Research Article Volume 4 Issue 2
^{1}Department of Physics Engineering, Ankara University, Turkey
^{2}Department of Physics, Gazi University, Turkey
^{3}Department of Electric and Energy, Ahi Evran University, Turkey
^{4}Department of Physics, Middle East Technical University, Turkey
^{5}Photonics Application and Research Center, Gazi University, Turkey
Correspondence: Mogulkoc Y, Department of Physics Engineering, Ankara University, 06100, Ankara, Turkey, Tel +90 312 2033550
Received: April 16, 2018  Published: April 30, 2018
Citation: Mogulkoc Y, Ciftci YO, Surucu G. Study of electronic and lattice dynamical properties of antiperovskitetype nitrides XNNi 3 (X= Pd, Sn and Sb).Int J Biosen Bioelectron. 2018;4(2):80–85. DOI: 10.15406/ijbsbe.2018.04.00102
First principles study of electronic and lattice dynamical properties of the XNNi_{3} (X=Pd, Sn and Sb) ternary nitrides with E2_{1} structure (space group Pm3 (221)) has been reported using the planewave pseudopotential 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 XNNi_{3} (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 XNNi_{3} (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: firstprinciples, lattice dynamical properties, electronic properties, ternary nitrides, antiperovskite
The antiperovskite type Nirich ternary nitrides XNNi_{3} researches have been increased since the discovery of superconductivity (8 Kelvin) for cubic antiperovskite MgCNi_{3} compound.^{1} This discovery has strongly motivated to study antiperovskite series. The studies of the family of Nirich 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 Nirich ternary nitrides, in particular, the mechanical properties of XNNi_{3}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 XNNi_{3} (X= Pd, Sn and Sb). One of interesting study is about a new Nibased antiperovskite nitride, CuNNi_{3} that shows the superconductivity at 3.2 K and it is reported with Xray diffraction, magnetization, resistivity and heat capacity measurements.^{16} The structural and mechanical properties of the antiperovskite XNNi_{3} (X=Zn, Mg, Al) with pressure effect are studied by HongCun et al.,^{26} by using CASTEP code. Optical functions of SnNNi_{3}, ZnNNi_{3} and CuNNi_{3} compounds are studied until 40 eV.^{29} In this study, by means of the abinitio calculations, we have analyzed in details the comparative study of electronic and elastic properties of the XNNi_{3} (X=Pd, Sn and Sb). Optimized lattice parameters and electronic band structures are reported by using ultra soft pseudo potential.^{31,32}In addition, anisotropic independent second order elastic constants (C_{ij}). These constants give permissions us to get the mechanical parameters of XNNi_{3} (X=Pd, Sn and Sb). Additionally, vibrational properties of XNNi_{3} (X=Pd, Sn and Sb) compounds are investigated and summarized.
The density functional theory (DFT)^{33,34}has successfully been applied to the abinitio calculations of the groundstate properties. In view of these circumstances, we have applied to the Generalized Gradient Approximation (GGA)^{35}for the exchangecorrelation functional. All properties of calculations are investigated by using the Vienna Abinitio Simulation Package (VASP).^{36–39} The calculations are performed for Pd(4d^{10}), Sn(5s^{2}5p^{2}), Sb(5s^{2}5p^{3}), N(2s^{2}2p^{3}), Ni(4s^{1}3d^{9}). In our calculations, planewave basis sets with cutoff energy 500 eV and the 12x12x12 Monkhorst and Pack^{40} kpoints are used in the Brillouin zone for XNNi_{3} (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 stressstrain method.^{42,43}
Structural and elastic properties
The unit cell of SnNNi_{3} compound is shown in Figure 1. The crystal structures of SbNNi_{3} and PdNNi_{3} compounds are the same with SnNNi_{3} compound as shown in Figure 1. In our case E2_{1}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 BirchMurnaghan equation of states (eos)^{44} and the calculation results are given in Table 1 for cubic perovskite (E2_{1}) structure (space group Pm3 (221)) of XNNi_{3} (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 XNNi_{3} (X=Pd, Sn and Sb) compounds. Our lattice constants are very good convenient parameters with the other theoretical studies. Our lattice constants in E2_{1}type crystal structure for SnNNi_{3} is nearly 0.127% lower, for SbNNi_{3} is nearly 0.051% higher and for PdNNi_{3} is nearly 0.132% higher than the reference value.^{28} These deviations may stem from the using of GGA approximations with different abinitio codes. Additionally, the volume values and bulk modulus values of SnNNi_{3} compound are in convenient with other theoretical values and also the same values are in very good agreement of SbNNi_{3} and PdNNi_{3} 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 XNNi_{3} (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 XNNi_{3} (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 XNNi_{3} (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)
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 XNNi_{3} (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. SbNNi_{3} shows the lowest value of formation enthalpy, which indicates that SbNNi_{3} 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 secondorder 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 secondorder derivatives of the total energy:
${C}_{ij}=\frac{1}{{V}_{0}}\frac{{\partial}^{2}{E}_{total}}{\partial {\xi}_{i}\partial {\xi}_{j}}$ (2)
Material 
acal (Å) 
V (Å3) 
B (GPa) 
dB/dP 
ΔH (eV) 
PdNNi3

3.808 
55.219

190.4 217.40d 
4.44

2.977

SnNNi3

3.905 
59.547 
159.203 182.30b 
4.399

3.095

SbNNi3

3.944 
61.349 
156.405

4.59

3.109

Table 1 Calculated lattice parameter (acal), volume (V), bulk modulus (B), pressure derivative of bulk modulus (dB/dP), formation energy (ΔH)
Abbreviations: a, numerical study according to empirical model,8; b, theoretical study with APW+lo (FLAPW) implemented in WIEN2k code, GGAPBE,28; c, theoretical study with CASTEP code, GGA,29; d, theoretical study with CASTEP code, GGAPBE30
The cubic crystal has only three independent parameters, C_{11}, C_{12} and C_{44}. The traditional rules on the elastic constants: C_{11}>0, C_{11}C_{12}>0, C_{44}>0, C_{11}+2C_{12}>0 and C_{11}>B>C_{12}. These traditional mechanical stability conditions (called that Born’s stability criteria) (P=0 GPa)^{46} are investigated by using the obtained secondorder elastic constants all our three nitride compounds. The calculated values of C_{i}_{j} are summarized and given in the table for XNNi_{3} (X=Pd, Sn and Sb), respectively (Table 2). Secondorder elastic constants of XNNi_{3} (X=Pd, Sn and Sb) meant to Born’s stability conditions Table 2. According to Table 2, it is obvious that XNNi_{3} (X=Pd, Sn and Sb) compounds satisfy stability conditions. For SbNNi_{3} compound in reference,^{28} C_{44} is found as 8.6 GPa although our present calculated value is 34.79 GPa for C_{44}. 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 posttransition 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}=(C_{11}–C_{12}+3C_{44})/5 and 5/G_{R}= 4/(C_{11}C_{12})+3/C_{44}, resopectively. The calculated an isotropic shear modulus, Young’s modulus, Poisson ratios, Zener anisotropy factor, and compressibility (β) of the XNNi_{3} (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 code^{41} 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 PdNNi_{3}, SnNNi_{3} and SbNNi_{3} 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 XNNi_{3} (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 XNNi_{3} (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 PdNNi_{3} compound has the biggest one in our ternary nitrides systems. From the literature it is wellknown 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 SbNNi_{3} is calculated as zero and maximum value Poisson’s ratio of SbNNi_{3} is calculated as 0.14. It might have originated from directions or maximum stability. The other values of Poisson’s ratios of XNNi_{3} (X=Pd, Sn and Sb) are obtained similar values for each of three nitride compounds. Three ternary E2_{1} structure nitrides show metalliclike systems as indicated in reference.^{13}In 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}
Material 
C11 [GPa] 
C12 [GPa] 
C44 [GPa] 
B [GPa] 
Stability 
PdNNi3

324.67 
164.5 
53.71 
190.4 
Stable

SnNNi3 
272.33 
138.75 
39.28 
159.203 
Stable

SbNNi3

257.48 
142.52 
34.79 
156.405 
Stable 
Table 2 Secondorder elastic constants (Cij), bulk modulus (B), stability
Abbreviations: a, GGAPBE28; b, GGA29; c, GGAPBE30

PdNNi3 
SnNNi3 
SbNNi3 
GV [GPa] 
152.89 
129.86 
130.05 
GR [GPa] 
151.52 
128.91 
128.17 
G/B 
0.8 
0.81 
0.83 
EV [GPa] 
338.8 
284.35 
279.15 
ER [GPa] 
336.54 
282.83 
276.25 
υmax 
0.16 
0.14 
0.14 
υmin 
0.05 
0.04 
0 
A

1.21 
1.19

1.28

β[GPa1]

0.0023 
0.0029 
0.0031

Table 3 Voigt’s shear modulus (GV), Reuss’s shear modulus (GR), voigt’s young’s modulus (EV), reuss’s Young’s modulus (ER), Maximum and minimum poisson ratios (υ), zener anisotropy factor (A), compressibility (β)
Abbreviations: a28, b30
$\beta =\frac{{C}_{11}{C}_{12}}{\Omega}$ (6)
$\Omega =({C}_{11}+{C}_{12}){C}_{11}2{C}_{12}^{2}$(7)
The calculated compressibility values are found as 0.0023 GPa^{1}, 0.0029 GPa^{1} and 0.0031 GPa^{1}, respectively, for PdNNi_{3}, SnNNi_{3} and SbNNi_{3} compounds. The calculated present values of compressibility are compatible with other theoretical data for SbNNi_{3} and PdNNi_{3} compounds.
Electronic properties
In this section, the main features of electronic properties of XNNi_{3} (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 SbNNi_{3} with other compounds, its DOS is lower at Fermi energy level. The DOS of PdNNi_{3} are above at Fermi energy level according to SbNNi_{3} compound. These all compounds exhibit metallic character in consideration of rate for impletion at Fermi energy levels. The metallic behavior of XNNi_{3} (X=Pd, Sn and Sb) compounds are mostly owing to the addition of Nid states at the Fermi level and a little addition of Pdd states for PdNNi_{3} compound. It is clearly seen that Sns state and Np state contribute at Fermi level, and this emerges to a sphybridization between metals state and Np 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 2sstates electrons for XNNi_{3} (X=Pd, Sn and Sb) compounds. The charge densities of XNNi_{3} (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 sphybridization that is also confirmed by partial density of states plots. It is easy to observe that from Figures 3 and 4, the SbNNi_{3} compound is much more covalent according to PdNNi_{3} and SnNNi_{3} compounds. The charges are more accumulated between atoms. Moreover, a high ratio of G/B is related with brittleness. Considering that SbNNi_{3} compound has the greatest value of G/B, consolidates that SbNNi_{3} compound has more covalent character than PdNNi_{3} and SnNNi_{3}.
Additionally, stability of XNNi_{3} 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 antibonding 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 pseudogaps (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 XNNi_{3} 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 SbNNi_{3} compound. It is obvious that SbNNi_{3} 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.
Materials 
Wp 
Wocc 
Wb 
Wocc/Wb 
n 
PdNNi3 
0.773 
9.453 
10.185 
0.928 
7.829 
SnNNi3 
0.481 
11.677 
12.158 
0.96 
4.162 
SbNNi3 
0.166 
13.082 
13.248 
0.987 
1.526 
Table 4 The calculated pseudogap Wp (eV), the width of occupied states Wocc (eV), bonding states Wb (eV), electron numbers at fermi levels n (Fermi) for XNNi3 compounds
Vibrational properties
The phonon dispersion curves of XNNi_{3} (X=Pd, Sn and Sb) were obtained using by the PHONOPY code.^{53} The partial atomic phonon density of states (DOS) for XNNi_{3} ternary nitrides were calculated along the high symmetry directions using a 2x2x2 super cell and given in Figure 5A5C. 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 XNNi_{3} contains 5 atoms with 15 phonon branches have 3 acoustic and 12 optical modes. For PdNNi_{3} 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 SnNNi_{3} and PdNNi_{3} 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 XNNi_{3} (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 lowlying optical phonon modes have interactions with phonon modes for SnNNi_{3} and SbNNi_{3} as the acoustic. For the phonon DOS of XNNi_{3} 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. PdNNi_{3} has upper phonon energies than SnNNi_{3} and SbNNi_{3} at Gamma point. The main distinction of the three nitrides is due to difference in the chemical bonding and masses for XNNi_{3} (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 firstprinciples 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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