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First principles study on the phase stability and mechanical properties of MoSi2 alloyed with Al, Mg and Ge
Intermetallics 67 (2015) 26e34
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First principles study on the phase stability and mechanical properties of MoSi2 alloyed with Al, Mg and Ge* Hai Hu a, Xiaozhi Wu a, b, *, Rui Wang a, Weiguo Li c, Qing Liu b a
Institute for Structure and Function, Chongqing University, 401331, China College of Materials Science and Engineering, Chongqing University, 400044, China c College of Aerospace Engineering, Chongqing University, 400044, China b
a r t i c l e i n f o
a b s t r a c t
Article history: Received 23 April 2015 Received in revised form 25 June 2015 Accepted 13 July 2015 Available online xxx
The phase stability, mechanical properties and electronic structure of C11b and C40 MoSi2 with alloying elements Al, Mg and Ge were systematically investigated using first principles methods. The calculated lattice constants and elastic constants of C11b and C40 MoSi2 are in good agreement with the previous results. It is found that there is a phase transition from C11b to C40 when the concentrations of Al and Mg reach ~7 at.% and ~6 at.%, respectively. Based on the elastic constants, the anisotropy, ductility, hardness and melting temperatures are presented for MoSi2 with alloying elements. For C11b, the ductility will be enhanced by increasing the concentrations of Al or Mg. Simultaneously, hardness will be reduced by the increasing of Al or Mg. Ge have a reverse effects. For C40, the ductility is reduced weakly by Al or Mg. In addition, the effects of substitution of Mo by Nb with Si substitution of Si by Al, Mg and Ge are also investigated. Nb and Mg codoping can improve the ductility of MoSi2. Finally, the density of states is used to analysis the effects of alloying elements on the mechanical properties, and the results are in consistent with the predictions based on elastic constants. © 2015 Elsevier Ltd. All rights reserved.
Keywords: Molybdenum silicides Elastic properties Mechanical properties Theory Ab initio calculations
1. Introduction Molybdenum disilicide (MoSi2) has attracted much attention due to its high melting point, superior high-temperature stability and low density [1e4]. There are two known polymorphs of MoSi2, the stable C11b structure and the metastable C40 structure (see Fig. 1). Due to the mixed covalent and metallic bonding, MoSi2 displays its specific dual characteristics of ceramic and metal. But the inherent brittleness limit its technological assignment [5,6]. Therefore, it is considered to be one of the most suitable candidate for high temperature structural applications if the ductility can be improved. Generally, alloying by chemical substitutions for Mo and Si is regarded as an effective strategy to address this deficiency [1,7]. Sharif et al. have suggested that the hardness of MoSi2 decreases
* The work is supported by the Natural Science Foundation of China (11104361) and Projects supported by the Fundamental Research Funds for the Central Universities (CDJZR14328801 and CQDXWL2014003). * Corresponding author. Institute for Structure and Function, Chongqing University, 401331, China. E-mail addresses: [email protected] (X. Wu), [email protected] (Q. Liu).
http://dx.doi.org/10.1016/j.intermet.2015.07.008 0966-9795/© 2015 Elsevier Ltd. All rights reserved.
obviously even at room temperature when doping with Al [8]. Under 773 K, Al doping (1.5 wt.% ~ 3.0 wt.%) can also reduce the micro-hardness of MoSi2 [9]. The effects of substitutional alloying on the ductility of MoSi2 are also studied using first principles methods based on fracture mechanics theory [10e14]. It is found that substitution of Mo by V or Nb, and substitution of Si by Al or Mg are particularly beneficial to the enhancement of ductility. Al is one of the earliest and the most efficient alloying elements proved by experiment and theory research [1,15]. The alloying elements can weaken the strong covalent bonds of SieSi and MoeSi in MoSi2 crystal, while the ductility is improved with the increase of Al content. Experimental results show that the structure of MoSi2 will be changed from C11b to C40 with the increasing of doping concentrations. The compositional phase analysis indicates that the solid solubility limit of Al in C11b phase is 3 at.%, and the experiment reported that a biphasic mixture of C11b and C40 phases will coexistence for higher concentrations, however, when the doping concentration ups to about 10 at.% there is only C40 phase [16]. Besides, Sharifet al. suggest that the Nb alloyed MoSi2 could be deformed in compression even at room temperature [17]. Instead of a single element alloying, two or more elements codoping MoSi2 is also the common way to improve the ductility. Dasgupta proved
H. Hu et al. / Intermetallics 67 (2015) 26e34
27
Fig. 1. The structure cell of (a) C11b type and (b) C40 type for MoSi2 crystal.
that Nb and Al codoping is an effective way to overcome the low temperature brittleness [18]. In this paper, we will focus on the mechanical properties and electronic structures of MoSi2 with substitution of Si by Al, Mg and Ge, and Nb and Al codoping. The phase transformation between C11b and C40 are investigated based on different doping concentrations for Al, Mg and Ge. The article is organized as follows: Section 2 covers the computational details based on the firstprinciples quantum mechanical method. The structure stability and phase transition were studied in Section 3. In Section 4, some important mechanical behaviors such as ductileebrittle properties, hardness, melting temperatures are investigated systematically. The electronic mechanism behind the effects of the doped elements on MoSi2 was presented in Section 5. Finally, some conclusions are given in last section. 2. Computational details In this paper, the various concentrations of alloying elements Al, Mg and Ge and Nb codoping with Al, Mg and Ge on the mechanical
properties are systematically studied. Based on C11b and C40 structure, the supercells are built (see Table 1), and the alloying elements are injected to substitute Si atoms in these supercells to build 2.08 at.%, 4.17 at.%, 8.33 at.%, 16.67 at.% and 33.33 at.% doping concentration, respectively. The selection of the supercell models are given in Table 1. The present calculations were performed within the density functional theory framework based on VASP code. A plane-wavepseudo-potential (PAW) method is used [19,20]. The generalized gradient approximation (GGA) was adopted with Perdew-BurkeErnzerh (PBE) exchange correlation functional [21]. A convergence tests guided the choice of 450 eV for the cutoff energy. A 17 17 17 Monkhorst-Pack grid of k-points is adopted for sampling the Brillouin zone, and the proper grids are applied on corresponding supercells. The total energy is converged to within 1.0 105 eV/atom and the force components on each atom are set to 1.0 103 eV/Å. The elastic constants were calculated by the ’stress-strain’ method and the spin polarization was adopted for the calculation of electronic density of states (DOS).
Table 1 Lattice parameters a and c (in parentheses) for MoSi2 with various doping concentrations of Al, Mg and Ge (in units of Å).
C11b Mo(Si1x Alx)2 Mo(Si1x Mgx)2 Mo(Si1x Gex)2
0
2.08 at%
4.17 at%
8.33 at%
16.67 at%
33.33 at%
111 3.220 7.889 3.202a, 3.215b 7.851a, 7.866b 3.2047c, 3.2056c 7.8498c, 7.8450c
222 3.228 7.921 3.221 7.928 3.223 7.905
221 3.221 7.927 3.228 7.979 3.227 7.901
1 1 2 3.228 7.985 3.238 8.034 3.230 7.933
111 3.243 8.091 3.274 8.213 3.242 7.967
111 3.278 8.295 3.319 8.712 3.265 8.049
C40
111
441
222
221
112
111
Mo(Si1x Alx)2
4.615 6.623 4.5321c, 4.6016d 6.7981c, 6.5700d 4.605e, 4.622f 6.559e, 6.646f
4.625 6.622 4.636 6.625 4.624 6.636
4.636 6.618 4.658 6.621 4.633 6.650
4.656 6.616 4.678 6.623 4.650 6.641
4.692 6.640 4.756 6.681 4.681 6.662
4.759 6.710 4.919 6.898 4.737 6.707
Mo(Si1x Mgx)2 Mo(Si1x Gex)2 a b c d e f
See See See See See See
[22]. [23]. [24] and references therein. [25]. [26]. [27].
28
H. Hu et al. / Intermetallics 67 (2015) 26e34
when the doped Al ups to about ~10 at.%. This means that our theoretical results are reasonable.
3. Heat of formation and phase stability As is well known, MoSi2 has more than one crystal structure, among them C11b is the prevailing structure. Furthermore, with the doping increasing, it can be transformed into the C40 type. In this part, we will focus on the phase transition between C11b and C40 type with different concentrations of alloying elements Al, Mg and Ge according to the formation energy. The supercells used for calculations are listed in Table 1. Si in the supercells of MoSi2 are substituted by Al, Mg and Ge, respectively. The different supercell means the different concentrations. The lattice constants of the C11b and C40 MoSi2 for various concentrations are also listed in Table 1. MoSi2 alloyed with Al, Mg and Ge can be represented by Mo(Si1x Mx)2, in which x is the molar ratio of M (M ¼ Al, Mg, Ge) to Si of the compound. The heat of formation was calculated because it can directly reflect the structure stability. The heat of formation is defined as
DE ¼ E MoðSi1x Mx Þ2 EðMoÞ 2ð1 xÞEðSiÞ 2xEðMÞ
(1)
where E[Mo(Si1xMx)2] is the total energy of Mo(Si1x Mx)2 crystal at the equilibrium lattice, E(Mo), E(Si) and E(M) are the total energies of Mo, Si and M the solid phase, respectively. The calculated heat of formation as well as previous theoretical and experimental values for pure C11b and C40 MoSi2 are listed in Table 2. Both of them have negative formation energy. This suggests that C11b and C40 MoSi2 are thermodynamically stable. But the heat of formation of C11b are lower than C40. This means C11b are more stable structure than C40. The simulation results show that the heat of formation, which was calculated by different methods produced different results, especially for C11b type. The result of our calculation for C11b type is 1.49 eV/f.u., which is in good agreement with previous results, GGA-PBE (1.53 eV/f.u.), GGA-PW92 (1.38 eV/f.u.) or LAPW (1.45 eV/f.u.) data, while is about 25% larger than that LMTO-FP value (1.87 eV/f.u.). These deviations could be caused by different pseudopotential selection, k-points or cutoff energy. The heat of formation with variation impurity concentration is shown in Fig. 2. It is easy to see that with the increment of impurities, the heat of formation will tend to positive values, quickly. This suggests that the structural stability is getting poorer. It is shown that there will be a structure phase transition from C11b type to C40 type with increasing doping concentration up to 33.3 at.% except Ge doping. It is worth mentioning that when the doped Mg ups to ~20 at.% the Mo(SixMg1x)2 will not be stable exist. It is easy to see that the phase transition point for Mg is ~6 at.%, and for Al is ~7 at.% which is good agreement with Dasgupta's experiment [16]. As suggested by Dasgupta's experiment, there is only C40 phase Table 2 The formation energies for C11b and C40 MoSi2. MoSi2 C11b
C40
This work GGA-PBE [28] GGA-PW92 [28] LMTO-FP [6] LAPW [27] Experiments [31] This work GGA-PW92 [28] LAPW [27]
DE (eV/f.u.)
DE (kJ mol1)
1.49 1.53 1.38 1.87 1.45
143.68 148.08 133.05 180.42 139.90 135.80 ± 4.5 127.54 126.78 129.29
1.32 1.31 1.34
The abbreviation of calculation method: GGA, generalized gradient approximation; PBE, Perdew-Burke-Ernzerhof functional; PW92, Perdew-Wang 92 functional; LMTO-FP, linear muffin-tin-orbitals first-principles; NC-PP, norm-conserving pseudopotential; LAPW, linear augmented-plane-wave.
4. Mechanical properties Elastic behavior is one of the most basic properties to reflect the essential attribute of materials. In this part, the elastic behaviors of C11b and C40 MoSi2 and the extension of the other mechanical properties with different concentrations will be investigated systematically. Generally, there are 21 non-zero independent elastic stiffness tensors. And if a crystal has a higher symmetry, it can be further reduced. For a C11b or C40 type MoSi2, there are six independent elastic constants (C11, C12, C13, C33, C44, C66). Additional energy will be gained by the deformation of the lattice. The elastic constants C11b and C40 MoSi2 are listed in Table 3. The previous and theoretical results are also presented for comparison. The results for different concentrations of Al, Mg and Ge are given in Fig. 3. The phase transformation point is marked by the dashed line. From Fig. 3, we can see clearly that the elastic constants have similar varying tendency with the increase of impurity concentrations. But there is no much significant effect for Ge doping. Besides the thermodynamical viewpoint above, the mechanical stability, which reflects the specific deformation by the applied stress into a particular direction. As early as 1954, systematic and embedded research has been done in this dissertation by Born and Huang [33], and a stability criterion has been presented in terms of the elastic constants. Different crystal system correspond to different elastic stiffness constants. There is a mechanical stability criterion [34] for C11b type, 2 C44 > 0; C66 > 0; C11 > jC12 j; ðC11 þ C12 ÞC33 > 2C13 ;
(2)
and for C40 type, the criterion reads as 2 C11 > 0; C44 > 0; C11 > jC12 j; ðC11 þ C12 ÞC33 > 2C13 :
(3)
The calculated elastic constants meet all these criterias, which implies that the two type are mechanically stable. Unfortunately, there are no experimental data to compare for C40 type. The calculation results show that C33 is larger than C11, which suggests stronger directional property exists in MoSi2 crystal. It can be seen that C33 is mainly dominated by the bonds parallel to c-axis, while C11 is dominated by the bonds parallel to a-axis from the lattice structure. This suggests that the structure stiffer along c-axis. There are two MoeSi and one SieSi bonds parallel to c-axis in C11b primitive cell, which indicates the parallel MoeSi bonds is the key factor impacting the hardness of MoSi2. However, there is no parallel bonds in C40 primitive cell, that is the reason why the elastic constant C33 of C40 MoSi2 is much smaller than C11b. So we could get that the C11b type has much higher hardness than C40 type MoSi2. Overall, the calculation results show that the larger shear elastic constants for C44 and C66 however smaller data for C12 and C13 suggest remarkable brittleness of MoSi2. From Fig. 3, we can see that C33 decreases obviously with the increasing of the concentrations of Al and Mg. But Ge does not have transparent effects on C33. This means that the Al or Mg can reduce the hardness of MoSi2. Nonetheless, C11b type should have higher hardness and brittleness than C40 type. Based on the calculated Cij, the bulk mechanical properties, such as bulk modulus (B), shear modulus (G), Young's modulus (Y) and Poisson's ratio (n), can be evaluated using the Voigt-Reuss-Hill (VRH) scheme [35]. The theoretical upper bounds Voigt approximation which assuming uniform strain throughout a polycrystal are given as follows
H. Hu et al. / Intermetallics 67 (2015) 26e34
29
Fig. 2. The hear of formation with various impurity concentration. It is clear to see that there will be a structure phase transition from body centered tetragonal C11b type to a hexagonal C40 type with increasing doping concentration up to 33.3 at.% except Ge doping. It is worth mentioning that when the doped Mg ups to about 20 at.% the Mo(SixMg1x)2 will not be stable exist.
Table 3 Elastic constants for C11b and C40 MoSi2 (in units of GPa). MoSi2
Cij
C11
C12
C13
C33
C44
C66
C11b
This work GGA-PBE [23] GGA-PW92 [28] GGA-PWP [29] GGA-PWP [30] Exp [32]. This work GGA-PW92 [28]
393.6 407.8 384.63 393.0 392.32 417.0 401.3 394.4
106.9 108.9 99.96 117.0 104.19 104.2 77.1 68.2
89.3 92.2 85.88 87.0 91.33 83.8 126.3 120.2
493.4 505.7 473.78 505.0 482.44 514.5 418.2 403.8
192.3 196.8 186.95 194.0 188.44 204.2 151.1 145.1
182.5 192.8 178.55 178.0 178.10 193.6 166.3 163.1
C40
BV ¼
1 ð2C11 þ 2C12 þ 4C13 þ C33 Þ 9
(4)
GV ¼
1 ð2C11 C12 2C13 þ C33 þ 6C44 þ 3C66 Þ 15
(5)
and the theoretical lower bounds Reuss approximation which assuming uniform stress are written as
BR ¼ ½S11 þ S22 þ S33 þ 2ðS12 þ S13 þ S23 1 GR ¼ ½4ðS11 þ S22 þ S33 Þ þ 4ðS12 þ S13 þ S23 Þ þ 3ðS44 þ S55 þ S66 Þ1
(6)
(7)
Where Sij are elastic compliances and their values can be given by the inversion of elastic constants matrix (Sij ¼ Cij1 ). We estimated these bulk values by the Voigt-Reuss-Hill approximation:
9BG ; 3B þ G
(10)
n¼
3B 2G : 6B þ 2G
(11)
4.1. Elastic anisotropy Because of the low symmetry of C11b and C40 lattice and hence it is necessary to study elastic anisotropy properties to looking for effective ways to improve material properties. Through the comparison of the previous elastic constants, it is also found that strong elastic orientation of MoSi2 crystal. Based on these characteristics, a percentage anisotropy in compressibility and shear [36] are proposed as
AB ¼
ðBV BR Þ ; ðBV þ BR Þ
(12)
AG ¼
ðGV GR Þ : ðGV þ GR Þ
(13)
GV (BV) and GR (BR) are Voigt and Reuss shear (bulk) modules, respectively. A minimum value of zero corresponds to isotropic materials and a maximum 1 represents the greatest possible anisotropy. In order to quantitatively describe the anisotropy of single crystal, a universal index is used [37].
AU ¼ 5
1 B ¼ ðBR þ BV Þ; 2
(8)
1 ðG þ GV Þ; 2 R
(9)
G¼
Y¼
GV BV þ 6 0: GR BR
(14)
The isotropic single crystal corresponds to minimum value zero, and the more deviates from the value of zero, the stronger anisotropy. The results are given in Table 4 and the anisotropy with various doping concentration are given in Fig 3. It is quite clear that they are all obvious anisotropy, and C11b type exhibits more
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H. Hu et al. / Intermetallics 67 (2015) 26e34
Fig. 3. The calculated elastic constants for C11b type and C40 type of Mo(SixX1x)2 (X ¼ Al, Mg, Ge) with various doping concentration are presented in (a), (b) and (c). Calculated percentage anisotropy in shear AG, compressibility AB and the universal anisotropy AU are presented in (d).
obviously. Simultaneously, growth of different degrees was seen for AG, AB and AU with the increase of Al or Mg doping in C11b type, but little influence on C40 type, except for AB. But, it has little effect on them with Ge doped in all range of doping concentration. Notwithstanding, there is a small fluctuation of AB with Ge doping. On the whole, the anisotropy is enhanced to some extent with Al and Mg doping for C11b, however little impact for Ge. This could be because of the similar electronic structure between Si and Ge. As we know, Nb or Nb-M codoping MoSi2 is also a practical way to improve material properties. The effects of material physical properties due to Nb or Nb-M codoping with 4.17 at.% was discussed too. It is not difficult to find that large influence on the shear
anisotropy due to the addition of Nb atoms, by the comparison of Table 5 and Fig. 3(d). 4.2. Ductility and hardness Precise machining for MoSi2 is always in difficulty and also a research hot spot due to its intrinsic brittleness, which has greatly hindered the development in industry. The essence of brittleness is that the brittleness crack occurred in materials before dislocation emission under deformations [38]. But for its complexity, to solve this problem has always been the object of the researchers. We may recall from our earlier analysis of elastic constants that
Table 4 Elastic anisotropy factors (A), Poisson's ratio n, Pugh ratio G/B, Vichers hardness Hv (GPa) and melting point Tm (K) for C11b and C40 MoSi2. MoSi2 C11b
C40
This work GGA-PBE [23] GGA-PW92 [28] Exp [32]. This work GGA-PW92 [28]
AB
AG
AU
n
G/B
Hv
Tm
0.324 0.299 0.291 0.259 0.288 0.265
0.686 0.625 0.736 0.574 0.329 0.392
0.076 0.069 0.080 0.063 0.039 0.045
0.163 0.162 0.159 0.151 0.206 0.202
0.871 0.874 0.882 0.911 0.733 0.743
32.304 33.161 32.368 35.756 23.321 23.387
2274.9 2336.0 2218.6 2376.8 2185.2 2142.9
H. Hu et al. / Intermetallics 67 (2015) 26e34
31
Table 5 Elastic anisotropy factors (A), Poisson's ratio n, Pugh ratio G/B, Vichers hardness Hv (GPa) and melting point Tm (K) for Nb-M (M ¼ Al, Mg, Ge) codoping. 4.17 at%
AB
AG
AU
n
G/B
Hv
Tm
Mo0.875 Mo0.938 Mo0.938 Mo0.938
0.306 0.360 0.320 0.205
2.164 9.268 2.762 0.664
0.227 1.029 0.291 0.071
0.172 0.161 0.498 0.165
0.838 0.877 0.809 0.863
30.081 32.356 27.781 31.847
2179.1 2099.1 2094.1 2267.8
Nb0.125 Si2 Nb0.062(Si0.969 Al0.031)2 Nb0.062(Si0.969 Mg0.031)2 Nb0.062(Si0.969 Ge0.031)2
the ductile or brittle should be related to shear and bulk properties. Based on this consideration, an empirical Pugh [39] criterion is used to give a phenomenological analysis. According to the Pugh's criterion, the ratio of shear modulus to bulk modulus presents a measurement for ductile-brittle behavior. A high G/B value is associated with brittleness, whereas a low value corresponds ductility. The critical value that separates ductile and brittle behaviors is around 0.5. And, the higher the ratio, the more brittleness. Besides, good ductile materials correspond to the larger Poisson's ratio (n), which provides insight into the change of bond angle and bond length [40]. Furthermore, the large Young's modulus is due to stronger directional boding betweens atoms. The results are given in Tables 4 and 5 and Fig. 4. The calculated results show that the Pugh's ratio is always more than 0.5, this indicates intrinsic brittleness of C11b and C40 type. In addition, larger Pugh's ratio for C11b type suggests stronger brittleness than C40 type. Similar results show that there are downward trends for G/B but upward trends for n as the increasing doping concentration Al and Mg for C11b type, however small fluctuations for Ge doping except in 4.17 at.% doping concentration. Although a weak reduction of ductility with additional elements for C40 type, the influence is little. Which suggests that Al and Mg can enhance the difficulty for C11b type. Overall, the ductility of C11b MoSi2 can be improved by Al and Mg. This is in good agreement with previous experimental and theoretical results [4,12,28]. And the same methods have been used to investigate the Nb doping and Nb-M codoping. The results show that both of them can effectively
improve the ductility of MoSi2. But NbeAl and NbeGe codoping have weak effect. Hardness is one of the most basic mechanical properties of the material, which represents the ability of solids resistance to elastic and plastic deformation or brittle failure [41,42], and closely related to practical applications of functional materials. Previous analysis also suggests that the hardness is closely associated with elastic properties and lattice structure [41,42]. It is generally believed that there exists a necessary inherent relationship between hardness and elastic modulus. Up to date the best correspondence between them has been achieved in the recent papers. Based on this thought, an empirical criterion has been proposed by Teter [43], and based on this proposed an improved formula was given by Chen et al. [44,45]. The new theoretical formula to calculate Vicker's hardness,
0:585 HV ¼ 2 k2 G 3
(15)
with k¼G/B the Pugh ratio. The calculated results showed that it can reduce the hardness of materials in different extent, whether a single element doping or Nb-M codoping for C11b type. It might be the reduction of elastic constant of C33 to a great extent. Also, the effects for Nb doping and Nb-M codoping are studied. The results show that the hardness can significantly reduced with NbeMg codoping, but little influenced for NbeGe and NbeAl. Which may be the reason of a weak influence under low concentration.
Fig. 4. Mechanical properties in C11b and C40 type with various doping concentration; (a) Young's modulus E and Poisson's ratio n; (b) Pugh ratio G/B, Vichers hardness Hv and melting temperatures Tm.
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H. Hu et al. / Intermetallics 67 (2015) 26e34
Fig. 5. The calculated DOS of the pure C11b (a1) and C40 type (a2); The doping concentration of 4.17 at% (C11b type) for Al and Ge are given in Fig (a2) and (a3) respectively; Fig (b2) and (b3) represent the C40 type with doping concentration of 33.33 at.% for Al and Ge.
H. Hu et al. / Intermetallics 67 (2015) 26e34
obviously with Nb doping and Nb-M codoping.
4.3. Melting temperatures It is known that the melting temperatures of metals and intermetallic compounds are closely related to their elastic constants C11 and C33 based on the empirical formula of Fine et al. [46]. For C11b and C40 MoSi2, the empirical formula is [47,48].
Tm ¼ 354 þ
4:5ð2C11 þ C33 Þ : 3
33
(16)
where Tm is in K, C11 and C33 in GPa. The melting temperatures of MoSi2 with alloying elements can be obtained when elastic constants are substituted into this expression. The results are given in Table 4. The melting temperature of pure C11b MoSi2 (2274.9 K) is in an agreement with experimental data (2376.8 K). Usually, the melting temperature is a kind of outside performance of the binding force between atoms [47]. The results show that the estimated melting temperature of C40 is slightly lower than C11b, which is consistent with the stronger bonding force for C11b type. From Fig. 4, a general downward trend with Al and Mg doping, but not very obviously for Ge doping. According to the calculation of formation energy and modulus properties, the bonding force reduced significantly for the Al and Mg doping. The same as previous discussed, the melting point is increased more or less, but not
5. Density of states Mechanical properties of materials is closely related to the electronic structure, such as electronic density of states (DOS). The distribution of total density of states (TDOS) and partial density of states (PDOS) for MoSi2 are shown in Fig. 5(a1) and (b1). The dominant feature of the DOS of pure C11b and C40 is the existence of a wide pseudogap and the Fermi level (Ef) is located in the bottom of this valley, which implies strong covalent characteristic. Generally, the pseudogap is the dividing line between the bonding states and the antibonding states. However, a small number of electrons occupying at Fermi level, which indicates some extent of metallic characteristic for MoSi2. By contrast, C11b has much wider pseudogap means stronger covalent, which may be one reason for higher melting temperature, hardness and lower ductility of C11b. In addition, relative to C11b type, the distribution of electronic is more localized for C40 type, which can also indicates that stronger covalent bonding for C11b type. It is found from PDOS that the electronic is mainly contributed Mo-d orbitals, which hybridize weakly with Si-s, p or other orbitals. This suggests that the occupation of d orbitals electrons is the key factor in dominating the
Fig. 6. The calculated DOS of the Nb-M (M ¼ Al, Mg, Ge) codoping.
34
H. Hu et al. / Intermetallics 67 (2015) 26e34
mechanical behaviors of materials. In order to explore the influence of the alloying elements on MoSi2 crystal, the TDOS and PDOS are calculated under various doping concentrations. The TDOS and PDOS are shown in Fig. 5(a2), (a3) and (b2), (b3). Note that only the DOS with 4.17 at.% concentration is given for C11b due to the negligible effects on the DOS of the lower concentration. There exists a correlation between structural stability and the position of Ef with respect to the pseudogap [49,50]. We observe that the Fermi level falls below the pseudogap in MoSi2 with the increasing of Al (Mg) doping. This indicates that not all the bonding states are filled and some extra electrons are required to reach maximum stability in this compound. This may be the reason for the presence of metastable phase hexagonal C40 under higher Al (Mg) doping. But minimally changed for Ge doping. From the PDOS, this position displacement should be caused by the weak hybridization between Al-p and Mop (2 eV ~ 0 eV), while the hybridization between Al-s and Si-p (7.5 eV ~ 5 eV) also has weak contributions. In order to get a further explanation, the DOS with a high doping concentration for C40 type are obtained. There is an obvious shift to the left of the Fermi level with the increasing of doped Al (Mg). This suggests that the stability of this type will get worse, which is good agreement with previous phase diagram analysis (see Fig. 2). The DOS can also reflects the bonding characters. In general, the stronger the covalent bonding and the worse the ductility. It is evident from Fig. 5(a1)that the covalent combination between Mop and Si-p at 5 eV ~ 1.5 eV, while the peak become sharper of Sip (Fig. 5a2) means an enhancement of localization of Si-p which weakens the covalent combination of them. As a result, a better ductile and lower melting point and hardness of C11b type with the increase of Al (Mg) doped. For C40 type, strong covalent binding occurs at all range from 13.5 eV to 5 eV between M and Si, which leads to the poor ductility of MoSi2. The DOS and PDOS for Nb doped and Nb-M codoped MoSi2 are given in Fig. 6. Similarly, the Fermi level is moved to a lower energy position with Nb doping and Nb-M codoping. According to the PDOS, there is an obvious orbital hybridization between Si-s and Als for Nb and Al codoping (see Fig. 6(b)). In addition, more electrons are located at Fermi level, which suggests a weaker stability comparing to pure MoSi2. It is transparent that the mechanical properties are nearly the same for Nb doped and Nb-M codoped MoSi2 with the same concentrations 4.17 at% (see Table 5). Form Fig. 6, we can see that the pseudogaps of Mo d-bands are also nearly the same. These results in the same bonding characteristic and same mechanical properties. 6. Conclusions In this paper, the phase stability, mechanical properties and electronic structure of C11b and C40 MoSi2 with Si substituted by Al, Mg and Ge were systematically investigated using first principles methods. The calculated lattice constants and elastic constants of C11b and C40 MoSi2 are in good agreement with the previous results. It is found that there is a phase transition from C11b to C40 when the concentrations of Al and Mg reach ~7 at.% and ~6 at.%, respectively. The anisotropy, ductility, hardness and melting temperatures are also presented for MoSi2 with alloying elements using
the elastic constants. For C11b, the ductility (hardness) will be enhanced (reduced) by increasing the concentrations of Al or Mg. Ge have a reverse effects. For C40, the ductility is reduced weakly by Al or Mg. In addition, the effects of substitution of Mo by Nb with Si substitution of Si by Al, Mg and Ge are also investigated. Nb and Mg codoping can improve the ductility of MoSi2. Finally, the density of states is used to analysis the effects of alloying elements on the mechanical properties, and the results are in consistent with the predictions based on elastic constants.
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Intermetallics journal homepage: www.elsevier.com/locate/intermet
First principles study on the phase stability and mechanical properties of MoSi2 alloyed with Al, Mg and Ge* Hai Hu a, Xiaozhi Wu a, b, *, Rui Wang a, Weiguo Li c, Qing Liu b a
Institute for Structure and Function, Chongqing University, 401331, China College of Materials Science and Engineering, Chongqing University, 400044, China c College of Aerospace Engineering, Chongqing University, 400044, China b
a r t i c l e i n f o
a b s t r a c t
Article history: Received 23 April 2015 Received in revised form 25 June 2015 Accepted 13 July 2015 Available online xxx
The phase stability, mechanical properties and electronic structure of C11b and C40 MoSi2 with alloying elements Al, Mg and Ge were systematically investigated using first principles methods. The calculated lattice constants and elastic constants of C11b and C40 MoSi2 are in good agreement with the previous results. It is found that there is a phase transition from C11b to C40 when the concentrations of Al and Mg reach ~7 at.% and ~6 at.%, respectively. Based on the elastic constants, the anisotropy, ductility, hardness and melting temperatures are presented for MoSi2 with alloying elements. For C11b, the ductility will be enhanced by increasing the concentrations of Al or Mg. Simultaneously, hardness will be reduced by the increasing of Al or Mg. Ge have a reverse effects. For C40, the ductility is reduced weakly by Al or Mg. In addition, the effects of substitution of Mo by Nb with Si substitution of Si by Al, Mg and Ge are also investigated. Nb and Mg codoping can improve the ductility of MoSi2. Finally, the density of states is used to analysis the effects of alloying elements on the mechanical properties, and the results are in consistent with the predictions based on elastic constants. © 2015 Elsevier Ltd. All rights reserved.
Keywords: Molybdenum silicides Elastic properties Mechanical properties Theory Ab initio calculations
1. Introduction Molybdenum disilicide (MoSi2) has attracted much attention due to its high melting point, superior high-temperature stability and low density [1e4]. There are two known polymorphs of MoSi2, the stable C11b structure and the metastable C40 structure (see Fig. 1). Due to the mixed covalent and metallic bonding, MoSi2 displays its specific dual characteristics of ceramic and metal. But the inherent brittleness limit its technological assignment [5,6]. Therefore, it is considered to be one of the most suitable candidate for high temperature structural applications if the ductility can be improved. Generally, alloying by chemical substitutions for Mo and Si is regarded as an effective strategy to address this deficiency [1,7]. Sharif et al. have suggested that the hardness of MoSi2 decreases
* The work is supported by the Natural Science Foundation of China (11104361) and Projects supported by the Fundamental Research Funds for the Central Universities (CDJZR14328801 and CQDXWL2014003). * Corresponding author. Institute for Structure and Function, Chongqing University, 401331, China. E-mail addresses: [email protected] (X. Wu), [email protected] (Q. Liu).
http://dx.doi.org/10.1016/j.intermet.2015.07.008 0966-9795/© 2015 Elsevier Ltd. All rights reserved.
obviously even at room temperature when doping with Al [8]. Under 773 K, Al doping (1.5 wt.% ~ 3.0 wt.%) can also reduce the micro-hardness of MoSi2 [9]. The effects of substitutional alloying on the ductility of MoSi2 are also studied using first principles methods based on fracture mechanics theory [10e14]. It is found that substitution of Mo by V or Nb, and substitution of Si by Al or Mg are particularly beneficial to the enhancement of ductility. Al is one of the earliest and the most efficient alloying elements proved by experiment and theory research [1,15]. The alloying elements can weaken the strong covalent bonds of SieSi and MoeSi in MoSi2 crystal, while the ductility is improved with the increase of Al content. Experimental results show that the structure of MoSi2 will be changed from C11b to C40 with the increasing of doping concentrations. The compositional phase analysis indicates that the solid solubility limit of Al in C11b phase is 3 at.%, and the experiment reported that a biphasic mixture of C11b and C40 phases will coexistence for higher concentrations, however, when the doping concentration ups to about 10 at.% there is only C40 phase [16]. Besides, Sharifet al. suggest that the Nb alloyed MoSi2 could be deformed in compression even at room temperature [17]. Instead of a single element alloying, two or more elements codoping MoSi2 is also the common way to improve the ductility. Dasgupta proved
H. Hu et al. / Intermetallics 67 (2015) 26e34
27
Fig. 1. The structure cell of (a) C11b type and (b) C40 type for MoSi2 crystal.
that Nb and Al codoping is an effective way to overcome the low temperature brittleness [18]. In this paper, we will focus on the mechanical properties and electronic structures of MoSi2 with substitution of Si by Al, Mg and Ge, and Nb and Al codoping. The phase transformation between C11b and C40 are investigated based on different doping concentrations for Al, Mg and Ge. The article is organized as follows: Section 2 covers the computational details based on the firstprinciples quantum mechanical method. The structure stability and phase transition were studied in Section 3. In Section 4, some important mechanical behaviors such as ductileebrittle properties, hardness, melting temperatures are investigated systematically. The electronic mechanism behind the effects of the doped elements on MoSi2 was presented in Section 5. Finally, some conclusions are given in last section. 2. Computational details In this paper, the various concentrations of alloying elements Al, Mg and Ge and Nb codoping with Al, Mg and Ge on the mechanical
properties are systematically studied. Based on C11b and C40 structure, the supercells are built (see Table 1), and the alloying elements are injected to substitute Si atoms in these supercells to build 2.08 at.%, 4.17 at.%, 8.33 at.%, 16.67 at.% and 33.33 at.% doping concentration, respectively. The selection of the supercell models are given in Table 1. The present calculations were performed within the density functional theory framework based on VASP code. A plane-wavepseudo-potential (PAW) method is used [19,20]. The generalized gradient approximation (GGA) was adopted with Perdew-BurkeErnzerh (PBE) exchange correlation functional [21]. A convergence tests guided the choice of 450 eV for the cutoff energy. A 17 17 17 Monkhorst-Pack grid of k-points is adopted for sampling the Brillouin zone, and the proper grids are applied on corresponding supercells. The total energy is converged to within 1.0 105 eV/atom and the force components on each atom are set to 1.0 103 eV/Å. The elastic constants were calculated by the ’stress-strain’ method and the spin polarization was adopted for the calculation of electronic density of states (DOS).
Table 1 Lattice parameters a and c (in parentheses) for MoSi2 with various doping concentrations of Al, Mg and Ge (in units of Å).
C11b Mo(Si1x Alx)2 Mo(Si1x Mgx)2 Mo(Si1x Gex)2
0
2.08 at%
4.17 at%
8.33 at%
16.67 at%
33.33 at%
111 3.220 7.889 3.202a, 3.215b 7.851a, 7.866b 3.2047c, 3.2056c 7.8498c, 7.8450c
222 3.228 7.921 3.221 7.928 3.223 7.905
221 3.221 7.927 3.228 7.979 3.227 7.901
1 1 2 3.228 7.985 3.238 8.034 3.230 7.933
111 3.243 8.091 3.274 8.213 3.242 7.967
111 3.278 8.295 3.319 8.712 3.265 8.049
C40
111
441
222
221
112
111
Mo(Si1x Alx)2
4.615 6.623 4.5321c, 4.6016d 6.7981c, 6.5700d 4.605e, 4.622f 6.559e, 6.646f
4.625 6.622 4.636 6.625 4.624 6.636
4.636 6.618 4.658 6.621 4.633 6.650
4.656 6.616 4.678 6.623 4.650 6.641
4.692 6.640 4.756 6.681 4.681 6.662
4.759 6.710 4.919 6.898 4.737 6.707
Mo(Si1x Mgx)2 Mo(Si1x Gex)2 a b c d e f
See See See See See See
[22]. [23]. [24] and references therein. [25]. [26]. [27].
28
H. Hu et al. / Intermetallics 67 (2015) 26e34
when the doped Al ups to about ~10 at.%. This means that our theoretical results are reasonable.
3. Heat of formation and phase stability As is well known, MoSi2 has more than one crystal structure, among them C11b is the prevailing structure. Furthermore, with the doping increasing, it can be transformed into the C40 type. In this part, we will focus on the phase transition between C11b and C40 type with different concentrations of alloying elements Al, Mg and Ge according to the formation energy. The supercells used for calculations are listed in Table 1. Si in the supercells of MoSi2 are substituted by Al, Mg and Ge, respectively. The different supercell means the different concentrations. The lattice constants of the C11b and C40 MoSi2 for various concentrations are also listed in Table 1. MoSi2 alloyed with Al, Mg and Ge can be represented by Mo(Si1x Mx)2, in which x is the molar ratio of M (M ¼ Al, Mg, Ge) to Si of the compound. The heat of formation was calculated because it can directly reflect the structure stability. The heat of formation is defined as
DE ¼ E MoðSi1x Mx Þ2 EðMoÞ 2ð1 xÞEðSiÞ 2xEðMÞ
(1)
where E[Mo(Si1xMx)2] is the total energy of Mo(Si1x Mx)2 crystal at the equilibrium lattice, E(Mo), E(Si) and E(M) are the total energies of Mo, Si and M the solid phase, respectively. The calculated heat of formation as well as previous theoretical and experimental values for pure C11b and C40 MoSi2 are listed in Table 2. Both of them have negative formation energy. This suggests that C11b and C40 MoSi2 are thermodynamically stable. But the heat of formation of C11b are lower than C40. This means C11b are more stable structure than C40. The simulation results show that the heat of formation, which was calculated by different methods produced different results, especially for C11b type. The result of our calculation for C11b type is 1.49 eV/f.u., which is in good agreement with previous results, GGA-PBE (1.53 eV/f.u.), GGA-PW92 (1.38 eV/f.u.) or LAPW (1.45 eV/f.u.) data, while is about 25% larger than that LMTO-FP value (1.87 eV/f.u.). These deviations could be caused by different pseudopotential selection, k-points or cutoff energy. The heat of formation with variation impurity concentration is shown in Fig. 2. It is easy to see that with the increment of impurities, the heat of formation will tend to positive values, quickly. This suggests that the structural stability is getting poorer. It is shown that there will be a structure phase transition from C11b type to C40 type with increasing doping concentration up to 33.3 at.% except Ge doping. It is worth mentioning that when the doped Mg ups to ~20 at.% the Mo(SixMg1x)2 will not be stable exist. It is easy to see that the phase transition point for Mg is ~6 at.%, and for Al is ~7 at.% which is good agreement with Dasgupta's experiment [16]. As suggested by Dasgupta's experiment, there is only C40 phase Table 2 The formation energies for C11b and C40 MoSi2. MoSi2 C11b
C40
This work GGA-PBE [28] GGA-PW92 [28] LMTO-FP [6] LAPW [27] Experiments [31] This work GGA-PW92 [28] LAPW [27]
DE (eV/f.u.)
DE (kJ mol1)
1.49 1.53 1.38 1.87 1.45
143.68 148.08 133.05 180.42 139.90 135.80 ± 4.5 127.54 126.78 129.29
1.32 1.31 1.34
The abbreviation of calculation method: GGA, generalized gradient approximation; PBE, Perdew-Burke-Ernzerhof functional; PW92, Perdew-Wang 92 functional; LMTO-FP, linear muffin-tin-orbitals first-principles; NC-PP, norm-conserving pseudopotential; LAPW, linear augmented-plane-wave.
4. Mechanical properties Elastic behavior is one of the most basic properties to reflect the essential attribute of materials. In this part, the elastic behaviors of C11b and C40 MoSi2 and the extension of the other mechanical properties with different concentrations will be investigated systematically. Generally, there are 21 non-zero independent elastic stiffness tensors. And if a crystal has a higher symmetry, it can be further reduced. For a C11b or C40 type MoSi2, there are six independent elastic constants (C11, C12, C13, C33, C44, C66). Additional energy will be gained by the deformation of the lattice. The elastic constants C11b and C40 MoSi2 are listed in Table 3. The previous and theoretical results are also presented for comparison. The results for different concentrations of Al, Mg and Ge are given in Fig. 3. The phase transformation point is marked by the dashed line. From Fig. 3, we can see clearly that the elastic constants have similar varying tendency with the increase of impurity concentrations. But there is no much significant effect for Ge doping. Besides the thermodynamical viewpoint above, the mechanical stability, which reflects the specific deformation by the applied stress into a particular direction. As early as 1954, systematic and embedded research has been done in this dissertation by Born and Huang [33], and a stability criterion has been presented in terms of the elastic constants. Different crystal system correspond to different elastic stiffness constants. There is a mechanical stability criterion [34] for C11b type, 2 C44 > 0; C66 > 0; C11 > jC12 j; ðC11 þ C12 ÞC33 > 2C13 ;
(2)
and for C40 type, the criterion reads as 2 C11 > 0; C44 > 0; C11 > jC12 j; ðC11 þ C12 ÞC33 > 2C13 :
(3)
The calculated elastic constants meet all these criterias, which implies that the two type are mechanically stable. Unfortunately, there are no experimental data to compare for C40 type. The calculation results show that C33 is larger than C11, which suggests stronger directional property exists in MoSi2 crystal. It can be seen that C33 is mainly dominated by the bonds parallel to c-axis, while C11 is dominated by the bonds parallel to a-axis from the lattice structure. This suggests that the structure stiffer along c-axis. There are two MoeSi and one SieSi bonds parallel to c-axis in C11b primitive cell, which indicates the parallel MoeSi bonds is the key factor impacting the hardness of MoSi2. However, there is no parallel bonds in C40 primitive cell, that is the reason why the elastic constant C33 of C40 MoSi2 is much smaller than C11b. So we could get that the C11b type has much higher hardness than C40 type MoSi2. Overall, the calculation results show that the larger shear elastic constants for C44 and C66 however smaller data for C12 and C13 suggest remarkable brittleness of MoSi2. From Fig. 3, we can see that C33 decreases obviously with the increasing of the concentrations of Al and Mg. But Ge does not have transparent effects on C33. This means that the Al or Mg can reduce the hardness of MoSi2. Nonetheless, C11b type should have higher hardness and brittleness than C40 type. Based on the calculated Cij, the bulk mechanical properties, such as bulk modulus (B), shear modulus (G), Young's modulus (Y) and Poisson's ratio (n), can be evaluated using the Voigt-Reuss-Hill (VRH) scheme [35]. The theoretical upper bounds Voigt approximation which assuming uniform strain throughout a polycrystal are given as follows
H. Hu et al. / Intermetallics 67 (2015) 26e34
29
Fig. 2. The hear of formation with various impurity concentration. It is clear to see that there will be a structure phase transition from body centered tetragonal C11b type to a hexagonal C40 type with increasing doping concentration up to 33.3 at.% except Ge doping. It is worth mentioning that when the doped Mg ups to about 20 at.% the Mo(SixMg1x)2 will not be stable exist.
Table 3 Elastic constants for C11b and C40 MoSi2 (in units of GPa). MoSi2
Cij
C11
C12
C13
C33
C44
C66
C11b
This work GGA-PBE [23] GGA-PW92 [28] GGA-PWP [29] GGA-PWP [30] Exp [32]. This work GGA-PW92 [28]
393.6 407.8 384.63 393.0 392.32 417.0 401.3 394.4
106.9 108.9 99.96 117.0 104.19 104.2 77.1 68.2
89.3 92.2 85.88 87.0 91.33 83.8 126.3 120.2
493.4 505.7 473.78 505.0 482.44 514.5 418.2 403.8
192.3 196.8 186.95 194.0 188.44 204.2 151.1 145.1
182.5 192.8 178.55 178.0 178.10 193.6 166.3 163.1
C40
BV ¼
1 ð2C11 þ 2C12 þ 4C13 þ C33 Þ 9
(4)
GV ¼
1 ð2C11 C12 2C13 þ C33 þ 6C44 þ 3C66 Þ 15
(5)
and the theoretical lower bounds Reuss approximation which assuming uniform stress are written as
BR ¼ ½S11 þ S22 þ S33 þ 2ðS12 þ S13 þ S23 1 GR ¼ ½4ðS11 þ S22 þ S33 Þ þ 4ðS12 þ S13 þ S23 Þ þ 3ðS44 þ S55 þ S66 Þ1
(6)
(7)
Where Sij are elastic compliances and their values can be given by the inversion of elastic constants matrix (Sij ¼ Cij1 ). We estimated these bulk values by the Voigt-Reuss-Hill approximation:
9BG ; 3B þ G
(10)
n¼
3B 2G : 6B þ 2G
(11)
4.1. Elastic anisotropy Because of the low symmetry of C11b and C40 lattice and hence it is necessary to study elastic anisotropy properties to looking for effective ways to improve material properties. Through the comparison of the previous elastic constants, it is also found that strong elastic orientation of MoSi2 crystal. Based on these characteristics, a percentage anisotropy in compressibility and shear [36] are proposed as
AB ¼
ðBV BR Þ ; ðBV þ BR Þ
(12)
AG ¼
ðGV GR Þ : ðGV þ GR Þ
(13)
GV (BV) and GR (BR) are Voigt and Reuss shear (bulk) modules, respectively. A minimum value of zero corresponds to isotropic materials and a maximum 1 represents the greatest possible anisotropy. In order to quantitatively describe the anisotropy of single crystal, a universal index is used [37].
AU ¼ 5
1 B ¼ ðBR þ BV Þ; 2
(8)
1 ðG þ GV Þ; 2 R
(9)
G¼
Y¼
GV BV þ 6 0: GR BR
(14)
The isotropic single crystal corresponds to minimum value zero, and the more deviates from the value of zero, the stronger anisotropy. The results are given in Table 4 and the anisotropy with various doping concentration are given in Fig 3. It is quite clear that they are all obvious anisotropy, and C11b type exhibits more
30
H. Hu et al. / Intermetallics 67 (2015) 26e34
Fig. 3. The calculated elastic constants for C11b type and C40 type of Mo(SixX1x)2 (X ¼ Al, Mg, Ge) with various doping concentration are presented in (a), (b) and (c). Calculated percentage anisotropy in shear AG, compressibility AB and the universal anisotropy AU are presented in (d).
obviously. Simultaneously, growth of different degrees was seen for AG, AB and AU with the increase of Al or Mg doping in C11b type, but little influence on C40 type, except for AB. But, it has little effect on them with Ge doped in all range of doping concentration. Notwithstanding, there is a small fluctuation of AB with Ge doping. On the whole, the anisotropy is enhanced to some extent with Al and Mg doping for C11b, however little impact for Ge. This could be because of the similar electronic structure between Si and Ge. As we know, Nb or Nb-M codoping MoSi2 is also a practical way to improve material properties. The effects of material physical properties due to Nb or Nb-M codoping with 4.17 at.% was discussed too. It is not difficult to find that large influence on the shear
anisotropy due to the addition of Nb atoms, by the comparison of Table 5 and Fig. 3(d). 4.2. Ductility and hardness Precise machining for MoSi2 is always in difficulty and also a research hot spot due to its intrinsic brittleness, which has greatly hindered the development in industry. The essence of brittleness is that the brittleness crack occurred in materials before dislocation emission under deformations [38]. But for its complexity, to solve this problem has always been the object of the researchers. We may recall from our earlier analysis of elastic constants that
Table 4 Elastic anisotropy factors (A), Poisson's ratio n, Pugh ratio G/B, Vichers hardness Hv (GPa) and melting point Tm (K) for C11b and C40 MoSi2. MoSi2 C11b
C40
This work GGA-PBE [23] GGA-PW92 [28] Exp [32]. This work GGA-PW92 [28]
AB
AG
AU
n
G/B
Hv
Tm
0.324 0.299 0.291 0.259 0.288 0.265
0.686 0.625 0.736 0.574 0.329 0.392
0.076 0.069 0.080 0.063 0.039 0.045
0.163 0.162 0.159 0.151 0.206 0.202
0.871 0.874 0.882 0.911 0.733 0.743
32.304 33.161 32.368 35.756 23.321 23.387
2274.9 2336.0 2218.6 2376.8 2185.2 2142.9
H. Hu et al. / Intermetallics 67 (2015) 26e34
31
Table 5 Elastic anisotropy factors (A), Poisson's ratio n, Pugh ratio G/B, Vichers hardness Hv (GPa) and melting point Tm (K) for Nb-M (M ¼ Al, Mg, Ge) codoping. 4.17 at%
AB
AG
AU
n
G/B
Hv
Tm
Mo0.875 Mo0.938 Mo0.938 Mo0.938
0.306 0.360 0.320 0.205
2.164 9.268 2.762 0.664
0.227 1.029 0.291 0.071
0.172 0.161 0.498 0.165
0.838 0.877 0.809 0.863
30.081 32.356 27.781 31.847
2179.1 2099.1 2094.1 2267.8
Nb0.125 Si2 Nb0.062(Si0.969 Al0.031)2 Nb0.062(Si0.969 Mg0.031)2 Nb0.062(Si0.969 Ge0.031)2
the ductile or brittle should be related to shear and bulk properties. Based on this consideration, an empirical Pugh [39] criterion is used to give a phenomenological analysis. According to the Pugh's criterion, the ratio of shear modulus to bulk modulus presents a measurement for ductile-brittle behavior. A high G/B value is associated with brittleness, whereas a low value corresponds ductility. The critical value that separates ductile and brittle behaviors is around 0.5. And, the higher the ratio, the more brittleness. Besides, good ductile materials correspond to the larger Poisson's ratio (n), which provides insight into the change of bond angle and bond length [40]. Furthermore, the large Young's modulus is due to stronger directional boding betweens atoms. The results are given in Tables 4 and 5 and Fig. 4. The calculated results show that the Pugh's ratio is always more than 0.5, this indicates intrinsic brittleness of C11b and C40 type. In addition, larger Pugh's ratio for C11b type suggests stronger brittleness than C40 type. Similar results show that there are downward trends for G/B but upward trends for n as the increasing doping concentration Al and Mg for C11b type, however small fluctuations for Ge doping except in 4.17 at.% doping concentration. Although a weak reduction of ductility with additional elements for C40 type, the influence is little. Which suggests that Al and Mg can enhance the difficulty for C11b type. Overall, the ductility of C11b MoSi2 can be improved by Al and Mg. This is in good agreement with previous experimental and theoretical results [4,12,28]. And the same methods have been used to investigate the Nb doping and Nb-M codoping. The results show that both of them can effectively
improve the ductility of MoSi2. But NbeAl and NbeGe codoping have weak effect. Hardness is one of the most basic mechanical properties of the material, which represents the ability of solids resistance to elastic and plastic deformation or brittle failure [41,42], and closely related to practical applications of functional materials. Previous analysis also suggests that the hardness is closely associated with elastic properties and lattice structure [41,42]. It is generally believed that there exists a necessary inherent relationship between hardness and elastic modulus. Up to date the best correspondence between them has been achieved in the recent papers. Based on this thought, an empirical criterion has been proposed by Teter [43], and based on this proposed an improved formula was given by Chen et al. [44,45]. The new theoretical formula to calculate Vicker's hardness,
0:585 HV ¼ 2 k2 G 3
(15)
with k¼G/B the Pugh ratio. The calculated results showed that it can reduce the hardness of materials in different extent, whether a single element doping or Nb-M codoping for C11b type. It might be the reduction of elastic constant of C33 to a great extent. Also, the effects for Nb doping and Nb-M codoping are studied. The results show that the hardness can significantly reduced with NbeMg codoping, but little influenced for NbeGe and NbeAl. Which may be the reason of a weak influence under low concentration.
Fig. 4. Mechanical properties in C11b and C40 type with various doping concentration; (a) Young's modulus E and Poisson's ratio n; (b) Pugh ratio G/B, Vichers hardness Hv and melting temperatures Tm.
32
H. Hu et al. / Intermetallics 67 (2015) 26e34
Fig. 5. The calculated DOS of the pure C11b (a1) and C40 type (a2); The doping concentration of 4.17 at% (C11b type) for Al and Ge are given in Fig (a2) and (a3) respectively; Fig (b2) and (b3) represent the C40 type with doping concentration of 33.33 at.% for Al and Ge.
H. Hu et al. / Intermetallics 67 (2015) 26e34
obviously with Nb doping and Nb-M codoping.
4.3. Melting temperatures It is known that the melting temperatures of metals and intermetallic compounds are closely related to their elastic constants C11 and C33 based on the empirical formula of Fine et al. [46]. For C11b and C40 MoSi2, the empirical formula is [47,48].
Tm ¼ 354 þ
4:5ð2C11 þ C33 Þ : 3
33
(16)
where Tm is in K, C11 and C33 in GPa. The melting temperatures of MoSi2 with alloying elements can be obtained when elastic constants are substituted into this expression. The results are given in Table 4. The melting temperature of pure C11b MoSi2 (2274.9 K) is in an agreement with experimental data (2376.8 K). Usually, the melting temperature is a kind of outside performance of the binding force between atoms [47]. The results show that the estimated melting temperature of C40 is slightly lower than C11b, which is consistent with the stronger bonding force for C11b type. From Fig. 4, a general downward trend with Al and Mg doping, but not very obviously for Ge doping. According to the calculation of formation energy and modulus properties, the bonding force reduced significantly for the Al and Mg doping. The same as previous discussed, the melting point is increased more or less, but not
5. Density of states Mechanical properties of materials is closely related to the electronic structure, such as electronic density of states (DOS). The distribution of total density of states (TDOS) and partial density of states (PDOS) for MoSi2 are shown in Fig. 5(a1) and (b1). The dominant feature of the DOS of pure C11b and C40 is the existence of a wide pseudogap and the Fermi level (Ef) is located in the bottom of this valley, which implies strong covalent characteristic. Generally, the pseudogap is the dividing line between the bonding states and the antibonding states. However, a small number of electrons occupying at Fermi level, which indicates some extent of metallic characteristic for MoSi2. By contrast, C11b has much wider pseudogap means stronger covalent, which may be one reason for higher melting temperature, hardness and lower ductility of C11b. In addition, relative to C11b type, the distribution of electronic is more localized for C40 type, which can also indicates that stronger covalent bonding for C11b type. It is found from PDOS that the electronic is mainly contributed Mo-d orbitals, which hybridize weakly with Si-s, p or other orbitals. This suggests that the occupation of d orbitals electrons is the key factor in dominating the
Fig. 6. The calculated DOS of the Nb-M (M ¼ Al, Mg, Ge) codoping.
34
H. Hu et al. / Intermetallics 67 (2015) 26e34
mechanical behaviors of materials. In order to explore the influence of the alloying elements on MoSi2 crystal, the TDOS and PDOS are calculated under various doping concentrations. The TDOS and PDOS are shown in Fig. 5(a2), (a3) and (b2), (b3). Note that only the DOS with 4.17 at.% concentration is given for C11b due to the negligible effects on the DOS of the lower concentration. There exists a correlation between structural stability and the position of Ef with respect to the pseudogap [49,50]. We observe that the Fermi level falls below the pseudogap in MoSi2 with the increasing of Al (Mg) doping. This indicates that not all the bonding states are filled and some extra electrons are required to reach maximum stability in this compound. This may be the reason for the presence of metastable phase hexagonal C40 under higher Al (Mg) doping. But minimally changed for Ge doping. From the PDOS, this position displacement should be caused by the weak hybridization between Al-p and Mop (2 eV ~ 0 eV), while the hybridization between Al-s and Si-p (7.5 eV ~ 5 eV) also has weak contributions. In order to get a further explanation, the DOS with a high doping concentration for C40 type are obtained. There is an obvious shift to the left of the Fermi level with the increasing of doped Al (Mg). This suggests that the stability of this type will get worse, which is good agreement with previous phase diagram analysis (see Fig. 2). The DOS can also reflects the bonding characters. In general, the stronger the covalent bonding and the worse the ductility. It is evident from Fig. 5(a1)that the covalent combination between Mop and Si-p at 5 eV ~ 1.5 eV, while the peak become sharper of Sip (Fig. 5a2) means an enhancement of localization of Si-p which weakens the covalent combination of them. As a result, a better ductile and lower melting point and hardness of C11b type with the increase of Al (Mg) doped. For C40 type, strong covalent binding occurs at all range from 13.5 eV to 5 eV between M and Si, which leads to the poor ductility of MoSi2. The DOS and PDOS for Nb doped and Nb-M codoped MoSi2 are given in Fig. 6. Similarly, the Fermi level is moved to a lower energy position with Nb doping and Nb-M codoping. According to the PDOS, there is an obvious orbital hybridization between Si-s and Als for Nb and Al codoping (see Fig. 6(b)). In addition, more electrons are located at Fermi level, which suggests a weaker stability comparing to pure MoSi2. It is transparent that the mechanical properties are nearly the same for Nb doped and Nb-M codoped MoSi2 with the same concentrations 4.17 at% (see Table 5). Form Fig. 6, we can see that the pseudogaps of Mo d-bands are also nearly the same. These results in the same bonding characteristic and same mechanical properties. 6. Conclusions In this paper, the phase stability, mechanical properties and electronic structure of C11b and C40 MoSi2 with Si substituted by Al, Mg and Ge were systematically investigated using first principles methods. The calculated lattice constants and elastic constants of C11b and C40 MoSi2 are in good agreement with the previous results. It is found that there is a phase transition from C11b to C40 when the concentrations of Al and Mg reach ~7 at.% and ~6 at.%, respectively. The anisotropy, ductility, hardness and melting temperatures are also presented for MoSi2 with alloying elements using
the elastic constants. For C11b, the ductility (hardness) will be enhanced (reduced) by increasing the concentrations of Al or Mg. Ge have a reverse effects. For C40, the ductility is reduced weakly by Al or Mg. In addition, the effects of substitution of Mo by Nb with Si substitution of Si by Al, Mg and Ge are also investigated. Nb and Mg codoping can improve the ductility of MoSi2. Finally, the density of states is used to analysis the effects of alloying elements on the mechanical properties, and the results are in consistent with the predictions based on elastic constants.
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