Theoretical Investigations of a BN Polymorph with sp² + sp³ Hybridizations

Abstract

The crystal structure, mechanical anisotropy, elastic properties and electronic characteristics, as well as the stability, of P4/m BN are predicted by means of density functional theory. In this work, BN in the P4/m phase demonstrates mechanical and dynamical stability. Compared with the values of bulk B, E and G in the P4/m phase, the B of BN in the P4/m phase is greater than that of dz4 BN, while the G and E of P4/m BN are greater than those of Pnc2 BN and dz4 BN. The ratio of the bulk-to-shear modulus for P4/m BN is less than 1.75 and dz4 BN, dz2 BN and lzlz2 BN, indicating that P4/m BN is more brittle than dz4 BN, dz2 BN and lzlz2 BN. P4/m BN exhibits stronger mechanical anisotropy in G and E than Pbca BN, P4₂/mnm BN and Pm-3m BN but much weaker mechanical anisotropy than P4/mbm BN, B₇N₇, B₁₁N₁₁ and B₁₅N₁₅. In addition, P4/m BN is a quasi-direct bandgap semiconductor, and the difference between the direct and the indirect bandgap is 0.008 eV. In order to obtain further characteristics of P4/m BN for future synthetic verification, the X-ray diffraction (XRD) patterns for P4/m BN are also calculated. Given its properties, P4/m BN is a good candidate for photoelectric devices.

1. Introduction

As an isoelectronic structure in carbon materials, boron nitride has many polymorphs, such as sp³ hybridizations [1,2,3,4], sp² hybridizations [5,6,7,8] and sp² + sp³ hybridization BN [9], similar to carbon structures having many allotropes, including sp³ hybridizations [10,11,12,13,14], sp² hybridizations [15,16,17,18] and sp² ₊ sp³ hybridizations [19,20,21]. Cubic boron nitride (c-BN) is potentially a third-generation semiconductor material with properties that make it suitable for electronic devices working under extreme circumstances, as well as deep UV luminescence emitters and detectors.

BN polymorphs are attracting increasing attention, not only because they have excellent wear resistance and are super hard, such as Pbca-BN [22], c-BN, P-BN [23], O-BN [24], BC8-BN [25], etc. A completely tetrahedral-bonded boron nitride within an orthorhombic system, namely Pbca-BN (space group: Pbca), was proposed by Zhang et al. [26] and investigated by Fan et al. using first-principles calculations [22]. According to Fan et al., the B, G and hardness of Pbca-BN are 344 GPa, 316 GPa and 60.1 GPa, respectively. Therefore, Pbca-BN presents super hard characteristics and may be used in important applications in technology and the industry. A novel polymorph of boron nitride with monoclinic symmetry, m-BN, was established by Ma et al. [27]. The B, G, E and hardness of m-BN are 329 GPa, 328 GPa, 729 GPa and 56.1 GPa, respectively, the bulk modulus of m-BN is slightly smaller than that of Pbca-BN and the G is slightly greater than that of Pbca-BN. BN polymorphs have attracted increased attention from researchers, because they are essentially broad-bandgap semiconductors, and they have excellent physical properties, such as high thermal conductivity, great resistivity, high mobility, a small dielectric constant and strong breakdown electric field. They are recommended for the production of electronic equipment employed under extreme circumstances. Regarding Pnma-BN, the bandgap is 7.18 eV [28], which is larger than that of c-BN (6.1–6.4 eV [29,30]); thus, it may be the largest bandgap BN polymorph in theory. In contrast, the bandgaps of Pbca-BN, m-BN and P4/mbm BN are 5.399 eV, 4.629 eV and 4.8 eV [31], respectively. In addition to its super hard properties and wide bandgap, the BN polymorph has many other interesting properties, such as ductility in the case of P2₁3 BN [32]. The B/G ratio of P2₁3 BN is 2.220, which is close to that of dz4-BN [6] and larger than that of lz2-BN [6], while it is lower than that of cT8-BN [5].

In this work, we propose an all sp² + sp³ hybridization BN polymorph, P4/m BN, with dynamical stability and mechanical stability, and the physical characteristics (including crystal structure, mechanical anisotropy, elastic properties and electronic characteristics) of P4/m BN are analyzed by means of density functional theory [33,34].

2. Materials and Methods

As in the majority of crystal structure and physical characteristics predictions, the Perdew–Burke–Ernzerhof (PBE) functional of the exchange correlation potential and the generalized gradient approximation (GGA) [35] are used in this work, and the density functional theory (DFT) calculations are performed by utilizing the ultrasoft pseudopotentials [36] under the Cambridge Sequential Total Energy Package (CASTEP) code [37]. The valence electron structures of N and B are 2s²2p³ and 2s²2p¹, respectively. A high k-point separation, smaller than or approximately equal to 0.025 Å⁻¹ × 2π, is used for P4/m BN, and 6 × 6 × 10 Monkhorst-Pack meshes [38] are employed for a conventional cell of P4/m BN. The Broyden–Fletcher–Goldfarb–Shanno (BFGS) minimization scheme [39] is employed for crystal geometric majorization. The plane wave cut-off energy for structural majorization is 500 eV, and this was adopted for the crystal property calculations for P4/m BN. The electronic bands were investigated utilizing the Heyd–Scuseria–Ernzerhof (HSE06) hybrid functional [40]. All the phonon spectra of P4/m BN were investigated based on the density functional perturbation theory (DFPT) method [41].

3. Results and Discussion

3.1. Crystal Structure

The crystal structures of the BN polymorphs are shown in Figure 1. The green and gray spheres represent B atoms and N atoms, respectively. According to Figure 1, in the crystal structure of P4/m BN, one conventional cell contains 12 N and 12 B atoms, and these N and B atoms are connected by sp² and sp³ hybridizations, including eight nitrogen atoms and eight boron atoms connected by sp³ hybridizations and four nitrogen atoms and four boron atoms connected by sp² hybridizations. The P4/m BN involves a 4-membered BN ring, 6-membered BN ring, 8-membered BN ring and 16-membered BN ring. These ring structures are composed of alternating connections of N atoms and B atoms, and there is only one 4-, 6- and 16-membered BN ring each, as well as two eight-membered rings. The lattice parameters of the crystal structures of P4/m BN are presented in

Table 1. Lattice constants (Å) and volumes per BN unit (ų) in the P4/m, Pnc2, dz4, dz2 and lzlz2 phases and c-BN.

Materials	Methods	a	b	c	β	V	ρ
P4/m	GGA	7.2455		3.9257		17.1740	2.3995
	LDA	7.1143		3.8736		16.3376	2.5334
Pnc2	GGA 1	10.9536	6.3020	4.9544		17.1001	2.4099
	LDA 1	10.8877	6.0603	4.9076		16.1899	2.5454
dz4	GGA	4.9850	3.4268	16.6081		17.7321	2.3240
dz2	GGA	4.9349	3.2640	8.0329		16.1736	2.5480
lzlz2	GGA	13.0946	2.5104	4.3953	89.3	18.0596	2.2819
c-BN	GGA	3.6217				11.8762	3.4699
	LDA	3.5779				11.4505	3.5989
	Experimental 2	3.6200				11.8595	3.4748

¹ Reference [9]. ² Reference [42].

, together with the lattice constants of other BN polymorphs. The theoretical results regarding the GGA level for c-BN are in better agreement with the experimental results [42] than those of the LDA method. Thus, the results presented later in this work are based on the GGA level. The volumes per BN unit of BN in the P4/m, Pnc2, dz4, dz2 and lzlz2 phases and c-BN are also shown in Table 1. All the BN polymorphs with sp² + sp³ hybridizations and sp² hybridizations are larger than the sp³ hybridizations (c-BN), so the bulk modulus of the BN polymorphs with sp² + sp³ hybridizations and sp² hybridizations is less than that of c-BN.

3.2. Stability

Stability is a significant physical property of new phases. In order to investigate the dynamical stability of P4/m BN, the phonon spectra of P4/m BN were generated, as presented in Figure 2. There was no hypothetical frequency observed in the entire Brillouin zone of P4/m BN, which proves that P4/m BN is dynamically stable. To study the mechanical stability of P4/m BN, the elastic constants of P4/m BN and other BN polymorphs were consulted, and these are shown in

Table 2. The elastic parameters (GPa) and E (Young’s modulus), B (bulk modulus) and G (shear modulus) (GPa) of BN polymorphs with the GGA functional.

Space Group	C11	C22	C33	C44	C55	C66	C12	C13	C23	C16	C26	B	G	E	B/G
P4/m	261	261	539	138	138	69	90	65	65	3	−3	160	117	282	1.368
Pnc2	659 1	122	661	36	136	56	117	149	68			176	93	237	1.892
dz4	810	60	672	20	261	28	26	197	79			148	92	229	1.609
dz2	786	107	629	86	290	72	64	229	131			183	127	309	1.441
lzlz2	548	835	335	132	32	214	97	47	51			207	133	329	1.556
c-BN	779			447			165					370	384	856	0.964
	820 2			480			190					400

¹ Reference [9]. ² Reference [43].

. The theoretical results of the elastic parameter C_ij of c-BN are closer to the experimental results [43]; this also indicates that our prediction of the theoretical results of the elastic constants of P4/m BN is also reliable. The space group of P4/m BN is P4/m; it can be classified as having tetragonal symmetry, and the mechanical stability can be verified by [44]

$$C_{11}>0,C_{33}>0,C_{44}>0,C_{66}>0,$$

(1)

$$(C_{11}-C_{12})>0,(C_{11}+C_{33}-2C_{13})>0,$$

(2)

$$[2(C_{11}+C_{12})+C_{33}+4C_{13}]>{0.}_{{}_{}}$$

(3)

The elastic constants (see Table 2) of P4/m BN satisfy the above Born criteria, thus proving that P4/m BN are mechanically stable.

3.3. Mechanical and Mechanical Anisotropy Properties

The elastic moduli and the B/G of P4/m BN are also shown in Table 2. The Voigt–Reuss-Hill approximations [45,46,47] are estimated for the values of B and G. For the tetragonal structure, B_V, B_R, G_V and G_R can be taken as [48]

$$B_v=(1/9)[2(C_{11}+C_{12})+C_{33}+4C_{13}],$$

(4)

$$B_R=C^2/M,$$

(5)

$$C^2=(C_{11}+C_{12})C_{33}-2C_{13}^2.$$

(6)

$$M=C_{11}+C_{12}+2C_{33}-4C_{13},$$

(7)

$$G_V=(1/30)(M+3C_{11}-3C_{12}+12C_{44}+6C_{66}),$$

(8)

$$G_R=15{\{(18B_V/C^2)+[6/(C_{11}-C_{12})]+(6/C_{44})+(3/C_{66})\}}^{-1},$$

(9)

For the B, G and E of P4/m BN, the following equations are used:

$$B=\frac{1}{2}(B_V+B_R),$$

(10)

$$G=\frac{1}{2}(G_V+G_R),$$

(11)

$$E=9BG/(3B+G),$$

(12)

According to Table 2, the B of the BN polymorphs with sp² + sp³ hybridizations and sp² hybridizations, P4/m BN, Pnc2 BN, dz4 BN, dz2 BN and lzlz2 BN, are all smaller than that of c-BN; this is consistent with our previous conclusions in the crystal structure section. In addition, the B of P4/m BN are greater than those of dz4 BN but less than those of Pnc2 BN, dz2 BN, lzlz2 BN and c-BN. In contrast, the G and E of P4/m BN are greater than those of Pnc2 BN and dz4 BN and smaller than those of dz2 BN, lzlz2 BN and c-BN. The B/G values of the BN materials are also shown in Table 2. It is well-known that B/G < 1.75 indicates ductility; otherwise, the material is brittle [49]. According to Table 2, the values for P4/m BN, dz2 BN, dz4 BN, lzlz2 BN and c-BN are smaller than 1.75, i.e., their brittleness is confirmed; moreover, the B/G of Pnc2 BN exceeds 1.75, which means that the ductility for Pnc2 BN is verified. Additionally, the ratio of the B/G of P4/m BN indicates its greater ductility compared to the other BN polymorphs mentioned in this work.

The mechanical anisotropy of crystals is very important in the study of their physical characteristics [50]. A directional dependence can visually show the anisotropic characteristics due to the varying Young’s moduli generated in different planes. The three-dimensional (3D) surface is a regular sphere, which means that the material is isotropic; otherwise, the material shows anisotropic characteristics [8,51,52,53,54,55], and the smaller the resemblance to a sphere, the larger the anisotropy, to evaluate the mechanical anisotropy properties of E.

For P4/m BN in distinct directions, the variations of E of P4/m BN in different planes were plotted, as shown in Figure 3a. In Figure 3a, different colors represent different value ranges: the minimum value and maximum value of E in P4/m BN are indicated by the curved surface of the solid cyan line and the solid purple line, respectively. The ratio between the maximum value and the minimum value (E_max/E_min = 514.72/193.57 = 2.66) of E is greater than that of the Pbca phase [22], P4₂/mnm phase [50] and Pm-3m phase [56], while it is much weaker than that of P4/mbm BN, B₇N₇, B₁₁N₁₁ and B₁₅N₁₅ [57]. The anisotropy in E of P4/mbm BN, B₇N₇, B₁₁N₁₁ and B₁₅N₁₅ is approximately 4, 6, 8 and 24 times that of P4/m BN, respectively. Therefore, the anisotropy in E of P4/m BN is much lower than that of P4/mbm BN, B₇N₇, B₁₁N₁₁ and B₁₅N₁₅. The two-dimensional (2D) extreme values of E in the (001), (101), (100), (110), (010) and (111) planes for P4/m BN are presented in Figure 3b, respectively. The distribution of E in the (100) plane and (010) plane is the same, including the E_max and E_min, and the same situation also occurs in the (101) plane and (011) plane. The anisotropy in E of P4/m BN at the (001) plane (E_max/E_min = 228.00/193.57 = 1.18) is the lowest, and the anisotropy in E at the (110) plane (E_max/E_min = 514.72/194.54 = 2.65) is the strongest.

The variations of the G_max and G_min of P4/m BN in different directions are displayed in Figure 4a,b, which indicate the mechanical anisotropy properties of G for P4/m BN in distinct directions. The shape of the 3D-structured surface of G for P4/m BN is a very close sphere; the G_max/G_min for P4/m BN is 2.08, while it is larger than that for the Pbca phase [22], P4₂/mnm phase [50] and Pm-3m phase [56]. Regarding the Young’s modulus, the anisotropy of the shear modulus of P4/m BN is still weaker than that of P4/mbm BN, B₇N₇, B₁₁N₁₁ and B₁₅N₁₅. The G_max/G_min of P4/mbm BN, B₇N₇, B₁₁N₁₁ and B₁₅N₁₅ is still many times that of P4/m BN, and the anisotropy of G of P4/mbm BN, B₇N₇, B₁₁N₁₁ and B₁₅N₁₅ is approximately 7, 8, 14 and 44 times that of P4/m BN, respectively. The 2D representations of the shear modulus in the major planes for P4/m BN are provided in Figure 4c,d, respectively. As demonstrated in Figure 4c,d, as in the case of the Young’s modulus, the distribution of G in the (100) and (010) planes is identical, and the distribution of G in the (101) plane and (011) plane is the same. The anisotropy of G of P4/m BN in the (110) plane (G_max/G_min = 138.27/85.62 = 1.61) is the lowest, and the anisotropy of the shear modulus in the (101) and (011) planes (G_max/G_min = 141.94/68.22 = 2.08) is the greatest.

The variations of the extremum value of v for P4/m BN in different directions are demonstrated in Figure 5a,b, and the 2D representations of the Poisson’s ratio in the (001), (101), (100), (110), (010) and (111) planes for P4/m BN are displayed in Figure 5c,d, respectively. The shape of the three-dimensional (3D) contour map of the P4/m BN shear modulus deviates far from the sphere; the v_max/v_min for P4/m BN is 7.00, while it is larger than that of the Pbca phase [22], P4₂/mnm phase [50] and Pm-3m phase [56]. Since all these materials are BN polymorphs, the reason for the differences in their anisotropy is not the different constituent elements. The anisotropy of the Young’s modulus, Poisson’s ratio and shear modulus is due to the different stacking modes of these BN polymorphs. Different stacking modes form different element rings. The larger the element ring, the larger the number of pore structures in these BN polymorphs. The more pore structures present, the greater the variations in the Young’s modulus, shear modulus and Poisson’s ratio in different directions. According to Figure 5c,d, for E and the shear modulus, the distribution of G in the (100) plane and (010) plane is the same, and the distribution of the Young’s modulus and shear modulus in the (101) plane and (011) plane is the same. The anisotropy of the Poisson’s ratio of P4/m BN in the (110) plane and (111) plane (v_max/v_min = 0.42/0.06 = 7.00) is the largest, and the anisotropy in the v ratio in the (100) and (010) planes (v_max/v_min = 0.32/0.08 = 4.00) is the smallest.

3.4. Electronic Band Structures

The electronic band structures and partial density of states (PDOS) using the HSE06 hybrid functional of P4/m BN and c-BN are plotted in Figure 6a,b, respectively. From Figure 6, one can observe that the P4/m BN is a semiconductor material with a quasi-direct bandgap, as it has a direct bandgap of 2.043 eV; while the direct bandgap of P4/m BN is 2.051 eV, the conduction band minimum (CBM) is located at the Z point, and the valence band maximum (VBM) is located at the A point. For other BN polymorphs, most of them are semiconductor materials and have a wide bandgap of more than 4 eV, such as P4₂/mnm BN (6.13 eV [50]), Pbca BN (6.81 eV [22] and 6.79 eV [3]), Pm-3m BN (5.87 eV [56]), m-BN (4.629 eV [27]), Pnma BN (7.18 eV [28]), B₄N₄-II (5.32 eV [3]) and B₄N₄-I (4.86 eV [3]). Although the bandgap of P4/m BN is smaller than that of the above BN polymorphs, it is still larger than that of P2₁3 BN (1.826 eV [32]). For P4/m BN, there are two hybridization modes of nitrogen atoms and boron atoms: sp² hybridization and sp³ hybridization. The PDOS of atoms with different hybridization modes were studied, and the results are shown in Figure 6a. According to the partial density of the states in Figure 6a, the contribution of 2p orbitals from B atoms with sp² hybrids is larger than that of B atoms with sp³ hybrids in the lower energy range (−8 to −4 eV), while the contribution of 2p orbitals from B atoms with sp² hybrids is less than that of B atoms with sp³ hybrids in the range near the Fermi level (0 to 2 eV). For N atoms, the contribution from N 2p orbitals is greater than that of B 2p orbitals from 0 to 12 eV. In the range of 18–20 eV, the contribution of 2p orbitals from N atoms with sp² hybrids is greater than that of N atoms with sp³ hybrids, while the contribution of 2p orbitals from N atoms with sp² hybrids is less than that of N atoms with sp³ hybrids at 15–18 eV and 20–22 eV. While, in c-BN, boron atoms and nitrogen atoms are all combined by sp³ hybridization, in the energy range of −12 eV to the Fermi level, 6–20 eV, the contributions of the N 2p orbitals and B 2p orbitals are greater than those of the B 2s orbitals and N 2s orbitals; in the energy range from −22 eV to −15 eV, the contribution of B 2p orbitals is greater than that of B 2s orbitals, while the contribution of N 2s orbitals is greater than that of N 2p orbitals.

In addition, the electron localization functions (ELF) of P4/m BN and the band decomposed charge densities (BDCD) of P4/m BN are shown in Figure 7; the ELF with an isosurface level was set to 0.88, and the isosurface of the BDCD at the CBM and VBM was set to 0.005 eV/ų, respectively. The electron localization at the center of the sp³ B–B bonds was stronger than that of the sp² + sp² BN bonds and stronger than that of the sp³ − sp³ BN bonds. The band decomposed charge densities (BDCD) of the CBM and VBM for P4/m BN are shown in Figure 7b,c, respectively. As shown in Figure 6 and Figure 7b, the CBM is composed mainly of sp³ N 2p orbitals and sp² B atom 2p orbitals to a lesser extent, while the VBM is composed mainly of sp³ B 2p orbitals and, to a lesser extent, sp³ N atom 2p orbitals. Due to the quasi-direct bandgap of the P4/m BN, P4/m BN is a good candidate for photoelectric devices.

3.5. X-ray Diffraction

In order to obtain further characteristics of P4/m BN for future synthetic verification, the simulation of X-ray diffraction (XRD) patterns for P4/m BN and other BN polymorphs was performed, as illustrated in Figure 8. Strong peaks of c-BN appeared between 40° and 90°, while strong peaks of P4/m BN appeared between 5° and 40°, respectively. The strongest peak of c-BN was (-1-1-1), which was located at around 43.21°, while the strongest peak of P4/m BN was (100), (110) and (200) for P4/m BN. In addition to the main diffraction peaks in Figure 7, the XRD spectrum also included several other weak diffraction peaks. For the (100) peak, the diffraction angles were 12.21° and 8.06° for P4/m BN and Pnc2 BN, respectively; for the (110) peak, the diffraction angles were 17.29°, 8.06° and 32.98° for P4/m BN, Pnc2 BN and dz2 BN, respectively. For the (011) peak, the diffraction angles were 22.81°, 29.62° and 26.79° for Pnc2 BN, dz2 BN and dz4 BN, respectively. For the (111) peak, the diffraction angles were 28.62°, 24.23° and 32.34° for P4/m BN, Pnc2 BN and dz4 BN, respectively. These XRD characteristics are crucial for distinguishing the structures of P4/m BN in future experiments.

4. Conclusions

In summary, three novel boron nitride polymorphs, P4/m BN, with wide bandgap properties are proposed theoretically through first-principles calculations. The structural characteristics, dynamical stability, mechanical stability, mechanical anisotropy, mechanical properties and electronic characteristics of P4/m BN are investigated in this work. P4/m BN has the strongest brittleness compared to that of dz2 BN, lzlz2 BN and dz4 BN. The B of P4/m BN is greater than that of dz4 BN but less than that of Pnc2 BN, dz2 BN, lzlz2 BN and c-BN. The G and E of P4/m BN are larger than those of Pnc2 BN and dz4 BN but smaller than those of dz2 BN, lzlz2 BN and c-BN. The anisotropy of the shear and Young’s modulus of P4/m BN is greater than that of Pbca BN, P4₂/mnm BN and Pm-3m BN, and it is less than that of P4/mbm BN, B₇N₇, B₁₁N₁₁ and B₁₅N₁₅. The G_max/G_min and E_max/E_min of P4/mbm BN, B₇N₇, B₁₁N₁₁ and B₁₅N₁₅ are still many times that of P4/m BN. In addition, P4/m BN is a quasi-direct and wide bandgap semiconductor material. P4/m BN may play an important role in the manufacture of photoelectric devices. Finally, it is possible that the XRD characteristics will be very helpful in determining the crystal structure of P4/m BN in future experiments.