A. The structural properties

Although MgSnT₂ (T = P, As, Sb) has been not already synthesized, it is most commonly known that II–IV–V₂ type compounds crystallize in a body centered tetragonal phase (space group no:122, $I - \bar 42d$, D2d) and this structure is derived from the zinc blended binary semiconductors, but contain two different cations instead of one in binary.^{Reference Jaffe and Zunger4,Reference Sharma, Verma and Jindal8,Reference Sainia, Singh, Reshak and Kashyap9} Therefore, MgSnT₂ (T = P, As, Sb) has been identified throughout this study as D2d space group with Mg atom at 4a (0 0 0), Sn atoms at 4c (0.0 0.0 0.5) and T atoms at 8d (u 0.25 0.125). The body centered tetragonal structure is simply defined by three lattice parameters: a and c and internal parameter u.

DFT is an extremely successful theoretical approach for the description of ground state properties of molecules and solids. DFT is formally exact, but in practice approximations are needed to define the unknown exchange-correlation functional. The choice of energy functional is an important consideration in DFT calculations. As mentioned in before, six different approximations for the exchange-correlation energy are used in this study. The LDA functional is based on accurate calculations of the exchange-correlation density in a homogeneous electron gas. Here, the implementation of LDA follows the Ceperley and Alder recipe.^{Reference Ceperley and Alder22} In addition, we use the PW91,^{Reference Perdew and Wang23} the PBE,^{Reference Perdew, Burke and Ernzerhof24} the GGA-RPE,^{Reference Zhang and Yang25} most recent GGA-AM05^{Reference Armiento and Mattsson26–Reference Mattsson, Armiento, Paier, Kresse, Wills and Mattsson28} and GGA-PBEsol^{Reference Perdew, Ruzsinszky, Csonka, Vydrov, Scuseria, Constantin, Zhou and Burke29} gradient-corrected functionals. Both PW91 and PBE approximations are free of empirical input and satisfy sum rules and analytically known limits. Furthermore, PBE incorporates an accurate description of the linear response of the uniform electron gas, generates a smoother potential and has the correct behavior under uniform scaling. The pseudopotentials used in the calculations were constructed consistently by using the same approximation for the exchange-correlation energy in the generating scheme which has been found to be important in obtaining accurate geometries and energetics. Several attempts to improve current GGA functionals for solids, either through a reparameterization or a refinement of their mathematical expressions, have recently appeared in the literature. Perdew and co-workers reanalyzed the performance of the PBE functional and proposed a modification denoted as PBEsol.^{Reference Perdew, Ruzsinszky, Csonka, Vydrov, Scuseria, Constantin, Zhou and Burke29}

Due to the electronic, elastic, lattice dynamical properties based on a structural properties, accurate calculation of the structural properties of a material requires a precise reproduction of the lattice parameters using first-principles calculations. We, therefore, carried out detailed calculations for several exchange-correlation functionals for these compounds (see Table I). First of all, we computed the total energy versus lattice constant a for selected exchange-correlation functionals as shown in Fig. 1 It is clearly seen from Fig. 1 that the order of ground state energy from low to high is: LDA < PW91 <RPBE < PBE < PBEsol < AM05 for each compound. Obviously, the lattice parameters from LDA give a smaller value than the other considered functionals. The cell volume increases from the P to Sb compound as expected. All of the relevant equilibrium structural parameters and tetragonal distortion (n = c/2a) for all considered compounds are given in Table I, including other theoretical values.^{Reference Jaffe and Zunger4,Reference Shaposhnikov, Krivosheeva and Borisenko15} Our computed lattice parameters show excellent agreement, in particular compared with the results of Ref. Reference Jaffe and Zunger4 and also the agreement with values from other computation^{Reference Shaposhnikov, Krivosheeva and Borisenko15} is found good. The order of lattice constant a from low to high is: LDA < PBEsol < AM05 < PW91 < PBE < RPBE for all considered compounds.

TABLE I.

Calculated structural parameters of MgSnT₂ (T = P, As, Sb). The lattice constants (a and c) are expressed in Å, the internal parameter (u) and c/2a are dimensionless.

FIG. 1.

Lattice constants a versus the ground state energy of MgSnT₂ (T = P, As, Sb) compounds using different exchange correlation functionals.

B. Elastic properties

The values of six independent elastic constants for these body centered tetragonal phases (C ₁₁, C ₁₂, C ₁₃, C ₃₃, C ₄₄, and C ₆₆ see Table II) were estimated by calculating the stress tensors on strains applied to equilibrium structure.^{Reference Page and Saxe30,Reference Mehl, Osburn, Papaconstantopoulos and Klein31} Our obtained elastic constants satisfy all the stability conditions given earlier.^{Reference Abdellaouia, Ghaffour, Bouslama, Benalia, Ouerdane, Abidri and Monteil32–Reference Mouhat and Coudert35} Clearly, the absent negative values for six independent elastic constants for all selected exchange functionals suggest that all the investigated compounds are mechanically stable in this phase. So far, elastic constants have been neither measured nor calculated in the literature for these compounds. Thus, the present work is the first theoretical prediction for elastic constants. Using the calculated elastic constants, the polycrystalline elastic modulus like bulk modulus (B), shear modulus (G), and Young's modulus (E) and Poisson's ratio (ν) can be calculated using the Voigt-Reuss-Hill approximation.^{Reference Reuss and Angew36,Reference Hill37} Our calculated data were obtained by the relations given in Refs. Reference Sahin, Ciftci, Colakoglu and Korozlu10, Reference Abdellaouia, Ghaffour, Bouslama, Benalia, Ouerdane, Abidri and Monteil32, and Reference Shein and Ivanovskii38–Reference Peng, Chen and Yang40. The predicted valuable elastic parameters are presented in Tables III and IV allowing us to make the following conclusions:

TABLE II.

The calculated elastic constants (C _ij in GPa) for MgSnT₂ (T = P, As, Sb).

TABLE III.

The calculated bulk modulus (B in GPa), shear modulus (G, in GPa), Young's modulus (E, in GPa), B/G ratio and Poisson's ratio (ν) for MgSnT₂ (T = P, As, Sb).

TABLE IV.

The calculated bulk modulus of B _a and B _c (in GPa), shear anisotropic factors of A _a and A _c, the percentage of elastic anisotropy for bulk modulus A _comp and shear modulus A _shear, universal anisotropy index A ^U and melting temperature (T _m in K) for MgSnT₂ (T = P, As, Sb).

(i) It can be seen that our obtained values of the elastic constants for LDA are higher than the all GGA values. It is also expressed that elastic constants are in order of LDA > GGA-PBEsol > GGA-AM05 > GGA-PW91 > GGA-PBE > GGA-RPBE for each compound. Our GGA-PW91 elastic constants are also almost the same that of GGA-PBE as we achieved the close results for lattice constants. It is noticeable that the obtained values of elastic constants decrease with the size of anion atom from P → Sb. For each compound, C ₁₁ has bigger value than C ₃₃ which means that the c-axis is more compressible than the a-axis. It is also pointed out that C ₄₄ is lower than C ₆₆, indicating that the shear along the (001) plane is easier relative to the shear along the (100) plane.

(ii) The bulk modulus measures the resistance of a solid to volume change by applied pressure.^{Reference Young, Sanloup, Gregoryanz, Scandolo, Hemley and Mao41} The bulk moduli of these compounds increase in the sequence: B(MgSnP₂) > B(MgSnAs₂) > B(MgSnSb₂). The present values of the bulk moduli for several exchange correlation functionals are ordered same with elastic constants. It should be stressed that MgSnSb₂ has a higher cell volume than MgSnAs₂ and MgSnSb₂ which may result in the lower bulk moduli of MgSnSb₂.

(iii) The shear modulus is a measurement of resistance to shape change and is more pertinent to hardness.^{Reference Young, Sanloup, Gregoryanz, Scandolo, Hemley and Mao41} Because of the relationship between the elastic constants and shear modulus, the values of shear modulus are of the same order.

(iv) The Young's modulus E, the ratio between stress and strain is required to provide information about the measure of the stiffness of the solids.^{Reference Peng, Chen, Fu and Gao42,Reference Peng, Peng, Fu and Yang43} MgSnP₂ possesses the largest Young modulus for LDA functional, thus MgSnP₂ is the stiffest one. It is also noticed that Young modulus are in order of E(MgSnP₂) > E(MgSnAs₂) > E(MgSnSb₂).

(v) According to criterion,^{Reference Peng, Chen and Yang45} a material is brittle/ductile if the B/G ratio is less/higher than 1.75. The values of the B/G for these compounds are higher than 1.75; hence, all of them exhibited in a ductile manner.^{Reference Peng, Peng, Fu and Yang43} Furthermore, the ductile manner is also related to Cauchy pressure. For tetragonal crystals C ₁₃–C ₄₄ > 0, C ₁₂–C ₆₆ > 0 means ductile, while negative Cauchy pressure denotes brittle manner.^{Reference Pettifor44} In this case, these compounds exhibit positive Cauchy pressure, suggesting ductility.

(vi) Poisson's ratio is the ratio of transverse contraction strain to longitudinal extension strain in the direction of stretching force. Tensile deformation is considered positive and compressive deformation is considered negative. The definition of Poisson's ratio contains a minus sign so that normal materials have a positive ratio.^{Reference Sokolnikoff46} Poisson's ratio v is an important mechanical property which related to the compressibility. And it is also interpreted to chemical bonding forces than any of the other elastic constants.^{Reference Peng, Peng, Fu and Yang43,Reference Peng, Chen and Yang45} The lower limit and upper limit of Poisson's ratio ν are given as 0.25 and 0.5 for central forces in ionic solids, respectively.^{Reference Peng, Chen and Yang45} Calculated ν values for different approximations change between 0.293–0.324, 0.293–0.340, and 0.295–0.344 for MgSnP₂, MgSnAs₂, and MgSnSb₂, respectively. These results demonstrate that interatomic forces are central in nature for all considered compounds. In our case, we found Poisson's ratio the sequence in ν(MgSnP₂) < ν(MgSnAs₂) < ν(MgSnSb₂). On the other hand, Poisson's ratio v > 0.25^{Reference Aydin and Simsek47} corresponds to the large deformation of volumetric change and high anisotropy. This means that the MgSnT₂ compounds exhibit larger deformation to the volume changing and they would have higher anisotropy.

(vii) To gain more information about the mechanical properties, we predicted the bulk modulus and anisotropic factors along the a axis (Ba) and the c axis (Bc)^{Reference Fu, Li, Peng, Gao and Cheng34,Reference Ravindran, Fast, Korzhavyi and Johansson48} as listed in Table IV. It can be concluded that the bulk moduli along the c-axis are smaller than those along the a-axis for all investigated compounds. Therefore, the c-axis is harder to be compressed compared with the a-axis.

The shear anisotropic factors provide a measure of the degree of anisotropy in the bonding between atoms in different planes. Our calculated results are listed in Table IV. For tetragonal structure, the shear anisotropy factors in the {100} and {001} planes are as follows in equations:^{Reference Fu, Li, Peng, Gao and Cheng34,Reference Ravindran, Fast, Korzhavyi and Johansson48}

(1)$${{A^{\left\{ {100} \right\}}} = {{4{C_{44}}} \over {{C_{11}} + {C_{33}} - 2{C_{13}}}}}\quad {{A^{\left\{ {001} \right\}}} = {{2{C_{66}}} \over {{C_{11}} - 2{C_{12}}}}}$$

Our obtained values of shear anisotropy factors suggest that these compounds have strong elastic anisotropy of the chalcopyrite phase, both {100} and {001} planes.

(viii) We have also calculated the percentage elastic anisotropy for polycrystalline materials that is defined as A _comp = (B _V − B _R)/(B _V + B _R) and A _shear = (G _V − G _R)/(G _V + G _R) in compressibility and shear, respectively.^{Reference Peng, Peng, Fu and Yang43,Reference Peng, Chen and Yang45} In these expressions, the value of zero corresponds to the elastic isotropy and the value of 100% identifies the largest elastic anisotropy. The calculated results for A _B and A _G are collected in Table IV. The values of A_comp and A_shear in Table IV indicate that these compounds exhibit large elastic isotropy in compressibility and small elastic anisotropy in shear. Another important elastic anisotropy factor is known as universal anisotropy index A ^{UReference Ranganathan and Ostoja-Starzewski49} which is given by the following relation: A ^U = [5 (G _V/G _R) + (B _V/B _R)] − 6. If A ^U is equal to zero, elastic isotropy exists. On the other hand, the deviations of A ^U from zero also show elastic anisotropy. It is noticed that three compounds exhibit elastic anisotropy.

(ix) Finally, our calculated elastic constants can be used to estimate a melting temperature for each of the compounds for various exchange-correlation functionals. For tetragonal structure, the empirical equation of melting temperature^{Reference Fine, Brown and Marcus50,Reference Alouani, Albers and Methfessel51} is given by (T _m/K) = 354 + 4.50[(2C ₁₁ + C ₃₃)/3 GPa] ± 500. Our calculated results indicate that the melting points decrease with the size of anion atom from P → Sb (Table IV).

C. Electronic properties

We now turn our attention to analyze electronic band structure, total and partial density of states (PDOS) as well as an electronic charge density of MgSnT₂ (T = P, As, Sb) compounds. Exchange correlation energy functionals play a vital role in the efficiency of DFT calculations, more soundly in the calculation of fundamental electronic energy band gap.

In Fig. 2, we presented the electronic band structure along high-symmetry points in the Brillouin zone of the body centered tetragonal for these compounds under zero pressure and temperature. Since the valance band top and the conduction band bottom are located at Γ-point, these compounds are a direct semiconductor compounds and the obtained value of energy band gap is found to be 1.304, 0.550, and 0.407 eV using LDA scheme of MgSnT₂ (T = P, As, Sb) compounds, respectively. The computed band gap values at chosen exchange correlation functionals are presented in Table V with the existing studies. According to Table II, our finding values of LDA scheme are somewhat smaller than experimental values^{Reference Suh and Rajan13} (2.48 eV for MgSnP₂ and 1.65 eV for MgSnAs₂). It is roughly 47 and 66.7 % smaller than experimental study^{Reference Suh and Rajan13} for MgSnP₂ and MgSnAs₂, respectively. Our GGA-PBE predictions are consistent with Ref. Reference Shaposhnikov, Krivosheeva and Borisenko15 using the GGA-PBE calculation, while these results show an intermediate error compared to the GW. It is also seen that GW significantly change both LDA and GGA picture of the band gap.^{Reference Shaposhnikov, Krivosheeva and Borisenko15} The computed values are slightly different from those given in Ref. Reference Jaffe and Zunger4 and Reference Zhaochun, Ruiwu and Nianyi52 and the experiment results^{Reference Suh and Rajan13} are very high as compared to our values for all compounds as expected from DFT results. This is most likely related to DFT which cannot be properly described in the band gap values.

FIG. 2.

LDA band structures of MgSnT₂ (T = P, As, Sb) compounds along some high-symmetry directions of the brillouin zone.

TABLE V.

The calculated band gap energies of MgSnT₂ (T = P, As, Sb) compounds.

Comparing GGA functionals with those obtained by us, GGA-AM05 is bigger than other functionals. It can be seen also from Table V that if we compare the experimental measurements with our results calculated with different exchange-correlation functionals, the best agreement is obtained within the LDA scheme. Thus, from now on we have decided to use the LDA functional for the electronic properties as well as optical and lattice dynamical calculations. The usage of PAW formalism chosen for various exchange correlation potential seems also influenced the results of band gap energy significantly. More specifically, it is observed that the computed value of band gap decreases from MgSnP₂ to MgSnSb₂. Generally, it should be expressed that the behavior of the electronic band structures is very similar except for the values of their band gaps.

To clarify the contribution of each orbital of these atoms, we computed PDOS of MgSnT₂ (T = P, As, Sb) using LDA scheme as given in Figs. 3(a)–(c). The DOS graphs of the semiconducting MgSnT₂ (T = P, As, Sb) compounds show the PDOS nearly with same shape. The lowest valence band between about −12.5 eV and −9.5 eV consists mainly of the anion (P, As, and Sb)-s states with a minor contribution coming from the mixture of Sn-s and -p states. The PDOS between about −7.5 and −5 eV is due to mainly the Sn-s states. The electronic structure below the Fermi energy between −4.5 and 0 eV comes from the mixture of Sn-p with the anion (P, As and Sb)-p states. The conduction band is composed of Sn-s and -p states and anion T (P, As and Sb) -p states.

To have an explicit understanding of the interaction among different orbitals of atoms, the charge densities of MgSnT₂ (T = P, As, Sb) are calculated. The contour maps of the charge density in (111) plane are given in Fig. 4 using VESTA (Visualization for Electronic and Structural Analysis). It can be noticed from Fig. 4 that the strong covalent bonding occurs between Sn atom and anions (P, As, and Sb) and the weak ionic bonding exists between Mg and anions (P, As and Sb). The main contributions of covalence come from sp hybridization of Sn and anions.

FIG. 3.

The total and partial electronic DOS for (a) MgSnP₂, (b) MgSnAs₂, and (c) MgSnSb₂ as calculated using the LDA.

FIG. 4.

Electronic charge density of a) MgSnP₂, b) MgSnAs₂ and c) MgSnSb₂ compounds in the (111) plane within LDA (prepared by VESTA^{Reference Momma and Izumi53}).

D. Optical properties

To learn more about the technological importance of these compounds, we focus our attention on optical properties using LDA functional, which gives reasonably satisfactory results for electronic band structure between all the exchange-correlation functional studied here. The calculated real and imaginary parts of the dielectric function ε(ω) as a function of photon energy are illustrated in Figs. 5(a) and (b) for the energy range up to 15 eV. As the crystal structure of MgSnT₂ (T = P, As, Sb) compounds is body centered tetragonal, they are characterized by two independent tensor components (perpendicular and parallel to c axis) of the dielectric tensor. The static perpendicular real part of the dielectric function, ε_1⊥(0) are found to be 11.357, 13.085, and 16.164 for MgSnT₂ (T = P, As, Sb), respectively. And the static parallel real part of dielectric function ε_1∥(0) 11.617, 13.654, and 16.996 for MgSnP₂, MgSnAs₂, and MgSnSb₂, respectively. One can notice that from P to Sb, the peaks in the ε₁(ω) increase and shift to the low energy region. Using δε = [ε_1∥(0) − ε_1⊥(0)/ε_total(0)] relation,^{Reference Reshak, Alahmed and Azam54} we calculated uni-axial anisotropy and they were found to be about 0.011, 0.021, and 0.025 for MgSnP₂, MgSnAs₂, and MgSnSb₂, respectively. It means that the difference between perpendicular and parallel real part of dielectric function suggest the anisotropic behavior of optical property for MgSnT₂ (T = P, As, Sb) compounds in chalcopyrite phase. For larger photon energies about 4.83 eV, 4.43 eV, and 3.89 eV the real part becomes negative for MgSnP₂, MgSnAs₂, and MgSnSb_2, respectively. As it can be seen from Fig. 5(b), these compounds have two major groups of the peaks. The first important peak of the imaginary part of the dielectric function ε_2⊥(ω) is located at 2.56, 2, 1.63 eV and ε_2∥(ω) is also found to be 2.414, 2, and 1.5 eV for MgSnP₂, MgSnAs₂, and MgSnSb₂, respectively, which are related to interband transitions between the valence and conduction bands. These results can be compared favorably with experimental results of band gap energy. Shaposhnikov et al.^{Reference Shaposhnikov, Krivosheeva and Borisenko15} investigated the optical properties of MgSnP₂ and MgSnAs₂ compounds. They found that the imaginary part of the dielectric function ε₂(ω) are at about 2.0 eV for MgSnP₂ and 1.2 eV for MgSnAs₂ which agree with our results. With the comparisons of MgSnP₂, MgSnAs₂, and MgSnSb₂, we can see that the peak of MgSnP₂ shifts to a higher energy, which is the related to the band-gap energy changed from P to Sb. While the calculated maximum peak values of perpendicular imaginary part of the dielectric function ε_2⊥(ω) are 4.26, 3.85, and 3.42 eV for MgSnP₂, MgSnAs₂, and MgSnSb₂, the parallel imaginary parts of dielectric function ε_2∥(ω) are located at 3.69, 3.42, and 2.89 eV for MgSnP₂, MgSnAs₂, and MgSnSb₂, respectively.

FIG. 5.

The computed (a) real and (b) imaginary part of the dielectric function of MgSnT₂ (T = P, As, Sb) compounds versus photon energy at 0 GPa and 0 K.

The calculated refractive index n(ω), extinction coefficients k(ω), reflectivity R(ω), and energy loss function L(ω) are estimated by Kramer–Kroning relations.^{Reference Adachi55,Reference Ravindran, Delin, Johansson, Eriksson and Wills56} Our obtained results are plotted in Figs. 6(a)–(d). The calculated refractive index is displayed in Fig. 6(a) for three compounds. While the predicted values of perpendicular static refractive index n _⊥(0) are 3.37, 3.617, and 4.02, the parallel static refractive index values n _∥(0) are 3.408, 3.695, and 4.123 for MgSnP₂, MgSnAs₂, and MgSnSb₂, respectively. The main peak values of refractive index for MgSnP₂, MgSnAs₂, and MgSnSb₂ are 4.2 at 2.27 eV, 4.25 at 1.713 eV and 4.65 at 1.2 eV, respectively. We note that the refractive index reaches maximal values for the energies near the band gap energy. From Fig. 6(b), we have predicted the extinction coefficients for MgSnP₂, MgSnAs₂, and MgSnSb₂ to be 2.8, 2.64, and 2.80, respectively. As it can be seen from Fig. 6(c), while the static perpendicular reflectivity R _⊥(0) is 0.294, the maximum values of that is about 0.76 at 15.625 eV. While the static parallel reflectivity R _∥(0) is 0.298, the maximum values of that is about 0.618 at 15.34 eV for MgSnP₂. For MgSnAs₂, the static reflectivity of R _⊥(0) and R _∥(0) are 0.321 and 0.33, but the maximum values of that is about 0.637 at 14.73 eV and 0.65 at 14.49 eV, respectively. While the calculated static reflectivity of R _⊥(0) and R _∥(0) are 0.362 and 0.372, the maximum values of that are about 0.654 at 13.92 eV and 0.67 at 14.35 eV of MgSnSb₂. One further point of interest is energy loss functions, which is an important factor describing the energy loss of a fast electron traversing in a material, as shown in Fig. 6(d) for these compounds. The maximum peaks in the energy-loss function indicate that plasmon resonance occurs at around 17.05, 15.625, and 14.56 eV for MgSnP₂, MgSnAs₂, and MgSnSb₂, respectively. It can be pointed out that the plasma frequency of MgSnP₂ is the largest one.

FIG. 6.

The computed (a) refractive index n(ω), (b) extinction coefficient k(ω), (c) reflectivity R(ω), and (d) electron energy loss spectrum L(ω) of MgSnT₂ (T = P, As, Sb) compounds versus photon energy at 0 GPa and 0 K.

Figure 7 displays the variation of birefringence Δn (ω) with the photon energy for MgSnT₂ (T = P, As, Sb) compounds. It is defined that the difference between the index of refraction for an electric field oriented along the c-axis (n _∥) and the index of refraction for an electric field perpendicular (n _⊥) to the c-axis (Δn = n _∥ − n _⊥). It is also related to fulfill phase-matching condition.^{Reference Chiker, Kebbab, Miloua and Benramdane12,Reference Reshak and Auluck57} The static values of the birefringence Δn(0) are 0.038, 0.078, and 0.102 eV for MgSnP₂, MgSnAs₂, and MgSnSb₂, respectively. These results indicate that they are a phase matchable. It can be noticed that the general tendency of Δn(ω) for these compounds is similar features, but MgSnSb₂ has higher value than others. The positions of the peaks also shift to lower energy regions as we replace the anion element from P to Sb.

FIG. 7.

The calculated birefringence Δn(ω) versus photon energy of MgSnT₂ (T = P, As, Sb) compounds.

E. Vibrational and thermal properties

Knowledge of phonon characteristics is required to describe mechanical, acoustic, dynamic, spectroscopic, and thermodynamic properties of crystals at finite temperature. The easiest and most convenient way to calculate phonons from first-principles is to use existing and well-tested DFT softwares,^{Reference Kresse and Joubert20,Reference Kresse and Hafner21,Reference Blaha, Schwarz, Madsen, Kvasnicka and Luitz58,Reference Artacho, Gale, García, Junquera, Ordejón, Sánchez-Portal and Soler59} which can calculate crystalline structures, electronic bands, magnetic properties, and so called Hellmann–Feynman forces.

The density-functional perturbation theory employing the force constant method has been used to calculate the phonon dispersion relation and phonon DOS for MgSnT₂ (T = P, As, Sb) compounds using both LDA and new functional of GGA-AM05. Throughout this calculation, we used PHONOPY code^{Reference Togo60} with 2 × 1 × 1 supercell. The phonon dispersion curves along the Z–Γ–X–P–N–Γ high-symmetry directions together with the total phonon density are displayed in Fig. 8 and phonon PDOS are also shown in Fig. 9 for each compound. For the chalcopyrite structure ($I\bar 42d$) primitive cell, there are eight atoms. Thus, there should be twenty one optical modes and three acoustic modes. Unfortunately, there is no available experimental data for MgSnT₂ (T = P, As, Sb) compounds based on phonon dispersion curves. Therefore, we cannot conclude that LDA or GGA exchange-correlation functional gives better results to compare our calculated results of phonon spectrum. On the other hand, our estimated main features of the phonon dispersion and partial phonon DOS can be compared with previous ab-initio calculations^{Reference Shi, Hu, Qin, Duan, Wu, Yang and Tang17} for BeSiV₂ and MgSiV₂ (V = P, As, Sb) compounds. Their calculation revealed that the character of vibrational spectra of these crystalline compounds is found to be governed by both chemical bonding strengths and masses of structural units. They noticed that the Si–V bonding is remarkably stronger than the Mg–V bonding in Mg-containing chalcopyrite semiconductors. Their suggestion also consists well with our prediction. Generally, it should be stressed that the calculated phonon spectra and phonon DOS for these chalcopyrites are qualitatively similar to each other. It is also important to notice that all calculated optical and acoustic modes have no soft mode at any wave vectors, suggesting their vibration stability for MgSnT₂ (T = P, As, Sb). While the lower acoustic frequencies for LDA and GGA-AM05 functional have almost given similar results, the higher optic frequencies LDA is bigger than that of GGA-AM05 values. When looking at the total DOS in Fig. 8, it looks as if it was simply shifted below when changing the exchange-correlation functionals from LDA to GGA-AM05.

FIG. 8.

Phonon dispersion curves and total phonon DOS for (a) MgSnP₂, (b) MgSnAs₂, and (c) MgSnSb₂. The solid lines represent the LDA and dashed lines are GGA-AM05.

FIG. 9.

Phonon PDOS of MgSnT₂ (T = P, As, Sb) as calculated using the LDA.

As it can be seen in Fig. 9(a), the phonon PDOS of MgSnP₂ consist of two groups of regions: the lower modes with frequencies up to ∼7 THz are mixed states composed of the Mg, Sn, and P atoms and from 8 to 11.5 THz are predominately Mg and P atoms. MgSnAs₂ and MgSnSb₂ compounds divided into three groups of regions (see Figs. 9(b)–9(c)). The first region with frequencies up to ∼5.7 THz for MgSnAs₂ and ∼4.25 THz for MgSnSb₂ mixed of Mg, Sn, and anions (As and Sb). The main contribution of the middle region for MgSnAs₂ and MgSnSb₂ between 5.8–7 THz and 4.6–6 THz comes from Sn and anions (As and Sb). The vibrations of the Mg atom are dominant in the upper range of 8 THz for MgSnAs₂ and 7 THz for MgSnSb₂.

Phonon dispersion curve plays a key parameter in dynamical behaviors and thermal properties such as constant volume heat capacity (C _v), Helmholtz free energy (F), and entropy (S), internal energy (E). As a final remark, the temperature dependence of the C _v and S of MgSnT₂ (X = P, As, Sb) is estimated using the quasi harmonic approximation as is given following equation.^{Reference Togo and Tanaka61}

(2)$${C_{\rm{V}}} = \sum\limits_{qj} {{C_{qj}}} = \sum\limits_{qj} {{k_{\rm{B}}}} {\left( {{{\hbar {\omega _{qj}}} \over {{k_{\rm{B}}}T}}} \right)^2}{{{\rm{exp}}\left( {{{\hbar {\omega _{qj}}} \over {{k_{\rm{B}}}T}}} \right)} \over {{{\left[ {{\rm{exp}}\left( {{{\hbar {\omega _{qj}}} \over {{k_{\rm{B}}}T}}} \right) - 1} \right]}^2}}}$$

(3)$$S = {1 \over {2T}}\sum\limits_{qj} {\hbar {\omega _{qj}}\;{\rm{coth}}\left[ {{{\hbar {\omega _{qj}}} \over {2{k_{\rm{B}}}T}}} \right]} - {k_{\rm{B}}}\sum\limits_{qj} {{\rm{ln}}\left[ {2\;\sinh \left( {{{\hbar {\omega _{qj}}} \over {2{k_{\rm{B}}}T}}} \right)} \right]}$$

In Fig. 10(a), we denote the calculated heat capacity at constant as a function of temperature for these compounds. Clearly, the heat capacity increases linearly with the temperature (∼T < 300 K), but the rate of increase is very moderate especially at high pressures as the heat capacity reaches the classical Dulong Petit limit. The computed values of heat capacity at constant volume at 300 K are 176.29, 185.75, and 189.58 J/mol K for MgSnT₂ (T = P, As, Sb), respectively.

FIG. 10.

Variation of (a) heat capacity at constant volume (C _v) and (b) entropy (S) as a function of temperature for MgSnT₂ (T = P, As, Sb) compounds within LDA approximation.

The variations of entropy of all considered compounds from 0 K to 1000 K are presented in Fig. 10(b) under zero pressure within LDA functional. This figure revealed that their entropy's are zero at 0 K, when the entropy increases considerably with increasing temperature in low temperature and then the variation of entropy is small above about 300 K. The values of entropy at 300 K are found to be 226.45, 280.39, and 326. 67 J/mol K for MgSnT₂ (T = P, As, Sb), respectively. Obviously, it is observed that the entropy increases as we replace anion T from P to Sb. Generally, it is noticeable that the temperature variation of thermo dynamical functions exhibits a similar trend for these compounds.