A novel superhard boron nitride polymorph with monoclinic symmetry

Qing-Yang Fan,Chen-Si Li,Ying-Bo Zhao,Yan-Xing Song and Si-Ning Yun

1 College of Information and Control Engineering,Xi’an University of Architecture and Technology,Xi’an 710055,China

2 Shaanxi Key Laboratory of Nano Materials and Technology,Xi’an 710055,China

3 School of Mechanical and Electrical Engineering,Xi’an University of Architecture and Technology,Xi’an 710055,China

4 Key Laboratory of Ministry of Education for Wide Band-Gap Semiconductor Materials and Devices,School of Microelectronics,Xidian University,Xi’an 710071,China

5 Functional Materials Laboratory (FML),School of Materials Science and Engineering,Xi’an University of Architecture and Technology,Xi’an 710055,China

Abstract

Keywords:boron nitride polymorph,superhard material,wider band gap semiconductor,firstprinciples calculations

1.Introduction

More and more materials composed of light elements have been found by researchers [1–31].Among them,carbon allotrope [1–10]and boron nitride polymorph [11–16]are representatives,and BN polycrystalline materials can be used not only as superhard materials but also as two-dimensional magnets [17].Recently,a metastable carbon allotrope with sp3-hybridized,denoted as Penta-C20,was proposed and studied by Zhang et al[1].The Penta-C20structure consists of seven carbon pentagons connected by bridge bonds to form atomic groups.Then,after the combination of atomic groups,the Penta-C20structure with four-membered rings,fivemembered rings,and twelve-membered rings was formed.Penta-C20has a band gap of 2.89 eV and is a direct band semiconductor material;it is slightly smaller than that of Diamond.Fan et al [2]proposed three allotropes of carbon,oP-C16,oP-C20,and oP-C24,respectively.These three kinds of carbon allotropes are built in three-dimensional form by sp2+sp3hybrid bonding networks.These three kinds of carbon allotropes are superhard materials with metallic properties and their ideal shear strengths are greater than those of common metals,such as Al,Fe,and Cu.oP-C16,oP-C20,and oP-C24have excellent physical properties such as superhardness and electrical conductivity,and have great potential for multifunctional devices in extreme conditions,as well as potential materials for electronic devices and mechanical tools.The crystal structure of another superhard material,P2/m C54 [3],was obtained from V-carbon,which is like a reduction of the existing structure of V-carbon into a new lattice system.

As an isoelectronic of carbon,boron nitride also has many polymorphs [11–16,32–36].Since 2013,the 3D metal BN structure has been predicted in three studies [13–15].Unfortunately,the T-B3N3of the four structures proved to be mechanically unstable in subsequent studies [35].Very recently,III-V nitride compounds in the Pm-3n phase were proposed by Zhang et al [36];however,Pm-3n BN is an indirect band gap semiconductor material,and the hardness of Pm-3n BN is only 37.7 GPa according to Chen’s model[37],so the hardness of Pm-3n BN does not meet the minimum requirement of a superhard material.M-BN,an sp2/sp3-hybridized metallic monoclinic 3D BN structure,was proposed by Xiong et al [34].Interestingly,M-BN is a threedimensional material with metallic electronic properties.Its Fermi surface gathers p orbital electrons of boron and nitrogen atoms.Like Pm-3n BN,its hardness is 33.7 and 35.4 GPa using Chen’s model and Gao’s model [38].These hardness values for M-BN do not reach 40 GPa,so it is developed as a hard material.

In this work,the structure of Pm BN is designed and proposed based on C14 carbon [4].The structural properties,electronic properties,stability,mechanical anisotropy properties,and mechanical properties of Pm BN are investigated.

2.Computational detail

Mechanical property prediction calculations and structural optimization were performed using density functional theory with ultrasoft pseudopotentials [39]according to the Cambridge Sequential Total Energy Package [40]code.The Broyden–Fletcher–Goldfarb–Shanno (BFGS) [41]minimization scheme was used to perform the geometric optimization of Pm BN and C14 carbon.The exchange correlation potentials are used with the Perdew–Burke–Ernzerhof functional of the generalized gradient approximation (GGA) [42]and the local density approximation (LDA) [43,44]and are used in the Structural optimization and mechanical property predictions.A high k-point separation of ～0.025 ?×2n was utilized for Pm BN and C14 carbon;the conventional cell of Pm BN is 11×16×4,and for C14 carbon,it is 16×4×11.Additionally,the plane-wave cutoff energy of 500 eV was adopted for structural optimization and mechanical property predictions for Pm BN and C14 carbon.For the phonon spectra of Pm BN,we adopted the density functional perturbation theory approach[45].The Heyd-Scuseria-Ernzerhof(HSE06)hybrid functional[46]was applied to the estimation of the electronic band structures.For the electronic properties of Pm BN,the plane-wave cutoff energy of 910 eV and the norm-conserving pseudopotentials [47]were adopted.

Figure 1.Crystal structure of C14 carbon (a),Pm BN (b) and m-BN (c).

3.Results and discussion

The crystal structure of Pm BN comes from C14 carbon,the new boron nitride polymorph denoted as Pm BN in this work,Pm is the space group.The crystal structures of C14 carbon and Pm BN are displayed in figures 1(a)and(b),respectively.C14 carbon has four inequivalently positioned carbon atoms.After replacing carbon atoms in different positions with boron and nitrogen atoms,the symmetry of Pm BN decreases.The space of Pm BN is Pm,which belongs to monoclinic symmetry.Both boron and nitrogen atoms in Pm BN contain seven inequivalent positions.The seven inequivalent positions of B atoms are 1b(0.08204,0.50000,0.58655),1b (0.40344,0.50000,0.28387),1b (0.10462,0.50000,0.85659),1b (0.38722,0.50000,0.00275),1a (0.90845,1.00000,0.41193),1a (0.58358,1.00000,0.71617),and 1a (0.90562,1.00000,0.14133),respectively;the seven inequivalent positions of N atoms are 1b(0.14344,0.50000,0.41382),1b (0.34558,0.50000,0.71384),1b (0.15164,0.50000,0.14489),1a (0.84703,1.00000,0.58706),1a (0.64518,1.00000,0.28157),1a (0.84763,1.00000,0.85660),and 1a (0.64453,1.00000,0.00302),respectively.The lattice constants of Pm BN,c-BN,and C14 carbon are shown in table 1.Compared to c-BN,the lattice constants within the GGA level are close to the experimental value (3.6200 ?) [67],and according to the above all the predictions in this work regarding Pm BN are based on the GGA level.

Figure 2.Phonon spectra of Pm BN(a)and the enthalpies of Pm-3n BN,Pbca BN,Pnma BN,P42/mnm BN,and Pm BN (b).

The stability of Pm BN was verified using the phonon spectrum,relative enthalpies,and elastic constants.As shown in figure 2(a),the phonon spectrum of Pm BN has no frequencies below zero,indicating that Pm BN is dynamically stable.Pm BN,c-BN,and diamond are adopted sp3hybridizations.The calculated maximum phonon frequency of B–N bond stretching mode in Pm BN is 37 THz,which is very close to the frequency of diamond (40 THz) [48,49],indicating that the B–N bond in Pm BN is strong.The average B–N bond length for Pm BN is 1.5720 ?,thus the average B–N bond length of Pm BN is much closer to c-BN(1.5687 ?)and diamond(1.5445 ?).This is also proof of the bond strength of Pm BN.The related enthalpies of Pm BN,Pm-3n BN [33],Pbca BN [11],and P42/mnm BN (pct-BN) [50],and the enthalpy of c-BN are equal to 0.Compared to the other BN allotropes,the enthalpy of Pm BN is 0.683 eV/f.u.higher than that of c-BN,which is lower than that of T-B3N3(which is higher by 0.947 eV/f.u.than that of c-BN),T-B7N7(which is higher by 0.965 eV/f.u.than that of c-BN),T-B11N11(which is higher by 0.841 eV/f.u.than that of c-BN),Imm2 BN[25],tP24 BN[34],NiAs BN,and rocksalt BN[51].The mechanical stabilities were also studied.The structure of Pm BN belongs to the monoclinic system,which has 13 independent elastic constants.Table 2 lists the main elastic constants of Pm BN,and appendix A shows the whole elastic constants of Pm BN at GGA and LDA levels.The Born stability criteria of monoclinic symmetry for Pm BN are also described in [52–54].Since the elastic constants of Pm BN satisfy the Born stability criteria of monoclinic symmetry,Pm BN is mechanically stable.

Table 1.Calculated lattice constants (?) for Pm BN,C14 carbon,and c-BN.

Table 2.Calculated elastic constants of Cij(GPa),the bulk modulus B(GPa),the shear modulus G(GPa),the Young’s modulus E(GPa)of Pm BN,C14 carbon,Pm-3n BN,Pbca BN,Pnma BN,and c-BN.

The equations for calculating BV,BR,GV,and GRare as follows:BV=(1/9)[C11+C22+C33+2(C12+C13+C23)],GV=(1/15)[C11+C22+C33+3(C44+C55+C66)-(C12+C13+C23)],BR=A[(C33C55-C325)(C11+C22-2C12)+(C23C55-C25C35) (2C12-2C11-C23)+(C13C35-C15C33)(C15-2C25)+(C13C55-C15C35)(2C12+2C23-C13-2C22)+2(C13C25-C15C23)(C25-C15)+m]-1,GR=15{4[(C33C55-C325)(C11+C22+C12)+(C23C55-C25C35) (C11-C12-C23)+(C13C35-C15C33) (C15+C25)+(C13C55-C15C35) (C22-C12-C23-C13)+(C13C25-C15-C15(C12C25-C15C22)+C25(C23C35-C25C33),n=C11C22C33-C11C223-C22C123-C33C122+2C12C13C23,A=2[C15C25(C33C12-C13C23)+C15C35(C22C13-C12C23)+C25C35(C11C23-C12C13)]-[C125(C22C33-C223)+C225(C11C33-C123)+C325(C11C22-C122)+(C11C22C33-C11C223-C22C123-C33C122+2C12C13C23)C55.The G (shear modulus) and B (bulk modulus) are the arithmetic mean of the Voigt–Reuss–Hill approximation [55–57].Compared with Pbca BN,c-BN and C14 carbon,the bulk and shear moduli of Pm BN are slightly lower than that of C14 carbon,c-BN and Pbca BN,except for G of Pm BN and Pbca BN,in which the G of Pm BN is slightly larger than that of Pbca BN.However,the bulk modulus and the shear modulus of Pm BN are larger than those of Pm-3n BN and Pnma BN.The Young’s modulus and Poisson’s ratio are given by E=9BG/(3B+G) [57–60].The distribution of the Young’s modulus of these materials is the same as that of the bulk modulus and shear modulus,and the Young’s modulus of Pm BN is slightly smaller than those of C14 carbon,c-BN,and Pbca BN,while larger than those of Pm-3n BN and Pnma BN.In this work,the hardness of these materials is also investigated using Chen’s model [37].Chen’s model is described as-3,and the hardnesses calculated according to the equations and for several other materials are listed in table 2.Compared to the other materials,the hardness of Pm BN (55.33 GPa) is smaller than that of C14 carbon and c-BN,while it is slightly greater than that of Pbca BN (49.51 GPa).This is because the shear modulus of Pm BN is larger than that of Pbca BN.According to the empirical formula of Chen’s model,it can be determined that Pbca BN,C14 carbon,c-BN,and Pm BN are all superhard materials.

The stress–strain curve is important data for studying the tensile strength of a material,and the shape of the curve responds to various deformation processes such as brittleness,plasticity,yielding,and fracture of the material under the action of external forces.In order to illustrate the deformation of the new BN material (Pm BN) with sp3hybridization,the strain–stress curve was studied and the results are shown in figure 3,from which the stress magnitude of the Pm BN in each direction can be obtained.At 0.18 along the [100]direction,the stress of the Pm BN is about 98 GPa,which is twice to three times larger than that in the hP24 BN,hP18-I BN and hP18-II BN [61].In the direction along [010],the stress of Pm BN is about 78 GPa when the strain is 0.22.It is twice as large as the stress of hP24 BN,hP18-II BN.As for the strain of 0.08 along the [001]direction,the stress of Pm BN is 12.53 GPa,which is larger than that of hP24 BN,hP18-I BN,mP36 BN [61]and hP18-II BN,especially twice as large as that of hP18-I BN,mP36 BN.This indicates that Pm BN can be used for superhard materials in these directions.

Figure 3.The strain–stress curves of Pm BN.

The mechanical anisotropy in Young’s modulus of Pm BN and C14 carbon are investigated in this work.The threedimensional distributions and the two-dimensional representations of Young’s modulus for Pm BN and C14 carbon are shown in figure 4.From figures 4(a) and (c),the threedimensional distribution of Young’s modulus is not a sphere.From other references [62–64]and our understanding of common sense,the three-dimensional distribution of Young’s modulus of isotropic materials is a sphere.Therefore,both Pm BN and C14 carbon exhibit the mechanical anisotropy in Young’s modulus,while the three-dimensional distribution of Young’s modulus of C14 carbon is closer to a sphere,so the mechanical anisotropy of Young’s modulus of C14 carbon is lower than that of Pm BN.From figures 4(b) and (d) we can see the two-dimensional representations of Young’s modulus for Pm BN and C14 carbon,and the two-dimensional representations of Young’s modulus of C14 carbon is closer to circular than that of Pm BN.In order to quantify the magnitude of this mechanical anisotropy,we use the ratio of the maximum value to the minimum value to measure this mechanical anisotropy.We get the maximum and minimum values of the three-dimensional distributions of Young’s modulus of the two materials,and the maximum and minimum values of the Young’s modulus in the three main planes,so as to get the ratio of the maximum and the minimum values.The corresponding results are shown in figure 5.It is worth noting that the ratio of the maximum value to the minimum value of Young’s modulus of Pm BN is the same in the three-dimensional space and in the (010) plane,that is to say,the maximum value (895.66 GPa) and the minimum value (596.85 GPa) of our whole material also appear in the(010)plane.A similar situation also occurs in C14 carbon,the difference is that the maximum(1092.13 GPa)and minimum(853.44 GPa) Young’s modulus are different from Pm BN.By quantifying the ratio of the maximum to the minimum value of Young’s modulus,we also verified the mechanical anisotropy of Young’s modulus of C14 carbon and Pm BN from the three-dimensional distributions and two-dimensional representations of Young’s modulus.For the mechanical anisotropy of Young’s modulus of diamond and graphite,the magnitude of Young’s modulus of Pm BN is larger than that of C14 carbon,and in the same plane,the mechanical anisotropy of Young’s modulus of C14 carbon is smaller than that of Pm BN.In addition,for the three main planes,the(001),(010),and (100) planes,the greatest mechanical anisotropy of Young’s modulus of Pm BN is the(010)plane,while the greatest mechanical anisotropy of C14 carbon is the(001) plane.

Figure 4.The three-dimensional distributions of Young’s modulus for Pm BN (a) and C14 carbon (c),and the two-dimensional representations of Young’s modulus for Pm BN (b) and C14 carbon (d).

Figure 5.The ratio of Emax/Emin for Pm BN and C14 carbon.

Figure 6.The three-dimensional distributions of shear modulus for Pm BN(a)and C14 carbon(c),and the two-dimensional representations of Poisson’s ratio for Pm BN (b) and C14 carbon (d).

In order to better understand the mechanical anisotropy of C14 carbon and Pm BN,we also studied the spatial distribution of shear modulus and Poisson’s ratio for C14 carbon and Pm BN,the related results are shown in figure 6.The light colored surfaces represent the maximum values of shear modulus and Poisson’s ratio for C14 carbon and Pm BN in all directions,while the dark colored surfaces represent the minimum values of shear modulus and Poisson’s ratio.Similar to Young’s modulus,the spherical shape of the shear modulus and Poisson’s ratio represents isotropy,and any deviation from the spherical shape indicates the mechanical anisotropy of the shear modulus and Poisson’s ratio.The maximum and the minimum values in all directions and in(001) plane,(010) plane,(100) plane of shear modulus are shown in figure 7(a),it is clear that of the maximum and the minimum values of C14 carbon in all directions and in(001)plane,(010) plane,(100) plane of shear modulus is greater than that of Pm BN,respectively.By quantifying the ratio of the maximum to the minimum value of shear modulus and Poisson’s ratio,we can compare the anisotropy between C14 carbon and Pm BN.TheGmax/Gminratio andvmax/vminratio of C14 carbon and Pm BN are shown in figures 7(b) and (c),respectively.The mechanical anisotropy of shear modulus of C14 carbon and Pm BN in (001) plane,(010) plane,(100)plane are slightly smaller than that of all directions,while the mechanical anisotropy of Poisson’s ratio of C14 carbon and Pm BN in (100) plane are slightly smaller than that of all directions.Compared with the structure of space group which is also Pm,the mechanical anisotropy of shear modulus for Pm BN in (001) plane (Gmax/Gmin=1.481),(010) plane(Gmax/Gmin=1.481),(100)plane(Gmax/Gmin=1.551),and all directions (Gmax/Gmin=1.566) are slightly smaller than that of m-BN [65].

Figure 7.The maximum value and the minimum value of shear modulus of BN and C14 carbon(a),Gmax/Gmin ratio for Pm BN and C14 carbon(b),and vmax/vmin ratio for Pm BN and C14 carbon(c).

Figure 8.The electronic band structures of Pm BN.

The electronic band structures of Pm BN are shown in figure 8(a).From figure 8(a),the band gap of Pm BN obtained by the HSE06 hybrid functional is 4.54 eV,and it is close to the band gap of C14 carbon(4.60 eV).It can be seen that the valence band top(VBM)and conduction band bottom(CBM)of Pm BN are located at point Z and point C,while both the VBM and CBM located at point X,suggesting that Pm BN and C14 carbon are an indirect band gap semiconductor and a direct band gap semiconductor,respectively.Compared with m-BN whose space group is also Pm,the band gap of Pm BN is slightly smaller than that of m-BN (4.629 eV) [66].In addition,both the band gaps of Pm BN and C14 carbon are smaller than that of diamond (5.45 eV) within the HSE06 hybrid functional.The Brillouin zone and the high symmetry points are shown in figure 8(b).The coordinate of high symmetry points in the Brillouin zone of Pm BN are Z (0.0,0.0,0.5),G(0.0,0.0,0.0),Y(0.0,0.5,0.0),A(0.5,0.5,0.0),B (0.5,0.0,0.0),D (0.5,0.0,0.5),E (0.5,0.5,0.5),and C(0.0,0.5,0.5),respectively.

4.Conclusion

On the basis of density functional theory,we designed a wider band gap and superhard semiconductor material,named Pm BN after the space group.Pm BN is dynamically stable and mechanically stable.The average B–N bond length(1.5720 ?) of Pm BN is close to c-BN (1.5687 ?) and diamond(1.5445 ?),indicating a relatively strong B–N bonding in Pm BN,as shown in the phonon spectra.The shear modulus,bulk modulus,and Young’s modulus of Pm BN are 327 GPa,331 GPa,and 738 GPa,respectively,and all the elastic moduli of Pm BN are greater than that of Pnma BN and Pm-3n BN.In addition,according to Chen’s model,it can be determined that Pm BN is a superhard material.According to the three-dimensional distributions and the two-dimensional representations of Young’s modulus for Pm BN and C14 carbon,the mechanical anisotropy of Young’s modulus of Pm BN is larger than that of C14 carbon via the ratio of the maximum value to the minimum value.In the three main planes of the two materials,the plane where Young’s modulus shows the maximum anisotropy is different:for C14 carbon,it is the (001) plane,and for Pm BN,it is the (010)plane.

Acknowledgments

This study was supported by the National Natural Science Foundation of China (Grant No.61804120);China Postdoctoral Science Foundation (Nos.2019TQ0243,2019M663646);Natural Science Basic Research Program of Shaanxi (2021JQ-515);Key scientific research plan of Education Department of Shaanxi Provincial Government (Key Laboratory Project) (No.20JS066);Young Talent fund of University Association for Science and Technology in Shaanxi,China(No.20 190 110);National Key Research and Development Program of China (No.2018YFB1502902);and Key Program for International S&T Cooperation Projects of Shaanxi Province (No.2019KWZ-03).