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

Authors: Liang Sun; Yimin Gao; Yangzhen Liu; Guoliang Wang; Yiran Wang; Wenyan Zhai; Wen Wang
Journal: Advances in Condensed Matter Physics
 (2016)

---

## 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. C
V of the three compounds is also calculated.


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## Body

## 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
n
+
1
A
X
n (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 n
=
1
–
3). When n
=
1, more than 50 M
2
A
X (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 M
6
X octahedral interleafed with layers of A atoms [5]. However, there are six 312 phases when n
=
2 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 n
=
3) 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 a
=
b
=
0.30753 nm and c
=
1.8578 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 9.0
±
0.2
×
10
-
6 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 4
f (1/3, 2/3, 0.132493) and the titanium atoms occupy 2
a (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 Ti
I and those at 4f positions as T
i
I
I. 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 Ti
I and those at 4f positions as T
i
I
I. 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 10
×
10
×
2 Monkhorst–Pack mesh [53] and the total energy of self-consistent convergence was at 1.0
×
10
-
9 eV/atom and the maximum force on the atom was below 10
-
5 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]:(1)
P
=
P
x
x
P
x
y
P
x
z
P
y
x
P
y
y
P
y
z
P
z
x
P
z
y
P
z
z
.Table 1
The calculated cell parameters (a, b, and c, in Å; V
0 in Å
3; ρ in g/m3) of (M2/3Ti1/3)3AlC2 (M = Cr, Mo, and Ti).
Species
Space group
Opt method
Lattice parameters
ρ
V
0
a
b
c
Cr2TiAlC2
P63MMC
GGA-PW91SAED [31]Cal. [31]
2.9022.9062.922
2.9022.9062.922
17.8117.80317.806
5.1865.1755.118
129.94130.2131.66
Mo2TiAlC2
P63MMC
GGA-PW91Exp. [46]
3.0012.997
3.0012.997
18.74118.661
6.6066.653
146.18145.16
Ti3AlC2
P63MMC
GGA-PW91Exp. [22]Cal. [24]Cal. [47]
3.0793.0753.0723.066
3.0793.0753.0723.066
18.65618.57818.73218.526
4.224.254.224.29
153.22152.13153.09150.84Each element of the tensor is the force that acts on the surface of the structure that has edges parallel to thex-, y-, and z-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 E
=
0 and it can be clearly seen that there is a certain value of Fermi energy (E
f) 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 E
f 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.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.
(a)
(b)From the partial DOS of each element shown in Figures2–4, it is found that there is mainly a contribution of C 2s states for the energy range of -
15


~


-
7.5 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 E
f of the others because of Cr 3d states. Meanwhile, strongp-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 ofV/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 P 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.Figure 5
Variation of the relative volumeV/V
0 (a) and bond length (b) of three quaternary (M2/3Ti1/3)3AlC2 (M = Cr, Mo, and Ti) compounds with pressure.
(a)
(b)More insight into the chemical bonds will be gained by investigating the relative bonds length evolving with pressure and the results are shown in Figure5(b). The relative bonds lengths are decreasing when the pressure increases. Ti
I
-C 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 Ti
II
-C 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 constantsC
i
j
i
,
j
=
1,2
,
3,4
,
5,6 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(2)
σ
1
σ
2
σ
3
τ
4
τ
5
τ
6
=
C
11
C
12
C
13
0
0
0
C
12
C
11
C
13
0
0
0
C
13
C
13
C
33
0
0
0
0
0
0
C
44
0
0
0
0
0
0
C
44
0
0
0
0
0
0
C
11
-
C
12
2
ε
1
ε
2
ε
3
γ
4
γ
5
γ
6
.Table 2
Theoretically calculated elastic properties including elastic constants (C
i
j in GPa), bulk modulus (B in GPa), shear modulus (G in GPa), Young’s modulus (E in GPa), and Poisson’s ratio (σ) for M2TiAlC2 (M = Cr, Mo, and Ti).
Spaces
Pressure
C
11
C
12
C
13
C
33
C
44
=
C
55
C
66
B
G
B
/
G
E
σ
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
0.316In (1), C
i
j represents the nonzero elastic constants; σ
i and τ
i are the normal and shear stresses, and the corresponding uniaxial and shear strains are given by ε
i and τ
i, respectively. Then, the following two strain modes were applied to compute C
i
j in the present paper: (3)
ε
T
=
η
0
0
1
0
0
0
ε
T
=
η
1
0
0
1
0
0
.Here,ε
T is the transpose of the strain matrix, and η 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 ofk
c
/
k
a of a hexagonal crystal can be expressed as [58](4)
k
c
k
a
=
C
11
+
C
12
-
2
C
13
C
33
-
C
13
.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 k
c
/
k
a 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 k
c
/
k
a increases, accompanied with pressure of Cr2TiAlC2 and Mo2TiAlC2 while decreasing for Ti3AlC2.Based on these elastic constants, the polycrystalline bulk modulusB (the resistance of a material to hydrostatic pressure), shear modulus G (the resistance of a material to shear), and Young’s modulus E (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 B, G, and E 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 σ
c
d
<
σ
b
c
c
<
σ
f
c
c
,
h
c
p (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.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.
(a)
(b)The value ofB
/
G is generally applied to indicate the compound with ductility or brittleness. It is supposed that the ductility (brittleness) compound, B
/
G, 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. B
/
G 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 indexA
U [63] is introduced to present anisotropy of the compounds and their shear anisotropic factors:(5)
A
U
=
5
G
V
G
R
+
B
V
B
R
-
6
≥
0
.Here,B
V, G
V, B
R, and G
R are the bulk modulus and shear modulus estimation within the Voigt and Reuss methods, respectively. The anisotropic index (A
U) is the appropriate indicator of the mechanical anisotropic properties of compound. The larger A
U is, the stronger the anisotropy of a compound is. For some single crystal data obtained from Ledbetter, A
U 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 A
U 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 A
U 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]:(6)
1
B
=
S
11
+
S
12
+
S
13
-
S
11
+
S
12
-
S
13
-
S
33
l
3
2
1
E
=
1
-
l
3
2
S
11
+
l
3
4
S
33
+
l
3
2
1
-
l
3
2
2
S
13
+
S
44
.In the equations above,S
11, S
12, S
13, S
33, and S
44 represent elastic compliance constants and l
1, l
2, and l
3 are the direction cosines. The relationships of the direction cosines in spherical coordinates with respect to θ and ϕ are as follows: l
1
=
sin
⁡
θ
cos
⁡
ϕ,  l
2
=
sin
⁡
θ
sin
⁡
ϕ, and l
3
=
cos
⁡
ϕ 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].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).
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)From Figure7, 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 Section3.2 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](7)
H
=
∏
u
H
v
u
n
u
1
/
∑
n
u
H
v
u
G
P
a
=
740
P
u
v
b
u
-
5
/
3
v
b
u
=
d
u
3
∑
v
d
v
3
N
b
v
=
d
u
3
∑
v
d
v
3
N
v
/
Ω
=
d
u
3
Ω
∑
v
d
v
3
N
v
,where H is the hardness of a compound; H
v
u is the hardness of u type bond; d
u is the bond length; v
b
u is the volume of u type bond; N
b
v refers to the v type bond density per cubic angstroms; and N
v is the total number of v type bonds in the cell; Ω and P
u are the cell volume and overlap population of u 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
d
u (Å)
P
u
N
u
Ω (Å
3)
v
b
u
H
v
u (GPa)
H
v (GPa)
Cr2TiAlC2
0 GPa
C-CrC-TiAl-Cr
1.962.142.68
1.330.650.37
444
129.94
6.698.7017.09
41.4413.072.41
10.93
100 GPa
C-CrC-TiAl-Cr
1.841.972.34
1.190.44
-
0.3
444
98.38
5.744.567.65
47.8425.91
-
7.47
21
Mo2TiAlC2
0 GPa
C-MoC-TiAl-Mo
2.112.162.79
1.200.750.41
444
146.18
8.338.9419.27
25.9214.412.19
9.35
100 GPa
C-MoC-TiAl-Mo
1.991.982.47
1.050.6
-
0.15
444
112.5
7.215.310.2
28.8327.56
-
2.28
12.19
Ti3AlC2
0 GPa
C-TiC-TiAl-Ti
2.072.202.91
1.160.870.48
444
153.22
7.699.2421.38
28.6515.822.16
9.93
100 GPa
C-TiC-TiAl-Ti
1.922.02.47
0.920.73
-
0.22
444
110.58
6.495.3710.11
30.1532.81
-
3.44
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,N:(8)
N
=
n
+
i
k
.In vacuum,N is real and equal to unity. For transparent materials, it is purely real, with the imaginary part being related to the absorption coefficient by(9)
η
=
2
k
ω
c
.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:(10)
R
=
1
-
N
1
+
N
2
=
n
-
1
2
+
k
2
n
+
1
2
+
k
2
.The complex dielectric constantε
(
ω
) and the correlation between the dielectric function and refractive index can be shown as follows:(11)
ε
=
ε
1
+
i
ε
2
=
N
2
.The relation between the real and imaginary parts of the refractive index and dielectric constant is as(12)
ε
1
=
n
2
-
k
2
ε
2
=
2
n
k
.Another frequently used quantity for expressing optical properties is the optical conductivityσ
(
ω
):(13)
σ
=
σ
1
+
i
σ
2
=
-
i
ω
4
π
ε
-
1
.Here,ω is the angular frequency. The imaginary part ε
2 of the dielectric function can be thought of as detailing the real transition between occupied and unoccupied electric states and given by [73](14)
ε
=
v
e
2
2
π
ħ
m
2
∫
d
3
K
∑
K
n
P
K
n
′
2
×
f
K
n
1
-
f
K
n
′
δ
E
K
n
-
E
K
n
′
-
ħ
ω
.ħ
ω is the energy of the incident photon, P is the momentum operator, and f
(
K
n
) is the Fermi function. The real part ε
1 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: T
i
3
A
l
C
2
a [74] and T
i
3
A
l
C
2
b [75].When all the phases are under 0 GPa (the calculated absorption spectrums are shown in Figure9(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 atomsd top states and M atomsd top 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 n
0 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 n and k 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 L
ω is related to the trailing edge in the reflection spectra. The main peak in L
ω spectra represents the characteristic associated with the plasma response and corresponding frequency is plasma frequency ω
p and Mo2TiAlC2 has the highest loss function at 15 eV. The calculated real ε
1and imaginary ε
2 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 p and C s states or the Ti p and M d states, where the ambiguous origin is attributed mainly to the similar peak intervals. The real part ε
1 vanishes at about 7.5 eV. Metallic reflectance characteristics are exhibited in the range of ε
1
<
0. The peak of the imaginary part of the dielectric function is related to the electron excitation. For the imaginary part of ε
2, the peak for <1.5 eV is due to the intraband transitions.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.
(a)
(b)When under 100 GPa, it can be seen that the absorption index increases with the pressure from Figure9(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 n
0 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 ε
1 and imaginary ε
2 part of the dielectric function are shown in Figure 9(b)(6); the real part ε
1vanishes at about 9.5 eV.


### 3.6. Thermal Properties

The Debye temperature could be estimated by the mean sound velocity ofv
m, using the following equation [19]:(15)
θ
D
=
h
k
3
n
4
π
V
0
1
/
3
v
m
,where θ
D is the Debye temperature, h represents Plank’s constant, k represents Boltzmann’s constant, n is the number of atoms in unit cell, V
0 is the volume of unit cell, and π is the circumference ratio. The polycrystalline structure of mean sound velocity is calculated by (12) [51, 76]: (16)
v
m
=
1
3
1
v
l
3
+
1
v
t
3
1
/
3
.v
l and v
t represent the longitudinal and transverse sound velocity, respectively. It is determined that v
l and v
t can be calculated by the bulk modulus B and shear modulus G from Navier’s equations: (17)
v
t
=
G
ρ
v
l
=
B
+
4
G
/
3
ρ
.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 (θ
D in K) and sound velocity (v
l, v
s, and v
m in m/s) under various pressure (GPa) for Al4Si2C5.
Pressure (GPa)
Cr2TiAlC2
Mo2TiAlC2
Ti3AlC2
Debye (θ
D)
v
s
v
l
v
m
Debye (θ
D)
v
s
v
l
v
m
Debye (θ
D)
v
s
v
l
v
m
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
7.24Figure 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 volumeC
V as a function of temperature can be calculated by the Debye QHA [78]:(18)
C
V
T
=
9
n
N
A
k
T
θ
D
3
∫
0
θ
D
x
4
e
x
e
x
-
1
2
d
x
.In the equation above,N
A is Avogadro’s constant and k 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,C
V of each compound decrease slightly with pressure and increase dramatically with temperature. When temperature is below 500 K, C
V increases rapidly with T
3 well accompanied with the well-known Debye model; when temperature is above 500 K, C
V increases slowly with the temperature approaching the Dulong–Petit limit as 149.652 J·mol−1 K−1. In our case, C
V
=
3
n
R, where R is the gas constant equal to 8.314 J·mol−1 K−1.


## 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 E
=
0 and it can be clearly seen that there is a certain value of Fermi energy (E
f) 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 E
f 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.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.
(a)
(b)From the partial DOS of each element shown in Figures2–4, it is found that there is mainly a contribution of C 2s states for the energy range of -
15


~


-
7.5 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 E
f of the others because of Cr 3d states. Meanwhile, strongp-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 ofV/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 P 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.Figure 5
Variation of the relative volumeV/V
0 (a) and bond length (b) of three quaternary (M2/3Ti1/3)3AlC2 (M = Cr, Mo, and Ti) compounds with pressure.
(a)
(b)More insight into the chemical bonds will be gained by investigating the relative bonds length evolving with pressure and the results are shown in Figure5(b). The relative bonds lengths are decreasing when the pressure increases. Ti
I
-C 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 Ti
II
-C 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 constantsC
i
j
i
,
j
=
1,2
,
3,4
,
5,6 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(2)
σ
1
σ
2
σ
3
τ
4
τ
5
τ
6
=
C
11
C
12
C
13
0
0
0
C
12
C
11
C
13
0
0
0
C
13
C
13
C
33
0
0
0
0
0
0
C
44
0
0
0
0
0
0
C
44
0
0
0
0
0
0
C
11
-
C
12
2
ε
1
ε
2
ε
3
γ
4
γ
5
γ
6
.Table 2
Theoretically calculated elastic properties including elastic constants (C
i
j in GPa), bulk modulus (B in GPa), shear modulus (G in GPa), Young’s modulus (E in GPa), and Poisson’s ratio (σ) for M2TiAlC2 (M = Cr, Mo, and Ti).
Spaces
Pressure
C
11
C
12
C
13
C
33
C
44
=
C
55
C
66
B
G
B
/
G
E
σ
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
0.316In (1), C
i
j represents the nonzero elastic constants; σ
i and τ
i are the normal and shear stresses, and the corresponding uniaxial and shear strains are given by ε
i and τ
i, respectively. Then, the following two strain modes were applied to compute C
i
j in the present paper: (3)
ε
T
=
η
0
0
1
0
0
0
ε
T
=
η
1
0
0
1
0
0
.Here,ε
T is the transpose of the strain matrix, and η 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 ofk
c
/
k
a of a hexagonal crystal can be expressed as [58](4)
k
c
k
a
=
C
11
+
C
12
-
2
C
13
C
33
-
C
13
.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 k
c
/
k
a 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 k
c
/
k
a increases, accompanied with pressure of Cr2TiAlC2 and Mo2TiAlC2 while decreasing for Ti3AlC2.Based on these elastic constants, the polycrystalline bulk modulusB (the resistance of a material to hydrostatic pressure), shear modulus G (the resistance of a material to shear), and Young’s modulus E (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 B, G, and E 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 σ
c
d
<
σ
b
c
c
<
σ
f
c
c
,
h
c
p (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.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.
(a)
(b)The value ofB
/
G is generally applied to indicate the compound with ductility or brittleness. It is supposed that the ductility (brittleness) compound, B
/
G, 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. B
/
G 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 indexA
U [63] is introduced to present anisotropy of the compounds and their shear anisotropic factors:(5)
A
U
=
5
G
V
G
R
+
B
V
B
R
-
6
≥
0
.Here,B
V, G
V, B
R, and G
R are the bulk modulus and shear modulus estimation within the Voigt and Reuss methods, respectively. The anisotropic index (A
U) is the appropriate indicator of the mechanical anisotropic properties of compound. The larger A
U is, the stronger the anisotropy of a compound is. For some single crystal data obtained from Ledbetter, A
U 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 A
U 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 A
U 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]:(6)
1
B
=
S
11
+
S
12
+
S
13
-
S
11
+
S
12
-
S
13
-
S
33
l
3
2
1
E
=
1
-
l
3
2
S
11
+
l
3
4
S
33
+
l
3
2
1
-
l
3
2
2
S
13
+
S
44
.In the equations above,S
11, S
12, S
13, S
33, and S
44 represent elastic compliance constants and l
1, l
2, and l
3 are the direction cosines. The relationships of the direction cosines in spherical coordinates with respect to θ and ϕ are as follows: l
1
=
sin
⁡
θ
cos
⁡
ϕ,  l
2
=
sin
⁡
θ
sin
⁡
ϕ, and l
3
=
cos
⁡
ϕ 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].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).
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)From Figure7, 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 Section3.2 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](7)
H
=
∏
u
H
v
u
n
u
1
/
∑
n
u
H
v
u
G
P
a
=
740
P
u
v
b
u
-
5
/
3
v
b
u
=
d
u
3
∑
v
d
v
3
N
b
v
=
d
u
3
∑
v
d
v
3
N
v
/
Ω
=
d
u
3
Ω
∑
v
d
v
3
N
v
,where H is the hardness of a compound; H
v
u is the hardness of u type bond; d
u is the bond length; v
b
u is the volume of u type bond; N
b
v refers to the v type bond density per cubic angstroms; and N
v is the total number of v type bonds in the cell; Ω and P
u are the cell volume and overlap population of u 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
d
u (Å)
P
u
N
u
Ω (Å
3)
v
b
u
H
v
u (GPa)
H
v (GPa)
Cr2TiAlC2
0 GPa
C-CrC-TiAl-Cr
1.962.142.68
1.330.650.37
444
129.94
6.698.7017.09
41.4413.072.41
10.93
100 GPa
C-CrC-TiAl-Cr
1.841.972.34
1.190.44
-
0.3
444
98.38
5.744.567.65
47.8425.91
-
7.47
21
Mo2TiAlC2
0 GPa
C-MoC-TiAl-Mo
2.112.162.79
1.200.750.41
444
146.18
8.338.9419.27
25.9214.412.19
9.35
100 GPa
C-MoC-TiAl-Mo
1.991.982.47
1.050.6
-
0.15
444
112.5
7.215.310.2
28.8327.56
-
2.28
12.19
Ti3AlC2
0 GPa
C-TiC-TiAl-Ti
2.072.202.91
1.160.870.48
444
153.22
7.699.2421.38
28.6515.822.16
9.93
100 GPa
C-TiC-TiAl-Ti
1.922.02.47
0.920.73
-
0.22
444
110.58
6.495.3710.11
30.1532.81
-
3.44
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,N:(8)
N
=
n
+
i
k
.In vacuum,N is real and equal to unity. For transparent materials, it is purely real, with the imaginary part being related to the absorption coefficient by(9)
η
=
2
k
ω
c
.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:(10)
R
=
1
-
N
1
+
N
2
=
n
-
1
2
+
k
2
n
+
1
2
+
k
2
.The complex dielectric constantε
(
ω
) and the correlation between the dielectric function and refractive index can be shown as follows:(11)
ε
=
ε
1
+
i
ε
2
=
N
2
.The relation between the real and imaginary parts of the refractive index and dielectric constant is as(12)
ε
1
=
n
2
-
k
2
ε
2
=
2
n
k
.Another frequently used quantity for expressing optical properties is the optical conductivityσ
(
ω
):(13)
σ
=
σ
1
+
i
σ
2
=
-
i
ω
4
π
ε
-
1
.Here,ω is the angular frequency. The imaginary part ε
2 of the dielectric function can be thought of as detailing the real transition between occupied and unoccupied electric states and given by [73](14)
ε
=
v
e
2
2
π
ħ
m
2
∫
d
3
K
∑
K
n
P
K
n
′
2
×
f
K
n
1
-
f
K
n
′
δ
E
K
n
-
E
K
n
′
-
ħ
ω
.ħ
ω is the energy of the incident photon, P is the momentum operator, and f
(
K
n
) is the Fermi function. The real part ε
1 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: T
i
3
A
l
C
2
a [74] and T
i
3
A
l
C
2
b [75].When all the phases are under 0 GPa (the calculated absorption spectrums are shown in Figure9(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 atomsd top states and M atomsd top 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 n
0 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 n and k 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 L
ω is related to the trailing edge in the reflection spectra. The main peak in L
ω spectra represents the characteristic associated with the plasma response and corresponding frequency is plasma frequency ω
p and Mo2TiAlC2 has the highest loss function at 15 eV. The calculated real ε
1and imaginary ε
2 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 p and C s states or the Ti p and M d states, where the ambiguous origin is attributed mainly to the similar peak intervals. The real part ε
1 vanishes at about 7.5 eV. Metallic reflectance characteristics are exhibited in the range of ε
1
<
0. The peak of the imaginary part of the dielectric function is related to the electron excitation. For the imaginary part of ε
2, the peak for <1.5 eV is due to the intraband transitions.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.
(a)
(b)When under 100 GPa, it can be seen that the absorption index increases with the pressure from Figure9(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 n
0 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 ε
1 and imaginary ε
2 part of the dielectric function are shown in Figure 9(b)(6); the real part ε
1vanishes at about 9.5 eV.


## 3.6. Thermal Properties

The Debye temperature could be estimated by the mean sound velocity ofv
m, using the following equation [19]:(15)
θ
D
=
h
k
3
n
4
π
V
0
1
/
3
v
m
,where θ
D is the Debye temperature, h represents Plank’s constant, k represents Boltzmann’s constant, n is the number of atoms in unit cell, V
0 is the volume of unit cell, and π is the circumference ratio. The polycrystalline structure of mean sound velocity is calculated by (12) [51, 76]: (16)
v
m
=
1
3
1
v
l
3
+
1
v
t
3
1
/
3
.v
l and v
t represent the longitudinal and transverse sound velocity, respectively. It is determined that v
l and v
t can be calculated by the bulk modulus B and shear modulus G from Navier’s equations: (17)
v
t
=
G
ρ
v
l
=
B
+
4
G
/
3
ρ
.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 (θ
D in K) and sound velocity (v
l, v
s, and v
m in m/s) under various pressure (GPa) for Al4Si2C5.
Pressure (GPa)
Cr2TiAlC2
Mo2TiAlC2
Ti3AlC2
Debye (θ
D)
v
s
v
l
v
m
Debye (θ
D)
v
s
v
l
v
m
Debye (θ
D)
v
s
v
l
v
m
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
7.24Figure 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 volumeC
V as a function of temperature can be calculated by the Debye QHA [78]:(18)
C
V
T
=
9
n
N
A
k
T
θ
D
3
∫
0
θ
D
x
4
e
x
e
x
-
1
2
d
x
.In the equation above,N
A is Avogadro’s constant and k 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,C
V of each compound decrease slightly with pressure and increase dramatically with temperature. When temperature is below 500 K, C
V increases rapidly with T
3 well accompanied with the well-known Debye model; when temperature is above 500 K, C
V increases slowly with the temperature approaching the Dulong–Petit limit as 149.652 J·mol−1 K−1. In our case, C
V
=
3
n
R, where R 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 E
f of total DOS at the Fermi level which exhibits metallic property, such as metallic conductivity. Cr2TiAlC2 shows the biggest value of E
f because of Cr 3d states. For the strongp-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. C
V of each compound decreases slightly with pressure and increases dramatically with temperature.



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*Source: 3231027-2016-09-19.xml*