The Properties of Elasticity, Thermology, and Anisotropy in Pd-Based Alloys

Kuankuan Chen, Meng Hu, Chunmei Li, Guannan Li, Zhiqian Chen^*

Faculty of Materials and Energy, Southwest University, Chongqing, China.

DOI: 10.4236/msce.2017.53002                

Abstract

This work is devoted to investigate the elasticity, anisotropy, plastic properties, and thermal conductivity of PdSnYb, PdSn₂Yb and Heusler alloy Pd₂SnYb via employing the first-principles. The magnetic properties of Pd₂SnYb, PdSnYb and PdSn₂Yb are obtained by the geometry optimization combining with spin polarization. And the stability of these three kinds of materials is ensured by comparing with the enthalpy of formation and binding energy. The Fermi energy has same trend with stability. The details of bulk and Young’s modulus are demonstrated in 3D plots, embodied the elastic anisotropies of PdSnYb, PdSn₂Yb, and Pd₂SnYb. The calculations of plastic properties are also anisotropic. And the minimum thermal conductivities are small enough for these three materials to be used as thermal barrier coatings.

Keywords

First-Principles, Pd-Based Alloys, Elasticity, Thermal Conductivity, Anisotropy

1. Introduction

Heusler alloys are composed of a series of intermetallics. In recent years, plenty of magnetic properties that Heusler alloys presented and their applications in spintronic devices had aroused wide concern [1] [2] [3] . The properties of Heusler alloys were well diversified, such as non-ferromagnetic elements could exhibit ferromagnetism after highly ordered [4] [5] , 100% spin polarization were presented in materials which were called half-metallic ferromagnets (HMF) [6] [7] , only a minority of Heusler alloys containing rare earth had been reported to be superconductors [8] [9] [10] , etc. These above features elucidated their potential for future applications in different fields.

As early as 1903, Cu₂MnAl became the prototype of Heusler alloys, since F. Heusler [11] firstly reported Cu₂MnAl and high magnetic ordered alloy of Cu₂MnAl series. In 1969, P. Webster [12] discussed magnetic and structure properties of Heusler alloys systematically. Liu et al. [13] discovered another highly ordered Heusler alloy, which named Hg₂CuTi. Up to now, more than one hundred kinds of Heusler alloys were found both in theory and experiments, such as Mn-based alloys [14] [15] [16] , Co-based alloys [17] [18] , Cu-based alloys [19] [20] [21] , Ni-based alloys [22] [23] . But the reports about Pd group Heusler alloys were relatively less and the majority of them were related to experiments. Kierstead et al. [24] , Aoki et al. [25] [26] and Stanley et al. [8] studied the Heusler compound Pd₂SnYb and Pd₂SnEr, whose superconductivity and antiferromagnetism were concomitant. Novel properties of thermodynamics and transmission were shown in Pd₂SnYb obviously. And the superconductivity presented at T_c = 2.3 K, along with a synchronous phase of antiferromagnetism and superconductivity yielding at T_N = 220 mK. The testing of elastic and inelastic neutron scattering for Pd₂SnEr was carried out, which proved that Pd₂SnEr turned into superconductor at T_c = 1.17 K. Only when temperature conditions met T > T_c, the antiferromagnetic correlations would occur. The maximum critical temperature was found in Pd₂YSn, which was revealed as the Heusler alloy [27] .

However, in the aspect of theoretical calculation, there is no systematic research on elasticity, thermal properties and anisotropy of Pd-based alloys PdSnYb, PdSn₂Yb and Pd₂SnYb so far. In this work, we provide the overall calculation and analysis of these properties. Especially, once the thermal conductivity is smaller, the heat-shielding performance will be better. The computed minimum thermal conductivities of Pd₂SnYb, PdSnYb and PdSn₂Yb are all less than 0.5 W・m⁻¹・K⁻¹. This minimum thermal conductivity is small enough to be applied to thermal barrier coatings and many other fields. Hence, the thorough discussion carried on the three materials is essential, which inspires our passion on studying these materials. And it makes great sense to explore the microstructure and properties of Pd-based alloys.

2. Calculation Model and Parameters

2.1. Model Details

Pd-based system used in this work includes three alloys: Pd₂SnYb, PdSnYb and PdSn₂Yb. The symmetry group and international table number of Pd₂SnYb are and 216. PdSnYb and PdSn₂Yb are orthorhombic system. Their space groups are PNMA (No. 62) and CMCM (No. 63). In Pd₂SnYb, Pd possesses the 8c site (0.25, 0.25, 0.25), Sn perches the 4a site (0, 0, 0), Yb possesses the 4b site (0.5, 0.5, 0.5). In PdSnYb, atoms of Pd, Sn, and Yb respectively possess the 4c site (0.28675, 0.25, 0.3988), (0.17329, 0.25, 0.07563) and (0.0176, 0.25, 0.69161). In PdSn₂Yb, Pd and Yb perch the 4c site (0, 0.70228, 0.25), (0, 0.42899, 0.25), Sn possesses the 8f site (0, 0.14024, 0.04483). In order to obtain reliable structural optimization results, the lattice constants we employed are all from experiments.

2.2. Parameters Setting

CASTEP code [28] was used for this work, which grounded on the density functional theory [29] . The exchange correlation functional employed the PBE method in the generalized gradient approximation (GGA) [30] . Ultra soft pseudo potential (USPP) [31] was chosen for interaction potential between ionic potential and valence electrons. The atom orbits Pd 4d¹⁰, Sn 5s²5p², and Yb 4f¹⁴5s²5p⁶6s² were considered as valence electrons in the calculation of pseudo potential. The cut-off energy of 450 eV was set for plane waves in the wave-vector K space. For Brillouin regions k-point sampling, the Monkhors-Pack mesh was set as 4 × 4 × 4 [32] . The lattice parameters of Pd₂SnYb, PdSnYb and PdSn₂Yb were optimized successively by using the BFGS scheme [33] [34] [35] [36] . On this basis, the magnetic, alloy corrosion resistance, elastic, thermal conductivity and anisotropy are being computed.

3. Calculation Results and Discussions

3.1. Magnetic Property

The equilibrium lattice constants of Pd₂SnYb, PdSnYb, and PdSn₂Yb are obtained by geometry optimization with spin polarization. The paramagnetic (NM), ferromagnetic (FM) and anti-ferromagnetic (AFM) coupling between Yb atoms are taken into account in the calculations. Atomic initial magnetic order affects the convergence of ground state. Therefore, the different magnetic orders of Yb atoms are considered to ensure the convergence of ground state. In the condition of different magnetic orders, the curves of the relative energy are drawn out in Figure 1, whose minimum energy is set up to be the ground state (0 eV).

As the Figure 1 shown, the energy of AFM-2 (each layer of Yb atoms spin in the opposite manner along the crystal orientation [001]) in Pd₂SnYb is higher than other magnetic orders. And this proves spin polarization displaying in Pd₂SnYb. However, the energy of NM in PdSnYb and PdSn₂Yb are the highest. It demonstrates the ground state of these three materials, which is in accordance

Figure 1. The energy curve of NM, FM and AFM in Pd₂SnYb, PdSnYb and PdSn₂Yb.

with the experiments [37] [38] . The calculated magnetic moment value of each Yb is 2.4 μ_B for the alloy composition Pd₂SnYb. The total magnetic moment of per formula unit for Pd₂SnYb and atoms Yb, Pd, Sn are all 0 μ_B. As we can see, Pd₂SnYb is barely the magnetic one among the three Pd-based alloys. According to Mulliken’s bond population and length shown in Table 1, Pd₂SnYb contains the bond type Pd-Yb, which the other two alloys don’t. And the lack of Pd-Yb bonding maybe indicates the reason for appearing a transition from non-mag- netism (Pd₂SnYb, Pd₂SnYb) to anti-ferromagnetism (Pd₂SnYb). The bond population means the distribution of overlapping electron charge between two atoms. It is usually used to evaluate the ionicity or covalency of a bond. Compared with the positive values of Pd-Sn, the negative of Pd-Yb displays its iconicity, which is also connected with the magnetism in Pd₂SnYb.

3.2. Structural Parameters

Based on the calculation of magnetic ground state in 3.1 Magnetic properties, lattice constant, volume, density, total energy, cohesive energy, formation enthalpy and partial experiment values of Pd₂SnYb, PdSnYb and PdSn₂Yb are listed in Table 2. As we all know, GGA calculation usually overestimates lattice constants. On the contrary, the elastic constants are underestimated. Therefore, lattice parameters calculated by GGA are slightly larger. While the error is negligible, and the computed results still agree well with the experiment data.

For further details of the bonding properties in these alloys, the cohesive energy and formation enthalpy per atom of Pd, Sn and Yb atoms are defined

Table 1. Mulliken’s bond population and length (Ǻ) of the Pd-based alloys.

Table 2. The calculated lattice constants a, b, c (Å), volume V (Å3), density ρ (g・cm⁻³), E_tot (eV/atom), ΔH (eV), ΔE_coh (eV) and partial experiment values of Pd-Sn-Yb.

as the calculated Equation (1) and (2) [42] [43] .

(1)

(2)

Here, ΔH and ΔE_coh respectively represent the formation enthalpy and cohesive energy of Pd-based compounds. E_tot stands for the energy of a unit cell., , are the energy of each Pd, Sn and Yb atom in the bulk state, and, , show the total energy of insular Pd, Sn, Yb atom, respectively. x, y and z are the number of Pd, Sn, and Yb atom in unit cell.

It is clear that the calculated formation enthalpy and the cohesive energy given in Table 2 are negative: 0 > Pd₂SnYb > PdSnYb > PdSn₂Yb. The results show that Pd₂SnYb, PdSnYb and PdSn₂Yb are all thermally stable. Among them, PdSn₂Yb is the easiest to synthesis and the most stable alloy. And Pd₂SnYb, which has the poorest stability and reacts easily with Cl⁻ or H⁺ resulting in corrosion, is just on the contrary.

3.3. Fermi Energy

Fermi level (E_f) also can be known as the Fermi energy. If the electrons accumulation in semiconductor is regarded as a thermodynamic system, the statistic theory has been proved that Fermi energy is the electronic chemical potential of this system.

(3)

in which μ is the chemical potential, F is the free energy, N represents the total number of electrons, T is temperature.

The corrosion behavior on alloys is complicated. In the light of the electron theory, each fermion obeys Fermi-Dirac statistics. According to Pauli exclusion- principle, the minimal energy principle, and Hund rule, fermion occupies the quantum state respectively. On behalf of the top level of electron filling, Fermi energy loses electron in the first. And the higher Fermi level reaches, the easier outermost shells are to lose.

Fermi energy of Pd₂SnYb, PdSnYb, and PdSn₂Yb are shown in Figure 2. Due to the different types and structures of the system, Fermi level is different in the ground state. Corrosion potential is bound on with the Fermi level. So the higher Fermi level reaches, the smaller corrosion potential will be. As shown in Figure 2, the Fermi energy (E_f) values of these compounds with E_f (PdSnYb) > E_f (Pd₂SnYb) > E_f (PdSn₂Yb) indicate that PdSnYb is most likely to lose electrons, while PdSn₂Yb is difficult. Their corrosion potential and complexity of corroding are in the order of PdSn₂Yb > Pd₂SnYb > PdSnYb.

3.4. Elastic Property

The reaction to external stress in the elastic limit of crystal lattice can be charac-

Figure 2. Fermi energy of Pd₂SnYb, PdSnYb and PdSn₂Yb.

terized by elastic constants. It’s of important significant on the stability and stiffness of materials. Table 3 lists the elastic constants of these three alloys. The elastic constants of cubic and orthorhombic system need to satisfy the generalized stability criteria which can be expressed as [44] :

For cubic phase (Pd₂SnYb):

(4)

For orthorhombic phase (PdSnYb and PdSn₂Yb):

(5)

As mentioned in Table 3, the mentioned Pd-based alloys are stable in mechanics, due to the elastic constants satisfy the corresponding stability criterions.

It’s well known that the elastic constants C₁₁ and C₃₃ are depicted as the ability to resist linear compression along x and z-axis [45] . The present C₁₁ is equal to C₃₃ in Pd₂SnYb, indicating that the compression of x and z-axis is isotropy. The largest C₁₁ of PdSnYb implies that it is the most incompressible material along x-axes obviously. For PdSn₂Yb, the value of C₃₃ is slightly higher than the C₁₁, which indicates that the z-axis is less compressible than x-axis. The calculated elastic constants of Pd₂SnYb follow the order:, suggesting the anisotropies of shear moduli for Pd-based intermetallics are relatively weak. The PdSnYb is in the order of. Thus, the bonding strength of adjacent atoms are the highest in (1 0 0) plane. The present C₂₂ is higher than the C₁₁ and C₂₂ for PdSn₂Yb. Therefore, the bonding strength of (0 1 0) plane is higher than (1 0 0) and (0 0 1) planes. In conclusion, all the three compounds have the highest binding strength in (1 0 0) plane.

Additionally, C₄₄, which measures the ability to resist monoclinic shear strain

Table 3. Elastic constants (GPa) of the three Pd-Sn-Yb.

in (1 0 0) plane, is a vital parameter indirectly affecting the indentation hardness [46] . The highest C₄₄ for PdSnYb indicates that it has the strongest resistance to shear deformation in (1 0 0) plane. The equation (C₁₂-C₄₄) is a classical representation of Cauchy pressure. When the value of Cauchy pressure is positive, it reveals the material is ductile, whereas the negative value represents brittleness [47] . The computed Cauchy pressure for Pd-based intermetallics follows this order: PdSnYb (232 GPa) > PdSn₂Yb (93 GPa) > PdSn₂Yb (60 GPa) > 0. The largest value of Cauchy pressure for PdSnYb and the smallest one for PdSn₂Yb manifest PdSnYb is the most ductile structure and PdSn₂Yb is the least one.

For the polycrystalline system, elastic modulus can be got via independent elastic constants. In order to obtain the bulk modulus and shear modulus, we consult the Voigt and Reuss models. Ref. [44] sums up the expressions of bulk and shear modulus for different systems:

For cubic phase (Pd₂SnYb):

(6)

(7)

(8)

For orthorhombic phase (PdSnYb and PdSn₂Yb):

(9)

(10)

(11)

(12)

where, and, , which represent the bulk modulus and shear modulus respectively, are calculated by Voigt and Reuss approximation.

According to the extreme value principle, the Reuss’s and Voigt’s models have been proved to be the lower and upper limits of the elastic constant by Hill [48] . The formula called Voigt-Reuss-Hill (VRH) agrees well with the experiments:

(13)

(14)

where B and G represent the bulk and shear modulus.

The value of bulk modulus and shear modulus, Young’s modulus and Poisson’s ratio using Hill’s models are obtained:

(15)

(16)

Melting point, characterizing the thermodynamic stability of alloy, has always been considered as an important parameter. Deduced from Ref. [49] , the melting temperature of materials, which is closely related to elastic constants, is estimated as follows:

(17)

Table 4 lists the elastic modulus (GPa), bulk modulus (GPa), shear modulus (GPa), Poisson’s ratio ν, Pugh modules ratio and the melting temperature (˚C) of Pd₂SnYb, PdSnYb, and PdSn₂Yb.

Generally, the bulk modulus reflects the average values of bonding strength and the ability to resist volume change. Shear modulus measures the resistance to plastic deformation. As Table 4 shown, the bulk modulus is in the sequence of Pd₂SnYb (106 GPa) > PdSnYb (71 GPa) > PdSn₂Yb (45 GPa), indicating Pd₂SnYb is the least compressible material in all structures. However, the shear moduli of them are almost the same. Young’s modulus serves as a measure of the stiffness. The higher the Young’s modulus is, the stiffer the material will be.

Poisson’s ratio ν and Pugh modules ratio further confirm the brittleness and ductility of materials. Poisson’s ratio v reflects the elastic parameter of uniaxial deformation, especially in atom binding force. With 0.5 as the critical point, Poisson’s ratio v = 0.5 suggests the constancy of volume. When the variation of v is between 0.25 - 0.5, the atom binding force is central force. The v value

Table 4. The calculated values for elastic modulus (GPa), bulk modulus (GPa), shear modulus (GPa), Poisson’s ratio and Pugh modules ratio, melting temperature T_m (˚C).

for Pd₂SnYb, PdSnYb and PdSn₂Yb is higher than 0.25, which shows the atomic forces are remarkably central forces. Pd₂SnYb presents the largest v, reflecting its resistance of shear strain is the weakest. In accordance with Pugh’s criterion [50] , material with is brittle; whereas material with shows ductility. In Table 4, the G/B ratios of Pd₂SnYb, PdSnYb, and PdSn₂Yb are less than 0.57. Thus, these three alloys are deemed to be ductility. The melting point of alloys is implied its thermodynamic stability. In Table 4 the melting temperature of Pd₂SnYb, PdSnYb, and PdSn₂Yb are 984˚C, 1230˚C, 747˚C, respectively, which further verifies the stability.

3.5. Anisotropy

3.5.1. Elastic Anisotropy

It is well known that single crystal is anisotropic, which has great influence on the performance of thin-film materials. So how to characterize the degree of anisotropic is necessary. To obtain the anisotropic degree of Pd-based alloys, the two-dimensional images of shear modulus of Pd₂SnYb, PdSnYb, and PdSn₂Yb are described in Figure 3, the two quarter circles with radius of 50 and 100 in

Figure 3. Two-dimensional graphs of the shear modulus (GPa) in Pd-Sn-Yb alloys. (a) Pd₂SnYb, (b) PdYbSn, (c) YbSn₂Yb.

Figure 3, which are labeled in black solid line, mean isotropy and play a supporting role in estimating the anisotropic degree.

The share modulus G on different plane along different directions can be expressed as [51] :

(001) plane from [100] to [010]: (18)

(100) plane from [100] to [010]: (19)

(010) plane from [100] to [010]: (20)

() plane from [001] to [110]:

(21)

(110) plane from [001] to []:

(22)

in which θ represent the angle between [uvw] direction and [HKL] direction.

As we can see in Figure 3(a), the shear modulus of Pd₂SnYb in (001), (100), and (010) trajectory planes are similar to the quarter circles, which imply that Pd₂SnYb shows almost isotropy in these planes. On the curves of () plane from [001] to [110] and (110) plane from [001] to [], shear modulus show obviously anisotropy of Pd₂SnYb. For PdSnYb and PdSn₂Yb, shear modulus are all anisotropy due to the noncircular plots on the mentioned planes along different directions. To take a panoramic view of Figure 3, shear modulus is the smallest on (110) plane, which suggests it may be the glide plane of Pd-based alloys.

In order to clearly illustrate the anisotropies of mechanical modulus for Pd₂SnYb, PdSnYb, and PdSn₂Yb, we plot three dimensional surfaces of modulus in Figure 4. For bulk modulus and Young’s modulus, the 3D plots can be more intuitive to determine the ability to withstand external stress. Their formulas are as follow:

bulk modulus [52] :

(23)

Young’s modulus [53] :

(24)

where is the elastic compliance coefficient and, , , re- present the directional cosine.

In Figure 4, the bulk modulus of Pd₂SnYb is spherical, reflecting isotropy of the bulk modulus. The three dimensional graphs of bulk modulus of PdSnYb and PdSn₂Yb, and Young’s modulus of Pd₂SnYb are irregularly. Thus, they express anisotropic nature, as well as the PdSn₂Yb performs the strongest anisotropy. Conversely, Pd₂SnYb is isotropy, which is in good agreement with the

Figure 4. Three-dimensional stereograms of the bulk modulus and Young’s modulus (GPa) of Pd-Sn-Yb.

calculated anisotropy of shear modulus in the present work. Compared with bulk modulus, the projections of Young’s modulus on the (100), (010) and (001) planes show a more pronounced anisotropy. Therefore, a stronger directional dependence of Young’s modulus has displayed on these planes.

3.5.2. Ideal Strength of Tensile and Shear Deformation

It is essential to comprehend the causation of the structural stability for the design and application of these Pd-based alloys, especially the response of lattice stress to the applied strain. To analysis the mechanism of mechanical deformation, the stress-strain curves of tensile and shear deformation are performed in Figure 5.

For tensile deformation, the strain directions [100], [010], [001] are parallel to the coordinate axis of the corresponding unit cell. From Figures 5(a)-(c), the tensile strengths of PdSnYb and PdSn₂Yb show anisotropy. And the strongest ideal tensile strengths of these two alloys exist in the strain direction [001]. Due to the different symmetry of Pd₂SnYb compared with PdSnYb and PdSn₂Yb, it’s isotropic along the strain direction [100], [010], and [001]. The yielding stage of Pd₂SnYb, PdSnYb, and PdSn₂Yb in different orientations all occurs in 2% strains.

It can be seen in the Figures 5(d)-(f) that the shear moduli can be obtained from the strains less than 2% [54] . On the basis of this linear parts, the computed

Figure 5. The tensile and shear stress-strain curves of Pd-based alloys along different directions.

shear moduli values are

for Pd₂SnYb,

, for PdSnYb and, , for PdSn₂Yb, respectively. In contrast with the results using Voigt-Reuss-Hill method, they are not accord, which proves that they are all anisotropic in the whole crystal of the three structures.

3.6. The Minimum Value of Thermal Conductivity K_min

The thermal conductivity is a measure of material’s heat conduction ability. Therefore, the research on it of Pd-based alloys in this work is significant.

Owing to the lattice vibration influences the crystal macroscopic thermodynamic properties, the lattice vibration becomes important factors we want to know. And lattice vibration is determined by phonon system. Thus it has great significance to the materials’ thermal conductivity. The transverse acoustic wave velocity (v_t), longitudinal acoustic wave velocity (v_l), and wave velocity (v_m) are calculated [55] :

(25)

(26)

(27)

In the condition of high temperature, the value of thermal conductivity will decrease with increasing temperature [56] . Hence the minimum thermal conductivity value for materials in the applications of high temperature is extremely important. The minimum thermal conductivity of Pd₂SnYb, PdSnYb, and PdSn₂Yb is calculated on the basis of Clark’s model [56] and Cahill’s model [57] :

Clark’s Model: (28)

Cahill’s Model: (29)

where k_B represents Boltzmann’s constant, Ma is the average mass of atoms, E is the Young’s modulus, ρ is density, v_n (n = 1, 2, 3) is acoustic wave velocity, p is the number of atoms in unit volume. All the indexes are calculated in Table 5. The thermal conductivity for cubic ZrO₂ is also calculated, aiming to compare the value with the experimental value to confirm the accuracy of the calculation method.

As shown in Table 5, the calculated thermal conductivity using the Cahill’s model is lightly greater than that computed by the Clark’s model. This is due to the atom number density and phonon spectrum are both considered in Cahill’s model, whereas the Clark’s model does not [58] . Thus, the Clark’s model underestimates the thermal conductivity, however the value adopting Cahill’s model gets closer to the real values of thermal conductivity. In comparison with Clark’s model, the Cahill’s value of ZrO₂ is closer to the experimental value, which confirms this calculation method is credible. As for Pd₂SnYb, the minimum thermal conductivity is largest, and that for PdSn₂Yb is the smallest. Compared to the results in the present work, the increasing content of Sn atoms cause the decreasing in minimum thermal conductivity, when the proportion of Pd/Sn ratios modify. As is known to all, the Y₂O₃-stabilized ZrO₂ (~2.2 W・m⁻¹・K⁻¹) are investigated for application as materials for thermal barrier coatings. Based on the accuracy of the calculation method, the calculated minimum thermal conductivities of Pd₂SnYb, PdSnYb and PdSn₂Yb are all at least a quarter less than ZrO₂,

Table 5. Transverse speed v_t (km・s⁻¹), longitudinal speed v₁ (km・s⁻¹), acoustic speed v_m (km・s⁻¹) for Pd-Sn-Yb, and the minimum thermal conductivities K_min (W・m⁻¹・K⁻¹) of Cahill’s Model, Clark’s Model for Pd-Sn-Yb and ZrO₂.

which show Pd₂SnYb, PdSnYb and PdSn₂Yb can be used for high-temperature- resistant materials, aerospace field, and many other fields.

4. Conclusions

The calculated results showed that the AFM-2 state of Pd₂SnYb and the NM state of PdSnYb, PdSn₂Yb are found to be the ground state, which are agreed with experimental reports. The obtained enthalpy of formation and binding energy are in the order: 0 > Pd₂SnYb > PdSnYb > PdSn₂Yb, indicating that the Pd-based alloys are mechanically stable. The Fermi energy (E_f) values of these compounds with E_f (PdSnYb) > E_f (Pd₂SnYb) > E_f (PdSn₂Yb) imply that PdSnYb is most likely to lose electrons while PdSn₂Yb is difficult. In line with the Cauchy pressure, values of Poisson’s ratio ν, and Pugh modules ratio, these three alloys are deemed to be ductility. The three compounds are all elastic anisotropic, and the anisotropic sequence is PdSn₂Yb > PdSnYb > Pd₂SnYb. The ideal strength of tensile and shear deformation are inconformity in different crystal orientations, implying that Pd₂SnYb, Pd₂SnYb, and Pd₂SnYb are plastic anisotropic. Moreover, the calculated minimum thermal conductivities of Pd₂SnYb, PdSnYb and PdSn₂Yb are all at least a quarter less than that of ZrO₂, the usual thermal barrier coatings materials. That implies these Pd-based alloys can be candidates for high-temperature-resistant materials.

Acknowledgements

This work was supported by Fundamental Research Funds for the Central Universities (XDJK2016D043).

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	Felser, C., Fecher, G.H. and Balke, B. (2007) Spintronics: A Challenge for Materials Science and Solid-State Chemistry. Angewandte Chemie International Edition, 46, 668-699. http://dx.doi.org/10.1002/anie.200601815
[2]	Barth, J., Fecher, G.H., Balke, B., Graf, T., Shkabko, A., Weidenkaff, A. and Ueda, S. (2011) Anomalous Transport Properties of the Half-Metallic Ferromagnets Co2TiSi, Co2TiGe and Co2TiSn. Philosophical Transactions of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 369, 3588-3601. http://dx.doi.org/10.1098/rsta.2011.0183
[3]	Sharma, V., Solanki, A.K. and Kashyap, A. (2010) Electronic, Magnetic and Transport Properties of Co2TiZ (Z= Si, Ge and Sn): A First-Principle Study. Journal of Magnetism and Magnetic Materials, 322, 2922-2928. http://dx.doi.org/10.1016/j.jmmm.2010.05.006
[4]	Pierre, J., Skolozdra, R.V., Gorelenko, Y.K. and Kouacou, M. (1994) From Nonmagnetic Semiconductor to Itinerant Ferromagnet in the TiNiSn-TiCoSn Series. Journal of Magnetism and Magnetic Materials, 134, 95-105. http://dx.doi.org/10.1016/0304-8853(94)90078-7
[5]	Wang, L.L., Miao, L., Wang, Z.Y., Wei, W., Xiong, R., Liu, H.J. and Tang, X.F. (2009) Thermoelectric Performance of Half-Heusler Compounds TiNiSn and TiCoSb. Journal of Applied Physics, 105, Article ID: 013709. http://dx.doi.org/10.1063/1.3056384
[6]	Raphael, M.P., Ravel, B., Huang, Q., Willard, M.A., Cheng, S.F., Das, B.N. and Harris, V.G. (2002) Presence of Antisite Disorder and Its Characterization in the Predicted Half-Metal Co2MnSi. Physical Review B, 66, Article ID: 104429. https://doi.org/10.1103/PhysRevB.66.104429
[7]	Umetsu, R.Y., Kobayashi, K., Kainuma, R., Fujita, A., Fukamichi, K., Ishida, K. and Sakuma, A. (2004) Magnetic Properties and Band Structures of Half-Metal-Type Co2Cr Ga Heusler Alloy. Applied physics letters, 85, 2011-2013. http://dx.doi.org/10.1063/1.1790029
[8]	Stanley, H.B., Lynn, J.W., Shelton, R.N. and Klavins, P. (1987) Antiferromagnetic Structure of the Cubic Superconductor ErPd2Sn. Journal of Applied Physics, 61, 3371-3373. http://dx.doi.org/10.1063/1.338775
[9]	Dönni, A., Fischer, P., Fauth, F., Convert, P., Aoki, Y., Sugawara, H. and Sato, H. (1999) Antiferromagnetic Ordering in the Cubic Superconductor YbPd2Sn. Physica B: Condensed Matter, 259, 705-706. http://dx.doi.org/10.1016/S0921-4526(98)01081-3
[10]	Jeong, T. and Kwon, Y. (2007). Ab-Initio Studies on the Electronic Structure of Y bPd2Sn. Solid State Communications, 143, 429-431. http://dx.doi.org/10.1016/j.ssc.2007.06.008
[11]	Heusler, F. (1904) Über Manganbronze und über die Synthese magnetisierbarer Legierungen aus unmagnetischen Metallen. Angewandte Chemie, 17, 260-264. https://doi.org/10.1002/ange.19040170903
[12]	Webster, P.J. (1969) Heusler Alloys. Contemporary Physics, 10, 559-577. https://doi.org/10.1080/00107516908204800
[13]	Liu, G.D., Dai, X.F., Yu, S.Y., Zhu, Z.Y., Chen, J.L., Wu, G.H., Xiao, J.Q., et al. (2006) Physical and Electronic Structure and Magnetism of Mn2NiGa: Experiment and Density-Functional Theory Calculations. Physical Review B, 74, Article ID: 054435. https://doi.org/10.1103/PhysRevB.74.054435
[14]	Weht, R. and Pickett, W.E. (1999) Half-Metallic Ferrimagnetism in Mn2VAl. Physical Review B, 60, Article ID: 13006. https://doi.org/10.1103/PhysRevB.60.13006
[15]	Özdogan, K., Galanakis, I., Sasioglu, E. and Aktas, B. (2006) Search for Half-Metallic Ferrimagnetism in V-Based Heusler Alloys Mn2VZ (Z = Al, Ga, In, Si, Ge, Sn). Journal of Physics: Condensed Matter, 18, 2905. https://doi.org/10.1088/0953-8984/18/10/013
[16]	Xing, N., Li, H., Dong, J., Long, R. and Zhang, C. (2008) First-Principle Prediction of Half-Metallic Ferrimagnetism of the Heusler Alloys Mn2CoZ (Z = Al, Ga, Si, Ge) with a High-Ordered Structure. Computational Materials Science, 42, 600-605. https://doi.org/10.1016/j.commatsci.2007.09.007
[17]	Ishida, S., Akazawa, S., Kubo, Y. and Ishida, J. (1982) Band Theory of Co2MnSn, Co2TiSn and Co2TiAl. Journal of Physics F: Metal Physics, 12, 1111. https://doi.org/10.1088/0305-4608/12/6/012
[18]	Barth, J., Fecher, G.H., Balke, B., Ouardi, S., Graf, T., Felser, C., Yoshikawa, H., et al. (2010) Itinerant Half-Metallic Ferromagnets Co2Ti Z (Z = Si, Ge, Sn): Ab Initio Calculations and Measurement of the Electronic Structure and Transport Properties. Physical Review B, 81, Article ID: 064404. https://doi.org/10.1103/PhysRevB.81.064404
[19]	Sprungmann, D., Westerholt, K., Zabel, H., Weides, M. and Kohlstedt, H. (2010) Evidence for Triplet Superconductivity in Josephson Junctions with Barriers of the Ferromagnetic Heusler Alloy Cu2MnAl. Physical Review B, 82, Article ID: 060505. https://doi.org/10.1103/PhysRevB.82.060505
[20]	Baek, K.H., Kim, J.H., Woo, H.J., Lee, G.J., Lee, Y.P. and Yoon, C.S. (2009) Magnetic Grating Produced by Localized Crystallization of Amorphous Cu2MnSn Thin Film Using Femtosecond Laser Pulses. Journal of Applied Physics, 105, Article ID: 083927. https://doi.org/10.1063/1.3103582
[21]	Ko, V., Han, G., Qiu, J. and Feng, Y.P. (2009) The Band Structure-Matched and Highly Spin-Polarized Co2CrZ/Cu2CrAl Heusler Alloys Interface. Applied Physics Letters, 95, Article ID: 202502. https://doi.org/10.1063/1.3263952
[22]	Simon, E., Vida, J.G., Khmelevskyi, S. and Szunyogh, L. (2015) Magnetism of Ordered and Disordered Ni2MnAl Full Heusler Compounds. Physical Review B, 92, Article ID: 054438. https://doi.org/10.1103/PhysRevB.92.054438
[23]	Galanakis, I. and Sassoilu, E. (2011) Structural-Induced Antiferromagnetism in Mn-Based Full Heusler Alloys: The Case of Ni2MnAl. Applied Physics Letters, 98, Article ID: 102514. https://doi.org/10.1063/1.3565246
[24]	Kierstead, H.A., Dunlap, B.D., Malik, S.K., Umarji, A.M. and Shenoy, A.G. (1985) Coexistence of Ordered Magnetism and Superconductivity in Pd2YbSn. Physical Review B, 32, 135. https://doi.org/10.1103/PhysRevB.32.135
[25]	Aoki, Y., Sato, H.R., Matsuda, T.D., Sugawara, H. and Sato, H. (1998) Coexistence of and Competition between, Superconductivity and Magnetism in YbPd2Sn. Journal of Magnetism and Magnetic Materials, 177, 559-560. https://doi.org/10.1016/S0304-8853(97)00385-5
[26]	Aoki, Y., Sato, H.R., Sugawara, H. and Sato, H. (2000) Anomalous Magnetic Properties of Heusler Superconductor YbPd2Sn. Physica C: Superconductivity, 333, 187- 194. https://doi.org/10.1016/S0921-4534(00)00100-3
[27]	Wernick, J.H., Hull, G.W., Geballe, T.H., Bernardini, J.E. and Waszczak, J.V. (1983) Superconductivity in Ternary Heusler Intermetallic Compounds. Materials Letters, 2, 90-92. https://doi.org/10.1016/0167-577X(83)90043-5
[28]	Hohenberg, P. and Kohn, W. (1964) Inhomogeneous Electron Gas. Physical Review, 136, B864. https://doi.org/10.1103/PhysRev.136.B864
[29]	Segall, M.D., Lindan, P.J., Probert, M.A., Pickard, C.J., Hasnip, P.J., Clark, S.J. and Payne, M.C. (2002) First-Principles Simulation: Ideas, Illustrations and the CASTEP Code. Journal of Physics: Condensed Matter, 14, 2717. https://doi.org/10.1088/0953-8984/14/11/301
[30]	Perdew, J.P., Burke, K. and Ernzerhof, M. (1996) Generalized Gradient Approximation Made Simple. Physical Review Letters, 77, 3865. https://doi.org/10.1103/PhysRevLett.77.3865
[31]	Vanderbilt, D. (1990) Soft Self-Consistent Pseudopotentials in a Generalized Eigenvalue Formalism. Physical Review B, 41, 7892. https://doi.org/10.1103/PhysRevB.41.7892
[32]	Monkhorst, H.J. and Pack, J.D. (1976) Special Points for Brillouin-Zone Integrations. Physical review B, 13, 5188. https://doi.org/10.1103/PhysRevB.13.5188
[33]	Broyden, C.G. (1970) The Convergence of a Class of Double-Rank Minimization Algorithms 2. The New Algorithm. IMA Journal of Applied Mathematics, 6, 222- 231. https://doi.org/10.1093/imamat/6.3.222
[34]	Fletcher, R.A. (1970) A New Approach to Variable Metric Algorithms. Computer Journal, 13, 317.
[35]	Goldfarb, D. (1970) A Family of Variable-Metric Methods Derived by Variational Means. Mathematics of Computation, 24, 23-26. https://doi.org/10.1090/S0025-5718-1970-0258249-6
[36]	Shanno, D.F. (1970) Conditioning of Quasi-Newton Methods for Function Minimization. Mathematics of Computation, 24, 647-656. https://doi.org/10.1090/S0025-5718-1970-0274029-X
[37]	Giudicelli, P., Roessli, B., Stunault, A., Ollivier, J., Amato, A., Sugawara, H. and Bernhoeft, N. (2004) Low Energy Magnetic Excitations in Superconducting YbSnPd2. Journal of Magnetism and Magnetic Materials, 272, E141-E142. https://doi.org/10.1016/j.jmmm.2003.11.086
[38]	Adroja, D.T. and Malik, S.K. (1992) Magnetic-Susceptibility and Electrical-Resis- tivity Measurements on RPdSn (R = Ce-Yb) Compounds. Physical Review B, 45, 779. https://doi.org/10.1103/PhysRevB.45.779
[39]	Malik, S.K., Umarji, A.M. and Shenoy, G.K. (1985) Depression of the Superconducting Transition Temperature of the Heusler Alloy Pd2YSn with the Addition of Magnetic Rare-Earth Metals. Physical Review B, 32, 4426. https://doi.org/10.1103/PhysRevB.32.4426
[40]	Mullmann, R. and Mosel, B.D. (1998) Dimorphic YbPdSn with ZrNiAl and TiNiSi Type Structure. Z. Kristallogr, 213, 356-363.
[41]	Simkin, M.V. and Mahan, G.D. (2000) Minimum Thermal Conductivity of Superlattices. Physical Review Letters, 84, 927. https://doi.org/10.1103/PhysRevLett.84.927
[42]	Zhou, Z., Zhou, X. and Zhang, K. (2016) Phase Stability, Electronic Structure and Mechanical Properties of IrBx (X = 0.9, 1.1): First-Principles Calculations. Computational Materials Science, 113, 98-103. https://doi.org/10.1016/j.commatsci.2015.11.033
[43]	Song, Y., Guo, Z.X., Yang, R. and Li, D. (2001) First Principles Study of Site Substitution of Ternary Elements in NiAl. Acta Materialia, 49, 1647-1654. https://doi.org/10.1016/S1359-6454(01)00052-0
[44]	Wu, Z.J., Zhao, E.J., Xiang, H.P., Hao, X.F., Liu, X.J. and Meng, J. (2007) Crystal Structures and Elastic Properties of Superhard Ir N2 and Ir N3 from First Principles. Physical Review B, 76, Article ID: 054115. https://doi.org/10.1103/PhysRevB.76.059904
[45]	Gao, X., Jiang, Y., Zhou, R. and Feng, J. (2014) Stability and Elastic Properties of Y-C Binary Compounds Investigated by First Principles Calculations. Journal of Alloys and Compounds, 587, 819-826. https://doi.org/10.1016/j.jallcom.2013.11.005
[46]	Ozisik, H., Deligoz, E., Colakoglu, K. and Surucu, G. (2013) Structural and Mechanical Stability of Rare-Earth Diborides. Chinese Physics B, 22, Article ID: 046202. https://doi.org/10.1088/1674-1056/22/4/046202
[47]	Lewandowski, J.J., Wang, W.H. and Greer, A.L. (2005) Intrinsic Plasticity or Brittleness of Metallic Glasses. Philosophical Magazine Letters, 85, 77-87. https://doi.org/10.1080/09500830500080474
[48]	Hill, R. (1952) The Elastic Behaviour of a Crystalline Aggregate. Proceedings of the Physical Society. Section A, 65, 349. https://doi.org/10.1088/0370-1298/65/5/307
[49]	Fine, M.E., Brown, L.D. and Marcus, H.L. (1984) Elastic Constants versus Melting Temperature in Metals. Scripta Metallurgica, 18, 951-956. https://doi.org/10.1016/0036-9748(84)90267-9
[50]	Pugh, S.F. (1954) XCII. Relations between the Elastic Moduli and the Plastic Properties of Polycrystalline Pure Metals. The London, Edinburgh and Dublin Philosophical Magazine and Journal of Science, 45, 823-843.
[51]	Yan, H., Zhang, M., Wei, Q. and Guo, P. (2013) Ab Initio Studies of Ternary Semiconductor BeB2C2. Computational Materials Science, 68, 174-180. https://doi.org/10.1016/j.commatsci.2012.10.013
[52]	Nye, J.F. (1985) Physical Properties of Crystals: Their Representation by Tensors and Matrices. Oxford University Press, Oxford.
[53]	Zhang, Y., Franke, P., Seifert, H.J. and Wang, J. (2015) Polymorphism of M3AlX Phases (M = Ti, Zr, Hf; X = C, N) and Thermomechanical Properties of Ti3AlN Polymorphs. Journal of the American Ceramic Society, 98, 2570-2578. https://doi.org/10.1111/jace.13602
[54]	Zhang, X., Luo, X., Li, J., Hu, P. and Han, J. (2010) The Ideal Strength of Transition Metal Diborides TMB2(TM = Ti, Zr, Hf): Plastic Anisotropy and the Role of Prismatic Slip. Scripta Materialia, 62, 625-628. https://doi.org/10.1016/j.scriptamat.2010.01.009
[55]	Ravindran, P., Fast, L., Korzhavyi, P.A., Johansson, B., Wills, J. and Eriksson, O. (1998) Density Functional Theory for Calculation of Elastic Properties of Orthorhombic Crystals: Application to TiSi2. Journal of Applied Physics, 84, 4891-4904. https://doi.org/10.1063/1.368733
[56]	Clarke, D.R. (2003) Materials Selection Guidelines for Low Thermal Conductivity Thermal Barrier Coatings. Surface and Coatings Technology, 163, 67-74. https://doi.org/10.1016/S0257-8972(02)00593-5
[57]	Cahill, D.G., Watson, S.K. and Pohl, R.O. (1992) Lower Limit to the Thermal Conductivity of Disordered Crystals. Physical Review B, 46, 6131. https://doi.org/10.1103/physrevb.46.6131
[58]	Li, C.X., Duan, Y.H. and Hu, W.C. (2015) Electronic Structure, Elastic Anisotropy, Thermal Conductivity and Optical Properties of Calcium Apatite Ca5(PO4)3X (X = F, Cl or Br). Journal of Alloys and Compounds, 619, 66-77. https://doi.org/10.1016/j.jallcom.2014.09.022
[59]	Vassen, R., Cao, X., Tietz, F., Basu, D. and Stöver, D. (2000) Zirconates as New Materials for Thermal Barrier Coatings. Journal of the American Ceramic Society, 83, 2023-2028. https://doi.org/10.1111/j.1151-2916.2000.tb01506.x