The investigation of structural, electronic, elastic and thermodynamic properties of Gd1−xYxAuPb alloys: A first principle study

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The investigation of structural, electronic, elastic and thermodynamic properties of Gd1−x Yx AuPb alloys: A first principle study

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Parviz Saeidi , Shahram Yalameha , Zahra Nourbakhsh ∗

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Department of Physics, Faculty of Sciences, University of Isfahan, Isfahan, Iran

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First principle calculations have been employed to investigate the effects of Y concentration, pressure and temperature on various properties of Gd1−x Yx AuPb (x = 0, 0.25, 0.5, 0.75, 1) alloys using density functional theory (DFT). The full potential linearized augmented plane wave (FP-LAPW) method within a framework of the generalized gradient approximation (GGA) is used to perform the calculated results of this paper. Phase stability of Gd1−x Yx AuPb alloys is studied using the total energy versus unit cell volume calculations. The equilibrium lattice parameters of these alloys are in good agreement with the available experimental results. The mechanical stability of Gd1−x Yx AuPb alloys is proved using elastic constants calculations. Also, the influence of Y concentration on elastic properties of Gd1−x Yx AuPb alloys such as Young modulus, shear modulus, Poisson’s ratio and anisotropy factor are investigated and analyzed. By considering both Pugh’s ratio and Poisson’s ratio, the ductility and brittleness of these alloys are studied. In addition, the total density of states and orbital’s hybridizations of different atoms are investigated and discussed. Moreover, the effect of pressure and temperature on some important thermodynamic properties is investigated. © 2018 Elsevier B.V. All rights reserved.

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Article history: Received 5 August 2018 Received in revised form 15 October 2018 Accepted 16 October 2018 Available online xxxx Communicated by L. Ghivelder Keywords: Density functional theory Gd1−x Yx AuPb alloys Structural properties Thermodynamic properties Electronic properties Elastic properties

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1. Introduction

Ternary half Heusler compounds have the XYZ chemical formula, where X and Y are transition or rare earth elements and Z is a heavy element. The half Heusler compounds can be regarded as a hybrid XZ rock-salt structure with XY and YZ zinc-blend structure [1]. Cubic XYZ compounds crystallize in the zinc-blend structure type [2]. Half Heusler compounds attract much attention due to their structural, magnetic, electronic and transport properties. Due to these different physical properties, they play an important role in telecommunication and electronics [3]. The RAuPb (R = Gd, Tb, Dy, Ho, Er, Y) compounds crystallize in the face centered cubic MgAgAs-type structure with the F 43m space group (with 4 Au atoms located at (0, 0, 0), 4 R at (0.25, 0.25, 0.25) and 4 Pb at (0.75, 0.75, 0.75)) [4]. There are some experimental and theoretical studies of the physical properties of YAuPb and GdAuPb compounds. Marazza et al. [4] determined the lattice parameters and crystal structure of GdAuPb and YAuPb using X-ray diffraction. They found that these materials have half Heusler MgAgAs structure at normal condition. Lekhal et al. [2] studied the band order, structural and electronic prop-

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* 1

Corresponding author. E-mail address: [email protected] (Z. Nourbakhsh). The first two authors contributed equally to this work.

https://doi.org/10.1016/j.physleta.2018.10.014 0375-9601/© 2018 Elsevier B.V. All rights reserved.

erties of YAuPb compound. They found that it has metallic behavior. Lin et al. [5] investigated the topological phase of YAuPb compound within GGA approach. Al-Sawai et al. [6] studied the electronic properties of YAuPb using the modified Becke and Johnson potentials on the local density approximation (MBJLDA) [7]. They found that it is semi-metal. Also, Singh et al. [8] investigated the structural, electronic and thermoelectric properties of YAuPb compound. They announced that it is semiconductor with zero energy band gap. Also, Kandpal et al. [9] obtained lattice parameter and bulk modulus of YAuPb. Finally, Parviz Saeidi and Zahra Nourbakhsh [10] investigated the topological phase of Gd1−x Yx AuPb alloys under hydrostatic pressure. Up to now, there are no theoretical and experimental results regarding Gd1−x Yx AuPb alloys except Ref. [11], so the present studies can be references for future investigations which can be fruitful and helpful for the experimental and theoretical studies. In this paper, we have followed four important purposes. The first purpose is to determine the phase stability of Gd1−x Yx AuPb alloys. The second purpose is to investigate the structural properties of Gd1−x Yx AuPb alloys for different values of x (x = 0, 0.25, 0.5, 0.75, 1) and to explain the effect of Y concentration on these properties. The third purpose is to investigate the mechanical stability of Gd1−x Yx AuPb alloys using elastic constants calculations and to study the effect of Y concentration on the elastic properties of these alloys. The fourth

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aim of this article is to study the effect of temperature and pressure on thermodynamic properties of these alloys. The paper is organized as follows: the methodology is described in section 2. Results and discussions are presented in section 3 and finally, conclusions are drawn in section 4.

stress to linear normal (shear) strain is defined as the Young (shear)’s modulus. The definition of Poisson’s ratio is the ratio of shear strain to normal strain. Calculation details of these quantities in the cubic structure, are explained in reference [22]. The simple forms of relations between these quantities are as follows:

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n+1 , 2

n 2

and n f ↑ = n f ↓ = − 1, where n is the total number of 4 f electrons of Gd atom. Consequently, with the aid of Madsen and Novak approach, the calculated effective Hubbard parameter of Gd 4 f orbital of GdAuPb compound is 5.77 eV. The second-order elastic tensor and its analysis are performed to determine the elastic properties using the Elam codes [20] within the GGA approach. The symmetry reduction of strain (ε ) and the stress (σ ) can be indicated by a 2nd order tensor, with 6 independent coordinates. The general relation between ε and σ using Voigt notation is as follows [20,21]:

σi = C i j ε j ;

εi = S i j σ j ,

(1)

where, C i j and S i j are the fourth-order tensors, named stiffness and compliance tensor, respectively. The ratio of normal (shear)

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3(1 − 2ν )

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∗

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∗ ( V , T ) are the hydrostatic pressure, volume where P , V and F vib of unit cell and non-equilibrium vibrational Helmholtz free en∗ ( V , T ) can be written ergy [23]. According to the QHD model, F vib as [23]:

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π K B Θ D + 3nK B T ln 1 − e−Θ D /T 

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G ( V ; P , T ) = E ( V ) + P V + F vib ( V , T )

∗ F vib (V ; T ) =

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where B, G, E and ν are bulk, shear, Young’s modulus and Poisson’s ratio, respectively. Furthermore, the thermodynamic properties of Gd1−x Yx AuPb alloys are investigated using quasi-harmonic Debye (QHD) model in the framework of the Gibss2 code [23]. The calculations of these properties based on quasi-harmonic Debye (QHD) model are performed using the energy-volume calculations, which are performed using the WIEN2k code. In the Debye model, a linear dispersion curve of ω D = vk D is used where v is the average velocity of sound waves and ω D is the Debye frequency. The non-equilibrium Gibbs free energy in the QHD model is given by [23]: ∗

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where n is the number of atom per formula unit, D (Θ D / T ) is the Debye integral and Θ D is Debye temperature. By minimizing the G ∗ ( V ; P , T ) with respect to volume, V , the mechanical equilibrium can be achieved. Thus, some thermodynamic properties can be obtained by the following equations [25]:

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The first-principle calculations of Gd1−x Yx AuPb alloys are performed based on density functional theory (DFT) which is implemented in WIEN2k code [11]. The calculated results are obtained using linearized augmented plane wave plus local orbital (LAPW + Lo) method [12]. The Perdew–Burke–Ernzerhof generalized gradient approximation (PBE-GGA) is used for exchange correlation functional [13]. In this work, the radii of the muffin-tin spheres are selected as R Gd = 2.25 R Y = 2.3 R Au = 2.4 R Pb = 2.6 (a.u .). These muffin-tin radii are selected to ensure no charge leakage from the core and to obtain better energy convergence [14]. A set of 343 k-point is generated in the irreducible of Brillouin zone of Gd1−x Yx AuPb alloys. The largest k-vector in the plane wave expansion in interstitial region is limited to K max = 8.5/ R MT (a.u .)−1 , where R MT is the smallest muffin-tin radius in the unit cell. Also, the Fourier expansion of charge density and potential in the interstitial region is confined to G max = 16 ( R y )1/2 . The crystal structure of Gd1−x Yx AuPb alloys is achieved with substituting the Gd atoms of 2 × 2 × 2 super cell with Y to get alloys with 25%, 50%, 75% and 100% concentrations. The atomic positions of Gd1−x Y x AuPb alloys are relaxed to reduce the force of each atom to less than 1.0 mR y /a.u. The first step in the alloy’s physical properties investigation is determination of the minimum energy structures. The earlier studies on the materials with d and f unfilled orbitals such as Gd used compounds show that these materials are stable in magnetic phase. Although density functional theory within GGA is a successful method for the structural properties of many materials, it is insufficient to describe the electronic properties of materials which have unfilled d or f orbitals [15]. For such systems, the generalized gradient approximation plus Hubbard parameter (GGA + U ) is recommended. It is well known that the GGA + U approach enhances the excited state characteristics like energy band gap and ground state properties of the correlated systems [16]. In GGA + U approach, the strong Coulomb repulsion of the localized electrons is computed in a static mean field like approximation [17]. There is a useful effective Hubbard parameter (U eff = U − J ) calculation method introducing by Anisimov and Gunnarsson [18], where U and J are the Coulomb repulsion term and exchange integral, respectively. We use the Madsen and Novak [19] approach based on Anisimov method to find the suitable value of U eff for half full f orbital of Gd atom. In this approach, in order to find U eff , the hopping integrals between f orbital of central atom and the rest of atoms are calculated using a 2 × 2 × 2 super cell, so the two values 1 of occupation of central atom f orbital are: u f ↑ = n+ , n f ↓ = n2 2

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C v = C v V ( p , T ), T =





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(e −ω j /k B T − 1)2

∂2 F BT (P , T ) = V ; ∂V 2 T     ∗ F ≡ E V ( P , T ) + F vib V ( P , T ), T ,   − V ∂(− T S ) γ CV α = th ; γth = − , V BT CV T ∂V T C P = C V (1 + αγth T ),

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where C V , B T and γth are the constant-volume heat capacity, isothermal bulk modulus and thermodynamic Grüneisen parameter, respectively. To calculate γth the volume derivative of − T S is utilized.

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3. Results and discussions

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3.1. Structural aspects

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In this work, to study the phase stability of Gd1−x Yx AuPb alloys, the total energy versus unit cell volume of these alloys in

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Experimental, [4].

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the presence of spin–orbit interaction in ferromagnetic and nonmagnetic phases is calculated. The calculated results indicate that the energy-volume curve of Gd1−x Yx AuPb (0, 0.25, 0.5, 0.75) alloys in ferromagnetic phase lies lower than the nonmagnetic phase, so the ferromagnetic phase of these alloys is the most stable phase, while for YAuPb (x = 1) alloy the nonmagnetic phase is stable. The calculated results of Gd0.75 Y0.25 AuPb (x = 0.25) alloy as a sample of ferromagnetic alloys and also YAuPb (x = 1) alloy which is nonmagnetic are shown in Fig. 1. The equilibrium lattice parameters of these alloys are calculated and tabulated in Table 1. As it can be observed in Table 1, the calculated lattice parameters of these alloys are in good agreement with the available experimental results. It is well known that the GGA overestimate the lattice parameters, so the computational method used in this work is reliable. Also with the increase of x, the lattice parameters of these alloys decrease. Therefore, it seems that with the increase of Y concentration there is a tendency to lattice parameters reduction. Moreover, the lattice parameter of GdAuPb is calculated 6.977 angstrom using GGA + U approach. The calculated lattice parameter of GdAuPb within GGA is closer to the experimental result compare to GGA + U . It is well known fact that GGA + U mostly gives the lattice parameters more than GGA [24] and GGA is enough for investigation of structural properties, so the elastic and thermodynamic properties of these alloys are only investigated within GGA approach, while the electronic properties of these alloys are investigated within both GGA and GGA + U approaches.

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Table 1 Calculated lattice parameters (a0 ), elastic constants, Bulk moduli’s, B, Shear moduli’s, G, Young’s moduli’s, E, Pugh’s ratio (B /G), Poisson’s ratio (v) and Anisotropy factor ( A) of GaAuPb compounds as a function of Y concentration.

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YAuPb

6.891

6.870

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103.772 49.372 49.045 67.505 40.307 100.848 1.674 0.251 0.4292

107.548 50.305 48.867 69.386 40.768 102.273 1.710 0.254 0.3515

106.964 50.528 36.467 69.340 33.167 85.818 2.09 0.293 0.0790

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Fig. 1. The total energy versus volume of nonmagnetic (non-sp) and ferromagnetic (sp) of YAuPb (x = 1) and Gd0.75 Y0.25 AuPb alloys in the presence of spin–orbit coupling.

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Elastic constants can be used as a criterion to evaluate the response of material to applied forces, determining by bulk, young, shear’s modulus, Poisson’s ratio and Pugh’s ratio are associated to strength of material [25]. Moreover, elastic constants can provide important information between neighboring planes of atoms and anisotropic property of chemical bonding and structural constancy [25]. Crystal symmetry plays a vital role to decrease the number of elastic constants from 81 to 3, 5, 6, 13 for cubic, hexagonal, tetragonal and monoclinic, respectively [26]. Therefore, the Gd1−x Yx AuPb (x = 0, 0.25, 0.5, 0.75, 1) alloys have three independent elastic constants, i.e., C 11 , C 12 , C 44 due to their cubic symmetry. In order to investigate the mechanical stability of Gd1−x Yx AuPb alloys, the elastic constants of these alloys are calculated and given in Table 1. As it is observed in Table 1, all of Gd1−x Yx AuPb alloys satisfy the conditions of mechanical stability showing below [27,28]:

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(11)

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so, these alloys are mechanically stable. Each of these three independent elastic constants describes specific physical phenomena. C 11 is used to describe the elasticity in length and measures the elastic stiffness of solids with respect to (1 0 0) uniaxial strain [25]. Also, both C 12 and C 44 are related to elasticity

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C 11 − C 12 > 0,

C 11 > 0,

C 11 + 2C 12 > 0,

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Fig. 2. Elastic constants, Bulk moduli’s, B, Shear moduli’s, G, Young’s moduli’s, E, Pugh’s ratio (B /G), Poisson’s ratio (v) and Anisotropy factor ( A) of Gd1−x Yx AuPb alloys as a function of Y concentration.

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to x = 0.25, then increases up to x = 0.75 and again decreases up to x = 1, so the strongest and weakest resistance to volume deformation belong to Gd0.25 Y0.75 AuPb and Gd0.75 Y0.25 AuPb alloys. Shear modulus is an appropriate quantity to evaluate the plastic deformation (i.e., shape change) [25]. The larger shear modulus indicates that the material is more able to resist to shape change. As indicated in Fig. 2, with the ascending Y concentration, shear modulus decreases up to x = 0.25, then it increases up to x = 0.75 and again it finds a decreasing trend up to x = 1. Therefore, it is true to say that Gd0.25 Y0.75 AuPb and YAuPb alloys have the strongest and weakest resistance to shape change, respectively. Young’s modulus is utilized to evaluate the resistance to longitudinal tensions. In fact, young’s modulus reflects the stiffness of materials [25], so harder materials have high young’s modulus. In these alloys, as it is illustrated in Fig. 2, the Gd0.25 Y0.75 AuPb and YAuPb alloys have maximum and minimum hardness. Bulk modulus to shear modulus (B /G) is known the Pugh’s ratio [30] which is used to identify the brittle or ductile materials. The threshold value of B /G is 1.75, which separates brittle materials from ductile one. Materials with the B /G larger than 1.75 are considered as ductile materials while materials which have B /G smaller than 1.75 are brittle materials. As it is indicated in Fig. 2, with the increasing of Y concentration, there is a decreasing trend of B /G up to x = 0.25 and then the B /G increases up to x = 1. However, except YAuPb alloy which has the value of B /G larger than 1.75, the other alloys have the value of B /G smaller

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in shape. C 12 is associated to pure shear stress in (1 0 0) plane along [1 0 0] direction and C 44 corresponds to the shear stress in (0 1 0) plane in the [0 0 1] direction [25], so the shear deformation is related to both shear stresses. The variation of elastic constants of Gd1−x Yx AuPb alloys versus Y concentration is indicated schematically in Fig. 2. As it can be seen in Fig. 2, with the increase of Y concentration, C 11 reduces from x = 0 to x = 0.25, then increases up to x = 0.75 and again decreases for x = 1, so it is evident that Gd0.75 Y0.25 AuPb and Gd0.25 Y0.75 AuPb alloys have the lowest and highest elastic stiffness than the other alloys. In addition, the increase of Y concentration causes that C12 increases up to x = 0.5 and then decreases up to x = 1, so Gd0.5 Y0.5 AuPb and YAuPb alloys have the maximum and minimum shear stress in (1 0 0) plane along [1 0 0] direction. Finally, by considering the behavior of C 44 in Fig. 2, it is obvious that with the escalating of Y concentration, C 44 descends from x = 0 to x = 0.25 and then ascends up to x = 1 showing that Gd0.75 Y0.25 AuPb and YAuPb alloys have the lowest and highest shear stress in (0 1 0) plane in the [0 0 1] direction. Elastic properties such as bulk, shear and Young’s modulus, Pugh’s ratio (B /G), Poisson’s ratio (ν ) and anisotropy factor ( A) of these alloys are calculated and shown in Fig. 2. Bulk modulus is a criterion to investigate the volume compression [29]. In the other words, bulk modulus is a quantity which can measure the volume change resistance, so that the larger the bulk modulus, the better resistance to volume deformation. With increase of Y concentration, the bulk modulus of Gd1−x Yx AuPb alloys reduces up

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where G V /G R and B V / B R are the shear/bulk modulus which are calculated by Voigt–Reuss approximation, respectively. As it can be observed in Table 1, the calculated anisotropy of these alloys has deviation from 0 and 1, so all of Gd1−x Yx AuPb alloys are elastically anisotropic.

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3.3. Electronic properties

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In order to investigate the electronic properties of Gd1−x Yx AuPb alloys, the total and partial density of states (TDOS and PDOS, respectively) of these alloys are calculated. The TDOS and PDOS of Gd1−x Yx AuPb alloys are calculated using GGA and GGA + U approaches and plotted in Fig. 4. As it can be observed in Fig. 4, the non-zero DOS of these alloys at the Fermi energy in both GGA and GGA + U approaches exhibits the metallic behavior of Gd1−x Yx AuPb alloys. Also, by paying attention to small charts focused around the Fermi energy which are located in TDOS charts, the metallic characteristic of these alloys is remarkably revealed. In addition, the d orbital of Gd atom and d orbital of Y atom have major contribution around the Fermi energy of Gd0.75 Y0.25 AuPb, Gd0.5 Y0.5 AuPb and Gd0.25 Y0.75 AuPb alloys, while p orbital of Pb atom and d orbital of Gd and Y atoms play dominant role around the Fermi energy of GdAuPb and YAuPb alloys, so these atoms have major contribution to electronic properties of Gd1−x Yx AuPb alloys. Moreover, to better understand of the role of f orbital of Gd atom in Gd1−x Yx AuPb alloys, the electronic density of states of this orbital using GGA and GGA + U approaches are calculated and illustrated in Fig. 5. It is seen that the role of f orbital of Gd atom using GGA approach around the Fermi energy especially for electrons with spin down is considerable, so this orbital has considerable effect in electronic properties of Gd1−x Yx AuPb (x = 0, 0.25, 0.5, 0.75) alloys and can’t be ignored. However, when the GGA + U approach is used, the f orbital partial density of states of Gd atom is located far from the Fermi energy, so this orbital plays less important role in electronic properties of these alloys in comparison with the GGA approach. In addition, in order to study the hybridization of Gd1−x Yx AuPb alloys different orbitals, the partial electron density of states (PDOS) of these alloys are calculated and shown in Fig. 4. It is well known that there must be two condition for the orbitals which may appropriate for hybridization. First, the hybrided orbitals must relate to two atoms which are located in neighboring to each other. Second, hybrided orbitals must have similar behavior in one energy interval. Electronic configurations of Y, Gd, Au and Pb atoms are

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than 1.75, so YAuPb alloy is ductile alloy while the rest of alloys (x = 0, 0.25, 0.5, 0.75) are considered as brittle alloys. Poisson’s ratio (ν ) is a suitable quantity, which is used for determining various properties of solids. First, it can be used as a criterion to assess brittle or ductile materials. The threshold value of ν is 0.26, so materials with ν -value smaller than 0.26 are brittle while ductile material has ν -value larger than 0.26 [31]. The Gd1−x Yx AuPb alloys with x = 0, 0.25, 0.5 and 0.75 are brittle and YAuPb (x = 1) alloy is ductile which is in agreement with the obtained results from B /G. Second, Poisson’s ratio is a good tool to anticipate the nature of chemical bonding. Materials with purely covalent bonding have ν = 0.1 while materials which have pure metallic bonding have ν = 0.33 [32]. The ν -values of our alloys are located between these two values, so it is predicted that the chemical bonding of Gd1−x Yx AuPb alloys would be the mixture of covalent and metallic bonding [25,33]. The charge density distribution is an important tool which provides useful information about the chemical bonding [34]. The distribution of charge density in the (1 0 1) plane is indicated in Fig. 3. The intensity of charge density is represented in thermo-scale in which the dark blue and white colors show low and high intensity, respectively. The lines of charge density represented by pale blue and the rate of their compaction show that there is a covalent bonding between Au–Pb and Au–Gd (Y) atoms. Also, the interstitial regions which have non-zero charge density shown by dark blue. These regions show the weak metallic bonding. Therefore, both metallic and covalent bonding are observed in Gd1−x Yx AuPb alloys. However, the compaction of charge density’s lines around these atoms for x = 0.25, 0.5 and 0.75 alloys is less than x = 0 and 1 alloys, shows that the covalent bonding of these alloys is weaker than GdAuPb (x = 0) and YAuPb (x = 1) alloys. Due to close similarity between the distribution of charge density of x = 0.25 with x = 0.5 and 0.75, the charge distribution of x = 0.25 is only shown in Fig. 3. Third, Poisson’s ratio can help us to determine the nature of interatomic forces [35]. Materials which have the Poisson’s ratio between 0.25 and 0.5 are considers as central force solids, whereas materials which have the Poisson’s ratio less than 0.25 or greater than 0.5 are maintained by the non-central forces of atoms. Therefore, all of Gd1−x Yx AuPb alloys except Gd0.75 Y0.25 AuPb are kept by the central forces of atoms. The 0 and 1 values of universal anisotropy ( A) represent the perfect isotropic and anisotropic materials [36,37], respectively. Each deviation from these two values shows the degree of anisotropy. The calculated formulas of universal anisotropy factor A U is defined below:

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Fig. 3. The distribution of charge density for GdAuPb (x = 0), YAuPb (x = 1) and Gd0.75 Y0.25 AuPb (x = 0.25) in the (1 0 1) plane. The zones with white and blue colors show highest and lowest of charge density’s distribution, respectively. (For interpretation of the colors in the figures, the reader is referred to the web version of this article.)

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5d , [Xe]6s 4 f 5d 6p , respectively. As it can be seen in Fig. 4, for GdAuPb (x = 0) alloy, there is a roughly strong hybridization between Pb-6p orbital and Gd-5d orbital in the energy range of −2 eV to 0 and weak hybridization between 0 eV to 1.5 eV. By paying attention to PDOS of Gd0.75 Y0.25 AuPb, Gd0.5 Y0.5 AuPb and Gd0.25 Y0.75 AuPb alloys in Fig. 3, we observe that the strong hybridization between Y-4d and 2

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Fig. 4. The calculated total and partial density of state of Gd1−x Yx AuPb alloys.

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Gd-5d orbitals in the energy interval of −3 eV to 3 eV, so that d orbital of Y and Gd plays an important role in electronic properties of these three alloys. Finally, there is a hybridization between Pb-6p and Y-4d orbitals of YAuPb (x = 1) alloy in the energy range of −2.5 eV to 1.5 eV which is weak between −2.5 eV to about −0.7 eV and 0.5 to 1.5 eV and becomes strong between −0.7 eV to 0.5 eV. Therefore, it can be concluded that with the substituting

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Fig. 5. The partial density of states of f orbital of Gd atom in Gd1−x Yx AuPb alloys.

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of Y concentration instead of Gd concentration, the strength of hybridization between Y-4d and Gd-5d orbitals increases especially in Gd0.75 Y0.25 AuPb, Gd0.5 Y0.5 AuPb and Gd0.25 Y0.75 AuPb alloys. In fact, the hybridization of Pb-6p and Gd-5d orbitals becomes weaker

and weaker and Gd-5d orbitals tend to be hybrided with Y-4d, so it is true to say that the role of Y concentration is to improve hybridization of these three alloys rather than GdAuPb (x = 0) and YAuPb (x = 1) alloys.

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In order to study thermodynamic properties, some important quantities such as specific heat at constant volume and pressure (C v and C p ), Debye temperature (Θ D ), thermal expansion coefficient (α ) and Gruneisen parameter (γ ) of Gd1−x Yx AuPb alloys at various pressures (0–20 GPa) and temperatures (0–800 K) are calculated. The phonon-based quasi-harmonic Debye model is valid in these temperatures and pressures [25]. The effect of temperature

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Fig. 7. The temperature and pressure dependence of the Debye temperature, thermal expansion coefficient and Grüneisen parameter of Gd1−x Yx AuPb alloys.

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and pressure on C v is calculated and indicated in Fig. 6. At low temperatures, the rate of C v increasing is high and this may be related to exponential increasing of the number of excited phonon modes. As indicated in the small chart of C v versus temperature, the C v versus T 3 is linear, demonstrating Debye law. Therefore, the T 3 Debye law [38] at low temperatures is satisfied for all of Gd1−x Yx AuPb alloys at low temperatures. At high temperatures, the value of C v of all alloys approaches to classical asymptotic limit i.e., Dulong–Petit limit [39]. The variation of C v versus pressure

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References

[1] W. Feng, D. Xiao, Y. Zhang, Y. Yao, Phys. Rev. B 82 (2010) 235121. [2] A. Lekhal, F. Benkhelifa, S. Méçabih, B. Abbar, B. Bouhafs, Bull. Mater. Sci. 39 (2016) 195–200. [3] S. Talakesh, Z. Nourbakhsh, J. Supercond. Nov. Magn. (2017) 1–16. [4] R. Marazza, D. Rossi, R. Ferro, J. Less-Common Met. 138 (1988) 189–193. [5] H. Lin, L.A. Wray, Y. Xia, S. Xu, S. Jia, R.J. Cava, A. Bansil, M.Z. Hasan, Nat. Mater. 9 (2010) 546–549. [6] W. Al-Sawai, H. Lin, R. Markiewicz, L. Wray, Y. Xia, S.-Y. Xu, M. Hasan, A. Bansil, Phys. Rev. B 82 (2010) 125208. [7] F. Tran, P. Blaha, Phys. Rev. Lett. 102 (2009) 226401. [8] S. Singh, K. Kaur, R. Kumar, Appl. Surf. Sci. (2016). [9] H.C. Kandpal, C. Felser, R. Seshadri, J. Phys. D, Appl. Phys. 39 (2006) 776. [10] P. Saeidi, Z. Nourbakhsh, J. Magn. Magn. Mater. 451 (2018) 681–687. [11] P. Blaha, K. Schwarz, J. Luitz, WIEN97: A Full Potential Linearized Augmented Plane Wave Package for Calculating Crystal Properties, Techn. Universitat Wien, Austria, ISBN 3-9501031-0-4, 1999. [12] P. Blaha, K. Schwarz, G.K.H. Madsen, D. Kvasnicka, J. Luitz, University of Technology, Vienna, 2014. [13] J.P. Perdew, K. Burke, M. Ernzerhof, Phys. Rev. Lett. 77 (1996) 3865. [14] I.H. Bhat, S. Yousuf, T.M. Bhat, D.C. Gupta, J. Magn. Magn. Mater. 395 (2015) 81–88. [15] B. Sahli, H. Bouafia, B. Abidri, A. Abdellaoui, S. Hiadsi, A. Akriche, N. Benkhettou, D. Rached, J. Alloys Compd. 635 (2015) 163–172. [16] V.I. Anisimov, F. Aryasetiawan, A. Lichtenstein, J. Phys. Condens. Matter 9 (1997) 767. [17] M. Noorafshan, Z. Nourbakhsh, J. Supercond. Nov. Magn. 1 (8) (2017). [18] V. Anisimov, O. Gunnarsson, Phys. Rev. B 43 (1991) 7570. [19] G.K. Madsen, P. Novák, Europhys. Lett. 69 (2005) 777. [20] A. Marmier, Z.A. Lethbridge, R.I. Walton, C.W. Smith, S.C. Parker, K.E. Evans, Comput. Phys. Commun. 181 (12) (2010) 2102–2115. [21] C. Kittel, Introduction to Solid State Physics, John Wiley & Sons, 2008. [22] H. Fu, L. Yao, Z. Hou, J. Fu, Y. Ma, Eur. Phys. J. B 3 (87) (2014) 1–6. [23] A. Otero-de-la-Roza, D. Abbasi-Pérez, V. Luaña, Comput. Phys. Commun. 182 (2011) 2232–2248. [24] M. Yazdani-Kachoei, S. Jalali-Asadabadi, I. Ahmad, K. Zarringhalam, Sci. Rep. 6 (2016) 31734. [25] M. Hadi, M. Roknuzzaman, A. Chroneos, S. Naqib, A. Islam, R. Vovk, K. Ostrikov, Comput. Mater. Sci. 137 (2017) 318–326. [26] O. Pavlic, W. Ibarra-Hernandez, I. Valencia-Jaime, S. Singh, G. Avendaño-Franco, D. Raabe, A.H. Romero, J. Alloys Compd. 691 (2017) 15–25. [27] S. Yalameha, A. Vaez, Int. J. Mod. Phys. B 32 (2018) 1850129. [28] C. Kittel, P. McEuen, P. McEuen, Introduction to Solid State Physics, 1996. [29] W.L. Bond, J. Phys. Chem. Solids 3 (1957) 338. [30] S. Pugh, Philos. Mag. J. Sci. 45 (1954) 823–843. [31] G. Vaitheeswaran, V. Kanchana, A. Svane, A. Delin, J. Phys. Condens. Matter 19 (2007) 326214. [32] A. Savin, H.J. Flad, J. Flad, H. Preuss, H.G. von Schnering, Angew. Chem., Int. Ed. Engl. 31 (1992) 185–187. [33] N.W. Ashcroft, N.D. Mermin, Phys. Solid State 403 (2005).

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stability of Gd1−x Yx AuPb alloys is proved using elastic constants. By considering bulk modulus, it is found that Gd0.25 Y0.75 AuPb and Gd0.75 Y0.25 AuPb alloys have the strongest and weakest resistance to volume deformation. Further, all Gd0.25 Y0.75 AuPb and YAuPb alloys show the strongest and weakest resistance to shape change between Gd1−x Yx AuPb alloys due to their shear modulus. According to both Pugh’s ratio and Poisson’s ratio of Gd1−x Yx AuPb alloys, all of magnetic alloys (i.e., x = 0, 0.25, 0.5 and 0.75) are construed as brittle alloys, while YAuPb (x = 1) is ductile alloy. Further, all Gd1−x Yx AuPb alloys show metallic behavior due to their TDOS. Moreover, with the increase of Y concentration, the strength of hybridization between Y-4d and Gd-5d orbitals increases particularly in Gd0.75 Y0.25 AuPb, Gd0.5 Y0.5 AuPb and Gd0.25 Y0.75 AuPb alloys. By investigating C v as a function of temperature, both T 3 Debye law at low temperatures and Dulong–Petit limit at high temperatures are satisfied for all Gd1−x Yx AuPb alloys. By studying Debye temperature versus temperature, it is determined that YAuPb and GdAuPb alloys have the highest and lowest thermal conductivity and also covalent bonds among Gd1−x Yx AuPb alloys. Finally, the harmony of the forces which acts on the structures of Gd1−x Yx AuPb alloys is studied using Grüneisen parameter. It is revealed that the harmony of the forces enhances as pressure increases and decreases when temperature increases.

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shows that C v of all alloys decreases linearly with the increase of pressure. Also, it is obvious that for all pressures, the value of C v of YAuPb (x = 1) alloy is smaller than the corresponding value of other alloys. The behavior of C p as a function of temperature is similar to C v at low temperatures but at high temperatures, it is observed that there is a linear relationship between C p and temperature which is proportional to T . Moreover, the variation of C p versus pressure shows a linear decrease for all alloys below about 5.5 K and exponential reduction after 5.5 K. In addition, the GdAuPb and Gd0.5 Y0.5 AuPb alloys have the largest and smallest C p at zero pressure respectively. Debye temperature is an appropriate quantity to express the solid state phenomena that is related to thermal conductivity and specific heat [40–42]. As it can be seen in Fig. 7, with the temperature increasing, the value of Debye temperature of Gd1−x Yx AuPb alloys reduces and at all temperatures ranging the YAuPb and GdAuPb alloy have the largest and smallest value among the Gd1−x Yx AuPb alloys, so these two alloys have stronger and weaker thermal conductivity rather than the other alloys, respectively. In addition, Debye temperature is a criterion to specify the strength of covalent bond of compounds [43]. In fact, the compound with larger Debye temperature has the stronger covalent bonds. In these alloys, the strength of covalent bonds decreases with increasing the temperature, so GdAuPb (x = 0) alloy has the weakest covalent bonds among the Gd1−x Yx AuPb alloys in all temperatures. On the other hand, the variation of Debye temperature versus pressure shows that the increase of pressure leads to improve thermal conductivity and covalent bonds of all Gd1−x Yx AuPb alloys. Therefore, the temperature and pressure have the opposite effect on the Debye temperature of Gd1−x Yx AuPb alloys. The influence of temperature and pressure on the thermal expansion coefficient (α ) is depicted in Fig. 7. It is observed that for all Gd1−x Yx AuPb alloys, α increases rapidly at low temperatures (below 100 K) and then increases linearly as temperature increases. Further, α has an exponential decreasing trend as pressure increases. The Grüneisen parameter (γ ) is associated to important properties of crystal lattice dynamic [44] indicating a measure of anharmonicity of the forces acting in a crystal and reflects the features of the distribution of phonon frequencies and their variation under pressure. It is well known that with the increase of pressure, γ decreases in case of solids [45,46] and increases in case of liquids [47]. The variation of Grüneisen parameter of Gd1−x Yx AuPb alloys versus pressure and temperature is illustrated in Fig. 7. It is observed that with the increase of pressure, γ reduces linearly for all alloys, so it is reasonable to say that the anharmonicity of the forces tends to be decreased by pressure increasing. In the other words, the harmony of the forces which act on the structure of these alloys enhances as pressure increases. However, the influence of temperature on Grüneisen parameter is opposite to the effect of pressure on it. As it is seen in Fig. 6, with the increase of temperature at zero pressure, γ increases for all Gd1−x Yx AuPb alloys leading to increasing the anharmonicity of forces.

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The effects of Y concentration, pressure and temperature on different properties of Gd1−x Yx AuPb alloys have been investigated within GGA approach based on density functional theory. Using the total energy curve versus volume, the phase stability of Gd1−x Yx AuPb alloys is studied and we found that x = 0, 0.25, 0.5, 0.75 alloys are stable in ferromagnetic phase while x = 1 alloy is stable in nonmagnetic phase. The equilibrium lattice parameters of Gd1−x Yx AuPb alloys are in good agreement with the available experimental data. Also, with the increase of Y concentration, the lattice parameters of these alloys decrease. Moreover, mechanical

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[34] M. Shafiq, S. Arif, I. Ahmad, S.J. Asadabadi, M. Maqbool, H.R. Aliabad, J. Alloys Compd. 618 (2015) 292–298. [35] O.L. Anderson, H.H. Demarest, J. Geophys. Res. 76 (1971) 1349–1369. [36] S.I. Ranganathan, M. Ostoja-Starzewski, Phys. Rev. Lett. 101 (2008) 055504. [37] Z. Wen, Y. Zhao, H. Hou, B. Wang, P. Han, Mater. Des. 114 (2017) 398–403. [38] P. Debye, Ann. Phys. 39 (1912) 789. [39] A.T. Petit, P.L. Dulong, Ann. Chim. Phys. 10 (1819) 395. [40] L. Qi, Y. Jin, Y. Zhao, X. Yang, H. Zhao, P. Han, J. Alloys Compd. 621 (2015) 383–388. [41] W.-C. Hu, Y. Liu, D.-J. Li, X.-Q. Zeng, C.-S. Xu, Comput. Mater. Sci. 83 (2014) 27–34. [42] Y. Zhao, H. Hou, Y. Zhao, P. Han, J. Alloys Compd. 640 (2015) 233–239.

[43] Q. Chen, Z. Huang, Z. Zhao, C. Hu, Comput. Mater. Sci. 67 (2013) 196–202. [44] T. Barron, Grüneisen parameters for the equation of state of solids, Ann. Phys. 1 (1957) 77–90. [45] R. Boehler, Adiabats of quartz, coesite, olivine, and magnesium oxide to 50 kbar and 1000 K, and the adiabatic gradient in the Earth’s mantle, J. Geophys. Res., Solid Earth 87 (1982) 5501–5506. [46] M. Kumari, N. Dass, On the pressure dependence of Grüneisen parameter in solids, Phys. Status Solidi B 133 (1986) 101–110. [47] R. Boehler, G.C. Kennedy, Pressure dependence of the thermodynamical Grüneisen parameter of fluids, J. Appl. Phys. 48 (1977) 4183–4186.

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