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Three-dimensional hexagonal boron nitride foam containing both sp2 and sp3 hybridized bonds
Materials Chemistry and Physics 217 (2018) 5–10
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Three-dimensional hexagonal boron nitride foam containing both sp2 and sp3 hybridized bonds
T
Jiacheng Shang, Jiamin Xiong, Xuechun Xu, Yingxiang Cai∗ Department of Physics, Nanchang University, Jiangxi, Nanchang 330031, PR China
A R T I C LE I N FO
A B S T R A C T
Keywords: Boron nitride Electronic structure Hexagonal BN foam Density functional theory
A family of semiconducting multiporous hexagonal boron nitride foams (HBNFs) with both sp2 and sp3 hybridized bonds is presented, which can be achieved by linking armchair BN nanoribbons (A-BNNRs). Based on density functional theory (DFT) calculations, we find that most of these HBNFs are dynamically, thermally and mechanically stable. Their electronic bandgaps vary in a wide range and exponentially change with the ribbon width. Interestingly, the electronic bandgaps of HBNFs can be classified into three groups. We also find that the edge effects affect not only the electronic band structure but also the length of B-N chemical bonds. In addition, the electron localization function of HBNFs are also been evaluated.
1. Introduction As a III-V compound and isoelectronic material to carbon, boron nitride (BN) has many different polymorphs since B and N atoms can form chemical bonds by means of sp , sp2 and sp3 hybridizations or combining them together. Numerous BN structures have been theoretical predicted or experimental synthesized in the past decades, such as zero-dimensional (0D) nanocages [1–3], one-dimensional (1D) nanotubes [4–7], two-dimensional (2D) nanosheets [8–11] and three-dimensional (3D) crystals [12–15]. Due to excellent thermal and chemical stability as well as unique electronic and optical properties, BN materials exhibit significant application potentials in the fields of hydrogen storage [15–17], water cleaning [18–20], flexible resistive memory [21], ultraviolet laser devices [22,23], nanoscale spintronic devices [24], nanomedicine [25,26] and cutting and polishing tools [27,28]. Interestingly, most BN polymorphs have similar structures to wellknown carbon materials, such as graphite-like hexagonal BN (h-BN) [29,30], diamond-like cubic BN (c-BN) [31,32] and superhard carbon materials-like BN (bct-BN and z-BN) [33,34]. Theoretical or experimental studies have also indicated that BN materials sometimes have similar properties to known carbon materials. For instance, nanotwinned c-BN exhibits extremely high hardness which is even competitive with the diamond [31]. However, the physical properties of BN and carbon materials are quite different in most cases even if they have similar structures. For example, graphene is a zero bandgap semiconductor but BN nanosheet is a wide bandgap semiconductor. Tunable properties in a wide range and practical application potentials in many fields inspire researchers to prepare and search new BN materials. ∗
Corresponding author. E-mail address: [email protected] (Y. Cai).
https://doi.org/10.1016/j.matchemphys.2018.06.041 Received 19 April 2018; Received in revised form 15 June 2018; Accepted 20 June 2018
Available online 22 June 2018 0254-0584/ © 2018 Elsevier B.V. All rights reserved.
Using precursors to prepare new BN polymorphs has been proved to be a feasible method. Graphite-like and turbostratic onion-like BN have been successfully used to generate nanograined and nanotwinned c-BN, respectively [31]. Nevertheless, the synthesis of high-quality precursor BN nanomaterials is the previous major obstacle in designing highly ordered BN nanostructures. Fortunately, the successful preparation of single-walled BN nanotubes (SWBNNT) [35,36], multi-walled BN nanotubes (MWBNNT) [37,38] and BN nanoribbons (BNNRs) [39,40] provides the fundamental building blocks for 3D BN polymorphs. Experiments have verified that BNNRs can be synthesized by unzipping BN nanotubes [40–42]. Therefore, the major obstacle of the synthesis of high-quality BNNRs with a specified width has been overcome by the synthesization of high quality single-walled BN nanotubes [43–45]. A few new BN structures built by either BN nanosheets, nanotubes or nanoribbons have been presented [12,13,15,46–51]. It is worth noting that these BN polymorphs exhibit unusual physical properties. For instance, multiporous BN, such as T-B3N3, T-B7N7, (P-6M2)-BN and (IMM2)-BN [12,13], can even be metallic. All these 3D multiporous BN structures as well as dz2-BN and lz1-BN [15] are obtained by linking zigzag BNNRs (Z-BNNRs). However, less studies have been performed for 3D multiporous BN polymorphs built by linking armchair BNNRs (ABNNRs). In this paper, we present a new family of multiporous BN foam consisting of both sp2 and sp3 hybridizations. These new BN structures are suggested to be achieved by linking A-BNNRs. We not only examine their thermodynamic and dynamic stabilities but also evaluate their mechanical properties. Intrinsic dependence of electronic bandgap on the hexagonal pore size of HBNF are found. The effect of edge effects on
Materials Chemistry and Physics 217 (2018) 5–10
J. Shang et al.
Fig. 1. (a) Perspective view of a hexagonal BN foam 3 × 3 × 4 supercell with Na = 6 (i.e. 6-HBNF). (b)∼(d) are the top, side and front views of 6-HBNF supercell, respectively. (e) The armchair BN nanoribbon used to build Na -HBNF. Green and gray spheres denote boron and nitrogen atoms, respectively. The parameter wa is the ribbon width. The da is the length of Na -HBNF unit cell along pore direction, i.e. lattice constant c. The la (parallel to the da ) denotes the B-N bond length. (f) Formation of Na -HBNFs by linking armchair BN nanoribbons. (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.) Table 1 Space group (SG), lattice parameters (LP: Å), density (ρ: g⋅cm−3), electronic bandgaps (Eg : eV), bulk modulus (B: GPa), shear modulus (G: GPa) and Young's modulus (Y: GPa) for 2, 3, 4-HBNF and several known BN polymorphic structures (c-BN, w-BN, bct-BN, z-BN and h-BN) at zero pressure. System
SG
LP (a, b, c)
ρ
Eg
B
G
Y
Ref.
2-HBNF 3-HBNF 4-HBNF c-BN
P6mm P63mc P6mm F4 3m
w-BN
P63mc
bct-BN
P42/mnm
z-BN
Pbam
h-BN
P6/mmc
4.869, 4.233 6.917, 4.325 9.102, 4.317 3.600 3.62 3.615 2.538, 4.198 2.55, 4.20 2.536, 4.199 4.393, 2.536 4.380, 2.526 8.750, 4.223, 2.517 8.891, 4.293, 2.555 2.511, 6.650 2.494, 6.666
2.791 2.164 1.722 3.534 3.48 3.450 3.520 3.49 3.485 3.688 – 3.545 3.25 2.270 2.271
1.35 2.46 3.67 4.440 – – 5.207 – – 4.782 4.676 5.180 5.27 4.778 4.65
221 178 143 407 400 – 420 – – 395 360 430 – 281 –
181 105 82 409 – – 345 – – 316 390 370 – 193 –
427 262 206 919 – – 812 – – 749 – 849 – 471 –
This work This work This work This work & Ref [68]a Experimental [65,69] Calculated [70] This work & Ref [68]a Experimental [65] Calculated [70] This work & Ref [68]a Calculated [34] This work & Ref [68]a Calculated [33] This work & Ref [68]a Calculated [70,71]
a
B, G and Y are taken from the references.
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Fig. 2. Phonon dispersions of 1-HBNF, 2-HBNF and 3-HBNF. Γ (0, 0, 0), K(-1/3, 2/3, 0), H(-1/3, 2/3, 1/2), A(0, 0, 1/2), M(0, 1/2, 0) and L(0, 1/2, 1/2) are the high symmetric points in the first Brillouin zone.
modulus, shear modulus and Young's modulus are evaluated using the Voigt-Reuss-Hill average scheme [60]. 3. Results and discussion The structure of a typical honeycomb BN foam (see Fig. 1 (a)-(d)) can be obtained by linking the A-BNNRs. Following the conventional naming rules of nanoribbon [61–63], the BN nanoribbons can be classified according to the number (Na ) of B-N chemical bonds along its width (zigzag direction) as shown in Fig. 1 (e). Accordingly, the BN family built by them can be named as Na -hexagonal BN foam (Na -HBNF). Fig. 1 (f) demonstrates the formation of Na -HBNFs. It can be found that Na -HBNFs are sp2 +sp3 hybridized structures. Three ABNNRs bonds together through sp3 hybridizations, which results in the formation of a sp3 chain. Table .1 lists the space group, equilibrium lattice parameters, density, electronic bandgaps, bulk modulus, shear modulus and Young's modulus of three Na -HBNFs and a few typical BN polymorphs including c-BN [31,32], w-BN [64,65], bct-BN [34], z-BN [33] and h-BN [29,30]. As far as the symmetry is concerned, the Na -HBNFs can be classified into two groups. When Na is even, the space group of Na -HBNF is P6mm (C6V-1). Otherwise, the space group is P63mc (C6V-4) for odd Na . Bulk modulus (B) is determined by the formula of B = (2c11 + c33 + 2c12 + 4c13)/9 , shear modulus (G) is given by G = (3.5c11 + c33 − 2.5c12 − 2c13 + 6c44 )/15 and Young's modulus (Y) is calculated by Y = 9G⋅B /(G + 3B ) [60,66,67], where cij are the elastic constants. Due to their multiporous structures, both bulk modulus and shear modulus of Na -HBNFs are less than other typical BN structures listed in Table .1. With the increasing of Na , the density of Na -HBNF decreases and thus bulk modulus, shear modulus and Young's modulus decrease too. To evaluate the dynamic stability of Na -HBNFs, we investigate their phonon dispersion in first Brillouin zone. In view of that the phonon calculation is very time consuming and there are numerous members in HBNFs family, only the phonon dispersions of 1-HBNF, 2-HBNF and 3HBNF are taken into account as shown in Fig. 2. We find the 2-HBNF is dynamically instable since distinct imaginary frequencies (i.e. soft phonon mode) mainly occur along Γ -K, Γ -A and Γ -M directions. In contrast, there are no imaginary frequencies in the whole first Brillouin zone for both 1-HBNF and 3-HBNF, and thus they are dynamically stable. It should be noted that the imaginary frequencies in Na -HBNFs might be inhibited due to the existence of gases in their multiporous structures. Unlike 2-HBNF, there is no phonon gap in 1-HBNF and 3HBNF. The highest phonon frequency is 1241 cm−1 and 1578 cm−1 along Γ -A direction for 1-HBNF and 3-HBNF, respectively. However, the highest phonon frequency at Γ is only 1127 cm−1 for 1-HBNF and 1535 cm−1 for 3-HBNF. To verify the thermodynamic stability of Na -HBNFs, we calculate the total energies of per pair BN and compare them with a few well known BN materials including c-BN, w-BN, p-BN, z-BN, bct-BN, BN sheet, dz2-BN [15], (P-6M2)-BN [13], T-B3N3 [12], M-BN and BC8-BN [14]. In view of that the c-BN is the most stable structure among all BN
Fig. 3. Relative stability of BN polymorphs. The energy of the most stable BN (i.e. c-BN) is taken as a reference and ΔE denotes the energy difference between any BN polymorph and c-BN.
electronic structure and chemical bonds are clarified. Three groups of bandgaps and their hierarchy are presented. Furthermore, the electron localization function (ELF) is also evaluated. 2. Computational details Our calculations are carried out using density functional theory (DFT) with general gradient approximation (GGA) [52,53] as implemented in the Vienna ab initio simulation package (VASP) [54,55]. The interactions between the nucleus and valence electrons of boron and nitrogen atoms are described by the projector augmented wave (PAW) method [56]. The pseudopotentials with 3 and 5 valence electrons for the B (2s 2 2 p1) and N (2s 2 2 p3 ) atoms are used. A plane-wave basis with a cutoff energy of 400 eV is used to expand the wave functions for all BN structures investigated in this work. The geometry for these new type and other BN polymorphic structures are fully relaxed including the atomic position and lattice parameters until the residual forces on each atom is less than 0.0001 eV⋅Å−1. The relative stability of different BN polymorphs is evaluated by PBEsol exchange-correlation functional [57]. For other calculations, Perdew-Burke-Ernzerhof (PBE) exchange-correlation functional is used [52]. The Brillouin zone sample meshes are set to be dense enough in this study and the k point spacing in reciprocal space is less than 0.3 Å−1. The phonon band structures are determined by the direct supercell method as implemented in the Phonopy program [58]. Since phonon calculations are time consuming, 2 × 2 × 2 supercells are used in this study. For these new BN polymorphs, their elastic constants (cij ) are determined by the formula of cij 2
∂ E = 1 ⎡ ∂ε ∂ε ⎤, where the E is total energy, the εi and εj are the strains. More V⎣ i j ⎦ computational details can be found in reference. [59] And their bulk
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Fig. 4. (a)–(c) Electronic band structures of Na -HBNFs at Na = 8, 9 and 10. (d) Electronic bandgap as functions of Na . The bandgaps can be classified into three groups which correspond to the Na = 3n-1, 3n and 3n+1, respectively. The bandgap of 1-HBNF is highlighted using an asterisk symbol because it is an all sp3 hybridized structure unlike other Na -HBNFs with both sp2 and sp3 hybridizations.
Fig. 5. (a) Relative B-N bond length (Δl ) as a function of bond number (nb ). Δl is calculated by the formula of Δl = la -l 0 , where the l 0 (1.450 Å) is the B-N bond length of ideal BN sheet and the la is the B-N bond length of Na -HBNF (see Fig. 1 (e)). The nb denotes the B-N bond number from l1 to la + 1. (b) 2D contour plot of electron localization function (ELF) through the (21̄1̄0) plane of Na -HBNF (Na =1, 2, …, 12, 13 from top to bottom). The A-BNNR structures are shown in the ELFs of 7-HBNF and 8-HBNF.
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with the increase of their pore sizes. Electronic structure calculations show that Na -HBNFs are semiconductors with a direct Eg at Γ point. Interestingly, the Eg variations versus Na , except for the Na = 1, exhibit three distinct family behaviors. The B-N bond length at the edge of ABNNRs change evidently after they polymerize into Na -HBNF. In addition, the electron localization function shows the covalent nature of B-N bonds in Na -HBNFs.
polomophs, we take its energy as a reference. The relative energies of other BN polymorphs to c-BN are shown in Fig. 3. It can be found that the Na -HBNFs, except for the all sp3 hybridized 1-HBNF, become more stable with the increasing of Na because the total energy gradually decreases. At larger Na , the stability of Na -HBNFs approaches to that of BN sheet. It can also be found that all Na -HBNFs are less stable than denser c-BN and w-BN. However, Na -HBNFs are more stable than lightweight BN polymorphs, such as dz2-BN, (P-6M2)-BN, T-B3N3 when the Na is more than two. In addition, the relative stability of above BN polymorphs is not changed even though the contribution of entropy is taking into account. The mechanical stability is investigated based on the well-known Born-Huang criteria [72]. For a stable Na -HBNF structure, five independent elastic constants (c11, c33 , c44 , c12 and c13 ) 2 . should satisfy: c11 > 0, c33 > 0, c11 > c12, c44 > 0, (c11 + c12)⋅c33 > 2c13 In this study, we calculate the elastic constants of 2-HBNF (c11 = 397, c33 = 677, c44 = 186, c12 = 138 and c13 = 60 GPa), 3-HBNF (c11 = 205, c33 = 582, c44 = 138, c12 = 169 and c13 = 68 GPa) and 4-HBNF (c11 = 156, c33 = 487, c44 = 109, c12 = 140 and c13 = 52 GPa). We find that all these elastic constants (cij ) are satisfied with the above BornHuang stability criteria. Therefore they are mechanically stable. The electronic band structures of Na -HBNFs are investigated for Na from 1 to 25. Three typical electronic band structures for Na = 8, 9 and 10 are shown in Fig. 4 (a)-(c). We find all of them are semiconductors with a direct band gap (Eg ) at Γ point and the Eg is 3.60, 3.68 and 3.77 eV for Na = 8, 9 and 10, respectively. Similar to A-BNNRs [73], the Eg variations versus Na , except for the Na = 1, exhibit three distinct family behaviors [see Fig. 4 (d)] which are well characterized by three functions of Eg 3n + 1 = − 6.72e (−Na /1.98) + 3.80 , Eg 3n = − 4.84e (−Na /2.33) + 3.80 and Eg 3n + 1 = − 0.42e (−Na /3.80) + 3.80 , where n is a positive integer. Each group of Eg increases and the Eg hierarchy is Eg 3n + 1 > Eg 3n > Eg 3n − 1. For instance, the Eg at n = 3 satisfies the hierarchy of Eg 10 > Eg 9 > Eg 8 . At Na = 1, i.e. 1-HBNF, its Eg is 5.26 eV close to the bandgap (5.21 eV) of wBN [68]. It doesn't belong to any groups because 1-HBNF is an all sp3 hybridized structure but other Na -HBNFs have both sp2 and sp3 hybridizations. The hybridization of the edge atoms of A-BNNRs becomes sp3 when they link together to form Na -HBNFs. The edge effects affect not only electronic bandgap but also the bond length. The relative length of B-N bonds parallel to the c axis of Na -HBNF unit cell are investigated as shown in Fig. 5 (a). The B-N bond length (l 0 ) of ideal BN sheet is taken as a reference and the relative B-N bond length (Δl ) is calculated by the formula of Δl = l-l 0 . For instance, the Δl for the seventh (nb =7) B-N bond is the length difference between l 7 and l 0 . We find that the Δl is evidently lengthened due to the change of hybridization from sp2 to sp3 for l1 and la + 1. In contrast, Δl is tiny for B-N bonds in the middle of ABNNRs especially at Na≥12. We also evaluate the ELF because it provides a good description of electron delocalization in molecules and solids and is also a useful tool for chemical bond classification. Fig. 5 (b) shows the ELF of Na -HBNFs for Na from 1 to 13. The values of 1.0 and 0.5 correspond to fully localized and fully delocalized electrons, respectively. And the value 0.0 refers to very low charge density. It is found that the electron is more localized in the middle of B-N bond which imply its covalent nature. Around N atoms, the electrons are in between delocalized and localized states. The slight delocalized electron gas is still surrounding N atoms, which results in the semiconducting electronic structure of Na -HBNFs.
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4. Conclusions In summary, a family of semiconducting multiporous hexagonal boron nitride foams (Na -HBNFs) built using A-BNNRs has been proposed. Their stabilities, elastic properties and electronic structures have been systematically investigated by first-principles calculations. Our study indicates that most Na -HBNF structures are dynamically, thermodynamically and mechanically stable except for the 2-HBNF. Their density, bulk modulus, shear modulus and Young's modulus decrease 9
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Three-dimensional hexagonal boron nitride foam containing both sp2 and sp3 hybridized bonds
T
Jiacheng Shang, Jiamin Xiong, Xuechun Xu, Yingxiang Cai∗ Department of Physics, Nanchang University, Jiangxi, Nanchang 330031, PR China
A R T I C LE I N FO
A B S T R A C T
Keywords: Boron nitride Electronic structure Hexagonal BN foam Density functional theory
A family of semiconducting multiporous hexagonal boron nitride foams (HBNFs) with both sp2 and sp3 hybridized bonds is presented, which can be achieved by linking armchair BN nanoribbons (A-BNNRs). Based on density functional theory (DFT) calculations, we find that most of these HBNFs are dynamically, thermally and mechanically stable. Their electronic bandgaps vary in a wide range and exponentially change with the ribbon width. Interestingly, the electronic bandgaps of HBNFs can be classified into three groups. We also find that the edge effects affect not only the electronic band structure but also the length of B-N chemical bonds. In addition, the electron localization function of HBNFs are also been evaluated.
1. Introduction As a III-V compound and isoelectronic material to carbon, boron nitride (BN) has many different polymorphs since B and N atoms can form chemical bonds by means of sp , sp2 and sp3 hybridizations or combining them together. Numerous BN structures have been theoretical predicted or experimental synthesized in the past decades, such as zero-dimensional (0D) nanocages [1–3], one-dimensional (1D) nanotubes [4–7], two-dimensional (2D) nanosheets [8–11] and three-dimensional (3D) crystals [12–15]. Due to excellent thermal and chemical stability as well as unique electronic and optical properties, BN materials exhibit significant application potentials in the fields of hydrogen storage [15–17], water cleaning [18–20], flexible resistive memory [21], ultraviolet laser devices [22,23], nanoscale spintronic devices [24], nanomedicine [25,26] and cutting and polishing tools [27,28]. Interestingly, most BN polymorphs have similar structures to wellknown carbon materials, such as graphite-like hexagonal BN (h-BN) [29,30], diamond-like cubic BN (c-BN) [31,32] and superhard carbon materials-like BN (bct-BN and z-BN) [33,34]. Theoretical or experimental studies have also indicated that BN materials sometimes have similar properties to known carbon materials. For instance, nanotwinned c-BN exhibits extremely high hardness which is even competitive with the diamond [31]. However, the physical properties of BN and carbon materials are quite different in most cases even if they have similar structures. For example, graphene is a zero bandgap semiconductor but BN nanosheet is a wide bandgap semiconductor. Tunable properties in a wide range and practical application potentials in many fields inspire researchers to prepare and search new BN materials. ∗
Corresponding author. E-mail address: [email protected] (Y. Cai).
https://doi.org/10.1016/j.matchemphys.2018.06.041 Received 19 April 2018; Received in revised form 15 June 2018; Accepted 20 June 2018
Available online 22 June 2018 0254-0584/ © 2018 Elsevier B.V. All rights reserved.
Using precursors to prepare new BN polymorphs has been proved to be a feasible method. Graphite-like and turbostratic onion-like BN have been successfully used to generate nanograined and nanotwinned c-BN, respectively [31]. Nevertheless, the synthesis of high-quality precursor BN nanomaterials is the previous major obstacle in designing highly ordered BN nanostructures. Fortunately, the successful preparation of single-walled BN nanotubes (SWBNNT) [35,36], multi-walled BN nanotubes (MWBNNT) [37,38] and BN nanoribbons (BNNRs) [39,40] provides the fundamental building blocks for 3D BN polymorphs. Experiments have verified that BNNRs can be synthesized by unzipping BN nanotubes [40–42]. Therefore, the major obstacle of the synthesis of high-quality BNNRs with a specified width has been overcome by the synthesization of high quality single-walled BN nanotubes [43–45]. A few new BN structures built by either BN nanosheets, nanotubes or nanoribbons have been presented [12,13,15,46–51]. It is worth noting that these BN polymorphs exhibit unusual physical properties. For instance, multiporous BN, such as T-B3N3, T-B7N7, (P-6M2)-BN and (IMM2)-BN [12,13], can even be metallic. All these 3D multiporous BN structures as well as dz2-BN and lz1-BN [15] are obtained by linking zigzag BNNRs (Z-BNNRs). However, less studies have been performed for 3D multiporous BN polymorphs built by linking armchair BNNRs (ABNNRs). In this paper, we present a new family of multiporous BN foam consisting of both sp2 and sp3 hybridizations. These new BN structures are suggested to be achieved by linking A-BNNRs. We not only examine their thermodynamic and dynamic stabilities but also evaluate their mechanical properties. Intrinsic dependence of electronic bandgap on the hexagonal pore size of HBNF are found. The effect of edge effects on
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Fig. 1. (a) Perspective view of a hexagonal BN foam 3 × 3 × 4 supercell with Na = 6 (i.e. 6-HBNF). (b)∼(d) are the top, side and front views of 6-HBNF supercell, respectively. (e) The armchair BN nanoribbon used to build Na -HBNF. Green and gray spheres denote boron and nitrogen atoms, respectively. The parameter wa is the ribbon width. The da is the length of Na -HBNF unit cell along pore direction, i.e. lattice constant c. The la (parallel to the da ) denotes the B-N bond length. (f) Formation of Na -HBNFs by linking armchair BN nanoribbons. (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.) Table 1 Space group (SG), lattice parameters (LP: Å), density (ρ: g⋅cm−3), electronic bandgaps (Eg : eV), bulk modulus (B: GPa), shear modulus (G: GPa) and Young's modulus (Y: GPa) for 2, 3, 4-HBNF and several known BN polymorphic structures (c-BN, w-BN, bct-BN, z-BN and h-BN) at zero pressure. System
SG
LP (a, b, c)
ρ
Eg
B
G
Y
Ref.
2-HBNF 3-HBNF 4-HBNF c-BN
P6mm P63mc P6mm F4 3m
w-BN
P63mc
bct-BN
P42/mnm
z-BN
Pbam
h-BN
P6/mmc
4.869, 4.233 6.917, 4.325 9.102, 4.317 3.600 3.62 3.615 2.538, 4.198 2.55, 4.20 2.536, 4.199 4.393, 2.536 4.380, 2.526 8.750, 4.223, 2.517 8.891, 4.293, 2.555 2.511, 6.650 2.494, 6.666
2.791 2.164 1.722 3.534 3.48 3.450 3.520 3.49 3.485 3.688 – 3.545 3.25 2.270 2.271
1.35 2.46 3.67 4.440 – – 5.207 – – 4.782 4.676 5.180 5.27 4.778 4.65
221 178 143 407 400 – 420 – – 395 360 430 – 281 –
181 105 82 409 – – 345 – – 316 390 370 – 193 –
427 262 206 919 – – 812 – – 749 – 849 – 471 –
This work This work This work This work & Ref [68]a Experimental [65,69] Calculated [70] This work & Ref [68]a Experimental [65] Calculated [70] This work & Ref [68]a Calculated [34] This work & Ref [68]a Calculated [33] This work & Ref [68]a Calculated [70,71]
a
B, G and Y are taken from the references.
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Fig. 2. Phonon dispersions of 1-HBNF, 2-HBNF and 3-HBNF. Γ (0, 0, 0), K(-1/3, 2/3, 0), H(-1/3, 2/3, 1/2), A(0, 0, 1/2), M(0, 1/2, 0) and L(0, 1/2, 1/2) are the high symmetric points in the first Brillouin zone.
modulus, shear modulus and Young's modulus are evaluated using the Voigt-Reuss-Hill average scheme [60]. 3. Results and discussion The structure of a typical honeycomb BN foam (see Fig. 1 (a)-(d)) can be obtained by linking the A-BNNRs. Following the conventional naming rules of nanoribbon [61–63], the BN nanoribbons can be classified according to the number (Na ) of B-N chemical bonds along its width (zigzag direction) as shown in Fig. 1 (e). Accordingly, the BN family built by them can be named as Na -hexagonal BN foam (Na -HBNF). Fig. 1 (f) demonstrates the formation of Na -HBNFs. It can be found that Na -HBNFs are sp2 +sp3 hybridized structures. Three ABNNRs bonds together through sp3 hybridizations, which results in the formation of a sp3 chain. Table .1 lists the space group, equilibrium lattice parameters, density, electronic bandgaps, bulk modulus, shear modulus and Young's modulus of three Na -HBNFs and a few typical BN polymorphs including c-BN [31,32], w-BN [64,65], bct-BN [34], z-BN [33] and h-BN [29,30]. As far as the symmetry is concerned, the Na -HBNFs can be classified into two groups. When Na is even, the space group of Na -HBNF is P6mm (C6V-1). Otherwise, the space group is P63mc (C6V-4) for odd Na . Bulk modulus (B) is determined by the formula of B = (2c11 + c33 + 2c12 + 4c13)/9 , shear modulus (G) is given by G = (3.5c11 + c33 − 2.5c12 − 2c13 + 6c44 )/15 and Young's modulus (Y) is calculated by Y = 9G⋅B /(G + 3B ) [60,66,67], where cij are the elastic constants. Due to their multiporous structures, both bulk modulus and shear modulus of Na -HBNFs are less than other typical BN structures listed in Table .1. With the increasing of Na , the density of Na -HBNF decreases and thus bulk modulus, shear modulus and Young's modulus decrease too. To evaluate the dynamic stability of Na -HBNFs, we investigate their phonon dispersion in first Brillouin zone. In view of that the phonon calculation is very time consuming and there are numerous members in HBNFs family, only the phonon dispersions of 1-HBNF, 2-HBNF and 3HBNF are taken into account as shown in Fig. 2. We find the 2-HBNF is dynamically instable since distinct imaginary frequencies (i.e. soft phonon mode) mainly occur along Γ -K, Γ -A and Γ -M directions. In contrast, there are no imaginary frequencies in the whole first Brillouin zone for both 1-HBNF and 3-HBNF, and thus they are dynamically stable. It should be noted that the imaginary frequencies in Na -HBNFs might be inhibited due to the existence of gases in their multiporous structures. Unlike 2-HBNF, there is no phonon gap in 1-HBNF and 3HBNF. The highest phonon frequency is 1241 cm−1 and 1578 cm−1 along Γ -A direction for 1-HBNF and 3-HBNF, respectively. However, the highest phonon frequency at Γ is only 1127 cm−1 for 1-HBNF and 1535 cm−1 for 3-HBNF. To verify the thermodynamic stability of Na -HBNFs, we calculate the total energies of per pair BN and compare them with a few well known BN materials including c-BN, w-BN, p-BN, z-BN, bct-BN, BN sheet, dz2-BN [15], (P-6M2)-BN [13], T-B3N3 [12], M-BN and BC8-BN [14]. In view of that the c-BN is the most stable structure among all BN
Fig. 3. Relative stability of BN polymorphs. The energy of the most stable BN (i.e. c-BN) is taken as a reference and ΔE denotes the energy difference between any BN polymorph and c-BN.
electronic structure and chemical bonds are clarified. Three groups of bandgaps and their hierarchy are presented. Furthermore, the electron localization function (ELF) is also evaluated. 2. Computational details Our calculations are carried out using density functional theory (DFT) with general gradient approximation (GGA) [52,53] as implemented in the Vienna ab initio simulation package (VASP) [54,55]. The interactions between the nucleus and valence electrons of boron and nitrogen atoms are described by the projector augmented wave (PAW) method [56]. The pseudopotentials with 3 and 5 valence electrons for the B (2s 2 2 p1) and N (2s 2 2 p3 ) atoms are used. A plane-wave basis with a cutoff energy of 400 eV is used to expand the wave functions for all BN structures investigated in this work. The geometry for these new type and other BN polymorphic structures are fully relaxed including the atomic position and lattice parameters until the residual forces on each atom is less than 0.0001 eV⋅Å−1. The relative stability of different BN polymorphs is evaluated by PBEsol exchange-correlation functional [57]. For other calculations, Perdew-Burke-Ernzerhof (PBE) exchange-correlation functional is used [52]. The Brillouin zone sample meshes are set to be dense enough in this study and the k point spacing in reciprocal space is less than 0.3 Å−1. The phonon band structures are determined by the direct supercell method as implemented in the Phonopy program [58]. Since phonon calculations are time consuming, 2 × 2 × 2 supercells are used in this study. For these new BN polymorphs, their elastic constants (cij ) are determined by the formula of cij 2
∂ E = 1 ⎡ ∂ε ∂ε ⎤, where the E is total energy, the εi and εj are the strains. More V⎣ i j ⎦ computational details can be found in reference. [59] And their bulk
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Fig. 4. (a)–(c) Electronic band structures of Na -HBNFs at Na = 8, 9 and 10. (d) Electronic bandgap as functions of Na . The bandgaps can be classified into three groups which correspond to the Na = 3n-1, 3n and 3n+1, respectively. The bandgap of 1-HBNF is highlighted using an asterisk symbol because it is an all sp3 hybridized structure unlike other Na -HBNFs with both sp2 and sp3 hybridizations.
Fig. 5. (a) Relative B-N bond length (Δl ) as a function of bond number (nb ). Δl is calculated by the formula of Δl = la -l 0 , where the l 0 (1.450 Å) is the B-N bond length of ideal BN sheet and the la is the B-N bond length of Na -HBNF (see Fig. 1 (e)). The nb denotes the B-N bond number from l1 to la + 1. (b) 2D contour plot of electron localization function (ELF) through the (21̄1̄0) plane of Na -HBNF (Na =1, 2, …, 12, 13 from top to bottom). The A-BNNR structures are shown in the ELFs of 7-HBNF and 8-HBNF.
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with the increase of their pore sizes. Electronic structure calculations show that Na -HBNFs are semiconductors with a direct Eg at Γ point. Interestingly, the Eg variations versus Na , except for the Na = 1, exhibit three distinct family behaviors. The B-N bond length at the edge of ABNNRs change evidently after they polymerize into Na -HBNF. In addition, the electron localization function shows the covalent nature of B-N bonds in Na -HBNFs.
polomophs, we take its energy as a reference. The relative energies of other BN polymorphs to c-BN are shown in Fig. 3. It can be found that the Na -HBNFs, except for the all sp3 hybridized 1-HBNF, become more stable with the increasing of Na because the total energy gradually decreases. At larger Na , the stability of Na -HBNFs approaches to that of BN sheet. It can also be found that all Na -HBNFs are less stable than denser c-BN and w-BN. However, Na -HBNFs are more stable than lightweight BN polymorphs, such as dz2-BN, (P-6M2)-BN, T-B3N3 when the Na is more than two. In addition, the relative stability of above BN polymorphs is not changed even though the contribution of entropy is taking into account. The mechanical stability is investigated based on the well-known Born-Huang criteria [72]. For a stable Na -HBNF structure, five independent elastic constants (c11, c33 , c44 , c12 and c13 ) 2 . should satisfy: c11 > 0, c33 > 0, c11 > c12, c44 > 0, (c11 + c12)⋅c33 > 2c13 In this study, we calculate the elastic constants of 2-HBNF (c11 = 397, c33 = 677, c44 = 186, c12 = 138 and c13 = 60 GPa), 3-HBNF (c11 = 205, c33 = 582, c44 = 138, c12 = 169 and c13 = 68 GPa) and 4-HBNF (c11 = 156, c33 = 487, c44 = 109, c12 = 140 and c13 = 52 GPa). We find that all these elastic constants (cij ) are satisfied with the above BornHuang stability criteria. Therefore they are mechanically stable. The electronic band structures of Na -HBNFs are investigated for Na from 1 to 25. Three typical electronic band structures for Na = 8, 9 and 10 are shown in Fig. 4 (a)-(c). We find all of them are semiconductors with a direct band gap (Eg ) at Γ point and the Eg is 3.60, 3.68 and 3.77 eV for Na = 8, 9 and 10, respectively. Similar to A-BNNRs [73], the Eg variations versus Na , except for the Na = 1, exhibit three distinct family behaviors [see Fig. 4 (d)] which are well characterized by three functions of Eg 3n + 1 = − 6.72e (−Na /1.98) + 3.80 , Eg 3n = − 4.84e (−Na /2.33) + 3.80 and Eg 3n + 1 = − 0.42e (−Na /3.80) + 3.80 , where n is a positive integer. Each group of Eg increases and the Eg hierarchy is Eg 3n + 1 > Eg 3n > Eg 3n − 1. For instance, the Eg at n = 3 satisfies the hierarchy of Eg 10 > Eg 9 > Eg 8 . At Na = 1, i.e. 1-HBNF, its Eg is 5.26 eV close to the bandgap (5.21 eV) of wBN [68]. It doesn't belong to any groups because 1-HBNF is an all sp3 hybridized structure but other Na -HBNFs have both sp2 and sp3 hybridizations. The hybridization of the edge atoms of A-BNNRs becomes sp3 when they link together to form Na -HBNFs. The edge effects affect not only electronic bandgap but also the bond length. The relative length of B-N bonds parallel to the c axis of Na -HBNF unit cell are investigated as shown in Fig. 5 (a). The B-N bond length (l 0 ) of ideal BN sheet is taken as a reference and the relative B-N bond length (Δl ) is calculated by the formula of Δl = l-l 0 . For instance, the Δl for the seventh (nb =7) B-N bond is the length difference between l 7 and l 0 . We find that the Δl is evidently lengthened due to the change of hybridization from sp2 to sp3 for l1 and la + 1. In contrast, Δl is tiny for B-N bonds in the middle of ABNNRs especially at Na≥12. We also evaluate the ELF because it provides a good description of electron delocalization in molecules and solids and is also a useful tool for chemical bond classification. Fig. 5 (b) shows the ELF of Na -HBNFs for Na from 1 to 13. The values of 1.0 and 0.5 correspond to fully localized and fully delocalized electrons, respectively. And the value 0.0 refers to very low charge density. It is found that the electron is more localized in the middle of B-N bond which imply its covalent nature. Around N atoms, the electrons are in between delocalized and localized states. The slight delocalized electron gas is still surrounding N atoms, which results in the semiconducting electronic structure of Na -HBNFs.
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4. Conclusions In summary, a family of semiconducting multiporous hexagonal boron nitride foams (Na -HBNFs) built using A-BNNRs has been proposed. Their stabilities, elastic properties and electronic structures have been systematically investigated by first-principles calculations. Our study indicates that most Na -HBNF structures are dynamically, thermodynamically and mechanically stable except for the 2-HBNF. Their density, bulk modulus, shear modulus and Young's modulus decrease 9
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