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.


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 sp 2 and sp 3 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], BC 8 -BN [15], T-B x N x [16], Z-BN [17], I-BN [18], cT 8 -BN [19], B 4 N 4 [20], o-BN [21], bct-BN [22], zeolite-like microporous BN [23,24], turbostratic BN [25], and BN fiber [10].  Table 1. For Pnma-BN, Pbca-BN, and F43m-BN, the calculated lattice parameters are in excellent agreement with the reported calculated results [38][39][40], and the calculated lattice parameters of F43m-BN are in excellent agreement with the experimental results [41]. With the pressure increasing to 50 3 3.6224 11.8835 LDA 3.5692 11.3672 LDA 3 3.5764 11.4364 Experiment 4 3.6200 11 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,  -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.  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, F43m-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 F43m-BN is better than Pnma-BN. So we can expect the bulk modulus of Pnma-BN is smaller than that of Pbca-BN and F43m-BN. For F43m-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.

Stability
The orthorhombic phase has nine independence elastic constants Cij (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]: 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

Stability
The orthorhombic phase has nine independence elastic constants C ij (C 11 , C 12 , C 13 , C 22 , C 23 , C 33 , C 44 , C 55 , C 66 ), 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]: 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.
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   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 F43m-BN (c-BN). It can be clearly seen that P6 3 /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 P6 3 /mc-BN, hence Pnma-BN should be thermodynamically metastable.

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 C 66 . 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:    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  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 m F 3 4 -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.

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:

Elastic constants (GPa)
Pressure (GPa)   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 v ) 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] [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: S αβγδ = r ai r aj r ak r al S ijkl where Einstein's summation rule is adopted and where the r αi 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 ij = S ijkl σ kl 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 a. The determination of some elastic properties (shear modulus, Poisson's ratio) requires another unit vector b, perpendicular to unit vector a, 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: G(θ, ϕ, χ) = 1 4S 66 (θ, ϕ, χ) = 1 4r 1i r 2j r 1k r 2l S ijkl = 1 4a i a j a k a l S ijkl (9) Nanomaterials 2017, 7, 3

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The Poisson's ratio can be given in: v(θ, ϕ, χ) = − S 12 (θ, ϕ, χ) S 11 (θ, ϕ) = − r 1i r 1j r 2k r 2l S ijkl r 1i r 1j r 1k r 1l S ijkl = − a i a j b k b l S ijkl a i a j a k a l S ijkl (10) 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. 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.     The universal elastic anisotropy index A U proposes an anisotropy measure based on the Reuss and Voigt averages which quantifies the single crystal elastic anisotropy, and A U = 5GV/GR + BV/BR − 6 [50]. The universal elastic anisotropy index A U as a function of pressure is shown in Figure 4a. The universal elastic anisotropy index A U 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 A U (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 vm and Debye temperature ΘD can be approximately calculated by the following relations [51]: vl and vt are the longitudinal and transverse sound velocities, respectively, which can be obtained from Navier's equation [52]: where h is Planck's constant, kB 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, vi (θ, ϕ) can be obtained by solving a secular equation, and vm and ΘD can then be calculated by numerical integration over θ and The universal elastic anisotropy index A U proposes an anisotropy measure based on the Reuss and Voigt averages which quantifies the single crystal elastic anisotropy, and [50]. The universal elastic anisotropy index A U as a function of pressure is shown in Figure 4a. The universal elastic anisotropy index A U 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 A U (0.798). It is almost eight times that of Pbca-BN (0.095).
The interest in the calculation of the Debye temperature Θ D 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 m and Debye temperature Θ D can be approximately calculated by the following relations [51]: v l and v t are the longitudinal and transverse sound velocities, respectively, which can be obtained from Navier's equation [52]: 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 molecular weight, and is the density, (θ, ϕ) are angular coordinates and dΩ = sinθdθdϕ. If the elastic constants of the crystal are known, v i (θ, ϕ) can be obtained by solving a secular equation, and v m and Θ D 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 (Θ D = 1734 K) at ambient pressure, and it is also smaller than F43m-BN (Θ D = 1896 K), the result of F43m-BN has a high credibility [55]. The longitudinal and transverse sound velocities of Pnma-BN are smaller than Pbca-BN [39] and F43m-BN, because Pnma-BN has the smaller elastic modulus.

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.

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. 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).

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 m F 3 4 -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.   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).

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 F43m-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.