Structural, Mechanical and Magneto-Electronic Properties of the Ternary Sodium Palladium and Platinum Oxides

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 [24] 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.

Figure 1:

Crystal structure of cubic NaPd₃O₄ and NaPt₃O₄.

Figure 2:

Optimisations curves variation in energy versus change in volume of NaPd₃O₄ and NaPt₃O₄.

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 [16], while for Pt−O is 2.02 Å [26], 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.

Figure 3:

Spin-dependent electron charge density of NaPd₃O₄ and NaPt₃O₄.

Cohesive energy is the energy required to rip a sample apart into widely separated constitute atoms. To discuss the relative stability of NaPd₃O₄ and NaPt₃O_4, we estimated their cohesive energies (E_Coh) using the following well-known formula as [27];

in (1) E_Total 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 E_Coh for NaPd₃O₄ is −19.49 Ry and for NaPt₃O₄ −62.38 Ry, (E_Coh (NaPt₃O₄)>E_Coh (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 [28].

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. [23], 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 [29]. 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 [30]. 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) [31] 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 [32]. 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.

Figure 4:

Electronic band structures of NaPd₃O₄ and NaPt₃O₄.

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 [33].

Figure 5:

Total density of states of NaPd₃O₄ and NaPt₃O₄.

Figure 6:

Partial density of states of NaPd₃O₄.

Figure 7:

Partial density of states of NaPt₃O₄.

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 d_{x2 − y2} is greatly destabilised while pairing in d_xy 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 d_{xz + yz} and d_xy states occur in the valance band while d_z2 and d_{xz + yz} states crosses the Fermi level makes these compounds metallic. It can also be seen that the density of d_z2 and d_{xz + yz} states at the Fermi level is greater for NaPt₃O₄ than NaPd₃O₄.

Figure 8:

d-state splitting of NaPd₃O₄ and NaPt₃O₄.

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=(E_para − E_FM) 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 [36] 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 [37], the Weiss constant (Θ) is zero for paramagnetic (PM), positive for ferromagnetic (FM), and negative for an antiferromagnetic (AFM) compound.

Figure 9:

Magnetic susceptibility and its inverse versus temperature of NaPd₃O₄ and NaPt₃O₄.

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 [37]. 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.