Ternary palladium (Pd) and platinum (Pt) oxides have attracted the focus of materials scientists because of their applications such as in fuel-cell electrocatalysts and chlor-alkali anodes . There are many known palladium compounds with oxidation state of 2+ and 4+ depends on the preparation methods. Divalent palladium compounds can be synthesised by hydroxide flux method or conventional solid-state reaction, such as Ba2PdO3 , MPd3O4 (M=Ca, Sr, and Cd) [3, 4] and Lu0.5Na0.5Pd3O3 , while compounds present in higher oxidation states can be obtained by high-pressure method, such as LnPd2O4 (Ln=La, Pr, Nd, Gd, and Y) , LaPdO3 , Zn2PdO4 , and M4PdO6 (M=Ca and Sr) .
Recently, it has been studied that the presence of alkali metals greatly facilitates the oxidation state of palladium [9–11]. However, extensive research required to explored APdO (A=alkali metals) systems, among them only NaPd3O4 was thoroughly characterised, other member still needs to be explored . For knowing the chemistry of palladium oxides, APdO systems (A=Na, K) are studied and complex oxides such as NaPd3O4 and Na2PdO3 are prepared and their structure characterisation was also carried out. Various structures of the mentioned compounds discussed through coordination chemistry of Pd in various oxidation states .
In noble metal oxides, Ax Pt3O4 is the most widely studied phase (A=Li, K, Na, Mg, Ca, Sr, Ba, Co, Ni, Zn, and Cd). In square planar coordination, platinum or palladium is synthesised with a variety of alkali or alkaline-earth metals and transition metals as counter ions. The A-site of these compounds need not to be totally occupied, but the value of x is found to be zero. Cahen and Ibers  demonstrated that NaPt3O4 is the active component in Adam’s catalyst. The bulk of this work has also concentrated on the synthesis methods, stability of these compounds. The compositions of AxPt3O4 are poorly defined with respect to both stoichiometry and the response of the structure to partial counter ion occupancies and wide variation in the size of the metal cations present .
Jorgenson  reported the preparation of NaPt3O4 by the fusion of sodium chloroplatinate with sodium carbonate. The compound obtained is a jet black powder which is reasonably stable at room temperature but decomposes slowly over a period of several months. The structure of the compound is characterised by isolated Pt-Pt chains parallel to the three cubic axes. Cahen et al.  and Waser and McClanahan  used X-ray powder diffraction techniques refined of atomic positions and space group Pm-3n of NaPd3O4 and NaPt3O4 compounds. Pd-O bond length (1.999 Å) in this compound is shorter comparing to that of the isostructural phases CaPd3O4 (2.048 Å), SrPd3O4 (2.055 Å), and CdPd3O4 (2.030 Å) containing Pd2+. The cubic coordination environment of B3O4 (B=Pd or Pt) is quite rigid and depends on the B-O bond length and oxidation state of Pd/Pt .
In this study, we present DFT studies of the structural properties such as lattice constant, ground state energy, bonding nature, mechanical, and magneto-electronic properties of the cubic palladium- and platinum-based oxides NaPd3O4 and NaPt3O4 for the possible advanced technological applications.
2 Method of Calculations
Density functional theory (DFT) [17, 18] as implemented the WIEN2k package  is utilised for this study. Structural properties are calculated by GGA-exchange correlation functional . In DFT, the exact solution of the exchange-correlation functional is a difficult task; as a result, the localisation of d/f states is usually underestimated. To overcome this problem, modified Becke–Johnson (mBJ)-exchange potential with GGA [20, 21] is used for the electronic properties. mBJ-exchange potential is extremely efficient for the treatment of highly correlated electron systems (d-orbitals) . The elastic properties are calculated using the cubic elastic package interfaces in Wien2k .The separation energy is −6.0 Ry selected to distinguish the core and valence states. The atomic spheres in FP-LAPW scheme are expanded in terms of spherical harmonics up to lmax=10, and in terms of plane waves, the cut-off value of kmax=8/RMT in the interstitial region is used. The number of k-points used in the Brillouin zone integration are 2300 for the present calculations.
3 3 Results and Discussion
3.1 Structural Properties
The experimental values of the lattice constants [16, 13], atomic coordinates, and space group Pm-3n (No. 223) of NaPd3O4 and NaPt3O4 are used to optimised these compounds (crystal structure is shown in Fig. 1). Optimisations curves, minimising the total energy with respect to unit cell volume for both compounds are presented in Figure 2. The lowest value of the energy for NaPd3O4 is −6238.4853 Ry which corresponds to the ground state volume 1215.2513 (a.u)3, while for NaPt3O4 the ground-state energy is −223197.1027 Ry and the ground-state volume is 1259.4246 (a.u)3. The fitted Birch–Murnaghan equation of state  is used to evaluate the ground-state configurations such as lattice constants, bulk moduli, and pressure derivative of bulk moduli and are presented in Table 1. It can be deduced from the table that the calculated lattice constants are differ from the experimental [16, 13, 15, 25] results by 0.028 % and 0.49 % for NaPd3O4 and NaPt3O4, respectively. These small differences between the experimental and theoretical lattice constants that are due to the electron-exchange correlation reveal the reliability of our theoretical work.
Bond lengths and bond angles play vital role in the symmetry of a crystal. Keeping in view the importance of these parameters, bond lengths and bond angles between different atoms in these crystals are calculated and listed in Table 1. The bond length for Na−O is 2.4866 Å, Na−Pd is 3.2102 Å, Pd−O is 2.0303 Å in NaPd3O4 and for Na-O is 2.5086 Å, Na−Pt is 3.2386 Å, Pt−O is 2.0482 Å in NaPt3O4, respectively. The experimental reported value of the Pd-O is 1.999 Å , while for Pt−O is 2.02 Å , these comparisons shows that our results are logical. The calculated bond angle between Pd−O−Pd and Pt−O−Pt is 120°. These data are also compared with experiments and found in good agreement.
To explain the bonding nature between different atoms in these compounds, electron charge densities in (100) and (110) planes are calculated as shown in Figure 3. It is clear from the Figure 3a and c (100) planes that covalent bond exist between Pd−Pd/ Pt-Pt due to the overlapping of the electronic clouds, while the nature of bonding between Na-Pd/ Pt are ionic. In (110) planes, Figure 3b and d, the overlapping of the orbitals at an angle of 120° between Pd/ Pt-O-Pd/ Pt shows the covalent bond, and hence, bond between Pd/ Pt-O are covalent. It can also be seen from the Figure 3b and d that metallic bond is formed between Pd/Pt and Pd/Pt atoms. The non-spherical electron cloud of Pd/Pt shows that the d-states of these elements are partially filled, whereas the spherical shape of Na shows that the valance state is completely filled by loosing electron making ionic bond.
Cohesive energy is the energy required to rip a sample apart into widely separated constitute atoms. To discuss the relative stability of NaPd3O4 and NaPt3O4, we estimated their cohesive energies (ECoh) using the following well-known formula as ;
in (1) ETotal is the net energy of NaPd3O4 and NaPt3O4 as obtained for the whole cells, whereas E(Na), E(Pd/Pt), and E(O) are the energies of Na, Pd/Pt, and O atoms and are calculated by GGA-exchange correlation functional. The calculated cohesive energies are listed in Table 1, which indicates that ECoh for NaPd3O4 is −19.49 Ry and for NaPt3O4 −62.38 Ry, (ECoh (NaPt3O4)>ECoh (NaPd3O4)). The lower value of the cohesive energy of NaPt3O4 shows that it is more stable than NaPd3O4.
3.2 Mechanical Properties
Elastic constants are vital in describing the response of an applied macroscopic stress and external forces as illustrated by Young’s modulus, shear modulus, bulk modulus, and Poisson’s ratio play central role in quantifying the mechanical strength and stability of compounds .
Elastic stiffness coefficients of NaPd3O4 and NaPt3O4 compounds are calculated with PBE-GGA as exchange-correlation approximation using the Cubic-Elast method developed by Jamal et al. , interfaced in Wien2k package. The independent elastics constants in a cubic symmetry (C11, C12, and C44) for NaPd3O4 and NaPt3O4 are listed in Table 2. To the best of our knowledge, there are no experimental neither theoretical literature available on the elastic and mechanical properties of these compounds. All the calculated elastic constants are positive and satisfy the elastic stability criteria of the cubic crystal at ambient conditions: C11−C12>0; C11 + 2C12>0; C44>0. The elastic constants of both the compounds are comparable in a small difference in details; this is due to the fact that both the compounds are lattice-matched.
The mechanical parameters, such as the bulk modulus, Young’s modulus, shear modulus, Poisson’s ratio, anisotropy factor, paugh ratio, and Chusshy pressure are calculated from the three independent elastic constants (C11, C12, and C44) listed in Table 2. These mechanical parameters are directly related to many beneficial physical properties such as fracture, internal strain, sound velocity, toughness, and thermoelastic stress. The bulk moduli of our cubic compounds increases from NaPd3O4 to NaPt3O4, shows that NaPt3O4 is harder than NaPd3O4. Bulk modulus of a material is used to measure hardness; in order to confirm it; elastic characteristics must be account as well. The calculated bulk moduli from elastic constants are nearly the same as obtained by Birch–Murnaghan equation of state (Tab. 1); this shows the accuracy of our elastic calculations. Shear modulus can provide a better relation of hardness than bulk modulus. Materials having covalent bonds generally have larger shear modulus.
Young’s modulus (Y) is an important parameter from technological applications point of view and is used to quantify the stiffness of the materials, that is, larger the value of Y, the stiffer is the material and will have covalent bonds . From Table 2, it is clear that NaPd3O4 has higher value Y than NaPt3O4 showing to be more covalent in nature.
Anisotropic factor (A) is another significant parameter, which computes the anisotropy of the elastic wave velocity in a crystal. It also gives the information about the structural stability and it is interrelated with the possibility of inducing microcracks in the materials. From the Table, it can be seen that the anisotropic factor for these compounds is less than 1, which designates that these compounds are not elastically isotropic.
The bond nature is explained in term of Cauchy pressure C″ (C″=C12 − C44). Solids having more positive Cauchy pressure tend to form metallic bond, whereas compounds having more negative Cauchy pressure forms bond which is more angular in character . Thus, the ductile nature of these compounds is related to their positive Cauchy pressures and thereby the metallic behaviour in their bonds. The higher positive Cauchy pressure of NaPt3O4 clarifies strong metallic bonding (ductility). Pugh ratio (B/G)  of these compounds is also calculated, which shows that if B/G>1.75; a material works in a ductile manner. The larger B/G of NaPt3O4 indicates that it has more ductile nature than NaPd3O4. The ratio B/C44 may be understood as a measure of plasticity . Due to the large B/C44 ratio of NaPt3O4 as listed in Table 2, it may possess lubricating properties.
3.3 Electronic Properties
To know about the electronic behaviour of these compounds, self-consistent field calculations (SCF) are performed using mBJ-GGA in order to treat the exchange correlation effect in these compounds properly. The calculated electronic band structures of NaPd3O4 and NaPt3O4 are plotted in Figure 4. It can be observed from Figure 4 that the band profiles of both compounds are almost similar to each other with a small difference in details. Figure also reveals that there is no band gap between the valence and conduction bands at the Fermi level. Both the valence and conduction bands cross each other at the Fermi level, which is the indication of the metallic nature, and hence, both materials are metals.
The origin of the electronic band structures and contribution of the different electrons energy levels in the band structure can be explained on the basis of density of states (DOS). The total and partial DOSs for both NaPd3O4 and NaPt3O4 are calculated and plotted in Figures 5–7. In total, the density of states (Fig. 5) shows almost symmetric densities for both compounds with a small difference in details. For both compounds, the densities covers the Fermi level and extended into the conduction bands hence both compounds are metallic in nature. Figure 6 shows partial density of states of NaPd3O4 in which the valence band is occupied by O-s state in the range from −19.0 to −17.5 eV the combined contribution of O-p state and Pd-d states occurs from −7.3 eV crossing the Fermi level and extended into the conduction band, the localisation of the Pd-d state occurs at −2.5 eV in the valance band. Some minor contribution Pd-p state in valence band exists at −1.4 eV. Similar results are observed for NaPt3O4 (Fig. 7) in the valence band O-s state occurs in range −20 to −19.4 eV and Na-p in the range from −21 to −18 eV. O-p and Pt-d states are overlapped from −9.0 to 6.0 eV making the material metal. Similar results are obtained for both compounds. Hence, our theoretical calculations show that both compounds NaPd3O4 and NaPt3O4 are metals and consistent with the reported experimental work .
NaPd3O4 and NaPt3O4 are present in square planar complexes due to the fact that Pd and Pt are d8 metal ions. The highest energy orbital dx2 − y2 is greatly destabilised while pairing in dxy orbital is more favorable. Figure 8 shows the d-states splitting of Pd/Pt atoms in these compounds. Figure 8 shows that the densities of dxz + yz and dxy states occur in the valance band while dz2 and dxz + yz states crosses the Fermi level makes these compounds metallic. It can also be seen that the density of dz2 and dxz + yz states at the Fermi level is greater for NaPt3O4 than NaPd3O4.
3.4 Magnetic Properties
To study the stable magnetic phase of these materials, we optimised the unit cell of each compound for non-magnetic phase (paramagnetic) and ferromagnetic phase like our previous works [34, 35]. The optimisation energies show that these compounds are stable in paramagnetic phase in which these systems lower their energy as compared to ferromagnetic phase. The energy difference between the paramagnetic and ferromagnetic phases (∇E=(Epara − EFM) are −0.005 Ry and −0.008 Ry for NaPd3O4 and NaPt3O4, respectively. Hence, our DFT calculations show that both compounds are paramagnetic. To confirm the magnetic stable phase, we also utilised the post-DFT calculations BoltzTraP code  is utilised to calculate the magnetic susceptibility versus temperature curves of these compounds are presented in Figure 9. The curves in figure can be explained on the bases of Curie Weiss law, which is a semiclassical approach to explain the magnetic properties of a material. The well-known Curie Weiss law explains the magnetic order of a material , the Weiss constant (Θ) is zero for paramagnetic (PM), positive for ferromagnetic (FM), and negative for an antiferromagnetic (AFM) compound.
Pauli paramagnetism is typically observed in metals and conductors, which are due to the conduction of electrons on the Fermi surface purely quantum mechanical effect because electrons are fermions and obey Fermi–Dirac statistics. Pauli paramagnetism is the weak magnetism and mostly temperature-independent . Figure 8 shows the magnetic susceptibility of these compounds, which varies with temperature. The magnetic susceptibility of NaPd3O4 and NaPt3O4 at room temperature is 2×10−3 emu/mol and 1.5×10−3 emu/mol, respectively. The inverse curve of the magnetic susceptibility show that the Weiss constant (Θ) is zero for both materials. Hence, both compounds are paramagnetic metals.
In summary, structural, mechanical, and magneto-electronic properties of the cubic NaPd3O4 and NaPt3O4 are investigated using DFT. The structural parameters and geometry are found consistent with the experiments. The cohesive energy reveals that NaPt3O4 is more stable than NaPd3O4. Mechanical properties show that both compounds are ductile in nature and show that NaPt3O4 is harder than NaPd3O4. Electronic cloud explain the chemical bonding between different atoms in these compounds, the bond between Na and Pd/Pt is ionic whereas between O and Pd/Pt is covalent. The electronic band profiles reveal that these materials are metallic in nature where as the d-state splitting explains the origin of the electronic nature of these compounds. The optimised magnetic energies and magnetic susceptibility confirm that these compounds are paramagnetic metals.
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About the article
Published Online: 2015-09-05
Published in Print: 2015-10-01