Indian Journal of Physics

, Volume 92, Issue 12, pp 1503–1513 | Cite as

Theoretical calculations of elastic, mechanical and thermal properties of REPt3 (RE = Sc, Y and Lu) intermetallic compounds based on DFT

  • V. Thakur
  • G. Pagare
Original Paper


The elastic, mechanical and thermal properties of isostructural and isoelectronic nonmagnetic REPt3 (RE = Sc, Y and Lu) intermetallic compounds, which crystallize in AuCu3-type structure, are studied using first principles density functional theory based on full potential linearized augmented plane wave method. The calculations are carried out within PBE-GGA, WC-GGA and PBE-sol GGA for the exchange correlation potential. Our calculated ground state properties such as lattice constant (ao), bulk modulus (B) and its pressure derivative (B′) are in good agreement with the available experimental and other theoretical results. We first time predict the elastic constants for these compounds using GGA approximations. All the compounds are found to be ductile in nature in accordance with Pugh’s criteria. The computed electronic band structures show metallic character of these compounds. The charge density plot and density of states of these compounds reveals that the chemical bond between RE and Pt is mainly ionic. The elastic properties including Poisson’s ratio (σ), Young’s modulus (E), shear modulus (GH) and anisotropy factor (A) are also determined using the Voigt–Reuss–Hill averaging scheme. The average sound velocities (vm), density (ρ) and Debye temperature (θD) of these REPt3 compounds are also estimated from the elastic constants. We first time report the variation of elastic constants, elastic moduli, sound velocities and Debye temperatures of these compounds as a function of pressure.


Intermetallic compounds Density functional theory Electronic properties Elastic constants Thermal properties 


71.20.Lp 71.15Mb 71.20.-b 65.40.-b 

1 Introduction

In recent years, the intermetallic compounds have attracted considerable attention because of their potential for technological applications at high temperature. Moreover, some of the intermetallics exhibit different valence states in compounds with metallic character, which gives elevate to intriguing structural, electronic and magnetic properties. Many rare earths (RE) and group IIIA or IVA elements (X) form stable REX3 compounds which crystallize in the AuCu3 structure. REX3 compounds with the AuCu3-type cubic structure exhibit a variety of phenomena [1, 2, 3, 4]. Compounds of this family are investigated because of phenomena such as magnetic moment formation, crystal field and Kondo effect or multiaxial magnetic structures, due to their incomplete 4f shell. Platinum and its alloys readily form intermetallics. The platinum group metal based intermetallics are of particular interest due to high temperature material, better oxidation or corrosive resistance than refractory metals. Arikan et al. [5] have reported the structural, electronic and elastic properties of scandium predicated compound ScX3 (X = Ir, Pd, Pt and Rh) utilizing ab-initio calculations. Meddar et al. [6] have studied the structure and morphology of Pt3Sc alloy thin film using pulsed laser deposition (PLD). Xing et al. [7] have studied the structural stability and enthalpy of formation of refractory intermetallics TM and TM3 (T = Ti, Zr, Hf; M = Ru, Rh, Pd, Os, Ir, Pt). Kumar et al. [8] studied the electronic structure and magneto-optical properties of XPt3 (X = V, Cr, Mn, Fe, Co, Ni) compounds. Archarya et al. [9] have reported the structure, electronic and Fermi properties of ScPt3 and YPt3 compounds. Chen et al. [10] have investigated the structural, elastic, electronic and thermodynamic properties of ScX3 (X = Ir, Pd, Pt and Rh) compounds under high pressure by using the ab-initio pseudo potential method implemented in CASTEP code. Giannozzi et al. [11] have been studied the structural and electronic properties of ScRh3. Sundareswari et al. [12] have reported the electronic structure of the Rhodium predicated intermetallic compounds (A3B) such as Rh3Sc, Rh3Y and Rh3La by the Self Consistent Tight Binding Linear Muffin Tin Orbital (TB-LMTO). Goncharuk et al. [13] have investigated the thermodynamic parameters of ScIr3 and ScIr2. The high-pressure structural (B1–B2) phase transition and the elastic properties of ScS and ScSe have studied by Maachou et al. [14] using the full-potential augmented plane wave plus local orbitals method (FP-APW+LO) with the generalized-gradient approximation (GGA) exchange–correlation functional. Platinum based compounds REPt3 (RE = Sc, Y and Lu) are binary intermetallics compounds with AuCu3 structure, which belongs to Pm-3m space group, with the RE atoms occupying the corners of the cube while the Pt atoms occupy the cube faces. It is well known that pressure plays a significant role in physical properties of materials. Hence the deformation behavior of materials under compression has become quite interesting as it can provide deep insight into the nature of solid-state theories and evaluate the values of fundamental parameters [15]. So the study of pressure effects on REPt3 (RE = Sc, Y and Lu) is necessary and significant. Hence in the present paper, we have made efforts to shed more light on structural, electronic, elastic, mechanical and thermal behavior of LuPt3, ScPt3 and YPt3 compounds using three different approaches under generalized gradient approximation (GGA) based on full-potential linear augmented plane-wave (FP-LAPW) method within the density functional theory (DFT), at ambient pressure as well as high pressures.

The paper is organized as follows: Sect. 2 is staunch to the description of our methods of calculation. In Sect. 3, the results are presented including comparison with experimental and theoretical data, followed by discussion. Conclusions are given in Sect. 4.

2 Computational method

The structural, electronic, elastic, mechanical and thermal properties of REPt3 (RE = Sc, Y and Lu) compounds are investigated utilizing first principle Full Potential Linearized Augmented Plane Wave Method (FP-LAPW) within the density functional theory [16]. The structure is presented in Fig. 1. The effects of exchange–correlation interaction are treated within three different forms of generalized gradient approximation (GGA) [17, 18, 19] for non-magnetic REPt3 (RE = Sc, Y and Lu) compounds. For a good compromise between computational efficiency and precision, we have considered a number of basis function up to RMT × Kmax = 7 for ScPt3, YPt3 and LuPt3 where RMT is the minimum radius of the muffin-tin spheres and K gives the magnitude of the largest k vector in the plane wave basis. Fourier expanded charge density is truncated at Gmax = 12 (Ryd)1/2. A dense mesh of 10 × 10 × 10 k points is used and the tetrahedral method [20] has been used for the Brillouin zone integration. The self-consistent calculations are considered to be converged when the total energy of the system is stable within 10−4 Ry. The total energies are calculated as a function of volume and fitted to Birch–Murnaghan equation of state [21] to obtain the ground state properties like zero-pressure equilibrium volume. Information on the influences of pressure and temperature on the elastic moduli and related aggregate properties of single crystals plays an essential role in predicting and understanding the interatomic interactions, strength, mechanical stability, phase transition mechanisms and dynamical response of materials. For a cubic crystal, the three elastic moduli C11, C12 and C44 plenary describe its elastic behavior.
Fig. 1

Crystal structure of REPt3 (RE = Sc, Y and Lu) compound

The longitudinal and transverse sound velocities (vl and vt) were obtained by using these elastic constants as follows:
$$v_{l} = \sqrt {\frac{{\left[ {C_{11} + \frac{2}{5}\left( {2C_{44} + C_{12} - C_{11} } \right)} \right]}}{\rho }}$$
$${v_{t} = \sqrt {\frac{{\left[ {C_{ 4 4} - \frac{ 1}{ 5}\left( { 2 {\text{C}}_{ 4 4} + C_{12} - C_{ 1 1} } \right)} \right]}}{\rho }} }$$
where C11, C12 and C44 are second-order elastic constants and ρ is mass density per unit volume, and the average sound velocity vm was approximately calculated from [22, 23]:
$$v_{m} = \left[ {\frac{1}{3}\left( {\frac{2}{{v_{t}^{3} }} + \frac{1}{{v_{l}^{3} }}} \right)} \right]^{{\frac{ - 1}{3}}}$$
One of the standard methods to calculate the Debye temperature was from elastic constants, since θD may be estimated from vm by the following equation [22, 24]:
$$\theta_{D} = \frac{h}{{k_{B} }}\left[ {\frac{3n}{{4\pi V_{a} }}} \right]^{{\frac{1}{3}}} v_{m}$$
where h is Planck’s constant, kB is Boltzmann’s constant and Va is the average atomic volume.

3 Results and discussions

3.1 Structural properties

In order to calculate the ground state properties of REPt3 (RE = Sc, Y and Lu) compounds, such as lattice constant (ao), bulk modulus (B) and its pressure derivative (B′), the total energies are calculated as a function of unit cell volume and fitted to Birch-Murnaghan equation of state [21] utilizing exchange correlation as PBE-GGA, Wu–Cohen (WC) GGA and PBE-sol GGA at zero-pressure equilibrium volume. The variation of total energy as a function of volume for these compounds is shown in Fig. 2(a)–(c). The calculated ground state properties are shown in Table 1. It can be optically discerned from Table 1 that our calculated values of lattice parameter ao with PBE-GGA are found to be in good acquiescent with the available experimental [25] and other theoretical data than other GGA approximations [26, 27, 28, 29]. The further calculations have been carried out using only PBE-GGA. The calculated bulk modulus of ScPt3 and YPt3 using PBE-GGA shows close agreement with the previous theoretical values [5, 9, 30]. Due to unavailability of the experimental value of bulk modulus and pressure derivative for LuPt3 compound, we could not compare our results with the experimental values; consequently, our results may accommodate as a presage for future studies.
Fig. 2

Variation of total energy with volume of (a) ScPt3 (b) YPt3 (c) LuPt3 in AuCu3 structure

Table 1

Calculated lattice constant (ao), bulk modulus (B), its pressure derivative (B′) of REPt3 (RE = Sc, Y and Lu) compounds at ambient pressure




ao (Å)



N(EF) states/eV














































































































aRef. [25]; bRef. [26]; cRef. [27]; dRef. [28]; eRef. [29]; fRef. [30]; gRef. [5]; hRef. [9]; iRef. [10]

Pre. present, Expt. experiment, Oth. other theoretical

3.2 Electronic properties

In order to get insight into the bonding character, we have calculated the self consistent non-spin polarized electronic band structure of REPt3 (RE = Sc, Y and Lu) compounds, along with the high symmetry directions in the first Brillouin zone utilizing (GGA) approach are shown in Fig. 3(a)–(c). The Fermi level is considered at zero and it is optically discerned from Fig. 3(a)–(c) that there is no band gap at the Fermi level which shows that all the studied REPt3 compounds exhibit metallic behavior. On the whole, band profiles are seen to be virtually homogeneous for all the studied compounds, except in LuPt3 with highly localized ‘4f’ states of Lu. The lowest lying bands in the valence region below the Fermi level are due to the ‘s’ like states of Pt in all the studied compounds. A flat band is obtained in LuPt3 around − 3.6 eV which is mainly due to ‘f’ like states of Lu as seen in Fig. 3(c). The bands spread in the conduction region above the Fermi level are mainly due to ‘d’ states of Sc, Y and Lu respectively. Our calculated electronic band structure for ScPt3 and YPt3 are in good acquiescent with previous reported works [9]. To further reveal the nature of the electronic band structures and the bonding nature, we have additionally calculated the total densities of states (TDOS) along with partial densities of states (PDOS) for REPt3 compounds as shown in Fig. 4(a)–(c). The metallic character of all these intermetallics compounds is pellucidly optically discerned from the finite DOS at the Fermi level. The peaks observed in the conduction region are mainly due to ‘d’ states of Sc, Y and Lu respectively, in all the compounds. Traditionally the stability in compounds is associated with low density of states (DOS) at the Fermi level EF [31]. We have calculated the density of states at the Fermi level N(EF) and presented in Table 1. The values of this parameter are 3.76, 2.91 and 3.53 states/eV for ScPt3, YPt3 and LuPt3 respectively, we found that the value of N(EF) for ScPt3 and YPt3 are in good agreement with the previous theoretical data [9]. We conclude that the YPt3 has lowest N(EF), this substantiates that YPt3 is more stable compound.
Fig. 3

Band structure in AuCu3 phase for (a) ScPt3 (b) YPt3 (c) LuPt3 at ambient pressure

Fig. 4

Density of states (total and partial) in AuCu3 phase for (a) ScPt3 (b) YPt3 (c) LuPt3

3.3 Electronic charge density

The charge density distribution is a consequential property of solid materials and provides good information about the chemical bonding. Figure 5(a)–(c) shows the charge density plots in (100) plane for REPt3 (RE = Sc, Y and Lu) compounds. The corner atoms are Sc, Y and Lu and the central atom is Pt respectively. The obtained charge density plots reveal the spherically symmetric concentration [32] centered on Sc, Y, Lu and Pt atoms with some distortions near the region of contact with neighboring atoms which indicates the possibility of the presence of ionic bonding along with weak covalent bonding in these compounds. Large difference in electronegativity is responsible for charge transfer among different atoms resulting in ionic bonding nature whereas small electronegativity difference results in charge sharing and responsible for covalent bonding nature. Electronegative values for Sc, Y, Lu and Pt are 1.36, 1.22, 1.27 and 2.28 respectively. We have found that the electronegativity differences in all the studied compounds are large, indicating the presence of ionic bonding. For covalent and ionic materials, the typical cognations between bulk and shear moduli are GH = 1.1B and GH = 0.6B [33] respectively. In our case, the ratio for ScPt3, YPt3 and LuPt3 is 0.29, 0.50 and 0.32 respectively, which confirm the ionic nature of bonding. All the evidence for the bonding charge densities indicates the ionic characteristics of REPt3 (RE = Sc, Y and Lu) compounds.
Fig. 5

Electronic charge density plots of (a) ScPt3 (b) YPt3 (c) LuPt3 at ambient conditions

3.4 Elastic properties

The elastic properties play a paramount role in providing valuable information about the binding characteristic between adjacent atomic planes and provide the important information on the mechanical and dynamical properties, such as interatomic potentials, equation of state and phonon spectra. The second order elastic constants (SOECs) for all the REPt3 (RE = Sc, Y and Lu) compounds have been calculated with PBE-GGA as exchange correlation functional in their AuCu3 structure at ambient pressure, and are given in Table 2 together with the other theoretical data. It can be noted that our calculated elastic constants satisfy the elastic stability criteria [34, 35] for a cubic crystal at ambient conditions, by using the following relation: C11–C12 > 0, C44 > 0, B0 > 0: which clearly indicate the stability of these compounds in AuCu3 structure. Our calculated elastic constants for ScPt3 are in closer agreement with available theoretical data [5, 30]. Though it slightly differ from other theoretical results which might be due to the different exchange correlation approximations used. In the absence of any available measured data in the literature, the elastic constants of LuPt3 and YPt3 compounds could not be compared. To visually perceive the effect of pressure on the elastic constants, the second order elastic constants are also investigated up to 16 GPa for these compounds and presented in Fig. 6(a)–(c), which implicatively insinuates that these compounds are stable in AuCu3 structure over the pressure range 0–16 GPa. The calculated positive values of the elastic constants (C11, C12 and C44) in Fig. 6(a)–(c) shows that these compounds are stable against pressure. Besides, the increment of C11 is higher than that of both C12 and C44. In a physical sense, C11 stands for the longitudinal elastic behavior, whereas C12 and C44 explain the off diagonal and elastic shear characteristic of cubic crystals, respectively. So, a longitudinal strain produces a change in volume without a change in shape. This volume change is related to pressure, and thus reflects a larger change in C11. In contrast, a transverse strain or shearing causes a change in shape without a change in volume. So, C12 and C44 are less sensitive to pressure than C11. To the best of our knowledge, no experimental values for the elastic constants of these compounds have been appeared in the literature, so our results can serve as a prediction for future investigations.
Table 2

Calculated elastic constants of REPt3 (RE = Sc, Y and Lu) compounds at ambient pressure




C11 (GPa)

C12 (GPa)

C44 (GPa)

























286.58 g




















fRef. [30]; gRef. [5]; iRef. [10]

Pre. present, Oth. other theoretical work

Fig. 6

Variation of elastic constants with pressure of (a) ScPt3 (b) YPt3 (c) LuPt3 in AuCu3 structure

3.5 Mechanical properties

Mechanical stability has been analyzed in terms of their elastic constants. In order to investigate the mechanical properties of REPt3 (RE = Sc, Y and Lu) compounds, we determine the Young’s modulus (E), shear modulus (GH), Poisson’s ratio (σ) and anisotropic ratio (A) for useful applications. These are fundamental parameters which are closely related to many physical properties like internal strain, thermo elastic stress, sound velocity, fracture, toughness. We have calculated these properties of REPt3 (RE = Sc, Y and Lu) compounds and presented in Table 3. The shear modulus GH describes the material’s response to shearing strain using the Voigt–Reuss–Hill (VRH) method [36, 37, 38].
Table 3

Calculated Young’s modulus (E), shear modulus (GH), anisotropic factor (A), Poisson’s ratio (σ), B/GH ratio and Cauchy pressure (C12–C44) of REPt3 (RE = Sc, Y and Lu) compounds at ambient pressure




E (GPa)

GH (GPa)




C12–C44 (GPa)




















































fRef. [30]; gRef. [5]; iRef. [10]

Pre present, Oth. other theoretical work

The bulk and shear modulus, defined as-
$$B = \frac{1}{3}\left( {C_{11} + 2C_{12} } \right)$$
$$G_{H} = \frac{{G_{V} + G_{R} }}{2}$$
$$G_{V} = \frac{{C_{11} - C_{12} + 3C_{44} }}{5}\quad {\text{is the Voigt shear modulus}}$$
$${G_{R} = \frac{{ 5 {\text{C}}_{ 4 4} (C_{ 1 1} - C_{ 1 2} )}}{{ 4 {\text{C}}_{ 4 4} + 3\left( {C_{ 1 1} - C_{ 1 2} } \right)}}}\quad {\text{is the Reuss shear modulus}}$$
Anisotropy factor of a material is another important parameter, which gives a measure of the anisotropy of the elastic wave velocity in a crystal and it is given as:
$${A = \frac{{ 2 {\text{C}}_{ 4 4} }}{{C_{ 1 1} - C_{ 1 2} }}}$$
and used to determine whether the structural properties remain the same in all directions or not.

For A = 1 means a completely isotropic material, whereas a value smaller or larger than unity indicates the degree of elastic anisotropy. The calculated elastic anisotropic factor for all REPt3 (RE = Sc, Y and Lu) compounds is larger than unity, indicating that these materials can be regarded as elastically anisotropic materials. The bulk modulus is usually assumed to be a measure of resistance to volume change by applied pressure [39].

As suggested by Pugh’s [40], if B/GH> 1.75; a material behaves in a ductile manner otherwise, the material behaves in a brittle manner. In the present work, the value of B/GH ratio indicates that all the studied compounds are ductile in nature. This is in good agreement with the results found for ScPt3 [25]. Ganeshan et al. [41] have established a correlation between the bonding properties and ductility. Pettifor [42] has suggested that the angular character of atomic bonding in metals and compounds, which also relates to the brittle or ductile characteristics, can be described by the Cauchy pressure (C12–C44). Compounds having more positive Cauchy pressure tend to form bonds which are primarily metallic in nature where as for directional bonding with angular character, the Cauchy pressure is negative, with larger negative pressure representing a more directional character. Based on the calculated equilibrium Cauchy pressure (see Table 3), we may conclude that REPt3 (RE = Sc, Y and Lu) belongs to class of ductile materials. Young’s modulus is defined as the ratio of stress and strain, and is used to provide a measure of the stiffness of the solid, i.e., the larger value of E, the stiffer is the material. And the stiffer solids have covalent bonds [43].

Young’s modulus (E) is given by
$${E = \frac{{ 9BG_{H} }}{{ 3 {\text{B}} + G_{H} }}}$$

It can be seen from Table 3 that the largest value E among the compounds is that of YPt3, so we conclude that YPt3 is the stiffest one.

Bulk modulus is still used as a preliminary measure of the hardness of material but in order to confirm it, other properties must also be taken into account. A high bulk modulus does not mean that a material is hard. Elastic characteristics must be considered as well, and shear modulus might even provide a better correlation with hardness than bulk modulus. Covalent materials generally have a high shear modulus. The nature of bond can be predicted using Poisson’s ratio and reflect the stability of a crystal against shear [44].

The Poisson’s ratio (σ) is given by-
$${\sigma = \frac{{ 3 {\text{B}} - 2 {\text{G}}_{H} }}{{ 2\left( { 3 {\text{B}} + G_{H} } \right)}}}$$

The value of Poisson’s ratio is found to be ≈ 0.1 for covalent materials and ≈ 0.33 for metallic materials [45, 46]. It is observed from Table 3 that the value of Poisson’s ratio lies in between 0.28 and 0.36 for all these intermetallic compounds, signifying metallic bonding of REPt3 compounds. Frantsevich et al. [47] devised a rule on the base of Poisson’s ratio ‘σ’ to make a distinction between ductile and brittle materials. With respect to his criterion, the critical value of ‘σ’ is 0.26. For brittle materials the value of ‘σ’ should be less than 0.26. The value of Poisson’s ratio ‘σ’ in our calculation as reported in Table 3 is 0.36, 0.28 and 0.35 for ScPt3, YPt3 and LuPt3 respectively, endorsing their ductile nature.

The ratio of bulk modulus (B) to C44 may be interpreted as a measure of plasticity [48]. This ratio is also calculated and given in Table 3, and we can see that this ratio is highest for LuPt3 and lowest for YPt3 which shows that LuPt3 contains more delocalized bonds. We have also observed the dependency of elastic moduli with the pressure for all these REPt3 compounds and presented in Fig. 7(a)–(c). It is found that all the three elastic moduli (bulk modulus B, Young's modulus E and shear modulus GH) increase linearly with the applied pressure.
Fig. 7

Variation of elastic moduli with pressure (a) ScPt3 (b) YPt3 (c) LuPt3 at ambient pressure

3.6 Thermal properties

As a consequential thermodynamic index, the Debye temperature is proximately cognate to many fundamental solid-state properties, such as elastic modulus, expansion thermal coefficient and melting point. At low temperature, the vibrational excitation arises solely from acoustic mode. Hence, at low temperature Debye temperature calculated from elastic constants. There are various methods to obtain the values of Debye temperature. A conventional method for estimating the θD value can be obtained from the values of the elastic constants and the wave velocities such as average wave velocity, transverse and longitudinal elastic wave velocities. The predicted sound velocities, Debye temperature and density are given in Table 4. We have studied first time the variation of sound velocities with pressure and presented in Fig. 8(a)–(c). Because of the lack of availability of experimental as well as theoretical data of these compounds, we could not compare them. Hence, our results can be considered as a prediction for these properties and it will testify future experimental work.
Table 4

Calculated density (ρ), longitudinal (vl), transverse (vt), average elastic wave velocities (vm) and Debye temperature (θD) of REPt3 (RE = Sc, Y and Lu) compounds at ambient pressure


ρ*103 (kg/m3)


vt (m/s)

vm (m/s)

θD (K)



















Fig. 8

Variation of longitudinal (vl), transverse (vt) and average (vm) sound velocities with pressure of (a) ScPt3 (b) YPt3 (c) LuPt3 in AuCu3 structure

4 Conclusion

In this paper, we have systematically studied the structural, electronic, elastic, mechanical and thermal properties of states of REPt3 (RE = Sc, Y and Lu) compounds by utilizing FP-LAPW method based on density functional theory as the exchange correlation energy. The lattice constants and bulk moduli are determined and are in excellent agreement with the available experimental data and theoretical values. We have additionally studied the mechanical properties of these compounds. The elastic constants have been calculated using the approach, the energy-strain method. The values obtained for ScPt3 are in reasonable agreement with those obtained previously. The calculated elastic constants satisfy the mechanical stability criterion and the ductility of REPt3 (RE = Sc, Y and Lu) is predicted by Pugh’s criterion. In addition to that the ductility properties of REPt3 (RE = Sc, Y and Lu) determined by Poisson’s ratio σ criterion are in good agreement with the results estimated by the B/G ratio. The band structure and densities of states DOS for these compounds are analyzed and compared. They showed that all materials exhibit metallic character. We have also calculated thermal properties for these compounds. We also report the variation of elastic constants and elastic moduli of these compounds with pressure over the range 0–16 GPa. For the lack of experimental as well as other theoretical results, we could not compare them. Hence our results can be considered as a prediction for these properties, which would be tested in the future both by experimentally and theoretically.



The authors are thankful to MPCST for the financial support.


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Copyright information

© Indian Association for the Cultivation of Science 2018

Authors and Affiliations

  1. 1.Department of PhysicsSarojini Naidu Government Girls P. G. Autonomous CollegeBhopalIndia

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