J. Cent. South Univ. (2017) 24: 1551-1559

DOI: 10.1007/s11771-017-3560-3

Alloying effects of V on stability, elastic and electronic properties of TiFe₂ via first-principles calculations

NONG Zhi-sheng(农智升)¹, CUI Pu-chang(崔普昌)¹, ZHU Jing-chuan(朱景川)², ZHAO Rong-da(赵荣达)³

1. School of Materials Science and Engineering, Shenyang Aerospace University, Shenyang 110136, China;

2. School of Materials Science and Engineering, Harbin Institute of Technology, Harbin 150001, China;

Central South University Press and Springer-Verlag Berlin Heidelberg 2017

Abstract: The alloying effects of V on structural, elastic and electronic properties of TiFe₂ phase were investigated by the first-principles calculations based on the density functional theory. The calculated energy properties including cohesive energy and formation enthalpy indicate V atom would preferentially substitute on 6h sites of Fe atoms in the lattice of TiFe₂ to form the intermetallic Ti₄Fe₇(V). The calculated results of polycrystalline elastic parameters confirm that the plasticity of TiFe₂ would be improved with the addition of V. By discussing the percentage of elastic anisotropy, anisotropy in linear bulk modulus and directional dependence of elastic modulus, it is revealed that the anisotropy of TiFe₂ and Ti₄Fe₇(V) is small. Finally, the density of states, charge density distribution and Mulliken population for TiFe₂ and Ti₄Fe₇(V) were calculated, suggesting there is a mixed bonding with metallic, covalent and ionic nature in TiFe₂ and Ti₄Fe₇(V) compounds. These results also clarify that the reason for the improvement of plasticity with the addition of V in TiFe₂ is the weakened bonding of covalent feature between Ti and V atoms.

Key words: first-principles; elastic properties; alloying effect; TiFe₂ phase

1 Introduction

Nowadays there are more and more Laves phases with AB₂ structure being applied in aerospace structure materials, heat resisting materials and hydrogen storage materials, which is due to their excellent structural stability and designability[1-5]. The intermetallic compound TiFe₂, which meets the stability range of atomic ratio for the C14 structure, has caught widely attention because its structural degrees of freedom can effectively combine with the magnetic degrees of freedom. Hence, the study on magnetic measurements of the hexagonal TiFe₂ Laves phase (Cl4 structure) has been increased. Based on the lattice constants of TiFe₂, the magnetic ground state energy was investigated by HOFFMANN et al [6] using first-principles calculations. It was found that as the antiferromagnetic (AF) state is slightly favored, TiFe₂ shows continually subdued AF and ferromagnetic (FM) ground states.

Although TiFe₂ phase has attractive and potential magnetic applications, its room temperature brittleness enormously limits its development. It is well known that alloying is an effective method to improve the mechanical properties of alloys by changing elastic constants of alloys. The elastic constants of materials play an important role in deciding the reaction of the crystal resisting the external stress, and are necessary parameters for determining the mechanical properties of solid materials [7]. The experimental results have revealed that the addition of V plays a very important role in optimizing the microstructures and mechanical properties of C14 TiX₂ (X=Cr, Ti and Mn) alloys [8]. However, there is little theoretical investigation on the elastic and electronic properties of TiFe₂ phase, and the study of alloying with V in TiFe₂ phase at the atomic and electronic level is not sufficient. Moreover, the effect of V atom on the plasticity, especially the mechanisms of the interaction between TiFe₂ and V atom are not clear.

In this work, the stability, elastic constants and electronic structure of TiFe₂ alloyed with V atom were obtained by the first-principles calculations. The elastic anisotropy, bonding behavior and interaction between atoms were discussed in detail.

2 Calculation method

For theoretically analyzing the influence of V atom on structure and energy of TiFe₂, the density functional theory (DFT) with ultrasoft pseudo-potentials [9] was carried out in the cambridge sequential total energy package (CASTEP) code. In this code, first-principles quantum mechanics calculations were operated by the plane-wave pseudo-potential. The generalized gradient approximation (GGA) of the Perdew-Wang 91 version (PW91) [10, 11] was adopted as the exchange and correlation terms. The cutoff energies of 350 eV were set for all calculations to ensure the precision. The convergence conditions of these calculations can be described as: the maximum force, maximum displacement, maximum stress and energy change were below 0.01 eV/, 5.0×10^-4 , 0.02 GPa and 5.0×10^-6 eV/atom, respectively.

TiFe₂ phase crystallizes in C14 structure with symmetry of P63/mmc (No. 194) and contains 12 atoms per unit cell. In this unit cell, 4 Ti atoms are located on 4f sites and 8 Fe atoms on 2a and 6h sites. The Fe (6h) sites take effective magnetic moments, but there are any molecular fields in the Fe (2a) sites due to structural symmetry [12-14]. The direction of the magnetic moments for Fe (6h) sites is along the c-axis, as shown in Fig. 1. The structures, in which the alloying atom V is placed at Fe (2a) and (6h) sites, substituting for Fe atoms, are named as Ti₄Fe₇(V)-1 and Ti₄Fe₇(V)-2, respectively. In addition, the alloying atom V is placed at Ti (4f) site, which is Ti₃(V)Fe₈.

Fig. 1 Crystal structure of Laves phase TiFe₂ (Arrows represent direction of magnetic moments for Fe atoms)

3 Results and discussion

3.1 Structure and stability of TiFe₂ alloyed with V

The optimized crystal structures of TiFe₂, Ti₄Fe₇(V)-1, Ti₄Fe₇(V)-2 and Ti₃(V)Fe₈ are obtained by using the Broyden-Fletcher-Goldfarb-Shanno (BFGS) methods [15], and then the equilibrium lattice constants can also be calculated. Table 1 shows the calculated results of equilibrium structural parameters for TiFe₂, Ti₄Fe₇(V)-1, Ti₄Fe₇(V)-2 and Ti₃(V)Fe₈ at zero pressure, where V₀ is the volume of unit cell, E_c is the cohesive energy and H_f is the formation enthalpy. Although there are lack of experimental lattice constants for TiFe₂ alloyed with V, the consistent values between calculated and experimental lattice constants of TiFe₂ measured by WERTHEIM et al [12] (a=b=4.778 and c=7.761 ) indicate that the precision for these calculations meets the requirements of analysis. In addition, there is an obvious increase of cell volumes (or lattice distortion) when Fe atom was substituted by V atom in the crystal structure of TiFe₂, which results mainly from the V atom with larger atomic radius (1.71 ) occupying the position of Fe atom with lesser atomic radius (1.56 ).

For further investigating the structural stability of TiFe₂, Ti₄Fe₇(V)-1, Ti₄Fe₇(V)-2 and Ti₃(V)Fe₈ from the view of energy, the formation enthalpy H_f and cohesive energy E_c are introduced to discuss the influence of different substitution positions of V atom on the structural stability of TiFe₂. H_f and E_c are usually used to describe the alloying ability and structural stability of a compound, respectively. H_f is defined as the energy difference that the total energy of final compound subtracts that of initial constituents in proportion [16], and can be expressed by Eq. (1). Similarly, E_c is the energy to decompose the crystal structure of initial compound into free atoms by the ratio of chemical formula [17], and can be calculated by Eq. (2):

             (1)

             (2)

Table 1 Calculated and experimental results equilibrium structural parameters

where E_tot is the calculated total energy of TiFe₂, Ti₄Fe₇(V)-1, Ti₄Fe₇(V)-2 or Ti₃(V)Fe₈, n, m and k are atom numbers of Ti, Fe and V in these compounds, respectively. and (X=Ti, Fe and V) are the energy per X element in corresponding initial stability states and under each isolated states, respectively. The energy of isolated state for an atom is obtained by calculating the crystal total energy of corresponding elements in a large box.

The calculated formation enthalpy H_f and cohesive energy E_c for TiFe₂ and TiFe₂ alloyed with V listed in Table 1 show that TiFe₂, Ti₄Fe₇(V)-1, Ti₄Fe₇(V)-2 and Ti₃(V)Fe₈ all have stable crystal structures due to their lower cohesive energies. It can be easily seen that TiFe₂ has the lowest value of formation enthalpy and cohesive energy, suggesting that the alloying ability and structural stability of TiFe₂ all obviously weaken with the addition of V in the crystal structure of TiFe₂. The reason for this result is probably that the magnetic symmetry of TiFe₂ compound is broken with the addition of V, and the magnetic disorder would increase the energy of structure resulting in the decline of alloying ability and structural stability. Moreover, it should be noted that the structures in which the alloying atom V substitutes for the Fe atom in the structure of TiFe₂ present a lower value of formation enthalpy than that of Ti₃(V)Fe₈, which indicates that V atom would substitute on the 6h sites of Fe atoms in the lattice of TiFe₂ easily, to form the intermetallic Ti₄Fe₇(V)-2.

3.2 Elastic properties of TiFe₂ alloyed with V

The elastic constants (C_ij) of a compound reflect the ability of the crystal resisting the applied stress, and the change of C_ij is important for investigating the influence of V atom on the plasticity of TiFe₂. In this calculation section, the elastic constants are estimated by the method of calculating the fluctuations of total energy of a crystal under different small strains to the equilibrium structure. The elastic strain energy can be calculated by [18]

                      (3)

where ΔΕ is the total energy of deformed structure minus that of initial one, V₀ is the volume of structure at the equilibrium state, C_ij represents the elastic constants, and e_i and e_j are applied strains. For TiFe₂ with hexagonal structure, there are five independent elastic constants, and Table 2 lists the corresponding five types of strains for the hexagonal structure [19]. In Table 2, 5 different strain types have corresponding applied strains parameters represented by using δ, and the unlisted strains are all zero. Based on Eq. (3), types of elastic constants C_ij are also obtained when ΔE/V₀ is equal to 0. In order to obtain the fluctuations of total energy for a hexagonal structure, five appropriate deformations according with each type of strains listed in Table 2 are employed. The calculated elastic constants of TiFe₂ and TiFe₂ alloyed with V are listed in Table 3. The values of C₁₁ and C₃₃ reflect the ability that structure resists external uniaxial compression along the x and z axes, respectively. The larger the value is, the better the ability of resistance is. It can be easily seen that Ti₃(V)Fe₈ and TiFe₂ show a larger values of C₁₁ and C₃₃, while there is an obvious decrease of C₁₁ or C₃₃ when alloying atom V substitutes for the Fe atom in the structure of TiFe₂. It suggests that comparing with TiFe₂, the ability of resistance to be uniaxial compressed would weaken when Fe atom is replaced by V atom in the crystal.

Table 2 Strains used to calculate elastic constants of TiFe₂ with hexagonal structure

The mechanical stability of a compound is associated with the elastic constants, and the Born stability criteria [20] can be an effective measure to judge whether the structure is stable or not from the values of C_ij. The Born stability criteria for a hexagonal structure at zero pressure is given by

C₁₂>0, C₄₄>0, C₁₁-C₁₂>0,     (4)

From Table 3, the calculated values of elastic constants for TiFe₂ and TiFe₂ alloyed with V satisfy well the stability criteria. It indicates that these compounds are all with mechanical stable structure at zero pressure.

Table 3 Elastic constants C_ij of TiFe₂ and TiFe₂ alloyed with V

Based on calculated values of C_ij, bulk modulus (B), shear modulus (G), Young’s modulus (E) and Poisson ratio (ν), these polycrystalline mechanical properties for TiFe₂ and TiFe₂ alloyed with V can be easily obtained by using the Voigte-Reusse-Hill (V-R-H) approximations [21], and the computational formulas are listed as follows:

     (5)

where subscripts V and R express the model of Voigt and Reuss, respectively.

;

;

;

       (6)

The elastic properties for polycrystalline hexagonal TiFe₂ and TiFe₂ alloyed with V are calculated and listed in Table 4. The B and G reflect the ability that structure resists volume change under external compression and reversible deformations in the direction of shear stress, respectively [22]. The present calculated results indicate that TiFe₂ displays larger B and G than Ti₄Fe₇(V)-1, Ti₄Fe₇(V)-2 and Ti₃(V)Fe₈, suggesting that the resistance to volume change and reversible deformations all weaken as V atom enters into the crystal structure of TiFe₂. Furthermore, B and G show the lowest values when V atom substitutes on the 6h sites of Fe atoms in the lattice of TiFe₂. In addition, the larger E corresponds to the stiffer material. The calculated results indicate that TiFe₂ is much stiffer than the others, and there is a similar change rule between E and B in TiFe₂ and TiFe₂ alloyed with V compounds.

The plasticity of materials plays an important role in industry application, and is usually evaluated by the Poisson ratio ν. The bigger value of ν corresponds to the better plasticity of materials. It can be seen from Table 5 that comparing with TiFe₂, the compound that V atom substitutes on the 6h sites of Fe atoms in the lattice of TiFe₂ shows a higher Poisson ratio of 0.31. Based on the results of formation enthalpy, V atom would substitute on the 6h sites of Fe atoms in the lattice of TiFe₂ easily, to form the intermetallic Ti₄Fe₇(V)-2. It is clear that the addition of V atom within the TiFe₂ lattice can indeed improve the plasticity of TiFe₂ alloy. Moreover, according to Pugh’s assumption, the ratio of bulk to shear modulus B/G is an effective parameter to predict brittle or ductile behavior of materials [22]. When B/G> 1.75 is satisfied, the bigger value of B/G is, the better the plasticity is. In the present work, the calculated value of B/G for Ti₄Fe₇(V)-2 is 2.23, which is higher than that of TiFe₂ (B/G=2.06), implying that the compound that V atom substitutes on the 6h sites of Fe atoms in the lattice of TiFe₂ is more ductile than TiFe₂. These results are also in agreement with the conclusions of Poisson ratio discussed above.

Consequently, in the present work, taking the obtained formation enthalpy and elastic properties of TiFe₂ and TiFe₂ alloyed with V into consideration, the structure of Ti₄Fe₇(V)-2, in which V atom substitutes on the 6h sites of Fe atoms in the lattice of TiFe₂, is chosen for further investigation on the elastic anisotropy and electronic properties, and is named as Ti₄Fe₇(V).

3.3 Elastic anisotropy of TiFe₂ alloyed with V

The microcracks of materials, which are closely related to the elastic anisotropy of crystals, are an important facet to evaluate the application of materials in engineering [23]. The elastic anisotropy including the compressibility anisotropy (A_B) and shear anisotropy (A_G) are widely used to investigate the elastic anisotropy, and can be calculated as [24]:

                (7)

where B and G refer to bulk and shear modulus, respectively. When the values of A_B and A_G are 100%, it means that the structure is maximum anisotropic, while 0 suggests that the isotropy of structure shows the feature of isotropy. As shown in Table 5, TiFe₂ and Ti₄Fe₇(V) all display small anisotropy in shear and compression. TiFe₂ is more isotropic in compression, while Ti₄Fe₇(V) is more isotropic in shear.

Table 4 Elastic modulus for TiFe₂ and TiFe₂ alloyed with V by using V-R-H approximations

Table 5 Elastic anisotropy A_B, A_G and anisotropy in linear bulk modulus of TiFe₂ and Ti₄Fe₇(V)

For a hexagonal structure, there is another effective method to express the elastic anisotropy, and it is the anisotropy in linear bulk modulus. B_a and B_c are the linear bulk modulus along the a-axis and c-axis, respectively, and can be taken as [25]:

             (8)

                     (9)

where β reflects the influence of deformation function for a-axis on the c-axis, thus, 1/β means the anisotropy in linear bulk modulus along the c-axis regarding the a-axis. If the value of B_c/B_a is closer to 1, it implies the better isotropic compressibility of materials. Table 5 shows the calculated values of B_a and B_c for TiFe₂ and Ti₄Fe₇(V). It can be easily seen that the anisotropy in linear bulk modulus of TiFe₂ is smaller than that of Ti₄Fe₇(V) because the ratio of B_c/B_a for TiFe₂ is closer to unity, which is consistent with the conclusion of the elastic anisotropy for bulk modulus A_B.

Besides above A_B, A_G and B_c/B_a which are three dimensionless quantities, another significant method of describing elastic anisotropy is the directional dependence of E, which is the three-dimensional surface representation and can be used to describe the in-plane elastic anisotropy. The directional dependence of E for a hexagonal symmetry crystal can be calculated by [26]

  (10)

where l₃ is the direction cosine to the c axis, and S_ij are the elastic compliance constants. Based on the calculated elastic compliance constants and direction cosine, the directional dependence of E for TiFe₂ and Ti₄Fe₇(V) can be obtained by Eq. (10), as shown in Fig. 2. The spherical shape of the three-dimensional surface representation for the directional dependence of E implies a fully isotropic system of crystal. It is clearly seen that the obvious non-spherical nature of elastic modulus for TiFe₂ and Ti₄Fe₇(V) shown in Fig. 2 indicates the anisotropy in the plane, which is consistent with the above results on the percentage elastic anisotropy. In addition, This in-plane elastic anisotropy of TiFe₂ and Ti₄Fe₇(V) can be further understood by the interaction between atoms, and also the feature of chemical bonding, which would be investigated in the section of electronic structure.

Fig. 2 Calculated directional dependence of elastic modulus E for TiFe₂ (a) and Ti₄Fe₇(V) (b) compounds

3.4 Electronic structure of TiFe₂ alloyed with V

The density of states, which is associated with the bonding interaction within compounds, is usually employed to reveal the structural stability and elastic properties. In this section, the total density of states (TDOS) and partial density of states (PDOS) at equilibrium structures for TiFe₂ and Ti₄Fe₇(V) are calculated and presented in Fig. 3. The TDOS of TiFe₂ and Ti₄Fe₇(V) are all mainly constituted by p and d states of Ti and Fe (and V element for Ti₄Fe₇(V)) elements, and the s states of these elements also have some contributions at lower energy level. These compounds all form metallic bonding due to the positive values of TDOS for TiFe₂ and Ti₄Fe₇(V) at the Fermi level. For the antiferromagnetic TiFe₂, as shown in Fig. 3(a), it can be seen that spin-up and down states have the same TDOS while there is difference of direction between these spin-up and down states. The reason can be attributed to the symmetric magnetic structure of TiFe₂, because only Fe (6h) sites carry an effective and symmetric magnetic moment (shown in Fig. 1) in TiFe₂ compound [12-14]. Regarding Ti₄Fe₇(V), owing to the alloying atom V replacing the 6h sites of Fe atoms in the lattice of TiFe₂, the symmetry of structure is broken, resulting in the broken symmetry of the spin-up and down TDOS for Ti₄Fe₇(V), as shown in Fig. 3(b). Hence, Ti₄Fe₇(V) presents an obvious ferromagnetic characteristic. The origin of magnetism as well as hybridization process for TiFe₂ and Ti₄Fe₇(V) can also be revealed by analyzing the feature of PDOS. Figure 3(a) shows that the d-d hybridizations between Ti and Fe atoms in the energy range from about -4 to -2 eV and 0 to 2 eV are clear, and the Fe-p states also play a role in the hybridization within the range of -2 to 2 eV. These d-d hybridizations, which are the common character in the hexagonal structure, imply that some covalent interaction exists between Fe and Ti atoms in TiFe₂ compound [27]. Hence, the TiFe₂ presents a pronounced structural stability. Comparing with TiFe₂, there are four notable peaks at about -3.5, -1.7, 0.3 and 1.2 eV in the up-spin PDOS of Ti₄Fe₇(V). In particular, the peaks at -3.5 eV and 0.3 eV have strong hybridizations between Fe-d, Ti-d and V-d. In addition, the down-spin PDOS of Ti₄Fe₇(V) has four peaks located at about -3.8, -2.8, -0.9 and 0.9 eV, and only the peak at -2.8 eV has stronger hybridization between the down-spin electrons of Fe-d and V-d. The different orbital hybridizations between the spin-up and spin-down TDOS in Ti₄Fe₇(V) which can lead to the charge transfer between different elements are an important reason for bringing the ferromagnetic characteristic of Ti₄Fe₇(V).

Besides density of states, the charge density distribution can also reveal the feature of bonding from the view of electron density. Figure 4 shows the charge density distributions maps on (0 1 -1 0) plane for TiFe₂ and Ti₄Fe₇(V). The core electron distributions of Fe, Ti and V atoms, which are visualized in the form of higher density regions, usually contribute little to the bonding. Moreover, there is obvious metallic bonding between these metal atoms forming in TiFe₂ and Ti₄Fe₇(V) due to the appearance of the delocalized valence electron clouds in these charge density distributions maps. It also can be seen that the shapes of metal atoms including Fe, Ti and V atoms are slightly deformed, and obvious overlaps of electron densities are found between Fe1 and Fe2 atoms, which indicates that there is covalent bonding forming between the nearest Fe atoms. In addition, in contrast to the covalent bonding of Fe1-Fe2, obvious high density regions, which are found between Fe (or V shown in Fig. 4(b)) and its nearest Ti atoms, are firmly localized around these Fe (or V) atoms. Obviously, the Fe-Ti and V-Ti bonding in these compounds are confirmed to be with covalent and somewhat ionic feature [28].

Fig. 3 Total density of states (TDOS) and partial density of states (PDOS)

Fig. 4 Charge density distribution maps on plane

Consequently, there is a mixed bonding of metallic, covalent and ionic nature in TiFe₂ and Ti₄Fe₇(V) compounds. It is the main reason for the high strength and stability, and also a common feature for the electronic structure of C14 AB₂ type Laves phases [18]. In particular, it can be seen from Fig. 4(b) that the density of charge localization around V atom is obviously lower than that around Fe3 atom, and this V atom exactly is the atom that replaces on the 6h sites of Fe atoms in the lattice of TiFe₂ to form the Ti₄Fe₇(V), suggesting that comparing with TiFe₂, the formation of V-Ti bonding in Ti₄Fe₇(V) makes the ionic and covalent interaction weak. Generally speaking, a stronger bonding directionality is a disadvantage for ductility in a compound [29, 30]. Therefore, the ductility of TiFe₂ may be less than that of Ti₄Fe₇(V), which agrees with the results of elastic properties. The addition of V in TiFe₂ can indeed increase the plasticity by weakening the bonding interaction between Ti and alloying atoms V.

As a necessary complementary to the charge density distributions maps, the atomic Mulliken populations are widely used to analyze the bonding behavior from the view of the change of charge distribution on atoms. The atomic Mulliken population on plane for TiFe₂ and Ti₄Fe₇(V) are calculated, and the results are listed in Table 6. Valence states of these atoms in this calculation are Ti 3s²3p⁶3d²4s², Fe 3d⁶4s² and V 3s²3p⁶3d³4s².

Table 6 Atomic Mulliken population on plane for TiFe₂ and Ti₄Fe₇(V) (Ions represent different positions of atoms shown in Fig. 4)

The integrated charge intensities of Ti atoms in TiFe₂ and Ti₄Fe₇(V) are all positive, suggesting that Ti atoms all lose electron, and there is charge transferring from Ti to other atoms. Parts of Ti-4s states are the main orbit of losing valence electrons in these compounds, and these electrons partly transfer to 3p and 3d states of Ti atoms as the incremental free-electron, while the others contribute to other atoms after bonding. Regarding TiFe₂ compounds, the change of charge distribution on Fe atoms which are on 2a sites has the same rule with the Ti atoms, while Fe atoms on 6h sites obtain these lost electron and show negative charge. In contrast to TiFe₂ compound, Fe atoms in Ti₄Fe₇(V) all obtain electron because there is a large number of charge transferring from V atom to other atoms. Obviously, the bonding of ionic feature slightly strengthens while the bonding of covalent feature weakens with the addition of V atom in TiFe₂. It is the main source of the improvement of plasticity for TiFe₂, and consistent with the results of charge density distribution. By the way, it should be noted that the sum of charge for all atoms in Table 6 is not equal to 0, because these atoms are only on the plane of TiFe₂ (or Ti₄Fe₇(V)), not all the atoms in TiFe₂ (or Ti₄Fe₇(V)) compound.

4 Conclusions

The first-principles calculations based on DFT about the addition of V on C14 TiFe₂ compound have been performed. The consistent values between calculated equilibrium lattice constants and experimental ones of TiFe₂ confirm the precision of calculations. These compounds all present better structural stability due to their lower E_c. The results of formation enthalpy show that V atom would preferentially substitute on the 6h sites of Fe atoms in the lattice of TiFe₂, to form the intermetallic Ti₄Fe₇(V). The calculated results of polycrystalline elastic parameters show that the addition of V atom within the TiFe₂ lattice can indeed improve the plasticity of TiFe₂ alloy. Based on the calculated results of electronic structure for TiFe₂ and Ti₄Fe₇(V), it is confirmed that there is a mixed bonding of metallic, covalent and ionic nature in TiFe₂ and Ti₄Fe₇(V) compounds, and the bonding of covalent feature weakens with the addition of V atom in TiFe₂, which is the main reason for the improvement of the plasticity. Overall, these results would provide useful data for further investigating the magnetic properties of TiFe₂ phase, and for future design of new structural alloys.

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(Edited by FANG Jing-hua)

Cite this article as: NONG Zhi-sheng, CUI Pu-chang, ZHU Jing-chuan, ZHAO Rong-da. Alloying effects of V on stability, elastic and electronic properties of TiFe₂ via first-principles calculations [J]. Journal of Central South University, 2017, 24(7): 1551-1559. DOI: 10.1007/s11771-017-3560-3.

Foundation item: Project(51401099) supported by the National Natural Science Foundation of China; project(201501079) supported by the Doctor Startup Foundation of Liaoning Province, China

Received date: 2016-03-14; Accepted date: 2016-05-09

Corresponding author: NONG Zhi-sheng, Lecturer, PhD; Tel: +86-24-89724198; E-mail: nzsfir@163.com