Newly synthesized Zr2AlC, Zr2(Al0.58Bi0.42) - arXiv

Newly synthesized Zr2(Al0.58Bi0.42)C, Zr2(Al0.2Sn0.8)C, and Zr2(Al0.3Sb0.7)C MAX phases: A first-principles study M. A. Ali1, M. M. Hossain1, N. Jahan1, S. H. Naqib2, A. K. M. A. Islam3 1

Department of Physics, Chittagong University of Engineering and Technology, Chittagong4349, Bangladesh. 2 Department of Physics, University of Rajshahi, Rajshahi-6205, Bangladesh. 3 International Islamic University Chittagong, 154/A College Road, Chittagong, Bangladesh.

ABSTRACT We have investigated the structural, elastic and electronic properties of Zr2(Al0.58Bi0.42)C, Zr2(Al0.2Sn0.8)C, and Zr2(Al0.3Sb0.7)C for the first time using first-principles calculations based on density functional theory (DFT) by means of the plane-wave pseudopotential method. The structural properties are calculated and found to be in good agreement with experimental data. The single crystal elastic constants Cij and other polycrystalline elastic parameters of these compounds have been calculated. The mechanical stability of the compounds with different doping concentrations has also been studied. The electronic band structure and density of states are calculated and the effect of doping on these properties is also analyzed. This study represents significant results of the new compounds belong to the MAX phase family, since this is the first theoretical study of the antimony and bismuth containing MAX phase. The results are compared with experimental data and also with Zr2AlC where available. 1. Introduction A family of ternary carbides and nitrides is known to scientific community as MAX phase with general formula Mn+1 AXn where n is an integer, M represents an early transition metal, A represents an A group (group 13–16) element and X represents C and/or N [1]. A compound belongs to MAX phase should be crystallized with hexagonal structure with space group P63/mmc. The MAX phases reveal unique properties combining the advantages of metals and ceramics due to their nanolaminated crystal structure with A planes metallic layers between the MX ceramic layers [1,8]. These notable properties are very important in high temperature device applications. As a result, many theoretical and experimental research works on MAX phases are performed to investigate the characteristics. The MAX phases with a M2AC stoichiometry has been synthesized for the first time in the 1960s, although, more than half of 211 MAX phases known as “H-phases” studied by Nowtony et al [2] were almost ignored for two decades. Later on, in 1996, Barsoum and El-Raghy [3] have reported some excellent properties of the ceramic compound of Ti3SiC2 to highlight their characteristics. Similar properties were also showed by the other MAX phases [4,5]. The MAX phases family have drawn special attention due to the report on Ti4 AlN3 [6] and the existence of a number of 312 phases, became extended and recognized as the Mn+1AXn

(MAX) phases [7]. The member belongs to the MAX phase family are increasing continuously due to the interesting properties mentioned above where n is not limited to 3 but up to 5 [9]. Hu et al. have reported a partial list of about 70 new MAX phases in the period 2004–2013 [10]. Other two 211 important phases Nb2GeC [11] and Mn2GaC [12] are also added. Recently, Naguib et al. [13] reported a large number of works (68 quaternary MAX phases) on the solid solutions where existing phases are mixed. These approaches yield the new phases (where neither end member exists) [9,14] and/or ordered quaternary phases [15-18]. Furthermore, the recent discovery of Mo2Ga2C (with double “A” layers) “may be the first of a distinct family of MAX-related phases” [19]. More recently, Lapauw et al. have synthesized the compounds Zr2AlC and Zr3AlC2. The latter one is the first experimentally produced MAX phase in the ZrAl-C system [20, 21]. The new member of MAX phases reported are mainly either with higher n values or forming a solid solutions [10, 13, 22], in which the M, A or X elements are partially substituted which increases a number of possible combinations. In addition, Quaternary MAX phases suggest the opportunity of including new elements which are not exist in a bulk ternary MAX phase, for example, recently Mockute et al. [23] have reported (Cr1−xMnx)2GaC (0 ≤ x ≤ 0.3) where, Mn is partially substituted for Cr2. In the very first of 2016, Horlait et al. synthesized the first Bi containing solid solutions Zr2(Al1−xBix)C (0 ≤ x ≤ 1) in the family of MAX phase [24] and also produced other solid solutions such as Zr2(Al0.2Sn0.8)C, Zr2(Al0.35Pb0.65)C, and Zr2(Al0.3Sb0.7)C [25]. It is well known that Zr-based materials become important for the nuclear industry due to the small cross-section of Zr atoms for thermal neutrons. In addition to this, it is also very important for the fuel cladding materials used in next generation (Gen-III+) light water reactors (LWRs) to hold up severe working conditions. By considering the above facts, Zr 2AlC as well as Zr containing alloys could be potential candidates for fuel cladding applications. However, no complete theoretical studied on Zr2(Al0.58Bi0.42)C, Zr2(Al0.2Sn0.8)C and Zr2(Al0.3Sb0.7)C have been reported, so far. This background motivates us to investigate the properties of these newly synthesized compounds Zr2(Al0.58Bi0.42)C, Zr2(Al0.2Sn0.8)C and Zr2(Al0.3Sb0.7)C by means of first principles calculations and compared with that of Zr2AlC. In present work, therefore, the structural, elastic and electronic properties of these newly synthesized compounds will be carefully studied. 2. Computational methodology The CASTEP code [26] is used to calculate the structural, elastic, and electronic properties of Zr2(Al0.58Bi0.42)C, Zr2(Al0.2Sn0.8)C, Zr2(Al0.3Sb0.7)C and Zr2AlC. The method used in this code is the plane wave pseudopotential approach based on the density functional theory (DFT) [27]. During the calculations the exchange-correlation potential are treated within the GGA using Perdew-Burke-Ernzerhof (PBE) [28]. A 10 × 10 × 2 k-point mesh of Monkhorst-Pack scheme was used for intregration over the first Brillouin zone [29]. The convergence of the planewave is done with a kinetic energy cutoff of 500 eV. Excellent convergence is guaranteed by testing the

Brillouin zone sampling and the kinetic energy cutoff which makes the tolerance for selfconsistent field, energy, maximum force, maximum displacement, and maximum stress to be 5.0×10-7 eV/atom, 5.0×10-6 eV/atom, 0.01 eV/Å, 5.0×10-4 Å, and 0.02 GPa, respectively. The structural parameters of Zr2(Al0.3Sb0.7)C were determined using the BFGS [30] minimization technique. 3. Results and discussion 3.1 Structural Properties The ternary layered carbide Zr2AlC compound with space group P63/mmc belongs to hexagonal system like other MAX phase compounds. Its unit cell contains two formula units. The atomic positions are Zr atoms at (1/3, 2/3, z), Al atoms at (2/3, 1/3, 1/4) and C atoms at (0, 0, 0). Bi/Sn/Sb atoms are substituted for Al at (2/3, 1/3, 1/4). Fig. 1 shows the unit cell of Zr2AlC. The equilibrium crystal structure of compound is first obtained by minimizing its total energy. Optimum structural parameters are presented in Table 1 along with the parameters of Zr2AlC and compared. The results are in good agreement with the reported experimental values. From Table 1, it is seen that the lattice constants of substituted compounds are not same to that of the Zr2AlC. An increase in lattice constant a is found while a decreased in c is found due to the Bi/Sn/Sb substitution for Al in Zr2AlC. The change in lattice constants can be attributed to the ionic size differences between Bi/Sn/Sb and Al.

Fig. 1. Crystal structure of Zr2AlC

Table 1 Optimized lattice parameters (a and c, in Å), hexagonal ratio c/a, internal parameter zM., unit cell volume V (Å3) for MAX phases.

Phases a Zr2(Al0.58Bi0.42)C 3.375 3.344 Zr2(Al0.2Sn0.8)C 3.352 3.345 Zr2(Al0.3Sb0.7)C 3.357 3.367 Zr2AlC 3.319 3.324 3.319

c 13.719 14.510 13.747 14.567 14.311 14.620 14.604 14.570 14.606

c/a 4.06

zM 0.0923

4.10

0.0922

4.26

0.0885

4.40

0.0864 0.0871 0.0864

4.40

V Ref 135.372 This Expt.[24] 133.796 This Expt.[25] 139.699 This Expt.[25] 139.300 This Expt.[22] Theo.[31]

2

*Calculated using V =0.866a c

3.2 Elastic properties The elastic constants of materials give a relationship connecting the dynamical and mechanical nature of crystalline materials, and provide an important contribution to explore the research regarding the nature of the forces acting in solids [32]. The elastic constants of Zr2(Al0.58Bi0.42)C, Zr2(Al0.2Sn0.8)C, Zr2(Al0.3Sb0.7)C and Zr2 AlC at ambient pressure have been calculated and are given in Table 2. Since we have calculated these constants of the substituted compounds for the first time i.e. neither theoretical nor experimental data are available; therefore our calculated data are not compared with any other results but compared with that of the Zr2AlC. A compound said to be mechanically stable if the single crystal constants Cij are all positive and satisfy the well known criteria [33]: C11> 0, (C11-C12) > 2 0, C44> 0 and (C11+C12)C33> 2𝐶13 . The compounds under consideration satisfy these criteria’s and therefore mechanically stable. Table 2 The elastic constants Cij (GPa), bulk modulus, B (GPa), shear modulus, G (GPa), Young’s modulus, Y (GPa), Poisson ratio, ν, and Pugh ratio G/B, Vickers hardness Hv, A, kc/ka. Compound Zr2(Al0.58Bi0.42)C Zr2(Al0.2Sn0.8)C Zr2(Al0.3Sb0.7)C Zr2AlC

C11 C12 242 99 253 108 220 98 258 67

C13 124 110 103 63

C33 C44 284 129 293 109 243 94 221 91

B 161 161 143 124

G 90 88 73 91

Y 227 223 187 219

G/B 0.55 0.54 0.51 0.73

ν Hv A kc/ka 0.26 10.81 1.85 0.58 0.27 10.34 1.34 0.77 0.28 8.19 1.46 0.80 0.24 16.37 1.03 1.25

The bulk modulus B, shear modulus G, Young’s modulus Y, the Poisson’s ratio v and Pugh ratio are also calculated and given in Table 2. Y and v are calculated from B & G by using the relationships: Y = 9BG/ (3B + G), ν = (3B-Y)/6B [34, 35]. The arithmetic average of the Voigt (BV, GV) and the Reuss (BR, GR) bounds are used to estimate the polycrystalline modulus B and G [36] . Furthermore, we have calculated the elastic anisotropy, A and another anisotropic factor, kc/ka for the hexagonal crystals defined by the ratio between the linear compressibility coefficients along the c- and a-axis of these compounds. We have computed the anisotropy factor for {100} shear planes, A = 4C44/(C11+C33-2C13) [37] and kc/ka = (C11 + C12 - 2C13)/(C33 - C13) from the present values of the elastic constants. From Table 2 it is obvious that due to the Bi/Sn/Sb for Al in Zr2AlC causes a significant increase of most of the single elastic constants except C11 that represents the elasticity in length and generally a longitudinal strain produces a change in C11. The value of bulk modulus is found to be increased due to the substitution of Bi/Sn/Sb for Al in Zr2AlC while the value of shear modulus decreased. Where the shear modulus describes the material’s response to shearing strains i.e., the resistance to change in shape and the bulk modulus describes the material’s response to uniform pressure. The lower value of Young’s modulus Y, indicate the lowering of stiffness of the substituted (Bi/Sn/Sb for Al in Zr2AlC) materials. The critical value of Poisson’s ratio for a material is 0.33 that separates brittle (less than 0.33) and ductile (greater than 0.33) materials [38]. Another parameter suggested by Pugh for ductility-brittle transition is the G/B [39] ratio. If G/B > 0.5, the material behaves as a brittle material, otherwise it is ductile. According to these two conditions, the compounds under considerations are brittle in nature but the effect of of Bi/Sn/Sb substitution is noticeable. It is found from Table 2 that the value of Poisson’s ratio is increased and G/B ratio is decreased with of Bi/Sn/Sb substitution for Al, indicating a tendency towards ductility behavior. In order to study the effect of Bi/Sn/Sb substitution on the mechanical hardness of Zr2AlC, we have calculated and given in Table 2 the theoretical Vickers hardness using Chen’s formula [40], which is expressed as: Hv = 2(k2G)0:585 – 3, where, k = G/B, from which we see that the hardness is related with both the bulk modulus and shear modulus. Physically, the bulk modulus only measures isotropic resistance to volume change under hydrostatic strain, whereas shear modulus responses to resistance to anisotropic shear strain. Even though it was thought that the direct relation between the bulk modulus and hardness is less compared to shear modulus [41], the Pugh’s ratio (k=G/B) clearly shows the contribution of both B and G to the Vickers hardness. The Equation shows that, hardness would be decreased with increasing bulk modulus until the shear modulus will remain unchanged, and vice versa. The value of Pugh’s ratio (G/B) will get smaller with increasing bulk modulus and hence the material would become more ductile. As a result the hardness value will become lower. From Table 2, we see that the value of B is increased for substituted compounds and the value of G is decreased. The theoretical Vickers hardness of the substituted compounds is decreased and is in good agreement with the above explanation. Anisotropic factor of any crystal is very important in engineering science and also in crystal Physics. Therefore, it is very important to describe properly the anisotropic behavior of any materials for engineering application. The degree of anisotropy in the bonding between atoms in

different planes is described by the shear anisotropic factors which is calculated and given in Table 2. For a completely isotropic material, A is equal to 1, while the deviations of Zenger’s anisotropy from unity measure the degree of elastic anisotropy. It is found that Zr2(Al0.58Bi0.42)C, Zr2(Al0.2Sn0.8)C and Zr2(Al0.3Sb0.7)C are more anisotropic than Zr2AlC. In addition, the calculated value of kc/ka of Zr2(Al0.58Bi0.42)C, Zr2(Al0.2Sn0.8)C and Zr2(Al0.3Sb0.7)C is less than 1, indicating the compressibility along the c-axis is smaller than that along the a-axis but reverse for Zr2AlC. 3.3 Electronic properties

c

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Fig. 2. Electronic band structures of (a) Zr2(Al0.58Bi0.42)C, (b) Zr2(Al0.2Sn0.8)C (c) Zr2(Al0.3Sb0.7)C and (d) Zr2AlC compounds.

M L

H

The calculated energy band diagram of the Zr2(Al0.58Bi0.42)C, Zr2(Al0.2Sn0.8)C, Zr2(Al0.3Sb0.7)C and Zr2AlC compounds along high symmetry direction are shown in Fig. 2(a, b, c and d). The EF is chosen to be zero of the energy scale. From the figure, it is clear that the valence and conduction bands are overlapped considerably and there is no band gap at the EF, resulting the compounds exhibit metallic properties. There is a strong anisotropic characteristic with lower c-axis energy dispersion, which can be seen from the reduced dispersion along the HK and M-L directions. It is indicated that the compounds show anisotropic metallic conductivity. Total and partial DOS (PDOS) of Zr2(Al0.58Bi0.42)C, Zr2(Al0.2Sn0.8)C, Zr2(Al0.3Sb0.7)C and Zr2AlC compounds are represented in Fig. 3 (a, b, c and d). The value of DOS at the EF, are found to be 3.26, 3.22, 3.65 and 3.13 states/eV for Zr2(Al0.58Bi0.42)C, Zr2(Al0.2Sn0.8)C, Zr2(Al0.3Sb0.7)C and Zr2AlC, respectively. It is found that the DOS is increased due to Bi/Sn/Sb substitution for Al in Zr2AlC with largest value for Sb substitution, indicating the improved conduction properties of the substituted compounds. We observe that the Zr 4d electrons are mainly contributed to the DOS at the EF and determine its conduction properties. The contribution from C-2s, Al-3s, Zr-5s, Bi-6s, Sn-5s and Sb-5s states are also noticeable but an order of smaller in magnitude. The top energy band called the conduction band is mainly attributed from Zr 4d states. The top valence band is attributed from C-2p, Al-3p, Zr-4d, Bi-6p, Sn-5p and Sb-5p states. The key point of the band spectra of the substituted compound (Fig. 3) is the appearance of additional band at the Fermi level, which can clearly be seen in Fig. 3 results an increase in DOS. The additional band at the Fermi level is attributed from Bi/Sn/Sb-p states as shown in the partial DOS of Bi/Sn/Sb. Due to this additional band, the DOS at the Fermi level is increased.

a

12 8

c

EF

Zr 5s Zr 4d

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0 12

Al 3s Al 3p

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4 DOS (States/eV)

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0 Bi 6s Bi 6p

8 4 0 6

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8 4

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Zr 5s Zr 4d

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0 Sn 5s Sn 5p

8 4

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Al 3s Al 3p

8 4 0 6

C 2s C 2p

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Fig. 3. Total and partial densities of states for (a) Zr2(Al0.58Bi0.42)C, (b) Zr2(Al0.2Sn0.8)C (c) Zr2(Al0.3Sb0.7)C and (d) Zr2AlC compounds.

4. Conclusions The DFT calculations have been carried out to study the structural, elastic and electronic properties of the newly synthesized, Zr2(Al0.58Bi0.42)C, Zr2(Al0.2Sn0.8)C, Zr2(Al0.3Sb0.7)C for the first time. The calculated lattice parameters are in reasonable agreement with experimental values. The variations between the calculated and experimental values are not surprising due to the nature of GGA-PBE functional. The value of elastic constants confirmed that the technologically important materials under consideration are mechanically stable. The analysis of the elastic constants and other moduli confirmed that Zr2(Al0.58Bi0.42)C, Zr2(Al0.2Sn0.8)C, Zr2(Al0.3Sb0.7)C solid solutions show large anisotropic on elasticity and a tendency towards ductility behavior in compared to Zr2AlC. The increase of bulk modulus and decrease of shear modulus results the lower mechanical hardness due to the substitution of Bi/Sn/Sb for Al in Zr2AlC. The electronic band structures reveal that the solid solutions considered are all metallic conductivity. The conductivity of the alloy was increased due to the additional band crossing the Fermi level attributed from substituting elements. Based on our theoretical and reported experimental results, technologically promising Zr containing alloys could be potential candidates for fuel cladding applications.

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