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First-principles study on the structural, elastic and electronic properties of Ti2SiN under high pressure
Solid State Communications 237-238 (2016) 24–27
Contents lists available at ScienceDirect
Solid State Communications journal homepage: www.elsevier.com/locate/ssc
First-principles study on the structural, elastic and electronic properties of Ti2SiN under high pressure Hui Li a, Zhenjun Wang a,b,n, Guodong Sun a, Pengfei Yu a, Wenxue Zhang a a b
School of Materials Science and Engineering, Chang’an University, Xi’an 710064, Shaanxi, PR China Engineering Research Central of Pavement Materials, Ministry of Education of PR China, Chang’an University, Xi’an 710064, Shaanxi, PR China
art ic l e i nf o
a b s t r a c t
Article history: Received 26 January 2016 Received in revised form 13 March 2016 Accepted 23 March 2016 By Ralph Gebauer Available online 29 March 2016
The structural, elastic and electronic properties of Ti2SiN under pressure range of 0–50 GPa have been systemically investigated by first-principles calculations. It is found that both Poisson's ratio and shear anisotropy factor of Ti2SiN increase with pressure, and Ti2SiN is elastic anisotropic. The DOS and Mulliken population analysis have been explored, which indicts that Ti2SiN is metallic–covalent–ionic in nature. The present calculations may contribute preliminary results and a better understanding of Ti2SiN for its applications under high pressure environments. & 2016 Elsevier Ltd. All rights reserved.
Keywords: A. Ti2SiN D. Elastic properties D. Electronic properties D. High pressure
1. Introduction The MAX phases [1–6] are a group of compounds with the general formula of Mn þ 1AXn, where n varies from 1 to 3, M is an early transition metal, A is an IIIA or IVA elements, and X is either C and/or N. The MAX phases exhibit many excellent properties associated with metals and ceramics, including the low density, excellent thermal shock resistance, easy machinability, good oxidation resistance, durability and damage tolerance. These excellent properties make them mostly be used as structural material under extreme conditions such as high temperature, high pressure or nuclear radiation environments [6–9]. There are currently more than 70 MAX phases [6], with most of the synthesized or theoretically predicted ones being the 211 M2AX variety, e.g., Ti2AlC, Ti2AlN, Ti2SiC, and Ti2SiN. Manoun et al. [10] performed X-ray high-pressure study on Ti2AlN and Ti2AlC. Hug et al. [11] studied the electronic structure of Ti2AlC and Ti2AlN using X-ray absorption spectroscopy and full-potential augmented plane wave methods. Du et al. [12] theoretically investigated the elastic and thermodynamic properties of Ti2AlC and Ti2AlN. Li et al. [13] and Ghebouli et al. [14] theoretically studied the structural, elastic and electronic properties of Ti2SiC. However, Ti2SiN has not been as popularly studied as the previously mentioned 211 phases. Barsoum [1] studied the thermodynamically stable Mn þ 1AXn phases, and the n Corresponding author at: School of Materials Science and Engineering, Chang’an University, Xi’an 710064, Shaanxi, PR China. Tel./Fax: þ 86 29 82337340. E-mail address: [email protected] (Z. Wang).
http://dx.doi.org/10.1016/j.ssc.2016.03.019 0038-1098/& 2016 Elsevier Ltd. All rights reserved.
occurrence of Ti2SiN was not confirmed. Later, Keast et al. [3] predicted the stability of the MAX phases from the first-principles calculations, and the result of a small energy difference suggests that Ti2SiN has the potential to be fabricated as a metastable compound. Only Gan et al. [8] studied the structural, elastic and electronic properties of Ti2SiN by first-principle calculations. Up to now, the experimental observation of Ti2SiN and the influence of high pressure on the structural, elastic and electronic properties of Ti2SiN have not been reported. It is known that the study of pressure behavior of material is very important, because some specific characteristics and useful information can be gained during the studies [15]. Considering the possible application of Ti2SiN as structural material under high pressure [6], more theoretical research still needs to be focused on this compound.
2. Calculation methods The CASTEP code [16,17] is used to perform the present firstprinciples calculations, utilizing the ultrasoft pseudopotential [18]. The 3s23p63d24s2, 3s23p2 and 2s22p3 electronic configurations are performed for pseudo atomic calculation of Ti, Si and N, respectively. The CA-PZ [19,20] function within the local density approximation (LDA) and the PBE [21] functional within the generalized gradient approximation (GGA) [22] are treated for the exchange-correlation energy. For a primitive cell, the special points sampling integration over the Brillouin zone is employed by using the Monkhorst–Pack method [23] with 10 10 2. 400 eV
H. Li et al. / Solid State Communications 237-238 (2016) 24–27
25
and 5.0 10–7 eV/atom are chosen as the energy cutoff and the tolerance of self consistent field calculation, respectively. The elastic constants of (Cij) are calculated by the CASTEP code using finite strain method [24], such that σ i ¼ C ij ϵj for small stresses σ and strains ε. The Voigt [25] and Reuss [26] models result in the theoretical maximum and minimum values of the elastic modulus, respectively. For hexagonal lattice the Voigt (BV) and Reuss (BR) bulk modulus are determined by (Eqs. (1) and 2) BV ¼ ½2ðC 11 þ C 12 Þ þ C 33 þ 4C 13 =9
ð1Þ
ðC 11 þ C 12 ÞC 33 2C 213 : C 11 þ C 12 þ 2C 33 4C 13
ð2Þ
BR ¼
Similarly, the upper (GV ) and the lower (GR ) bounds for the shear modulus are given by (Eqs. (3) and 4) GV ¼ ½C 11 þ C 12 þ 2C 33 4C 13 þ 12C 55 þ 12C 66 =30 h i ðC 11 þ C 12 ÞC 33 2C 213 C 55 C 66 5 h i : GR ¼ 2 3B C C þ ðC þ C ÞC 2C 2 ðC þ C Þ V 55 66 55 11 12 33 66 13
ð3Þ ð4Þ
Fig. 1. Variations of relative lattice parameters and relative unit cell volume of Ti2SiN with pressure.
According to the Voigt–Reuss–Hill approximation [25–27], the Hill values of bulk modulus (BH ) and shear modulus (GH ) are defined as (Eqs. (5) and 6)
using the LDA functional are slightly smaller than the GGA results, and this is in agreement with the density functional theory result “GGA lattice constants are larger than the LDA ones” [29]. Manoun et al. [10] experimentally measured the pressure dependencies of the lattice parameters of Ti2AlN and Ti2AlC up to pressures E50 GPa. Contrastively, Ti2SiN is optimized at fixed values of applied hydrostatic pressure in the range of 0–50 GPa at 5 GPa intervals, to show the response of structural parameter to external pressure. Variations of relative lattice parameters and relative unit cell volume of Ti2SiN with applied quasi-hydrostatic pressure are plotted in Fig. 1. It shows that the lattice parameter a changes more than c in the whole pressure range. This means that Ti2SiN is more compressible along the a-axis than that along the caxis, that is to say, it is stiffer in the c direction than along the basal plane. So, it is concluded that Ti2SiN is anisotropic.
BH ¼ ðBR þBV Þ=2
ð5Þ
GH ¼ ðGR þ GV Þ=2:
ð6Þ
Then, Young's modulus (E) and Poisson's ratio (v) can be derived from the following relations: E¼
9GB 3B þ G
ð7Þ
ν¼
3B 2G : 2ð3B þGÞ
ð8Þ
To quantify the elastic anisotropy of Ti2SiN, the shear anisotropic factor (A) [24,28] is evaluated using the following Eq. (9): A¼
4C 44 : C 11 þ C 33 2C 13
ð9Þ
3. Results and discussion 3.1. Structural properties Ti2SiN crystallizes in the hexagonal structure with P63/mmc space group. There are two Ti2SiN formula units in the unit cell. The atomic positions are Ti: 4f-(1/3, 2/3, u) where u is the internal parameter, Si: 2d-(2/3, 1/3, 1/4), and N: 2a-(0, 0, 0). Convergence tests for the number of k points and energy cutoff have been performed firstly. It is shown that the 10 10 2 Monkhorst–Pack grid and 400 eV are good enough for the further calculations. The optimized structural properties for Ti2SiN at zero pressure are displayed in Table 1, which are in good accordance with the available theoretical results [3,8]. Unfortunately, there are no available experimental data for comparison. The results also reveal that the lattice constants Table 1 Lattice parameters (a, c), internal parameter (u) and volume (V) of Ti2SiN at zero pressure. Method
a (Å)
c (Å)
u
V (Å3)
LDA PBE Calc. [3] Calc. [8]
2.927 2.984 2.990 2.979
12.617 12.822 12.880 12.820
0.093 0.093 0.092 0.093
93.640 98.853
3.2. Elastic properties With the equilibrium geometries optimized under pressure in the above section, the elastic constants Cij under pressure could be calculated by the CASTEP code using finite strain method, which have been used in the previous studies for Zr2AlX and Ti2AlX (X ¼C and N) [24], M2AlC (M¼ V, Nb and Ta) [30], and WS2 [15]. Then, the following elastic properties such as bulk modulus (B), shear modulus (G), Young's modulus (E), Poisson's ratio (v) and shear anisotropy factor (A) could be calculated by using (Eqs. (1)–9). The elastic results of Ti2SiN at 0 GPa are listed in Table 2. Comparing Table 1 with Table 2, it can be seen that the LDA and GGA methods lead to slightly dispersive elastic constants and elastic modulus. For the shear anisotropic factor A, the magnitude of the deviation from 1 shows the degree of elastic anisotropy. From discrepancy of both bulk modulus and shear modulus in Table 2, it can also be seen that Ti2SiN exhibits a profound elastic anisotropy. The present elastic properties of Ti2SiN are in good accordance with the previous calculation results [8]. The pressure dependence of elastic properties for Ti2SiN is also considered, with the results provided in Fig. 2a–c. The trend of Cij under pressure in Fig. 2a is similar to that of Cr2AlN [31]. Fig. 2b and c reveals that both v and A increase with pressure. The mechanical stability criterion for hexagonal crystal under pressure (p) are expressed as follows [15,32]: ð10Þ C 044 4 0; C 011 4 C 012 ; C 011 þ C 012 C 033 2C 013 2 4 0 where C 0αα ¼ C aa p ðα ¼ 1; 3; 4Þ; C 012 ¼ C 12 þ p; C 013 ¼ C 13 þ p:
ð11Þ
26
H. Li et al. / Solid State Communications 237-238 (2016) 24–27
Table 2 Elastic constants Cij (in GPa), bulk modulus BV, BR, BH (in GPa), shear modulus GV, GR, GH (in GPa), Young's modulus E (in GPa), Poisson's ratio ν and the shear anisotropic factor A for Ti2SiN at zero pressure. Method
C11
C33
C44
C12
C13
BV
BR
BH
GV
GR
GH
E
ν
A
LDA PBE Calc. [8]
342.1 296.7 298.0
409.9 347.8 347.0
181.3 155.1 153.0
106.7 100.2 96.0
149.5 126.3 127.0
211.7 183.0
207.7 180.7
209.7 181.9 182.0
142.0 120.9
134.6 114.7
138.3 117.8 118.0
340.1 290.7 291.0
0.230 0.234 0.233
1.601 1.583 1.57.0
Fig. 2. Pressure dependence of elastic properties for Ti2SiN: (a) Cij, B, G; (b) v and (c) A.
PI
PII
PIII
PIV
Fig. 3. The DOS for Ti2SiN: (a) total and partial DOS at zero pressure and (b) variations of total DOS with pressure.
H. Li et al. / Solid State Communications 237-238 (2016) 24–27
27
4. Conclusions
Table 3 Mulliken population analysis of Ti2SiN. Method
Species
s
p
d
Total
Charge (e)
LDA
Ti Si N
2.18 1.35 1.67
6.68 2.63 4.03
2.81 0.00 0.00
11.66 3.97 5.70
0.34 0.03 –0.70
PBE
Ti Si N
2.19 1.36 1.68
6.71 2.61 4.04
2.75 0.00 0.00
11.66 3.97 5.72
0.34 0.03 –0.72
It is easy to find that the calculated Cij at different pressures in Fig. 2a satisfy the mechanical stability criteria of (Eqs. (10) and 11), indicating that Ti2SiN is mechanically stable under pressure range from 0 to 50 GPa. 3.3. Electronic density of state The electronic density of state (DOS) for Ti2SiN is calculated at the GGA level in the scheme of PBE, the total and partial DOS at 0 GPa and variations of total DOS with pressure are plotted in Fig. 3a and b, respectively. As Fig. 3a shows, there is no band gap at the Fermi level, indicating the metallic feature of Ti2SiN. The calculated total DOS for one Ti2SiN unit cell at the Fermi level is N (EF)¼3.59 eV–1, which is mainly contributed by the Ti-3d electrons and is in accordance with the report of 3.67 eV–1 [8]. As shown in Fig. 3a, the total DOS curve of Ti2SiN can be divided into four main peaks. Comparing the total and partial DOS of Ti2SiN, it can be concluded that: peak PI in the valence band is dominated by the N2s state; peak PII originates from N-2p and Ti-3d states; peak PIII is primarily determined by the hybridization of Ti–3d and Si-3p states; while peak PIV in the conduction band mainly originates from Ti-3d state, with a small contribution of Si–3p and N-2p states. Moreover, peaks PII and PIII reveal that the Ti(3d)–N(2p) bonds are located at a lower energy range than the Ti(3d)–Si(3p) bonds, which suggest that Ti–N bonds have higher binding energy and are stronger than Ti–Si bonds. This explanation for soft Ti–Si bonds is similar to that of Ti2SiC [13,14]. Fig. 3b reveals that the DOS shifts under pressure slightly to the higher energy zone in the conduction band while moves slightly to the lower energy zone in the valence band, but the deviation of total DOS at 10, 30 and 50 GPa is very small. That is to say, the high pressures have no significant influence on the DOS of Ti2SiN.
We have presented a first-principles study on the structural, elastic and electronic properties of Ti2SiN under pressure range from 0 to 50 GPa. It is found that the both Poisson';s ratio (v) and shear anisotropy factor (A) of Ti2SiN increase with pressure. Though the analysis of structural properties, DOS and Mulliken population, it is concluded that Ti2SiN is elastic anisotropic, and its bonding nature is metallic combined with some covalent and ionic. The present calculation results may guide its applications under high pressure environments.
Acknowledgments This work is supported by the National Natural Science Foundation of China (Nos. 51402023, 51402024 and 51301020), the Natural Science Basic Research Plan in Shaanxi Province of China (Nos. 2014JQ6217, 2014JQ6212 and 2015JQ5144), and the Fundamental Research Funds for the Central Universities of Chang’an University (Nos. 310831151081, 310831151084, 310831152020, and 310831153504). The authors also acknowledge the Northwestern Polytechnical University High Performance Computing Center for the allocation of computing time on their machines.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17]
3.4. Mulliken population analysis The Mulliken population [33–35] of Ti2SiN is calculated by the CASTEP code to quantify the bond overlap as well as the charge transfer among Ti, Si and N atoms, and the results at zero pressure are shown in Table 3. It is shown that the charge transfer from Ti and Si to N after bonding is 0.70–0.72 electron. In the PBE result, there is 0.68 electron transferred from Ti to N atoms, but only 0.03 electron from Si to N. This result shows an ionic bonding feature of Ti2SiN with a strong directional bonding between Ti and N atoms. From the results of the DOS and Mulliken population, it can be concluded that bonding nature of Ti2SiN is metallic combined with some covalent and ionic. This conclusion will identify the associated properties of both ceramic and metal for Ti2SiN, and will guide its practical applications.
[18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35]
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Contents lists available at ScienceDirect
Solid State Communications journal homepage: www.elsevier.com/locate/ssc
First-principles study on the structural, elastic and electronic properties of Ti2SiN under high pressure Hui Li a, Zhenjun Wang a,b,n, Guodong Sun a, Pengfei Yu a, Wenxue Zhang a a b
School of Materials Science and Engineering, Chang’an University, Xi’an 710064, Shaanxi, PR China Engineering Research Central of Pavement Materials, Ministry of Education of PR China, Chang’an University, Xi’an 710064, Shaanxi, PR China
art ic l e i nf o
a b s t r a c t
Article history: Received 26 January 2016 Received in revised form 13 March 2016 Accepted 23 March 2016 By Ralph Gebauer Available online 29 March 2016
The structural, elastic and electronic properties of Ti2SiN under pressure range of 0–50 GPa have been systemically investigated by first-principles calculations. It is found that both Poisson's ratio and shear anisotropy factor of Ti2SiN increase with pressure, and Ti2SiN is elastic anisotropic. The DOS and Mulliken population analysis have been explored, which indicts that Ti2SiN is metallic–covalent–ionic in nature. The present calculations may contribute preliminary results and a better understanding of Ti2SiN for its applications under high pressure environments. & 2016 Elsevier Ltd. All rights reserved.
Keywords: A. Ti2SiN D. Elastic properties D. Electronic properties D. High pressure
1. Introduction The MAX phases [1–6] are a group of compounds with the general formula of Mn þ 1AXn, where n varies from 1 to 3, M is an early transition metal, A is an IIIA or IVA elements, and X is either C and/or N. The MAX phases exhibit many excellent properties associated with metals and ceramics, including the low density, excellent thermal shock resistance, easy machinability, good oxidation resistance, durability and damage tolerance. These excellent properties make them mostly be used as structural material under extreme conditions such as high temperature, high pressure or nuclear radiation environments [6–9]. There are currently more than 70 MAX phases [6], with most of the synthesized or theoretically predicted ones being the 211 M2AX variety, e.g., Ti2AlC, Ti2AlN, Ti2SiC, and Ti2SiN. Manoun et al. [10] performed X-ray high-pressure study on Ti2AlN and Ti2AlC. Hug et al. [11] studied the electronic structure of Ti2AlC and Ti2AlN using X-ray absorption spectroscopy and full-potential augmented plane wave methods. Du et al. [12] theoretically investigated the elastic and thermodynamic properties of Ti2AlC and Ti2AlN. Li et al. [13] and Ghebouli et al. [14] theoretically studied the structural, elastic and electronic properties of Ti2SiC. However, Ti2SiN has not been as popularly studied as the previously mentioned 211 phases. Barsoum [1] studied the thermodynamically stable Mn þ 1AXn phases, and the n Corresponding author at: School of Materials Science and Engineering, Chang’an University, Xi’an 710064, Shaanxi, PR China. Tel./Fax: þ 86 29 82337340. E-mail address: [email protected] (Z. Wang).
http://dx.doi.org/10.1016/j.ssc.2016.03.019 0038-1098/& 2016 Elsevier Ltd. All rights reserved.
occurrence of Ti2SiN was not confirmed. Later, Keast et al. [3] predicted the stability of the MAX phases from the first-principles calculations, and the result of a small energy difference suggests that Ti2SiN has the potential to be fabricated as a metastable compound. Only Gan et al. [8] studied the structural, elastic and electronic properties of Ti2SiN by first-principle calculations. Up to now, the experimental observation of Ti2SiN and the influence of high pressure on the structural, elastic and electronic properties of Ti2SiN have not been reported. It is known that the study of pressure behavior of material is very important, because some specific characteristics and useful information can be gained during the studies [15]. Considering the possible application of Ti2SiN as structural material under high pressure [6], more theoretical research still needs to be focused on this compound.
2. Calculation methods The CASTEP code [16,17] is used to perform the present firstprinciples calculations, utilizing the ultrasoft pseudopotential [18]. The 3s23p63d24s2, 3s23p2 and 2s22p3 electronic configurations are performed for pseudo atomic calculation of Ti, Si and N, respectively. The CA-PZ [19,20] function within the local density approximation (LDA) and the PBE [21] functional within the generalized gradient approximation (GGA) [22] are treated for the exchange-correlation energy. For a primitive cell, the special points sampling integration over the Brillouin zone is employed by using the Monkhorst–Pack method [23] with 10 10 2. 400 eV
H. Li et al. / Solid State Communications 237-238 (2016) 24–27
25
and 5.0 10–7 eV/atom are chosen as the energy cutoff and the tolerance of self consistent field calculation, respectively. The elastic constants of (Cij) are calculated by the CASTEP code using finite strain method [24], such that σ i ¼ C ij ϵj for small stresses σ and strains ε. The Voigt [25] and Reuss [26] models result in the theoretical maximum and minimum values of the elastic modulus, respectively. For hexagonal lattice the Voigt (BV) and Reuss (BR) bulk modulus are determined by (Eqs. (1) and 2) BV ¼ ½2ðC 11 þ C 12 Þ þ C 33 þ 4C 13 =9
ð1Þ
ðC 11 þ C 12 ÞC 33 2C 213 : C 11 þ C 12 þ 2C 33 4C 13
ð2Þ
BR ¼
Similarly, the upper (GV ) and the lower (GR ) bounds for the shear modulus are given by (Eqs. (3) and 4) GV ¼ ½C 11 þ C 12 þ 2C 33 4C 13 þ 12C 55 þ 12C 66 =30 h i ðC 11 þ C 12 ÞC 33 2C 213 C 55 C 66 5 h i : GR ¼ 2 3B C C þ ðC þ C ÞC 2C 2 ðC þ C Þ V 55 66 55 11 12 33 66 13
ð3Þ ð4Þ
Fig. 1. Variations of relative lattice parameters and relative unit cell volume of Ti2SiN with pressure.
According to the Voigt–Reuss–Hill approximation [25–27], the Hill values of bulk modulus (BH ) and shear modulus (GH ) are defined as (Eqs. (5) and 6)
using the LDA functional are slightly smaller than the GGA results, and this is in agreement with the density functional theory result “GGA lattice constants are larger than the LDA ones” [29]. Manoun et al. [10] experimentally measured the pressure dependencies of the lattice parameters of Ti2AlN and Ti2AlC up to pressures E50 GPa. Contrastively, Ti2SiN is optimized at fixed values of applied hydrostatic pressure in the range of 0–50 GPa at 5 GPa intervals, to show the response of structural parameter to external pressure. Variations of relative lattice parameters and relative unit cell volume of Ti2SiN with applied quasi-hydrostatic pressure are plotted in Fig. 1. It shows that the lattice parameter a changes more than c in the whole pressure range. This means that Ti2SiN is more compressible along the a-axis than that along the caxis, that is to say, it is stiffer in the c direction than along the basal plane. So, it is concluded that Ti2SiN is anisotropic.
BH ¼ ðBR þBV Þ=2
ð5Þ
GH ¼ ðGR þ GV Þ=2:
ð6Þ
Then, Young's modulus (E) and Poisson's ratio (v) can be derived from the following relations: E¼
9GB 3B þ G
ð7Þ
ν¼
3B 2G : 2ð3B þGÞ
ð8Þ
To quantify the elastic anisotropy of Ti2SiN, the shear anisotropic factor (A) [24,28] is evaluated using the following Eq. (9): A¼
4C 44 : C 11 þ C 33 2C 13
ð9Þ
3. Results and discussion 3.1. Structural properties Ti2SiN crystallizes in the hexagonal structure with P63/mmc space group. There are two Ti2SiN formula units in the unit cell. The atomic positions are Ti: 4f-(1/3, 2/3, u) where u is the internal parameter, Si: 2d-(2/3, 1/3, 1/4), and N: 2a-(0, 0, 0). Convergence tests for the number of k points and energy cutoff have been performed firstly. It is shown that the 10 10 2 Monkhorst–Pack grid and 400 eV are good enough for the further calculations. The optimized structural properties for Ti2SiN at zero pressure are displayed in Table 1, which are in good accordance with the available theoretical results [3,8]. Unfortunately, there are no available experimental data for comparison. The results also reveal that the lattice constants Table 1 Lattice parameters (a, c), internal parameter (u) and volume (V) of Ti2SiN at zero pressure. Method
a (Å)
c (Å)
u
V (Å3)
LDA PBE Calc. [3] Calc. [8]
2.927 2.984 2.990 2.979
12.617 12.822 12.880 12.820
0.093 0.093 0.092 0.093
93.640 98.853
3.2. Elastic properties With the equilibrium geometries optimized under pressure in the above section, the elastic constants Cij under pressure could be calculated by the CASTEP code using finite strain method, which have been used in the previous studies for Zr2AlX and Ti2AlX (X ¼C and N) [24], M2AlC (M¼ V, Nb and Ta) [30], and WS2 [15]. Then, the following elastic properties such as bulk modulus (B), shear modulus (G), Young's modulus (E), Poisson's ratio (v) and shear anisotropy factor (A) could be calculated by using (Eqs. (1)–9). The elastic results of Ti2SiN at 0 GPa are listed in Table 2. Comparing Table 1 with Table 2, it can be seen that the LDA and GGA methods lead to slightly dispersive elastic constants and elastic modulus. For the shear anisotropic factor A, the magnitude of the deviation from 1 shows the degree of elastic anisotropy. From discrepancy of both bulk modulus and shear modulus in Table 2, it can also be seen that Ti2SiN exhibits a profound elastic anisotropy. The present elastic properties of Ti2SiN are in good accordance with the previous calculation results [8]. The pressure dependence of elastic properties for Ti2SiN is also considered, with the results provided in Fig. 2a–c. The trend of Cij under pressure in Fig. 2a is similar to that of Cr2AlN [31]. Fig. 2b and c reveals that both v and A increase with pressure. The mechanical stability criterion for hexagonal crystal under pressure (p) are expressed as follows [15,32]: ð10Þ C 044 4 0; C 011 4 C 012 ; C 011 þ C 012 C 033 2C 013 2 4 0 where C 0αα ¼ C aa p ðα ¼ 1; 3; 4Þ; C 012 ¼ C 12 þ p; C 013 ¼ C 13 þ p:
ð11Þ
26
H. Li et al. / Solid State Communications 237-238 (2016) 24–27
Table 2 Elastic constants Cij (in GPa), bulk modulus BV, BR, BH (in GPa), shear modulus GV, GR, GH (in GPa), Young's modulus E (in GPa), Poisson's ratio ν and the shear anisotropic factor A for Ti2SiN at zero pressure. Method
C11
C33
C44
C12
C13
BV
BR
BH
GV
GR
GH
E
ν
A
LDA PBE Calc. [8]
342.1 296.7 298.0
409.9 347.8 347.0
181.3 155.1 153.0
106.7 100.2 96.0
149.5 126.3 127.0
211.7 183.0
207.7 180.7
209.7 181.9 182.0
142.0 120.9
134.6 114.7
138.3 117.8 118.0
340.1 290.7 291.0
0.230 0.234 0.233
1.601 1.583 1.57.0
Fig. 2. Pressure dependence of elastic properties for Ti2SiN: (a) Cij, B, G; (b) v and (c) A.
PI
PII
PIII
PIV
Fig. 3. The DOS for Ti2SiN: (a) total and partial DOS at zero pressure and (b) variations of total DOS with pressure.
H. Li et al. / Solid State Communications 237-238 (2016) 24–27
27
4. Conclusions
Table 3 Mulliken population analysis of Ti2SiN. Method
Species
s
p
d
Total
Charge (e)
LDA
Ti Si N
2.18 1.35 1.67
6.68 2.63 4.03
2.81 0.00 0.00
11.66 3.97 5.70
0.34 0.03 –0.70
PBE
Ti Si N
2.19 1.36 1.68
6.71 2.61 4.04
2.75 0.00 0.00
11.66 3.97 5.72
0.34 0.03 –0.72
It is easy to find that the calculated Cij at different pressures in Fig. 2a satisfy the mechanical stability criteria of (Eqs. (10) and 11), indicating that Ti2SiN is mechanically stable under pressure range from 0 to 50 GPa. 3.3. Electronic density of state The electronic density of state (DOS) for Ti2SiN is calculated at the GGA level in the scheme of PBE, the total and partial DOS at 0 GPa and variations of total DOS with pressure are plotted in Fig. 3a and b, respectively. As Fig. 3a shows, there is no band gap at the Fermi level, indicating the metallic feature of Ti2SiN. The calculated total DOS for one Ti2SiN unit cell at the Fermi level is N (EF)¼3.59 eV–1, which is mainly contributed by the Ti-3d electrons and is in accordance with the report of 3.67 eV–1 [8]. As shown in Fig. 3a, the total DOS curve of Ti2SiN can be divided into four main peaks. Comparing the total and partial DOS of Ti2SiN, it can be concluded that: peak PI in the valence band is dominated by the N2s state; peak PII originates from N-2p and Ti-3d states; peak PIII is primarily determined by the hybridization of Ti–3d and Si-3p states; while peak PIV in the conduction band mainly originates from Ti-3d state, with a small contribution of Si–3p and N-2p states. Moreover, peaks PII and PIII reveal that the Ti(3d)–N(2p) bonds are located at a lower energy range than the Ti(3d)–Si(3p) bonds, which suggest that Ti–N bonds have higher binding energy and are stronger than Ti–Si bonds. This explanation for soft Ti–Si bonds is similar to that of Ti2SiC [13,14]. Fig. 3b reveals that the DOS shifts under pressure slightly to the higher energy zone in the conduction band while moves slightly to the lower energy zone in the valence band, but the deviation of total DOS at 10, 30 and 50 GPa is very small. That is to say, the high pressures have no significant influence on the DOS of Ti2SiN.
We have presented a first-principles study on the structural, elastic and electronic properties of Ti2SiN under pressure range from 0 to 50 GPa. It is found that the both Poisson';s ratio (v) and shear anisotropy factor (A) of Ti2SiN increase with pressure. Though the analysis of structural properties, DOS and Mulliken population, it is concluded that Ti2SiN is elastic anisotropic, and its bonding nature is metallic combined with some covalent and ionic. The present calculation results may guide its applications under high pressure environments.
Acknowledgments This work is supported by the National Natural Science Foundation of China (Nos. 51402023, 51402024 and 51301020), the Natural Science Basic Research Plan in Shaanxi Province of China (Nos. 2014JQ6217, 2014JQ6212 and 2015JQ5144), and the Fundamental Research Funds for the Central Universities of Chang’an University (Nos. 310831151081, 310831151084, 310831152020, and 310831153504). The authors also acknowledge the Northwestern Polytechnical University High Performance Computing Center for the allocation of computing time on their machines.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17]
3.4. Mulliken population analysis The Mulliken population [33–35] of Ti2SiN is calculated by the CASTEP code to quantify the bond overlap as well as the charge transfer among Ti, Si and N atoms, and the results at zero pressure are shown in Table 3. It is shown that the charge transfer from Ti and Si to N after bonding is 0.70–0.72 electron. In the PBE result, there is 0.68 electron transferred from Ti to N atoms, but only 0.03 electron from Si to N. This result shows an ionic bonding feature of Ti2SiN with a strong directional bonding between Ti and N atoms. From the results of the DOS and Mulliken population, it can be concluded that bonding nature of Ti2SiN is metallic combined with some covalent and ionic. This conclusion will identify the associated properties of both ceramic and metal for Ti2SiN, and will guide its practical applications.
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