Carbo-graphite: Structural, Mechanical, and Electronic Properties

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Carbo-Graphite : Structural, Mechanical and Electronic Properties Jean Marie Ducere, Christine Lepetit, and Remi Chauvin J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/jp4067795 • Publication Date (Web): 17 Sep 2013 Downloaded from http://pubs.acs.org on September 21, 2013

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The Journal of Physical Chemistry

Carbo-graphite : Structural, Mechanical and Electronic Properties

Jean-Marie Ducéré*a,b, Christine Lepetita,b, Remi Chauvin*a,b

[a] CNRS, LCC (Laboratoire de Chimie de Coordination), 205, route de Narbonne, BP 44099, F31077 Toulouse Cedex 4, France. Fax: (+33)5 61 55 30 03. E-mail: [email protected] [email protected] - [email protected].

[b] Université de Toulouse, UPS, INPT, F-31077 Toulouse Cedex 4, France.

Keywords: carbo-mer, graphynes, graphitynes, elastic constants, graphene-like band structures, effective masses.

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Abstract The bulk structure of total and partial carbo-mers of graphite, referred to as graphitynes, is investigated by first-principles calculations using the Rutgers-Chalmers nonlocal correlation functional vdW-DF2 in combination with the Cooper's exchange functional C09. This calculation level is shown to perform well for describing graphene and graphite references structures. The ABand ABC-graphityne stackings are predicted to be the most stable, with interlayer distances close to the one of the graphite parent. The atomic sparsity of the 2D- and 3D-α-graphyne materials resulting from the insertion of C2 units, makes them much softer than the parent graphene or graphite, respectively, but they exhibit the same large elastic anisotropy. The band structures, effective masses of charge carriers, Fermi velocities, and other electronic properties of various bulk graphyne-type carbon allotropes have been calculated and are shown to depend on the number of acetylenic-like linkages between the sp2 centers and on the stacking mode. Most of the graphitynes are predicted to be graphite-like semi-metals, except the ABC-α-graphityne exhibiting a graphenelike band-structure with two nonequivalent Dirac cones.


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Introduction 1

Non-amorphous carbon has long been known to exist under two natural allotropic forms: graphite and diamond. In 1985, Kroto et al. disclosed a cage-like third form corresponding to 2

3

fullerenes. This finding led to the discovery of other arrangements such as carbon nanotubes, and more recently, graphene could be isolated as a graphite monolayer.

4,5

The graphene pattern is

common to all these carbon materials. Fullerenes may be considered as wrapped-up graphene pieces where the pentagonal closures of the hexagonal pattern create positive curvature and a spherical arrangement. Carbon nanotubes may be regarded as various types of graphene rolls. These all-sp2-carbon materials have been extensively studied, and proved to exhibit unique electronic, mechanical, and thermal properties.

6 7

Graphynes are alternative carbon planar forms, postulated by Baughman et al. in 1987. They are constructed by partial or complete insertion of acetylene-like C2 units, into the C-C bonds of 8

graphene and may be therefore considered as partial or total carbo-mers of graphene. Because of their heterogenous content (both sp and sp2 carbon atoms are present), exceptional electronic and mechanical properties similar to those of graphene are expected for these still putative carbon allotropes. In the graphyne series, γ-graphyne and γ-graphdiynes containing isolated C6 benzenic cores (that are peri-condensed in graphene) have attracted more attention because of known efficient synthetic tools for fragments thereof. In γ-graphyne (Figure 1a), one-third of the C-C bonds of graphene have thus been replaced by but-2-yn-1,4-diylidene linkages, and substructures of γ9

graphyne have been synthesized and reported to be efficient for two-photon absorption.

Semiconducting films of γ-graphdiyne, showing a conductivity comparable to that of silicon, have been obtained on copper surfaces.

10

The total carbo-mer of graphene, α-graphyne (Figure 1b), has been less studied. It is constructed by insertion of C2 units in all the C-C bond of graphene. In contrast to the other graphyne isomers, it thus does not contain any of the Csp2-Csp2 bonds of the parent graphene. The two-dimensional (2D) structure and electronic properties of α-graphyne sheets have been studied at various levels of calculation and compared to those of graphene.

11

Moreover, the unit ring of α-

graphyne is the C18 hexagonal macrocycle of the so-called carbo-benzene molecules, which have been exemplified experimentally.

8b,12

It should be also reminded that the carbo-mer approach has 13

been previously addressed for other carbon allotropes like carbon nanotubes,


and fullerenes.

14

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a

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b

Figure 1. 2D-structures of carbo-meric graphene : a) γ-graphyne. b) α-graphyne. The black rhombus is the unit cell. Very few studies of the bulk structure of either α- or γ-graphyne stacks, here referred to as α15

and γ-“graphitynes” respectively, have been reported.

Until recently indeed, a relevant

description of the weak interactions between layers was only possible at the tight-binding level. At the density functional theory (DFT) level, standard LDA and GGA functionals are indeed unable to account for the weak interlayer interaction. In former first-principles calculations, the calculated interlayer distance was therefore fixed or constrained by intercalants such as lithium atoms.

16

In this work, three-dimensional (3D) structures of graphitynes are calculated for the first time from first-principles calculations and full relaxation of the unit cell, and compared to those of other all-carbon materials. The second version of the nonlocal Rutgers-Chalmers correlation functional (vdW-DF2), was indeed reported to describe accurately the graphite interlayer binding energy and 17

spacing,

and it is therefore expected to enable the same chemically accurate calculations for other

bulk carbon materials. In the present work, it has thus been used to compute the structure stabilities, elastic constants and effective masses of charge carriers in α- and γ-graphyne stacks. ACS Paragon Plus Environment

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Computational Details The calculations were performed with the DFT-based Quantum Espresso package.

18

In

order to properly describe the weak interactions between the layers in bulk materials, the Rutgers17

Chalmers nonlocal correlation functional (van der Waals density functional vdW-DF2) 19

in combination with the Cooper's exchange functional (C09).

was used

The valence electrons were

described by a plane-waves basis set, up to a cut-off energy of 40 Rydberg, while the C4+ ionic cores were described using the Projector Augmented Wave (PAW) method, 21

were obtained using the functional of Perdew, Burke and Ernzerhof

20

the data of which 22

from the PSlibrary.

The

Brillouin zone was sampled using Γ-centered Monkhorst-Pack grids: 16×16×x for graphene-based materials and 7×7×x for α- and γ-graphyne-based materials; x is here equal to 1 for isolated monolayers, to 12 for single-layer bulk structures, to 6 for two-layer bulk structures, and to 4 for three-layer bulk structures. Test calculations showed that results are fully converged with respect to k points. The positions of all the atoms were fully optimized by reducing the Hellman-Feynman forces down to 0.001 Rydberg atomic units. Two-dimensional periodic boundary conditions were applied to the single-layered materials while a vacuum spacing of 25 Å was set along the perpendicular direction to the sheets to avoid the mirror interaction.

Results and discussion

Structural and thermodynamic properties of graphitynes α- and γ-graphynes are hypothetical carbon sheets, which can stack as the α- and γgraphitynes multilayer materials. In order to find the most stable bulk structure, various stacking modes have been investigated, by reference to graphite polytypes: - “eclipsed” AA stacking, also referred to as 1H; - “compact” AB stacking (Figure 2), also referred to as 2H; - “compact” ABC stacking (Figure 3), also referred to as 3R for the primitive cell is no longer hexagonal but rhombohedral.


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Figure 2. Top view along the c axis of AB-α-graphityne (left) and AB-γ-graphityne (right). The black rhombus is the unit cell. Black circles are first layer atoms and white circles are atoms of the second layer.

Figure 3. Top view along the c axis of ABC-α-graphityne (left) and ABC-γ-graphityne (right). The black rhombus is the unit cell. White circles are first layer atoms, gray and black circles are atoms of the second and third layers respectively.

The reliability of the present calculation level was first tested through its ability to describe the structure and cohesive energies of diamond, graphite and graphene.

The calculated C-C bond length and lattice parameter of diamond are in very good agreement with the corresponding experimental data, (1.542 Å vs 1.545 Å and 3.562 Å vs 3.567 Å, respectively - Table 1).

23

The calculated energy of formation of diamond from graphite (the

standard state of carbon) is also in good agreement with experimental data: ∆Hf° = 0.34 kJ.mol-1 vs 24

2.4 kJ.mol-1 (Table 1).

At the present vdW-DF2-C09 level, in agreement with previous reports, the AB stacking mode of graphite is the most stable. AB-graphite is calculated to be slightly more stable than AAgraphite, and as stable as the ABC-stacking mode. This is consistent with the large amount of stacking faults previously observed and reported for graphite.

25

The calculated hexagonal lattice 23

parameters of AB-graphite are in good agreement with both experimental calculated data (Table 1).

26

and recently reported

The calculated interlayer distance in AB-graphite (3.284 Å) is slightly ACS Paragon Plus Environment

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underestimated as compared to either the experimental value (3.354 Å), or to previous vdW-DF2revPBE estimates (3.53 Å), or to very accurate Quantum Monte Carlo (QMC) calculations (3.426 Å).

27

As compared to revPBEx, the exchange functional C09x used in this work offers significant

improvements in both the lattice parameters and interlayer interaction energy of graphite,

19

and

performs almost as well as QMC calculations or B97-D extrapolations from graphene models 28

(Table 1).

C-C bond length

Diamond

1.542 a [1.545]

AB-graphite

1.422 a [1.421]

Interlayer distance Calcd.

3.284 c (3.426)

Interlayer distance Exptl.

3.354

e

(3.34)

Lattice constants Calcd.

Lattice constants Exptl.

a = 3.562

a = 3.567

a = 2.463 c = 6.568 d (c = 7.140)

a

a

a = 2.456

Energy of formation 0.34 b [∆Hf° = 2.4]

c = 6.696

0.0 [∆Hf° = 0.0]

a

AA-graphite

1.422

3.529

a = 2.463 c = 3.529

-

1.2

ABC-graphite

1.422

3.288

a = 2.463 c = 9.863

-

> 0.0

Table 1. Calculated geometric parameters (in Å) and energy of formation, with respect to the most stable AB-graphite carbon state (in kJ.mol-1), of reference structures for validation of the present calculation method (vdW-DF2-C09). All phases were calculated as hexagonal structures except the face-centered cubic lattice of diamond. Experimental bond length and energy of formation in a

b

square brackets and previously reported calculated values in parentheses. : from reference 23. : c

d

from reference 24. : QMC calculations from reference 27. : vdW-revPBE calculations from e

reference 28a. : B97-D calculations from reference 28b.

The reliability of the vdW-DF2-C09 calculation level is further demonstrated from the calculated geometries of α-graphyne and γ-graphyne sheets that are in good agreement with previous reports as illustrated below (Table 2).

15, 29-31

The heat of formation of the graphene and graphyne sheets are given in Table 2 relatively to AB-graphite. The energy of formation of graphene relative to graphite may be approximated by the exfoliation energy, if surface effects are neglected. Indeed, the value calculated in this work (5.3 26

kJ.mol-1) lies in the same range as the reported experimental (5.0 kJ.mol-1) ACS Paragon Plus Environment

or calculated (4.9 7


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6.4 kJ.mol-1)

17, 24

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values of the interlayer cohesive or exfoliation energy of graphite. The energy

of formation of γ-graphyne is 64.7 kJ.mol-1 higher than the one of graphene (Table 2), a value close to the 0.64 eV/atom value (61.7 kJ.mol-1) reported by Wang et al.

C-C bond length (this work) vdW-DF2C09

Graphene

C-C bond length a PBE (PW91) {LDA}

b c

Lattice constant (this work) vdW-DF2C09

1.422

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Lattice constant Energy of formation a PBE relative to b AB-graphite (PW91) 5.3 d [Eex = 5.0]

2.463

(Eex = 5.4) sp2-sp: 1.394 α-graphyne sp-sp: 1.229

sp2-sp: 1.40 sp-sp: 1.24

29

6.957

7.01 (PBE)

e

100.1

30

6.97 (PBE)

(sp2-sp2: 1.42) (sp2-sp: 1.40) sp2-sp2:1.422 (sp-sp: 1.22) γ-graphyne sp2-sp: 1.406 sp-sp: 1.221 {sp2-sp2:1.419} {sp2-sp: 1.401} {sp-sp: 1.221}

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6.86 (PW91) 6.877

31a

6.86 {LDA}

70.1 c {88.3}

31b

6.89 (PBE)

Table 2. Geometric parameters (in Å) and energy of formation relative to AB-graphite (in kJ.mol-1) a

of graphene and graphyne sheets (hexagonal lattice symmetry) calculated at various levels. : PBE b

c

functional from references 29, 30 and 31b. : PW91 calculations from reference 15 : LDA d

e

calculations from reference 31a. : Experimental exfoliation energy Eex from reference 26. : Exfoliation energy from QMC calculations in reference 27.

These findings suggest that the present vdW-DF2-C09 level of calculation provides accurate geometries and energies of graphene-like or γ-graphyne-like carbon materials, although the interlayer distance may be slightly underestimated. The 3D-structure of α- and γ-graphityne was therefore investigated at this level. The corresponding lattice constants and energy of formation relative to graphite are given in Table 3.

The C-C bond lengths are calculated to be almost the same in the 2D- and 3D-materials ACS Paragon Plus Environment

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(Table 3) and suggest that electron delocalization is more extended in α-graphityne than in γgraphityne. Like 2D-graphyne sheets, 3D-graphitynes (whatever their stacking mode) are calculated to be much less stable than graphene and graphite (Table 3). The energy of formation of γ-graphityne is about 66 kJ.mol-1 higher than that of graphite, and it is thus about 4 kJ.mol-1 more stable than γgraphyne (Table 2). Despite their large energies of formation, these carbon materials are considered as viable, as they are predicted to be more stable than the already-synthesized γ-graphdiyne by at 15,33

least 18 kJ.mol-1.

α-graphyne and α-graphityne are however less stable than γ-graphyne and γ-

graphityne by about 30 kJ.mol-1. This is consistent with the increasing sparsity and proportion of sp-hybridized carbon atoms in these materials: none in graphene, one per sp2 carbon atom in γgraphyne and γ-graphityne and three per sp2 carbon atom in α-graphyne and α-graphityne. It is also consistent with the a priori stabilizing high aromatic content of the γ-isomers, where half of the carbon atoms are sp2 hybridized and belong to a C6 benzene ring (γ-graphyne is just the partial 28b

carbo-mer of graphene).

AA-graphitynes exhibit longer interlayer distances and are calculated to be less stable than AB and ABC stacking modes, the latter beeing degenerated in energy (Table 3). Stacking faults are therefore expected to occur as commonly in both α- and γ- graphitynes as in graphite.

25

The stacking energy may be estimated from the difference of the energy of formation of graphynes with respect to graphitynes given in Table 3. The values decrease with the sparsity of the monolayer in the following order: 5.3 kJ.mol-1 for graphene, 4.5 kJ.mol-1 for γ-graphyne and 2.6 kJ.mol-1 for α-graphyne. The latter low value indicates that α-graphynes should be easily exfoliated from bulk αgraphitynes.


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C-C bond length Interlayer distance Lattice constants

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Energy of formation

AB-graphite

sp2-sp2: 1.422

3.284

a = 2.463 c = 6.568

0.0

AA-α-graphityne

sp2-sp: 1.394 sp-sp: 1.229

3.771

a = 6.956 c = 3.771

98.3

AB-α-graphityne

sp2-sp: 1.393 sp-sp: 1.229

3.266

a = 6.956 c = 6.531

97.6

ABC-α-graphityne

sp2-sp: 1.394 sp-sp: 1.229

3.201

a = 6.954 c = 9.604

97.5

AA-γ-graphityne

sp2-sp2: 1.422 sp2-sp: 1.405 sp-sp: 1.221

3.607

a = 6.876 c = 3.607

66.9 a (88.6)

AB-γ-graphityne

sp2-sp2: 1.421 sp2-sp: 1.405 sp-sp: 1.222

3.294

a = 6.875 c = 6.589

65.6 a (87.2)

ABC-γ-graphityne

sp2-sp2: 1.422 sp2-sp: 1.405 sp-sp: 1.222

3.293

a = 6.875 c = 9.879

65.6

Table 3. Geometric parameters (in Å) and energy of formation (at 0 K) relative to AB-graphite (in kJ.mol-1) of various stacking modes of α- and γ-graphityne calculated at the vdW-DF2-C09 level. a

: LDA calculations from reference 31a.

Elastic constants of graphitynes 34

Graphene has been reported by Lee et al.

as one of the strongest materials ever investigated

with ultra-high elastic stiffnesses. The Young’s modulus of 1.0 TPa measured by atomic force microscopy (AFM) for a free-standing monolayer graphene membrane, lies in the range of 35

previous predictions of Kundin et al.

Beyond these extreme mechanical properties, predicted and

measured for graphene, the elastic properties of graphynes have also been investigated using various theoretical approaches, but to the best of our knowledge, the mechanical properties of the bulk materials, namely graphitynes, have not been addressed hiterto. Elastic constants are critical parameters in finite element analysis for determining mechanical properties of materials. In general, a crystal deforms in a homogeneous linear elastic manner when subjected to sufficiently small strains εij (i, j = x, y, z). After a small uniform deformation of the solid, the axes of the unit cell (Figure 1b) are distorted in orientation and length. The axes of the deformed lattice (x’, y’, z’) can be obtained by multiplying the original axes by a distortion matrix, ACS Paragon Plus Environment

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where the coefficients εij are the elements of the strain tensor ε:

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x’ = (1+εxx) xˆ + εxy yˆ + εxz zˆ y’ = εyx xˆ + (1+εyy) yˆ + εyz zˆ z’ = εzx xˆ + εzy yˆ + (1+εzz) zˆ

In the linear regime of the Hooke's law, the stress tensor σ is the product of the strain tensor

ε with the elastic stiffness tensor referred to as C:

σ =C⊗ ε

(eqn.1)

σ ij = Cijklε kl The Voigt notation allows for reducing the order of these symmetrical tensors. The Voigt notation replaces xx by 1, yy by 2, zz by 3, zy (and yz) by 4, xz (and zx) by 5 and xy (and yx) by 6. The present materials fall in the category of orthotropic materials (i.e. having three orthogonal planes of symmetry) that have only six independent elasticity constants. Graphite, graphene and αgraphyne are also transversely isotropic materials (i.e. symmetric with respect to a rotation about a symmetry axis) and are left with only five independent elasticity constants (C66 = ½ (C11 – C12).

σ

σ 1    σ 2  σ 3  =  σ 4  σ 5    σ 6 

    ε =     

ε1 ε2 ε3 ε4 ε5 ε6

     C=     

         

C11 C12

C13

0

0

C12

C11

C13

0

0

C13

C13 C33

0

0

0

0

0

C44

0

0

0

0

0

C44

0

0

0

0

0

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0   0   0  0   0  C66 

By applying to the unit cell, a set of eight small deformations corresponding to the following distortion matrices where ε = ± 0.5 %,

1 + ε 0 0 1 0 0      0 1 0 0 1 0     0 0 1 0 0 1 + ε    

1 0 0   1 ε /2 0     0 1 ε /2 ε /2 1 0     0 ε /2 1 0 0 1    


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    and corresponding to the following strain vectors:     

±ε 0 0 0 0 0

        

        

        

0 0 ±ε 0 0 0

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        

0 0 0 ±ε 0 0

        

        

0 0 0 0 0 ±ε

        

the resulting stress tensors are calculated using Quantum Espresso. The elements of the elastic stiffness tensor are then derived by solving the set of linear equations arising from (eqn.1) using a least-squares method. For single-layers, the matrix elements along directions 3, 4 and 5 are set to zero. The results for bulk materials are displayed in Table 4, while the results for 2-D sheets are given in Table 5. The reported experimental values of graphite elastic stiffness constants spread over a large range, because they are very sensitive to the crystallinity of the samples and to the presence of defects.

38

The C11 value of 1075 GPa, calculated here for AB-graphite at the vdW-DF2-c09 level,

is in good agreement with the one reported by Bosak et al.

38

39

and Blakslee et al.,

namely 1109

GPa and 1060 GPa respectively (Table 4). The calculated C12 and C66 elastic stiffness constants (C66 = ½ (C11 – C12)) are very similar to the one obtained from the experimental measurements of Blakslee et al., while the value of C13 is closer to the one reported by Bosak et al.. Experimental and calculated values of C33 and C44 are also in very good agreement (Table 4). Elastic properties are commonly discussed on the basis of the Young’s modulii Ys and of the Poisson’s ratios ν.  are the constitutive elements of the elastic compliance matrix, namely the inverse of the elastic stiffness tensor. For the present materials, the in-plane Young’s modulus Ys and the Poisson ratio ν can be approximated using the following relationships:

C112 − C122 C Ys ≈ = C11 − ν •C12 with ν = 12 C11 C11 The C11 value and the Poisson's ratio calculated here for γ-graphyne (Table 5) are comparable 40

to recently reported values: 178 vs. 166 N.m-1 0.429

41

41

or 199 N.m-1.

40

for C11, and 0.479 vs. 0.417

or

for the Poisson’s ratio ν. Considering the in-plane stiffness C11, α-graphyne is softer than

γ-graphyne. The latter was reported to be twice softer than graphene and its Young’s modulus was shown to decrease faster with increasing temperature than the one of graphene.

42

The Young’s

modulus of 0.07 TPa (resp. 0.42 TPa) calculated here for α-graphyne (resp. γ-graphyne), is also significantly reduced as compared to graphene (1.08 TPa) in agreement with the recently reported 43

respective values of 0.12 TPa and 0.51 TPa obtained from molecular dynamics studies.

Graphitynes exhibit the same large anisotropy of their elastic constants as graphite, but are ACS Paragon Plus Environment

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easier to deform, consistently with their relative sparsity. It is noticeable however, that not only C11 but also the in-plane Young’s modulus of γ-graphityne, i. e. 498.3 GPa (resp. α-graphityne, i. e. 64.0 GPa) are larger than those of steel with Young’s modulus of ca 200 GPa (resp. comparable to 44

the one of aluminum with a Young’s modulus of 69 GPa).

A large value of the Young’s modulus,

i. e. 412 GPa, was also reported recently for 2D-graphdiyne.

28a

The C12 value is the only elastic

stiffness constant to be larger for graphitynes than for graphite, yielding large in-plane Poisson's ratios: 0.43 for AB-γ-graphityne and 0.88 for AB-α-graphityne. As for graphynes, these values are much larger than the one of graphene. Recall that a perfectly incompressible material (in all directions, like a liquid or rubber) has a Poisson ratio of 0.5. Hence graphynes and graphitynes are 41

expected to keep their volume unchanged under in-plane axial strain.

These very large Poisson's

ratios values originate in the in-plane atomic sparsity that allows for large contractions. In contrast, graphite and graphynes exhibit out-of-plane Poisson's ratios close to zero (see C13 values Table 4). They can therefore be stretched widely in the axial direction without shrinking much in the transverse direction. The small C13 values are consistent with the weak interaction between layers. Indeed, the constants corresponding to in-plane deformation: C11, C12 and C66, of the monolayers are very close to those of the corresponding bulk materials. See Table 5, where the same volumic unit is adopted, assuming that the layers have the same thickness whether isolated or stacked. The stacking mode, namely AB or ABC, has therefore no influence on the elastic properties of the bulk material.

Elastic constant (GPa)

C11

C12

C13

C33

C44

C66

1075 (1109) [1060]

197 (139) [180]

-4 (0) [15]

38 (39) [37]

6 (5) [4]

438 (485) [440]

AB-α-graphityne

291

257

-1

8

1

17

ABC-α-graphityne

292

258

-1

9

1

18

AB-γ-graphityne

610

261

-2

25

2

174

ABC-γ-graphityne

606

262

-4

25

1

170

AB-graphite

Table 4. Elastic constants (Voigt notations of the elastic stiffness tensor) of bulk materials calculated at the vdW-DF2-C09 level (in GPa). Experimental values in brackets from reference 33 and in square brackets from reference 34.


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Page 14 of 28

Elastic constant in N.m-1 or (GPa)

C11

C12

C66

graphene

363 (1105)

54 (164)

154 (467)

α-graphyne

95 (297)

83 (259)

6 (19)

γ-graphyne

178 (540)

85 (258)

58 (176)

Table 5. Calculated elastic constants (Voigt notations of the elastic stiffness tensor) of singlelayered materials (N.m-1). For comparison with bulk materials (Table 4), values in brackets are the corresponding elastic constants in GPa assuming a layer thickness equal to the interlayer distance given in Table 3: 3.284 Å for graphene, 3.201 Å for α-graphyne and 3.293 Å for γ-graphyne.

Electronic properties of graphitynes The unique electronic properties and high conductivity of graphene originate in the existence of so-called Dirac points and Dirac cones in the band structure. At a Dirac point, the valence and conduction bands meet in a single point at the Fermi level. In the vicinity of a Dirac point, the valence and conduction bands form a double cone, the Dirac cone, approaching the Fermi level 45

with zero curvature along at least one symmetry-driven direction.

Graphene can therefore be

considered as a semiconductor with a zero band gap or as a metal with a zero density of states (DOS).

46

The band structures and density of states of graphynes and graphitynes have been calculated at the vdW-DF2-C09 level and are are shown in Figures 4 -10. The band gaps and the effective masses of charge carriers are given in Table 6. It is noticeable that the valence part of the band structure of graphyne-type materials is segmented in several non-overlapping narrow bands while for graphene-type materials, it is constituted of a single band. This originates from the lower electron delocalization in graphynes due to the constitutive heterogeneity imposed by the presence of two different kinds of carbon atoms corresponding to different hybridization states.


14

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Band gap (eV)

me* (m0)

– mh* (m0)

Graphene

0

0

0

AB-graphite

0

0.090 (K → Γ) 0.093 (K → M) 12 (K → H)

0.11 (H → A) 0.11 (H → L) 4.0 (H → K)

α-graphyne

0

0

0

AB-α-graphityne

0

0.11 (H → A) 0.039 (H → L) 6.2 (H → K)

0.062 (K → Γ) 0.073 (K → M) 15 (K → H)

ABC-α-graphityne

0

0

0

γ-graphyne

0.46 (M)

0.20 (M → Γ) 0.080 (M → K)

0.21 (M → Γ) 0.084 (M → K)

AB-γ-graphityne

direct: 0.19 (M) indirect: 0.15

1.1 (U1/2 → ∆1/2) 0.034 (U1/2 → P1/2) 1.3 (U1/2 → L)

0.42 (M → Γ) 0.040 (M → K) 11 (M → L)

0.24 (M → Γ) 0.28 (M → Γ) 0.019 (M → K) 0.019 (M → K) 0.89 (M → L) 4.8 (M → L) Table 6: Band gap (in eV) and charge carrier (electrons and holes) effective masses (in units of ABC-γ-graphityne

0

electron mass m0). Graphene and α-graphyne exhibit Dirac cones at Κ points and thus their charge carrier effective mass is zero. Calculations at the vdW-DF2-C09 level.

The band structures calculated here for graphene (Figure 5) and for the most stable ABgraphite (Figure 4) are consistent with previous reports using tight-binding principles calculations.

48,49

47

or other first-

As expected, the valence and conduction bands of graphene cross the

Fermi level at the K point of the Brillouin zone with a linear curvature defining a Dirac cone (Figure 5). AB-graphite exhibits semi-metallic properties, as the conduction band and the valence band are slightly overlapping at the K point resulting in a small but non-zero DOS at the Fermi level (Figure 4). The conduction band and the valence band cross the Fermi level at the H point with a quasi-linear dispersion. Although it was not derived for this purpose, the vdW-DF2-C09 functional appears therefore as reliable for band structure calculations as are the well-established DFT levels such as LDA

50

or GGA-PBE.

21


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Figure 4. Band structure and density of states of AB-graphite. Energies in eV, the Fermi level is set to zero. Calculations at the vdW-DF2-C09 level.

Figure 5. Band structure and density of states of graphene (left) and α-graphyne (right). Energies in eV, the Fermi level is set to zero. Calculations at the vdW-DF2-C09 level.

As already shown from DFT (PBE) and tight-binding calculations, α-graphyne has a graphene-like electronic structure.

11

The band structure exhibits a Dirac cone at the K point of the

Brillouin zone (Figure 5). This property is therefore not restricted to homogeneous hexagonal 45

lattices with chemically equivalent carbon atoms.

Electronic properties similar to those of

graphene may thus be anticipated for α-graphynes, such as high conductivity due to massless charge carriers (Table 6). However, around the Fermi level, the DOS of α-graphyne is significantly larger (Figure 5) than the one of graphene. A higher charge carrier density, and consequently a ACS Paragon Plus Environment

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higher conductivity may thus be expected for α-graphyne. The case of bulk α-graphyne, αgraphityne, will be addressed farther below. In contrast to α-graphyne, γ-graphyne and bulk AB-γ-graphityne are not graphene-like materials. Both of them are narrow gap semiconductors. The band structure and DOS computed 15, 31, 51

here at the vdW-DF2-C09 fit very well with previous reports.

γ-Graphyne is thus a

semiconductor with a direct band gap of 0.46 eV. The maximum of the valence band and the minimum of the conduction band are both located at the same M point of the Brillouin zone (Figure 6) where the 0.46 eV band gap is minimum (Table 6). A second maximum (resp. minimum) of the valence (resp. conduction) band is located at the same point M* on the K-Γ branch yielding a larger band gap of 1.44 eV in very good agreement with the values reported by He et al. using 15

first-principles calculations.

Figure 6. Band structure and density of states of γ-graphyne. Energies in eV, the Fermi level is set to zero. Calculations at the vdW-DF2-C09 level.

Using a tight-binding fit of the band gaps and band lines of γ-graphyne, the same authors predicted a small band gap of 0.16 eV for bulk AB-γ-graphityne, of 0.19 eV reported at the LDA level,

31a

15

consistent with the indirect gap

and with the indirect gap of 0.15 eV calculated in this

work (Figure 7). For this material, the valence band maximum is still located at the M point but the conduction band minimum is now located at the middle of the M-L branch (Figure 7). Similarly to ACS Paragon Plus Environment

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28a

what was observed in the graphdiyne series,

Page 18 of 28

the band gap is found to be smaller in the bulk 3D-

material than in the 2D- material.

Figure 7. Band structure and density of states of AB-γ-graphityne. Energies in eV, the Fermi level is set to zero. Calculations at the vdW-DF2-C09 level.

The less stable ABC-γ-graphityne is a semi-metal: its valence and conduction bands are slightly overlapping at the M point resulting in a small but non-zero DOS at the Fermi level (Figure 8).

Figure 8. Band structure and density of states of ABC-γ-graphityne. Energies in eV, the Fermi level is set to zero. Calculations at the vdW-DF2-C09 level. ACS Paragon Plus Environment

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The properties of bulk assemblies of α-graphyne, the total carbo-mer and thus the closest expanded structural parent of graphene, are now addressed in detail. AB-α-graphityne is a graphite-like semi-metal. Its valence and conduction bands are slightly overlapping near the K point of the Brillouin zone resulting in a small but non-zero DOS at the Fermi level and crossing at point H below the Fermi level (Figure 9).

Figure 9. Band structure and density of states of AB-α-graphityne. Energies in eV, the Fermi level is set to zero. Calculations at the vdW-DF2-C09 level.

In contrast to the parent ABC-graphite (β-graphite) being a semi-metal,

52

ABC-α-graphityne

is calculated to be a zero-gap semiconductor like the 2D materials α-graphyne and graphene. The valence and conduction bands cross the Fermi level near H (towards the A point) and K (towards the M point) with a linear dispersion, defining two different Dirac cones (Figure 10). This finding deserves refinement and future investigations, as DFT is known to underestimate (or even to close) band gaps. This intriguing property was already predicted for rectangular 6,6,12-graphyne sheets at 45

the DFT (PBE) level.

To the best of our knowledge, this property (occurrence of two

nonequivalent Dirac cones) is very unusual in three-dimensional stacks. Moreover, only few threeACS Paragon Plus Environment

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53

dimensional zero-gap semiconductors have been reported. As predicted long ago,

intercalated AB-graphite was recently shown to be a zero indirect gap semiconductor. (α-Sn) is known as a zero-gap semiconductor having the diamond structure,

55

54

oxygenGray tin

and a report on

Ni(tmdt)2 (tmdt = trimethylenetetrathiafulvalenedithiolate), may also be related to this property.

56

Figure 10. Band structure and density of states of ABC-α-graphityne. Energies in eV, the Fermi level is set to zero. Calculations at the vdW-DF2-C09 level.

Electron (resp. hole) effective masses me* (resp. mh*) have been calculated from the equation 57 1 1 ∂2 E , where E is the energy of the charge carrier. They have been estimated from a = 2 m * h ∂k

parabola (quadratic) fit of the in-plane dispersion of the valence (conduction) band near the Fermi level. The values are given in Table 6. The calculation level proved to be reliable for the estimation of charge carrier effective masses. The results are indeed in very good agreement with literature data for graphite and γ-graphyne as discussed below. Electron effective masses around the Κ point of the Brillouin zone (Κ→ Γ) calculated here for graphite are comparable to previous estimations from tight-binding calculations (0.090 m0 vs 47

49

0.109 m0 ) and close to the experimental value of 0.06 m0.


The me* values in the direction 20

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perpendicular to the layers (12 m0) is also in good agreement with the value of 16 m0 estimated by 47

Gruneis et al.

Charge carrier effective masses calculated for γ-graphyne are the same as the ones

calculated at the PBE level by Li et al. by Narita et al.

31b

and are also in good agreement with the ones reported

51

: me* = 0.20 m0 vs 0.17 m0 (Μ→ Γ) and me* = 0.080 m0 vs 0.066 m0 (Μ→ Κ).

Effective masses of charge carriers for bulk γ-graphityne lie in the same range as the ones of 2D γ-graphyne (Table 6); me* = 0.24 (M → Γ), 0.019 (M → K), 0.89 (M → L) e.g. for ABC-γgraphityne. Comparable values have been calculated for γ-graphdiynes.

10c, 51, 58

At this level of

calculation, due to the kink at the Dirac cones, me* values for α-graphyne and ABC-α-graphityne are strictly zero. At the Dirac point, the linear dispersion E = h vF k is characterized by the 5

Fermi velocity vF, i.e. the slope of the cone. The value calculated here for α-graphyne (0.69 106 11,59

m/s) is in good agreement with previous reports.

The Fermi velocities related to both Dirac

cones of ABC-α-graphityne are calculated to be in the same range (0.75 106 m/s (K-M) and 0.65 106 m/s (H-A)). These values are slightly lower than those reported for graphene.

5,60

The effective masses of both α- and γ-graphitynes are very small as compared to those 51

occurring in typical semiconductors,

and make graphynes and graphitynes attractive for their

electronic properties. The graphene-like structure of bulk ABC-α-graphityne, exhibiting two nonequivalent Dirac cones of comparable Fermi velocities is particularly fascinating.

Conclusion The bulk structure of total and partial carbo-mers of graphite, α- and γ-graphitynes, has been investigated for the first time (by first-principles calculations using the Rutgers-Chalmers nonlocal correlation functional vdW-DF2 in combination with the Cooper's exchange functional C09). The latter is shown to perform well for describing graphene and graphite. In graphitynes, the AB stacking sequence is predicted to be the most stable as it is for graphite and the interlayer distance is also found to be close to the one in graphite. The atomic sparsity introduced by the acetylenic-like C2 linkages makes these materials mechanically much softer than graphene and graphite, but exhibiting the same large anisotropy. The band gaps and band lines of various graphyne-type 2D- and 3D-materials have been calculated and shown to depend on the number of acetylenic linkages and stacking type. Most of them are graphite-like semi-metal with low charge carrier mobilities. In ABC-α-graphityne, which is just slightly less stable than AB-α-graphityne, the existence of two nonequivalent Dirac cones is ACS Paragon Plus Environment

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reminiscent of what was reported for a non-hexagonal 6,6,12-graphyne sheet.

45

To the best of our

knowledge, it is however the first example among 3D-materials. The next challenge is to tune the band gap and corresponding electronic properties of these still elusive materials. Approaches similar to those employed for graphene such as doping, diatomic molecules and cations intercalation, can be envisaged.

30

In the same spirit, tight-binding

calculations have shown that the band gap of γ-graphyne may be tuned and that Dirac cones may 11

appear if the triple bonds are artificially elongated.

In contrast, addition of HF and corresponding

symmetry breaking, allows for opening of the band gap of α-graphyne.

11 61

The recently reported potentialities of γ- and α-graphyne in fuel cells design hydrogen storage after calcium or lithium decoration,

62-65

or for

might be extrapolated and further

enhanced in graphitynes. These prospects will be shortly addressed.

Acknowledgements Theoretical studies were performed using HPC resources from CALMIP (Grant 2011 and 2012 [0851]), and from GENCI-[CINES/IDRIS] (Grant 2011 and 2012 [085008]). J. -M.D. was supported by the ANR (11-BS07-016-01). Table of Contents (TOC) Image :

ABC-α α-graphityne as the first bulk zero-gap semiconductor with two nonequivalent Dirac cones


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Carbo-graphite: Structural, Mechanical, and Electronic Properties

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