ARTICLE pubs.acs.org/JPCC
Phase Stability and Physical Properties of Manganese Borides: A First-Principles Study Bing Wang, Xiang Li, Yuan Xu Wang,* and Yu Fei Tu Institute for Computational Materials Science, School of Physics and Electronics, Henan University, Kaifeng 475004, People’s Republic of China
bS Supporting Information ABSTRACT: The thermodynamic and mechanical stabilities for the MnB system are investigated using the first-principles calculations method with density functional theory. The negative formation enthalpies of Mn2BAl2Cu (Mn2BAl2Cu represents Mn2B in the Al2Cu structure type, the same hereinafter), MnBCrB, MnBFeB, MnB2ReB2, MnB2AlB2, MnB3TcP3, and MnB4 indicate that they are thermodynamically stable at zero pressure. It is found that MnB2ReB2 is more energetically favorable than synthetic MnB2AlB2, which indicates that experimental synthesized MnB2 is a metastable phase. Among these studied compounds, monoclinic MnB4 has the largest shear modulus, the largest Young’s modulus, and the smallest Poisson’s ratio. The results of density of states and Mulliken overlap population reveal the strong covalent bonding, which results in the high bulk and shear moduli as well as small Poisson’s ratio of MnB2ReB2 and MnB4. An analysis of the elastic constants, elastic moduli, formation enthalpy, electronic structure, and theoretical hardness shows that MnB2ReB2 and MnB4 are potential superhard materials.
I. INTRODUCTION Superhard materials are widely used in various industrial applications, such as cutting tools and hard coating. Great efforts have been devoted to search new types of hard materials. Recently, several transition-metal borides (TMBs, such as ReB2,1 OsB2,2 TaB2,3 and WB44) have been successfully synthesized. They show high bulk and shear moduli. What is more, TMBs can be synthesized under ambient pressure, which leads to the low-cost synthesis condition and is beneficial to their application. This makes TMBs good candidates as hard materials. Knowledge of phase stability and elastic and electronic structure are important for their applications. The borides of manganese are well known for their relatively high melting temperature, hardness, and brittleness.511 In the MnB binary phase diagram, there are five solid state phases: Mn2B (tetragonal structure of the Al2Cu type, No. 140, I4/ mcm),5 MnB (orthorhombic structure of the FeB type, No. 62, Pnma),6 Mn3B4 (orthorhombic structure of the Ta3B4 type, No. 71, Immm),12 MnB2 (hexagonal structure of the AlB2 type, No. 191, P6/mmm),7 and MnB4 (monoclinic structure of the MnB4 type, No. 12, C2/m).13 Fruchart et al.14 reported the MnB2 structure and found a new phase MnB4 formed by decomposition of MnB2 at low temperature: 4MnB2 f Mn3B4 + MnB4. Andersson et al.13,15 found that MnB4 is monoclinic and determined its crystal structure. Cely et al.7 obtained further information on the phase equilibria in the MnB system and measured their microhardness. Recently, a theoretical study revealed that r 2011 American Chemical Society
MnB2 with the ReB2-type structure is a potential superhard material, and its calculated hardness is 43.9 GPa.9 Armstrong16 examined the electronic structure of the first-row transitionmetal diborides and found that while moving from ScB2 to MnB2 the ionic bonding becomes weak. Moreover, electronic structure, bonding, and ground-state stability of MnB2 with the AlB2 structure have also been studied.17 Although the manganese borides have been discussed for decades, their structural stabilities and elastic and electronic properties are still far from being clear. Therefore, it is desirable to investigate the properties of manganese borides with different boron concentration. In this work, first-principles calculations were performed on the structural, elastic, and electronic properties of the MnB system. In order to reveal the relative stability of manganese borides, we calculated the formation enthalpies and drew the convex hulls at 0 and 50 GPa.
II. COMPUTATIONAL DETAIL In the present study, the experimental structures of Mn2B,5 MnB,6 Mn3B4,12 MnB2,7 and MnB413 were adopted. For the other borides, since no experimentally structural parameters have been found to date, we selected some possible structures. For Mn7B3, orthorhombic Mn7C3 (No. 62, Pnma)18 was the considered Received: August 2, 2011 Revised: September 27, 2011 Published: September 28, 2011 21429
dx.doi.org/10.1021/jp2073683 | J. Phys. Chem. C 2011, 115, 21429–21435
The Journal of Physical Chemistry C
ARTICLE
Table 1. Calculated Formation Enthalpy per Unit ΔH, Optimized Equilibrium Lattice Parameters a, b, and c (Å), and Cell Volume Per Formula Unit (V in Å3) of Manganese Borides structure
ΔH
a
Mn7B3
Mn7C3
2.74
4.4188
Mn2B
Al2Cu
1.21
5.0645
b
c
6.9463
12.0201
5.148a 5.0962b
4.1135
26.38
0.56
4.7381
8.8405
27.42
Pt2P
0.19
2.5916
9.5583
27.80 43.26
0.022
2.6185
Mn3N2
2.5725
13.0721
MnB
CrB
0.82
2.8828
15.5279
FeB
0.82
5.5384
5.5600a
2.8428
5.5604c
MnB2
92.24 4.208a 4.1511b
Re2P Mn3B2
Mn3B4
V
2.9770a
4.0913
2.9759c
16.13 4.1514a
16.11
4.1465c
NiAs WC
0.30 0.30
2.6389 2.6399
Ta3B4
2.39
2.9428
3.0320a
ReB2
1.10
2.7785
d
2.7690
6.9542
6.9490d
23.25
AlB2
0.45
3.0549
3.0070e
2.702
3.0370e
21.84
5.6504 2.8130 2.8864
2.9600a
12.8022
3.0089c RuB2
1.03
2.7852
WB2
0.91
2.9379
MnB3
TcP3 RuP3
0.89 0.51
9.4929 4.4481
MnB4
MnB4
1.18
5.4985
5.3689
54.34
5.3668c
22.99
4.5571 7.1972
30.14 32.46
2.9394
2.9511c
36.67
2.9490f
5.3670
5.2134
23.39
12.3004
f
5.5030 1.37
3.8392
2.7868 5.2388 f
WB4
12.860a
3.0384c 4.3744
5.5027c
17.04 16.98
6.0502
35.60
a
Reference 37, experiment. b Reference 44, PBE, CASTEP. c Reference 10, experiment. d Reference 9, PBE method with GGA and USPP method with LDA. e Reference 24, experiment. f Reference 13, experiment.
structure. For Mn2B, except the Al2Cu structure, the other two considered structures were orthorhombic Re2P (No. 62, Pnma)19 and hexagonal Pt2B (No. 194, P63/mmc).20 For MnB, besides the FeB (No. 62, Pnma) structure, other three structures were also considered: CrB (No. 141, I41/amd),21 NiAs (No. 194, P63/mmc),22 and WC (No. 187, P-6m2).23 For Mn3B2, the tetragonal Mn3N2 (No. 139, I4/mmm)24 structure was selected as the initial structure. For MnB2, besides the AlB2 structure, ReB2 (No. 194, P63/mmc),25 RuB2 (No. 59, Pnma),26 and WB2 (No. 194, P63/mmc)27 were also considered. The orthorhombic TcP3 (No. 62, Pnma)28 and the triclinic RuP3 (No. 2, P-1)29 structures were considered for MnB3. Calculations were performed with the projector-augmented wave (PAW) method3032 implemented in the Vienna Ab initio Simulation Package (VASP).3336 The 2s22p1 and 3p63d54s2 were treated as valence electrons for B and Mn, respectively. The generalized gradient approximation (GGA)37 was used to describe the exchange correlation function. Geometry optimization was performed using the conjugate gradient algorithm method with a plane-wave cutoff energy of 500 eV. For the hexagonal structure, Γ centered k mesh was used. The structures were relaxed with respect to both lattice parameters and atomic positions. Formation enthalpy was calculated by ΔH = E(MnmBn) - mE(solid Mn) nE(solid B). The α phase of Mn38 was used in the present calculations. A previous study has found that γ-B28 is dynamically stable and thermodynamically more favorable than any other known forms of boron between 19 and 89 GPa.40 Thus, α-B1239 and γ-B2840 were used at 0 and 50 GPa, respectively. The elastic constants were calculated with the strainstress method. The bulk modulus B and shear modulus G were estimated via the Voigt ReussHill (VRH) approximation.4143
III. RESULTS AND DISCUSSION Structural Features. The calculated lattice parameters, cell volume, and formation enthalpy of manganese borides with different boron concentration are listed in Table 1. The optimized lowest energy structures of the studied manganese borides are shown in Figures 1 and 2 (see Supporting Information for other selected structures). For Mn2B, the Mn2BAl2Cu structure is the most stable of the considered structures. The calculated lattice parameters are in good agreement with the previous theoretical values44 and the experimental values.45 In the Mn2BAl2Cu structure, each B atom is surrounded by eight Mn atoms and there are four B atoms around the Mn atom (Figure 1a). The bond length of the shortest MnB bond (2.17 Å) is slightly larger than the sum of the bonding radii (2.05 Å) of Mn (r = 1.17 Å) and B (r = 0.88 Å) atoms. For MnB (Figure 1b and 1c), the CrB-type structure is slightly lower in energy than the FeB-type one, indicating its higher stability. Of all considered structures of MnB2, the ReB2-type structure has the lowest energy. From Table 1, the calculated lattice constants of MnB2 with the ReB2 structure (Figure 2b) and the AlB2 structure (Figure 2a) are in excellent agreement with previous theoretical values.9,12,24 The MnB4 structure (Figure 2d) is a slightly distorted form of the orthorhombic CrB4 structure.13 A threedimensional network is formed by the boron atoms, and manganese atoms are situated in channels along the c direction in the boron network. The calculated lattice constants of MnB4 (a = 5.4985 Å, b = 5.3689 Å, and c = 2.9394 Å) are very close to the experimental values (a = 5.5027 Å, b = 5.3668 Å, and 2.9511 Å;10 a = 5.503 Å, b = 5.367 Å, c = 2.949 Å13), indicating the reliability of the present calculations. The shortest distance of the MnB 21430
dx.doi.org/10.1021/jp2073683 |J. Phys. Chem. C 2011, 115, 21429–21435
The Journal of Physical Chemistry C
ARTICLE
Figure 1. Crystal structures of manganese borides in MnB systems: (a) Mn2BAl2Cu, No. 140, the B atom is located at 4a (0, 0, 1/4) and the Mn atom is at 8h (0.1591, 0.6591, 0); (b) MnBFeB, No. 62, the B atom is at 4c (0.0342, 1/4, 0.1160) and the Mn atom is 4c (0.1806, 1/4, 0.6177); (c) MnBCrB, No. 141, the B atom is at 8e (0, 1/4, 0.0343) and the Mn atom is at 8e (0, 1/4, 0.1803); (d) Mn3B4Ta3B4, No. 71, the B1 atom is at 4j (1/2, 0, 0.3554), the B2 atom is at 4i (0, 0, 0.4328), the Mn1 atom is at 4j (1/2, 0, 0.1815), and the Mn2 atom is at 2a (0, 0, 0). The black and gray spheres represent Mn and B atoms, respectively.
bond in MnB4, 2.06 Å, is very close to the sum of the bonding radii, 2.05 Å, suggesting its strong covalent bonding feature. The interatomic distance of the shortest BB bond (1.71 Å) in MnB4 is shorter than the shortest BB bond (1.75 Å) in superhard WB4.46 Formation Enthalpy Considerations. It is known that application of the compounds requires the thermodynamic stability of all relevant phases, in particular, the phase stability.4749 The formation enthalpies of these borides were calculated and are summarized in Table 1. The negative values of the formation enthalpies indicate that they are thermodynamically stable and can be experimentally synthesized.50 Among these borides, the Mn7B3Mn7C3 phase has the lowest formation enthalpy, which suggests that it can be easily obtained under ambient condition. All considered MnB2 structures have negative formation enthalpies. For MnB2, the ReB2-type structure has the lowest formation enthalpy. For MnB3, the TcP3-type structure has negative formation enthalpy, which shows that it is thermodynamically stable. The calculated formation enthalpy of MnB4 with space group (C2/m) is also negative. However, the enthalpy value of MnB4WB4 is 1.37 eV, indicating that it is thermodynamically unstable at ambient condition. To understand the stability of these borides, we calculated the convex hull, which is defined as the formation enthalpy vs composition with the ground state at special composition such
that the phase corresponding to each such point is absolutely stable against decomposition into the neighboring phases.51 Any structure whose formation enthalpy lies on the convex hull is deemed stable and synthesizable in principle.52,53 Figure 3 presents the calculated convex hull for the studied MnB system at 0 and 50 GPa. At zero pressure, our calculations yield a ground state convex hull defined by four phases: Mn2BAl2Cu, MnBCrB, MnB2ReB2, and MnB4. Among the four borides, Mn2B Al2Cu and MnB4 have been observed experimentally and MnB2 ReB2 is more stable than MnB2AlB2, though it is not available experimentally. Therefore, the experimental exploration of MnB2ReB2 should be rather rewarding. The convex hull is nearly symmetric, and the lowest formation enthalpy is at a boron composition of 0.5 (MnBCrB). It shows that MnB with the CrB structure may be easier to be synthesized than the FeB structure at zero pressure. In addition, the formation enthalpy of MnB4WB4 lies above the line connecting the values of Mn + B, indicating that it is thermodynamically unstable with respect to the initial reactants.54 The high-pressure convex hull at 50 GPa is little different from that at zero pressure. From Figure 3b, it is shown that Mn2BAl2Cu, MnBFeB, Mn3B4Ta3B4, and MnB4 lie on the convex hull while others lie above the solid line of the convex hull. This figure shows that MnBFeB and 21431
dx.doi.org/10.1021/jp2073683 |J. Phys. Chem. C 2011, 115, 21429–21435
The Journal of Physical Chemistry C
ARTICLE
Figure 2. Crystal structures of manganese borides in MnB systems: (a) MnB2AlB2, No. 191, the B atom is located at 2d (1/3, 2/3, 1/2) and the Mn atom is at 1a (0, 0, 0); (b) MnB2ReB2, No. 194, the B atom is at 4f (1/3, 2/3, 0.5518) and the Mn atom is at 2c (1/3, 2/3, 1/4); (c) MnB3TcP3, No. 62, the B atoms have three sites, B1 at 4c (0.9462, 1/4, 0.01359), B2 at 4c (0.3242, 1/4, 0.1889), B3 at 4c (0.2796, 1/4, 0.5794), and the Mn atom is at 4c (0.1010, 1/4, 0.3350); (d) MnB4, No. 12, the B atom is at 2j (0.2024, 0.3405, 0.2024) and the Mn atom is at 2a (0, 0, 0). The black and gray spheres represent Mn and B atoms, respectively.
us to understand the mechanical properties and also provide very useful information to estimate the hardness of the material. The calculated elastic constants using the strainstress method are listed in Table 2. For comparison, the bulk moduli, shear moduli, and Poisson’s ratio of WB446 and MnB2ReB29 are also listed in the same table. The Young’s modulus Y (GPa) and Possion’s ratio ν were obtained by the following formulas
Figure 3. Convex hulls of the MnB system at a pressure of (a) 0 and (b) 50 GPa. The solid line denotes the ground state convex hull.
Mn3B4Ta3 B4 at 50 GPa are more thermodynamically stable than that at zero pressure, which may imply that they are stable in the course of synthesis. In addition, the negative formation enthalpy of MnB4WB4 at 50 GPa indicates that it is thermodynamically stable under high pressure. For further study, we only consider the thermodynamically stable structures. Elastic Properties. The mechanical stability is a necessary condition for a crystal to exist. Accurate elastic constants can help
Y ¼
9BG ð3B þ GÞ
ð1Þ
ν¼
3B 2G 2ð3B þ GÞ
ð2Þ
Table 2 shows that all studied compounds satisfy the mechanical stability criteria,55 indicating that they are elastically stable. The positive eigenvalues of the elastic constant matrix for each considered compound suggest that they are elastically stable. As is well known, elastic constants represent the ability to resist elastic deformation. From Table 2, it is seen that the values of C11, C22, and C33 for most manganese borides are larger than that of superhard material WB4, suggesting that they are extremely difficult to be compressed along the a axis, b axis, and c axis, respectively. The C44 values of the studied borides are also large, 21432
dx.doi.org/10.1021/jp2073683 |J. Phys. Chem. C 2011, 115, 21429–21435
The Journal of Physical Chemistry C
ARTICLE
Table 2. Calculated Elastic Constants (in GPa), Bulk Modulus B,B0 (in GPa), Shear Modulus G (in GPa), Young’s Modulus Y (in GPa), Poisson’s Ratio (ν), and Vickers Hardness (Hv, in GPa) of the MnB System
a
c11
c12
c13
c22
c33
c44
c55
c66
B0
B
G
Y
ν
B/G
Mn7B3Mn7C3
406
200
235
494
486
142
101
129
290
293
140
362
0.294
2.093
Mn2BAl2Cu MnBFeB
535 456
222 238
216 230
494
494 520
219 234
101
168 151
321 317
319 307
201 176
498 443
0.239 0.259
1.587 1.744
MnBCrB
402
279
230
442
161
253
303
301
144
373
0.294
2.090
2
Mn3B4Ta3B4
466
294
187
332
531
149
193
199
299
297
135
352
0.303
2.200
3.7
MnB2AlB2
510
263
241
272
168
124
288
288
135
350
0.297
2.133
16.5
MnB2ReB2
488
172
100
864
276
158
276
282
256
590
0.152
1.102
40.3
MnB2ReB2a
508
170
99
895
278
169
289
237
559
0.18
1.217
43.9
MnB3TcP3
434
120
208
465
435
215
278
232
253
245
226
519
0.147
1.084
MnB4 WB4b
545 389
63 280
124 224
963
522 437
237 151
244
176 55
273 293
281 297
273 104
619 279
0.133 0.34
1.029 2.86
Hv
1.8 14.9
49.9 41.1
Reference 9, USPP method with LDA. CASTEP. b Reference 46, GGA, VASP.
which is an important parameter indirectly governing the indentation hardness. Among them, MnB2ReB2 has the largest C44 (276 GPa), which is larger than that of superhard ReB2 (257 GPa),56 indicating its relatively strong shear strength. The C44 value of MnB4 is much larger than that of superhard WB4. Superhard materials should have high bulk modulus to resist the volume change caused by the applied load and high shear modulus to against the shape change. As seen from Table 2, all studied compounds have high bulk modulus, which indicates that they are difficult to be compressed. On the basis of the correlation between bulk modulus and valence electron density, we calculated the average valence electron density (VED). It is found that the calculated VED of Mn2BAl2Cu (1.10 e/Å3), MnB FeB (0.99 e/Å3), Mn3B4Ta3B4 (0.94 e/Å3), MnB2ReB2 (0.82 e/Å3), and MnB3TcP3 (0.73 e/Å3) decrease gradually. With increasing of boron composition, the general trend of bulk modulus is the same as that of VED, that is, the bulk modulus follows the order Mn2BAl2Cu > MnBFeB > Mn3B4Ta3B4 > MnB2ReB2 > MnB3TcP3. Therefore, the increasing boron content induces the increasing of VED, and consequently, the bulk modulus decreases with increasing boron content. Mn2B Al2Cu has the largest bulk modulus (319 GPa), which indicates that it is the most incompressible. For MnB, although the CrB-type structure is the most stable phase, its bulk modulus (301 GPa) and shear modulus (144 GPa) are smaller than those of the FeB-type structure. In order to confirm the reliability of the present calculation method, the third-order BirchMurnaghan equation of state is employed to calculate the bulk moduli (B0). The calculated results are also listed in Table 2. As seen in this table, the bulk moduli (B) are in good agreement with the measured B0, demonstrating the reliability of the present theoretical method. As we know, shear modulus provides a much better correlation with hardness than bulk modulus. For the most stable phases at ambient pressure, the shear modulus G and Young’s modulus Y first decrease with an increase in boron composition and then increase. MnB4 has the largest shear modulus (273 GPa), suggesting that it can withstand the shear strain to the largest extent. The shear modulus of MnB4 is also larger than that of superhard WB4,46 suggesting that the hardness of MnB4 is possibly higher than that of WB4. For MnB2, the elastic properties of the ReB2-type structure are superior to that of the AlB2-
Figure 4. Calculated total and partial DOSs of (a)Mn2BAl2Cu, (b)MnBFeB, (c)MnB3TcP3, (d)MnB2AlB2, (e)MnB2ReB2, and (f)MnB4. The Fermi level is at zero.
type one. It is in agreement with the previous report.9 The shear modulus of the ReB2-type structure is nearly two times larger than that of the AlB2-type one. The high shear moduli of MnB2ReB2 and MnB4 indicate their strong ability to restrict deformation. From Table 2, the trend of Poisson’s ratio with the increase of boron composition is opposite for the shear modulus G and Young’s modulus Y. Poisson’s ratio is an important parameter to describe the degree of directionality for the covalent bonding. The small Poisson’s ratios of MnB2ReB2, MnB3 TcP3, and MnB4 imply their strong degree of covalent bonding. The high (low) B/G ratio of a material indicates that it is ductile (brittle), and the critical value is about 1.75.57 The calculated B/G ratios for Mn7B3Mn7C3, MnBCrB, Mn3B4 Ta3B4, and MnB2AlB2 are 2.093, 2.090, 2.200, and 2.133, respectively, which indicates that they are ductile. Other manganese borides have smaller B/G values than the critical value, which suggests that they are brittle. The B/G value of MnB2 ReB2 (1.102) is much smaller than that of MnB2AlB2 (2.133), which shows that MnB2AlB2 is ductile and MnB2ReB2 is brittle. In a word, the large elastic modulus, low B/G ratio, and small Poisson’s ratio show that MnB2ReB2 and MnB4 are potential superhard materials. We also calculated the theoretical Vickers hardness (Hv) of manganese borides on the basis of the theoretical hardness model described in refs 5860, and the results are listed in Table 2. The 21433
dx.doi.org/10.1021/jp2073683 |J. Phys. Chem. C 2011, 115, 21429–21435
The Journal of Physical Chemistry C
Figure 5. Calculated electron density differences contours for MnB2 ReB2 (left) and MnB4 (right). Contours plot on the plane including the shortest MnB and BB bonds. The value of the contours for MnB2 ReB2 is from 0 to 0.13 e/Å3 with an increment of 0.01 e/Å3. The value of the contours for MnB4 is from 0 to 0.6 e/Å3 with an increment of 0.03 e/Å3.
ionicity used in this hardness model was estimated by referring to eqs 22b and 23 in ref 61. Previous work indicated that the hardness of complex compounds can be expressed as an average of hardness of all binary systems in the solid. We applied this rule to calculate their hardness. The calculated Hv of MnB2ReB2 and MnB4 is 40.3 and 49.9 GPa, which are larger than the superhard limit (40 GPa). The calculated hardness of MnB2ReB2 agrees well with the previous result (43.9 GPa).9 From Table 2, it can be seen that the calculated hardness of Mn2BAl2Cu, MnBCrB, and Mn3B4Ta3B4 is small. For the four borides, the high density of the state of the Mn atom at the Fermi level results in their strong metallicity. Consequently, their strong metallicity weaken their theoretical hardness. Electronic Structure Analysis. The electronic structure is crucial to understand the origin of the elastic and physical properties of these borides. The total and partial density of states (DOSs) of Mn2BAl2Cu, MnBFeB, MnB3TcP3, MnB2 AlB2, MnB2ReB2, and MnB4 were calculated at zero pressure and are plotted in Figure 4. As seen in this figure, they are all metallic due to their finite electron DOS at the Fermi level. It is found that the B-s electrons in these compounds are localized, and naturally its effect on bonding is very small. In the vicinity of the Fermi level, the DOSs are mainly composed of the Mn-3d and B-2p states. The typical feature of the total DOSs of Mn2BAl2Cu, MnB3TcP3, and MnB2ReB2 is the presence of what is termed as a pseudogap (a sharp valley around the Fermi level) as shown in Figure 4a, 4c, and 4e, a borderline between the bonding and the antibonding states.17 The presence of a pseudogap will surely increase the stability of the compounds. Two mechanisms were proposed for formation of a pseudogap in the binary alloys: one is of ionic origin, and the other is owing to hybridization effects.17 The electronegativity difference between Mn and B is small, and hence, the ionicity does not play a major role on the bonding behavior of these compounds. Consequently, the present pseudogap is mainly due to strong hybridization between the Mn and the B atoms. From Figure 4, we can see that the hybridization increases with further addition of boron concentration between the Mn-3d and B-2p states. The MnB3TcP3 structure has the lowest value of N(Ef) among all studied borides on the Fermi level, indicating its relatively strong covalent bonding behavior. It is clearly seen that the DOSs of Mn-3d and B-2p have a similar shape in the MnB2 ReB2 phase and the MnB4 phase, which indicates that there is a strong hybridization between the Mn-3d and the B-2p in them. Therefore, strong covalent bonding exists in the MnB2ReB2 structure and MnB4 structure. The strong covalent bonding would be beneficial to their high bulk and shear moduli. The strong covalent
ARTICLE
bonding is also conformed by the calculated average Mulliken overlap population (MOP). A high overlap corresponds to a high degree of covalency on the bond, and a value close to zero indicates little interaction between atoms.62 The calculated MOP of the shortest BB in the MnB2ReB2 structure is 0.75, and the value of the shortest BB bond in MnB4 is 0.80, which are both larger than that of the shortest BB bond in WB4 (0.71). The previous study has shown that the BB network in WB4 is the key for its high hardness,46 so we predict that the strong covalent BB bonds in MnB2ReB2 and MnB4 should be helpful to their hardness. The traditional quantum mechanical description of covalent bonding is characterized by charge build up between atoms. Thus, an intuitive approach to locating covalent bonds is to determine the location of the increased electron density. This approach is pursued here through postprocessing of DFT calculations, which aids in the visualization of electron density reorganization. Electron density difference (EDD) maps have been generated by subtracting the superpositioned, noninteracting, atomic electron densities (procrystal) from the calculated electron density of MnB2ReB2 and MnB4, shown in Figure 5. Charge accumulation between atoms indicates the form of a covalent bond. From this figure it is clear seen that two neighboring B atoms form strong covalent bonds, which indicates that formation of these directional covalent bonds leads to their high hardness. The neighboring B atom and Mn atom also form strong covalent bonds, which is in good agreement with the result of the density of states.
IV. CONCLUSION Using first-principles calculations, the lattice parameters, structural stability, elastic moduli, Young’s moduli, Poisson’s ratios, and theoretical hardness of manganese borides with different boron concentration have been investigated. Considering the convex hulls, Mn2BAl2Cu, MnB-CrB, MnB2ReB2, and MnB4 lie on the constructed ground state at zero pressure. At 50 GPa, the ground state phases are Mn2BAl2Cu, MnB FeB, Mn3B4Ta3B4, MnB2ReB2, and MnB4. As a result, Mn3B4Ta3B4 is more stable at high pressure than at zero pressure. Among these studied compounds, Mn2BAl2Cu has the largest bulk modulus, showing its high incompressibility. Experimentally synthesized MnB2AlB2 is a metastable phase. MnB2ReB2 and MnB4 are both thermodynamically and elastically stable. The calculated Vickers hardness of MnB2ReB2 and MnB4 are 40.3 and 49.9 GPa. Analysis of the electronic structure, Mulliken overlap population, and electron density difference shows that strong covalent bonding exists in MnB2 ReB2 and MnB4. In addition, the larger shear moduli, larger Young’s moduli, lower Poisson’s ratios, and smaller B/G ratios of MnB2ReB2 and MnB4 indicate that they are also potential superhard materials. We hope that these calculations will stimulate extensive experimental work on these technologically important manganese borides. ’ ASSOCIATED CONTENT
bS
Supporting Information. The space group, cell parameters, and atom positions of Mn7B3Mn7C3, Mn2BRe2P, Mn2BPt2B, MnBNiAs, MnBWC, Mn3B2Mn3N2, MnB2 RuB2, and MnB2WB2 in the MnB system.This material is available free of charge via the Internet at http://pubs.acs.org.
21434
dx.doi.org/10.1021/jp2073683 |J. Phys. Chem. C 2011, 115, 21429–21435
The Journal of Physical Chemistry C
’ AUTHOR INFORMATION Corresponding Author
*E-mail:
[email protected].
’ ACKNOWLEDGMENT This research was sponsored by the National Natural Science Foundation of China (No. 21071045), the Program for New Century Excellent Talents in University (No. NCET-10-0132), and the fund of Henan University (No.SBJ090508). ’ REFERENCES (1) Chung, H. Y.; Weinberger, M. B.; Levine, J. B.; Kavner, A.; Yang, J. M.; Tolbert, S. H.; Kaner, R. B. Science 2007, 316, 436–439. (2) Cumberland, R. W.; Weinberger, M. B.; Gilman, J. J.; Clark, S. M.; Tolberks, S. H.; Kamer, R. B. J. Am. Chem. Soc. 2005, 127, 7264–7265. (3) Zhang, X. H.; Hilmas, G. E.; Fahrenholtz, W. G. Math. Lett. 2008, 62, 4251–4253. (4) Gu, Q. F.; Krauss, G.; Steurer, W. Adv. Mater. 2008, 20, 3620–3626. (5) Kadomatsu, H.; Ishii, F.; Fujiwara, H. J. Phys. Soc. Jpn. 1979, 47, 1078–1085. (6) Wong-Ng, W.; McMurdie, H. F.; Paretzkin, B.; Zhang, Y.; Davis, K. L.; Hubbard, C. R.; Dragoo, A. L.; Stewart, J. M. Powder Diffr. 1987, 2, 191–201. (7) Cely, A.; Tergenius, L. E.; Lundstr€om, T. J. Less-Common Met. 1978, 61, 193–198. (8) Carter, G. C.; Swartz, J. C. J. Phys. Chem. Solids 1971, 32, 2415–2421. (9) Aydin, S.; Simsek, M. Phys. Rev. B 2009, 80, 134107. (10) Liao, P. K.; Spear, K. E. Bull. Alloy Phase Diag. 1986, 7, 543–549. (11) Sun, W. H.; Du, Y.; Liu, S. H.; Huang, B. Y.; Jiang, C. J. Phase Equilib. Diff. 2010, 31, 357–364. (12) Ishii, T.; Shimada, M.; Koizumi, M. J. Appl. Phys. 1983, 54, 6907. (13) Andersson, S.; Carlsson, J. O. Acta Chem. Scand. 1970, 24, 1791–1799. (14) Fruchart, B.; Micael, A. C. R. Acad. Sci. 1960, 251, 2953–2954. (15) Andersson, S. Acta Chem. Scand. 1969, 23, 687–688. (16) Armstrong, D. R. Theor. Chim. Acta 1983, 64, 137–152. (17) Vajeeston, P.; Ravindran, P.; Ravi, C.; Asokamani, R. Phys. Rev. B 2001, 63, 045115. (18) Karen, P.; Fjellvag, H.; Kjekshus, A.; Andresen, A. F. Acta Chem. Scand. 1991, 45, 549–557. (19) R€uhl, R.; Fl€orke, U.; Jeitschko, W. J. Solid State Chem. 1984, 53, 55–63. (20) Hassler, E.; Lundstr€om, T.; Tergenius, L. E. J. Less-Common Met. 1979, 67, 567–572. (21) Papesch, G.; Nowotny, H.; Benesovsky, F. Monatsh. Chem. 1973, 104, 933–942. (22) Guerin, R.; Guivarch, A. J. Appl. Phys. 1989, 66, 2122. (23) Krawitz, A. D.; Reichel, D. G.; Hitterman, R. L. J. Am. Ceram. Soc. 1989, 72, 515–517. (24) Kreiner, G.; Jacobs, H. J. Alloys Compd. 1992, 183, 345–362. (25) Telegus, V. S.; Kuzma, Y. B.; Stefanishina, T. K. Sov. Powder Metall. Met. Ceram. 1969, 8, 133–136. (26) Aronsson, B. Acta Chem. Scand. 1963, 17, 2036–2050. (27) Otani, S.; Ohashi, H.; Ishizawa, Y. J. Alloys Compd. 1995, 221, 8–10. (28) Ruhl, R.; Jeitschko, W. Acta Crystallogr., Sect. B 1982, 38, 2784–2788. (29) H€onle, W.; Kremer, R.; Von Schnering, H. G. Z. Kristallogr. 1987, 179, 443–453. (30) Bl€ochl, P. E. Phys. Rev. B 1994, 50, 17953–17979. (31) Kresse, G.; Joubert, D. Phys. Rev. B 1999, 59, 1758–1775. (32) Hohenberg, P.; Kohn, W. Phys. Rev. B 1964, 36, 864–867. (33) Kresse, G.; Hafner, J. Phys. Rev. B 1993, 47, 558–561. (34) Kresse, G.; Hafner, J. Phys. Rev. B 1994, 49, 14251–14269. (35) Kresse, G.; Furthm€uller, J. Comput. Mater. Sci. 1996, 6, 15–50. (36) Kresse, G.; Furthm€uller, J. Phys. Rev. B 1996, 54, 11169–11186.
ARTICLE
(37) Perdew, J. P.; Burke, K.; Ernzerhof, M. Phys. Rev. Lett. 1996, 77, 3865–3868. (38) Lawson, A. C.; Larson, A. C.; Aronson, M. C.; Johnson, S.; Fisk, Z.; Canfield, P. C.; Thompson, J. D.; Von Dreele, R. B. J. Appl. Phys. 1994, 76, 7049. (39) Will, G.; Kiefer, B. Z. Anorg. Allg. Chem. 2001, 627, 2100–2104. (40) Oganov, A. R.; Chen, J. H.; Gatti, C.; Ma., Y. Z.; Ma., Y. M.; Glass, C. W.; Liu, Z. X.; Yu, T.; Kurakevych, O. O.; Solozhenko, V. L. Nature 2009, 457, 863–867. (41) Voigt, W. Lehrburch der Kristallphysik; Teubner: Leipzig, 1928. (42) Reuss, A. Z. Angew. Math. Mech. 1929, 9, 49–58. (43) Hill, R. Proc. Phys. Soc.(London) 1952, 65, 349–354. (44) Zhou, C. T.; Xing, J. D.; Xiao, B.; Feng, J.; Xie, X. J.; Chen, Y. H. Comput. Mater. Sci. 2009, 44, 1056–1064. (45) Kiessling, R. Acta Chem. Scand. 1950, 4, 209–227. (46) Wang, M.; Li, Y. W.; Cui, T.; Ma, Y. M.; Zou, G. T. Appl. Phys. Lett. 2008, 93, 101905. (47) Mishin, Y.; Mehl, M. J.; Papaconstantopoulos, D. A. Acta Mater. 2005, 53, 4029–4041. (48) Ghosh, G.; Delsante, S.; Borzone, G.; Asta, M.; Ferro, R. Acta Mater. 2006, 54, 4977–4997. (49) Ghosh, G.; Asta, M. Acta Mater. 2005, 53, 3225–3252. (50) Shein, I. R.; Ivanovskii, A. L. Phys. Rev. B 2006, 73, 144108. (51) Liu, J. Z.; Zunger, A. Phys. Rev. B 2008, 77, 205201. (52) Ghosh, G.; Van de Walle, A.; Asta, M. Acta Mater. 2008, 56, 3202–3221. (53) Zhang, X. W.; Trimarchi, G.; Zunger, A. Phys. Rev. B 2009, 79, 092102. (54) Zhao, E. J.; Wang, J. P.; Meng, J.; Wu, Z. J. J. Comput. Chem. 2010, 31, 1904–1910. (55) Wu, Z. J.; Zhao, E. J.; Xiang, H. P.; Hao, X. F.; Liu, X. J.; Meng, J. Phys. Rev. B 2007, 76, 054115. (56) Zhang, R. F.; Veprek, S.; Argon, A. S. Appl. Phys. Lett. 2007, 91, 201914. (57) Pugh, S. F. Philos. Mag. 1954, 45, 823–843. (58) Gao, F. M.; He, J. L.; Wu, E. D.; Liu, S. M.; Yu, D. L.; Li, D. C.; Zhang, S. Y.; Tian, Y. J. Phys. Rev. Lett. 2003, 91, 015502. (59) Guo, X. J.; Li, L.; Liu, Z. Y.; Yu, D. L.; He, J. L.; Liu, R. P.; Xu, B.; Tian, Y. J.; Wang, H. T. J. Appl. Phys. 2008, 104, 023503. (60) Gao, F. M. Phys. Rev. B 2006, 73, 132104. (61) Levine, B. F. J. Chem. Phys. 1973, 59, 1463–1486. (62) Segall, M. D.; Shah, R.; Pickard, C. J.; Payne, M. C. Phys. Rev. B 1996, 54, 16317–16320.
21435
dx.doi.org/10.1021/jp2073683 |J. Phys. Chem. C 2011, 115, 21429–21435
