Nanoporous Graphene and H-BN from BCN Precursors: First

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Nanoporous Graphene and H-BN From BCN Precursors: First-Principles Calculations Rafael Freitas Dias, Jonathan da Rocha Martins, Helio Chacham, Alan Barros de Oliveira, Taíse Matte Manhabosco, and Ronaldo J.C. Batista J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.7b09908 • Publication Date (Web): 29 Jan 2018 Downloaded from http://pubs.acs.org on February 3, 2018

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The Journal of Physical Chemistry

Nanoporous Graphene and H-BN from BCN Precursors: First-Principles Calculations Rafael F. Dias,† Jonathan da Rocha Martins,‡ Hélio Chacham,¶ Alan B. de Oliveira,§ Taíse M. Manhabosco,§ and Ronaldo J. C. Batista∗,§ †Departamento de Física, Universidade Federal de Viçosa, 36570-000, Viçosa, MG, Brazil ‡Departamento de Física, Universidade Federal do Piauí, 64049-550, Teresina,PI, Brazil ¶Departamentode Física, Universidade Federal de MinasGerais, 30123-970, Belo Horizonte, MG, Brazil §Departamento de Física, Universidade Federal de Ouro Preto, 35400-000, Ouro Preto, MG, Brasil E-mail: [email protected] Phone: +55 31 99311 2842

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Abstract We propose, based on results of first-principles calculations, that nanoporous graphene and h-BN might be efficiently produced from B-C-N layers as precursors. In our calculations, we find that the removal of the h-BN islands that naturally occur in BN-doped graphene, forming nanoporous graphene, requires less energy than if pristine graphene is used as a precursor. The same reduction ∆Ef in pore formation energy is found for nanoporous h-BN obtained from graphene-doped BN as a precursor. ∆Ef is found to increase linearly as a function of the number of B-C and N-C bonds at the island boundary, with the slope being nearly the same for either porous graphene or porous h-BN. This is explained by an analytical bond-energy model. In the case of porous graphene, we find that the pore formation energy would be further reduced by passivation by pyridinic and quaternary remnant nitrogen atoms at the pore edges, a mechanism that is found to be more effective than the passivation by hydrogen atoms. Both mechanisms for pore formation energy reduction should lead to a possibly efficient method for nanoporous graphene production.

Introduction Graphene and single-layer hexagonal boron nitride (h-BN) are 2D materials with outstanding chemical and mechanical properties. Their characteristics lead to a range of possible technological applications such as supercapacitors, ion lithium batteries, fillers to increase the mechanical resistance of materials, and others. 1–5 Recent experimental research has been made on porous graphene 2,4,6,7 and porous boron nitride 8,9 for technological applications such as capacitive devices 2,10,11 and water cleaning. 6 Recently, Liao et al. 8 developed a simple method to produce nanoporous boron nitride with defined hole morphology and hole edge structures through oxidative etching. It was shown that the controllable nanostructures exhibited significant changes in electronic properties, turning them suitable for applications like catalysis and molecular transport/separation. 2


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The methods currently used to produce porous graphene and h-BN membranes involve either ion bombardment or chemical etching of graphene and h-BN precursors. 12–15 In the present work, we propose, based on results of first-principles calculations, that nanoporous graphene and h-BN might be produced from B-C-N layers as precursors. Such B-C-N layers can be produced 16 as atomic sheets, via thermal catalytic CVD, containing hybridized bonds involving elements B, N and C over a wide compositional range. These B-C-N 2D materials possess electronic properties intermediate between those of graphene and h-BN due to partial segregation of the 2D alloy into graphene and h-BN nano-islands and regions. 16–19 The size of those regions can vary over a considerably wide range depending on experimental conditions.16−17 Sub-nanometer C islands in a h-BN matrix, which may be composed of 2 C atoms only, have been observed by the means of dark-field electron microscopy measurements

17

. In another experiment, elemental boron mapping measurements shows that the

dimensions of h-BN domains in B-C-N layers are in the range between 2 and 10 nm. It is worth mentioning that the ability to control the size of dopant island in B-C-N precursors may lead to a better control of nanoporous size in porous graphene/hBN, which is important in devices applications. In our calculations, we find that the removal of the h-BN islands that naturally occur in BN-doped graphene to form nanoporous graphene requires less energy than if pristine graphene is used as a precursor. The same reduction ∆Ef in pore formation energy is found for nanoporous h-BN from graphene-doped BN as a precursor. We find that ∆Ef increases linearly with the number of B-C and N-C bonds at the island boundary, with the slope being nearly the same for either porous graphene or porous h-BN, which is explained by an analytical bond-energy model. In the case of porous graphene, we find that the pore formation energy would be further reduced by passivation by pyridinic and quaternary remnant nitrogen atoms at the pore edges, which is found to be more effective than by hydrogen atoms. Additionally, we propose a bond model in the lines of the bond model of Mazzoni et al. 20 capable of reproducing the total energies obtained by means of ab initio

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calculations on nanoporous graphene and nanoporous h-BN. The present paper is organized as follows: in section two the models, methodology and computational details are presented; the energetics of pore formation in graphene and h-BN from 2D B-C-N precursors is presented in section three; section four describes the effects of border termination on the energetic stability of triangular pores; in section five we introduce the bond model for nanoporous graphene and h-BN; and the conclusions are presented in section six.

Models and Computational Details (Nd=2; Nα−C =4) (Nd=4; Nα−C =6)Arm (Nd=4;Nα−C =6)Zig

Figure 1: The top three panels show optimized geometries of BN doped graphene (B-CN with low BN content). B, C and N atoms are represented by green, gray and blue circles, respectively. The bottom panels show optimized geometries of nanoporous graphene structures obtained by removing B and N atoms from the precursor structures shown in the top panels. Nα−C is the number of N-C plus B-C bonds and N d is the number of dopant atoms. The models employed in this work to represent BN-doped-graphene, graphene-doped-hBN, nanoporous graphene, and nanoporous h-BN consist of periodic rectangular supercells, as shown in Fig. 1, 2 and 3. The lattice vectors of the supercells shown in Fig. 1 and √ 2 are a~1 = 12aˆi and a~2 = 6a 3ˆj, where a is an intermediate value between the C-C and B-N bond lengths in their respective hexagonal lattices. The lattice vectors of the larger √ supercells shown in Fig. 3 are a~1 = 18aˆi and a~2 = 10a 3ˆj. To obtain the BN-doped4


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(Nd=6; Nα−C =6)

(Nd=6; Nα−C =8)

(Nd=8; Nα−C =8)

(Nd=8; Nα−C =10)

(Nd=10; Nα−C =8)

(Nd=10; Nα−C =10)

(Nd=10; Nα−C =12)

(Nd=12; Nα−C =10)

(Nd=12; Nα−C =12)

(Nd=12; Nα−C =14)

(Nd=14; Nα−C =10)

(Nd=14; Nα−C =12)

(Nd=14; Nα−C =14)

(Nd=16; Nα−C =10)

(Nd=16; Nα−C =12)

(Nd=16; Nα−C =14)

Figure 2: BN-doped-graphene precursor structures considered in this work. Nanoporous graphene is obtained by removing the B and N atoms. Nα−C is the number of N-C plus B-C bonds and N d is the number of dopant atoms. 5 ACS Paragon Plus Environment


(C-N(Q)) (Ef = 2.1 eV )

(C-N) (Ef = 3.3 eV )

(C-H) (Ef = 4.2 eV )

(C-C) (Ef = 10.2 eV )

(C-NB) (Ef = 17.0 eV ) (C-BN) (Ef = 18.9 eV )

(C-B) (Ef = 29.7 eV ) (C-B(Q)) (Ef = 45.2 eV )

Figure 3: Geometries of triangular holes (tetra-vacancies) in graphene with different border terminations. The gray, blue and green circles represent C, N and B atoms, respectively. The small white circles represent H atoms. The hole border terminations are labeled as C-X, where X is the atom (H, N, B or C) or the pair of atoms (BN or NB) at the hole border. The geometries with a quaternary B or N atom at the triangle vertex are indicated with the superscript (Q). The values of formation energy, calculated through Eq.7 are shown between parenthesis.

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graphene precursor geometries shown in Fig. 2 and in the top panels of Fig. 1, we took into consideration that, to form a single pore per supercell in the porous structure, the h-BN phase in the precursor graphene matrix must be continuous. In addition, we considered equal proportions of B and N atoms, and that high energy cost homopolar B-B and N-N bonds 21–24 are absent. Thus, for a graphene cell doped with a single pair of B and N atoms there is only one possible geometry, which is shown in the top left panel of Fig. 1. There are two nonequivalent configurations for the graphene cell containing 2 pairs of B and N atoms with the constraints cited above: one where B and N atoms lie along the armchair direction (top middle panel of Fig. 1) and another where those atoms lie on the zigzag direction (top right panel of Fig. 1). The number of possible configurations grows exponentially with the number of dopant atoms. For instance, by exchanging a pair of C by B and N atoms in the structures with 4 dopant atoms shown in Fig. 1, it is possible to obtain 26 h-BCN configurations that are continuous and without homopolar bonds. 25 Each of those 26 geometries can lead to 12 up to 24 new configurations by exchanging an additional pair of C atoms by a BN pair. Therefore, it is not feasible to perform a study considering all possible configurations, even for a graphene sheet doped with a small number of B and N atoms. So, we considered, for a given number of dopant atoms, a single geometry for each number of C-B bonds (which is always equal to the number C-N bonds due to the stoichiometry and absence of homopolar B-B and N-N bonds). It is important to mention that at first we employed a simulated annealing process 18,19 to obtain low energy geometries for the structures containing 8, 10, 12, 14 and 16 dopant atoms. Then, we exchanged the positions of atoms in order to obtain the geometries with varying number of B-C(N-C) bonds shown in Fig. 2. The geometries for graphene-doped-h-BN structures are obtained by exchanging the graphene with the h-BN phases in Fig. 1 and 2. The concentrations of dopant h-BN regions in Fig. 1 and 2 are in the range between 2% and 16%, which in principle would lead to porous graphene with the same concentrations of pores. A large concentration of pores may have a strong effect on structural stability of porous graphene and h-BN. However, pores concentrations up to 50%

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in stable superhoneycomb networks have been reported, 26 which suggests that porous layers with concentrations between 2% and 16% are feasible. Our calculations have been performed within the framework of the density functional theory (DFT) 27 as implemented in the SIESTA program. 28 We have used Troullier-Martin, norm conserving relativistic pseudopotentials in the Kleinman-Bylander non-local form. 29 The exchange and correlation energies are treated within the generalized gradient approximation (GGA) according to the Perdew, Burke and Ernzerhof (PBE) parametrization. 30 It is important to mention that semi-local functionals such as the PBE functional fail to describe binding energies resulting from van der Waals interactions. It has recently shown that these interactions can have non-trivial effects in low dimensional materials, 31,32 in particular, effects that extend to separation distances of up to 10-20 nm have been reported. Such long range may allow non-negligible interaction between the pore and its periodic images. The Kohn-Sham eigenfunctions were expanded in a linear combination of numerical atomic orbitals 33 of finite range. The range of the orbitals were determined by the a common energy shift (0.015 Ry ) in the energy eigenvalues imposed by confining potentials in the pseudoatom problem. The split valence method was employed to generate a double-zeta basis for each angular momenta. The Brillouin zone was sampled using a 2 × 2 × 1 Monkhorst-Pack grid and a minimal 150 Ry mesh-cutoff energy was used to determine real space grid fineness. All geometries were optimized using the conjugated gradient algorithm, where the lattice is relaxed together with the atomic coordinates. The cell parameters and atomic positions were optimized until both the maximum component of the stress tensor and force on atoms were smaller than 1.0 GPa and 0.05 eV/Å, respectively.

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Energetics of Conversion of B-C-N into Nanoporous Graphene and Nanoporous h-BN As previously mentioned, B-C-N layers undergo partial segregation into graphene and BN regions. 16–19 In the case of B-C-N with low carbon content, or graphene-doped h-BN, this will lead to small graphene islands in h-BN, 17 and vice-versa in the case of of B-C-N with low BN content, or BN-doped graphene. 19 Motivated by that, we will investigate the energetics of the conversion of those islands in B-C-N into nanopores in graphene or h-BN, and compare it with the energetics of conversion of pristine graphene and h-BN into the same nanoporous structures. In order to address this question, we calculate the energies that are necessary to form nanoporous h-BN and graphene from pristine (h-BN and graphene) or (pristine/doped)

doped (BN-doped-graphene or graphene-doped-h-BN) precursors, Ef orm

, according

to the following expression:

(pristine/doped)

Ef orm (missing)

+NN (porous)

where Etotal

(porous)

= Etotal

(missing)

µ N + NC

(missing)

+ NB

µB

(pristine/doped)

µC − Etotal

(1)

,

is the total energy obtained through supercell calculations for nanoporous (missing)

graphene or nanoporous h-BN structures, Nα

(α = B, N and C) is the number of miss-

ing atoms of the type α in the nanoporous structure in comparison to the respective pristine (pristine/doped)

precursor structure, µα is the chemical potentials for atoms of type α, and Etotal

is the total energy obtained through supercell calculations for the precursor. µC is defined as the total energy per carbon atom of a pure graphene sheet without defects, that is, (graphene)

µC = Etotal

/Ntotal = −154.67 eV , where Ntotal is the number of carbon atoms in the (missing)

graphene supercell. In this section, NN

(missing)

= NB

for all structures, thus we defined

µB +µN = µBN , where µBN is the total energy per pair of atoms of a pure h-BN sheet without defects (µBN = −349.61 eV ). We then rewrite Eq. 1 as follows: 9



(pristine/doped)

Ef orm

(porous)

= Etotal

(missing) +NC µC

(missing)

+ NB −

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µBN +

(2)

(pristine/doped) Etotal .

(pristine/doped)

In the calculations of nanoporous h-BN formation energies, Etotal

can be ei-

ther the total energy of pristine h-BN sheet or the total energy of a graphene-doped-h-BN sheet. Thus, the two possibilities of formation energy obtained through Eq. 2 (pristine or doped) reflect two different experimental situations: (i) nanoporous h-BN is formed by the ejection/etching of B and N atoms from a h-BN lattice (as in the electron-beam radiation experiments performed by Ryu et al); 34 or (ii) it is formed by the ejection/etching of C atoms from a graphene-doped-h-BN matrix. Similarly, in the calculations of nanoporous graphene (pristine/doped)

formation energies, Etotal

can be either the total energy of pristine graphene or the

total energy of a BN-doped-graphene sheet. Using Eq. (2), we obtained the formation energies of pores with geometries that were obtained by removing B and N dopant atoms from the BN-doped-graphene geometries shown in Fig. 1 and 2. Fig. 4-(a) shows the dependence of pore formation energy with the number of removed atoms, N (missing) . The red and black lines shown in such a panel are fit√ (BN −doped−graphene/graphene) tings of Ef orm ∝ N (missing) , which roughly describes the behavior of (BN −doped−graphene/graphene)

Ef orm

(N (missing) ). Such an expression is based on a simple assumption:

(BN −doped−graphene/graphene)

∝ P , where P is the pore perimeter. Because N (missing) ∝ A √ √ (BN −doped−graphene/graphene) (pore area) and P ∝ A, we find Ef orm ∝ N (missing) . It can be seen

Ef orm

in Fig. 4-(a) and in Table 1 35 that BN-doped-graphene precursors, in comparison to pristine graphene precursor, significantly reduces the formation energy of porous graphene. The re(Graphene)

duction in formation energy (Ef orm

(BN −doped−graphene)

− Ef orm

= ∆E (Graphene) ) should also

be roughly proportional to the pore permiter. A linear-like behavior of ∆E (Graphene) (P ) is observed in Fig. 4-(b), where the pore perimeter was assumed to be proportional to the number of N-C and B-C bonds at the boundary between graphene and h-BN phases. The linear 10


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correlation coefficient between ∆E (Graphene) (P ) and the number of N-C and B-C bonds has a value close to unity (r = 0.988), which indicates an almost total positive linear correlation. It is important to mention that because of the superposition of points, not all calculated points are clearly visible in Fig. 4-(b), which emphasizes the dispersion of some points. It is useful to write ∆E (Graphene) and ∆E (h−BN ) as a function of the number of bonds because in this form they can be directly compared with the predictions of bond models such as that of Mazzoni et al. 20 Within such a model, the total energy of any h-BCN structure is given by:

(pristine/doped)

Etotal

=

X

nαβ εαβ ,

(3)

α,β

where α, β = B,C, or N, nα,β is the number of αβ bonds in the structure, and εαβ are energy parameters associated with the α − β bonds. Using Eqs. (2) and (3), the diference between the two possible values of formation energies obtained through Eq. (2) (pristine-doped) is given by: ∆E (Graphene) = (NC(missing) µC − ntotal εCC ) −[NB(missing) µBN − nCC εCC − nBN εBN

(4)

−nBC (εBC + εN C )], where ntotal is the total number of bonds in the precursor structures, that is, ntotal =nCC +nBN +nBC +nN C for the structures shown in Fig. 1 (top panels) and 2. Due to the geometry of the honeycomb lattice, ntotal = 3Ntotal /2, where Ntotal is the total number of atoms. For pristine graphene (graphene)

nCC =ntotal and Eq. (3) reads Etotal (graphene)

because µC = Etotal

= nCC εCC = (3Ntotal /2)εCC . On the other hand, (graphene)

/Ntotal we can eliminate Etotal

in the previous equations to ob-

tain: µC = 3εCC /2. Similarly, µBN = 3εBN is obtained through the application of Eq. (3) to pristine h-BN. 36 In addition, the honeycomb structure also implies that, for the BN-dopedgraphene precursor, nBN = 3NB − nBC . Because the number of atoms removed from the pristine and doped precursors to form the nanoporous graphene sheet are the same we have

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NC(missing) = NB(missing) + NN(missing) = 2NB(missing) . Thus, Eq. (4) becomes: ∆E (Graphene) = nBC (εBC + εN C − εBN − εCC ) (n + nN C ) (εBC + εN C − εBN − εCC ). = BC 2

(5)

Similarly, it is possible to obtain: (h−BN )

∆E (hBN ) = Ef orm

(graphene−doped−h−BN )

− Ef orm

(n + nN C ) = BC (εBC + εN C − εBN − εCC ). 2

(6)

The two previous equations indicate that, according to the model of Mazzoni et al., 20 the use of B-C-N precursors will always reduce the pore formation energy, and that the reduction is given by the same formula for both porous graphene and porous h-BN. More specifically, using the εαβ values parametrized from first principles in Ref. 20, we obtain ∆E Graphene = ∆E hBN = (nBC + nN C ) × 0.61 eV. From the linear fittings of Fig. 4 (b) and (d) we obtain ∆E Graphene = (nBC + nN C ) × 0.56 eV and ∆E hBN = (nBC + nN C ) × 0.44 eV. That is, the model has a predictive power, without any fitting to the present calculations, within an accuracy of about 30%. The dispersion of points of E hBN over the linear behavior (r = 0.93) is larger than that of E graphene (r = 0.99). Such a difference could be related to quantum confinement effects on graphene islands with the same number of B-C and N-C, in which shape (linear or rounded) and size (number of C atoms) can be very different. For instance, there are structures in Table 1 with total number of N-C and B-C bonds equals to 10 whose number of dopant C atoms varies from 8 up to 16. The process of converting B-C-N into nanoporous graphene/h-BN could be obtained through the chemical etching of one phase, which may result in passivated hole borders (specially in the etching by efficient stabilisers such as H, which have been reported for the production of nanoporous h-BN). 37 In this scenario, new chemical potentials should be defined in order to reflect the chemical environment. Nevertheless, our aim in this section

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Table 1: Formations energies of porous graphene and porous h-BN considering h-B-C-N layers as precursors. In parenthesis are the formation energies of pores considering pristine graphene or h-BN as precursors. Pores are obtained by either removing dopant atoms from h-B-C-N precursors or by removing the equivalent atoms in pristine graphene or h-BN. Figures 1 and 2 show the geometries of h-B-C-N precursors employed to obtain porous graphene, where Nd is the number of dopant atoms and Nα−C is the number of N-C plus B-C bonds. BN-doped-graphene Precursor (Nd =2; Nα−C =4) (Nd =4; Nα−C =6)Armchair (Nd =4; Nα−C =6)Zigzag (Nd =6; Nα−C =6) (Nd =6; Nα−C =8) (Nd =8; Nα−C =8) (Nd =8; Nα−C =10) (Nd =10; Nα−C =8) (Nd =10; Nα−C =10) (Nd =10; Nα−C =12) (Nd =12; Nα−C =10) (Nd =12; Nα−C =12) (Nd =12; Nα−C =14) (Nd =14; Nα−C =10) (Nd =14; Nα−C =12) (Nd =14; Nα−C =14) (Nd =16; Nα−C =10) (Nd =16; Nα−C =12) (Nd =16; Nα−C =14) Graphene-doped-h-BN Precursor (Nd =2; Nα−C =4) (Nd =4; Nα−C =6)Armchair (Nd =4; Nα−C =6)Zigzag (Nd =6; Nα−C =6) (Nd =6; Nα−C =8) (Nd =8; Nα−C =8) (Nd =8; Nα−C =10) (Nd =10; Nα−C =8) (Nd =10; Nα−C =10) (Nd =12; Nα−C =10) (Nd =12; Nα−C =12) (Nd =12; Nα−C =14) (Nd =14; Nα−C =10) (Nd =14; Nα−C =12) (Nd =14; Nα−C =14) (Nd =16; Nα−C =10) (Nd =16; Nα−C =12) 13

Porous graphene Ef orm (eV) 5.22 (6.99) 11.67 (14.60) 10.43 (14.22) 16.06 (19.10) 14.07 (18.30) 14.87 (19.90) 18.73 (24.00) 20.79 (25.40) 20.88 (26.20) 18.31 (25.10) 23.36 (28.60) 24.81 (31.10) 22.54 (29.69) 24.85 (30.30) 23.60 (30.80) 23.80 (31.50) 25.60 (31.74) 27.38 (33.97) 28.80 (36.70) Porous h-BN Ef orm (eV) 6.29 (8.26) 11.07 (13.70) 10.96 (13.60) 12.34 (14.70) 12.87 (16.70) 14.76 (19.90) 17.56 (22.20) 18.66 (21.60) 23.52 (28.10) 18.03 (22.20) 23.14 (28.10) 21.50 (26.00) 19.60 (25.00) 24.00 (29.50) 23.00 (29.20) 24.60 (27.90) 25.20 (29.90)



is to shed light on the role of the precursor (perfect graphene/h-BN versus B-C-N) in the energetics of pore formation, which can be performed provided that both precursors lead to the same pore. The effects of chemical species as stabilizing agents for pore edges will be discussed in the next section. 8

(eV)

(a)

30

Graphene

20 10 0 -10

∆E

E form (eV)

40

Graph.→Porous Graph. hBN Doped Graph.→ Porous Graph.

0

10

5

15

4 2

y=0.56x-0.07; r=0.99 0

20

0

8

(eV) hBN

20

∆E

10 0 -10

10

5

12

16

6 4 2

hBN→Porous hBN Graph. doped hBN→Porous hBN

0

8

(d)

(c)

30

4

Number of C-B + N-C Bonds

N

40

(b)

6

(missing)

E form (eV)

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

Page 14 of 29

15

y=0.44x-0.09; r=0.93

20

0

0

4

8

12

16

Number of C-B + N-C Bonds

(missing)

N

Figure 4: Panels (a), (c): Formation energies of nanoporous graphene (a) and nanoporous hBN (c) from either pristine (black circles) or B-C-N (red diamonds) precursors, as functions of the number of atoms removed to form the pore. Panels (b), (d): the difference between the energies shown in the panel at the left as a function of the number of bonds √ at h-BNgraphene boundaries. The curves in panels (a) and (c) are fittings of E ∝ N (missing) . The correlation coefficients of the lines shown in panels (b) and (d) are r=0.99 and r=0.93, respectively.

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Energetics of Nanoporous Graphene with Passivated Pore Edges Even in situations where the amount of hydrogen atoms (or any other stabilizing agent) in the environment is low, the conversion of h-BCN into nanoporous graphene could result in pores with B and N atoms at the edges. In order to address the question of the energetic stability of those modified (doped) edges, we calculate the formation energy of nanoporous graphene with doped edges as follows:

(Doped−pore)

Ef = Etotal

− NC µC − NN µN − NB µB − NH µH ,

(7)

doped−pore where Etotal is the total energy of doped pore structure, µN and µH are defined as half

the total energy of N2 and H2 , and µB is obtained by the constraint µB + µN = µBN . As in the previous sections, µBN is defined as the total energy per BN pair of a perfect h-BN sheet. In this section, rather than employing the geometries shown in Fig. 1 and 2, we study pores with triangular shapes. Such a choice is based on the fact that triangular shapes allow us to investigate separately each of the possible edge terminations, which is not possible using the general shapes shown in Fig. 1 and 2. Besides, such kind of triangular holes is consistent with experimental evidence. 34 The smallest studied triangular pore is the D3-symmetry tetra-vacancy, 38 shown in Fig. 3. In such small pores, reconstructions may prevent the presence of dangling bonds. 38 The reconstructed hole preserves the D3-symmetry. It consists of a nonagon sharing sides with 3 pentagons and 6 hexagons. In Fig. 3-(C-C) and 3-(C-BN) the nonagons are almost regular, with bond lengths 1.6 Å(at sides in contact with hexagons) and 1.5 Å(at side in contact with pentagons). As for the other terminations, the homopolar B-B and N-N bonds are larger than B-N or B-C bonds. In fact, in the N terminated tetra-vacany, Fig. 3 (C-N(Q) ) and (C-N), the N-N bond length (2.45 Å) shows that reconstructions do not occur in those structures, which is consistent with previous results on non-reconstructed N-doped mono15



vacancy in graphene. 39 Such structures present values of formation energy smaller than those of other geometries shown in Fig. 3. In particular, the formation energy of the (C-N(Q) ) border is only 1/5 of the value of the formation energy of the native tetra-vacant in graphene, Fig. 3-(C-C), which suggests that triangular pores in graphene with pyridinic N borders can be observed experimentally. In addition, the formation energies of pores with N-terminated borders are smaller than the formation energies of pores with hydrogenated borders, which suggest that N terminated borders are stable even in the presence of efficient stabilizer agents like H atoms. It is important to mention that the presence of H atoms decrease by 6 eV the formation energy of the native tetra-vacancy, which indicates that H atoms should react with reconstructed defects in graphene where dangling bonds are absent. Interestingly, the formation energy of triangular pores in graphene with pyridinic N borders decreases when C atoms at triangle vertex are replaced by quaternary N atoms, as seen in Fig. 3-(C-N) and (C-N(Q) ) . Such a result is not intuitive because the C atoms at triangle vertex in Fig. 3-(C-N) are surrounded by other three C atoms, which are not at the hole edge. Then, pyridinic N atoms at the pore edges can strongly affect second-neighbor atoms. It is important to verify if the results described above are also valid for triangular pores larger than tetra-vacancies. Fig. 5 shows the formation energy (Ef orm ) as a function of the length of the triangle side (L), where a linear dependence of the formation energy with L is observed for all types of border terminations. Then, we can separate the investigated hole terminations in three groups based on the angular coefficient (α) of Ef orm (L). A group containing the C-N, C-N(Q) and C-H terminations (0.92 ≤ α ≤ 0.98 eV/Å), another group containing the C-NB, C-C and C-BN terminations ( 4.1 ≤ α ≤ 5.1 eV/Å), and a third group containing the C-B and the C-B(Q) terminations (α = 6.9 and α = 7.5 eV/Å, respectively). Because the groups with smaller angular coefficients also have smaller linear coefficients no change in stability trends between groups as L varies is observed. However, the energetic stability trends between elements of the same group can change with increasing L. A change in the stability trends of BN-B structures in comparison to BN-N structures is observed for

16


Page 16 of 29

Page 17 of 29

L ≈ 9 Å. In fact, because of the N terminations BN-N structures tend to be energetically more stable, but in case of tetra-vacancies the longer B-B bonds in comparison to N-N enables reconstruction with lower strain. The right panel of Fig. 5 shows Ef vs L in an energy scale where only the group with smallest angular coefficient is shown. It can be seen that the family of triangular holes with C-N(Q) presents, for all values of L, the smallest values of formation energy. The process of pore formation depends on environmental conditions such as the presence of solvents eventually used in an etching process. Therefore, effects of kinetics of pore formation or the effect possible solvents would be important in further works on the subject. Recent experimental works on nanoporous graphene 40 have used oxygen plasma etching to produce the porous structures. Therefore, the study of possible sites for oxygen adsorption in BCN might be a possible route to investigate the kinetics of pore formation. For example, quantum chemical calculations 41 indicate that in BCN nanotubes the oxygen atom might tend to adsorb between carbon and boron atoms.

140 120 100

E form (eV)

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


80

15

(Q)

C-N C-N C-H C-C CN-B CB-N C-B (Q) C-B

C-N C-N C-H

(Q)

10

60 5

40 20 0

8

10

12

14

16

18

Triangle Side (Å)

0

8

10

12

14

16

18

Triangle Side (Å)

Figure 5: Left panel: Formation energies Ef of triangular pores in graphene calculated through Eq. 7 as functions of the triangle side length (L). The several types of border termination are the same as the ones depicted in Fig. 3. Right panel: Ef vs L at an energy scale that shows the most stable borders.

17



Bond Models for Nanoporous Graphene and Nanoporous h-BN

(Bond Model)

(eV)

40

E

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

Page 18 of 29

30

20

10

0 0

Porous Graphene Porous hBN 10

20 (ab initio)

E

30

40

(eV)

Figure 6: Comparison between the proposed bond models and the ab initio results shown in Table 1. The black circles refer to the formation energies of nanoporous graphene, while the blue diamonds refer to the formation energies of nanoporous h-BN. The red line is the line y=x. The mean values of the quantity E (BondM odel) /E (abinitio) are 0.99 and 1.03 for nanoporous graphene and h-BN, respectively. The standard deviations of E (BondM odel) /E (abinitio) from unity are 0.09 and 0.08 for nanoporous graphene and h-BN, respectively.

Bond models have been employed in simulated annealing calculations that are able to predict structural properties of heterostructures composed of h-BN and graphene, such as phase segregation. 18 Also, as shown in Section 3, bond models provide the possibility of analytical results that can explain the phenomenology of ab initio results. In this section, we present an extension of the bond model of Mazzoni et al. 20 that can reasonably predict the energies of ab initio calculations on nanoporous graphene and nanoporous h-BN. Because bonds derive from the electronic structure, changes in the coordination of atoms or in bond angles due to the presence of pores result in changes on the electronic structure. If the concentration of pores is not large, pores can be seen as defects that modify the electronic structure of graphene/h-BN near the Fermi level. 42 In this case, the bond model

18


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of Mazzoni et al. 20 parameterized for pristine graphene/h-BN should correctly account for (porous)

most of Etotal

(porous)

. Therefore, it is convenient to write Etotal

in terms of the bond model of

Mazzoni with additional corrections due to pore-induced changes in the electronic structure, as (porous) Etotal

Z∞ =A

εg(ε)(porous) F (ε)dε =

X

Zεf nαβ εαβ + A

α,β

−∞

εg(ε)(porous) F (ε)dε,

(8)

ε0

where α, β = B,C, or N, nα,β is the number of αβ bonds in the structure, and εαβ are energy parameters associated with the α − β bonds. The second term of the right side of the equation above is a correction due to changes in the electronic structure within the window between ε0 and εf . 43 ε0 is the energy of the highest occupied level in pristine graphene or h-BN and εf is the Fermi level of the porous structures. g(ε)(porous) is the electron density of states, F (ε) is the Fermi-Dirac distribution and A is the area of the nanoporous sheet (A ∝ NA , where NA is the number of atoms in the nanoporous structure). g(ε)(porous) depends on the presence of dangling bonds and lattice distortions due to pores. Thus, it is convenient to separate g(ε)(porous) in a term that depends essentially on the presence of dangling bonds and another that depends essentially on the lattice distortions, that is: g(ε)(porous) = g(ε)(f ixed

geometry)

+ ∆g(ε), where g(ε)(f ixed

geometry)

is the density of states of

porous graphene/h-BN whose atom coordinates were not allowed to relax after pores were made. Then, we rewrite Eq. 8 as:

(porous)

Etotal

=

X

C/BN

nαβ εαβ + εR

Zεf +A

α,β

C/BN

where εR

=A

Rεf

εg(ε)(f ixed

geometry)

F (ε)dε,

(9)

ε0

(porous)

ε∆g(ε)F (ε)dε is the contribution to Etotal

due to lattice distortions.

ε0

The magnitude of the lattice distortions, like the magnitude of distortions resulting from dislocations in 3D crystals lattices ( whose magnitude and direction are represented by the Burgers vector 44 ), should not scale with the pore size. Thus, for the sake of simplicity, we C/BN

assume that εR

is constant for any pore and null for the perfect lattice. 19



Page 20 of 29

To evaluate the integral in Eq. (8), we have to take into account that the electronic structure of graphene and h-BN, and their response to the presence of defects, is different. Let us first discuss the case of nanoporous graphene structures. We assume that the concentration of pores is not large enough to modify the linear dependence of graphene’s density of states, g(ε) =

2ε π¯ hvf2

= c1 ε, but it can change the position of the Fermi level relative to the Dirac

point ε0 . Using g(ε)(f ixed

geometry)

≈ c1 ε and F (ε) = 1 (ab initio calculations are usually

performed at/ zero Kelvin) we can rewrite Eq. 9 as follows: (porous)

Etotal

= nCC εCC + εC R + Ac1

(ε3f − ε30 ) . 3

The number of electrons within the integration window (ne = A

Rεf

(10)

g(ε)(f ixed

geometry)

F (ε)dε =

ε0

Ac1

(ε2f −ε20 ) ) 2

depends on the number of dangling bonds, nd . If we assume ne = c2 nd , where c2

is a constant, we can eliminate εf in Eq. 10 to obtain: " (porous)

Etotal

= nCC εCC + εC R + a0 NC 1 −

a1 n d 1+ NC

!3/2 # ,

(11)

where a0 and a1 are combinations of constants of proportionality. 45 εC R = 6.99 eV was defined as the formation energy of the divacancy. The values of a0 (39.57 eV) and a1 (0.81 eV) (Graphene)

are obtained by means of the best fit of Ef orm

(missing)

vs. NC

(the number of removed C (missing)

atoms) to a subset of the nanoporous geometries that obeys the relations: nd = NC (missing)

and nCC = 145−2NC

−2

. As previously mentioned in Ref. 20, εCC = 2µC /3 = −103.24 eV.

Fig. 6 shows a comparison between the values of formation energy obtained through ab initio calculations (shown in Table 1) and through the proposed bond model. Despite its simplicity, the bond model describes reasonably well the ab initio results, with the mean value of the quantity E (BondM odel) /E (abinitio) = 0.99 being very close to unity, and with small standard deviation from unity (0.09). It is important to mention that the quality of the model may be improved by discriminating between dangling bonds at zigzag and armchair sites or by a better description of term εC R. 20


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Let us now to discuss the bond model for nanoporous h-BN. Defects in h-BN tend to introduce localized states in its wide band gap. In this sense, we assume that the density of states with energy between ε0 and εf is composed of Dirac deltas, that is, g(ε)(f ixed P i ai δ(ε − εi ). Thus Eq. (9) leads to:

geometry)

≈

(h−BN )

Ef orm

= nBN εBN + nBB εBB + nN N εN N X +A εi ai + εBN R − NB µB − NN µN .

(12)

i

Again, we assume that the number of electrons with energy between ε0 and εf is propor Rεf P N tional to the number of dangling bonds, that is, A g(ε)dε = A i ai = aB nB d + aN nd /ε, ε0 N where nB d and nd are the number of dangling bonds at B and N sites, respectively. Assuming P P i ε i ai ≈ ε i ai we obtain:

(hBN )

Ef orm = nBN εBN + nBB εBB + nN N εN N +aB nB d

+

aN n N d

+

εBN R

(13)

− NB µB − NN µN ,

which is the same result we obtain by adding two new bonds (aB and aN ) and the term εBN to the bond model of Mazzoni et al. 20 The values of the constants εBN (−116.53 eV), R εBB (−50.40 eV) and εN N (−178.49 eV) were previously obtained by Mazzoni et al. 20 To obtain εBN R (−3.49 eV) we equal Eq. (13) to the formation energy of the divacancy in a h-BN matrix, shown in Table 1, which contains one N-N bond, one B-B bond and no dangling bonds. To obtain the constants aB (−22.81 eV) and aN (−88.87 eV), Eq. (13) is made equal to the values N of formation energies show in Table 1, which results in a set of equations where nB d = nd and N another set where nB d 6= nd . By adding up equations of each set we obtain two equations

that can be solved to obtain aB and aN . Fig. 6 shows the formations energies predicted by the bond model for nanoporous h-BN

21



as a function of the values obtained by means of ab initio calculations (Table 1). Similarly to the case of nanoporous graphene bond model, the bond model for nanoporous h-BN describes reasonably well the ab initio results. The mean value of E (BondM odel) /E (abinitio) is 1.03 and the standard deviation of such a quantity from unity is 0.08.

Conclusions In summary, we addressed the effects of the precursor (either B-C-N or pristine graphene and h-BN) in the formation energies of nanopores in graphene and h-BN. The B-C-N precursors lead to smaller formation energies in comparison to the values obtained considering pristine hBN or graphene precursors. The difference between the values of formation energy calculated considering the two types of precursors is proportional to number of B-C and C-N bonds at the boundary between graphene and h-BN phases of the B-C-N precursors. Such a result is explained in terms of an analytical bond model. The model also explains why the gain in formation energy with the use of a B-C-N precursor is approximately the same for nanoporous graphene and h-BN. We also studied the effects of several types of pore edge terminations in the pore energetic stability, in the case of triangular pores in graphene. We find that pores with N-pyridinic borders are energetically more stable than pores with hidrogenated borders, which suggests that nanoporous graphene obtained from B-C-N precursors might present Npyridinic borders even if efficient border stabilizers, such as hydrogen, are present in the synthesis process. The energetic stability of nanoporous graphene with N-pyridinic borders is increased with the substitution of a C atom at the triangle vertex by a N-quaternary, which shows that the N atoms at the border have a strong effect on second-neighbor atoms. Finally, we propose an extension of the bond model of Mazzoni et al. 20 that is able to reasonably well our ab-initio results on nanoporous graphene and nanoporous h-BN.

22


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Acknowledgement We acknowledge Brazilian science agencies CNPQ, FACEPE, FAPEMIG, CAPES, and the project INCT de Nanomateriais de Carbono. RJCB, ABO, and TMM thank PROPP-UFOP for the Auxílio Financeiro a Pesquisador-Custeio 2016.

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Storr, K.; Balicas, L. et al. Atomic layers of hybridized boron nitride and graphene domains. Nature Materials 2010, 9, 430. (17) Krivanek, O. L.; Chisholm, M. F.; Nicolosi, V.; Pennycook, T. J.; Corbin, G. J.; Dellby, N.; Murfitt, M. F.; Own, C. S.; Szilagyi, Z. S.; Oxley, M. P. et al. Atomby-atom structural and chemical analysis by annular dark-field electron microscopy. Nature 2012, 464, 571–574. (18) da Rocha Martins, J.; Chacham, H. Disorder and Segregation in B-C-N Graphene-Type Layers and Nanotubes: Tuning the Band Gap. ACS Nano 2011, 5, 385–393. (19) Nascimento, R.; da Rocha Martins, J.; Batista, R. J. C.; Chacham, H. Band gaps of BNdoped graphene: Fluctuations, trends, and bounds. The Journal of Physical Chemistry C 2015, 119, 5055–5061. (20) Mazzoni, M. S. C.; Nunes, R. W.; Azevedo, S.; Chacham, H. Electronic structure and energetics of B x C y N z layered structures. Physical Review B 2006, 73, 073108. (21) Blase, X.; De Vita, A.; Charlier, J.-C.; Car, R. Frustration effects and microscopic growth mechanisms for BN nanotubes. Physical Review Letters 1998, 80, 1666. (22) Batista, R. J. C. Stoichiometric boron nitride fullerenes with homopolar B–B and N–N bonds. Chemical Physics Letters 2010, 488, 209–212. (23) Batista, R. J. C.; Mazzoni, M. S. C.; Chacham, H. A theoretical study of the stability trends of boron nitride fullerenes. Chemical Physics Letters 2006, 421, 246–250. (24) Batista, R. J. C.; Mazzoni, M. S. C.; Chacham, H. Boron nitride fullerene B36N36 doped with transition metal atoms: First-principles calculations. Physical Review B 2007, 75 . (25) Some of those 26 geometries are equivalent by symmetry operations.

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(26) Bieri, M.; Treier, M.; Cai, J.; Ait-Mansour, K.; Ruffieux, P.; Groening, O.; Groening, P.; Kastler, M.; Rieger, R.; Feng, X. et al. Porous graphenes: two-dimensional polymer synthesis with atomic precision. CHEMICAL COMMUNICATIONS 2009, 6919–6921. (27) Kohn, W.; Sham, L. J. Self-consistent equations including exchange and correlation effects. Physical Review 1965, 140, A1133. (28) Soler, J. M.; Artacho, E.; Gale, J. D.; García, A.; Junquera, J.; Ordejón, P.; SánchezPortal, D. The SIESTA method for ab initio order-N materials simulation. Journal of Physics: Condensed Matter 2002, 14, 2745. (29) Kleinman, L.; Bylander, D. M. Efficacious form for model pseudopotentials. Physical Review Letters 1982, 48, 1425–1428. (30) Perdew, J. P.; Burke, K.; Ernzerhof, M. Generalized gradient approximation made simple. Physical Review Letters 1996, 77, 3865. (31) Ambrosetti, A.; Silvestrelli, P. L.; Tkatchenko, A. Physical adsorption at the nanoscale: Towards controllable scaling of the substrate-adsorbate van der Waals interaction. PHYSICAL REVIEW B 2017, 95 . (32) Ambrosetti, A.; Ferri, N.; DiStasio, R. A., Jr.; Tkatchenko, A. Wavelike charge density fluctuations and van der Waals interactions at the nanoscale. SCIENCE 2016, 351, 1171–1176. (33) Junquera, J.; Paz, Ó.; Sánchez-Portal, D.; Artacho, E. Numerical atomic orbitals for linear-scaling calculations. Physical Review B 2001, 64, 235111. (34) Ryu, G. H.; Park, H. J.; Ryou, J.; Park, J.; Lee, J.; Kim, G.; Shin, H. S.; Bielawski, C. W.; Ruoff, R. S.; Hong, S. et al. Atomic-scale dynamics of triangular hole growth in monolayer hexagonal boron nitride under electron irradiation. Nanoscale 2015, 7, 10600–10605. 26


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(35) Two initial porous h-BN geometries leads to h-BN with pentagon-heptagon defects after geometry optimization. Therefore, Table 1 contains only 17 values of formation energy for porous h-BN. (hBN )

(36) By definition: µBN = 2Etotal /Ntotal . (37) Sharma, S.; Kalita, G.; Vishwakarma, R.; Zulkifli, Z.; Tanemura, M. Opening of triangular hole in triangular-shaped chemical vapor deposited hexagonal boron nitride crystal. Scientific Reports 2015, 5 . (38) Bao, Z.-q.; Shi, J.-j.; Yang, M.; Zhang, S.; Zhang, M. Magnetism induced by D3symmetry tetra-vacancy defects in graphene. Chemical Physics Letters 2011, 510, 246– 251. (39) Wang, H.; Maiyalagan, T.; Wang, X. Review on Recent Progress in Nitrogen-Doped Graphene: Synthesis, Characterization, and Its Potential Applications. ACS Catalysis 2012, 2, 781–794. (40) Surwade, S. P.; Smirnov, S. N.; Vlassiouk, I. V.; Unocic, R. R.; Veith, G. M.; Dai, S.; Mahurin, S. M. Water desalination using nanoporous single-layer graphene. NATURE NANOTECHNOLOGY 2015, 10, 459–464. (41) Rupp, C. J.; Rossato, J.; Baierle, R. J. First-principles study of oxidized BC2N nanotubes. INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY 2012, 112, 3312–3319. (42) Defects in graphene tend to shift the position of the Fermi level relative to the Dirac point and defects in h-BN tend to introduce localized states in its wide band gap. (43) Brito, W. H.; da Silva-Araujo, J.; Chacham, H. g-C3N4 and Others: Predicting New Nanoporous Carbon Nitride Planar Structures with Distinct Electronic Properties. JOURNAL OF PHYSICAL CHEMISTRY C 2015, 119, 19743–19751. 27



(44) Ashcroft, N.; Mermin, N. Solid State Physics; HRW international editions; Holt, Rinehart and Winston, 1976. (45) a0 = c0 c1 |ε0 |3 /3, where c0 = NC /A, and a1 = 2c2 /c0 ε20 .

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Graphical TOC Entry

B

C

N

1

29


Nanoporous Graphene and H-BN from BCN Precursors: First

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