Advances in Condensed Matter Physics
Volume 2016 (2016), Article ID 3231027, 18 pages
http://dx.doi.org/10.1155/2016/3231027

Research Article

Pressure Prediction of Electronic, Anisotropic Elastic, Optical, and Thermal Properties of Quaternary (M2/3Ti1/3)₃AlC₂ (M = Cr, Mo, and Ti)

Liang Sun,¹ Yimin Gao,¹ Yangzhen Liu,¹ Guoliang Wang,² Yiran Wang,¹ Wenyan Zhai,¹ and Wen Wang³

¹State Key Laboratory for Mechanical Behavior of Materials, Xi’an Jiaotong University, Xi’an 710049, China
²Xi’an Research Institute of Hi-Tech, Xi’an 710025, China
³State Key Laboratory Base of Ecohydraulic Engineering in Arid Area, Xi’an University of Technology, Xi’an 710049, China

Received 29 May 2016; Revised 11 July 2016; Accepted 11 July 2016

Academic Editor: Veer P. S. Awana

Copyright © 2016 Liang Sun et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

The electronic, mechanical, anisotropic elastic, optical, and thermal properties of quaternary (M2/3Ti1/3)₃AlC₂ (M = Cr, Mo, and Ti) under different pressure are systematically investigated by first-principles calculations. The bonding characteristics of these compounds are the mixture of metallic and covalent bonds. With an increase of pressure, the heights of total density of states (TDOS) for these compounds decrease at Fermi level. The highest volume compressibility among three compounds is Mo₂TiAlC₂ for its smallest relative volume decline. The relative bond lengths are decreasing when the pressure increases. The bulk and shear modulus of the one doped with Cr or Mo are larger than those of Ti₃AlC₂ with pressure increasing. With an increase of pressure, the anisotropy of these compounds also increases. Moreover, Mo₂TiAlC₂ has the biggest anisotropy among the three compounds. The results of optical functions indicate that the reflectivity of the three compounds is high in visible-ultraviolet region up to ~10.5 eV under ambient pressure and increasing constantly when under pressure. Mo₂TiAlC₂ has the highest loss function. The calculated sound velocity and Debye temperature show that they all increase with pressure. of the three compounds is also calculated.

1. Introduction

Layered ternary compounds combine both the merits of ceramic and metal with high electrical and thermal conductivities, damage tolerance, excellent thermal shock resistance, high temperature strength, and well oxidation and corrosion resistance properties [1–5]. The standard formula of those compounds is (M represents an early transition metal, A is an A-group element which mostly is IIIA and IVA element, X means C or N atom, and ). When , more than 50 (211 phases) compounds have already been founded. They are all layered hexagonal structures with space group P₆₃/mmc consisting of alternate near-close-packed layers of octahedral interleafed with layers of A atoms [5]. However, there are six 312 phases when has been determined, that is, Ti₃AC₂ (A = Si, Ge, Sn, and Al) [6], V₃AlC₂, and Ta₃AlC₂, respectively. Only one polycrystalline, fully dense, predominantly single-phase bulk sample of Ta₄AlN₃ (when ) compound was synthesized using hot isostatic pressing [7]. While being based on theoretical first principle, the properties of several new phases like Ta₄AlC₃ [8], V₄SiC₃ [9], and Ti₄SiN₃ [10] are calculated but could not be factually fabricated in experiments.

Since 1996, high-purity Ti₃SiC₂ was firstly synthesized by Barsoum and El-Raghy [11] using HIP process and the experimental and theoretical studies on frication of 312 MAX phases were conducted [11–19]. Ti₃SnC₂ was discovered by Dubois et al. in 2007 [20]. Ti₃AlC₂ was first synthesized and fabricated by Pietzka and Schuster in 1994 and the unit cell is hexagonal with lattice parameters of  nm and  nm and its theoretical density is 4.25 g/cm³ [21, 22]. Both compounds were found to be isostructural with Ti₃SiC₂ and shared their many similar characteristics. Fully dense, predominantly single-phase samples of Ti₃Al1.1C1.8 were fabricated by Tzenov and Barsoum [23]. Thier thermal expansion coefficient (TEC) in the temperature range of 25–1200°C is  K⁻¹. The density of Ti₃AlC₂ is lower than that of Ti₃SiC₂ and Ti₃GeC₂, which is attractive as it is used as a structural material or reinforcement for polymers and metals. The macroscopic properties of Ti₃AlC₂ were thought to be strongly related to its electronic and structural properties [24].

First-principles calculations were supported to understand the electronic structure, elastic anisotropy, and other theoretical properties of this family of phases and to predicate new phases [6, 25–37]. Electrical structural, mechanical, and thermal properties of 312-phase like elastic constants have also been investigated by means of first-principles theory especially for Ti₃AlC₂ [23–27, 31, 32, 37–49]. Among 312 phases, some Al-containing ones like Ti₂AlC and Ti₃AlC₂ attract the most attention for their exceptional oxidation resistance to form a thin protective alumina layer on the surface [40, 41] and their crack self-healing characteristics [42]. In previous studies, only Ti, V, Cr, Nb, and Ta elements can be combined Al-containing MAX phase, while some exceptions as Zr, Nb Cr, Mn, and Mn atoms can be replaced the certain M element by solid solution to form quaternary Al-containing MAX phases like (Nb0.8Zr0.2)₂AlC [43], (Nb0.6Zr0.4)₂AlC [44], and (Cr0.7Mn0.3)₂AlC [45]. And the mass fraction of except M element is less than 0.5 of total M-content [46].

Until recently, two novel quaternary MAX phases have been synthesized: (Cr2/3Ti1/3)₃AlC₂ which reported on synthesizing by Cr₂AlC and TiC and (Mo2/3Ti1/3)₃AlC₂ synthesized by powders of Mo, Ti, Al, and graphite which have the similar crystallographic characteristics. Liu et al. [31] synthesized the (Cr2/3Ti1/3)₃AlC₂ and calculated its possessed ferromagnetism by first-principles calculations. Anasori et al. [46] firstly synthesized powder of (Mo2/3Ti1/3)₃AlC₂ and then calculated its space group and lattice parameters by Rietveld analysis of powder XRD (X-ray diffraction). Then, HSTM (high-resolution scanning transmission electron microscopy) was introduced to study the structural characteristics and its ordered phase with Ti layers sandwiched between two Mo layers in a Ti₃AlC₂ type structure. While there is no record on the electronic, optical, anisotropic elastic, and thermal properties of quaternary (M2/3Ti1/3)₃AlC₂ (M = Cr, Mo), also the behavior of these phases under high pressure is still unknown. In present paper, the authors’ purpose is to investigate relations of electronic, optical, anisotropic elastic, and thermal properties of these compounds under ambient and static high pressure which have rarely been reported.

2. Calculation Details

(M2/3Ti1/3)₃AlC₂ (M = Cr, Mo, and Ti) MAX phases have a crystal structure of isotopic like Ti₃SiC₂ and Ti₃AlC₂ with a hexagonal symmetry. The space group is P6₃/mmc which consists of periodic stacking of Al atoms layers. The pertinent features of the structure for the computations are that M atoms occupy the (1/3, 2/3, 0.132493) and the titanium atoms occupy (0, 0, 0) Wyckoff sites, respectively, while Al atoms occupy the 2b (0, 0, 0.25) and the carbon atoms are situated at the 4f (1/3, 2/3, 0.574905). For M = Ti atoms, we define the Ti atoms located at 2a positions as and those at 4f positions as . The crystal structure and calculated cell parameters of (M2/3Ti1/3)₃AlC₂ (M = Gr, Mo and Ti) are shown in Figure 1.

Figure 1: Crystal structures of M₂TiAlC₂ (M = Cr, Mo, and Ti) cell structures and Ti atoms located at 2a positions as and those at 4f positions as . Color of atoms: Ti: gray, M: red, C: black, and Al: blue.

All calculations in present work were performed based on the plane-wave pseudopotential density function theory (DFT) by CASPTEP code [48, 49]. The CASPTEP code employed special point integration over the Brillouin zone and a plane-wave basis set for the expansion of the wave functions which has been well illustrated in the literature [50]. Pseudoatomic calculations were performed for Cr: 3s² 3p⁶ 3d⁵ 4s¹, Mo: 4s² 4p⁶ 4d⁵ 5s¹, Ti: 3d² 4s², Al: 3s² 3p¹, and C: 2s² 2p², respectively. A gradient-corrected form of the exchange-correlation functional Perdew-Wang generalized-gradient approximation (GGA-PW91) was used [51]. The calculations have been performed using a plane-wave cutoff of 600 eV [52]. The Brillouin-zone sampling was performed using Monkhorst–Pack mesh [53] and the total energy of self-consistent convergence was at  eV/atom and the maximum force on the atom was below  eV/Å. The Broyden-Fletcher-Goldfarb-Shannon (BFGS) algorithm is introduced to relax the whole structure to reach the ground state where both cell parameters and fractional coordinates of atoms were optimized simultaneously. The lattice parameters of (M2/3Ti1/3)₃AlC₂ (M = Cr, Mo, and Ti) compounds and calculation method of our work are listed in Table 1. The lattice parameters of these compounds are in good agreement with other available experimental results. The induced pressure in this paper is defined as the force per unit area. A single number for the pressure implies that pressure is a scalar quantity, while, in fact, pressure can be seen as a tensor of the more general form [54]:

Table 1: The calculated cell parameters (, , and , in ; in ; in g/m³) of (M2/3Ti1/3)₃AlC₂ (M = Cr, Mo, and Ti).

Species
Space group
Opt method
Lattice parameters

Cr₂TiAlC₂
P₆₃MMC
GGA-PW91
SAED [31]
Cal. [31]
2.902
2.906
2.922
2.902
2.906
2.922
17.81
17.803
17.806
5.186
5.175
5.118
129.94
130.2
131.66

Mo₂TiAlC₂
P₆₃MMC
GGA-PW91
Exp. [46]
3.001
2.997
3.001
2.997
18.741
18.661
6.606
6.653
146.18
145.16

Ti₃AlC₂
P₆₃MMC
GGA-PW91
Exp. [22]
Cal. [24]
Cal. [47]
3.079
3.075
3.072
3.066
3.079
3.075
3.072
3.066
18.656
18.578
18.732
18.526
4.22
4.25
4.22
4.29
153.22
152.13
153.09
150.84

Each element of the tensor is the force that acts on the surface of the structure that has edges parallel to the -, -, and -axis. The changes in unit cell lattice parameters and volume resulting from the stress can be obtained from an analysis of dynamics trajectory data.

3. Results and Discussion

3.1. Electronic Properties under Pressure

The total and atomic site partial densities of states (PDOS) of (M2/3Ti1/3)₃AlC₂ (M = Cr, Mo, and Ti) under 0 GPa and 100 GPa pressure have been calculated, as shown in Figures 2–4, respectively. The atomic bonding characteristics are clearly illustrated in these PDOS and for C atoms do not display significantly the TDOS at the Fermi energy. Therefore, C atoms are not involved in electronic transport. Figures 2(a), 3(a), and 4(a) show the electronic structure of three compounds at 0 GPa while Figures 2(b), 3(b), and 4(b) display these of under 100 GPa. The Fermi level is set at and it can be clearly seen that there is a certain value of Fermi energy () of total DOS at the Fermi level. Therefore, three compounds in this work exhibit metallic properties, such as metallic conductivity. When pressure increased, the height of TDOS peaks decreases and of total DOS at the Fermi level also went down constantly. According to the results of Li et al. [55], it can be learnt that the minimum (maximum) of the DOS qualitatively indicates their stability (instability). Thus, the stability of these compounds becomes higher with an increase of pressure.

(a)

(b)

Figure 2: The total and atomic site partial densities of states for Cr₂TiAlC₂: (a) 0 GPa and (b) 100 GPa. The dot lines represent the position of the Fermi surface.

(a)

(b)

Figure 3: The total and atomic site partial densities of states for Mo₂TiAlC₂: (a) 0 GPa and (b) 100 GPa. The dot lines represent the position of the Fermi surface.

(a)

(b)

Figure 4: The total and atomic site partial densities of states for Ti₃AlC₂: (a) 0 GPa and (b) 100 GPa. The dot lines represent the position of the Fermi surface.

From the partial DOS of each element shown in Figures 2–4, it is found that there is mainly a contribution of C 2s states for the energy range of  eV. From −7.5 to 0 eV, the DOS of three compounds are dominated by hybridizing Ti 3d, M (M = Cr, Mo, and Ti) 3d, Al 3s/3p, and C 2p states. Particularly, the total DOS at Fermi level manly originates from Ti or M (M = Cr, Mo, and Ti) 3d states. Cr₂TiAlC₂ shows the biggest of the others because of Cr 3d states. Meanwhile, strong p-d hybridization can be seen between Ti-C and M (Cr, Mo, or Ti)-C atoms. Thus, there are covalent bonding contributions between Ti-C and M-C atoms. In the pervious work [13, 56], the electronic structure of M₂InC (M = Ti, Zr, and Hf) and Ti₃AC₂ (A = Si, Ge, and Sn) phases is affected by the M element.

3.2. Structural Anticompressibility

The relative volume of V/V ₀ and bond length of three quaternary (M2/3Ti1/3)₃AlC₂ atoms are calculated with a function of pressure in order to investigate their physical change and the results are shown in Figure 5. The relative volume is defined as the volume anticompressibility which decreases with the increase of pressure from 0 to 100 GPa, as shown in Figure 5(a). And the highest volume anticompressibility among three compounds is Mo₂TiAlC₂ for the smallest relative volume decline.

(a)

(b)

Figure 5: Variation of the relative volume V/V ₀ (a) and bond length (b) of three quaternary (M2/3Ti1/3)₃AlC₂ (M = Cr, Mo, and Ti) compounds with pressure.

More insight into the chemical bonds will be gained by investigating the relative bonds length evolving with pressure and the results are shown in Figure 5(b). The relative bonds lengths are decreasing when the pressure increases. bond in Cr₂TiAlC₂ and Mo₂TiAlC₂ reflects almost the same tendency which is stronger than the one in Ti₃AlC₂. The curve of Mo-C bond is above that of Cr-C and which shows the highest covalent strength and also the Mo-Al bond strength. Thus, Mo₂TiAlC₂ shows the highest volume anticompressibility because of its high strength of Mo-C and Mo-Al bond.

3.3. Anisotropic Elastic Properties under Pressure

The calculated full set of second-order elastic constants of (M2/3Ti1/3)₃AlC₂ under 0 GPa and 100 GPa via stress versus strain approach are included in Table 2 [57]. The method can be shown as follows: one can calculate all independent elastic constants by Hooker’s law. For hexagonal structure, the criterion can be written as

Table 2: Theoretically calculated elastic properties including elastic constants ( in GPa), bulk modulus ( in GPa), shear modulus ( in GPa), Young’s modulus ( in GPa), and Poisson’s ratio () for M₂TiAlC₂ (M = Cr, Mo, and Ti).

Spaces
Pressure

Cr₂TiAlC₂
0 GPa
376.3
98.2
119
352.6
146.4
139
197.5
137.1
1.44
315
0.234

100 GPa
850.8
432.6
440.1
814.3
362.5
209.1
571.2
256.7
2.23
538.2
0.343

Mo₂TiAlC₂
0 GPa
381.7
141.6
131.4
372.6
149.7
120
216
131.9
1.64
307.7
0.263

100 GPa
843.6
479
467.6
788.8
352.9
182.3
588.9
235.4
2.5
482.4
0.364

Ti₃AlC₂
0 GPa
353.3
74.1
67.6
297.6
134.1
139.6
157.5
134.3
1.17
310.5
0.173

0 GPa [47]
368
81
76
313
130
143
168
135
1.24
320
0.18

0 GPa [12]

165
124
1.35
297.5
0.2

100 GPa
805.5
327.9
371.7
727.6
292.6
238.8
497.8
245.6
2.03
551.1
0.316

In (1), represents the nonzero elastic constants; and are the normal and shear stresses, and the corresponding uniaxial and shear strains are given by and , respectively. Then, the following two strain modes were applied to compute in the present paper:

Here, is the transpose of the strain matrix, and is the magnitude of Lagrangian strain. The abovementioned two different strain patterns are applied to the optimized crystal structure by varying the strain amplitude for each strain pattern. The number of steps for each strain is set as four in this work, and the maximum strain amplitude is 0.003. Then, the stress tensor can be evaluated as a function of strain.

Those parameters determine the response of the crystal to external forces which plays a crucial role in several engineering application as abrasive resistance phases. Several fundamental physical properties, like elastic anisotropic properties, specific heat, and Debye temperature, are related to those parameters. For anisotropic elasticity, the linear anisotropic ratio of of a hexagonal crystal can be expressed as [58]

In the present work, the linear anisotropic ratio at 0 GPa is found to be 1.01 for Cr₂TiAlC₂, 1.08 for Mo₂TiAlC₂, and 1.27 for Ti₃AlC₂. Thus, the ratio increases from Cr and Mo to Ti for M element which reflects the increase of the anisotropic elasticity. When pressure increases to 100 GPa, the ratio is changed to be 1.08 for Cr₂TiAlC₂, 1.21 for Mo₂TiAlC₂, and 1.09 for Ti₃AlC₂. Clearly, the ratio increases, accompanied with pressure of Cr₂TiAlC₂ and Mo₂TiAlC₂ while decreasing for Ti₃AlC₂.

Based on these elastic constants, the polycrystalline bulk modulus (the resistance of a material to hydrostatic pressure), shear modulus (the resistance of a material to shear), and Young’s modulus (resistance against uniaxial tensions or compression) can be given by Reuss and Voigt methods [59, 60]. The calculated elastic constants at 0 GPa and 100 GPa are also shown in Table 2 and the real polycrystalline values are estimated by Hill’s average [61]. The calculated , , and under pressure are displayed in Figure 6. From the figure, all elastic moduli increase with pressure and Mo₂TiAlC₂ has both the highest bulk modulus and smallest Young’s modulus while Ti₃AlC₂ has the smallest bulk modulus and highest Young’s modulus. The bulk modulus and shear modulus doped with Cr or Mo are larger than Ti₃AlC₂. At 0 GPa, Poisson’s ratio of these compounds is much smaller than 0.3. With an increase of pressure, Poisson’s ratio is close to 0.3. As Greaves et al. mentioned [62], Poisson’s ratio tends to increase with the atomic packing density, so that (where cd, bcc, fcc, and hcp represent cubic-diamond, body-centered cubic, face-centered cubic, and hexagonal close-packed crystalline structures, resp.). Moreover, for a certain crystalline structure and valence, Poisson’s ratio mostly increases with atomic number, and also the electronic band structure and the valence electron density come into play. As a result, the increase of Poisson’s ratio means the increase of atomic packing density and electronic band structure.

(a)

(b)

Figure 6: Variation of the bulk modulus, shear modulus (a), and Young’s modulus (b) of three quaternary (M2/3Ti1/3)₃AlC₂ (M = Cr, Mo, and Ti) compounds with pressure.

The value of is generally applied to indicate the compound with ductility or brittleness. It is supposed that the ductility (brittleness) compound, , is larger (lower) than 1.75. In our case, the compounds are brittle at 0 GPa. With an increase of pressure, the compounds present ductile. doped with Cr or Mo are larger than Ti₃AlC₂ either at 0 GPa or at 100 GPa.

All single crystals exist anisotropy in this work, anisotropic index [63] is introduced to present anisotropy of the compounds and their shear anisotropic factors:

Here, , , , and are the bulk modulus and shear modulus estimation within the Voigt and Reuss methods, respectively. The anisotropic index () is the appropriate indicator of the mechanical anisotropic properties of compound. The larger is, the stronger the anisotropy of a compound is. For some single crystal data obtained from Ledbetter, of Mg is 0.04 and α- Ti is 0.19 and for the majority of hexagonal crystals, they have the least anisotropy index among all the systems [63, 64]. Based on the abovementioned discussion, the values at 0 GPa are 0.033, 0.058, and 0.023 for Cr₂TiAlC₂, Mo₂TiAlC₂, and Ti₃AlC₂, respectively. Thus, the highest anisotropy is Mo₂TiAlC₂ among three compounds. Moreover, the values at 100 GPa are 0.418, 0.584, and 0.151 for Cr₂TiAlC₂, Mo₂TiAlC₂, and Ti₃AlC₂, respectively. With an increase of pressure, the anisotropy of compounds increases.

Herein, a three-dimensional plot of the mechanical modulus as a function of crystallographic orientation can straightly show mechanical anisotropic properties of each compound. The elastic compliance constants of each compound are used to study anisotropy properties of bulk and Young’s modulus from different directions of solid. For hexagonal crystal, the directional dependence of the bulk modulus and Young’s modulus could be calculated as follows [65]:

In the equations above, , , , , and represent elastic compliance constants and , , and are the direction cosines. The relationships of the direction cosines in spherical coordinates with respect to and ϕ are as follows: ,  , and into (3). We obtain the equations to plot three-dimensional anisotropic mechanical figures which are shown in Figures 7 and 8; the details of our method have been presented in our previous work [66].

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

Figure 7: The surface construction of bulk modulus with different pressure for (M2/3Ti1/3)₃AlC₂ (M = Cr, Mo, and Ti) and Cr₂TiAlC₂ (a–c); Mo₂TiAlC₂ (d–f); Ti₃AlC₂ (g–i).

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

Figure 8: The surface construction of bulk modulus with different pressure for (M2/3Ti1/3)₃AlC₂ (M = Cr, Mo, and Ti), Cr₂TiAlC₂ (a–c); Mo₂TiAlC₂ (d–f); Ti₃AlC₂ (g–i).

From Figure 7, the three-dimensional plot of bulk modulus can straightly show that there are little anisotropic properties of bulk modulus. All the plots tend to be sphere and the size of the sphere reflects the value of bulk modulus. When the pressure increases, the size of the sphere also increases. The anisotropic properties of Young’s modulus are very important for the layer-like structure MAX phases and the properties under pressure are unknown. It can be seen from Figure 8 that Mo₂TiAlC₂ has the biggest anisotropy at 0 GPa with the increasing pressure; it tends to be stronger anisotropy than that of other two compounds, when at 50 GPa and 100 GPa, Mo₂TiAlC₂ shows the strongest anisotropy which means its layer-like structure. The results are in excellent agreement with the aforementioned discussion.

3.4. Theoretical Vickers Hardness under Pressure

The bond compressibility was discussed in Section 3.2 while the Mulliken bond populations can also be used to calculate the bonding behavior and the theoretical Vickers hardness could be obtained under pressure of 0 GPa and 100 GPa. The relevant formula for the hardness is given as [67, 68]

where is the hardness of a compound; is the hardness of type bond; is the bond length; is the volume of type bond; refers to the type bond density per cubic angstroms; and is the total number of type bonds in the cell; Ω and are the cell volume and overlap population of type bond, respectively.

With the help of Mulliken population analysis, the theoretical Vickers hardness values of M₂TiAlC₂ are shown in Table 3. At 0 GPa, the calculated result for Cr₂TiAlC₂ is 10.93 GPa, Mo₂TiAlC₂ is 9.35 GPa, and Ti₃AlC₂ is 9.93, respectively. Several experimental results have already been obtained: The intrinsic hardness for Ti₃AlC₂ measured by nanoindentation experiments is about 11.4 GPa [69]. The hardness values measured by microindentations are almost in the range of 3–7 GPa for Ti₃AlC₂ [70–72]. Because gain boundaries and impurities affect the measured hardness, the experimental results of hardness are always underestimated. Thus, our result seems reasonable, while for Cr₂TiAlC₂ and Mo₂TiAlC₂, it is suggested that they also have higher hardness. When all the phases are under 100 GPa, it is clear to see that the theoretical Vickers hardness values of M₂TiAlC₂ all increased according to the pressure. Based on the abovementioned discussion, the bond length decreased with an increase of pressure. The results indicated that the hardness increases with an increase of pressure, which is in excellent agreement with our results. The hardness of the compound doped with Cr and Mo is larger than that of the compound doped with Ti.

Table 3: Calculated theoretical Vickers hardness values of M₂TiAlC₂.

Species
Pressure
Bond
()

Ω ()

(GPa)
(GPa)

Cr₂TiAlC₂
0 GPa
C-Cr
C-Ti
Al-Cr
1.96
2.14
2.68
1.33
0.65
0.37
4
4
4
129.94
6.69
8.70
17.09
41.44
13.07
2.41
10.93

100 GPa
C-Cr
C-Ti
Al-Cr
1.84
1.97
2.34
1.19
0.44

4
4
4
98.38
5.74
4.56
7.65
47.84
25.91

21

Mo₂TiAlC₂
0 GPa
C-Mo
C-Ti
Al-Mo
2.11
2.16
2.79
1.20
0.75
0.41
4
4
4
146.18
8.33
8.94
19.27
25.92
14.41
2.19
9.35

100 GPa
C-Mo
C-Ti
Al-Mo
1.99
1.98
2.47
1.05
0.6

4
4
4
112.5
7.21
5.3
10.2
28.83
27.56

12.19

Ti₃AlC₂
0 GPa
C-Ti
C-Ti
Al-Ti
2.07
2.20
2.91
1.16
0.87
0.48
4
4
4
153.22
7.69
9.24
21.38
28.65
15.82
2.16
9.93

100 GPa
C-Ti
C-Ti
Al-Ti
1.92
2.0
2.47
0.92
0.73

4
4
4
110.58
6.49
5.37
10.11
30.15
32.81

15.04

3.5. Optical Properties

The study of optical functions of solids helps to gain a better understanding of the electronic structure. CASTEP code is used to calculate the optical properties of solids due to electronic transitions.

In general, the difference in the propagation of an electromagnetic wave through vacuum or other materials can be described by a complex refractive index, :

In vacuum, is real and equal to unity. For transparent materials, it is purely real, with the imaginary part being related to the absorption coefficient by

Then, the reflection coefficient can be obtained for the simple case of normal incidence onto a plane surface by matching both the electric and magnetic fields at the surface:

The complex dielectric constant and the correlation between the dielectric function and refractive index can be shown as follows:

The relation between the real and imaginary parts of the refractive index and dielectric constant is as

Another frequently used quantity for expressing optical properties is the optical conductivity :

Here, is the angular frequency. The imaginary part of the dielectric function can be thought of as detailing the real transition between occupied and unoccupied electric states and given by [73]

is the energy of the incident photon, is the momentum operator, and is the Fermi function. The real part of the dielectric function can be obtained by the Kramers-kronig transform which links both the real and imaginary parts. The optical properties of three compounds under 0 GPa and 100 GPa are calculated in the present paper. All the calculations in this work used 0.5 eV Gaussian smearing and the results at 0 GPa agree well with the pervious work: [74] and [75].

When all the phases are under 0 GPa (the calculated absorption spectrums are shown in Figure 9(a)(1)), there are three peaks for Mo₂TiAlC₂ (6.7 eV, 13.4 eV, and 36.5 eV) and Ti₃AlC₂ (5.5 eV, 12.4 eV, and 35.5 eV) as well as four peaks for Cr₂TiAlC₂ (5.9 eV, 11.8 eV, 35.1 eV, and 44.2 eV), respectively. All the peaks are related to the transition from Ti atoms d to p states and M atoms d to p states due to their metallic nature. The photoconductivity reflects the electrical conductivity of material increase as a result of absorbing photons. In Figure 9(a)(2), there are two peaks for Mo₂TiAlC₂ (3.8 eV and 35.8 eV) and Ti₃AlC₂ (3.0 eV and 34.9 eV) as well as four peaks for Cr₂TiAlC₂ (0.74 eV, 3.8 eV, 35 eV, and 43.9 eV), respectively. The reflectivity as a function of photon energy is presented in Figure 9(a)(3). It can be found that the reflectivity of three compounds is high in visible and ultraviolet region. In the visible region (0~4 eV), Mo₂TiAlC₂ has the highest reflectivity comparing to Ti₃AlC₂ which has the smallest one, while in the ultraviolet region, three compounds reach their maximum at around 9~11 eV and Mo₂TiAlC₂ is in the front rank which provides us with a well potential material as a coating film. The refractive index and extinction coefficient are illustrated in Figure 9(a)(4). It is found that the static refractive index is 15.5 eV for Cr₂TiAlC₂, 12.9 eV for Mo₂TiAlC₂, and 9.3 eV for Ti₃AlC₂, respectively. Generally, the refractive index and extinction coefficient are in direct proportion to the real part and the imaginary part of the dielectric constant, respectively. Comparing with and for all compounds, they have the same trend which displays that the origin of the structures in the imaginary part of the dialectic function also explains the structures in the refractive index. The calculated loss function is shown in Figure 9(a)(5) and the peak of is related to the trailing edge in the reflection spectra. The main peak in spectra represents the characteristic associated with the plasma response and corresponding frequency is plasma frequency and Mo₂TiAlC₂ has the highest loss function at 15 eV. The calculated real and imaginary part of the dielectric function are shown in Figure 9(a)(6). The dielectric function presents similar variation trends with those of refractive index; its giant real part is clearly seen. The peaks at about 1 eV correspond probably to the transition between the Al and C states or the Ti and M states, where the ambiguous origin is attributed mainly to the similar peak intervals. The real part vanishes at about 7.5 eV. Metallic reflectance characteristics are exhibited in the range of . The peak of the imaginary part of the dielectric function is related to the electron excitation. For the imaginary part of , the peak for <1.5 eV is due to the intraband transitions.

(a)

(b)

Figure 9: Optical properties of (M2/3Ti1/3)₃AlC₂ (M = Cr, Mo, and Ti) under 0 GPa and 100 GPa (1) 0 GPa, and (2) 100 GPa.

When under 100 GPa, it can be seen that the absorption index increases with the pressure from Figure 9(b)(1). The number of peaks for each compound is the same as that under 0 GPa pressure and the energy also increases constantly. Clearly, accompanied with the pressure, their metallic nature increases. Figure 9(b)(2) shows the photoconductivity of each compound increases with energy and pressure. For Cr₂TiAlC₂, there is one more peak appearing around 6.3 eV. As in Figure 9(b)(3), from the range of 0~30 eV, Mo₂TiAlC₂ has the highest conductivity and reflectivity. From the range of 30~60 eV, Ti₃AlC₂ has the highest conductivity and reflectivity. It is found that the reflectivity of three compounds is the highest in visible and ultraviolet parts up to ~12 eV, and for Mo₂TiAlC₂ when pressure is high the peaks of reflectivity will appear in the ultraviolet range. The refractive index and extinction coefficient are illustrated in Figure 9(b)(4). The static refractive index is found and is 12 eV for Cr₂TiAlC₂, 11.9 eV for Mo₂TiAlC₂, and 13.3 eV for Ti₃AlC₂. It means the refractive index of Cr₂TiAlC₂ and Mo₂TiAlC₂ decreases and that of Ti₃AlC₂ increases comparing with that at ambient pressure. In Figure 9(b)(5), as discussed above, Mo₂TiAlC₂ has the highest loss function under 100 GPa pressure. The calculated real and imaginary part of the dielectric function are shown in Figure 9(b)(6); the real part vanishes at about 9.5 eV.

3.6. Thermal Properties

The Debye temperature could be estimated by the mean sound velocity of , using the following equation [19]:

where is the Debye temperature, represents Plank’s constant, represents Boltzmann’s constant, is the number of atoms in unit cell, is the volume of unit cell, and is the circumference ratio. The polycrystalline structure of mean sound velocity is calculated by (12) [51, 76]:

and represent the longitudinal and transverse sound velocity, respectively. It is determined that and can be calculated by the bulk modulus and shear modulus from Navier’s equations:

The calculated sound velocity and Debye temperature at ambient pressure and under pressure of M₂TiAlC₂ are shown in Table 4. Clearly, the sound velocity and Debye temperature of each compound increase with pressure. Bai et al. [56] calculated the sound velocity and Debye temperature of Ti₃SiC₂ and Ti₃GeC₂, while the results agreed well with that measured in experiments [12, 77]. It is suggested that the method used for estimating the elastic properties and Debye temperature of ternary triennium carbide is accurate. Figure 10 shows the Debye temperature of three compounds as a function of pressure. Ti₃AlC₂ has the largest Debye temperature under pressure.

Table 4: Theoretically calculated Debye temperature ( in K) and sound velocity (, , and in m/s) under various pressure (GPa) for Al₄Si₂C₅.

Pressure (GPa)
Cr₂TiAlC₂
Mo₂TiAlC₂
Ti₃AlC₂

Debye ()

Debye ()

Debye ()

0
764.4
5.15
8.57
5.69
641.1
4.47
7.71
4.96
789.6
5.65
8.94
6.21

10
808.8
5.35
9.13
5.94
677.6
4.66
8.18
5.18
828.1
5.79
9.45
6.4

20
853.6
5.58
9.55
6.18
706.1
4.79
8.55
5.33
865.7
5.95
9.92
6.58

30
879.8
5.68
9.86
6.3
730.5
4.9
8.85
5.46
904.9
6.14
10.32
6.79

40
896.7
5.72
10.13
6.36
747.3
4.96
9.11
5.53
927.6
6.21
10.62
6.88

50
920.2
5.81
10.42
6.47
763.9
5.02
9.35
5.61
948.7
6.28
10.9
6.97

60
938.4
5.87
10.71
6.55
780.8
5.09
9.57
5.69
977.3
6.4
11.2
7.11

70
955.7
5.93
10.89
6.62
794
5.13
9.76
5.74
983.2
6.38
11.36
7.1

80
977
6.02
11.15
6.72
806.2
5.17
9.95
5.79
993.6
6.39
11.53
7.12

90
993.3
6.08
11.33
6.79
816.9
5.21
10.11
5.83
1001.9
6.38
11.67
7.12

100
1008.7
6.13
11.56
6.85
827.7
5.24
10.27
5.87
1025
6.49
11.89
7.24

Figure 10: Debye temperature of (M2/3Ti1/3)₃AlC₂ (M = Cr, Mo, and Ti) compounds as a function of pressure.

The specified heat capacity at constant volume as a function of temperature can be calculated by the Debye QHA [78]:

In the equation above, is Avogadro’s constant and is the Boltzmann constant. Using the calculated Debye temperatures, the difference in specific heat for each compound with pressure in the series calculated is shown in Figure 11. The pressure range is 0, 20, 40, 60, 80, and 100 GPa.

Figure 11: Heat capacity of (M2/3Ti1/3)₃AlC₂ (M = Cr, Mo, and Ti) compounds under different temperature.

From the figure, of each compound decrease slightly with pressure and increase dramatically with temperature. When temperature is below 500 K, increases rapidly with well accompanied with the well-known Debye model; when temperature is above 500 K, increases slowly with the temperature approaching the Dulong–Petit limit as 149.652 J·mol⁻¹ K⁻¹. In our case, , where is the gas constant equal to 8.314 J·mol⁻¹ K⁻¹.

4. Conclusions

The pressure prediction of electronic, anisotropic elastic, optical, and thermal properties of quaternary (M2/3Ti1/3)₃AlC₂ (M = Cr, Mo, and Ti) were investigated under pressure from 0 GPa to 100 GPa by first-principles calculations. There is a certain value of total DOS at the Fermi level which exhibits metallic property, such as metallic conductivity. Cr₂TiAlC₂ shows the biggest value of because of Cr 3d states. For the strong p-d hybridization, there are covalent bonding contributions between Ti-C and M-C atoms. Besides, the highest volume compressibility among three compounds is Mo₂TiAlC₂ for its smallest relative volume decline. The relative bonds lengths are decreasing when the pressure increases while Mo-C bond and Mo-Al bond have high strength.

All elastic moduli increase with pressure and Mo₂TiAlC₂ got the highest bulk modulus but the smallest Young’s modulus while Ti₃AlC₂ has the smallest bulk modulus and the highest Young’s modulus. Also, Cr₂TiAlC₂ and Mo₂TiAlC₂ have higher hardness by the theoretical hardness calculations. When under 100 GPa, it is clearly seen that the theoretical Vickers hardness values of all compounds increased according to the pressure. From the analysis of optical functions of all compounds, it is found that the reflectivity of three compounds is high in visible-ultraviolet region up to ~10.5 eV under ambient pressure and increasing constantly when under pressure. Mo₂TiAlC₂ has the highest loss function. The calculated sound velocity and Debye temperature at ambient pressure and under pressure show they all increase with pressure. of each compound decreases slightly with pressure and increases dramatically with temperature.

Competing Interests

The authors declare that they have no competing interests.

Acknowledgments

The authors would like to thank Professor Y. H. Cheng for providing the CASTEP code. This work was supported by the Natural Science Foundation of China (51501139), the Science and Technology Project of Guangdong Province in China (Special Foundation for Additive Manufacturing Technologies, no. 2015B010122003, and Special Foundation for Practical Science and Technology Research and Development Project, no. 2015B090926009), and the Postdoctoral Science Foundation Funded Project of China (no. 2014M552434).

References

W. Jeitschko, H. Nowotny, and F. Benesovsky, “Kohlenstoffhaltige ternäre Verbindungen (H-phase),” Monatshefte für Chemie, vol. 94, no. 4, pp. 672–676, 1963.

Y. C. Zhou and Z. M. Sun, “Microstructure and mechanism of damage tolerance for Ti₃SiC₂ bulk ceramics,” Materials Research Innovations, vol. 2, no. 6, pp. 360–363, 1999.

M. W. Barsoum, H.-I. Yoo, I. K. Polushina, V. Y. Rud', Y. V. Rud', and T. El-Raghy, “Electrical conductivity, thermopower, and Hall effect of Ti₃AlC₂, Ti₄AlN₃, and Ti₃SiC₂,” Physical Review B—Condensed Matter and Materials Physics, vol. 62, no. 15, pp. 10194–10198, 2000.

C. Hu, Y. Sakka, H. Tanaka, T. Nishimura, and S. Grasso, “Low temperature thermal expansion, high temperature electrical conductivity, and mechanical properties of Nb₄AlC₃ ceramic synthesized by spark plasma sintering,” Journal of Alloys and Compounds, vol. 487, no. 1-2, pp. 675–681, 2009.

M. W. Barsoum, “The phases: a new class of solids: thermodynamically stable nanolaminates,” Progress in Solid State Chemistry, vol. 28, no. 1–4, pp. 201–281, 2000.

M. Y. Gamarnik and M. W. Barsoum, “Bond lengths in the ternary compounds Ti₃SiC₂, Ti₃GeC₂ and Ti₂GeC,” Journal of Materials Science, vol. 34, no. 1, pp. 169–174, 1999.

M. W. Barsoum, T. El-Raghy, and A. T. Procopio, “Synthesis of Ti₄AIN₃ and phase equilibria in the Ti-AI-N system,” Metallurgical and Materials Transactions A: Physical Metallurgy and Materials Science, vol. 31, no. 2, pp. 373–378, 2000.

M. Dahlqvist, B. Alling, and J. Rosén, “Stability trends of MAX phases from first principles,” Physical Review B, vol. 81, no. 22, Article ID 220102, 2010.

F. Parvin, M. M. Hossain, and A. Islam, “Phase stability and physical properties of hypothetical V₄SiC₃,” https://arxiv.org/abs/1110.4452

M. M. Hossain, M. S. Ali, and A. K. M. A. Islam, “Structural, elastic, electronic and optical properties of a newly predicted layered-ternary Ti₄SiN₃: a first-principles study,” https://arxiv.org/abs/1108.0553

M. W. Barsoum and T. El-Raghy, “Synthesis and characterization of a remarkable ceramic: Ti₃SiC₂,” Journal of the American Ceramic Society, vol. 79, no. 7, pp. 1953–1956, 1996.

M. Radovic, M. W. Barsoum, A. Ganguly et al., “On the elastic properties and mechanical damping of Ti₃SiC₂, Ti₃GeC₂, Ti₃Si0.5Al0.5C₂ and Ti₂AlC in the 300–1573 K temperature range,” Acta Materialia, vol. 54, no. 10, pp. 2757–2767, 2006.

X. D. He, Y. L. Bai, Y. B. Li, C. C. Zhu, and M. Li, “Ab initio calculations for properties of MAX phases Ti₂InC, Zr₂InC, and Hf2InC,” Solid State Communications, vol. 149, no. 13-14, pp. 564–566, 2009.

S. Li, R. Ahuja, M. W. Barsoum, P. Jena, and B. Johansson, “Optical properties of Ti₃SiC₂ and Ti₄AlN₃,” Applied Physics Letters, vol. 92, no. 22, Article ID 221907, 2008.

T. Goto and T. Hirai, “Chemically vapor deposited Ti₃SiC₂,” Materials Research Bulletin, vol. 22, no. 9, pp. 1195–1201, 1987.

Y. C. Zhou, Z. M. Sun, X. H. Wang, and S. Q. Chen, “Ab initio geometry optimization and ground state properties of layered ternary carbides Ti₃MC₂ (M = Al, Si and Ge),” Journal of Physics: Condensed Matter, vol. 13, no. 44, pp. 10001–10010, 2001.

R. Ahuja, O. Eriksson, J. M. Wills, and B. Johansson, “Electronic structure of Ti₃SiC₂,” Applied Physics Letters, vol. 76, no. 16, pp. 2226–2228, 2000.

P. Finkel, M. W. Barsoum, and T. El-Raghy, “Low temperature dependencies of the elastic properties of Ti₄AlN₃, Ti₃Al1.1C 1.8, and Ti₃SiC₂,” Journal of Applied Physics, vol. 87, no. 4, pp. 1701–1703, 2000.

Z. M. Sun, S. Li, R. Ahuja, and J. M. Schneider, “Calculated elastic properties of M₂AlC (M=Ti, V, Cr, Nb and Ta),” Solid State Communications, vol. 129, no. 9, pp. 589–592, 2004.

S. Dubois, T. Cabioc'h, P. Chartier, V. Gauthier, and M. Jaouen, “A new ternary nanolaminate carbide: Ti₃SnC₂,” Journal of the American Ceramic Society, vol. 90, no. 8, pp. 2642–2644, 2007.

M. Pietzka and J. C. Schuster, “The ternary boundary phases of the quaternary system Ti-Al-CN,” in Concerted Action on Materials Science, Leuven Proceedings, Part A, Commission of the European Communities, 1992.

M. A. Pietzka and J. C. Schuster, “Summary of constitutional data on the Aluminum-Carbon-Titanium system,” Journal of Phase Equilibria, vol. 15, no. 4, pp. 392–400, 1994.

N. V. Tzenov and M. W. Barsoum, “Synthesis and characterization of Ti₃AlC₂,” Journal of the American Ceramic Society, vol. 83, no. 4, pp. 825–832, 2000.

Y. C. Zhou, X. H. Wang, Z. M. Sun, and S. Q. Chen, “Electronic and structural properties of the layered ternary carbide Ti₃AlC₂,” Journal of Materials Chemistry, vol. 11, no. 9, pp. 2335–2339, 2001.

A. Bouhemadou, “Calculated structural, electronic and elastic properties of M₂GeC (M=Ti, V, Cr, Zr, Nb, Mo, Hf, Ta and W),” Applied Physics A: Materials Science and Processing, vol. 96, no. 4, pp. 959–967, 2009.

A. Bouhemadou, “Prediction study of structural and elastic properties under pressure effect of M₂SnC (M=Ti, Zr, Nb, Hf),” Physica B: Condensed Matter, vol. 403, no. 17, pp. 2707–2713, 2008.

A. Bouhemadou and R. Khenata, “Prediction study of structural and elastic properties under the pressure effect of M₂GaC (M=Ti, V, Nb, Ta),” Journal of Applied Physics, vol. 102, no. 4, Article ID 043528, 2007.

C. Hu, H. Zhang, F. Li, Q. Huang, and Y. Bao, “New phases' discovery in MAX family,” International Journal of Refractory Metals and Hard Materials, vol. 36, pp. 300–312, 2013.

M. Magnuson, J.-P. Palmquist, M. Mattesini et al., “Electronic structure investigation of Ti₃AlC₂, Ti₃SiC₂, and Ti₃GeC₂ by soft x-ray emission spectroscopy,” Physical Review B, vol. 72, no. 24, Article ID 245101, 2005.

Y. Zhou and Z. Sun, “Electronic structure of the layered compound Ti₃GeC₂,” Journal of Applied Physics, vol. 86, no. 3, pp. 1430–1432, 1999.

Z. Liu, E. Wu, J. Wang et al., “Crystal structure and formation mechanism of (Cr2/3Ti1/3)₃AlC₂ MAX phase,” Acta Materialia, vol. 73, pp. 186–193, 2014.

C. M. Fang, R. Ahuja, O. Eriksson et al., “General trend of the mechanical properties of the ternary carbides M₃SiC₂ (M = transition metal),” Physical Review B, vol. 74, no. 5, Article ID 054106, 2006.

W. Luo, C. M. Fang, and R. Ahuja, “Nanolayered max phases from ab initio calculations,” in Condensed Matter Theories, pp. 207–211, World Scientific Publishing, 2009.

J. Etzkorn, M. Ade, and H. Hillebrecht, “Ta₃AlC₂ and Ta₄AlC₃—single-crystal investigations of two new ternary carbides of tantalum synthesized by the molten metal technique,” Inorganic Chemistry, vol. 46, no. 4, pp. 1410–1418, 2007.

Y. Mo, P. Rulis, and W. Y. Ching, “Electronic structure and optical conductivities of 20 MAX-phase compounds,” Physical Review B, vol. 86, no. 16, Article ID 165122, 2012.

C. Li, B. Wang, Y. Li, and R. Wang, “First-principles study of electronic structure, mechanical and optical properties of V₄AlC₃,” Journal of Physics D: Applied Physics, vol. 42, no. 6, Article ID 065407, 2009.

C. Li, J. Kuo, B. Wang, Y. Li, and R. Wang, “A new layer compound Nb₄SiC₃ predicted from first-principles theory,” Journal of Physics D: Applied Physics, vol. 42, no. 7, Article ID 075404, 2009.

Z. M. Sun and Y. C. Zhou, “Ab initio calculation of titanium silicon carbide,” Physical Review B, vol. 60, no. 3, pp. 1441–1443, 1999.

Y. Zhou and Z. Sun, “Electronic structure and bonding properties in layered ternary carbide Ti₃SiC₂,” Journal of Physics Condensed Matter, vol. 12, no. 28, pp. L457–L462, 2000.

X. H. Wang and Y. C. Zhou, “High-temperature oxidation behavior of Ti₂AlC in air,” Oxidation of Metals, vol. 59, no. 3-4, pp. 303–320, 2003.

M. Sundberg, G. Malmqvist, A. Magnusson, and T. El-Raghy, “Alumina forming high temperature silicides and carbides,” Ceramics International, vol. 30, no. 7, pp. 1899–1904, 2004.

H. J. Yang, Y. T. Pei, J. C. Rao, and J. T. M. De Hosson, “Self-healing performance of Ti₂AlC ceramic,” Journal of Materials Chemistry, vol. 22, no. 17, pp. 8304–8313, 2012.

M. Naguib, G. Bentzel, J. Shah et al., “New solid solution MAX phases: (Ti0.5, V0.5)₃AlC₂,(Nb0.5, V0.5)₂AlC,(Nb0.5, V0.5)₄AlC₃ and (Nb0.8, Zr0.2)₂AlC,” Materials Research Letters, vol. 2, no. 4, pp. 233–240, 2014.

H. Nowotny, P. Rogl, and J. C. Schuster, “Structural chemistry of complex carbides and related compounds,” Journal of Solid State Chemistry, vol. 44, no. 1, pp. 126–133, 1982.

A. Mockute, J. Lu, E. J. Moon et al., “Solid solubility and magnetism upon Mn incorporation in the bulk ternary carbides Cr₂AlC and Cr₂GaC,” Materials Research Letters, vol. 3, no. 1, pp. 16–22, 2015.

B. Anasori, J. Halim, J. Lu, C. A. Voigt, L. Hultman, and M. W. Barsoum, “Mo₂TiAlC₂: a new ordered layered ternary carbide,” Scripta Materialia, vol. 101, pp. 5–7, 2015.

X. He, Y. Bai, C. Zhu, Y. Sun, M. Li, and M. W. Barsoum, “General trends in the structural, electronic and elastic properties of the M₃AlC₂ phases (M = transition metal): a first-principle study,” Computational Materials Science, vol. 49, no. 3, pp. 691–698, 2010.

P. Hohenberg and W. Kohn, “Inhomogeneous electron gas,” Physical Review, vol. 136, pp. B864–B871, 1964.

W. Kohn and L. J. Sham, “Self-consistent equations including exchange and correlation effects,” Physical Review A, vol. 140, pp. A1133–A1138, 1965.

M. C. Payne, M. P. Teter, D. C. Allan, T. A. Arias, and J. D. Joannopoulos, “Iterative minimization techniques for ab initio total-energy calculations: molecular dynamics and conjugate gradients,” Reviews of Modern Physics, vol. 64, no. 4, pp. 1045–1097, 1992.

J. P. Perdew, J. A. Chevary, S. H. Vosko et al., “Atoms, molecules, solids, and surfaces: applications of the generalized gradient approximation for exchange and correlation,” Physical Review B, vol. 46, no. 11, pp. 6671–6687, 1992.

E. Sjöstedt, L. Nordström, and D. J. Singh, “An alternative way of linearizing the augmented plane-wave method,” Solid State Communications, vol. 114, no. 1, pp. 15–20, 2000.

H. J. Monkhorst and J. D. Pack, “Special points for Brillouin-zone integrations,” Physical Review B: Solid State, vol. 13, no. 12, pp. 5188–5192, 1976.

D. A. McQuarrie, Statistical Mechanics, chapter 3, Harper & Row, New York, NY, USA, 1976.

Y. F. Li, Y. M. Gao, B. Xiao et al., “Theoretical study on the stability, elasticity, hardness and electronic structures of W–C binary compounds,” Journal of Alloys and Compounds, vol. 502, no. 1, pp. 28–37, 2010.

Y. L. Bai, X. D. He, Y. Sun, C. Zhu, M. Li, and L. Shi, “Chemical bonding and elastic properties of Ti₃AC₂ phases (A = Si, Ge, and Sn): a first-principle study,” Solid State Sciences, vol. 12, no. 7, pp. 1220–1225, 2010.

J. Zhao, J. M. Winey, and Y. M. Gupta, “First-principles calculations of second- and third-order elastic constants for single crystals of arbitrary symmetry,” Physical Review B, vol. 75, no. 9, Article ID 094105, 7 pages, 2007.

J. M. Wang, J. Y. Wang, Y. C. Zhou, and C. F. Hu, “Phase stability, electronic structure and mechanical properties of ternary-layered carbide Nb₄AlC₃: an ab initio study,” Acta Materialia, vol. 56, no. 7, pp. 1511–1518, 2008.

S. K. R. Patil, S. V. Khare, B. R. Tuttle, J. K. Bording, and S. Kodambaka, “Mechanical stability of possible structures of PtN investigated using first-principles calculations,” Physical Review B, vol. 73, no. 10, Article ID 104118, 8 pages, 2006.

Z. J. Wu, E. J. Zhao, H. P. Xiang, X. F. Hao, X. Liu, and X. J. Meng, “Crystal structures and elastic properties of superhard IrN₂ and IrN₃ from first principles,” Physical Review B, vol. 76, no. 5, Article ID 054115, 2007.

W. Zhou, L. J. Liu, B. L. Li, P. Wu, and Q. G. Song, “Structural, elastic and electronic properties of intermetallics in the Pt-Sn system: a density functional investigation,” Computational Materials Science, vol. 46, no. 4, pp. 921–931, 2009.

G. N. Greaves, A. L. Greer, R. S. Lakes, and T. Rouxel, “Poisson's ratio and modern materials,” Nature Materials, vol. 10, no. 11, pp. 823–837, 2011.

S. I. Ranganathan and M. Ostoja-Starzewski, “Universal elastic anisotropy index,” Physical Review Letters, vol. 101, no. 5, Article ID 055504, 4 pages, 2008.

H. Ledbetter and S. Kim, Handbook of Elastic Properties of Solids, Liquids, and Gases, Academic Press, New York, NY, USA, 2001.

K. B. Panda and K. S. Ravi Chandran, “Determination of elastic constants of titanium diboride (TiB₂) from first principles using FLAPW implementation of the density functional theory,” Computational Materials Science, vol. 35, no. 2, pp. 134–150, 2006.

L. Sun, Y. Gao, B. Xiao, Y. Li, and G. Wang, “Anisotropic elastic and thermal properties of titanium borides by first-principles calculations,” Journal of Alloys and Compounds, vol. 579, pp. 457–467, 2013.

F. M. Gao, “Theoretical model of intrinsic hardness,” Physical Review B, vol. 73, no. 13, Article ID 132104, 4 pages, 2006.

H. Gou, L. Hou, J. Zhang, and F. Gao, “Pressure-induced incompressibility of ReC and effect of metallic bonding on its hardness,” Applied Physics Letters, vol. 92, no. 24, Article ID 241901, 2008.

G. P. Bei, V. Gauthier-Brunet, C. Tromas, and S. Dubois, “Synthesis, characterization, and intrinsic hardness of layered nanolaminate Ti₃ AlC₂ and Ti₃ Al0.8 Sn0.2 C₂ solid solution,” Journal of the American Ceramic Society, vol. 95, no. 1, pp. 102–107, 2012.

Y. C. Zhou, J. X. Chen, and J. Y. Wang, “Strengthening of Ti₃AlC₂ by incorporation of Si to form Ti₃Al1 − xSixC₂ solid solutions,” Acta Materialia, vol. 54, no. 5, pp. 1317–1322, 2006.

M. W. Barsoum and M. Radovic, “Elastic and mechanical properties of the MAX phases,” Annual Review of Materials Research, vol. 41, pp. 195–227, 2011.

J.-H. Han, S.-S. Hwang, D. Lee, and S.-W. Park, “Synthesis and mechanical properties of Ti₃AlC₂ by hot pressing TiCx/Al powder mixture,” Journal of the European Ceramic Society, vol. 28, no. 5, pp. 979–988, 2008.

H. Wang, Y. Zheng, M.-Q. Cai, H. T. Huang, and H. L. W. Chan, “First-principles study on the electronic and optical properties of BiFeO₃,” Solid State Communications, vol. 149, no. 15-16, pp. 641–644, 2009.

M. S. Ali, A. K. M. A. Islam, M. M. Hossain, and F. Parvin, “Phase stability, elastic, electronic, thermal and optical properties of Ti₃Al1 − xSixC₂ (0≤x≤1): first principle study,” Physica B: Condensed Matter, vol. 407, no. 21, pp. 4221–4228, 2012.

J. Jiu-Xing, J. Shan, W. Zhen-Hua, and T. Chang-Long, “Electronic structure and optical properties of layered ternary carbide Ti₃AlC₂,” Chinese Physics Letters, vol. 28, no. 3, Article ID 037101, 2011.

X. H. Deng, B. B. Fan, and W. Lu, “First-principles investigations on elastic properties of α- and β- Ta₄AlC₃,” Solid State Communications, vol. 149, no. 11-12, pp. 441–444, 2009.

P. Finkel, B. Seaman, K. Harrell et al., “Electronic, thermal, and elastic properties of Ti₃Si1 − xGexC₂ solid solutions,” Physical Review B, vol. 70, no. 8, Article ID 085104, 6 pages, 2004.

C. Kittel, Ed., Introduction to Solid State Physics, John Wiley & Sons, 3rd edition, 1966.