First principles study of electronic and lattice dynamical properties of the XNNi₃ (X=Pd, Sn and Sb) ternary nitrides with E2₁ structure (space group Pm3 (221)) has been reported using the plane-wave pseudo-potential technique based on density functional theory. The calculated equilibrium parameters are in good agreement with other works. The relationship between anisotropy and mechanical properties are also analyzed. Mechanical stability and stiffness of these materials are determined and XNNi₃ (X=Pd, Sn and Sb) ternary nitride compounds are found mechanically stable at zero pressure. Shear Modulus (G), Young’s Modulus (E), maximum and minimum Poisson ratios (υ), Zener anisotropy factor (A) and compressibility (β) values are calculated and evaluated in calculations of elastic properties. The electronic properties are studied and presented by plots with total and partial density of states with charge density distributions. The XNNi₃ (X=Pd, Sn and Sb) ternary nitrides are metallic behavior and have covalent bonding due to the hybridization. The vibrational properties are investigated to explain lattice dynamics of these types of ternary nitrides.

Keywords: first-principles, lattice dynamical properties, electronic properties, ternary nitrides, antiperovskite

The antiperovskite type Ni-rich ternary nitrides XNNi₃ researches have been increased since the discovery of superconductivity (8 Kelvin) for cubic antiperovskite MgCNi₃ compound.¹ This discovery has strongly motivated to study antiperovskite series. The studies of the family of Ni-rich carbides have been investigated theoretically and experimentally.^2–15 In recent years, the investigations on some new antiperovskite type nitrides have gained wide interest also some biosensor applications in bioelectronics industry.^16–30 Despite of the fact that there are number of studies related with their properties for some cubic antiperovskite type Ni-rich ternary nitrides, in particular, the mechanical properties of XNNi₃-type compounds with X= Al, Ga, In, Zn, Cd, Mg, Sn, Sb, Pd, Cu, Ag and Pt have been studied in theory,¹⁵ it is not mentioned on especially lattice dynamical and electronic properties of XNNi₃ (X= Pd, Sn and Sb). One of interesting study is about a new Ni-based antiperovskite nitride, CuNNi₃ that shows the superconductivity at 3.2 K and it is reported with X-ray diffraction, magnetization, resistivity and heat capacity measurements.¹⁶ The structural and mechanical properties of the antiperovskite XNNi₃ (X=Zn, Mg, Al) with pressure effect are studied by Hong-Cun et al.,²⁶ by using CASTEP code. Optical functions of SnNNi₃, ZnNNi₃ and CuNNi₃ compounds are studied until 40 eV.²⁹ In this study, by means of the ab-initio calculations, we have analyzed in details the comparative study of electronic and elastic properties of the XNNi₃ (X=Pd, Sn and Sb). Optimized lattice parameters and electronic band structures are reported by using ultra soft pseudo potential.^31,32In addition, anisotropic independent second order elastic constants (C_ij). These constants give permissions us to get the mechanical parameters of XNNi₃ (X=Pd, Sn and Sb). Additionally, vibrational properties of XNNi₃ (X=Pd, Sn and Sb) compounds are investigated and summarized.

The density functional theory (DFT)^33,34has successfully been applied to the ab-initio calculations of the ground-state properties. In view of these circumstances, we have applied to the Generalized Gradient Approximation (GGA)³⁵for the exchange-correlation functional. All properties of calculations are investigated by using the Vienna Ab-initio Simulation Package (VASP).^36–39 The calculations are performed for Pd(4d¹⁰), Sn(5s²5p²), Sb(5s²5p³), N(2s²2p³), Ni(4s¹3d⁹). In our calculations, plane-wave basis sets with cut-off energy 500 eV and the 12x12x12 Monkhorst and Pack⁴⁰ k-points are used in the Brillouin zone for XNNi₃ (X=Pd, Sn and Sb). To obtain mechanical anisotropic properties, EIAM code is used for calculations.⁴¹ The elastic properties are exploited to estimate with stress-strain method.^42,43

Structural and elastic properties

The unit cell of SnNNi₃ compound is shown in Figure 1. The crystal structures of SbNNi₃ and PdNNi₃ compounds are the same with SnNNi₃ compound as shown in Figure 1. In our case E2₁-type structure which is illustrated in Figure 1. The Wyckoff positions of atoms are located as follow: Sn (0, 0, 0); N (0.5, 0.5, 0.5); and Ni (0, 0.5, 0.5) (Figure 2). Firstly, the equilibrium lattice constants, bulk modulus and its pressure derivative have been obtained by minimizing the total crystal energy calculated for different values of lattice constants using the Birch-Murnaghan equation of states (eos)⁴⁴ and the calculation results are given in Table 1 for cubic perovskite (E2₁) structure (space group Pm3 (221)) of XNNi₃ (X=Pd, Sn and Sb). (Table 1) the present structural results are listed in the Table 1, along with the other theoretical and experimental works. The present lattice constants are obtained as 3.905 Å, 3.944 Å and 3.808 Å, respectively, for XNNi₃ (X=Pd, Sn and Sb) compounds. Our lattice constants are very good convenient parameters with the other theoretical studies. Our lattice constants in E2₁-type crystal structure for SnNNi₃ is nearly 0.127% lower, for SbNNi₃ is nearly 0.051% higher and for PdNNi₃ is nearly 0.132% higher than the reference value.²⁸ These deviations may stem from the using of GGA approximations with different ab-initio codes. Additionally, the volume values and bulk modulus values of SnNNi₃ compound are in convenient with other theoretical values and also the same values are in very good agreement of SbNNi₃ and PdNNi₃ compounds. The effect of hydrostatic pressure indicated by the derivative of bulk modulus under pressure (B´) is given in Table 1 for each of ternary nitride compounds. According to Table 1, the pressure derivatives of bulk modulus are calculated 4.440, 4.399 and 4.590, respectively, for XNNi₃ (X=Pd, Sn and Sb) compounds and sequenced like as. The derivative of bulk modulus is evaluated with anisotropy. From Table 3, the anisotropy values of XNNi₃ (X=Pd, Sn and Sb) compounds are, respectively, 1.19, 1.28 and 1.21. The magnitudes of anisotropy factors are arranged with. Due to both of these equalities, derivative of bulk modulus are confirmed by anisotropy as an expected. The thermodynamic stability of XNNi₃ (X=Pd, Sn and Sb) compounds can be reflected by the formation enthalpy (∆H). Negative formation enthalpy has been explained as an exothermic process, and formation energy in the lower ones shows the stability related with the decomposition to the constituents of an element. The formation enthalpy could be expressed by the relation:⁴⁵

∆H=(E_top- $(\Sigma n_i{{\rm E}}_i)/n$(1)

Figure 1 The unitcell of SnNNi₃.

Figure 2 Total energy versus volume curves of XNNi3 (X=Pd, Sn and Sb).

where E_tot is the total energy of the compound with n_i atoms of all i (X(Pd, Sn, Sb), N and Ni). n: total number of atoms in the primitive cell, E_i: total energy of a pure i the atom with equilibrium lattice constants. The calculated theoretical formation enthalpies of XNNi₃ (X=Pd, Sn and Sb) compounds are included in Table 1. As far as we known, there are no data for evaluation the formation energy in the literature to compare with ours. SbNNi₃ shows the lowest value of formation enthalpy, which indicates that SbNNi₃ compound has the highest stability of these nitride structures. It is important to investigate the second order elastic properties because of the fact that the calculations provide an accuracy and comparison of the calculations of mechanical properties. Herein, C_ij elastic constants are the second-order elastic constants of the structure and has been optimized under a given set of exchange–correlation (XC) potential functions and attained an equilibrium structure with a minimum total energy. The elastic parameters are obtained from the second-order derivatives of the total energy:

$C_{ij}=\frac{1}{V_0}\frac{{\partial }^2{E}_{total}}{\partial {\xi }_i\partial {\xi }_j}$ (2)

The cubic crystal has only three independent parameters, C₁₁, C₁₂ and C₄₄. The traditional rules on the elastic constants: C₁₁>0, C₁₁-C₁₂>0, C₄₄>0, C₁₁+2C₁₂>0 and C₁₁>B>C₁₂. These traditional mechanical stability conditions (called that Born’s stability criteria) (P=0 GPa)⁴⁶ are investigated by using the obtained second-order elastic constants all our three nitride compounds. The calculated values of C_ij are summarized and given in the table for XNNi₃ (X=Pd, Sn and Sb), respectively (Table 2). Second-order elastic constants of XNNi₃ (X=Pd, Sn and Sb) meant to Born’s stability conditions Table 2. According to Table 2, it is obvious that XNNi₃ (X=Pd, Sn and Sb) compounds satisfy stability conditions. For SbNNi₃ compound in reference,²⁸ C₄₄ is found as -8.6 GPa although our present calculated value is 34.79 GPa for C₄₄. All other theoretical references are compatible with present values. As it can be seen from Table 2, our ternary nitride compounds have different elastic constants due to their classifications of elements. Elastic properties of our nitride compounds are effected owing to the fact that Tin (Sn) is post-transition metal, antimony (Sb) is metalloid and palladium (Pd) is transition metal. The Zener anisotropy factor (A), Poisson ratio (), and Young’s modulus (E) that are important parameters to see all image of elastic properties are also calculated using by these formulas:⁴⁷

$A=\frac{{\text{2C}}_{\text{44}}}{C_{\text{11}}-C_{\text{12}}}$ (3)

$\upsilon =\frac{\text{1}}{\text{2}}[\frac{(B-\frac{\text{2}}{\text{3}}G)}{(B+\frac{\text{1}}{\text{3}}G)}]$ (4)

$E=\frac{\text{9}GB}{G+\text{3B}}$(5)

where G is the an isotropic shear modulus as a function of crystal orientation and is given like that G=(G_V+G_R)/2, herein G_V is Voigt’s shear modulus (it is related with the upper bound of G values) and G_R is Reuss’s shear modulus (it is related with the lower bound of G values) and can be written as G_V=(C₁₁–C₁₂+3C₄₄)/5 and 5/G_R= 4/(C₁₁-C₁₂)+3/C₄₄, resopectively. The calculated an isotropic shear modulus, Young’s modulus, Poisson ratios, Zener anisotropy factor, and compressibility (β) of the XNNi₃ (X=Pd, Sn and Sb) are presented in Table 3. The Shear and Young’s modulus are calculated with their Voigt and Reuss values and Poisson ratios with maximum and minimum values at zero pressure by ElAM code⁴¹ for anisotropic behaviors of three ternary nitrides (Table 3). Obtained by Voigt and Reuss values of isotropic shear modulus (G) are 152.21 GPa, 129.39 GPa and 129.11 GPa, respectively, for PdNNi₃, SnNNi₃ and SbNNi₃ compounds. Using ratio of isotropic shear modulus and bulk modulus, elastic manners of materials are estimated. Ratios of G/B that is called Pugh ratio of XNNi₃ (X=Pd, Sn and Sb) compounds are given in Table 3. Providing that G/B<0.5, the material exhibits in a ductile behavior, and while G/B>0.5, the material exhibits in a brittle behavior.^47,48 As can be seen in Table 3, all of XNNi₃ (X=Sn, Sb and Pd) ternary nitrides compounds indicate brittle manners due to the fact that their G/B ratios are greater than 0.5. In fact, they behave nearly at brittle/ductile border like in reference.¹³ It has also observed that for all of our three antiperovskite type nitrides B>G. As mentioned that parameters limit the mechanical stability of these materials. As a comparison, the Young’s modulus of PdNNi₃ compound has the biggest one in our ternary nitrides systems. From the literature it is well-known that, if the rate of Poisson is less than 0.25, the material shows covalent bond character, otherwise it is bigger than or equal to 0.25 it shows ionic bond character.⁴⁹ The minimum value Poisson’s ratio of SbNNi₃ is calculated as zero and maximum value Poisson’s ratio of SbNNi₃ is calculated as 0.14. It might have originated from directions or maximum stability. The other values of Poisson’s ratios of XNNi₃ (X=Pd, Sn and Sb) are obtained similar values for each of three nitride compounds. Three ternary E2₁ structure nitrides show metallic-like systems as indicated in reference.¹³In bulk materials, to see the elastic anisotropy behavior, the Zener anisotropy factor is using to determine the degree of anisotropy. Providing that it gives the value of 1, our compound shows entirely isotropic. Otherwise, this value exhibits anisotropic behavior. The values of our three nitrides are greater than 1. Our materials partially exhibit anisotropic behaviors. The compressibility is a measure of elasticity and is defined as following relations:⁵⁰

$\beta =\frac{C_{11}-C_{12}}{\Omega }$ (6)

$\Omega =(C_{11}+C_{12})C_{11}-2C_{12}^2$(7)

The calculated compressibility values are found as 0.0023 GPa^-1, 0.0029 GPa^-1 and 0.0031 GPa^-1, respectively, for PdNNi₃, SnNNi₃ and SbNNi₃ compounds. The calculated present values of compressibility are compatible with other theoretical data for SbNNi₃ and PdNNi₃ compounds.

Electronic properties

In this section, the main features of electronic properties of XNNi₃ (X=Pd, Sn and Sb) compounds are described by analyzing the density of states as total and partial with their related charge densities in Figure 3 & Figure 4. The energy zero is chosen to be at the Fermi energy E_F. All of the three total densities of states have nearly similar features. For all compounds conduction band minimum values are upper than from Fermi energy level. As a comparison of DOS of SbNNi₃ with other compounds, its DOS is lower at Fermi energy level. The DOS of PdNNi₃ are above at Fermi energy level according to SbNNi₃ compound. These all compounds exhibit metallic character in consideration of rate for impletion at Fermi energy levels. The metallic behavior of XNNi₃ (X=Pd, Sn and Sb) compounds are mostly owing to the addition of Ni-d states at the Fermi level and a little addition of Pd-d states for PdNNi₃ compound. It is clearly seen that Sn-s state and N-p state contribute at Fermi level, and this emerges to a sp-hybridization between metal-s state and N-p states. As it is seen from partial density of states explanations, owing to the covalent bonding, there is hybridization and clarifies the charge densities in Figure 4. The lower valance band is because of the 2s-states electrons for XNNi₃ (X=Pd, Sn and Sb) compounds. The charge densities of XNNi₃ (X=Pd, Sn and Sb) compounds are depicted in Figure 4. The computed charge density distributions are evident that the covalent bonding that nature of our three ternary nitrides is obtained as covalent due to the sp-hybridization that is also confirmed by partial density of states plots. It is easy to observe that from Figures 3 and 4, the SbNNi₃ compound is much more covalent according to PdNNi₃ and SnNNi₃ compounds. The charges are more accumulated between atoms. Moreover, a high ratio of G/B is related with brittleness. Considering that SbNNi₃ compound has the greatest value of G/B, consolidates that SbNNi₃ compound has more covalent character than PdNNi₃ and SnNNi₃.

Additionally, stability of XNNi₃ compounds is also confirmed by Band Filling Theory.^51,52 Considering the Band Filling Theory, the numbers of bonding states increase, the stability of material increases and anti-bonding states decrease the stability of compounds. If we called the ratio the width of the occupied states (W_occ) and the width of bonding states (W_b), we can explain the work about the material stability. If the ratio of W_occ/W_b is closer to 1.0, the stability increases. In this work, these quantities predict the structural stability, namely, the pseudo-gaps (W_p), gaps of occupation (W_occ), gaps of bonding (W_b) and the W_occ/W_b values are calculated for each compound and presented in Table 4 for XNNi₃ compounds. Also shown in Table 4, using this band theory formulation, the ratio of W_occ/W_b equals 0.987 and is closer to 1 for SbNNi₃ compound. It is obvious that SbNNi₃ compound is the most stable material. This result confirms that the previous presented partial density of states for our nitrides and charge density distributions for our nitrides.

Figure 4 Charge density distribution of (110) plane of XNNi3 (X=Pd, Sn and Sb) compounds.

Vibrational properties

The phonon dispersion curves of XNNi₃ (X=Pd, Sn and Sb) were obtained using by the PHONOPY code.⁵³ The partial atomic phonon density of states (DOS) for XNNi₃ ternary nitrides were calculated along the high symmetry directions using a 2x2x2 super cell and given in Figure 5A-5C. The 0.03 Å for displacement is adopted for each atom of the 2x2x2 supercell in to determine the forces of the atoms. The primitive cells of XNNi₃ contains 5 atoms with 15 phonon branches have 3 acoustic and 12 optical modes. For PdNNi₃ compound, a gap between acoustic and optic modes is found in the phonon dispersion curves owing to the bigger ratio of mass cation and anions. But, accordin to the SnNNi₃ and PdNNi₃ compounds there is not a gap between acoustic and optic modes. The lack of soft phonon imaginariy modes in the phonon spectra that supports the stable character as dynamically for the XNNi₃ (X=Pd, Sn and Sb) nitrides. In the literature, there is no study of the lattice dynamics of these compounds to compare with our data. It can be seen from Figure 5 that the low-lying optical phonon modes have interactions with phonon modes for SnNNi₃ and SbNNi₃ as the acoustic. For the phonon DOS of XNNi₃ ternary nitrides the acoustic modes are emerged by the vibrations of Sn, Sb and Pd atoms, while the optical modes are emerged by the vibrations of Ni atoms at low modes, with less addition from N and Ni atoms. At higher optical mode, the main contributions emerge from N atoms, with less contributions comes from Ni atoms. PdNNi₃ has upper phonon energies than SnNNi₃ and SbNNi₃ at Gamma point. The main distinction of the three nitrides is due to difference in the chemical bonding and masses for XNNi₃ (X=Pd, Sn and Sb) nitrides (Figure 5).

In this work, we have studies structural, elastic, electronic and vibrational properties of antiperovskite type nitrides XNNi3 (X=Pd, Sn and Sb) compounds with E21 crystal structure using the GGA. The found lattice constants, volume, bulk modulus as structural parameters at zero pressure are in convenient with the previous work. Mechanical stability of XNNi3 (X=Pd, Sn and Sb) compounds are predicted by Born’s stability criteria and found that three nitrides show stability at zero pressure. In elastic calculations, isotropic shear modulus, Poisson’s ratios and Young’s modulus were estimated using with Voigt and Reuss approximations. In addition to good understand of mechanical behaviors of these compounds, anisotropy factor and compressibility are determined. From our first-principles calculations, the stoichiometric XNNi3 (X=Pd, Sn and Sb) compounds are very similar in both structural and elastic properties. The three ternary nitrides have metallic behavior and exhibit covalent characters. The mechanical behavior of XNNi3 (X=Pd, Sn and Sb) compounds are corroborated with electronic properties as given in the results section. The calculated phonon spectra and phonon DOS indicate that XNNi3 (X=Pd, Sn and Sb) compounds are dynamically stable. To best of our knowledge is there is no experimental or theoretical study in vibrational properties of XNNi3 (X=Pd, Sn and Sb) ternary nitrides has been reported yet for comparison. We predict that, our results are good and qualified estimations for future investigations.

The authors acknowledge Ankara University for the high performance computing facility through the AYP under Grand No.17A0443001.

There are no conflicts to declare.

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