International Journal of Energetica (IJECA) https://www.ijeca.info ISSN: 2543-3717 Volume 4. Issue 1. 2019 Page 23-27 IJECA-ISSN: 2543-3717. June 2019 Page 23 Elastic, mechanical and thermodynamic properties of zinc blende III-X (X= As, Sb): ab-initio calculations Miloud IBRIR 1* , Moufdi HADJAB 2 , Gazi Yalcin BATTAL 3 , Fayçal BOUZID 2 , Abderrahim HADJ LARBI 2 , Samah BOUDOUR 2 , Hassene NEZZARI 2 1Laboratory of Materials Physics and its Applications, Department of Physics, University of M’sila, 28000, ALGERIA 2Thin Films Development and Applications Unit UDCMA- Setif / Research Center in Industrial Technologies CRTI, ALGERIA 3Department of Physics, Sakarya University, Esentepe Campus, 54187 Sakarya, TURKEY Email*: ibrirmiloud@yahoo.fr Abstract – In this work, density functional theory plane-wave full potential method, with local density approximation (LDA) are used to investigate the structural, mechanical and thermodynamic properties of of zincblende III-X ( X= As, Sb) compends. Comparison of the calculated equilibrium lattice constants and experimental data shows very good agreement. The elastic constants were determined from a linear fit of the calculated stress-strain function according to Hooke’s law. From the elastic constants, the bulk modulus B, shear modulus G, Young’s modulus E, Poisson’s ratio σ, anisotropy factor A, the ratio B/G and the hardness parameter H for zincblende III-X ( X= As, Sb) compound are obtained. Our calculated elastic constants indicate that the ground state structure of III-X ( X= As, Sb) is mechanically stable. The sound velocities and Debye temperature are also predicted from elastic constants. Keywords: Mechanics, Mechanics, energy, DFT, LDA Received: 03/03/2019 – Accepted: 15/05/2019 I. Introduction The method developed by Charpin (modified by Ferenc Karsai) and integrated in WIEN2k code [1] has been used to obtain elastic constants of considered binary compounds. The knowledge of elastic parameters of solids is very important because they provide important information about the stability and mechanical properties of solids such as sound velocities, load deflection, fracture toughness, thermoelastic stress and internal strain etc. II. Elastic and mechanical properties The elastic constants Cklmn (where the letter k, l, m, n refer to Cartesian components) are defined by the help of a Taylor expansion of the total energy of the system, E(V,ϵ), in accordance with a small strain ϵ of the lattice (V is the volume of the system). The energy E(V,δ) fit curve versus strain, δ, for the three different types of strains, namely the volume conserved, tetragonal and rhombohedral shear strains, are plotted in Figure 1 (a)- (c), respectively, for the studied binary compounds. The total energy has been calculated for five to seven different distortions for each of the three different deformations of the lattice. There are 21 independent elastic constants Cij, but symmetry of the cubic lattice reduces this number to only 3 independent constants (C11,C12, and C44) for cubic lattices. The calculated values of elastic constants are summarized in Table 1. The calculated elastic constant values of studied binary compounds are in good agreement with the results of other calculations [2, 4] and the available experimental data [3, 5].The obtained values for the elastic tensor constants satisfy the mechanical stability restrictionsfor cubic unit cells C11 - C12 > 0, C11 + 2C12 > 0 and C12 < B < C11 [6].Resistance to shear distortions of a cubic crystal is best characterized by the two moduli the tetragonal shear constant C' = (C11-C12)/2 and C44. The elastic constant C44 is related to an orthorhombic deformation whereas C' is related to a tetragonal deformation. At any volume V, the bulk modulus B for a cubic crystal is related to elastic constants by B0 = (C11+2C12)/3 [7]. The C11 and C12 can be obtained from the calculated bulk modulus and C'. Abder Image placée Miloud IBRIR Author et al IJECA-ISSN: 2543-3717. June 2019 Page 24 The Kleinmann parameter [8], ζ, is an important parameter describing the piezoelectric effect of solids [9]. It is given by the following relation: 11 12 11 12 8 7 2 C C C C     The obtained values of ζ for the materials are between 0.575 and 0.716 as shown Table 1. The obtained results for studied binary compounds fairly coincide with previous first-principles calculations [2, 10].We have also found the anisotropy factor A = 2C44/(C11 − C12). For an isotropic crystal, A is equal to 1, while any value smaller or larger than 1 indicates anisotropy. From Table 1, it is clearly seen that the calculated anisotropy factor for these compounds deviate from 1. The magnitude of the deviation from 1 is a measure of the degree of elastic anisotropy possessed by the crystal. The isotropic bulk modulus B0, which is related to C11 and C12, and shear modulus (G) are determined by the calculated elastic constants [11]. However, there is no distinct expression for the polycrystal-averaged shear modulus with respect to the Cij, but one can evaluate approximate averages of the lower and upper bounds given by various theories [12]. Voight [13] found upper bounds, while Reuss [14] discovered lower bounds for all lattice. The Kleinmann parameter [8], ζ, is an important parameter describing the piezoelectric effect of solids [9]. It is given by the following relation: 11 12 11 12 8 7 2 C C C C     The obtained values of ζ for the materials are between 0.575 and 0.716 as shown Table 1. The obtained results for studied binary compounds fairly coincide with previous first-principles calculations [2,10].We have also found the anisotropy factor A = 2C44/(C11 − C12). For an isotropic crystal, A is equal to 1, while any value smaller or larger than 1 indicates anisotropy. From Table 1, it is clearly seen that the calculated anisotropy factor for these compounds deviate from 1. The magnitude of the deviation from 1 is a measure of the degree of elastic anisotropy possessed by the crystal. The Kleinmann parameter [8], ζ, is an important parameter describing the piezoelectric effect of solids [9]. It is given by the following relation: 11 12 11 12 8 7 2 C C C C     The obtained values of ζ for the materials are between 0.575 and 0.716 as shown Table 1. The obtained results for studied binary compounds fairly coincide with previous first-principles calculations [2, 10].We have also found the anisotropy factor A = 2C44/(C11 − C12). For an isotropic crystal, A is equal to 1, while any value smaller or larger than 1 indicates anisotropy. From Table 1, it is clearly seen that the calculated anisotropy factor for these compounds deviate from 1. The magnitude of the deviation from 1 is a measure of the degree of elastic anisotropy possessed by the crystal. The isotropic bulk modulus B0, which is related to C11 and C12, and shear modulus (G) are determined by the calculated elastic constants [11]. However, there is no distinct expression for the polycrystal-averaged shear modulus with respect to the Cij, but one can evaluate approximate averages of the lower and upper bounds given by various theories [12]. Voight [13] found upper bounds, while Reuss [14] discovered lower bounds for all lattice. The upper bound due to Voight is calculated as 11 12 44( 3 ) / 5  VG C C C and the lower bound due to Reuss reads as 11 12 445 / 4 / ( ) 3 /  RG C C C According to Hill [15], the arithmetic average of the Voight and Reuss values can be used as an estimated of the average shear modulus G=1/2 (GV+GR). Another important mechanical parameter that is directly correlated to the ductility is the Poisson’s ratio and given by where Y is the Young’s modulus and is related to the bulk and shear moduli 9BG Y 3B G   Y and v are frequently measured for polycrystal materials when investigating their hardness. Young’s modulus is a measure of the stiffness of a given material, whereas Poisson’s ratio is the ratio (when a sample is stretched) of the contraction or transverse strain to the extension or axial strain.The calculated average shear modulus (G), Young’s modulus (Y) and Poisson’s ratio (v) are given in Table 1. The obtained results for studied binary compounds fairly coincide with previous first-principles calculations [2, 10] and experimental results [16, 17]. Materials with high G are likely to be hard materials. In studied compounds, GaAs exhibits the largest value of G (54.551 GPa) being the most incompressible of all. The Young modulus (Y) determines the stiffness of the material, i.e., the larger value of Y, the stiffer is the material [18] and the stiffer solids have covalent bonds [19].It can be seen from Table 1 that the largest value of Y (130.388 GPa) being the most stiffer of all occurs for GaAs implying it to be more covalent in nature as compared to other studied compounds. The proportion between bulk modulus and average shear modulus (B0/G) has been proposed by Pugh [20] to roughly determine the ductile or brittle character of a material. The critical value which separates ductile and brittle material is 1.75; i.e., if B0/G is smaller than 1.75, then the material behaves in a brittle manner; otherwise it Miloud IBRIR Author et al IJECA-ISSN: 2543-3717. June 2019 Page 25 will be of ductile nature [21]. The B0/G ratio of studied materials is presented in Table 1. It is clearly seen from this table that B0/G ratio of considered structures should be classified as brittle character. To obtain the stiffness of these compounds, the microhardness parameter (H) is also calculated using the following equation [22]: (1 2 )Y H 6(1 )     The calculated H values are 9.40GPa, 11.09GPa, 5.87GPa, 7.50GPa, 8.10GPaand 5.12forAlAs, GaAs, InAs, AlSb, GaSb and InSb at zero pressure, respectively. The Cauchy pressure is another interesting elastic parameter which describes the angular characteristic of atomic bonding in a material can be calculated by using the following relation [23]: C’= C12-C44 The positive value of Cauchy pressure is responsible for a ionic bonding while a negative Cauchy pressure, however, requires an angular or directional character in the bonding (covalent bonding). The more negative the Cauchy pressure, the more directional and of lower mobility the bonding. Moreover, a material with more negative value of Cauchy pressure will have more brittle nature. The calculated values of C’ are summarized in Table 2, which indicate that the sign of the Cauchy pressure is negative for all studied materials. The kind of bonds can be also determined by means of the value of Poisson’s ratio (υ). The value of Poisson’s ratio is nearly 0.25 or more for a typical ionic material, while it is much less than 0.25 (around 0.1) for a typical covalent Comp. C11 (GPa) C12 (GPa) C44 (GPa) C’ B0 (GPa) ƺ A G (GPa) Y (GPa) ν B0/G AlAs 110,442 56,982 78,132 26,730 74,802 0,638 2,923 50,867 124,402 0,22 1,471 113.1 [2] 55.5 [2] 54.7 [2] 28.8 [2] 0.592 [2] 1.899 [2] 77.14 [10] 175.39 [10] 30.329 [2] 1.043 [10] 119.9 [3] 57.5 [3] 56.6 [3] 0.481 [10] 0.136 [10] GaAs 113,589 50,115 78,401 31,737 71,273 2,470 54,551 130,388 0,195 1,307 115,1 [4] 51.5 [4] 56.8 [4] 36.4 [2] 72.7[4] 0,575 1.742 [2] 32.6 [16] 85.5 [16] 0.293 [2] 118.1 [5] 53.2 [5] 62.0 [5] 0.506 [2] 0.31 [17] InAs 81,547 49,809 60,838 15,869 60,388 0,716 3,834 35,683 89,433 0,253 1,692 92.2 [2] 46.5 [2] 44.4 [2] 22.9 [2] 0.598 [2] 1.943 [2] 0.335 [2] 83.3 [5] 45.3 [5] 39.6 [5] 84,615 42,727 60,737 20,944 56,690 0,629 2,900 39,665 96,490 0,216 1,429 AlSb 85.5 [2] 41.4 [2] 39.9 [2] 22.1 [2] 0.601 [2] 1.81 [2] 0.326 [2] 89.4 [5] 44.3 [5] 41.6 [5] GaSb 84,169 38,939 59,841 22,615 54,016 0,593 2,646 40,517 97,238 0,200 1,333 92.7 [2] 38.7 [2] 46.2 [2] 27 [2] 0.530 [2] 1.711 [2] 0.295 [2] 88.4 [5] 40.3 [5] 43.2 [5] InSb 65,062 38,348 50,121 13,357 47,253 0,699 3,752 29,636 73,535 0,241 1,594 72.0 [2] 35.4 [2] 34.1 [2] 18.3 [2] 0.603 [2] 1.863 [2] 0.487 [2] 66.7 [5] 36.5 [5] 30.2 [5] Table 1 Calculated elastic constants (C11, C12 and C44) and tetragonal shear constant (C'), Kleinman’s internal-strain parameter (ƺ), shear modulus anisotropy (A), the average shear modulus (G), Young’s modulus (Y) and Poisson’s ratio (v), and B0/G ratio of studied materials and the comparison of these quantities with available theoretical and experimental data. Miloud IBRIR Author et al IJECA-ISSN: 2543-3717. June 2019 Page 26 compound [24]. As indicated Table 1, our calculation shows that υ < 0.25 for all studied materials. Consequently, our Cauchy pressure calculations are consistent with our Poisson’s ratio values. The other interesting elastic parameters are Lame constants ( λ, μ) which depend on a material and its temperature. These parameters are related to the Young modulus and Poisson’s ratio by using the following equations: Y (1 )(1 2 )        and Y 2(1 )     where Y is the Young’s modulus and υ Poisson’s ratio. Our calculated values of λ and μ are summarized in Table 2. The two parameters together constitute a parametrization of the elastic moduli for homogeneous isotropic media, λ is known as Lame’s first constant and μ is Lame’s second constant. The Lame’s first modulus, λ, is related to a fraction of Young’s modulus. For an isotropic system one can show that λ=C12 and μ= C [25]. As shown from Table 1 (from anisotropy factor, A) the studied materials are strongly anisotropic character; therefore our obtained results do not satisfy the later relations, which are valid only for the isotropic systems, which is in agreement with the obtained results. III. Thermodynamic properties The Debye temperature (θD) which is a significant fundamental parameter closely related to many physical properties such as elastic constants, specific heat and melting point can be obtained from the average sound velocity (vm) by the following classical relation [26]: 1/3 2 0 6 D m B n k V            where V0 is atomic volume and vm the average wave velocity in the polycrystalline material is approximately calculated from the following equation [26]: 1/3 3 3 1 2 1 3 m t l               linear relationship between Tm and the C11 elastic constant. The scatter of all the different points falls within plus or minus 300 K of the following equation for Tm in units of K [28]: Tm = 553 K + (591/Mbar) C11 ± 300 K The calculated wave velocities (vt, vl, vm), Debye temperature (ΘD) and melting point (Tm) of studied binary compounds and ternary alloys estimated from elastic constants are listed in Table 2. It is clearly seen from Table 2, the Debye temperature of the considered compounds decrease with increasing atomic number.The high value of the Debye temperature for AlAs implies that its thermal conductivity is to be higher than other studied compounds. The sound velocities are related to the elastic moduli. Therefore, for a material having larger elastic moduli means higher sound velocity. Thermal conductivity, κ, is the property of a material that indicates its ability to conduct heat. So, in order to know if material is a potential candidate for thermal barrier coating application, its thermal conductivity need to be investigated. Based on the Debye model, Clarke [29] suggested that the theoretical minimum thermal conductivity can be calculated after replacing different atoms by an equivalent atom with a mean atomic mass M/n: 2/ 3 1/ 6 1/ 2 2/ 3 min B A 2/ 3 n Y 0.87k N M    where kB is the Boltzmann’s constant, M is the molecular mass and n is the number of atoms per molecule, NA the Avogadro’s number, ρ the density. The calculated minimum thermal conductivity of studied materials is summarized in Table 2. Table 2 Indicates that the value of minimum thermal conductivity decreases when one moves from Al to In in the compound XAs (Sb) (X=Al, Ga and In). The reduction can be attributed mainly to the difference in Young’s modulus, which is a measure of the second derivative of the bonding energy at the Comp. H (GPa) C” (GPa) λ (GPa) μ (GPa) vt (m/s) vl(m/s) vm (m/s) ΘD (K) Tm (K)±300 �min (WmK -1 ) AlAs GaAs InAs AlSb GaSb InSb 9,40 11,09 5,87 7,50 8,10 5,12 -21,150 -28,286 -11,029 -18,009 -20,901 -11,773 40,891 34,905 36,599 30,247 27,005 27,496 50,867 54,551 35,683 39,665 40,517 29,636 3680,935 3193,605 2501,806 3036,484 2678,678 2258,360 6163,647 5188,860 4351,768 5046,935 4374,136 3864,231 4073,853 3523,838 2778,513 3358,187 2957,205 2504,483 428,576 371,173 273,101 325,907 288,857 230,170 1205,712 1224,314 1034,942 1053,074 1050,442 937,519 1,769 1,534 1,053 1,241 1,107 0,830 Table 2.Calculated microhardness parameter (H), Cauchy pressure (C’’), and 1 st and 2nd Lame constants (λ, μ), wave velocities (vt, vl and vm), Debye temperature (θD), melting point (Tm) and the minimum thermal conductivity (κmin) of studied materials. Miloud IBRIR Author et al IJECA-ISSN: 2543-3717. June 2019 Page 27 equilibrium interatomic distance x0, between the studied binary compounds. Where vt and vl are the transverse and longitudinal elastic wave velocities, respectively, obtained using the shear modulus G and the bulk modulus B from Navier’s equations [27]: t G    3 4 3 l B G     where ρ is the density. Fine et al. [28] have studied many cubic metals and compounds and have obtained an approximate empirical. IV. Conclusion In this study, the structural, mechanical and thermodynamic properties of III-X ( X= As, Sb) compounds have been investigated by means of the DFT within Wien2k code. Our results for the optimized lattice parameters (a) and (c) are in good. Agreement with the available experimental data. The elastic constants Cij, and related polycrystalline mechanical parameters such as bulk modulus B, shear modulus G, Young’s modulus E and Poisson coefficient σ are calculated using Voigt– Reuss–Hill approximations. 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