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Metiri Wahiba, Cheikh Khaled. Ab initio study of structural, electronic, thermo-elastic and optical properties of Pt3Zr intermetallic compound. Chinese Physics B, 2020, 29(4): 047101  
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Ab initio study of structural, electronic, thermo-elastic and optical properties of Pt3Zr intermetallic compound
Metiri Wahiba1, †, Cheikh Khaled2, ‡
Département de Physique, Faculté des Sciences, Université 20 août 1955-Skikda BP 26, Route El-Hadaiek, 21000 Skikda, Algeria
Département de Technologie, Faculté de Technologie, Université 20 août 1955-Skikda BP 26, Route El-Hadaiek, 21000 Skikda, Algeria

 

† Corresponding author. E-mail: wahiba_metiri@yahoo.fr cheikh.khaled@yahoo.fr

Abstract

Structural, elastic, electronic and optical properties of the Pt3Zr intermetallic compound are investigated using first principles calculations based on the density functional theory (DFT) within the generalized gradient approximation (GGA) and the local density approximation (LDA). The Pt3Zr compound is predicted to be of cubic L12 and hexagonal D024 structures. The calculated equilibrium ground-state properties (lattice parameters a and c, bulk modulus B and its pressure derivative B′, formation enthalpy ΔH) of the Pt3Zr compound, for both cubic and hexagonal phases, show good agreement with the experimental results and other theoretical data. Elastic constants (C11, C12, C13, C33, C44, and C55) are calculated. The predicted elastic properties such as Young’s modulus E and shear modulus GH, Poisson ratio ν, anisotropic ratio A, Kleinman parameter ξ, Cauchy pressure (C12−C44), ratios B/C44 and B/G, and Vickers hardness Hv indicate the stiffness, hardness and ductility of the compound. Thermal characteristic parameters such as Debye temperature θD and melting temperature Tm are computed. Electronic properties such as density of states (DOS) and electronic specific heat γ are also reported. The calculated results reveal that the Fermi level is on the psedogap for the D024 structure and on the antibonding side for the L12 structure. The optical property functions (real part ε1(ω) and imaginary part ε2(ω) of dielectric function), optical conductivity σ(ω), refraction index n(ω), reflectivity R(ω), absorption α(ω) and extinction coefficients k(ω) and loss function L(ω)) are also investigated for the first time for Pt3Zr in a large gamme of energy from 0 to 70 eV.

PACS: ;71.15.Mb;;71.20.LP;;71.20.-b;
Keyword:;density functional theory;;intermetallic;;density of states;;L12;;D024;;optical properties;
1. Introduction

Platinum group metals (PGMs) and platinum-based alloys, as high temperature structural materials, have attracted considerable attention in industrial applications[1–6] (automotive, aeronautic, aerospatiale, catalysis etc.). They exhibit exceptional physical, chemical and mechanical properties such as high melting point, high density, high strength, low reactivity and good oxidation and corrosion resistance.[5–16] Melting point of Pt is about 1760 °C, which is higher than that of pure nickel (1450 °C), and a young modulus is about 164 GPa.[17–20] Alloyed Pt (early transition metal) with an element from groups 4B and 5B such as Ti, V, Zr, Nb, Hf, and Ta (late transition metals) could form very stable alloys (Engel–Brewer alloys).[21,22] Pt–Zr alloys, especially Pt3Zr, show enhanced thermal and mechanical properties compared with Ni-based alloys. The Pt3Zr melting temperature in the L12 cubic structure is up to 2600 °C and the Young modulus is about 260 GPa, which are much higher than those of other Pt-based compounds.[23,24]

Many experimental and theoretical works have been effectuated to investigate basic properties of Pt3Zr in the L12 and D024 structures. Bai et al. have studied the structural properties, formation enthalpies, elastic constants and bulk modulus of Pt–Zr alloys in stable and hypothetical phases using the CASTEP code.[25] First principles calculations have been implemented with the CASTEP code by Pan et al. to study a new structure and oxidation mechanism of the Pt3Zr compound.[26] Interfacial stability, electronic structure and bond characteristics of Pt3Zr (111)/Pt (111) interfaces have been investigated by Pan et al. using ab-initio calculations.[27] Using high resolution x-ray photoelectron spectroscopy, temperature programmed desorption, scanning tunneling microscopy and density functional theory, Li el al. studied growth of an ultrathin zirconia film on Pt3Zr.[28] Antlanger et al. investigated the surface of pure and oxidized Pt3Zr (0001) by scanning tunneling microscopy (STM).[29] However, there are insufficient experimental and computational data for realizing Pt3Zr electronic and elastic properties. Moreover, neither experimental nor theoretical results on their optical properties are available in the literature.

In this article, we investigate structural, elastic, thermal, electronic and optical properties of the Pt3Zr intermetallic compound in cubic and hexagonal structures. Our work is based on the first principles calculations considered as the most powerful method for research of materials properties.[30,31]

2. Computational details

In the present paper, computations of structural, elastic, electronic and optical properties of the Pt3Zr intermetallic compound were carried out using first principle calculations. All calculations are based on density functional theory (DFT) employing the full potential linearized augmented plane wave (FP-LAPW) method implemented in the WIEN2k code.[32,33] Exchange correlation functional was treated within the generalized gradient approximation (GGA) with the Perdew–Burke–Emzerhof (PBE) and the local density approximation (LDA).[34,35] The Pt3Zr compound is found to be crystalline in the L12 cubic structure (space group of ) with lattice parameter a = 4.051 Å[36] and in the D024 hexagonal structure (space group of P63/mmc) with lattice parameters a = 5.653 Å and b = 9.347 Å.[5] Muffin–Tin (MT) spheres radii RMT were taken as 2.5 a.u. for Pt and Zr atoms. Total energies are converged at 10−4 Ry. Integrations in the Brillouin zone were performed using special k-point generated with 12 × 12 × 12 and 10 × 10 × 5 mesh grids for cubic and hexagonal structures, respectively. The plane wave expansion parameter RMT × Kmax, which controls the size of the basis sets in our calculations, was taken to be 7. The wave function inside the MT spheres was chosen up to lmax = 10. The charge density was expanded up to Gmax = 12. The valence electron configurations were [Zr] = 4d25s2 and [Pt] = 5d96s1

3. Results and discussion
3.1. Structural properties

Crystal structures of Pt3Zr in Cubic and hexagonal phases are shown in Fig. 1. For the L12 structure, Pt and Zr occupying the Wyckoff site of 3c (0,1/2, 1/2) and 1a (0, 0, 0), respectively. For the D024 structure, the atomic positions are 6g (1/2, 0, 0), 6h (0.83, 2/3, 1/4) and 2a (0, 0, 0), 2c (1/3,2/3,1/4) for Pt and Zr, respectively. The equilibrium lattice constants of the Pt3Zr compound have been computed by optimizing volume through Murnaghan’s equation of state. The calculated total energies versus volume for cubic and hexagonal Pt3Zr using (a) PBE-GGA and (b) LDA approximations are plotted in Fig. 2. We show that the D024 phase of the Pt3Zr compound is slightly lower in energy than the L12 phase. The optimized ground state properties such as lattice parameters a and c, bulk modulus B and its pressure derivative B′, formation enthalpy ΔHf compared with the experimental and theoretical data are listed in Table 1. We find that there is an excellent agreement between our results and the available data. It can be seen that the calculated equilibrium lattice parameter a of the Pt3Zr L12 structure using GGA (LDA) deviates from the experimental value by +0.12% (−2.02%). However, the calculated equilibrium lattice parameters a and c of D024-Pt3Zr deviate from the experimental value by +1.23% (−0.61%) and −0.18% (−1.57%), respectively, using the GGA (LDA) method. We have not found experimental values of B′ to confirm our results.

Fig. 1. Crystal structures of Pt3Zr: (a) cubic L12 and (b) hexagonal D024.
Fig. 2. Total energy versus volume per unit cell of L12-Pt3Zr and D024-Pt3Zr using (a) PBE-GGA and (b) LDA.
Table 1.

Calculated equilibrium elastic constants (a, c), bulk modulus B and its derivative B′, and formation enthalpy ΔHf for cubic and hexagonal Pt3Zr.

.

From the ground-state total energy, the formation enthalpy of Pt3Zr is given as follows:[37]

where is the total energy of the Pt3Zr compound with ϕ structure, and are total energies per atom of Pt and Zr with ϕ and φ structures, respectively. The calculated values of ΔHf for the cubic structure are in good agreement with other theoretical data. However, our result for hexagonal structure is relatively smaller than those reported in Ref. [25]. We find that the ΔHf of the Pt3Zr compound in the D024 phase is lower than that in the L12 phase, which indicates that Pt3Zr in the hexagonal structure is more stable than that in the cubic structure.

3.2. Elastic properties

Elastic constants can be used to measure material’s resistance and mechanical stability under compression. Moreover, elastic constants of materials can offer important information about their mechanical and dynamical behavior. For cubic systems, there are three independent elastic constants Cij, namely C11, C12 and C44. However, there are five independent elastic constants Cij for hexagonal structures, namely C11, C12, C13, C33, and C55. Necessary mechanical stability criteria for cubic and hexagonal crystal systems are C11−C12 > 0, C11 + 2C12 > 0, C44 > 0 and C11 > |C12|, , C55 > 0, respectively.[40,41] The IRelast package contributed by Jamal et al.[42] has been used to calculate the equilibrium elastic constants of Pt3Zr using the GGA and LDA approximations. We use a dense mesh of 165 and 84 k-points in the irreducible Brillouin zone for cubic and hexagonal structures, respectively. The equilibrium calculated values of Cij for cubic and hexagonal Pt3Zr using the GGA and LDA methods are given in Table 2. It is clear that our results coincide very well with the experimental and theoretical data. Moreover, our results satisfy the mechanical stability conditions indicating the stability of the studied compound in cubic and hexagonal structures. It is known that C11 and C33 reflect the unidirectional compression resistance along the principle crystallographic direction while C44 (C55 for hexagonal phase) describes the crystal resistance to the shear strain. It is found that C11 in the cubic (hexagonal) system is 2.5 (5) times higher than C44 (C55), signifying that this compound presents a stronger resistance to the unidirectional compression compared to the pure shear deformation.

Table 2.

Calculated elastic constants of Pt3Zr (in units of GPa).

.

Elastic properties were investigated using the Voigt–Reuss–Hill (VRH) approximation.[43] Hill’s shear GH and bulk moduli BH can be expressed as the average between the Voigt[44] and Reuss[45] bounds as follows: , . For cubic phases, Gv, Bv, GR and BR can be written as

For the hexagonal system, the corresponding formulas are as follows:

where and M = C11 + 2C33 − 4C13.

The calculated bulk modulus BH and shear modulus GH of cubic and hexagonal Pt3Zr are listed in Table 2. The bulk moduli represent the material resistance to the volume change under pressure, however the shear moduli are measure of resistance to shape change under shear stress. Moreover, these parameters can measure the crystal hardness. Therefore, our results indicate that the Pt3Zr has higher hardness in the cubic structure than that in the hexagonal one. The shear modulus values imply that Pt3Zr has excellent shear deformation resistance.

Other significant elastic parameters such as Young’s modulus E, Poisson’s ratio ν, Pugh’s index G/B, machinability index B/C44, Cauchy pressure C12−C44 (C12−C55 for the hexagonal structure), anisotropic ratio A, Kleinman parameter ξ, and Vickers hardness Hv are investigated for the cubic and hexagonal structures using the GGA and LDA approximations. The calculated elastic properties together with other experimental and theoretical data are given in Table 3. From Table 3, the LDA calculations are larger than those of GGA. We can conclude that the smaller the lattice parameter, the higher the elastic properties. Young’s modulus and Poisson’s ratio ν are calculated using the following relations:[46,47]

Young’s modulus E, namely elasticity modulus, is defined as the ratio of the tensile stress to the corresponding tensile strain. It is an important property providing the material stiffness. The calculated Young moduli with the GGA (LDA) method are 300 GPa (335 GPa) for the L12 structure and 230 GPa (261 GPa) for the D024 phase, indicating that Pt3Zr has a strong stiffness higher than other[24] Pt–Zr alloys. Moreover, Pt3Zr in the cubic structure is stiffer than that in the hexagonal structure. Our result for the L12 structure coincides well with that obtained by Pan et al.[24,27] Poisson’s ratio, Pugh’s index and the Cauchy pressure are good indicators for ductile/brittle nature of materials. Poisson’s ratio is smaller than 0.26 for brittle materials, otherwise the material behaves in a ductile manner. Furthermore, Poisson’s ratio gives information about the characteristics of the bonding in materials. For covalent materials, Poisson’s ratio has a small value (ν = 0.1), whereas for ionic compounds, it has a typical value of 0.25. For metallic materials, the ν value is typically 0.33. In our case, the ν values for cubic and hexagonal structures are 0.29 and 0.33, respectively, indicating a considerable metallic contribution to intra-atomic bonding for the Pt3Zr compound. According to Pugh’s index, the material will be brittle (ductile) if the B/G ratio is smaller (higher) than 1.75. Our calculated B/G values are above 1.75 for the cubic and hexagonal structures, indicating a ductile character. The mechanibility index[48] B/C44 (C55 for the hexagonal structure) may be used as a measure of plasticity and a good indicator of lubricating properties.[49,50] For the cubic and hexagonal structures, the B/C44 values are 1.72 and 3.29, respectively. This means that Pt3Zr is a good lubricant in the hexagonal structure while it has a low lubricating ratio in the cubic structure (compared to gold which has a wonderful lubricating properties of 4.17).[51] Moreover, the calculated values of Cauchy pressure are positive, which proves the ductile and metallic behavior of the Pt3Zr compound.

Table 3.

Calculated bulk modulus B (GPa), shear modulus G (GPa), Young’s modulus E (GPa), Pugh’s index B/G, B/C44 ratio, Poisson’s ratio ν, anisotropic parameter A, Kleinman parameter ξ, and Vickers hardness Hv (GPa).

.

The Zener anisotropy factor A is an elastic parameter used to measure the degree of elastic anisotropy in solids. A is highly correlated with the possibility to introduce micro-cracks in materials. For completely isotropic materials, A takes the value of 1, however for anisotropic materials, A differs from the unity.[48] For the cubic structure, A reads[52]

However, for the hexagonal structure, there are three anisotropy factors A1, A2 and A3 corresponding to three shear planes: {1 0 0}, {0 1 0} and {0 0 1} between ⟨ 011 ⟩ and ⟨ 010 ⟩, ⟨ 101 ⟩ and ⟨ 001 ⟩, ⟨ 110 ⟩ and ⟨ 010 ⟩ directions, respectively,[53–55]

Values of A, A1, A2, and A3 differ from 1, indicating the anisotropic character of the cubic and hexagonal structures. The Kleinman parameter ξ is an important parameter to describe the relative position of cation and anion sub-lattices and the material resistance to bond bending against bond stretching. Here the ξ parameter can be expressed as[56]

The ξ value in the L12 structure is found to be very close to that in the D024 structure. To better define the hardness of our compound, Vickers hardness Hv is calculated using a theoretical model,[57–59]

The calculated hardness value for the cubic structure is 11.62 GPa (12.44 GPa) using the GGA (LDA) method. The values for the hexagonal structure are 6.99 GPa using the GGA and 6.86 GPa using the LDA, indicating that hardness of cubic Pt3Zr is the highest. The calculated Hv values for L12 and D024 are under the superhardness limit (≥ 40 GPa). There are no experimental or theoretical results available about the mechanibility index, anisotropy factor, Kleinman parameter and hardness for comparison.

3.3. Debye temperature

The Debye temperature θD is a characteristic parameter of numerous thermal solid properties such as specific heat, thermal conductivity, melting temperature and lattice vibration. Moreover, θD reflects the binding force between atoms. The higher the θD, the stronger the covalent bonding. Here θD can be calculated from the average acoustic velocity as fellows:[60]

where h is Plank’s constant, k Boltzmann’s constant, NA Avogadro’s number, n the number of atoms per formula unit, ρ is the density, M the molecular mass per formula unit, and vm the average acoustic velocity expressed as

where and vl = (G/ρ)1/2 are the shear and longitudinal acoustic wave velocities, respectively.[61,62]

The calculated vm, vl and vs are displayed in Table 4. Acoustic velocities for the cubic structure are the highest, indicating that acoustical wave’s propagation in the cubic structure is faster than that in the hexagonal structure. Debye temperature of cubic Pt3Zr is the largest. Large values of θD indicate the strong covalent bond Pt–Zr of the cubic and hexagonal Pt3Zr compound. The melting temperature is a significant parameter in engineering science. For the cubic structure, the calculated value of Tm is 2665 ∓ 300 K (2926 ∓ 300 K) using the GGA (LDA) method, which makes Pt3Zr promising for high temperature applications.

Table 4.

Calculated shear, longitudinal and average acoustic velocities (in m/s), i.e., vs, vl, and vm, as well as Debye temperature θD (in K), melting temperature (in K).

.
3.4. Density of states

To better understand the structural stability and the bonding characteristics of the Pt3Zr compound, total density of states (TDOS) and partial density of states (PDOS) are calculated using the GGA and LDA methods. Figure 3 gives the TDOS and PDOS of the Pt3Zr compound in the D024 and L12 structures with the GGA approximation. The Fermi level is situated at 0 eV. The TDOS histogram of L12-Pt3Zr (Fig. 3(a)) illustrates that the energy range from −10 eV to −0.67 eV at the bonding states is dominated by the Pt-5d states. However, in the above range the Fermi level from 0 to 9 eV at the anti-bonding states is dominated by the hybridization of Pt-5d and Zr-4d states. This result agree very well with those obtained by Pan et al.[24,26,27] The contribution of Zr-s, p and Pt-s, p in the PDOS of L12-Pt3Zr is negligible. Furthermore, the main bonding peak is situated at −3.66 eV and the main anti-bonding peak, coming from a strong hybridization between Pt-d and Zr-d states, is situated in the energy range 0.42–1.1 eV.

Fig. 3. Total and partial densities of states of Pt3Zr: (a) L12 structure, (b) D024 structure.

It is known that for the hexagonal structure, Pt and Zr atoms occupy two positions (Pt1, Pt2, Zr1 and Zr2). Figure 3(b) illustrates the TDOS and PDOS histograms of D024-Pt3Zr. It shows that the energy range from −9.25 eV to −0.21 eV is dominated by Pt1-5d and Pt2-5d states while the energy range from 0.70 eV to 9.28 eV is dominated by the hybridization of Pt1-d, Pt2-d, Zr1-d, and Zr2-d states. Moreover, the PDOS curves show that the orbital d of Zr1 and Zr2 atoms split into three levels: , (dxy, ) and (dxz − dyz); while Pt1-d and Pt2-d orbitals into five levels: , , dxy, dxz, and dyz.

Figures 3 illustrates the existence of a deep-valley-shaped pseudo gap separating the bonding and anti-bonding states. The pseudo gap is formed from the large hybridization between Zr-4d states and Pt-5d states. The existence of the pseudo gap reflects a strong metallic character of the Pt3Zr compound. Otherwise, The Fermi level is on the anti-bonding side for the cubic structure and exactly on the pseudo gap for the hexagonal system, indicating that D024-Pt3Zr is predicted to be more stable than the L12-Pt3Zr compound. This result is similar to that obtained by Popoola et al.[39,63] Furthermore, the total density of states at Fermi level, N(EF), reflects the material electronic conductivity. The values of N(EF) of cubic and hexagonal Pt3Zr are listed in Table 5. It is shown that N(EF) of D024-Pt3Zr (0.52 states/eV) is lower than that of L12-Pt3Zr (0.55 states/eV), indicating that the Pt3Zr hexagonal structure is slightly more stable than the cubic structure. Moreover, we can deduce that electrical conductivity in the cubic structure is better than that in the hexagonal structure.

Table 5.

Total and partial density of states at Fermi level (N(EF)), minimum of DOS (Nmin), energies at Nmin (E(Nmin), number of bonding electrons (nb) and electronic specific heat (γ) of cubic and hexagonal Pt3Zr.

.

Furthermore, N(EF) is correlated to the electronic specific heat as follows:

where kB is the Boltzmann constant. At low temperature, the electronic specific heat has a large contribution in the material heat capacity. Table 5 regroups the calculated values of γ for cubic and hexagonal system. It is clear that γ of L12-Pt3Zr is higher than that of D024-Pt3Zr, indicating that the heat capacity of cubic structure is the highest. The values of DOS (Nmin) in the minimum of pseudogap of L12-Pt3Zr (0.01 states/eV) is smaller than that of D024-Pt3Zr (0.47 states/eV), where this minimum is situated at E(Nmin) = −0.34 eV and −0.08 eV for the cubic and hexagonal structures, respectively. The calculated values of Nmin and E(Nmin) are listed in Table 3.

To further elucidate the bonding characteristics of the Pt3Zr compound, the numbers of bonding electrons per atom for cubic and hexagonal Pt3Zr, nb, have been calculated. Table 5 illustrates that the nb values for L12-Pt3Zr and D024-Pt3Zr are 7.26 and 5.63, respectively.

3.5. Optical properties

Optical properties of materials permit to understand their nature and describe their response to the electromagnetic radiations. These properties are strongly related to the electronic transition between occupied and unoccupied states. Study of materials optical nature provides good insight for their usage in optoelectronic devices. The complex dielectric function ε(ω) is known to determine the material response to the electromagnetic field and consists of two parts: ε(ω) = ε1(ω) + iε2(ω). The imaginary part ε2(ω) describes the material’s absorption behavior and related to the electronic band structure as follows:[64–67]

where M is the dipole matrix, i and j are the initial and final states respectively, fi is the Fermi distribution function for the ith state, Ei is the energy of electron in the ith state with wave vector k, and ω denotes the frequency of the incident wave. The real part ε1(ω) of the dielectric function gives the information about the polarizibility of the material and can be calculated from the imaginary part using the Kramers–Kronig relation:[67,68]

where P stands for the principal value of the integral. There are two contributions to the dielectric function: interband transitions describing transitions between valence and conduction bands and intraband transitions describing transitions occurring inside the valence or conduction band.

Optical properties of the Pt3Zr intermetallic compound have been calculated using ab-initio calculations implemented in the wien2k code with the GGA exchange-correlation method for the cubic and hexagonal structures. To calculate the dielectric function, a dense k-mesh (365 k-points) in the irreducible Brillouin zone has been used. This study has been carried out for a large range of energy (0–70 eV). For the hexagonal structure, ε(ω) have two independent components: εxx(ω) corresponding to a parallel phonon polarization to c-axis and εzz(ω) corresponding to a perpendicular light polarization. Figures 4(a) and 4(b) illustrate the ε2(ω) spectra of Pt3Zr as a function of photon energy for the cubic and hexagonal structures, respectively. We show that the threshold energy is 0 eV for L12-Pt3Zr and D024-Pt3Zr. For the cubic system, ε2(ω) exhibits seven peaks located at 1.61 eV, 2.61 eV, 7.70 eV, 17.5 eV, 29.10 eV and 54.11 eV, which indicate the high phonon absorption in the energy values. Obviously, ε2(ω) becomes zero in the energy ranges of 26.38–27.80 eV, 35.56–53.32 eV, and >62 eV, indicating that the cubic Pt3Zr becomes transparent in these energy ranges. For the hexagonal structure (Fig. 4(b)), ε2(ω)xx and ε2(ω)zz also exhibit seven peaks with a considerable anisotropy in the range 0–10 eV.

Fig. 4. Imaginary part of the dielectric function versus energy for the (a) cubic and (b) hexagonal Pt3Zr intermetallic compound with the PBE-GGA method.

Moreover, the imaginary part reflects the electronic structure of the material. At low energies (0–10 eV), the imaginary part is characterized by considerable intraband transitions of free electrons, where the density of states of Pt-5d is dominant. However, the interband behavior is dominant in high energy area (< 25 eV) with a transition from Pt-5d states of valence band below the Fermi level to Zr-4d states of conduction band above the Fermi level.

The real part of the dielectric function ε1(ω) is displayed in Fig. 5. For the cubic system (Fig. 5(a)), the static value ε1(ω) at 0 eV is 35.79. From this value, ε1(ω) starts increasing and reaches its maximum of 44.28 at 1.11 eV (1117 nm) in infrared (IR) region, then it decreases and becomes zero at 2.85 eV (435 nm). For the ranges 2.98–12.88 eV and 18.32–25.41 eV, ε1(ω) takes negative values. The ε1(ω) spectra of the hexagonal Pt3Zr is illustrated in Fig. 5(b). We show that the static constant value is 44.27 and 50.6 for ε2(0)xx and ε2(ω)zz, respectively. The maximum values of ε1(ω)xx and ε2(ω)zz appear at 0.58 eV (2137 nm) and 0.11 eV (11271 nm), respectively. Moreover, ε1(ω)xx becomes negative in the energy ranges 3.04–13.48 eV and 17.70–25.45 eV. In the energy ranges 3.71–13.48 eV and 18.05–25.45 eV, ε2(ω)zz takes negative values. We observe a considerable anisotropic behavior in the region 0–10 eV.

Fig. 5. Real part of dielectric function versus energy of the (a) cubic and (b) hexagonal Pt3Zr intermetallic compound with the PBE-GGA method.

Other optical parameters such as absorption coefficient α(ω), refractive index n(ω), energy loss function L(ω), reflectivity R(ω) and optical conductivity σ(ω) can be calculated by knowing ε1(ω) and ε2(ω) as follows:[69,70]

where K(ω) is the extinction coefficient given by

The absorption coefficient is an important optical parameter related to light intensity variation when light passes through the material. Here α(ω) is proportional to the imaginary part of the dielectric function. Peaks and valleys in the absorption spectra are corresponding to the possible transition between states from the valence band to the conduct band. Figure 6 illustrates the absorption curves of Pt3Zr for the cubic and hexagonal structures. For the cubic system (Fig. 5(a)), the maximum absorption occurs in the ultraviolet (UV) region at 29.29 eV (42.32 nm) with a high absorption power of 308.1 × 104 cm−1. For the hexagonal phase (Fig. 5(b)), we show an anisotropic behavior in the energy ranges 2.6–22.60 eV, 28.74–49.90 eV and 53–70 eV. The maximum absorptions for α(ω)xx and α(ω)zz are 29.25 eV (42.38 nm) and 30.61 eV (40.50 nm), respectively, in the UV region.

Fig. 6. Absorption coefficient of the (a) cubic and (b) hexagonal Pt3Zr intermetallic compound as a function of photon energy with the PBE-GGA method.

The refractive index n(ω) (displayed in Fig. 7) is a significant physical parameter corresponding to the measurement of the phase velocity and the attenuation of electromagnetic wave in a medium. At ω = 0, the static refractive index n(0) can be determined by[71] n(0) = [ε(0)]1/2. For L12-Pt3Zr, the refractive index increases with lower energies (0.1–1.35 eV) and attains a maximum value of 6.86 at 1.35 eV (918.40 nm) in IR region, while it exhibits decreasing tendency for higher energy values. Here n(ω) takes the values less than unity for energy ranges: 10.69–13.40 eV, 18.84–28 eV, 30–52.40 eV and above 54.23 eV. Furthermore, the obtained value of n(0) is 6.08.

Fig. 7. Refractive index of the (a) cubic and (b) hexagonal Pt3Zr intermetallic compound as a function of photon energy with the PBE-GGA method.

For D024-Pt3Zr, the maximum values of n(ω)xx and n(ω)zz are 7.15 and 6.73 given at 0.49 eV (2530.28 nm) and 0.10 eV (12398.42 nm) in IR region, respectively; n(0)xx and n(0)zz are 6.63 and 7.08, respectively. Moreover, the energy ranges corresponding to refractive index values less than 1 are 10.63–14.49 eV, ∼18.70–∼28 eV, 29.13–52.5 eV and above 54 eV.

The electron energy loss function is shown in Fig. 8. This important optical parameter is proportional to the energy, in a unit of length, of a fast electron transverse in a material. The main peak in the loss function spectra is known as the plasmon peak, corresponding to the plasma resonance. The maximum values of L(ω) is given in UV region at 26.20 eV (6.33 104 Hz) for the cubic structure. However, the resonant energy loss for the hexagonal system is seen at 25.78 eV (6.23 1015 Hz) in UV region. The resonant energy indicates the transition energy from metallic to dielectric properties (starting energy of interband transitions).

Fig. 8. Loss function of the (a) cubic and (b) hexagonal Pt3Zr intermetallic compound as a function of photon energy with the PBE-GGA method.

The optical reflectivity R(ω) is the ratio between the reflected wave energy to the incident wave energy. R(ω) spectra of Pt3Zr are displayed in Fig. 9. For the cubic structure, the maximum value of R(ω) is about 0.62, which occurs at 4.22 eV (293.80 nm) in UV region. Reflectivity becomes zero at 27.92 eV (44.40 nm) and in the region from 51 eV to 53 eV (24.31–23.39 nm), indicating that the material becomes transparent in these energies values. For the hexagonal structure, we observe an anisotropic structure. The main peak of R(ω)xx is obtained at 3.41 eV (363.6 nm) with a reflectivity of 0.61. The same reflectivity value for R(ω)ZZ occurs at 4.97 eV (249.46 nm). R(ω)xx and R(ω)ZZ tend to zero at 27.72 eV (44.72 nm). We can conclude that Pt3Zr with a strong reflectivity (of 60%) in UV region can be considered as a promising material for coatings in this range.

Fig. 9. Reflectivity of the (a) cubic and (b) hexagonal Pt3Zr intermetallic compound as a function of photon energy with the PBE-GGA method.

Plots of the optical conductivity versus energy are shown in Fig. 10. It is shown that optical conductivity starts from zero, indicating that Pt3Zr has no band gap. For the {L1}2 system, the highest peak is obtained at 2.93 eV (423.15 nm) with a magnitude of 12608.09 Ω−1·cm−1. For the D024 structure, the maximum photoconductivity is 12883.24 Ω−1·cm−1 and 11410.40 Ω−1· cm−1 for σ (ω)xx and σ(ω)ZZ, respectively, obtained at 3.04 eV (407.84 nm) and 2.65 eV (467.86 nm) energies.

Fig. 10. Conductivity of the (a) cubic and (b) hexagonal Pt3Zr intermetallic compound as a function of photon energy with the PBE-GGA method.

Furthermore, intraband transitions of free electrons at low photon energies correspond to the large values of the optical reflectivity, absorption coefficient and photoconductivity. When ε2(ω) attains a minimum, optical properties come from Pt-5d states with intraband transitions to interband transitions of Pt-5d states below the Fermi level to Zr-4d states at the Fermi level. This indicates the optical properties for transitions of metallic-to-dielectric behavior. Unfortunately, there are no experimental or theoretical data available in the literature for the optical properties of the Pt3Zr compound for comparison.

4. Conclusion

In summary, we have investigated structural, elastic, electronic and optical properties of the Pt3Zr intemetallic compound in cubic and hexagonal structures with the FP-LAPW method. The results predict that the hexagonal phase is more stable than the cubic phase. The calculated elastic constants illustrate the mechanical stability of our compound. Values of Young’s modulus indicate that cubic Pt3Zr is much stiffer than hexagonal Pt3Zr. Calculated bulk modulus, shear modulus and Vickers Hardness show that hexagonal Pt3Zr has the highest hardness. Poisson’s ratio reveals the metallic bonding behavior of cubic and hexagonal Pt3Zr. Pugh’s index and Cauchy pressure indicate the ductile nature of our material. Our compound exhibits an anisotropic character. Cubic Pt3Zr has the largest Debye and melting temperature, indicating the strong covalent nature of this compound. Computed density of states shows that hexagonal Pt3Zr electronically is more stable than cubic Pt3Zr. Considering N(EF) and electronic specific heat values, we can deduce that electrical conductivity, melting temperature and heat capacity in the cubic structure are better than those in the hexagonal structure. Optical properties of the Pt3Zr compound have been investigated using the real and imaginary parts of dielectric function. Maximum absorption and reflectivity are obtained in UV region for cubic and hexagonal Pt3Zr. The refractive index of hexagonal Pt3Zr (in IR region) is the highest. The electron energy loss function has also been computed. Optical conductivity σ(ω) spectrum illustrates that highest values of σ(ω) are obtained for the hexagonal structure according to the zz direction. Moreover, intra and inter-band transitions have been discussed. We can conclude that our study is valuable for the future applications of the Pt3Zr intermetallic compound as a structural material.

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