First principle study of mechanical stability, magneto-electronic and thermodynamic properties of double perovskites: A2MgWO6 (A = Ca, Sr)

Materials Science & Engineering B 250 (2019) 114434

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First principle study of mechanical stability, magneto-electronic and thermodynamic properties of double perovskites: A2MgWO6 (A = Ca, Sr) Shabir Ahmad Mir, Saleem Yousuf, Dinesh C. Gupta

T

⁎

Condensed Matter Theory Group, School of Studies in Physics, Jiwaji University Gwalior, 474011, India

A R T I C LE I N FO

A B S T R A C T

Keywords: Density functional theory Double perovskites Elastic constants Cation size and Thermodynamic properties

The first principle method within the framework of density functional theory is employed in order to explore the ground state structure along with electronic, elastic, and thermodynamic properties of double perovskites A2MgWO6 (A = Ca, Sr). Crystal structure optimization reveals nonmagnetic phase as stable state. The computed ground state parameters are consistent with the earlier results. Structural stability is further confirmed through computation of cohesive energy and elastic constants, where we have demonstrated the influence of cation size on various elastic constants. Band profile suggests indirect semiconducting behavior along Γ-X symmetric points for both perovskites. The contribution of different bands is explored through calculation of total and partial density of states. Effects of temperature and pressure on thermodynamic properties are predicted via the quasiharmonic Debye model to convey the thermodynamic stability of oxides.

1. Introduction ab-initio study within Kohn-Sham (K-S) framework has made it possible to analyze the physical properties of materials from atomic constituent and their spatial configurations using various standard codes [1]. Intensive theoretical and experimental research activities are done over crystalline materials like Heusler alloys [2], chalcogenides [3], skutterudites [4], perovskites [5], etc. possibly due to their promising technological applications. Researchers all over world are looking for smart materials that possess novel properties and are multifunctional. Double perovskite (DP) materials have remained very attractive due to their versatile properties ranging from high temperature superconductivity [6], colossal magnetoresistance [7], half metallic (HM) ferromagnetism [8], HM anti-ferromagnetism [9], piezoelectricity and ferroelectricity [10] to multiferrocity [11]. As a result, they have great technological interest with a wide range of possible applications in fields of spintronics, as an electrode material for solid oxide fuel cell, in thermoelectric as an active material for conversion of squander heat into useful energy [12–14]. A lot of research activities are done to explore new DP materials and understand their potential applications in futuristic technologies. DP (A2BB′O6) can be derived from simple perovskites (ABO3) by replacing half of B cations with a different B′ cation in 1:1 ratio. The B and B′ cations are ordered in a rock salt manner resulting in idealized Fm-3m cubic structure with lattice constant twice that of ABO3

⁎

perovskite sub cell. With the variations in A, B and B′ cations several structures with different space groups are possible. Empirically, the possible structure of DP can be explored through tolerance factor (t) defined as [15]

t=

rA + rO 2 (rBB′ + rO )

(1)

rA and rO are ionic radii of A cation and oxygen ion, respectively while rBB′ is average of ionic radii of B and B′ cations. If t is in the range of 0.9–1.0, it means that all atoms ideally fit to individual positions resulting in ideal cubic structure with Fm-3m space-group. When t = 0.71–0.9, A-cation is small due to which crystal undergo a distortion to orthorhombic or rhombohedral structure. However, for t > 1, A-site cations are oversized to fit their positions deforms structure from cubic to hexagonal. From literature survey, we found that investigation on DPs for functional applications dates back to 1961 when ferromagnetic behavior of Re-based DPs were reported by Longo and Ward [16]. Since then they have been explored for their properties, however more interest in this field has increased after discovery of half metallicity in Sr2FeMoO6 [7]. Tungsten based DPs have engrossed much attention because of their intriguing and diverse physical properties. Their crystal structure has been crucial; they show variation in crystalline structure and phase transition at higher temperature [17]. Sr2BWO6 (B = Ni, Mg) [18] show the transition from the less symmetric tetragonal phase (I4/m) to the

Corresponding author. E-mail address: sosfi[email protected] (D.C. Gupta).

https://doi.org/10.1016/j.mseb.2019.114434 Received 13 November 2018; Received in revised form 16 August 2019; Accepted 11 October 2019 Available online 25 October 2019 0921-5107/ © 2019 Elsevier B.V. All rights reserved.

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highly symmetric Fm-3m cubic phase with increase in temperature. Ba2MWO6 (M = Mg, Zn) [19] oxides are found to crystalline in cubic structure. A2BWO6 (A = Sr, Ba; B = Ni, Co, Zn) ceramic materials were reported to exhibit relatively better dielectric properties in microwave region as compared to other DPs and find application in microwave dielectric ceramics [20]. While using sacrificial agents Ba2CoWO6, Sr2CoWO6, Ba2NiWO6, and Sr2NiWO6 can be suitable candidates for photo catalysis [21]. Recently, A2MgWO6 (Ca, Sr) are found to be stable in ordered cubic structure and could be used as inert materials in chemically aggressive environments [22,23]. Earlier reports suggest Sr2MgWO6 show phase transition from less symmetric to highly symmetric cubic (Fm-3m) structure at high temperature [24,25]. However, by ab-initio study it is found to be stable in cubic structure with semiconducting nature [26–28]. To the best of our knowledge there are no detailed results regarding the elastic, thermoelectric and thermodynamic properties of Sr2MgWO6 oxide in literature, whereas Ca2MgWO6 is being investigated for first time. So, we concern the present study to understand the structural, elastic, magnetic, electronic, transport and thermodynamic properties of A2MgWO6 (A = Ca, Sr) oxides.

Fig. 1. Structural representation of A2MgWO6 (A = Ca, Sr) oxides wherein Mg and W lie in octahedra of O-atoms.

2. Computational method

headings as

The present results are obtained via the full potential linear augmented plane wave (FP-LAPW) formulism within the general rules of DFT using WIEN2k simulation code [1]. The calculations have been executed by optimization of the crystal energy to get the relaxed lattice constant followed by determination of electronic, elastic and other properties using the optimized parameters. In order to figure out, the exchange and correlation potential generalized gradient approximation with Perdew–Burke–Ernzerhof (GGA-PBE) parameterization is incorporated [29]. GGA formulism can underestimate exchange correlation effect. However, there are number of alternative ways to overcome GGA limitations like inclusion of Hubbard/Coulomb (U) or modified version of Becke-Johnson potentials (mBJ) to GGA. The U-parameter is set semi-empirically for the correlated system. Nevertheless, if one wants to stay within ab-inito framework that could lead to precise band profiles, then mBJ is a good choice [30]. So here we have opted mBJ potential to define more precise band profile. In FP-LAPW scheme, the unit cell is divided into muffin tin spheres with radii RMT and interstitial regions. Within non-overlapping muffin tin spheres, the wave function used are atomic like wave function (radial functions times spherical harmonics) and for interstitial space a plane wave basis set is engaged. Muffin tin radii are selected in such a way, to guarantee zero leakage of electrons from the core into interstitial region and thus convergence of total energy is ensured. Numerically controlled energy threshold of −6.0 Ry between the core and the valence states is used. The cut-off parameter RMTKmax, which determines the size of basis set was set to 7 for energy convergence, where RMT represents the smallest muffin tin sphere radii and Kmax is the largest k-value in momentum space used in the plane wave expansion i.e., it represents plane wave cutoff. Inside muffin tin spheres utmost value of orbital angular momentum (l) for wave function extension is lmax = 10. For Brillouin zone (BZ) integration in momentum space, a mesh of 1000 k points was used. The convergence criterion was set to 10−4 Ry for energy and charge difference between two successive cycles should be < 10−4 e. The mechanical nature of oxides is explored through computation of elastic constants determined via Cubic-elastic package as integrated in the Wien2k package. For determination of thermodynamic properties and their dependence on the pressure and temperature quasi-harmonic Debye model has been applied.

3.1. Structural properties The structural properties of A2MgWO6 (A = Ca, Sr), have been determined by relaxation and optimization of the ground state energy in magnetic and nonmagnetic configuration against the volume variation by Birch-Murnaghan’s equation of state [31]. The A2MgWO6 (A = Ca, Sr) are found to crystallize in an ideal cubic nonmagnetic (Fm-3m) structure, wherein the atoms Ca (Sr) have coordination 12 and lie within cage of 12 oxygen anions, while as Mg and W are coordinated by 6 oxygen anions as shown in Fig. 1. The atoms in unit cell are located as Ca (Sr) at 8c (0.25, 0.25, 0.25), Mg at 4a (0, 0, 0), W at 4b (0.5, 0.5, 0.5) and O at 24e (x, 0, 0) Wyckoff positions. The anion position plays a significant role in determining the stability and shaping the properties of DPs. Therefore in order to find the value x, we have relaxed the crystal structure by optimizing it with different anion positions. The optimization plots are shown in Fig. 2(a, b). The graphical variations convey that (0.26, 0, 0) is most stable anion position for present set of oxides as it has minimum energy compared to other positions. The calculated value of the lattice constant found for Sr2MgWO6 is consistent with earlier values, while for Ca2MgWO6 an overestimation by nearly 3% is seen. The relaxed lattice constants, total ground state energies, bulk modulus and derivative of bulk modulus with pressure in the stable phase are reported in Table 1. Tolerance factor (t) has also been employed to determine the stable phase of materials. The calculated tolerances factor both empirically (from ionic radii) as well as by DFT (from bond length) methods implies cubic phase to be stable. The variation of the volume with the applied pressures is represented in Fig. 3. The absence of any discontinuity in P-V plot suggests that the oxides are stable in the entire range of pressure and didn’t undergo any structural phase transition [32]. 3.2. Cohesive energy We have calculated the cohesive energy for A2MgWO6 (Ca, Sr) oxides which basically signifies how strongly atoms are held in solid and is helpful in determining the stability of any material. The cohesive energy per atom of a material Ax B y with x and y representing number of A and B atoms, respectively, can be determined via relation [33]

3. Results and discussion

A B

x y ECoh =

The discussion of the various calculated results like structural, chemical, electronic and mechanical properties are given under various

A B + yEatom [xEatom ] − E Ax By

x+y

For DP the relation would be; 2

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Fig. 2. Total energy as a function of unit cell volume for cubic A2MgWO6 (A = Ca, Sr) DPs in non-magnetic Fm-3m phase at different anion positions. Table 1 Calculated values of equilibrium lattice constant (a in Å), unit cell volume (V in Å3), bulk modulus (B in GPa), pressure derivative of bulk modulus B′, tolerance factor (t), ground state energy (E0 in eV) and cohesive energy per atom (Ecoh in eV/atom) of A2MgWO6 (A = Ca, Sr) alloys. Parameter

Present

Ca2MgWO6 a V t B B′ E0 Ecoh

7.7 456.53 1.0 172.04 4.36 −494477.93 6.18

Sr2MgWO6 a V t B B′ E0 Ecoh

7.93 498.67 0.99 169.28 4.57 −630449.20 5.74

Empirical

Others

A2 BB′O6 ECoh =

(2)

E A2 BB′O6 is equilibrium energy. The computed values for Ca2MgWO6 and Sr2MgWO6 are 6.20 and 5.79 eV, which clearly indicate that atoms are strongly held within the crystal lattice of the oxides.

Experimental

3.3. Elastic properties

7.43 [22]

Elastic properties reveal the potential of a material to regain original shape when distortion forces are ceased. The elastic constants of a material determine the response to an applied stress are important due to the fact that these parameters are related to behaviors of materials, including hardness, durability, strength, reliability, and performance required to designate the type of application and fabrication. The sufficient number of elastic constants required for a system is directly related to symmetry of structure. More symmetric the structure less are the number elastic constants required to explore mechanical behaviour of system. For cubic structures, the number of elastic constants reduces to three C11, C12 and C44. In order to determine them, we have used the rhombohedral and tetrahedral distortions on the cubic structure under volume conserving constrain. The calculated elastic parameters are reported in Table 2 and have been compared with reported values of similar type of oxide Ba2MgWO6 [34]. The longitudinal elastic constant (C11) directly represents the stiffness of the crystal while as the C12, is based on the transverse expansion. For the case of perovskites, distortion in cages around B sites influences the values of C12. The shear elastic parameter, (C44) is based on the shear modulus. Here, we have made an attempt to investigate their dependence on the cation size as shown in Fig. 4. The elastic constants are found sensitive to A-site cation. As, size of the A-cation increases from Ca to Ba, C11 and C12 decreases while as C44 shows an increasing trend. This behavior of elastic constants can be explained from the structural properties [35]. Oxygen anions within unit cell are shared among three cages in DPs (A2BB′O6) viz., a twelve oxygen anion cage around A-cation and two cages of six anions around B and B′-cations. Biggest cation in perovskite controls the rigidity of crystal; in that case only one cage contributes to hardness of material. However, the rigidity of crystal is increased when all the cages contribute, i.e., when size of Acation is ideal to B (B′)-cation. As we go from Ca to Ba atoms, difference in cation size occupying A and B (B′) sites increases, making only one cage to contribute to stiffness. Thus reduces C11. However, with the increase in size of A-cation, oxygen anion and A-cation come close to each other resulting strong bonding among them. Due to which it more difficult to shear constituents, therefore C44 increases. However,

0.93

7.95 [26]

B′ A B O [2Eatom ] − E A2 BB′O6 + Eatom + Eatom + 6Eatom 10

7.95 [24] 7.9 [25]

0.96

Fig. 3. Variation of volume with pressure. 3

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Table 2 Calculated values of elastic constants (C11, C12 and C44 in GPa), density ( ρ in g/cm3), Bulk modulus (B in GPa), Shear Modulus (G in GPa), Young’s modulus (Y in GPa) Pugh’s ratio (B/G) Cauchy’s pressure (CP in GPa) Anisotropic factor (A) and Poisson’s ratio (v) at P = 0 GPa and T = 0 K. Alloy

C11

C12

C44

ρ

B

G

Y

B/G

CP

A

v

Ca2MgWO6 Sr2MgWO6 Ba2MgWO6 [33]

326.82 316.36 265.67

102.47 98.27 93.64

93.79 102.78 104.28

5.98 6.45 –

177.37 171.05 150.6

100.75 107.68 96.8

254.13 267.01 216.4

1.76 1.58 –

8.86 −4.5 –

0.83 0.98 –

0.261 0.239 –

bonding with ductile nature, the CP is positive. Moreover, for brittle materials with directional and angular bonding character, it has negative value. We found positive value for Ca2MgWO6 showing a metallic bonding and ductile nature of the alloy while Sr2MgWO6 possess negative value indicates its brittle nature. Brittle and ductile nature can also be interpreted by Pugh’s ductility index (B/G) [40]. According to the Pugh’s criterion, the low (high) B/G ratio is coupled with the brittle (ductile) nature of the materials. The index values separating brittleness and ductility is 1.75, below this value material posses brittle nature and if B/G > 1.75 material has ductile nature. The calculated value for Ca2MgWO6 lie above the critical value indicates its ductile nature while for Sr2MgWO6 it lies below the critical value indicating its brittle nature. The ratio of transverse contraction strain to longitudinal extension strain in the direction of applied force is known as Poison’s ratio (ν) . It is an another criterion that has been verified experimentally to determine ductile or brittle nature of an alloy and is defined by relation, Fig. 4. Variation of elastic constants with A-site cation.

increasing bonding between A-site cation and oxygen makes BO6 and B′O6 to undergo distortion because of increasing bond length between oxygen anion and B (B′)-cations [36], as they share same set of oxygen anions. Hence, C12 decreases. For cubic crystals the necessary criteria for existence in stable or meta-stable phase is established in form stability condition [37];

C11 − C12 > 0, C11 > 0, C44 > 0, C11 + 2C12 > 0, C12 < B< C11

(C11 + 2C12 ) (C − C12 + 3C44 ) ; GV = 11 3 5

(3)

A=

5(C11 − C12 ) C44 4C44 + 3(C11 − C12 )

(4)

(9)

1

(5)

θD =

Using Hill’s average approximation the bulk and shear moduli are defined as;

B + BR G + GR B= V ;G= V 2 2

2C44 C11 − C12

For a completely isotropic material, A = 1, otherwise material is anisotropic. The calculated values for both materials suggest their anisotropic natures. Debye temperature, one of the important parameter of solids closely related to many physical properties such as specific heat and melting point and also provides the information about temperature variation of physical behaviors. Debye temperature of a material can be estimated using mean sound velocity vm [43]

However, the Reuss formulae for the bulk and shear moduli are:

BV = BR; GR =

(8)

The critical value of ν to differentiate brittle and ductile materials is 0.26. Materials with 0.26 < ν < 0.5 are regarded as ductile materials, for 0.12 < ν < 0.26 materials are supposed to be brittle [41]. The calculated value of ν in Table 2 indicates similar results for A2MgWO6 alloys as shown by CP and B/G values. The Zener anisotropy factor (A) [42], reveals how directionally dependent the physical properties like refractive index of a system are, whether they possess same value in all directions (isotropic) or not (anisotropic). It also determines the possibility of inducing micro-crakes in a crystal and thus from the mechanical application point of view, it is most important. It can be calculated via the relation;

A2MgWO6 (A = Ca, Sr) satisfy the required cubic stability condition signifying that these oxides are mechanically stable. C11 is larger than C44 for both materials, implies that these materials have higher resistance to the unidirectional compression in comparison to the pure shear deformation. Using the computed elastic constants, various other mechanical parameters can be defined. The bulk and shear moduli which can predict the hardness of material, can be determined via Viogt-Reuss-Hill averaging scheme method [38]. The Voigt limits of the bulk modulus (BV) and shear modulus (GV) for the cubic system are:

BV =

3B − Y 6B

ν=

h ⎛ 3nNA ρ ⎞ 3 vm k ⎝ 4πM ⎠

Here h is planks constant, k is Boltzmann’s constant, NA is Avogadro’s number, ρ is mass per unit volume vm is average sound velocity. Average velocity vm is approximated using longitudinal velocity vl and transverse velocity vt

(6)

Young’s modulus, determines the strength of material is ratio of linear stress and strain can be evaluated via relation.

−1 3

1 2 1 vm = ⎡ ⎛ 3 + 3 ⎞ ⎤ ⎢ 3 ⎝ vl vt ⎠ ⎥ ⎣ ⎦ ⎜

Y=

9BG 3B + G

(10)

(7)

⎟

(11)

vl and vt can be determined using Navier’s equation’s [44]

Ductile and brittle nature of an alloy can also be determined through the elastic constants (Cij). According to Pettifor [39], the nature of the atomic binding is related to the ductile or brittle nature and can be revealed by the Cauchy pressure (CP = C12 − C44). For metallic

vl =

(3B + 4G ) ; ρ

vt =

G ρ

(12)

The calculated values of longitudinal velocity, transverse velocity 4

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Fermi level is almost tangential to top of valance band located at Γsymmetry point while the bottom of conduction band are located at 2.55 eV and 2.48 eV on X-symmetry point of BZ for Ca2MgWO6 and Sr2MgWO6, respectively. This indicates there is indirect band gap along Γ-X symmetric points. Accordingly, energy required (band gap energy) to generate electron hole pair by exciting electron from valance band to conduction band is 2.57 eV for Ca2MgWO6 and 2.50 eV for Sr2MgWO6. Upon applying the mBJ potential, the separation between conduction band minima and valance band maxima in both materials increases and band gaps found are 3.47 eV and 3.58 eV for Ca2MgWO6 and Sr2MgWO6 respectively. The band gap for Sr2MgWO6 is found to be smaller than the previously reported theoretical values [27,28] with no experimental results to compare. The contribution of different electron states can be easily understood from DOS and projected density of state (pDOS) plots. DOS plots for both oxides shows absence of states at Fermi level, implies the semiconducting nature of both the oxides. The effect of incorporation of mBJ to GGA is made clear by plotting total density of states through the two methods shown in Fig. 7(a, b). With incorporation of mBJ the DOS peaks are significantly shifted which in turn increase the band gap. Crystal fields associated with WO6 cage, generated by Coulomb interactions are responsible for the splitting of degenerate 5d states of W resulting into non-degenerate doublet deg and triplet dt2g states. Atomic a resolved DOS shown from Figs. 5 and 6, revel that major contribution throughout the valance band comes due to oxygen anion. However, deep down in the valance band, there is hybridization between d-states of W with p states of oxygen as shown in Fig. 8(a, b). In conduction band minima, the contribution is mainly due to the hybridization of dt2g states of W with the p states of oxygen with a little from Sr and Ca atoms in respective oxides. In Ca2MgWO6, there are peaks at higher energy ranges of 4.5 eV onwards which are mainly due to Ca atom with little contribution of oxygen atom, while in Sr2MgWO6 there is a small peak by the hybridization of Sr and oxygen orbitals at the energy range of 5.5–6 eV. The effect of pressure on the electronic properties is elucidated by computation of DOS at different pressures shown in Fig. 9(a, b). With the application of pressure volume of the system decreases result in reduction of inter-atomic distances, due to which constituent atoms are strongly held. Consequently valance electrons get more strongly bound which in turn marks slight shift in DOS peaks as is observable from DOS plots under different pressure.

and mean velocity from elastic constants are 4248 m/s, 7468 m/s and 5519.3 m/s for Ca2MgWO6 while for Sr2MgWO6, the calculated values are 4085 m/s, 6943 m/s and 5251 m/s. Using these values in Eq. (10), Debye temperature for Ca2MgWO6 and Sr2MgWO6 are found to be 803 K and 744 K, respectively. Melting temperature Tm, determines the range of temperature over which solid can be utilized for application. It can be evaluated from elastic constants using relation [45]

Tm (k ) = [553 + (5.911) C11] ± 300

(13)

The calculated melting points of the Ca2MgWO6 and Sr2MgWO6 compounds are found to be 2484±300 K and 2423±300 K respectively. This indicates their applications up to a wide range of temperatures. 3.4. Electronic and magnetic properties From the application point of view, the study of electronic structure is highly desired. The electronic properties can be analyzed from band structure (BS) profile and density of states (DOS). The BS of a solid describes those ranges of energies in which an electron is forbidden or allowed, while the DOS depicts the number of states per unit energy. The self-consistent, spin polarized calculations were performed in order to explore the electronic properties of the cubic Ca2MgWO6 and Sr2MgWO6 alloys. From band profile, it is found that up and down spin channels show similar character (not shown here), which reveal the absence of spin polarization in these oxides. The electronic properties are sensitive to exchange and correlation potentials, needs to be approximated properly. For present study, GGA and mBJ potentials were used to approximate the exchange and correlation energies. The band gap of the semiconductor or insulator materials is underestimated by GGA approximation, mainly due to self-interaction of electrons and insufficient potential for highly correlated states. To overcome this issue, we have used the mBJ potential to get better band profile. It has been seen that on including the mBJ potential, electronic properties are expressed in better way than the one obtained by standard functional like, GGA, thus reducing inconsistency between the theoretical and experimental results [46]. The calculated BS along the high symmetric points in reduced zone scheme as well as total and atomic resolved DOS via GGA approximations is shown in Fig. 5(a, b) and by mBJ method is displayed in Fig. 6(a, b). From GGA calculation, it can be seen that

Fig. 5. Calculated band structure and density of states via GGA approximation a) Ca2MgWO6; b) Sr2MgWO6. 5

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Fig. 6. Obtained band structure and density of states by using mBJ approximation a) Ca2MgWO6; b) Sr2MgWO6.

Fig. 7. Relative shift in density of states peaks with inclusion of mBJ to GGA approximations of (a) Ca2MgWO6; (b) Sr2MgWO6.

much with induced pressure. Same conclusions can be drawn from DOS plots which also depict a minute variation with application of pressure. However, DOS and charge density are rough approximations which serves to capture the trends in hybridization i.e., provide qualitative rather than quantitative information regarding hybridization [47].

The A2MgWO6 (A = Ca, Sr) oxides show the nonmagnetic (NM) nature as it is found from structural optimization that the NM phase is energetically more stable as compared to magnetic phase. The constituents in DPs are in different oxidization states. For A2MgWO6 oxides the nominal valance states are of Ca2+, Sr2+, W6+ and Mg2+ and O2− ions. In the ionic ground state, each constituent atomic ion has a completely filled shell. The magnetic moment of any material is solely due to unpaired electrons. The unavailability of unpaired electrons indicates the absence of local magnetic moments on the cation sites, result the nonmagnetic behavior of oxides. The application of pressure has no change in spin states; it is because valance electrons get more bound with squeezing of inter-atomic distances. Also, with the application of the pressure there is no visible enhancement in hybridization of orbitals, which is made clear through determination of charge density at different pressures shown in Fig. 10(a, b). The charge sharing among the constituent atoms is well distinguished by color difference that changes around the atoms. From the contour, it is apparent that at different pressures the spatial distribution of electronic charge is almost invariant. The result signifies that p-d hybridization is not affected very

3.5. Thermodynamic properties In order to investigate the thermodynamic properties, we have employed quasi-harmonic approximation (QHA) wherein Gibbs G ∗ (V ; P , T ) Function is given by [48] G ∗ (V ; P , T ) = E (V ) + PV + Avib [θ (V ); T ]. Here E(V) represents the total energy, PV signifies constant hydrostatic condition, θ(V) is the Debye temperature and Avib corresponds to vibrational term. The model predicts various thermodynamic properties like specific heat at constant volume (CV), Grüneisen parameter (γ), Entropy (S) and thermal expansion coefficient (α) with their variation against the pressure and temperature. Heat capacity is a measure to show the capacity of a material to 6

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Fig. 8. Calculated partial density of W-deg, W-dt2g and O-p states via mBj approximations of (a) Ca2MgWO6; (b) Sr2MgWO6.

Fig. 13(a, b), displays the variation of γ with temperature and pressure, which demonstrates small variation in γ with temperature. With increase in temperature, sluggish increase in γ suggests rise in anharmonicity. However, the variation with volume is noteworthy in understanding the macroscopic behavior of solids and generally decreases with volume. The estimated value of γ at 0 GPa and 300 K is 2.03 and 2.14 for Ca2MgWO6 and Sr2MgWO6, respectively. The coefficient of thermal expansion (α) describes how dimensions of object changes with change in temperature. Precisely, it measures the fractional change in size per degree change in temperature at a constant pressure and is helpful in determining their mechanical applications. It usually decreases with increasing bond energy and so likely oxides possess lower thermal expansion. The relation of α with other thermodynamic coefficients is given by;

store heat as the temperature changes. From this, we can get the information about different properties like lattice vibration mandatory for many applications. The specific heat can be measured at either constant volume or at constant pressure and determined from one other. In solids volume does not change significantly with temperature. So here we report the specific heat at constant volume. However, for Sr2MgWO6 oxide CP has been determined empirically [49], our calculated values of CV are comparable to the reported values of CP. The variation in CV with temperature is shown in Fig. 11(a, b). From this graph, we can see that CV increases with temperature, it is due to increase in atomic vibrations. At high temperatures, CV tends towards a constant value showing the Dulong and Petit limit. However, at lower temperature CV changes abruptly with temperature as shown in Fig. 12(a, b) in the range of 0–38 K for Ca2MgWO6 and 0–35 K for Sr2MgWO6 obeying Debye T3 law. Grüneisen parameter (γ) is one among the thermodynamic quantities used to explore the relationship between thermal and elastic behavior of a solid. Many physical properties like the bulk modulus, volume dependence of Debye temperature, specific heat, and frequency of lattice vibrations and determination of logarithmic pressure derivative of diffusion coefficient are closely linked with Grüneisen parameter (γ).

α=

γCv 3BT

(15)

As bulk modulus and Grüneisen parameter have weak dependence on temperature so α may show a similar trend as CV. Fig. 14(a, b), depicts the variation of thermal expansion for A2MgWO6 (A = Ca, Sr) with temperature and pressure. Temperature variation is as similar as

Fig. 9. Density of states at different pressures of (a) Ca2MgWO6; (b) Sr2MgWO6. 7

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Fig. 10. Spatial distribution of charge at different pressures.

Fig. 11. Variation of specific heat at constant volume (CV) with temperature and pressure a) Ca2MgWO6; b) Sr2MgWO6.

CV, with T3 dependency at lower temperature while as pressure has a strong effect on α. With increasing pressure, the modulus of α decreases quickly at same temperature and may be due to decrease in interatomic spacing resulting in strong bonding. The calculated values of α for Ca2MgWO6 and Sr2MgWO6 at 0 GPa and 300 K are 1.4 × 10−5 K−1 and 1.6 × 10−5 K−1, respectively and are comparably near to the experimentally reported values [24]. Entropy (S) an extensive property measures the molecular disorder or randomness of a system. The variation of entropy with temperature and pressure is depicted in Fig. 15(a, b), where a linear rise at high temperatures can be seen which conveys molecular randomness

increases as shown by Grüneisen parameter while with the increase of pressure inverse effect is seen. The estimated value of S at 0 GPa and 300 K is 106 Jmol−1 and 130 Jmol−1 for Ca2MgWO6 and Sr2MgWO6, respectively. 4. Conclusion First-principles calculation within DFT frame work using FP-LAPW method are performed to investigate the structural, electronic, magnetic, elastic and thermodynamic properties of A2MgWO6 (A = Ca, Sr) double perovskites. Optimization, tolerance factor and elastic stability 8

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Fig. 12. Variation of CV with T3 at low temperature a) Ca2MgWO6; b) Sr2MgWO6.

Fig. 13. Variation of Grüneisen parameter (γ) with temperature and pressure a) Ca2MgWO6; b) Sr2MgWO6.

Fig. 14. Variation of thermal expansion coefficient with temperature and pressure a) Ca2MgWO6; b) Sr2MgWO6.

9

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Fig. 15. Variation of entropy (S) with temperature and pressure a) Ca2MgWO6; b) Sr2MgWO6.

criteria confirm the stability in cubic Fm-3m structural phase. The structural properties are well agreed with earlier reported results. Both mBJ as well as GGA approximation reveal the semiconducting behavior of oxides. The lack of unpaired electrons in the ionic state of constituent atoms leads to show nonmagnetic character. Impact of A site cation on elastic constants show that C11 and C12 decrease, whereas C44 increase with increase in size of A site cation. By analyzing the quotient of bulk to shear modulus, Cauchy pressure as well as Poisson’s ratio, the ductile nature of Ca2MgWO6 and brittle nature of Sr2MgWO6 is found. The thermodynamic properties of A2MgWO6 (A = Ca, Sr) viz., the specific heat, the thermal expansion, entropy and the Grüneisen parameter show their stability with respect to thermodynamics. The predicted results may prove as guide for future experimental study.

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