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Two novel superhard structures: Monoclinic BC3N
Journal Pre-proof Two novel superhard structures: Monoclinic BC3N Shuaiqi Li, Liwei Shi PII:
S0921-4526(20)30075-2
DOI:
https://doi.org/10.1016/j.physb.2020.412061
Reference:
PHYSB 412061
To appear in:
Physica B: Physics of Condensed Matter
Received Date: 29 September 2019 Accepted Date: 31 January 2020
Please cite this article as: S. Li, L. Shi, Two novel superhard structures: Monoclinic BC3N, Physica B: Physics of Condensed Matter (2020), doi: https://doi.org/10.1016/j.physb.2020.412061. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2020 Published by Elsevier B.V.
Two novel superhard structures: monoclinic BC3 N Shuaiqi Lia , Liwei Shia,∗ a School
of Physical Science and Technology, China University of Mining and Technology, Xuzhou 221116, China
Abstract Two potential superhard monoclinic BC3 N structures, namely, m-BC3 N-1, and m-BC3 N-2, have been proposed through first-principles calculations. The structural, mechanical, and electronic properties of m-BC3 N structures are investigated in detail. The phonon dispersions and calculated elastic constants show that two m-BC3 N phases are dynamic and mechanical stable. The formation energies of these m-BC3 N structures are positive, indicating that they are metastable which similar to most ternary B-C-N compounds. Based on the elastic calculations, both of m-BC3 N-1 and m-BC3 N-2 behave in brittle character, and show great anisotropy in Young’s modulus. In hardness simulation, a microscopic theoretical model is employed, and finally obtained the Vicker’s hardness of two m-BC3 N structures with 56.38 GPa and 52.96 GPa, respectively. Using a hybrid functional HSE03, we estimate that the two m-BC3 N structures are indirect semiconductors with a bang gap of 3.77 eV and 3.25 eV. Besides, m-BC3 N-2 can be converted from an indirect band gap semiconductor to a direct band gap semiconductor with band gap of 3.45 eV when exposed to a hydrostatic pressure of 30 GPa. Keywords: boron-carbon-nitrogen, computational physics, density functional theory, superhard material
∗ Corresponding
author Email address: [email protected] (Liwei Shi)
Preprint submitted to PHYSICA B-CONDENSED MATTER
September 29, 2019
1. INTRODUCTION The search for superhard materials has been an important direction in material science due to their immeasurable value in modern industry [1]. Originating from the nature, diamond is well known for its wide applications in industrial 5
as cutting, polishing tools and wear-resistant coating [2, 3]. However, its low oxidation temperature and chemical instability have greatly limited the practical application, especially for the processing of ferrous alloys [4, 5, 6]. Cubic boron nitride (c-BN) has excellent oxidation resistance and chemical inertness, thereby making it a substitute for diamond to a certain extent despite its hard-
10
ness is only approximately half that of diamond [7]. On account of their diverse structures and atomic compositions, the ternary boron-carbon-nitrogen (B-C-N) compounds family possess adjustable band gap with considerable Vicker’s hardness and superior oxidation resistance, significantly oriented the development of superhard materials[8, 9, 10, 11]. Starting from the graphitic BC2 N (g-BC2 N)
15
synthesized by Kouvetakis [12], considerable ternary B-C-N compounds have been predicted or synthesized for decades. Solozhenko et al. [13] synthesized cubic BC2 N from g-BC2 N under high pressure and high temperature, which had a hardness up to 76 GPa, harder than c-BN crystal but still softer than diamond. Zhou et al. [14, 15] predicted two ultrahard tetragonal t-BC2 N and
20
z -BC2 N structures that even reach a high hardness of 75.3 GPa and 75.9 GPa, respectively. Liu et al. [16] successively developed a novel BC2 N structure in hexagonal symmetry with large bandgap energy, showing a remarkable hardness of 71 GPa in simulation. Furthermore, metallic BC2 N compounds, diamondlike BCx N (x = 1 - 4) [17, 18, 19] and (P-4m2, P3m1) BC2 N [20] which can be
25
synthesised by high-pressure and high-temperature (up to 2500 K) conditions reached the hardness of 56 - 58 GPa, also drawing much interests of researchers in potential superhard and superconductive material. In addition to diamond-like ternary structures, numerous other kinds of BC-N structures are concerned by light atoms replacement or structural search,
30
such as CALYPSO [21, 22, 23]. Gao et al. [24] introduced three BC8 N struc-
2
tures with high carbon content, in which BC8 N-3 had a negative formation energy with a wide band-gap. The tetragonal C8 B2 N2 structure with hardness of 65 GPa, proposed by Wang et al. [25], have been studied systematically in our previous work [26]. Although BCx N (x = 3 or x ≈ 3) compounds are 35
rarely investigated because of the stoichiometry, its complex and potentially valuable nature still attract the enthusiasm of many researchers, such as the heterodiamond BC3 N synthesized using graphite-like B-C-N compounds [27] or graphite-like BCx N(H) (x = 2,3,5) [28], and orthorhombic BC3.3 N prepared by amorphous B-C-N precursor [29]. Li et al. [30] proposed 32 kinds of B3 C10 N3
40
monolayer, and based on this, a narrow gap semiconductor B3 C10 N3 was constructed. In this paper, we investigate two novel BC3 N structures containing twelve carbon atoms, four boron atoms, and four nitrogen atoms a unit cell. The ground-state properties of the BC3 N structures are investigated and discussed in detail through first-principles calculations are performed.
45
2. THEORETICAL METHODS The first-principle calculations are performed using the plane-wave pseudopotential method within the framework of DFT and performed by the CASTEP code [31, 32]. The Perdew-Berke-Ernzerhof form of the generalized gradient approximation (GGA) is used to describe the exchange correlation term [33]. The
50
2s 2 2p 1 , 2s 2 2p 2 , and 2s 2 2p 3 states are considered as valence electrons for B, C, and N atoms, respectively. The ultrasoft pseudopotential that described the interactions of electrons with ion cores [34], is expanded in a plane-wave basis set with a cutoff energy of 800 eV, where the self-consistent convergence of total energy is at 0.5×10−6 eV/atom. For the fine level k-point sampling, a
55
Monkhorst-Pack mesh of 7 × 16 × 8 is adopted in the Brillouin zone [35]. The phonon frequencies and dispersion curves are calculated by the linear-response method based on DFPT [36]. To ensure the accuracy of determined electronic properties, calculations are repeated using the hybrid Heyd-Scuseria-Ernzerhof functional (HSE03) [37, 38], which contains 75 % semi-local GGA-PBE and
3
60
25% nonlocal Hartree-Fock (HF) approach. In Vicker’s hardness simulation, a 12 times supercell is constructed, and a semi-empirical theoretical model of covalent crystal is used [39, 40].
3. RESULTS AND DISCUSSION 3.1. Structure Characters. 65
The monoclinic BC3 N (BC3 N-1 and BC3 N-2) compounds crystallize in the centered monoclinic structure with space group C/2m. Fig. 1 shows the crystal structures of m-BC3 N-1 and m-BC3 N-2 after structural optimization at ambient pressure, while the Table I lists their atomic Wyckoff positions. In their crystal structures, the five-membered rings have the same bond composition (C-C, C-B,
70
B-N, C-N); but compared to the six-membered rings in m-BC3 N-1 with C-C bonds and C-B bonds, m-BC3 N-2 has no C-C bonds in those, and instead are B-N bonds and C-N bonds. No N-N bonds and B-N bonds exist in the sevenmembered rings of m-BC3 N-2. Consequently, C-C, C-C, C-N, and B-N bonds coexist in both structures, but N-N bonds only present in m-BC3 N-1. Table 1: Atomic Wyckoff positions of the m-BC3 N-1, and m-BC3 N-2 structures.
BC3 N-1
75
BC3 N-2
Atom
Wyckoff positions
Atom
Wyckoff positions
C1
4i(0.0685,0,-0.522)
C1
4i(0.0783,0,-0.512)
C2
4i(-0.503,0,-0.333)
C2
4i(0.754,0,-0.0459)
C3
4i(0.748,0,-0.0524)
C3
4i(-0.994,0,-0.910)
B
4i(-0.995,0,-0.912)
B
4i(-0.510,0,-0.339)
N
4i(-0.364,0,-0.729)
N
4i(-0.361,0,-0.735)
The calculated equilibrium lattice constants, density, total energy, formation energy, elastic modulus, and elastic constants as well as the theoretical results of diamond and c-BN are shown in Table II. The bulk modulus of two types mBC3 N is 360 GPa and 338 GPa, respectively, lower than that of c-BN, 369 GPa.
4
Figure 1: Relaxed unit cells of (a)m-BC3 N-1, and (b)m-BC3 N-2. The carbon, boron, and nitrogen atoms are painted in dark, orange, and blue, respectively. There are twelve carbon atoms, four boron atoms, and four nitrogen atoms in a unit cell. (c) A 3 × 2 × 3 supercell of m-BC3 N-1.
The shear modulus of m-BC3 N-1 is 388 GPa, which is slightly higher than that 80
of c-BN. Actually, Bulk modulus and shear modulus characterize the resistance ability of incompressibility and plastic deformation of a material, respectively. According to Pughs theory [41], the value of B /G ratio is reckoned as a differentiation of ductile or brittle characters, that is, value overs 1.75 showed ductile character, otherwise it should be brittle. The ratio of m-BC3 N-1 and
85
m-BC3 N-2 are 0.927 and 0.921, respectively, suggesting their brittle behaviors. In addition, the positive formation energy E f of m-BC3 N, defined as E f = E BCN - ( 2E diamond + 3E c−BN )/4, indicates that the m-BC3 N structures are metastable. Nevertheless, structures with positive formation energies are quite common in ternary B-C-N system, except for the negative structures BC8 N-3
90
(-0.002 eV/atom) [24], h-BC4 N-1(-0.003 eV/atom) [42], and h-BC2 N-F (-0.017 eV/atom) [43].
5
Figure 2: Phonon spectra for (a)m-BC3 N-1, and (b)m-BC3 N-2.
3.2. Dynamic Stability and Mechanical Properties. To confirm the dynamic stability of m-BC3 N structures, the calculated phonon dispersion curves at ground-state are shown in Fig. 2. Apparently, 95
both of two structures are dynamically stable because no imaginary frequencies exist in the whole Brillouin zone. The elastic stiffness constants Cij (C11 , C22 , C33 , C44 , C55 , C66 , C12 , C13 , C15 , C23 , C25 , C35 , and C46 ) are calculated by solving the stress gradient equation and can be used to evaluate the structural stability, as summarized in Table II. For a monoclinic crystal, its thir-
100
teen independent elastic constants should satisfy the Born criteria [44], which can be expressed as: C11 > 0, C22 > 0, C33 > 0, C44 > 0, C55 > 0, C66 > 0, 2 2 )> ) > 0, (C44 C66 −C46 [C11 +C22 +C33 +2(C12 +C13 +C23 )] > 0, (C33 C55 −C35 2 2 2 C33 ] > C55 −C25 0, (C22 +C33 −C23 ) > 0, [C22 (C33 C55 −C35 )+2C23 C25 C35 −C23
0, {2[C15 C25 (C33 C12 − C13 C23 ) + C15 C35 (C22 C13 −C12 C23 )+C25 C35 (C11 C23 − 105
2 2 2 2 2 2 ) +C55 g]} > (C11 C22 −C12 )+C25 (C11 C33 −C13 )+C35 (C22 C33 −C23 C12 C13 )]−[C15
0. Clearly, the obtained elastic constants showing in Table II satisfy the above mechanical stability criteria, verifying the mechanical stability of two m-BC3 N structures.
6
Table 2: The calculated lattice constants, density (ρ), total energy (E t ), formation energy (E f ), elastic modulus (B, G, E ), and elastic constants of m-BC3 N-1, and m-BC3 N-2.
structure
m-BC3 N-1
m-BC3 N-2
c-BN
a (˚ A)
9.066
9.078
3.627
b (˚ A)
2.543
2.562
c (˚ A)
8.779
8.867
90, 143.01, 90
90, 143.34, 90
90
ρ (g/cm )
3.318
3.283
3.459
E t (eV/atom)
-162.596
-162.413
-140.281
E f (eV/atom)
0.514
0.696
B (GPa)
360.0
338.2
362.3
G (GPa)
388.2
367.1
381.6
E (GPa)
856.7
808.7
847.4
C11
935.7
985.2
768.4
C22
942.1
872.8
C33
872.1
715.7
C44
398.4
350.8
C55
341.6
358.1
C66
382.6
365.3
C12
55.1
42.0
C13
114.0
112.7
C15
21.0
27.1
C23
78.0
91.2
C25
27.2
21.8
C35
-4.2
15.2
C46
53.0
30.8
α, β, γ (deg) 3
7
443.7
159.2
3.3. Anisotropic Properties. 110
It is well known that anisotropic behavior of structure is also an important implication associated the engineering application and material science. The elastic anisotropy of crystal can be illustrated by a constructed three-dimensional surface for Young’s modulus E [45, 46]. As a consequence, we investigated the elastic anisotropy by using the compliance transformation [47] and ELAM [48, 49]
115
codes. For monoclinic structure, the mechanical components with directional dependence can be expressed in a spherical coordinate system via compliance transformation:
1 = S11 sin ϕ4 cos φ4 + S22 sin ϕ4 sin φ4 + S33 cos ϕ4 + E (S44 + 2S23 ) sin ϕ2 sin φ2 cos ϕ2 + (S55 + 2S13 ) cos ϕ2 sin ϕ2
(1)
cos φ2 + (S66 + 2S12 ) sin ϕ4 cos φ2 sin φ2 + 2(S15 ∗ sin ϕ2 cos φ2 + (S25 + 2S46 ) ∗ sin ϕ2 cos φ2 ) ∗ sin ϕ cos φ cos ϕ where Sij denote the independent constants in elastic compliance matrix. For a allotropic material, the 3D surface of Young’s modulus should be a perfect 120
sphere, and a degrees of ellipsoidal or non-spherical behaviors indicate the existence of anisotropy. Obviously, both structures of m-BC3 N have deviation in shape from the perfect sphere and show significant anisotropy, as shown in Fig. 3(a) and Fig. 3(b). To quantitatively describe the deviation of two 3D drawing surfaces, we presented their 2D projections of the directional depen-
125
dence of Young’s modulus at the xy plane, yz plane, and xz plane in Fig. 3(c) and Fig. 3(d). The maximal and minimum values for m-BC3 N-1 are 943 GPa and 756 GPa, and the maximal and minimum values for m-BC3 N-2 are 976 GPa and 691 GPa, respectively. Correspondingly, the Emax /Emin ratios of mBC3 N-1 and m-BC3 N-2 are 1.247 and 1.412, respectively. That is to say, both
130
two emphm-BC3 N structures show great anisotropy in Young’s modulus, which are correspond to the universal anisotropic index AU of 0.08 and 0.095.
8
Figure 3: The 3D structure of Young’s modulus E curved surface for (a)m-BC3 N-1, and (b)m-BC3 N-2. And the 2D representation of Young’s modulus for (c)m-BC3 N-1, and (d)mBC3 N-2. The black solid line, blue dash line, and red dash dot line represent Young’s modulus at the xy plane, yz plane, and xz plane, respectively.
3.4. Hardness Simulation. In Vicker’s hardness simulation, a covalent crystal model on the basis of microscopic theory is employed. In this semi-empirical theory, the hardness 135
of covalent crystal relates not only to the density, cell volume, but also to the ionicity of bonds and number of chemical bonds. The calculated formula ]1/ ∑ N µ [∏ µ µ Nµ (Hv ) , where the N µ is the numcan be expressed as: Hv = ber of µ type bond in a unit cell, and the Hvµ is the hardness of certain type bond. It is obvious that Hv is determined by that one of each type of bond:
140
Hvµ = 350(Neµ )2/3 exp(−1.191fiµ )(dµ )−2.5 , here the Neµ denotes the valence electron density; dµ denotes the bond length of binary bond; fiµ represents the Phillips ionicity of µ bonds that defined as fiµ = [1 − exp(−|Pc − Pµ |/Pµ )]0.735 . The calculated results and bond parameters of m-BC3 N crystals are listed in table 3 and table 4. Incidentally, all simulations are performed in a 2 × 3 ×
145
2 supercell composed of 240 atoms in order to ensure the accuracy of the ob-
9
Table 3: Chemical bond parameters and finally Vicker’s hardness (Hv , GPa) of m-BC3 N-1. dµ is the bong length (˚ A), Pµ is the bond overlap population in m-BC3 N, Pcµ is the bond overlap population in pure covalent crystal, Neµ is the valence electron density, and fiµ is the Phillips ionicity.
Bond type
dµ
Pµ
Pcµ
Neµ
fiµ
Hv
C-C (i)
1.514
0.84
0.84
0.552
0
56.38
C-C (ii)
1.543
0.82
0.83
0.522
0.039
C-C (iii)
1.554
0.85
0.83
0.511
0.063
C-C (iv)
1.156
0.79
0.81
0.444
0.066
C-N
1.539
0.61
0.78
0.591
0.353
C-B (i)
1.671
0.85
0.79
0.342
0.138
C-B (ii)
1.614
0.89
0.80
0.308
0.178
B-N
1.600
0.64
0.77
0.477
0.287
N-N
1.485
0.50
0.81
0.731
0.567
tained parameters. Finally, the calculated Vicker’s hardness of m-BC3 N-1 and m-BC3 N-2 are 56.38 GPa and 52.96 GPa, respectively. 3.5. Electronic Properties. In consideration the usual underestimation of band gap caused by DFT, 150
we adopt the screened hybrid functional HSE03 in band structure calculations, which is considered to be more accurate in describing the exchange-correlation terms. Fig. 4(a) and Fig. 4b show the band structures of two m-BC3 N phases. It can be seen that m-BC3 N-1, and m-BC3 N-2 are all indirect band gap semiconductors with a band gap energy of 3.77 eV and 3.25 eV at equilibri-
155
um, respectively. The conduction band minimum (CBM) is located at M point and the valence band maximum (VBM) is located at G point for m-BC3 N-1, while both of them are reversed in m-BC3 N-2 [red and green circles in Fig. 4(a) and 4(b)]. It is worth noting that the CBM of m-BC3 N-2 is close to the M point in the valence band. Further calculations show that the m-BC3 N-2 can
160
be converted from an indirect band gap semiconductor (M → G) to a direc-
10
Table 4: Chemical bond parameters and finally Vicker’s hardness (Hv , GPa) of m-BC3 N-2. dµ is the bong length (˚ A), Pµ is the bond overlap population in m-BC3 N, Pcµ is the bond overlap population in pure covalent crystal, Neµ is the valence electron density, and fiµ is the Phillips ionicity.
Bond type
dµ
Pµ
Pcµ
Neµ
fiµ
Hv
C-C (i)
1.608
0.74
0.76
0.407
0.069
52.96
C-C (ii)
1.555
0.79
0.82
0.514
0.089
C-C (iii)
1.510
0.82
0.78
0.562
0.106
C-C (iv)
1.503
0.86
0.80
0.570
0.137
C-N (i)
1.532
0.64
0.83
0.538
0.368
C-N (ii)
1.584
0.65
0.81
0.486
0.326
C-B (i)
1.566
0.87
0.81
0.441
0.120
C-B (ii)
1.583
0.87
0.81
0.426
0.136
B-N
1.745
0.58
0.75
0.364
0.365
t band gap semiconductor (M → M) under a hydrostatic pressure of 30 GPa while maintaining structural stability, as shown in Fig. 4 c. This result implies the potential research value of m-BC3 N-2 in high-pressure materials science. The calculated total density of states (DOS) and partial density of states (P165
DOS) for each kind of atom in m-BC3 N structures are plotted in Fig. 5. The two m-BC3 N structures have similar total DOS due to the homology of structures. As we all know, superhard material consist of light elements, including carbon, nitrogen, boron, and oxygen, generally have a high level of hybridization. From the comparison of the total DOS and PDOS, it can be observed that the entire
170
valence band and conduction band have a high degree of hybridization in two structures. In m-BC3 N unit cell, there are twelve carbon atoms, four nitrogen atoms, and four boron atoms, all the them are sp3 bonds, which serve as the basis for their potential as superhard materials.
11
Figure 4: Electronic structures of (a)m-BC3 N-1, and (b)m-BC3 N-2. The conduction band minimum and valence band maximum are represented by red and blue circles, respectively. (c) The band gap versus hydrostatic pressure for m-BC3 N-2.
Figure 5: Total density of states and partial density of states of the (a)m-BC3 N-1, and (b)mBC3 N-2 structures. The s and p states are represented by blue and red lines, respectively.
12
4. CONCLUSION 175
In summary, this paper reports the detailed investigations of the structural, elastic and electronic properties of two novel monoclinic BC3 N structures, namely, m-BC3 N-1 and m-BC3 N-2, using the DFT within the framework of firstprinciples calculations. The dynamic and mechanical stabilities of two m-BC3 N phases have been confirmed by the use of phonon spectra and elastic Born cri-
180
teria. In addition, both of them behave in good brittle character on the basis of B/G ratios, and show great anisotropy in Young’s modulus according to the ELAM codes. The simulated Vicker’s hardness of m-BC3 N-1 and m-BC3 N-2 are 56.38 GPa and 52.96 GPa, respectively. Electronic calculations show that all the two m-BC3 N structures are indirect band gap semiconductors at ground-state
185
with a bang gap of 3.77 eV and 3.25 eV, respectively. However, it is notable that m-BC3 N-2 can be converted from an indirect band gap semiconductor to a direct band gap semiconductor with band gap of 3.45 eV when a hydrostatic pressure of 30 GPa is applied. Combining the hardness, mechanical, and electronic properties, these m-BC3 N structures may have multiple industrial
190
applications as cutting tool, tunable semiconductor device, etc.
ACKNOWLEDGMENTS The work was supported by the National Natural Science Foundation of China under Grant No. 11774416 and No. 51702359 and sponsored by Qing Lan Project.
195
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Declaration of interests ☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. ☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests:
S0921-4526(20)30075-2
DOI:
https://doi.org/10.1016/j.physb.2020.412061
Reference:
PHYSB 412061
To appear in:
Physica B: Physics of Condensed Matter
Received Date: 29 September 2019 Accepted Date: 31 January 2020
Please cite this article as: S. Li, L. Shi, Two novel superhard structures: Monoclinic BC3N, Physica B: Physics of Condensed Matter (2020), doi: https://doi.org/10.1016/j.physb.2020.412061. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2020 Published by Elsevier B.V.
Two novel superhard structures: monoclinic BC3 N Shuaiqi Lia , Liwei Shia,∗ a School
of Physical Science and Technology, China University of Mining and Technology, Xuzhou 221116, China
Abstract Two potential superhard monoclinic BC3 N structures, namely, m-BC3 N-1, and m-BC3 N-2, have been proposed through first-principles calculations. The structural, mechanical, and electronic properties of m-BC3 N structures are investigated in detail. The phonon dispersions and calculated elastic constants show that two m-BC3 N phases are dynamic and mechanical stable. The formation energies of these m-BC3 N structures are positive, indicating that they are metastable which similar to most ternary B-C-N compounds. Based on the elastic calculations, both of m-BC3 N-1 and m-BC3 N-2 behave in brittle character, and show great anisotropy in Young’s modulus. In hardness simulation, a microscopic theoretical model is employed, and finally obtained the Vicker’s hardness of two m-BC3 N structures with 56.38 GPa and 52.96 GPa, respectively. Using a hybrid functional HSE03, we estimate that the two m-BC3 N structures are indirect semiconductors with a bang gap of 3.77 eV and 3.25 eV. Besides, m-BC3 N-2 can be converted from an indirect band gap semiconductor to a direct band gap semiconductor with band gap of 3.45 eV when exposed to a hydrostatic pressure of 30 GPa. Keywords: boron-carbon-nitrogen, computational physics, density functional theory, superhard material
∗ Corresponding
author Email address: [email protected] (Liwei Shi)
Preprint submitted to PHYSICA B-CONDENSED MATTER
September 29, 2019
1. INTRODUCTION The search for superhard materials has been an important direction in material science due to their immeasurable value in modern industry [1]. Originating from the nature, diamond is well known for its wide applications in industrial 5
as cutting, polishing tools and wear-resistant coating [2, 3]. However, its low oxidation temperature and chemical instability have greatly limited the practical application, especially for the processing of ferrous alloys [4, 5, 6]. Cubic boron nitride (c-BN) has excellent oxidation resistance and chemical inertness, thereby making it a substitute for diamond to a certain extent despite its hard-
10
ness is only approximately half that of diamond [7]. On account of their diverse structures and atomic compositions, the ternary boron-carbon-nitrogen (B-C-N) compounds family possess adjustable band gap with considerable Vicker’s hardness and superior oxidation resistance, significantly oriented the development of superhard materials[8, 9, 10, 11]. Starting from the graphitic BC2 N (g-BC2 N)
15
synthesized by Kouvetakis [12], considerable ternary B-C-N compounds have been predicted or synthesized for decades. Solozhenko et al. [13] synthesized cubic BC2 N from g-BC2 N under high pressure and high temperature, which had a hardness up to 76 GPa, harder than c-BN crystal but still softer than diamond. Zhou et al. [14, 15] predicted two ultrahard tetragonal t-BC2 N and
20
z -BC2 N structures that even reach a high hardness of 75.3 GPa and 75.9 GPa, respectively. Liu et al. [16] successively developed a novel BC2 N structure in hexagonal symmetry with large bandgap energy, showing a remarkable hardness of 71 GPa in simulation. Furthermore, metallic BC2 N compounds, diamondlike BCx N (x = 1 - 4) [17, 18, 19] and (P-4m2, P3m1) BC2 N [20] which can be
25
synthesised by high-pressure and high-temperature (up to 2500 K) conditions reached the hardness of 56 - 58 GPa, also drawing much interests of researchers in potential superhard and superconductive material. In addition to diamond-like ternary structures, numerous other kinds of BC-N structures are concerned by light atoms replacement or structural search,
30
such as CALYPSO [21, 22, 23]. Gao et al. [24] introduced three BC8 N struc-
2
tures with high carbon content, in which BC8 N-3 had a negative formation energy with a wide band-gap. The tetragonal C8 B2 N2 structure with hardness of 65 GPa, proposed by Wang et al. [25], have been studied systematically in our previous work [26]. Although BCx N (x = 3 or x ≈ 3) compounds are 35
rarely investigated because of the stoichiometry, its complex and potentially valuable nature still attract the enthusiasm of many researchers, such as the heterodiamond BC3 N synthesized using graphite-like B-C-N compounds [27] or graphite-like BCx N(H) (x = 2,3,5) [28], and orthorhombic BC3.3 N prepared by amorphous B-C-N precursor [29]. Li et al. [30] proposed 32 kinds of B3 C10 N3
40
monolayer, and based on this, a narrow gap semiconductor B3 C10 N3 was constructed. In this paper, we investigate two novel BC3 N structures containing twelve carbon atoms, four boron atoms, and four nitrogen atoms a unit cell. The ground-state properties of the BC3 N structures are investigated and discussed in detail through first-principles calculations are performed.
45
2. THEORETICAL METHODS The first-principle calculations are performed using the plane-wave pseudopotential method within the framework of DFT and performed by the CASTEP code [31, 32]. The Perdew-Berke-Ernzerhof form of the generalized gradient approximation (GGA) is used to describe the exchange correlation term [33]. The
50
2s 2 2p 1 , 2s 2 2p 2 , and 2s 2 2p 3 states are considered as valence electrons for B, C, and N atoms, respectively. The ultrasoft pseudopotential that described the interactions of electrons with ion cores [34], is expanded in a plane-wave basis set with a cutoff energy of 800 eV, where the self-consistent convergence of total energy is at 0.5×10−6 eV/atom. For the fine level k-point sampling, a
55
Monkhorst-Pack mesh of 7 × 16 × 8 is adopted in the Brillouin zone [35]. The phonon frequencies and dispersion curves are calculated by the linear-response method based on DFPT [36]. To ensure the accuracy of determined electronic properties, calculations are repeated using the hybrid Heyd-Scuseria-Ernzerhof functional (HSE03) [37, 38], which contains 75 % semi-local GGA-PBE and
3
60
25% nonlocal Hartree-Fock (HF) approach. In Vicker’s hardness simulation, a 12 times supercell is constructed, and a semi-empirical theoretical model of covalent crystal is used [39, 40].
3. RESULTS AND DISCUSSION 3.1. Structure Characters. 65
The monoclinic BC3 N (BC3 N-1 and BC3 N-2) compounds crystallize in the centered monoclinic structure with space group C/2m. Fig. 1 shows the crystal structures of m-BC3 N-1 and m-BC3 N-2 after structural optimization at ambient pressure, while the Table I lists their atomic Wyckoff positions. In their crystal structures, the five-membered rings have the same bond composition (C-C, C-B,
70
B-N, C-N); but compared to the six-membered rings in m-BC3 N-1 with C-C bonds and C-B bonds, m-BC3 N-2 has no C-C bonds in those, and instead are B-N bonds and C-N bonds. No N-N bonds and B-N bonds exist in the sevenmembered rings of m-BC3 N-2. Consequently, C-C, C-C, C-N, and B-N bonds coexist in both structures, but N-N bonds only present in m-BC3 N-1. Table 1: Atomic Wyckoff positions of the m-BC3 N-1, and m-BC3 N-2 structures.
BC3 N-1
75
BC3 N-2
Atom
Wyckoff positions
Atom
Wyckoff positions
C1
4i(0.0685,0,-0.522)
C1
4i(0.0783,0,-0.512)
C2
4i(-0.503,0,-0.333)
C2
4i(0.754,0,-0.0459)
C3
4i(0.748,0,-0.0524)
C3
4i(-0.994,0,-0.910)
B
4i(-0.995,0,-0.912)
B
4i(-0.510,0,-0.339)
N
4i(-0.364,0,-0.729)
N
4i(-0.361,0,-0.735)
The calculated equilibrium lattice constants, density, total energy, formation energy, elastic modulus, and elastic constants as well as the theoretical results of diamond and c-BN are shown in Table II. The bulk modulus of two types mBC3 N is 360 GPa and 338 GPa, respectively, lower than that of c-BN, 369 GPa.
4
Figure 1: Relaxed unit cells of (a)m-BC3 N-1, and (b)m-BC3 N-2. The carbon, boron, and nitrogen atoms are painted in dark, orange, and blue, respectively. There are twelve carbon atoms, four boron atoms, and four nitrogen atoms in a unit cell. (c) A 3 × 2 × 3 supercell of m-BC3 N-1.
The shear modulus of m-BC3 N-1 is 388 GPa, which is slightly higher than that 80
of c-BN. Actually, Bulk modulus and shear modulus characterize the resistance ability of incompressibility and plastic deformation of a material, respectively. According to Pughs theory [41], the value of B /G ratio is reckoned as a differentiation of ductile or brittle characters, that is, value overs 1.75 showed ductile character, otherwise it should be brittle. The ratio of m-BC3 N-1 and
85
m-BC3 N-2 are 0.927 and 0.921, respectively, suggesting their brittle behaviors. In addition, the positive formation energy E f of m-BC3 N, defined as E f = E BCN - ( 2E diamond + 3E c−BN )/4, indicates that the m-BC3 N structures are metastable. Nevertheless, structures with positive formation energies are quite common in ternary B-C-N system, except for the negative structures BC8 N-3
90
(-0.002 eV/atom) [24], h-BC4 N-1(-0.003 eV/atom) [42], and h-BC2 N-F (-0.017 eV/atom) [43].
5
Figure 2: Phonon spectra for (a)m-BC3 N-1, and (b)m-BC3 N-2.
3.2. Dynamic Stability and Mechanical Properties. To confirm the dynamic stability of m-BC3 N structures, the calculated phonon dispersion curves at ground-state are shown in Fig. 2. Apparently, 95
both of two structures are dynamically stable because no imaginary frequencies exist in the whole Brillouin zone. The elastic stiffness constants Cij (C11 , C22 , C33 , C44 , C55 , C66 , C12 , C13 , C15 , C23 , C25 , C35 , and C46 ) are calculated by solving the stress gradient equation and can be used to evaluate the structural stability, as summarized in Table II. For a monoclinic crystal, its thir-
100
teen independent elastic constants should satisfy the Born criteria [44], which can be expressed as: C11 > 0, C22 > 0, C33 > 0, C44 > 0, C55 > 0, C66 > 0, 2 2 )> ) > 0, (C44 C66 −C46 [C11 +C22 +C33 +2(C12 +C13 +C23 )] > 0, (C33 C55 −C35 2 2 2 C33 ] > C55 −C25 0, (C22 +C33 −C23 ) > 0, [C22 (C33 C55 −C35 )+2C23 C25 C35 −C23
0, {2[C15 C25 (C33 C12 − C13 C23 ) + C15 C35 (C22 C13 −C12 C23 )+C25 C35 (C11 C23 − 105
2 2 2 2 2 2 ) +C55 g]} > (C11 C22 −C12 )+C25 (C11 C33 −C13 )+C35 (C22 C33 −C23 C12 C13 )]−[C15
0. Clearly, the obtained elastic constants showing in Table II satisfy the above mechanical stability criteria, verifying the mechanical stability of two m-BC3 N structures.
6
Table 2: The calculated lattice constants, density (ρ), total energy (E t ), formation energy (E f ), elastic modulus (B, G, E ), and elastic constants of m-BC3 N-1, and m-BC3 N-2.
structure
m-BC3 N-1
m-BC3 N-2
c-BN
a (˚ A)
9.066
9.078
3.627
b (˚ A)
2.543
2.562
c (˚ A)
8.779
8.867
90, 143.01, 90
90, 143.34, 90
90
ρ (g/cm )
3.318
3.283
3.459
E t (eV/atom)
-162.596
-162.413
-140.281
E f (eV/atom)
0.514
0.696
B (GPa)
360.0
338.2
362.3
G (GPa)
388.2
367.1
381.6
E (GPa)
856.7
808.7
847.4
C11
935.7
985.2
768.4
C22
942.1
872.8
C33
872.1
715.7
C44
398.4
350.8
C55
341.6
358.1
C66
382.6
365.3
C12
55.1
42.0
C13
114.0
112.7
C15
21.0
27.1
C23
78.0
91.2
C25
27.2
21.8
C35
-4.2
15.2
C46
53.0
30.8
α, β, γ (deg) 3
7
443.7
159.2
3.3. Anisotropic Properties. 110
It is well known that anisotropic behavior of structure is also an important implication associated the engineering application and material science. The elastic anisotropy of crystal can be illustrated by a constructed three-dimensional surface for Young’s modulus E [45, 46]. As a consequence, we investigated the elastic anisotropy by using the compliance transformation [47] and ELAM [48, 49]
115
codes. For monoclinic structure, the mechanical components with directional dependence can be expressed in a spherical coordinate system via compliance transformation:
1 = S11 sin ϕ4 cos φ4 + S22 sin ϕ4 sin φ4 + S33 cos ϕ4 + E (S44 + 2S23 ) sin ϕ2 sin φ2 cos ϕ2 + (S55 + 2S13 ) cos ϕ2 sin ϕ2
(1)
cos φ2 + (S66 + 2S12 ) sin ϕ4 cos φ2 sin φ2 + 2(S15 ∗ sin ϕ2 cos φ2 + (S25 + 2S46 ) ∗ sin ϕ2 cos φ2 ) ∗ sin ϕ cos φ cos ϕ where Sij denote the independent constants in elastic compliance matrix. For a allotropic material, the 3D surface of Young’s modulus should be a perfect 120
sphere, and a degrees of ellipsoidal or non-spherical behaviors indicate the existence of anisotropy. Obviously, both structures of m-BC3 N have deviation in shape from the perfect sphere and show significant anisotropy, as shown in Fig. 3(a) and Fig. 3(b). To quantitatively describe the deviation of two 3D drawing surfaces, we presented their 2D projections of the directional depen-
125
dence of Young’s modulus at the xy plane, yz plane, and xz plane in Fig. 3(c) and Fig. 3(d). The maximal and minimum values for m-BC3 N-1 are 943 GPa and 756 GPa, and the maximal and minimum values for m-BC3 N-2 are 976 GPa and 691 GPa, respectively. Correspondingly, the Emax /Emin ratios of mBC3 N-1 and m-BC3 N-2 are 1.247 and 1.412, respectively. That is to say, both
130
two emphm-BC3 N structures show great anisotropy in Young’s modulus, which are correspond to the universal anisotropic index AU of 0.08 and 0.095.
8
Figure 3: The 3D structure of Young’s modulus E curved surface for (a)m-BC3 N-1, and (b)m-BC3 N-2. And the 2D representation of Young’s modulus for (c)m-BC3 N-1, and (d)mBC3 N-2. The black solid line, blue dash line, and red dash dot line represent Young’s modulus at the xy plane, yz plane, and xz plane, respectively.
3.4. Hardness Simulation. In Vicker’s hardness simulation, a covalent crystal model on the basis of microscopic theory is employed. In this semi-empirical theory, the hardness 135
of covalent crystal relates not only to the density, cell volume, but also to the ionicity of bonds and number of chemical bonds. The calculated formula ]1/ ∑ N µ [∏ µ µ Nµ (Hv ) , where the N µ is the numcan be expressed as: Hv = ber of µ type bond in a unit cell, and the Hvµ is the hardness of certain type bond. It is obvious that Hv is determined by that one of each type of bond:
140
Hvµ = 350(Neµ )2/3 exp(−1.191fiµ )(dµ )−2.5 , here the Neµ denotes the valence electron density; dµ denotes the bond length of binary bond; fiµ represents the Phillips ionicity of µ bonds that defined as fiµ = [1 − exp(−|Pc − Pµ |/Pµ )]0.735 . The calculated results and bond parameters of m-BC3 N crystals are listed in table 3 and table 4. Incidentally, all simulations are performed in a 2 × 3 ×
145
2 supercell composed of 240 atoms in order to ensure the accuracy of the ob-
9
Table 3: Chemical bond parameters and finally Vicker’s hardness (Hv , GPa) of m-BC3 N-1. dµ is the bong length (˚ A), Pµ is the bond overlap population in m-BC3 N, Pcµ is the bond overlap population in pure covalent crystal, Neµ is the valence electron density, and fiµ is the Phillips ionicity.
Bond type
dµ
Pµ
Pcµ
Neµ
fiµ
Hv
C-C (i)
1.514
0.84
0.84
0.552
0
56.38
C-C (ii)
1.543
0.82
0.83
0.522
0.039
C-C (iii)
1.554
0.85
0.83
0.511
0.063
C-C (iv)
1.156
0.79
0.81
0.444
0.066
C-N
1.539
0.61
0.78
0.591
0.353
C-B (i)
1.671
0.85
0.79
0.342
0.138
C-B (ii)
1.614
0.89
0.80
0.308
0.178
B-N
1.600
0.64
0.77
0.477
0.287
N-N
1.485
0.50
0.81
0.731
0.567
tained parameters. Finally, the calculated Vicker’s hardness of m-BC3 N-1 and m-BC3 N-2 are 56.38 GPa and 52.96 GPa, respectively. 3.5. Electronic Properties. In consideration the usual underestimation of band gap caused by DFT, 150
we adopt the screened hybrid functional HSE03 in band structure calculations, which is considered to be more accurate in describing the exchange-correlation terms. Fig. 4(a) and Fig. 4b show the band structures of two m-BC3 N phases. It can be seen that m-BC3 N-1, and m-BC3 N-2 are all indirect band gap semiconductors with a band gap energy of 3.77 eV and 3.25 eV at equilibri-
155
um, respectively. The conduction band minimum (CBM) is located at M point and the valence band maximum (VBM) is located at G point for m-BC3 N-1, while both of them are reversed in m-BC3 N-2 [red and green circles in Fig. 4(a) and 4(b)]. It is worth noting that the CBM of m-BC3 N-2 is close to the M point in the valence band. Further calculations show that the m-BC3 N-2 can
160
be converted from an indirect band gap semiconductor (M → G) to a direc-
10
Table 4: Chemical bond parameters and finally Vicker’s hardness (Hv , GPa) of m-BC3 N-2. dµ is the bong length (˚ A), Pµ is the bond overlap population in m-BC3 N, Pcµ is the bond overlap population in pure covalent crystal, Neµ is the valence electron density, and fiµ is the Phillips ionicity.
Bond type
dµ
Pµ
Pcµ
Neµ
fiµ
Hv
C-C (i)
1.608
0.74
0.76
0.407
0.069
52.96
C-C (ii)
1.555
0.79
0.82
0.514
0.089
C-C (iii)
1.510
0.82
0.78
0.562
0.106
C-C (iv)
1.503
0.86
0.80
0.570
0.137
C-N (i)
1.532
0.64
0.83
0.538
0.368
C-N (ii)
1.584
0.65
0.81
0.486
0.326
C-B (i)
1.566
0.87
0.81
0.441
0.120
C-B (ii)
1.583
0.87
0.81
0.426
0.136
B-N
1.745
0.58
0.75
0.364
0.365
t band gap semiconductor (M → M) under a hydrostatic pressure of 30 GPa while maintaining structural stability, as shown in Fig. 4 c. This result implies the potential research value of m-BC3 N-2 in high-pressure materials science. The calculated total density of states (DOS) and partial density of states (P165
DOS) for each kind of atom in m-BC3 N structures are plotted in Fig. 5. The two m-BC3 N structures have similar total DOS due to the homology of structures. As we all know, superhard material consist of light elements, including carbon, nitrogen, boron, and oxygen, generally have a high level of hybridization. From the comparison of the total DOS and PDOS, it can be observed that the entire
170
valence band and conduction band have a high degree of hybridization in two structures. In m-BC3 N unit cell, there are twelve carbon atoms, four nitrogen atoms, and four boron atoms, all the them are sp3 bonds, which serve as the basis for their potential as superhard materials.
11
Figure 4: Electronic structures of (a)m-BC3 N-1, and (b)m-BC3 N-2. The conduction band minimum and valence band maximum are represented by red and blue circles, respectively. (c) The band gap versus hydrostatic pressure for m-BC3 N-2.
Figure 5: Total density of states and partial density of states of the (a)m-BC3 N-1, and (b)mBC3 N-2 structures. The s and p states are represented by blue and red lines, respectively.
12
4. CONCLUSION 175
In summary, this paper reports the detailed investigations of the structural, elastic and electronic properties of two novel monoclinic BC3 N structures, namely, m-BC3 N-1 and m-BC3 N-2, using the DFT within the framework of firstprinciples calculations. The dynamic and mechanical stabilities of two m-BC3 N phases have been confirmed by the use of phonon spectra and elastic Born cri-
180
teria. In addition, both of them behave in good brittle character on the basis of B/G ratios, and show great anisotropy in Young’s modulus according to the ELAM codes. The simulated Vicker’s hardness of m-BC3 N-1 and m-BC3 N-2 are 56.38 GPa and 52.96 GPa, respectively. Electronic calculations show that all the two m-BC3 N structures are indirect band gap semiconductors at ground-state
185
with a bang gap of 3.77 eV and 3.25 eV, respectively. However, it is notable that m-BC3 N-2 can be converted from an indirect band gap semiconductor to a direct band gap semiconductor with band gap of 3.45 eV when a hydrostatic pressure of 30 GPa is applied. Combining the hardness, mechanical, and electronic properties, these m-BC3 N structures may have multiple industrial
190
applications as cutting tool, tunable semiconductor device, etc.
ACKNOWLEDGMENTS The work was supported by the National Natural Science Foundation of China under Grant No. 11774416 and No. 51702359 and sponsored by Qing Lan Project.
195
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Declaration of interests ☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. ☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests: