Influence of B Concentration on the Structural Stability and Mechanical

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The Influence of B Concentration on the Structural Stability and Mechanical Properties of Nb-B Compounds Yong Pan, and Yuanhua Lin J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.5b07877 • Publication Date (Web): 18 Sep 2015 Downloaded from http://pubs.acs.org on September 24, 2015

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The Journal of Physical Chemistry

The Influence of B Concentration on the Structural Stability and Mechanical Properties of Nb-B Compounds Yong Pan†, ‡*, Yuanhua Lin†,* †

School of Materials Science and Engineering, Southwest Petroleum University, Chengdu 610500,

China ‡

State Key Laboratory of Advanced Technologies for Comprehensive Utilization of Platinum Metals,

Kunming 650106, China ABSTRACT: Synthesizing and designing novel hard material is still a huge challenge to industrial applications. In this paper, the influence of B concentration on the structural stability, elastic

Nb 2.365Å B 2.340Å B 1.813Å Nb 1.790Å B

properties, hardness and chemical bonding of Nb-B compounds has been investigated by using first-principles approach. Two previously unreported crystal structures of Nb2B and Nb2B3 were predicted. In particular, the hard mechanism of TMBs has been explored in detail. The convex hull has indicated that the NbB with an orthorhombic structure is the most stable structure among those Nb-B compounds. The bulk modulus of Nb-B compounds is related to the B

concentration. However, Nb2B3 shows the highest shear deformation resistance and the strongest elastic stiffness. The calculated theoretical hardness of Nb2B3 is 33.5 GPa, can be considered as a potential superhard material. The high hardness and elastic properties are attributed not only to the B concentration and orientation of bonds, but also to the chemical bonding between layer-layer. Finally, we concluded that the bond strength at layer-layer plays a crucial role for hardness and elastic properties. and investigated intensively. However, metal Os

■ INTRODUCTION The search for novel superhard material is

is a toxic matter. In particular, the price of those

motivated by the need for the development of

5d- TMs is very expensive. Therefore, it is

industrial application. In recent years, transition

necessary to explore new TMBs materials with

metal

low cost and high hardness.

borides

(TMBs)

have

received

considerable attention due to the high hardness,

Recently, the CrB4 with a hardness of about

high elastic modulus, ultra-compressibility and

of 48 GPa is successfully synthesized by using

excellent thermal stability1-7. For those TMBs

arc-melting

materials, the important factor

indicating

that

the

design

potential superhard materials may be formed by

principle is that the transition metal (TM) has

the introduction of boron atom into TM (3d-)

high valence electron density (HVED) such as

with low valence electron density (LVED). This

Os, Re, Ir and Rh et al. It has been reported that

result provides a novel route for the search of

some potential superhard materials such as ReB2

new TMBs superhard material. According to the

8

of

method11,

9

(48 GPa by 0.49 N) , OsB2 (37 GPa) , IrB1.1 (43 GPa)

10

and RhB1.1(44 GPa)

10

have been found

design

principle,

Nb-based

borides

have

attracted attention because of low cost, high 1

ACS Paragon Plus Environment


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hardness, high elastic modulus and excellent 12-13

thermal stability

. For those Nb-B binary

Page 2 of 16

concluded that the chemical bonding and bonding force between layer-layer play a crucial

compounds, early studies have focused on the

role in hardness and elastic properties.

structure and mechanical properties of NbB2. For

■ THEORETICAL METHODS

example, the electronic structure and bonding

According to the Nb-B binary phase

state of NbB2 have been investigated by

diagram, the structures of Nb-B compounds with

Vajeeston and Ravindran14. It is found that the

various stoichiometries can be obtained: Nb3B2

calculated bulk modulus and shear modulus of

with a tetragonal structure (space group:

NbB2 is 299 GPa and 220 GPa, respectively15.

P4/mbm, No: 127)17, NbB with an orthorhombic

However, until the present works there have

structure (space group: Cmcm, No: 63)18, Nb5B6

been no investigations of the relationship

with an orthorhombic structure (space group:

between

mechanical

Cmcm, No: 65)19, Nb3B4 with an orthorhombic

properties. In particular, the trend of structural

structure (space group: Immm, No: 71)20 and

stability and mechanical properties varying with

NbB2 with a hexagonal structure (space group:

the B concentration are unknown.

P6/mmm, No: 191)21. However, other Nb-B

B

concentration

and

On the other hand, the nature of hardness

compounds such as Nb2B and Nb2B3, there are

for TMBs superhard materials remains a

not experimental structural parameters up to now.

challenge because the hardness of a solid is

Considering the elemental feature, the structures

attributed to the resistance to elastic and plastic

of Ta2B-type (space group: I4/mcm, No: 140) for

deformations. From the viewpoint of chemical

Nb2B and V2B3-type (space group: Cmcm, No:

bonding, the three-dimensional network covalent

63)22 for Nb2B3 are considered, respectively.

bonds (such as diamond) effectively resist the formation

and

propagation

of

The first-principles calculated total energy,

dislocation,

elastic properties and electronic structure were

resulting in high hardness. It is concluded that

carried out by using density functional theory

the hardness of TMBs is determined by B

(DFT), as implemented in the CASTEP code23.

concentration. However, the boron-poor borides

To describe the electronic function, ultrasoft

also have high hardness such as IrB1.1 (43 GPa),

pseudopotential24 was carried out and the

RuB1.1 (44 GPa) and Ru2B3 (49 GPa)16, which

electronic configurations of Nb and B atoms

are different from the previous point of view.

were 4p64d45s1 and 2s22p1, respectively. To

Obviously, the nature of hardness is related not

estimate

only to covalent bonding but also to other factors.

correlation functional was treated by the local

Unfortunately, the nature of hardness remains a

density

conundrum.

Ceperley-Alder (CA)25 and the generalized

In the present paper, the formation enthalpy,

the

calculated

approximation

gradient

results, (LDA)

approximation

exchange with

(GGA)

the with

26

elastic constants, elastic modulus, theoretical

Perdew-Burke-Ernzerhof (PBE) . To ensure the

hardness and chemical bonding of Nb-B

total energy at ground state to be converged, a

compounds are systematically investigated by

plane-wave basis set for electron wave function

first-principles density function theory (DFT).

with cutoff energy of 400 eV was used.

The relationship between B concentration and

Integrations

mechanical properties is discussed in detail.

performed by using special k-points generated

Moreover, the nature of hardness is explored. It

with 12×12×12, 8×8×15, 15×15×15, 15×15×15,

is found that the NbB is the most stable structure

15×15×15, 15×15×15 and 18×18×15 mesh grids

among those Nb-B compounds and the Nb2B3 is

for Nb2B, Nb3B2, NbB, Nb5B6, Nb3B4, Nb2B3

a potential superhard material. Finally, it is

and NbB2, respectively. During the structural

in

the

Brillouin

zone

were

2


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optimization, all atoms were fully relaxed using

of NbB is -0.833 eV/atom by LDA and -0.808

conjugate gradient method until the total energy,

eV/atom by GGA, respectively, which is lower

force and atomic displacement were less than

than that of other Nb-B compounds. The convex

1×10

-5

eV/Å, 0.03 eV/Å, and 0.001 Å,

hull indicates that NbB with an orthorhombic

respectively. Elastic constants of a solid were

structure is the most stable structure among

calculated by an efficient strain-stress method,

those

through a linear least square fit of first-principles

predicted

calculated stress vs strain.

possible exists because the calculated formation

■ RESULTS AND DISCUSSION

enthalpy of Nb2B3 is lower than that of NbB2.

Nb-B

compounds.

Nb2B3

with

Importantly,

V2B3-type

the

structure

Thermodynamic stability of a solid is

However, the obtained formation enthalpy of

evaluated by the knowledge of its ground state

Nb2B is much larger than that of other Nb-B

energy,

compounds,

which 27-29

is

defined

as

formation

implying

that

this

boride

is

. At the atomic level, the formation

energetically unstable compared with the other

enthalpy of a material is related to the atomic

Nb-B compounds. On the other hand, the

potential

obtained

enthalpy

between

atoms.

Therefore,

the

formation

enthalpy

of

Nb-B

structural stability is determined by atomic

compounds by LDA is slightly lower than GGA.

position and component in a material. The

It is suggest that this discrepancy is attributed to

negative

the choice of functional.

formation

enthalpy

indicates

the

thermodynamic stability at ground state. The general trend is, the lower the formation enthalpy, the more stability the material. To measure

the

compounds,

structural we

stability

calculated

the

of

Nb-B

formation

enthalpy of Nb-B compounds as a function of B concentration.

The

equation

of

formation

enthalpy is defined as: ∆H ( Nbx B y ) =

1 ( Etotal ( Nbx B y − xE Nb − yEB ) x+ y

(1)

where Etotal(NbxBy), ENb and EB are the total energy of NbxBy compounds, pure Nb atom and B atom at ground state; x and y are the number of Nb atom and B atom for a system, respectively. Figure calculated

1

shows

formation

the

first-principles

enthalpy

of

Nb-B

compounds as a function of B concentration. To measure the calculated results, the formation enthalpies within LDA and GGA are calculated and discussed. From Figure 1, the obtained formation enthalpy of Nb3B4 is -0.828 eV/atom by LDA and -0.806 eV/atom by GGA, respectively, which is in good agreement with the previous theoretical result30. It is worth noticing that the calculated formation enthalpy

Figure 1. The calculated formation enthalpy of Nb-B

compounds

as

a

function

of

B

concentration. It is obvious that the structural stability of a material is controlled by the structural type, chemical bonding and localized hybridization. Essentially, the crystal structure originated from the localized hybridization between atoms, which determines the strength, orientation and the number of chemical bonding. Therefore, we firstly discussed the structural information of Nb-B

compounds

as

a

function

of

B

concentration. The calculated lattice parameters, volume and density of Nb-B compounds within LDA and GGA are listed in Table 1. It can be

3



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Page 4 of 16

Table 1. Calculated lattice parameters, a-axis, b-axis and c-axis (Å), volume, V (Å3) and density, ρ (g/cm3) of Nb-B compounds as a function of B concentration. Phase

Method

Prototype

Space group

a

Nb2B

LDA

Ta2B-type

I4/mcm

GGA Nb3B2

NbB

Nb5B6

Nb3B4

Nb2B3

LDA

ρ

5.707

4.866

158.5

8.24

5.790

4.924

165.1

7.91

6.130

3.251

122.2

8.16

3.297

127.2

7.84

Exp17

6.178

3.280

3.259

8.653

3.132

88.3

7.80

3.300

8.758

3.167

91.5

7.53

Exp18

3.298

8.724

3.166

LDA

Cmcm

8.00

GGA

22.564

3.118

3.266

229.7

7.65

22.816

3.154

3.308

238.1

7.39

Exp19

22.768

3.154

3.299

LDA

Ta5B6-type

3.265

13.946

3.108

141.5

7.56

GGA

3.306

14.106

3.144

146.6

7.29

Exp20

3.305

14.080

3.137

LDA

V3B4-type

Cmcm

7.56

GGA

V2B3-type

Immm

Cmcm

7.32

3.267

19.288

3.093

194.9

7.44

3.308

19.503

3.132

202.1

7.17

3.079

3.279

26.7

7.11

GGA

3.105

3.323

27.7

6.86

Exp21

3.100

3.300

GGA NbB2

V

6.212 CrB-type

P4/mbm

c

GGA LDA

Ta3B2-type

b

LDA

AlB2-type

P6/mmm

seen that the calculated lattice parameters of

calculated bond length of Nb-B and Nb-Nb

Nb3B2, NbB, Nb5B6, Nb3B4 and NbB2 are in

bonds is 2.449 Å and 2.696 Å, respectively,

excellent agreement with the experimental

which are in good agreement with the previous

. The calculated lattice parameter of

theoretical results31. According to the structural

NbB2 along the a-axis is lower than that of

feature, the Nb-Nb metallic bond plays a key

c-axis in contrast to the lattice parameter of other

role in mechanical properties. Therefore, the

data

17-21

Nb-B compounds along the a-axis is larger than

structural stability and mechanical properties of

that of c-axis. This difference gives rise to

Nb2B are mainly determined by Nb-Nb metallic

different degree of localized hybridization along

bonds and part of Nb-B bonds. With increasing

the axis direction, to reveal the origin of the

B concentration, the number of Nb-Nb metallic

hardness. On the other hand, the obtained

bond decreases in contrast to the number of

density of Nb-B compounds decreases with

Nb-B and B-B bonds increases in those Nb-B

increasing B concentration because the density

compounds. Obviously, those Nb-B bond and

of Nb atom is larger than that of B atom.

B-B covalent bond markedly enhance bonding

In order to estimate the structural stability, the optimized structures of Nb-B compounds are

force and deformation resistance, resulting in high elastic modulus and hardness.

represented in Figure 2. For Nb2B, each B atom

For NbB, Nb5B6, Nb3B4 and Nb2B3, the

is surrounded by twelve Nb atoms, which form

alternative stacking of Nb layer and B layer can

the 3D-network Nb-Nb metallic bonds. The

be viewed along the crystallographic axis. The 4


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(a)

(b)

(e)

(c)

(d)

(g)

(f)

Figure 2. Crystal structure of Nb-B compounds. (a) Nb2B, (b) Nb3B2, (c) NbB, (d) Nb5B6, (e) Nb3B4, (f) Nb2B3, (g) NbB2. The blue and pink spheres represent the Nb atom and B atom, respectively. layered structure is constructed by graphite-like

properties of a superhard material. After

Nb layer and B layer with a clear sliding

examining the structural stability, the mechanical

between

properties of Nb-B compounds are discussed and

two

adjacent

layers.

The

slight

discrepancy about layer-layer determines the

analyzed following.

bonding force and structural stability. It can be

knowledge, the mechanical properties such as

seen that the obtained interlayer distance (2.890

bulk modulus (B), shear modulus (G), Young′s

Å) of Nb3B4 is larger than that of other Nb-B

modulus (E) and Poisson′s ratio (δ) are obtained

compounds,

interlayer

by elastic constants (Cij), which are defined as

interaction. It is worth noticing that the B atoms

the stiffness of a crystal against an externally

(in Nb2B3) alternative stacked in a sub-boundary

applied strain32. In addition, the mechanical

layers, forming the strong B-B covalent bond

stability of a solid is measured by elastic

(1.790 Å). The bonding state can effectively

constants. For those Nb-B compounds, there are

improve the structural stability and mechanical

three different structures such as tetragonal

properties. Importantly, the orientation of bond

(Nb2B and Nb3B2), orthorhombic (NbB, Nb5B6,

also plays a key role in mechanical properties

Nb3B4 and Nb2B3) and hexagonal (NbB2)

such as Nb5B6 (Figure 2(d)) and Nb2B3 (Figure

structures. According to the structural feature,

2(f)), determining the variation of elastic

the tetragonal, orthorhombic and hexagonal

indicating

a

weak

To the

best

of

our

modulus and hardness. For NbB2 with AlB2-type

structures have six, nine and five elastic

structure, the structural stability and mechanical

constants,

properties are attributed to the 3D-network Nb-B

assessment of mechanical stability is obtained by

bonds (2.414 Å), which are consistent with the

Wu and Zhao33.

previous theoretical results.

respectively.

Therefore,

the

Table 2 lists the calculated elastic constants

There is no doubt that the high hardness and large compression strength are the unique

of Nb-B compounds within LDA and GGA. It is observed that those Nb-B compounds are 5



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Page 6 of 16

Table 2. Calculated elastic constants, Cij (GPa) of Nb-B compounds as a function of B concentration. Phase

Method

C11

C12

C13

Nb2B

LDA

460

94

GGA

433

83

Nb3B2 NbB Nb5B6 Nb3B4

Nb2B3 NbB2

C22

C23

C33

C44

C55

C66

173

389

163

147

149

361

156

147

LDA

530

117

144

470

192

161

GGA

481

106

128

436

175

148

LDA

518

154

178

566

131

568

203

262

223

GGA

487

136

155

504

122

520

194

246

209

LDA

579

132

168

587

178

512

261

242

211

GGA

533

118

151

552

158

476

243

223

198

LDA

513

177

186

590

136

584

216

267

246

GGA

475

160

162

541

123

551

201

246

232

Theo30

473

157

168

546

126

544

201

242

236

LDA

523

178

177

601

127

624

239

271

256

GGA

480

159

160

536

129

553

221

245

233

LDA

647

117

199

491

245

GGA

587

109

190

428

210

Theo34

601

107

185

439

220

35

576

114

198

430

210

Theo

mechanical stability at ground state because the

main reason is that the large number of B-B

elastic constants of these systems obey the Born

covalent

36

bond

stability criteria . The obtained elastic constants

deformation

of Nb3B4 and NbB2 are in good agreement with

However,

the

reported

Furthermore,

results30,34-35.

theoretical

the

calculated

shear

elastic

effectively

resistance the

along

deformation

enhances the

the

a-axis.

resistance

is

determined not only by B-B covalent bond but also by other factors such as layered structure

constant C44 of Nb5B6 and Nb2B3 is 261 GPa and

and orientation of bonds. For Nb2B, Nb3B2,

239 GPa, respectively, which is larger than the

Nb5B6 and NbB2, the calculated elastic constant

corresponding elastic constant of WB2 (206

C11 is larger than that of elastic constant C33. On

GPa), and is close to the C44 of ReB2 (282

the contrary, the calculated elastic constant C11

GPa)37. Those results mean that those Nb-B

of NbB, Nb3B4 and Nb2B3 is lower than that of

compounds have strong shear deformation

elastic constant C33. Those results indicate that

resistance in comparison of WB2 and ReB2.

the incompressible of NbB, Nb3B4 and Nb2B3

Therefore, it is concluded that they are expected

along the c-axis is stronger than that of a-axis.

to have the high elastic modulus and hardness. It

This discrepancy derived from the structural

should be mentioned that this discrepancy is

feature (see Figure 2). It is suggested that the

attributed

atomic

deformation resistance of a solid is related to the

arrangement (see Figure 2(d) and Figure 2(f)).

applied load along the a-c plane. From Figure 2,

to

the

orientation

of

The discussion will be below in detail.

the chemical bonding in a-c plane for NbB,

As listed in Table 2, the elastic constant C11

Nb3B4 and Nb2B3 is different from the Nb5B6.

increases from 460 GPa to 647 GPa for Nb2B

Therefore, the strong chemical bonding indicates

and NbB2, with increasing B concentration. The

that the layered structure along the b-axis 6


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strengthens the deformation resistance along the

in high bulk modulus.

c-axis (the detailed discussion will be below).

However, the trend of shear modulus of

Although the structural feature of Nb5B6 is

Nb-B compounds is different from the bulk

similar to the NbB, Nb3B4 and Nb2B3, the

modulus. From Figure 3(b), the calculated shear

orientation of layered structure the former is

modulus of Nb-B compound increases with

different from the later. As a result, the above

increasing B concentration when B60.0 at%. The obtained

decreases

with

increasing

B

calculated elastic constant C33 of Nb2B3 is about

shear modulus of Nb2B3 is 235 GPa by LDA,

of 624 GPa, which is larger than that of other

which is larger than that of other Nb-B

Nb-B compounds. The convex hull indicates that

compounds. The convex hull implies that the

the Nb2B3 has the strongest incompressible

Nb2B3 has the strongest shear deformation

(c-axis) among those Nb-B compounds.

resistance in those Nb-B compounds. Therefore,

As mentioned above, the elastic modulus

it is suggested that the variation of shear

and hardness of Nb-B compounds should be

modulus

discussed. According to the structural symmetry,

concentration but also by the chemical bonding

the bulk modulus and shear modulus of Nb-B

such as bonding type and orientation of bond,

compounds are calculated by Voigt-Reuss-Hill

especially for bond strength between layer-layer.

38

is

determined

not

only

by

B

(VRH) approximation method . Therefore, the

According to the structural feature, the cohesive

Young′s modulus and Poisson′s ratio are

force for layered structure is crucial to

obtained through the relationship between bulk

understand the origin of elastic and plastic

39

modulus and shear modulus . Figure 3 shows the calculated bulk modulus and shear modulus of Nb-B compounds as a function of B concentration. It can be seen that the obtained bulk modulus and shear modulus of Nb3B4 and NbB2 are in good agreement with the previous theoretical results30,34. The calculated bulk modulus increases linearly with increasing B concentration. In going from Nb2B to NbB2, the increase in bulk modulus is about of 28.4 %. It is suggested that the increase in bulk modulus is related to the chemical bonding, especially for B-B covalent bond. As we known, the bond strength of B-B covalent bond is much stronger than that of Nb-B bond and Nb-Nb metallic bond. With increasing B concentration, a large number of B-B covalent bonds can be formed in those Nb-B compounds. On the other hand, the increased concentration of B decreases the

Figure 3. The calculated elastic modulus of

number of Nb-Nb metallic bond. Therefore, the

Nb-B

strong B-B covalent bonds obviously enhance

concentration, (a) bulk modulus, (b) shear

the resistance to volume deformation and result

modulus.

compounds

as

a

function

of

B

7



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deformations

in

comparison

with

the

2D-network chemical bonding.

covalent

bond.

However,

the

calculated hardness of Nb2B3 is larger than that

As shown in Figure 3, the calculated bulk modulus and shear modulus of Nb2B3 are larger than that of OsB2 and RuB2

3D-network

Page 8 of 16

40-41

of NbB2 with 3D-network Nb-B bonds. In particular,

the

ideal

diamond-structure

is

. Obviously, the

impossible to exist for all most of TMBs.

Nb2B3 may be expected to have high hardness

Therefore, the nature of hardness is related

among those Nb-B compounds. To evaluate the

determined not only by the 3D-network bonding

theoretical hardness, the hardness of Nb-B

state, but also by other factors.

compounds in this paper is calculated by 42

semi-empirical hard model :

To understand the hard discrepancy, we further calculated the Young′s modulus and Poisson′s ratio of Nb-B compounds as a function

H v = 2 ⋅ (k 2 ⋅ G ) 0.585 − 3

(2)

of B concentration. To our knowledge, the Young′s modulus of a solid indicates the degree

where k represents the B/G ratio and G is the

of elastic stiffness. The general trend is, the

shear modulus, respectively.

higher the Young′s modulus, the stronger the

Figure 4 depicts the calculated theoretical

elastic stiffness. On the other hand, the Poisson′s

hardness of Nb-B compounds as a function of B

ratio indicates the degree of covalent bonding.

concentration. It can be seen that the hard trend

The lower the Poisson′s ratio is, the stronger the

of Nb-B compounds is similar to the shear

covalent bonding is. Figure 5 shows the

modulus. The convex hull from Figure 4

calculated Young′s modulus and Poisson′s ratio

indicates that the Nb2B3 shows the highest

of Nb-B compounds as a function of B

hardness in those Nb-B compounds. It is

concentration. The trends of Young′s modulus

interesting to note that the calculated hardness of

and Poisson′s ratio are consistent with the shear

Nb2B3 is about of 33.5 GPa, which is close to the

modulus and hardness. The calculated Young′s

43

44

OsB2 (35.2 GPa) and WC (35.6 GPa) . That is

modulus of Nb2B3 is 559 GPa by LDA, which is

to say, the TMBs with low valence electronic

larger than that of OsB2 and RuB240. The convex

density also form the potential superhard

hull implies that the Nb2B3 has strongest elastic

materials. Obviously, this result is different from

stiffness in those Nb-B compounds.

45

design principle of TMBs superhard materials .

The calculated Poisson′s ratio of Nb2B3 is only about of 0.190, which is smaller than that of NbB2 (0.203), meaning that the Nb2B3 has strong covalent bonding. According to the structural feature (see Figure 2), the strong B-B covalent bond and Nb-B bond are observed in Nb2B3. However, the 3D-network Nb-B bonds play an important role in NbB2. It is explained why the elastic modulus and hardness of Nb2B3 are larger than that of NbB2. On the other hand, although

Figure 4. The calculated theoretical hardness of Nb-B

compounds

as

a

function

of

B

concentration.

the atomic arrangement of Nb5B6 is similar to the Nb2B3, the orientation of bonds the former is different from the latter. This result is very demonstrated by charge density distribution.

According the structural feature of diamond, the origin of high hardness comes from the

Moreover, the B/G ratio is used to describe the brittleness or ductility of a material within 8


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Figure 6. The calculated B/G ratio of Nb-B compounds as a function of B concentration. The solid line indicates the convex hull. chemical bonding are key factors to deeper understanding the origin of hardness and elastic properties. Following the density of states (DOS) and bond characteristic are calculated and analyzed, here. Considering the structural feature, Figure 5. The calculated Young′s modulus and

we select four typical structures such as Nb2B,

Poisson′s ratio of Nb-B compounds as a function

NbB, Nb2B3 and NbB2. Figure 7 represents the

of B concentration, (a) Young′s modulus, (b)

total and partial density of states of four different

Poisson′s.

Nb-B compounds, and the black vertical dashed of DOS indicates the Fermi level (EF). It is clear

1.75 as the critical value46. If B/G

Influence of B Concentration on the Structural Stability and Mechanical

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