Journal of ELECTRONIC MATERIALS
https://doi.org/10.1007/s11664-018-6163-3 Ó 2018 The Minerals, Metals & Materials Society
The Structural Stability and Mechanical Properties of Cu2MnAl and Cu2MnIn Under Pressure: First-Principles Study LILI LIU,1,2 XIANSHI ZENG,3,4 QINGDONG GOU,3 YUANXIU YE,3 YUFENG WEN,3,5,7 and PING OU5,6 1.—Department of Physics, Chongqing Three Gorges University, Chongqing 404100, China. 2.—Institute for Structure and Function, Chongqing University, Chongqing 401331, China. 3.—School of Mathematical Sciences and Physics, Jinggangshan University, Ji’an 343009, China. 4.—Research Center of Laser Fusion, China Academy of Engineering Physics, Mianyang 621900, China. 5.—School of Materials Science and Engineering, Shanghai Jiaotong University, Shanghai 200240, China. 6.—School of Materials Science and Engineering, Jiangxi University of Science and Technology, Ganzhou 341000, China. 7.—e-mail:
[email protected]
The full set of the independent second- and third-order elastic constants of Cu2 MnAl and Cu2 MnIn Heusler compounds have been obtained from the strain energy of the lattice by the first-principles calculations in combination with the homogeneous deformation theory. The ground-state lattice parameters and second-order elastic constants are found to be consistent with the available theoretical and experimental results. Among all the negative thirdorder elastic constants, the longitudinal modes C111 have larger values than their shear modes, showing that the contribution to lattice vibrations from the longitudinal modes is greater. The ductility, elastic anisotropy, melting point and Debye temperature of the two compounds under different pressures are investigated for the first time, and increase with increasing the pressure. Key words: Cu2 MnAl, Cu2 MnIn, mechanical properties, pressure, first-principles
INTRODUCTION Since the first Heusler compounds Cu2 Mn(Al, Sn, Zn, Bi, Sb or B) were discovered by Heusler,1 a large number of ternary intermetallic compounds with the generic formula X2 YZ have been developed, where X and Y are typically transition metals and Z is a III, IV or V group element.2 When the valence of X is larger than Y, crystallization of these compounds is reported in the well-known cubic structure with prototype Cu2 MnAl (Fm3m, space group no. 225) and the atomic sequence is X-Y-X-Z. However, if Y has a larger valence than X, an Inverse Heusler structure with prototype Hg2 CuTi (the space group of Fm3m and the number of 216) is observed and the atomic sequence is X-X-Y-Z. These intermetallic compounds possess good electrical and
(Received August 10, 2017; accepted February 20, 2018)
magnetic properties, which make them promising high-temperature structural materials for aviation, aerospace and automobile applications. Among these intermetallic compounds, Cu2 MnAl is of interest since it is magnetic in spite of its nonferromagnetic constituents.3–5 Until now, several experimental and theoretical studies have been carried out to research various properties of Cu2 MnAl. Kudryavtsev et al.6 experimentally investigated the structural dependence of the magnetic, transport, optical, and magneto-optical (MO) properties of Cu2 MnAl thin film and found a significant influence of the structural disorder on the optical and MO properties. Rai et al.7 researched the electronic, magnetic, optical and elastic properties of Cu2 MnAl by using the full potential linearized augmented plane wave method with the generalized gradient approximation. However, they actually obtained the values of electronic, optical and elastic constants of the Mn2 CuAl, not Cu2 MnAl. Later, Jalilian8 investigated all data again by using the
Liu, Zeng, Gou, Ye, Wen, and Ou
same method. All the results at the ground state are very important to further scientific and technical investigations. Elastic constants are important in many fields ranging from chemistry to physics and from geophysics to materials research. The pressure-dependent elastic constants are important for predicting and understanding the mechanical strength, stability, and phase transitions of a material. The elastic properties under pressure can be derived from the second-order elastic constants (SOECs) and the third-order elastic constants (TOECs). The SOECs and TOECs are liner and non-linear elastic constants, respectively. In addition, the SOECs and TOECs can be used to develop the ion–electron pseudopotentials9 or empirical interatomic potentials.10 Though many experiments have focused on determining the SOECs and high-order elastic constants, obtaining the complete non-linear elastic constants by experiment is very difficult. Recently, a simple method of homogeneous deformation using first-principles calculations has been employed to determine the TOECs for single crystals with arbitrary symmetry,11–13 and their results agree with the experiments. Following this, the present study has been conducted to predict the nonlinear elastic properties of Cu2 MnZ (Z=Al, In) using the first-principles total-energy calculations combined with the method of homogeneous elastic deformation. Phase stability and elastic constants of Cu2 MnZ (Z=Al, In) in the pressure range of 0–10 GPa have been investigated. In addition, various mechanical properties including bulk modulus, shear modulus, Poission’s ratio, Cauchy pressure, Young’s modulus, and melting point along with the Debye temperature at high pressure have been discussed in detail up to 10 GPa. THEORY METHOD All the lattice and elastic constants calculations reported in this work have been performed with the Vienna Ab initio Simulation Package (VASP) code, based on density functional theory (DFT).14–16 The ion–electron interaction is described by the projector augmented wave method.17,18 For the exchange– correlation functional, the generalized gradient approximation of Perdew–Burke–Ernzerhof19,20 has been applied. The atomic levels 3d10 4s1 of the Cu atom, 3d5 4s2 of the Mn atom, 3s2 3p1 of the Al atom, and 5s2 5p1 of the In atom are treated as valence electron states and the remaining electrons are kept frozen. The k-point meshes for the Brillouin zone sampling are represented by using the Monkhorst–Pack scheme.21 The k-point Monkhorst–Pack mesh was set as 9 9 9 9 9 for lattice and elastic constants calculations. The plane wave cutoff energy was chosen to be 600 eV and the energy convergence criterion was set to be 106 eV throughout the calculations. The lattice structures under different pressures were optimized by full
relaxation with respect to the volume, shape and internal atomic positions until the change in the total energy between the two ionic steps relaxation was less than 105 eV. Spin polarization has been considered in all the calculations. Here, the phonon calculations were carried out by using the supercell approach with the real-space force constants calculated in density-functional perturbation theory22 implemented in the VASP code, and the phonon frequencies calculated from the force constants using the PHONOPY package.23–25 Based on our tests, the 2 2 2 supercell with 128 atoms and the 5 5 5 k-points grid meshes were used for calculating the phonon spectrum. For Cu2 MnZ (Z=Al, In) with cubic structure, there are three independent SOECs (C11 , C12 , C44 ) and six dependent TOECs (C111 , C112 , C123 , C144 , C155 , C456 ). Here, the energy–strain approach has been implemented to determine these elastic constants, and the elastic constants are extracted from a polynomial fit to the energy versus strain data. After applying a finite strain g to a material, the elastic strain energy (DE) can be expanded in a Taylor series in terms of the strain tensor by using the contracted (Voigt) notation (11 ! 1; 22 ! 2; 33 ! 3, 23 ! 4; 13 ! 5, and 12 ! 6) as VX VX Cij gi gj þ Cijk gi gj gk þ Oðg4 Þ ð1Þ DE ¼ 2! 3! ij
ijk
where V is the volume of the unstrained lattice. To obtain the complete set of SOECs and TOECs of Cu2 MnZ (Z=Al, In), we used six Lagrangian strain tensors in terms of a single strain parameter n. Inserting these strain tensors into Eq. 1, the strain energy density U can be written as an expansion in the strain parameter n as DE 1 1 ð2Þ ¼ A2 n2 þ A3 n3 þ Oðn4 Þ V 2 6 where A2 and A3 are combinations of SOECs and TOECs, respectively. The six specific strain tensors labeled as ga (a ¼ 1; 2; . . . ; 6) can be expressed as: U¼
0
n B g1 ¼ @ 0 0 0 n B g4 ¼ @ 0 0
0 0 0 0 0 0 n
1
C 0 A; 0 1 0 C n A; 0
0
n B g2 ¼ @ 0 0 0 n B g5 ¼ @ 0 n
0 n 0 0 0 0
0
1
C 0 A; 0 1 n C 0 A; 0
0
n B g3 ¼ @ 0 0 0 0 B g6 ¼ @ n n
0 0
1
n
C 0 A;
0
n
n
n
1
0
C nA
n
0
ð3Þ Table I gives the relationship between the coefficients (A2 and A3 ) and the elastic constants (SOECs and TOECs) for the selected six strain tensors. For each strain tensor chosen, the strain parameter n varies from 0.08 to 0.08 in steps of 0.01. Based on the elastic constants, the bulk modulus B and shear modulus G for Cu2 MnZ (Z=Al, In) under different pressures are obtained using the Voigt–Reuss–Hill approximations.26 For the specific
