Qing-Yang Fan(樊庆扬)^,1,2,∗, Nan Wu(吴楠)¹, Shuai-Ming Chen(陈帅铭)¹, Li Jiang(江莉)¹, Wei Zhang(张伟)³, Xin-Hai Yu(于新海)⁴, Si-Ning Yun(云斯宁)^,5,∗1College of Information and Control Engineering, Xi'an University of Architecture and Technology, Xi'an 710055, China
²Shaanxi Key Laboratory of Nano Materials and Technology, Xi'an 710055, China
³School of Microelectronics, Xidian University, Xi'an 710071, China
⁴Department of Mechanical and Electrical Engineering, Hetao College, Bayannur, Inner Mongolia 015000, China
⁵Functional Materials Laboratory (FML), School of Materials Science and Engineering, Xi'an University of Architecture and Technology, Xi'an 710055, China

First author contact: ∗Authors to whom any correspondence should be addressed.
Received:2021-07-22Revised:2021-08-23Accepted:2021-08-25Online:2021-10-15


Abstract
A new boron nitride polymorph, P2₁3 BN (space group: P2₁3), is investigated by first-principles calculations, including its structural properties, stability, elastic properties, anisotropy and electronic properties. It is found that the new boron nitride polymorph P2₁3 BN is mechanically, dynamically and thermodynamically stable. The bulk modulus (B), shear modulus (G) and Young's modulus of P2₁3 BN are 91 GPa, 41 GPa and 107 GPa, respectively, all of which are larger than that of Y carbon and TY carbon. By comparing with c-BN, the Young's modulus, shear modulus and Poisson's ratio of P2₁3 BN show tiny anisotropy in the (001), (010), (100) and (111) planes. At the same time, in contrast with most boron nitride polymorphs, P2₁3 BN is a semiconductor material with a smaller band gap of 1.826 eV. The Debye temperature and the anisotropic sound velocities of P2₁3 BN are also investigated in this work.
Keywords： boron nitride polymorph;elastic properties;anisotropy;electronic properties


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Qing-Yang Fan(樊庆扬), Nan Wu(吴楠), Shuai-Ming Chen(陈帅铭), Li Jiang(江莉), Wei Zhang(张伟), Xin-Hai Yu(于新海), Si-Ning Yun(云斯宁). P2₁3 BN: a novel large-cell boron nitride polymorph. Communications in Theoretical Physics, 2021, 73(12): 125701- doi:10.1088/1572-9494/ac20ce

1. Introduction

Since the advent of boron nitride more than one hundred years ago, research on boron nitride has not stopped [1–7]. Because B and N atoms can bind together by sp, sp² and sp³ hybridizations, boron nitride can exist in many polymorphs stably [8, 9]. Boron nitride is isoelectronic to carbon and thus exists in various crystalline forms, such as hexagonal boron nitride (h-BN) [10], cubic boron nitride (c-BN) [11], wurtzite boron nitride (w-BN) [12, 13], rhombohedral boron nitride (r-BN) [14] and turbostratic boron nitride (t-BN) [15].

Light elemental carbon is the first choice for superhard materials [16–25]. However, with the development of experimental technology, an increasing number of boron nitride polymorphs have been theoretically proposed and systematically investigated for superhard materials. Fan et al [26]. have investigated the structural, elastic and electronic properties and elastic anisotropy of Pbca-BN by density functional theory (DFT); the BN phase has an orthorhombic structure and is an insulator with an indirect band gap of 5.399 eV. The investigation indicated that Pbca-BN is a superhard material with a hardness of 60.1 GPa. A hexagonal BN structure (HCBN) containing only sp² bonds was proposed by Wang et al, [27] which exhibits intriguingly intrinsic metallicity. It was found that the metallicity of HCBN is mainly ascribed to the delocalized B-2p electrons. While HCBN allotropes have porous structures with low density and large surface area per unit mass, they may also have potential application in hydrogen storage. Ma et al [28]. proposed a novel monoclinic phase (Pm space group) of boron nitride polymorph m-BN with a wide and indirect band gap semiconductor. The bulk modulus, shear modulus and Young's modulus of m-BN are 329 GPa, 328 GPa and 739 GPa, respectively, which are smaller than those of c-BN, while the value of hardness of m-BN is 56.1 GPa; thus, it is a prospective superhard material. Niu et al [29]. have investigated a high ductile material cT8-BN, and the value of B/G is about 2.67, which is much larger than those of other BN polymorphs. Meanwhile cT8-BN shows a transparent insulator with a band gap of 5.38 eV. Owing to the large interspaces between atoms, the hydrogen storage in cT8-BN could be expected.

At present, most of the investigated BN structures contain fewer than 100 atoms in each conventional cell [16–33]. A new boron nitrogen polymorph, P2₁3 BN, with 60 boron and 60 nitrogen atoms in the conventional cell, is proposed by space group and graph theory (RG²) [34, 35] in this work. The structural properties, stability, elastic properties, mechanical anisotropy and electronic properties are investigated systematically.

2. Computational detail

In this work, all studies are performed using the ultrasoft pseudopotentials [36] based on CASTEP (Cambridge Serial Total Energy package) code [37] according to DFT [38, 39]. Physical property predictions and structural geometric optimization calculations are performed using the generalized gradient approximation (GGA) parameterized by Perdew, Burke and Ernzerrof [40] and the local density approximation (LDA) [41]. The elastic constants are calculated by the strain–stress method, and the elastic moduli are calculated by Voigt–Reuss–Hill approximations [42]. The density functional perturbation theory [43] method is adopted to estimate the phonon spectra of P2₁3 BN. A cutoff energy of 500 eV and k point of 4 × 4 × 4 are used to sample the Brillouin region. Finally, for the electronic band structure calculations, the Heyd–Scuseria–Ernzerhof (HSE06) hybrid functional [44] and norm-conserving pseudopotentials [45] are also adopted for electronic band structure calculations in this work.

3. Results and discussion

3.1. Structural properties

The crystal structure of the P2₁3 BN is shown in figure 1(a). The red and blue balls represent boron and nitrogen atoms. The structure in figure 1(a) seems disordered, but in fact the atomic stacking laws can be found through careful analysis. After analyzing the crystal structure of P2₁3 BN, it was found that it also has the smallest repeating unit, which is shown in figure 1(b). From figure 1(b), it can be concluded that the unit consists of three identical six-membered rings named B, as shown in figure 1(c). The three six-membered rings are connected by a B–N bond, and a new six-membered ring A is formed at the center. The central ring A is close to planar structure while the edge rings B are close to spatial structure, and the B–N bond lengths of the A and B rings show some regularity. It is certain that the crystal structure can be obtained in this way. Table 1 shows the calculated lattice constants (Å) of P2₁3 BN, Pm-3n BN, [46] m-BN [27] and c-BN [47] with different functionals. The lattice constants (Å) of P2₁3 BN are 10.410 Å, which are larger than those of Pm-3n BN, m-BN and c-BN. By comparing with the theoretical and experimental values of the elastic constants for c-BN, it can be found that the theoretical values of lattice constants obtained by GGA level are closer to the experimental values than those obtained by LDA level. Thus the results of P2₁3 BN are obtained based on GGA level.

Figure 1.

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Figure 1.Crystal structure of P2₁3 BN (a), extension rings (b), central ring (c) and edge ring (d).



Table 1.
Table 1.Calculated lattice constants (Å), volumes of each BN unit (ų/f. u. ) of P2₁3 BN, Pm-3n BN, m-BN and c-BN with different functionals.

	Method	a	a	b	β	V
P213 BN	GGAa	10.413				18.817
	LDAa	10.202				17.701
Pm-3n BN	GGAb	4.438
m-BN	GGAc	6.143	2.562	4.145	74.74
	LDAc	6.058	2.527	4.091	74.64
c-BN	GGAd	3.622
	LDAd	3.567
	experimentale	3.620

Note.
^aThis work.^b[46].^c[28].^d[47].^e[24].

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3.2. Stability

The results of the elastic constants for P2₁3 BN are shown in table 2. The three necessary and sufficient Born stability criteria for cubic symmetry are as follows: C₁₁ − C₁₂ > 0, C₁₁ + 2C₁₂ > 0 and C₄₄ > 0. By analyzing the elastic constants of P2₁3 BN in table 2, the elastic constants satisfy the criteria, and thus P2₁3 BN is mechanically stable. The phonon spectra of P2₁3 BN are presented in figure 2(a). There is no negative frequency in the whole Brillouin zone, which means that P2₁3 BN is dynamically stable. To verify the stability of P2₁3 BN more clearly, the related enthalpies of P2₁3 BN and other BN polymorphs are investigated and the results are shown in figure 2(b). Simultaneously the enthalpy of c-BN is set to 0. From figure 2(b), the relative enthalpy of P2₁3 BN (0.609 eV/atom) is larger than that of c-BN, Pm-3n BN (0.324 eV/atom) and m-BN (0.227 eV/atom), while it is much lower than that of hP3-BN (1.184 eV/atom), rocksalt-BN (1.611 eV/atom) and NiAs-BN (1.648 eV/atom).

Figure 2.

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Figure 2.Phonon spectra of P2₁3 BN (a) and related enthalpies of P2₁3 BN and boron nitride polymorphs (b).



Table 2.
Table 2.Elastic constants C_ij (GPa), elastic moduli B, G, E (GPa) and Poisson's ratio v of P2₁3 BN, Pm-3n BN, m-BN, c-BN, HCBN-1 and HCBN-2.

	C11	C12	C13	C22	C23	C33	C44	C55	C66	B	G	E	v
P213 BN	144	64					42			91	41	107	0.300
Pm-3n BNa	700	85					209			290	244	572	0.171
m-BNb	803	53	108	837	102	804	375	307	254	329	328	739	0.142
c-BNc	779	165					446			370	384	856	0.120
c-BNd	820	190					480			400
HCBN-1e	244	177	84			512	42		33	188
HCBN-2e	144	131	52			424	28		68	131

Notes.
^a[46].^b[28].^c[47].^d[24] – experimental.^e[27].

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3.3. Elastic properties

The bulk modulus (B), shear modulus (G), Young's modulus (E) and Poisson's ratio (v) of P2₁3 BN, Pm-3n BN, m-BN and c-BN are presented in table 2. From table 2, the bulk modulus (B) of P2₁3 BN is 91 GPa and the shear modulus (G) of P2₁3 BN is 41 GPa. Obviously, the calculated bulk modulus and shear modulus of P2₁3 BN are both smaller than those of Pm-3n BN, m-BN and c-BN. Although the bulk modulus and shear modulus of P2₁3 BN are small, they are larger than those of some of the carbon allotropes. For example, the bulk modulus of P2₁3 BN is larger than that of HD C₄₀ (29.22 GPa), AHD C₄₀ (60.31 GPa), HL C₄₀ (30.25 GPa), [48] HD C₂₈ (45.54 GPa), HL C₂₈ (45.90 GPa), [49] and C₂₀₀ (76.4 GPa) [50]. Moreover, the shear modulus of P2₁3 BN is larger than that of zigzag carbon (3, 3) (8.7 GPa), [51] Y-II carbon (28.1 GPa) and TY-II carbon (1.2 GPa) [52]. The bulk modulus and the shear modulus of P2₁3 BN are both larger than those of Y carbon and TY carbon [53]. At the same time, Pugh proposes the ratio B/G as an indication of ductile or brittle characters [54]. If B/G > 1.75, the material behaves in a ductile manner; otherwise the material behaves in a brittle manner. The B/G ratio of P2₁3 BN is 2.220, which is closer to that of dz4-BN, greater than that of lz2-BN, while smaller than that of cT8-BN. In our current perception, it can be found that most of the BN polymorphs behave in a brittle manner, while in this work, the new BN polymorph P2₁3 BN behaves in a ductile manner, which is relatively rare. We believe that is due to a number of porous structures in P2₁3 BN. To perceive the ductile manner or brittle manner for P2₁3 BN in more detail, other 31 BN polymorphs are investigated besides P2₁3 BN, and the results are presented in figure 3. From figure 3, there are four polymorphs larger than 1.75, accounting for only 12.9% in figure 3. The largest value of B/G is cT8-BN (2.593), [28] and the value of dz4-BN (2.241) [29] is close to P2₁3 BN; moreover, the value of Iz2-BN [29] is 1.814, slightly larger than 1.75. Most other BN polymorphs are smaller than 1.75, implying that they are brittle materials. Meanwhile, the minimum value of B/G is BC₈-BN (0.928), [55] which is slightly smaller than w-BN (0.949) and c-BN (0.964), coinciding with their high brittleness. Because we have not been able to investigate all the boron nitride materials, there may exist other BN polymorphs with larger values of B/G than 1.75.

Figure 3.

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Figure 3.Value of B/G for BN polymorphs.


Young's modulus is a physical quantity representing the tensile or compressive resistance of material within the elastic limit. The larger the Young's modulus, the smaller the deformation of the material when the material is compressed or stretched. The Young's modulus (E) can be calculated by E = 9BG/(3B + G), [42] and the Young's modulus of P2₁3 BN is 107 GPa, which is larger than that of Y carbon (32 GPa) and TY carbon (8.8 GPa). This further shows that P2₁3 BN is a highly ductile material, consistent with the investigation of B/G. Poisson's ratio (v) is another important parameter of material deformation which can be calculated by v = (3B − 2G)/2(3B + G) [42]. High v values (>0.26) usually indicate ductile materials [56]. This coincides with the result of the ratio of B/G. The Poisson's ratio of P2₁3 BN is 0.300, which is much larger than that of Pm-3n BN (0.171), m-BN (0.142) and c-BN (0.120).

According to the elastic modulus and the density of the material, the Debye temperature can be expressed as follows: ${{\rm{\Theta }}}_{D}={v}_{m}(h/{k}_{B}){[3n/(4\pi )({N}_{A}\rho /M)]}^{1/3}$, [57] where h is Planck's constant, k_B is Boltzmann's constant, N_A is Avogadro's number, n is the number of atoms in the molecule, M is the molecular weight, and ρ is the density. The average sound velocity v_m can be calculated from ${v}_{m}\,={[(2/{v}_{l}^{3}+1/{v}_{t}^{3})/3]}^{-1/3}$, where the transverse wave velocity v_t and longitudinal wave velocity v_l are estimated through Navier' equations: v_t = (G/ρ)^1/2, v_l = [(B + 4G/3)/ρ]^1/2 [58]. In the main direction, the sound velocity of the tetragonal symmetry is given by the following expression: In the [111] propagation direction, the longitudinal wave velocity v_l in the [111] polarization direction is calculated by ${[({C}_{11}+2{C}_{12}+4{C}_{44})/\rho ]}^{1/2}$, and the transverse wave velocity v_t in the [11-2] polarization direction is calculated by ${[({C}_{11}-{C}_{12}+{C}_{44})/3\rho ]}^{1/2}$. At the same time, in the [110] propagation direction, the longitudinal wave velocity v_l in the [110] polarization direction is calculated by ${[({C}_{11}+{C}_{12}+2{C}_{44})/2\rho ]}^{1/2}$, and the transverse wave velocity v_t in the [1-10] polarization direction is calculated by ${[({C}_{11}-{C}_{12})/\rho ]}^{1/2}$. Moreover, in the [100] propagation direction, the longitudinal wave velocity v_l in the [110] polarization direction is calculated by ${({C}_{11}/\rho )}^{1/2}$, the transverse wave velocity v_t in the [010] polarization direction is calculated by ${({C}_{44}/\rho )}^{1/2}$, and the transverse wave velocity v_t2 in the [001] polarization direction is calculated by ${({C}_{12}/\rho )}^{1/2}$. The related results are presented in table 3. The [111], [110] and [100] directions in the first column of table 3 are the propagation directions, and the second column is the polarization direction. As is presented in table 3, the Debye temperature of P2₁3 BN is 684 K, which is smaller than that of Pm-3n BN and c-BN, and the average sound velocity, transverse wave velocity, and longitudinal wave velocity of P2₁3 BN are also smaller than those of Pm-3n BN and c-BN. The study of the sound velocity has revealed that it is also anisotropic. The largest value of the sound velocity for P2₁3 BN is 14 174 m/s in the [111]v_l direction, the same direction as Pm-3n BN and c-BN, while the value of P2₁3 BN is about three fifths that of Pm-3n BN and half of c-BN. The smallest value of P2₁3 BN is 4309 m s⁻¹ in the [11-2]v_t12 direction, different to the direction of Pm-3n BN and c-BN. The value of v_l in the [110] direction and the value of v_t1 in the [1-10] direction for Pm-3n BN and c-BN are about twice those of P2₁3 BN. Meanwhile, the value of sound velocity in the [010]v_t1 direction is the same as that in the [001] v_t2 direction. As for Pm-3n BN, the values of v_t1 in [1-10] and v_l in [100] even exceed the values of c-BN.


Table 3.
Table 3.Density (g/cm³), anisotropic sound velocities (m/s), average sound velocity (m/s) and Debye temperature (K) for P2₁3 BN, Pm-3n BN, and c-BN.

Propagation direction	Propagation direction	P213 BN	Pm-3n BN	c-BN
ρ		2.190	2.820	3.467
[111]	[111]vl	14174	24597	28887
	[11-2]vt12	4309	9870	10095
[110]	[110]vl	8165	14605	16272
	[1-11]vt1	6044	14768	13308
	[001]vt2	5406	5490	6899
[100]	[100]vl	8109	15756	14990
	[010]vt1	4379	8609	11342
	[001]vt2	4379	8609	11342
	vl	8154	14767	15947
	vt	4332	9298	10527
	vm	4841	10235	11522
	ΘD	684	1572	1896

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3.4. Elastic anisotropy

Elastic anisotropy is the study of the change of elastic modulus in different directions of materials; that is to say, the elastic modulus shows differences in different directions. In contrast, if the elastic modulus has no difference in all directions, the material will be isotropic. For comparison, the anisotropy of three cubic crystal system materials P2₁3 BN, c-BN and Pm-3n BN are investigated in this work, and the investigation of the Young's modulus, shear modulus and Poisson's ratio are shown in figure 4. Figures 4(a)–(c) show the three-dimensional (3D) surface distribution of the Young's modulus for P2₁3 BN, c-BN and Pm-3n BN respectively. It is generally acknowledged that any deviation from the shape of a sphere indicates that they are anisotropic materials [59]. The shape of the 3D surface distribution of the Young's modulus for P2₁3 BN is very close to a sphere, which means that P2₁3 BN has the smallest anisotropy in the Young's modulus. The Young's modulus of Pm-3n BN show the largest anisotropy because the 3D surface construction of Pm-3n BN deviates from the shape of a sphere. The 3D surface construction of the Young's modulus for c-BN is close to a cube, so the anisotropy of c-BN is between P2₁3 BN and Pm-3n BN. The maximum and minimum values of the Young's modulus for P2₁3 BN, c-BN and Pm-3n can be also studied in this work; the results are shown in table 4. The maximum and minimum values of Young's modulus for P2₁3 BN are both much smaller than those of Pm-3n and c-BN; meanwhile, the ${E}_{\max }/{E}_{\min }$ ratio of P2₁3 BN is 1.04, which is smaller than Pm-3n and c-BN, indicating that P2₁3 BN has the smallest anisotropy. To further investigate the anisotropy of Young's modulus for P2₁3 BN, the two-dimensional representation in the (001), (010), (100) and (111) planes are shown in figure 5(a). Because of the P2₁3 BN is belongs to the cubic crystal system, the graphics coincide in the (001), (010) and (100) planes. Then, from figure 5(a), the two-dimensional (2D) representation in the (001), (010) and (100) planes does not overlap exactly with the (111) plane; therefore it can be confirmed that the Young's modulus of P2₁3 BN has small anisotropy. The distribution of Young's modulus in the main planes of the 3D structure is also studied, such as the (100), (010), (001) and (111) planes. Meanwhile, the maximum value and the minimum value of Young's modulus in the main planes are presented in table 4. The maximum and minimum values of Young's modulus are the same in the (100), (010) and (001) planes for P2₁3 BN, c-BN and Pm-3n BN. The ${E}_{\max }/{E}_{\min }$ ratios of the (100), (010) and (001) planes for P2₁3 BN, c-BN and Pm-3n BN are larger than that for the (111) plane. The ${E}_{\max }/{E}_{\min }$ ratio of Pm-3n BN is largest in those polymorphs for the (100), (010) and (001) planes. The values of the ${E}_{\max }/{E}_{\min }$ ratio are equal to 1 in the (111) plane; that is to say, the Young's modulus for these polymorphs is isotropic.

Figure 4.

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Figure 4.Directional dependence of Young's modulus for P2₁3 BN (a), c-BN (b) and Pm-3n BN (c); shear modulus for P2₁3 BN (d), c-BN (e) and Pm-3n BN (f); and Poisson's ratio for P2₁3 BN (g), c-BN (h) and Pm-3n BN (i).

Figure 5.

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Figure 5.2D representation of the Young's modulus (a), the maximum shear modulus (b), the minimum shear modulus (c), the maximum Poisson's ratio (d) and the minimum Poisson's ratio (e) for P2₁3 BN in the (001), (010), (100) and (111) plane.



Table 4.
Table 4.Calculated maximum values, minimum values, and ratio of Young's modulus, shear modulus and Poisson's ratio for P2₁3 BN, Pm-3n BN, m-BN carbon and c-BN in the (001), (010), (100) and (111) planes.

	(100), (010), (001) planes			(111) plane			whole
	${E}_{\max }$	${E}_{\min }$	ratio	${E}_{\max }$	${E}_{\min }$	ratio	${E}_{\max }$	${E}_{\min }$	ratio
P213 BN	107.66	104.67	1.03	107.66	107.66	1.00	108.69	104.67	1.04
Pm-3n BN	681.86	539.60	1.26	539.60	539.60	1.00	681.86	504.72	1.35
c-BN	883.26	720.50	1.23	883.26	883.26	1.00	954.69	720.50	1.33
	${G}_{\max }$	${G}_{\min }$	ratio	${G}_{\max }$	${G}_{\min }$	ratio	${G}_{\max }$	${G}_{\min }$	ratio
P213 BN	41.79	40.02	1.04	41.79	40.02	1.04	41.79	40.02	1.04
Pm-3n BN	307.59	208.50	1.48	307.59	208.50	1.48	307.59	208.50	1.48
c-BN	446.77	306.62	1.46	446.77	306.62	1.46	446.77	306.62	1.46
	${v}_{\max }$	${v}_{\min }$	${v}_{\max }-{v}_{\max }$	${v}_{\max }$	${v}_{\min }$	${v}_{\max }-{v}_{\max }$	${v}_{\max }$	${v}_{\min }$	${v}_{\max }-{v}_{\max }$
P213 BN	0.32	0.29	0.03	0.32	0.29	0.03	0.32	0.29	0.03
Pm-3n BN	0.29	0.09	0.20	0.29	0.09	0.20	0.29	0.09	0.20
c-BN	0.21	0.00	0.21	0.21	0.00	0.21	0.21	0.00	0.21

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The 3D distributions of the shear modulus and Poisson's ratio for P2₁3 BN, c-BN and Pm-3n BN are shown in figures 4(d)–(i) respectively. The curved surface constructed by the dotted line shows the maximum value of the shear modulus and Poisson's ratio, and the curved surface constructed by the solid line shows the minimum value of the shear modulus and Poisson's ratio. From figures 4(d) and (g), the shape of the dotted line and the solid line are both nearly regular spheres, and moreover there is a small difference between the dotted line and the solid line; thus the shear modulus and Poisson's ratio of Pm-3n BN show small anisotropy. From analyzing the shape of the 3D distribution of the shear modulus and Poisson's ratio for c-BN and Pm-3n BN, they all show different degrees of anisotropy. From figures 5(b)–(e), it is certain that the shear modulus and Poisson's ratio of P2₁3 BN show small anisotropy in the (001), (010), (100) and (111) planes. The maximum and minimum values of the shear modulus and Poisson's ratio for the whole materials and four main planes are presented in table 4 of P2₁3 BN, c-BN and Pm-3n BN. From table 4, the ${G}_{\max }/{G}_{\min }$ ratio for P2₁3 BN is smaller than that of c-BN and Pm-3n BN in the (100), (010), (001) and (111) planes, and the value is approach to 1, implying that Pm-3n BN shows small anisotropy in shear modulus, which coincides with the result of the 3D distribution and the 2D representation. Meanwhile, the maximum and minimum values of the shear modulus for P2₁3 BN, c-BN and Pm-3n BN are the same in the main planes; thus the shear modulus shows the same anisotropy in different planes. Furthermore, the ${G}_{\max }/{G}_{\min }$ ratio of Pm-3n BN is almost equal to that of c-BN in the main planes, indicating that the anisotropy of Pm-3n BN is almost identical to that of c-BN. Because the minimum value of Poisson's ratio for c-BN is 0.00, the ratio of ${v}_{\max }$ and ${v}_{\min }$ is not suitable for the investigation of Poisson's ratio. For Poisson's ratio, we use the difference between the maximum and minimum to measure the anisotropy. The anisotropy of Poisson's ratio for P2₁3 BN is still the same in the main planes, and is also the smallest compared with c-BN and Pm-3n BN. Meanwhile, c-BN and Pm-3n BN show analogous anisotropy for Poisson's ratio in these planes.

3.5. Electronic properties

To study the electronic properties of P2₁3 BN, the electronic band structure of P2₁3 BN is investigated with the HSE06 hybrid functional, and the results are shown in figure 6. According to the different paths of electron transition from the valence band to conduction band, semiconductors can be divided into direct band gap semiconductors and indirect band gap semiconductors. From figure 6, the coordinates of the high system points across the Brillouin zone for P2₁3 BN are X (0.500, 0.000, 0.000) – R (0.500, 0.500, 0.500) – M (0.500, 0.500, 0.000) – G (0.000, 0.000, 0.000) – R (0.500, 0.500, 0.500). Meanwhile, the valence band maximum and conduction band minimum are not at the same point, and the band gap of P2₁3 BN is 1.826 eV; thus the P2₁3 BN shows an indirect semiconductor character. However, most of the boron nitride polymorphs have wide band gaps with values larger than 2 eV; the band gap of Pnma-BN is 7.180 eV, [32] which is nearly four times larger than that of P2₁3 BN. Other BN polymorphs with band gap larger than 2 eV are Ima2-BN (3.340 eV), [31] m-BN (4.629 eV), [27] Pm-3n BN (5.870 eV), [46] B4N4-I (4.860 eV), and B4N4-II (5.320 eV) [60].

Figure 6.

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Figure 6.Electronic band structures of P2₁3 BN with HSE06 hybrid functional.

4. Conclusion

In summary, based on first-principles calculations within the framework of DFT, the structural properties, stability, elastic properties, anisotropy and electronic properties of a new boron nitride polymorph, P2₁3 BN, were systemically investigated in this work. It can be concluded from the electronic band structures that P2₁3 BN is a semiconductor with an indirect band gap of 1.826 eV. The bulk modulus and shear modulus of P2₁3 BN are smaller than those of Pm-3n BN, m-BN and c-BN, while larger than those of some of the carbon allotropes, such as Y carbon and TY carbon, HD C₄₀, AHD C₄₀, HD C₂₈, HL C₄₀, HL C₂₈ and C₂₀₀, zigzag carbon (3, 3), Y-II carbon, and TY-II carbon. It is found that P2₁3 BN is mechanically and dynamically stable according to the elastic constants, the phonon spectra and the related enthalpies. Moreover, the calculated Pugh ratio and Poisson's ratio of P2₁3 BN are 2.219 and 0.300 respectively, which indicates that P2₁3 BN behaves in a highly ductile manner. Finally, P2₁3 BN shows tiny anisotropy in Young's modulus, shear modulus and Poisson's ratio compared with Pm-3n BN and m-BN.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (Nos. 61804120, 61803294 and 61901162); the China Postdoctoral Science Foundation (Nos. 2019TQ0243, 2019M663646); the Key Scientific Research Plan of the Education Department of Shaanxi Provincial Government (Key Laboratory Project) (No. 20JS066); the Young Talent Fund of the University Association for Science and Technology in Shaanxi, China (No. 20190110); the National Key Research and Development Program of China (No. 2018YFB1502902); and the Key Program for International S&T Cooperation Projects of Shaanxi Province (No. 2019KWZ-03).

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