Pnma-BN: Another Boron Nitride Polymorph with Interesting Physical Properties.

Structural, mechanical, electronic properties, and stability of boron nitride (BN) in Pnma structure were studied using first-principles calculations by Cambridge Serial Total Energy Package (CASTEP) plane-wave code, and the calculations were performed with the local density approximation and generalized gradient approximation in the form of Perdew-Burke-Ernzerhof. This BN, called Pnma-BN, contains four boron atoms and four nitrogen atoms buckled through sp³-hybridized bonds in an orthorhombic symmetry unit cell with Space group of Pnma. Pnma-BN is energetically stable, mechanically stable, and dynamically stable at ambient pressure and high pressure. The calculated Pugh ratio and Poisson's ratio revealed that Pnma-BN is brittle, and Pnma-BN is found to turn brittle to ductile (~94 GPa) in this pressure range. It shows a higher mechanical anisotropy in Poisson's ratio, shear modulus, Young's modulus, and the universal elastic anisotropy index AU. Band structure calculations indicate that Pnma-BN is an insulator with indirect band gap of 7.18 eV. The most extraordinary thing is that the band gap increases first and then decreases with the increase of pressure from 0 to 60 GPa, and from 60 to 100 GPa, the band gap increases first and then decreases again.

1. Introduction

In recent years, with the development of technology the interest in theoretical design and experimental synthesis of new superhard materials has increased. Such materials are in great demand in material science, electronics, optics, and even jewelry. Usually, borides, nitrides, and the covalent compounds of light elements (B, Be, O, C, N, etc.) are regarded as candidates of superhard materials [1,2,3,4,5]. Among these materials, boron nitrides are a typical group. c-BN is a superhard material. Boron nitride has various polymorphs, which are similar to structural modifications of carbon. Boron nitride (BN) can stably exist in many polymorphs because B and N atoms can bind together by sp2 and sp3 hybridizations. Hexagonal boron nitride (h-BN) is a graphite-like layered structure of the ABAB type, where each layer is rotated with respect to the previous one [6]. Also, there is a range of phases, usually referred to as turbostratic boron nitride (t-BN) [7,8], which are located between highly ordered h-BN and an amorphous material. Besides the well-known cubic diamond-like phase (c-BN) [9], wurtzite-like phase (w-BN) [7], layered graphite-like phase (h-BN or r-BN) [6,10,11], BN nanosheet [12], and BN nanotubes (BNNTs) [13], many new BN polymorphs have been experimentally prepared or theoretical predicted, including P-BN [14], BC8-BN [15], T-BN [16], Z-BN [17], I-BN [18], cT8-BN [19], B4N4 [20], o-BN [21], bct-BN [22], zeolite-like microporous BN [23,24], turbostratic BN [25], and BN fiber [10].

Dai et al. [23] found two types of highly stable porous BN materials using a Particle Swarm Optimization (PSO) algorithm as implemented in Crystal structure AnaLYsis by Particle Swarm Optimization (CALYPSO) code, and the first-principles calculations are utilized in properties calculations. In particular, type-II BN material lz3-BN with a relatively large pore size appears to be highly favorable for hydrogen adsorption as the computed average hydrogen adsorption energy is very close to the optimal adsorption energy suggested for reversible adsorptive hydrogen storage at room temperature. Li et al. [26] performed a systematic search for stable compounds in the BN system. They found a new stable N-rich compound with stoichiometry of B3N5 (C2221 phase), which at ambient pressure has a layered structure with freely rotating N2 molecules intercalated between the layers. Therefore, the C2221 phase is a potential high-energy-density material. Calculations also revealed C2221-B3N5 to be superhard.

In this paper, structural, mechanical, electronic properties, and stability of Pnma-BN were first studied using first-principles calculations by Cambridge Serial Total Energy Package (CASTEP) plane-wave code, and the calculations were performed with the local density approximation and generalized gradient approximation in the form of Perdew–Burke–Ernzerhof. Pnma-BN with space group 62 has four-, six-, and eight-membered rings, it is a three-dimensional structure, which is different from that of layered two-dimensional material (for example: h-BN).

2. Computational Methods

The total energy calculations were performed using density functional theory (DFT) with the Perdew–Burke–Ernzerhof (PBE) exchange correlation in the framework of the generalized gradient approximation (GGA) [27] and Ceperley and Alder data as parameterized by Perdew and Zunger (CA-PZ) in the framework of the local density approximation (LDA) [28] as implemented in the Cambridge Serial Total Energy Package (CASTEP) plane-wave code [29]. The equilibrium crystal structures were achieved by utilizing geometry optimization in the Broyden–Fletcher–Goldfarb–Shanno (BFGS) [30] minimization scheme. The interactions between the ionic core and valence electrons were described by the ultrasoft pseudo-potential [31], and the 2s22p1 and 2s22p3 were considered as valence electrons for B and N, respectively. The plane-wave basis set was truncated with an energy cutoff of 500 eV, and the Brillouin zone integration was generated using Monkhorst-Pack k-point meshes [32] with a high-quality grid of 0.025 Å−1 (8 × 15 × 9) for total-energy and elastic constants calculations, respectively. The elastic constants were calculated by the strain–stress method, which has been successfully utilized previously [33,34]. The bulk modulus, shear modulus, Young’s modulus, and Poisson’s ratio were estimated via Voigt–Reuss–Hill approximation [35,36,37].

3. Results and Discussion

3.1. Structural Properties

Pnma-BN adopts a Pnma symmetry with atoms occupying the B (−0.1601, 0.2500, 0.4087) and N (0.1773, 0.2500, 0.3909) positions. Pnma-BN has the lattice parameters a = 4.890 Å, b = 2.589 Å, c = 4.284 Å with GGA at ambient pressure. The crystal structure of Pnma-BN is shown in Figure 1. From Figure 1, Pnma-BN shares the configurations of four-, six-, and eight-membered sp3-bonded rings, and it is a three-dimensional structure. The calculated lattice parameters of Pnma-BN, Pbca-BN, and -BN (c-BN) are listed in Table 1. For Pnma-BN, Pbca-BN, and -BN, the calculated lattice parameters are in excellent agreement with the reported calculated results [38,39,40], and the calculated lattice parameters of -BN are in excellent agreement with the experimental results [41]. With the pressure increasing to 50 GPa, the B atoms’ positions change to (−0.1618, 0.2500, 0.4004), and the N atoms positions change to (0.2046, 0.2500, 0.4126); while under 100 GPa, the B atoms’ positions change to (−0.1621, 0.2500, 0.3950), the N atoms’ positions change to (0.2219, 0.2500, 0.4256). Compared to the boron atoms, the change of atom positions of the nitrogen atoms are much larger than that of the boron atoms.

Figure 1

Unit cell crystal structures of BN in Pnma structure.

Table 1

The calculated lattice parameters of BN polymorphs.

Space Group	Methods	a (Å)	b (Å)	c (Å)	V (Å3)
Pnma	GGA	4.8900	2.5890	4.2835	13.5574
	LDA	4.7954	2.5569	4.2432	13.0068
	LDA 1	4.7600	2.5800	4.2900	13.1712
Pbca	GGA	5.0987	4.4216	4.3981	12.3940
	GGA 2	5.1103	4.4336	4.3992	12.4591
	LDA	5.0412	4.3794	4.3316	11.9538
	LDA 2	5.0458	4.3800	4.3392	11.9873
F4¯3m	GGA	3.6258			11.9166
	GGA 3	3.6224			11.8835
	LDA	3.5692			11.3672
	LDA 3	3.5764			11.4364
	Experiment 4	3.6200			11.8595

1 Reference [38]; 2 Reference [39]; 3 Reference [40]; 4 Reference [41].

The structural properties, as well as the dependences of the normalized lattice parameters and volume on pressure up to 100 GPa for Pnma-BN, are shown in Figure 2. From Figure 2a, the lattice parameters of Pnma-BN decrease with increasing pressure, while for lattice parameter c, it decreases with a slightly smaller speed as pressure increases from 20 GPa to 40 GPa than other ranges. We noted that, when the pressure increases, the compression along the c-axis is much larger than those along the a-axis and b-axis in the basal plane. From Figure 2a, we can also easily see that the compression of c-axis is the most difficult. For the volumes on pressure up to 100 GPa of Pnma-BN, Pbca-BN, -BN, and diamond, it can be easily seen that the compression of diamond is the most difficult. From Figure 2b, it can be seen that the incompressibility of Pbca-BN and -BN is better than Pnma-BN. So we can expect the bulk modulus of Pnma-BN is smaller than that of Pbca-BN and -BN. For -BN, the calculated lattice parameters using GGA level are closer than that of experimental results (see Table 1), so we use the results of elastic constants and elastic modulus of Pnma-BN within the GGA level in this paper.

Figure 2

The lattice constants a/a0, b/b0, c/c0 compression as functions of pressure for Pnma-BN (a), and primitive cell volume V/V0 for Pbca-BN, Pnma-BN, c-BN, and diamond (b).

3.2. Stability

The orthorhombic phase has nine independence elastic constants C (C11, C12, C13, C22, C23, C33, C44, C55, C66), and the elastic constants and elastic modulus of Pnma-BN are listed in Table 2. The criteria for mechanical stability of the orthorhombic phase are given by [42]:

Table 2

The calculated elastic constants (GPa) and elastic modulus (GPa) within GGA level of Pnma-BN, Pbca-BN, and h-BN.

Materials	p	C11	C12	C13	C22	C23	C33	C44	C55	C66	B	G	E	v
Pnma-BN	0	392	99	256	770	116	675	299	272	187	298	227	543	0.196
	0 1	403	107	273	824	132	730	316	282	187	318	236	568	0.202
	10	397	117	282	836	150	756	316	281	185	326	234	566	0.210
	20	405	131	300	899	184	840	333	289	181	351	243	592	0.219
	30	412	147	318	961	220	923	348	297	171	369	245	602	0.228
	40	420	162	316	1023	258	1010	365	305	167	396	259	638	0.232
	50	477	177	311	1083	295	1089	381	314	165	430	274	678	0.237
	60	524	192	310	1138	331	1159	394	315	166	460	286	711	0.242
	70	584	209	312	1191	368	1217	408	325	168	494	299	746	0.248
	80	646	227	314	1243	400	1259	420	329	173	526	311	779	0.253
	90	716	247	317	1292	437	1296	430	334	181	559	322	810	0.258
	100	777	266	326	1347	468	1344	440	339	187	592	334	843	0.263
Pbca-BN	0	769	145	133	870	105	716	307	255	340	340	312	717	0.148
	0 2	772	135	139	885	92	716	312	257	357	344	316	718	0.140
c-BN	0	788	160				443				369	386	859	0.112
	0 2	779	165				446				370	384	856	0.120

1 local density approximation (LDA) level; 2 Reference [39].

The calculated elastic constants under ambient pressure and high pressure of Pnma-BN indicated that it is mechanically stable because of the satisfaction of the mechanical stability criteria. To confirm the stability of Pnma-BN, their dynamical stabilities should also be studied under ambient pressure and high pressures. Thus, the calculated the phonon spectra for Pnma-BN at 0 and 100 GPa are shown in Figure 3a,b. No imaginary frequencies are observed throughout the whole Brillouin zone, confirming the dynamical stability of Pnma-BN.

Figure 3

The phonon spectra of Pnma-BN at 0 GPa (a) and 100 GPa (b); Mixing enthalpy ΔH of BN alltropes calculated using PBE (c).

In an effort to assess the thermodynamic stability of Pnma-BN, enthalpy change curves with pressure for various structures were calculated, as presented in Figure 3c. The dashed line represents the enthalpy of the -BN (c-BN). It can be clearly seen that P63/mmc-BN has the lowest minimum value of enthalpy, which is in good agreement with previous reports and supports the reliability of our calculations. The minimum value of total energy per formula unit of BN is slightly larger than that of Pbam-BN and P63/mc-BN, hence Pnma-BN should be thermodynamically metastable.

3.3. Mechanical and Anisotropic Properties

The elastic constants and elastic modulus of Pnma-BN as a function of pressure are shown in Figure 4a, all elastic constants and elastic modulus of Pnma-BN are increasing with different rates as pressure increases, except for C66. It is well known that bulk modulus (B) represents the resistance to material fracture, whereas the shear modulus (G) represents the resistance to plastic deformation of a material, Young’s modulus (E) describes tensile elasticity. Young’s modulus E and Poisson’s ratio v are taken as: E = 9BG/(3B + G), v = (3B − 2G)/[2(3B + G)].

Figure 4

Elastic constants and elastic modulus (a) and B/G ratio (b); Poissons’ ratio v (c) of Pnma-BN as a function of pressure.

Hence, the Pugh ratio (B/G ratio) is defined as a quantitative index for assessing the brittle or ductile behavior of crystals. According to Pugh [43], a larger B/G value (B/G > 1.75) for a solid represents ductile, while a smaller B/G value (B/G < 1.75) usually means brittle. Moreover, Poisson’s ratio v is consistent with B/G, which refers to ductile compounds usually with a large v (v > 0.26) [44]. The value of Poisson’s ratio v and B/G various pressure as functions for Pnma-BN are shown in Figure 4b,c, respectively, which indicates that Pnma-BN is brittle when pressure less than around 94 GPa. The values of B/G and v for Pnma-BN are 1.312 and 0.196 at ambient pressure, respectively. Pnma-BN is found to turn from brittle to ductile in this pressure range (0–100 GPa).

Based on elastic modulus and other related values, the hardness (H) of Pnma-BN are evaluated using two different empirical models: Chen et al. model [45] and Lyakhov and Oganov’s et al. model [46,47], the calculated results of Chen et al. model and Ma et al. model are 31.8 GPa and 33.3 GPa. The results of Chen et al. model are slightly smaller than that of Lyakhov and Oganov’s model. The main reason for this situation is that an empirical formula may estimate the value of the material’s hardness as too high or too low. Most researchers agree on the definition according to which “superhard” materials are those with Hv exceeding 40 GPa [15]. Although there are slightly differences between the results of the two empirical models above, the hardness of Pnma-BN is slightly smaller than 40 GPa, indicating that Pnma-BN is a hard material.

The Poisson’s ratio v, shear modulus G and Young’s modulus E may have different values depending on the direction of the applied force with respect to the structure, so we continued to investigate the mechanical anisotropy properties of Pnma-BN. A fourth order tensor transforms in a new basis set following the rule: where Einstein’s summation rule is adopted and where the rαi is the component of the rotation matrix (or direction cosines). The Young’s modulus can be obtained by using a purely normal stress in ε = Sijklσkl in its vector form and it is given by the following form:

The Poisson’s ratio and shear modulus depending on two directions (if perpendicular, this corresponds to three angles) make them difficult to represent graphically. A convenient possibility is then to consider three representations: minimum, average, and maximum. For each θ and ϕ, the angle χ is scanned and the minimum, average, and maximum values are recorded for this direction. The transformation can be substantially simplified in calculation of specific modulus. The uniaxial stress can be represented as a unit vector, and advantageously described by two angles θ, ϕ, we choose it to be the first unit vector in the new basis set . The determination of some elastic properties (shear modulus, Poisson’s ratio) requires another unit vector , perpendicular to unit vector , and characterized by the angle χ. It is fully characterized by the angles θ (0, π), ϕ (0, 2π), and χ (0, 2π), as illustrated in Reference [48]. The coordinates of two vectors are:

The shear modulus in the vector form is obtained by applying a pure shear stress, then it can be expressed as:

The Poisson’s ratio can be given in:

The three-dimension surface representation of Poisson’s ratio v, shear modulus G, and Young’s modulus E for Pnma-BN are illustrated in Figure 5a–c, respectively. The green and purple surface representation denoted the minimum and the maximum values of Poisson’s ratio v and shear modulus G, respectively. For an isotropic system, the three-dimension directional dependence would exhibit a spherical shape, while the deviation degree from the spherical shape reflects the content of anisotropy [49]. From Figure 5a–c, one can note that the Poisson’s ratio, shear modulus, and Young’s modulus show different degree anisotropy of Pnma-BN. Pnma-BN shows the largest anisotropy in Poisson’s ratio than that of shear modulus and Young’s modulus.

Figure 5

The surface construction of Poisson’s ratio (a); shear modulus (b); and Young’s modulus (c) for the Pnma-BN.

In order to investigate the anisotropy of Pnma-BN in detail, the two-dimension representations of the shear modulus in the (001) plane, (010) plane, (100) plane, and (111) plane for Pnma-BN are illustrated in Figure 6a–c, and the calculated the maximum and minimum values of Poisson’s ratio v, shear modulus G and Young’s modulus E for Pnma-BN are listed in Table 3. From Figure 6a–c and Table 3, one can find that Pnma-BN shows a larger anisotropy in Poisson’s ratio v, shear modulus G and Young’s modulus E. For Poisson’s ratio v, the maximum value appears in the (010) and (100) planes, the minimum value are not appearing in these planes, but (100) plane (vmax/vmin = 14.431) shows the largest anisotropy. (100) plane shows the largest anisotropy in Poisson’s ratio v, while it shows the smallest anisotropy in shear modulus and Young’s modulus. For shear modulus, the maximum value of all directions for shear modulus is 310.85 GPa, while maximum values are not appearing in the (001), (100), (010), and (111) planes, the minimum value appear in the (010) and (111) planes. The anisotropies of shear modulus in all directions for Pnma-BN reduce in the sequence of (111) plane > (010) plane > (001) plane = (100) plane. The maximum value of Young’s modulus appears in the (001) and (100) planes, while the minimum value appears in the (010) and (100) planes, so the (001) plane shows the largest anisotropy in Young’s modulus. The anisotropies of Young’s modulus in all directions for Pnma-BN follow the order: (001) plane > (010) plane > (111) plane > (100) plane.

Figure 6

2D representation of Poisson’s ratio (a); shear modulus (b) and Young’s modulus (c) in the main plane for Pnma-BN, respectively.

Table 3

The calculated the maximum and minimum values of Poisson’s ratio v, Shear modulus G, and Young’s modulus E for Pnma-BN.

Surface	Poisson’s Ratio v			Shear Modulus (GPa)			Young’s Modulus (GPa)
	vmax	vmin	vmax/vmin	Gmax	Gmin	Gmax/Gmin	Emax	Emin	Emax/Emin
(001)	0.366	0.074	4.945	298.60	186.70	1.599	740.15	291.12	2.534
(010)	0.635	0.069	9.203	298.60	126.04	2.369	658.13	291.12	2.261
(100)	0.635	0.044	14.431	298.60	186.70	1.599	740.15	504.52	1.467
(111)	0.494	0.045	10.978	307.33	126.04	2.439	649.49	389.52	1.667
All	0.635	0.010	63.500	310.85	126.04	2.466	740.15	291.12	2.534

The universal elastic anisotropy index AU proposes an anisotropy measure based on the Reuss and Voigt averages which quantifies the single crystal elastic anisotropy, and AU = 5G/G + B/B − 6 [50]. The universal elastic anisotropy index AU as a function of pressure is shown in Figure 4a. The universal elastic anisotropy index AU increases with increasing pressure from 0 to 30 GPa, then it decreases with increasing pressure with 30 to 100 GPa. At ambient pressure, Pnma-BN has a larger universal elastic anisotropy index AU (0.798). It is almost eight times that of Pbca-BN (0.095).

The interest in the calculation of the Debye temperature ΘD has been increasing in both semiempirical and theoretical phase diagram calculation areas since the Debye model offers a simple but highly efficiency method to describe the phonon contribution to the Gibbs energy of crystalline phases. The average sound velocity v and Debye temperature ΘD can be approximately calculated by the following relations [51]: vl and v are the longitudinal and transverse sound velocities, respectively, which can be obtained from Navier’s equation [52]: where h is Planck’s constant, k is Boltzmann’s constant, NA is Avogadro’s number, n is the number of atoms in the molecule, M is molecular weight, and ρ is the density, (θ, ϕ) are angular coordinates and dΩ = sinθdθdϕ. If the elastic constants of the crystal are known, v (θ, ϕ) can be obtained by solving a secular equation, and v and ΘD can then be calculated by numerical integration over θ and ϕ [53,54]. The calculated sound velocities and Debye temperatures under pressure of Pnma-BN are listed in Table 4. The Debye temperature of Pnma-BN is 1502 K, it is smaller than that of Pbca-BN (ΘD = 1734 K) at ambient pressure, and it is also smaller than -BN (ΘD = 1896 K), the result of -BN has a high credibility [55]. The longitudinal and transverse sound velocities of Pnma-BN are smaller than Pbca-BN [39] and -BN, because Pnma-BN has the smaller elastic modulus.

Table 4

The calculated density ρ (g/cm3), sound velocities (m/s), and Debye temperature (K) of Pnma-BN.

p	ρ	vp	vs	vm	ΘD
0	3.040	14057	8642	9537	1502
10	3.144	14244	8627	9534	1518
20	3.248	14415	8649	9567	1540
30	3.354	14402	8547	9465	1540
40	3.460	14639	8653	9585	1576
50	3.559	14949	8774	9727	1614
60	3.653	15176	8848	9815	1643
70	3.740	15449	8941	9924	1674
80	3.822	15688	9020	10018	1702
90	3.899	15920	9087	10098	1728
100	3.973	16158	9169	10194	1755

3.4. Electronic Properties

The band structures with Heyd–Scuseria–Ernzerhof (HSE06) hybrid-functional [56,57] along high-symmetry direction in Brillouin zone under pressure of Pnma-BN are shown in Figure 7. At ambient pressure, Pnma-BN is an insulator with band gap of 7.18 eV. The band gap of Pnma-BN is slightly larger than that of h-BN at ambient pressure (LDA: 4.01 eV [58], Experiment: 5.97 eV [59]). When p = 30 GPa, the band gap of Pnma-BN is 7.51 eV, while the band gap is 7.30 eV when p = 60 GPa. More interestingly, with pressure increasing to 100 GPa, the band gap increases to 7.32 eV. Unusually, the band gap of Pnma-BN is not monotonically increasing or monotonically decreasing with increasing pressure. The band gap of Pnma-BN as a function of pressure is shown in Figure 8a. From 0 to 60 GPa, the band gap increases first and then decreases with the increase of pressure, and from 60 to 100 GPa, the band gap increases first and then decreases.

Figure 7

The band structures under pressure of Pnma-BN, (a) 0 GPa, (b) 30 GPa, (c) 60 GPa, (d) 100 GPa.

Figure 8

The band gap under pressure of Pnma-BN (a), the Fermi level and the energy of G high-symmetry points along valence band maximum (VBM) (b); the energy of T and Y high-symmetry points along conduction band minimum (CBM) (c).

Figure 8b,c shows the energies of Fermi level and G high-symmetry point along valence band maximum (VBM), the energies of T and Y high-symmetry points along conduction band minimum (CBM) as functions with pressure, respectively. From Figure 8b, it is clear that the Fermi levels are very close to G high-symmetry point along VBM. The energies of T and Y high-symmetry points along CBM both increase with increasing pressure. From 0 to 20 GPa, the energy of Y high-symmetry points along CBM is greater than that of T high-symmetry points, while when p = 20 GPa, the energy of Y high-symmetry points along CBM (15.97 eV) is very close to T high-symmetry points (15.94 eV). With increasing pressure (from 20 to 100 GPa), the energy of T high-symmetry points along CBM is greater than that of Y high-symmetry points (see Figure 7b–d).

4. Conclusions

The calculated lattice parameters agree very well with reported values in the literature, for all phases of both materials. The Pnma phase of BN is found to be metastable. The calculated Pugh ratio and Poisson’s ratio revealed that Pnma-BN is brittle, and Pnma-BN is found to turn from brittle to ductile (~94 GPa) in this pressure range. In addition, the mechanical anisotropy properties of Pnma-BN are investigated in this paper. Pnma-BN shows a larger anisotropy in Poisson’s ratio v, shear modulus G and Young’s modulus E, and its anisotropy is greater than that of Pbca-BN and -BN. The calculated band structure revealed that Pnma-BN is an insulator with band gap of 7.18 eV at ambient pressure. More interesting, the band gap of Pnma-BN is not monotonically increasing or monotonically decreasing with increasing pressure. From 0 to 60 GPa, the band gap increases first and then decreases with the increase of pressure, and from 60 to 100 GPa, the band gap increases first and then decreases. In addition, we will study nitride boron nitride (BN), aluminum nitride (AlN), and gallium nitride (GaN) [60] alloys, mainly researching some physical properties, such as mechanical properties, electronic properties, and mechanical anisotropy properties.