. 6). The reason for very small ∆ value is due to the fact that none of the carbon atoms in graphene layer are present perfectly above the atoms on PtSe2 layer. (a) Density (states/eV)

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Figure 5: Density of states (DOS) of isolated (a) h-BN, (b) h-C3 N4 and (c) C3 N monolayers, plotted with respect to their vacuum levels. The cyan and red-dashed vertical lines denote the Fermi levels of corresponding layers and isolated graphene (ED ), respectively. ED is also aligned with respect to its vacuum level.

Second criterion Besides Dirac cone splitting, it is also equally important that the graphene heterostructure system be semiconducting. The metallic nature of these heterostructures can arise due to the upward or downward movement of the absolute position of Dirac cone of graphene, with respect to the low energy bands of the substrate, because of the charge transfer from or to the graphene layer, respectively. The tendency for shift in the Dirac cone (termed as δ; Fig. 6) can be predicted from the relative band alignment of isolated graphene and substrate layers. The relative band alignment here refers to the relative energies of the Fermi level of isolated graphene layer and band edges (VBM or CBM) of the isolated substrate layers. This will aid in determining whether a charge transfer can occur between the Dirac cone of graphene and the band edges of substrate layer. For this purpose, the DOS of individual substrates have been plotted with respect to their respective vacuum levels in Fig. 5. If the Dirac cone (ED ) lies near the valence band of substrate, then there is charge transfer from the substrate to the empty Dirac cone of graphene and the Dirac cone shifts down, which will lead to a negative δ. A positive value of δ is obtained when ED moves up with respect to EF of the heterojunction, and it lies near the conduction band of the substrate. This arises due to the charge transfer from the filled Dirac cone of graphene to the empty conduction band of substrate. Only when

First criterion After analyzing the sublattice symmetry breaking and electronic bandstructures for all of the above cases, we find that the necessary condition for substrate-induced Dirac cone splitting in graphene is that more than 50 % of (A, B ) sublattice pairs in graphene should become inequivalent. This proposed hypothesis is further verified from the local density of states (LDOS) plots of A and B sublattices as shown in Fig. 4. In isolated graphene, the valence band maxima (VBM) and conduction band minima (CBM) originate from the overlap of C 2pz states and are degenerate at the Fermi level (EF ; the electrons in C atoms are filled up to the 2pz energy level). The VBM and CBM in isolated graphene correspond to π (molecular orbital (MO) with bonding character) and π ∗ (MO with antibonding character) states (Fig. S10(a) and S10(b)), respectively, due to the higher delocalized nature of VBM compared to the CBM. From the LDOS of isolated graphene layer (of same unit cell size as that of the substrate), we observe that the states of A and B sublattices are also degenerate at the Dirac cone. We find that whenever more than 50% (A,B ) pairs of C atoms become inequivalent, the A and B -projected states are shifted with respect to each other, near the Dirac cone splitting region


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-400 0 h-BN (AA) h-BN (AB) C3N (AA) C3N (AB) C3N3 (AA) C3N3 (AB) h-C3N4 (AA) h-C3N4 (AB) t-C3N4 (AA) t-C3N4 (AB) h-C4N3 (AA) h-C4N3 (AB) t-C4N3 (AA) t-C4N3 (AB) BC3 (AA) BC3 (AB) PtS2 PtSe2 (1) PtSe2 (2) MoS2 MoSe2 MoTe2 SnS2 WS2 HfO2 Ti2CO2 Phosphorene

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Dirac cone shift [d] (meV)

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(2D substrates) Figure 6: Bar plot of Dirac cone splitting (in dark grey color) and Dirac cone shift (in light grey color) observed in all 27 graphene-based heterostructures. cone under HSE06 calculation (Fig. S16(b)), while the Dirac cone splits in the case of AB stacked gr/C3 N, which has more than 50% inequivalent sublattice pairs (Fig. S16(b)). Similar observation is found in the two stackings of gr/PtSe2 heterostructures, which are shown in Fig. S17, although the bandgap for stacking 2 is more in the case of HSE06 (38 meV, Fig. S17(c)) than that in PBE calculation (18 meV, Fig. S17(d)). Hence, our proposed scheme is found to be correct even under highly accurate HSE06 calculations.

the Dirac cone lies within the bandgap of the substrate, there is no shift in ED (i.e., δ = 0). Therefore, as discussed earlier, there is a bandgap opening of graphene in gr/BC3 , while gr/C3 N is metallic in spite of the Dirac cone splitting. This is ascribed to the ED lying inside the valence band region of isolated C3 N (δ < 0), while it lies within the bandgap of isolated BC3 (δ = 0) (Fig. 5 and Fig. S12(o)). Utilizing the two criteria, we find that only the heterostructures of graphene with hBN (AA and AB stackings), h-C3 N4 (AA and AB stackings), t-C3 N4 (AA and AB stackings), BC3 (AB stacking), and PtSe2 (stacking 2) are semiconducting, as they have finite Dirac cone splitting, but no shift in the Dirac cone (Fig. 6).

Conclusion In summary, we have proposed a simple, robust and reliable scheme to quantify the amount of sublattice symmetry breaking in the graphene-based heterostructures. Subsequently, a critical amount of sublattice symmetry breaking in graphene was found to be necessary for splitting of the Dirac cone. Employment of both the DFT and TB calculations confirm that the splitting in Dirac cone does not occur until more than 50% of (A, B ) sublattice pairs become symmetrically inequivalent upon heterostructuring. In addition, to identify the semiconducting heterostructures of graphene, we have proposed that the position of the Dirac cone of graphene (ED ) should lie within the bandgap of the substrate. Therefore, we have developed a simple and robust approach, that can be very useful

Effect of functional We also studied electronic properties of a few heterostructures using HSE06 functional to verify the accuracy of our proposed scheme, since GGA-PBE functional is known to underestimate the bandgaps, and also inaccurately describes the electronic structure of strongly correlated systems (Fig. S13S16). We find that both gr/SnS2 (Fig. S14(a)) and gr/MoTe2 (Fig. S15(a)) heterostructures preserve the Dirac cone of pristine graphene, although they are shifted with respect to the PBE calculated bandstructures (Fig. S14(b) and Fig. S15(b)). The AA stacked gr/C3 N, in which less than 50% of sublattice pairs are inequivalent, has gapless Dirac


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References

to materials engineers, for the selection of appropriate substrates in the design of graphene-based heterostructures with tailored bandgap properties.

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SUPPORTING INFORMATION Derivation of tight-binding formalism, explanation of sublattice symmetry breaking of AA and AB stackings of gr/h-BN, gr/h-C3 N4 , and gr/C3 N from tight-binding analysis, phonon bandstructures of AA and AB stackings of gr/h-BN, gr/BC3 , and gr/C3 N, and stacking 1 and 2 of gr/PtSe2 , optimized structures of AA and AB stackings of gr/BC3 , gr/C3 N3 , gr/h-C4 N3 , gr/t-C4 N3 , and gr/t-C3 N4 , gr/HfO2 , gr/Ti2 CO2 , gr/MoS2 , gr/SnS2 , gr/MoSe2 , gr/MoTe2 , gr/WS2 , gr/PtS2 , stacking 1 and 2 of gr/PtSe2 , and gr/phos heterostructures, band-decomposed charge density of VBM and CBM of isolated graphene, LDOS of AA and AB stackings of gr/BC3 , gr/C3 N3 , gr/h-C4 N3 , gr/t-C4 N3 , and gr/t-C3 N4 , gr/HfO2 , gr/Ti2 CO2 , gr/MoS2 , gr/SnS2 , gr/MoSe2 , gr/MoTe2 , gr/WS2 , gr/PtS2 , stacking 1 and 2 of gr/PtSe2 , and gr/phos heterostructures, DOS of isolated C3 N3 , C3 N, h-C4 N3 , t-C4 N3 , t-C3 N4 , HfO2 , Ti2 CO2 , MoS2 , SnS2 , MoSe2 , MoTe2 , WS2 , PtS2 , PtSe2 , and phosphorene monolayers, PBE and HSE06 bandstructures of gr/SnS2 , gr/MoTe2 , stacking 1 and stacking 2 of gr/PtSe2 , and HSE06 bandstructures of AA and AB stackings of gr/C3 N.

(2) Berger, C.; Song, Z.; Li, T.; Li, X.; Ogbazghi, A. Y.; Feng, R.; Dai, Z.; Marchenkov, A. N.; Conrad, E. H.; First, P. N. et al. Ultrathin Epitaxial Graphite: 2D Electron Gas Properties and a Route Toward Graphene-based Nanoelectronics. J. Phys. Chem. B 2004, 108, 19912–19916. (3) Novoselov, K. S.; Geim, A. K.; Morozov, S. V.; Jiang, D.; Zhang, Y.; Dubonos, S. V.; Grigorieva, I. V.; Firsov, A. A. Electric Field Effect in Atomically Thin Carbon Films. Science 2004, 306, 666–669. (4) Son, Y.-W.; Cohen, M.; SG, L. Energy Gaps in Graphene Nanoribbons. Phys. Rev. Lett. 2006, 97, 216803. (5) Han, M.; Özyilmaz, B.; Zhang, Y.; Kim, P. Energy Band-Gap Engineering of Graphene Nanoribbons. Phys. Rev. Lett. 2007, 98, 206805. (6) Green, A. A.; Hersam, M. C. Emerging Methods for Producing Monodisperse Graphene Dispersions. J. Phys. Chem. Lett. 2009, 1, 544–549.

AUTHOR INFORMATION

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Corresponding Author ∗ Email: [email protected]

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Acknowledgement This work was financially supported by DST Nanomission. The authors thank Materials Research Center (MRC) and Supercomputer Education and Research Centre (SERC), and Solid State and Structural Chemistry Unit (SSCU), Indian Institute of Science, Bangalore for providing the required computational facilities. E. M. acknowledges financial support from Fondecyt Grant 1190361.

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Graphical TOC Entry Eg = 0 ninequivalent(A,B) > 50% ntotal(A,B)

Ai Bi

(No)

(d= 0)

Graphene heterostructure

(Y )

es (d

d

) >0

s)

(d= 0)

Carbon (Graphene) Atom 1 Atom 2

}

2D substrate

ED E’D


11

Metallic

(Ye

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Eg > 0

Critical Sublattice Symmetry Breaking: A Universal Criterion for Dirac

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