The obtained results of the study and the discussion over the results are presented below under different sections
Structural and Thermodynamic Stabilities
Double perovskites A2BB՛O6 are a modified form of simple perovskites ABO3 by replacing exactly half of the B-cations with different B՛-cation [12]. Ideal simple perovskites are most stable in Pm-3m cubic structure given in Fig. S1 (Supplementary Information). However, the transition from simple perovskite to double perovskite changes the prototype structure from Pm-3m to Fm-3m shown in Fig. S1. Also, the lattice constant in DPs is almost double of simple perovskites. The B and B՛ cations in Fm-3m structure are mostly ordered; occupy alternative sites or even can be layer-wise ordered. Like in the simple perovskites, the mismatch in the sizes of the constituents can distort the structure. Therefore, tolerance factor ‘t’ an empirical relation from ionic radii of constituents is most widely used to predict the structure of new double perovskites [25]. If ‘t’ is in the range of 0.9-1, the cation sizes are perfect for the ideal structure. However, when t < 0.9 and t > 1 the constituents are in an under bonded state, therefore BO6 and B՛O6 octahedra are distorted. The calculated t-factor is presented in Table 1 suggests the stability of titled perovskites in Fm-3m structure. To be more convinced about the stability of Ba2BNiO6 (M = Fe and Co) in a cubic structure, we have carried out optimization via spin-polarized and non-spin polarized calculations. The Brich-Murganian equation is used to make fit from energy-volume data and predict the optimized parameters [30,31].
Here, E0, V0, B0, and B՛0 represent energy, volume, bulk modulus, and pressure derivative of B0 respectively in the stress-free state. The optimization curves are presented in Fig. 1 and the optimized parameters are reported in Table 1. The parabolic character of the curves authenticates the stability in a cubic structure. On comparing the energy of the magnetic and non-magnetic phase it is evident that the magnetic phase is most stable.
Table 1: Optimized parameters of Ba2BNiO6 double perovskites
Material
|
Tolerance factor ( )
|
Lattice Constant
a (Å)
|
Bulk Modulus B (GPa)
|
Pressure derivative of bulk modulus
B` (GPa)
|
Total energy E (Ry)
|
Cohesive energy
EC (Ry)
|
Ba2FeNiO6
|
0.93
|
7.90
|
130.11
|
4.93
|
-38793.58
|
12.65
|
Ba2CoNiO6
|
0.95
|
7.89
|
132.11
|
4.79
|
-39289.46
|
13.32
|
Ba2FeMnO6[21]
|
0.99
|
7.97
|
148.17
|
5.05
|
|
|
The thermodynamic stability of these compounds is predicted by computing the cohesive energy (EC). The cohesive energy of a compound is defined by the difference between the total cell energy calculated at the equilibrium lattice constant, and the atomic energy calculated for the fundamental state configuration of Ba, B, Ni, and O according to the following formula [32]:
All these energies are computed by using the GGA-PBE approximation and the obtained values are given in Table 1. Positive values of magnitude 12.65 Ry and 13.52 Ry for FeNi and CoNi-perovskites support the stability of the materials. Besides cohesive energy, we have computed other thermodynamic properties and plotted their variation with temperature are shown in Fig. S2. Specific heat (Cv) is the amount of energy required to raise the temperature of that material by one degree. Therefore, specific heat represents the energy that can be stored in a material for a given temperature difference. If the temperature of that material is lowered back to the initial temperature the specific heat is converted back to energy. So, the higher the heat capacity more could be the energy store and likely material could be used as an efficient regenerator. The specific heat plot of the titled materials is given in Fig. S2(a). The Cv-variation with temperature indicates at low-temperature Cv follows T3 law only longwave phonon are excited in this range [33]. However, towards high temperature, all the phonons are thermally excited and Cv tends to Dulong limit value 3nR, R is gas constant [34]. Moreover, the plot of Cv, reflects there is no structural phase transition in the entire temperature range. The other thermodynamic parameters like Grüneisen parameter (γ) and Debye temperature (θD) are discussed in the Supplementary Information.
Electronic and Magnetic Behaviour
The band structure of Ba2BNiO6 double perovskites obtained by GGA is represented by Fig. S3. The GGA band structure indicates the metallic character of these perovskites as the Fermi level passes through the bands. However, by incorporating mBJ potential to GGA, the band structure changes effectively. The change in band structure is obvious because GGA underestimates the exchange-correlation potential and incorporation of mBJ potential sophisticatedly improves the results [34,35]. The GGA+mBJ band profile represented in Fig. 2, reveals Ba2CoNiO6 is semiconducting in both spin channels. While Ba2FeNiO6 is half-metallic, designates metallic character in the spin-up channel and poses semiconducting behavior in the spin-down channel. The half-metallic character of Ba2FeNiO6 thereby suggests 100% spin polarization around the Fermi level.
To further discuss the electronic properties and elucidate the band structure of the materials we have analyzed the distribution of energy states over an energy range of -8 eV to 6 eV. The total density of states (TDOS) obtained from GGA and GGA+mBJ methods are given in Fig. 3. With the implication of mBJ, the energy states sweep away and open the gap at the Fermi level. Moreover, the atomic projected density of the states is given by Fig. S4, the energy states of interest are Fe-d, Co-d, Ni-d, and O-p. In FeNi-based perovskites the distribution of states is as; In the spin-up channel, Fe-t2g and Ni-t2g states are filled happen below Fermi level but Fe-eg and Ni-eg state strongly hybridize with the O-p states occupy the Fermi level. While in the spin-down channel, Ni-t2g states are filled but Fe-t2g states are empty contribute towards conduction band formation as can be seen from Fig. 4. The Fe(eg)-O(p)-Ni(eg) states even in the spin-down state strongly hydride contributes one peak below fermi level while one above it. Also, in the Ba2CoNiO6 system, the eg-states {Co(eg)-O(p)-Ni(eg)} strongly hybridize via O-p states constitute two sub-bands one in valance band and others in conduction. The Co-t2g states are filled in the spin-up state and empty in the down channel, while Ni-t2g states are filled for both channels. No states are occupying the Fermi level in both spin orientations reflecting the semi-conducting type nature of Ba2CoNiO6. The bandgap values are provided in Table 2.
Table 2: Calculated band gap (Eg in eV) and Magnetic moment (μB) of Ba2MNiO6 materials
Material
|
Bandgap
|
Magnetic Moment
|
Ba2FeNiO6
|
Ba2CoNiO6
|
Ba2FeNiO6
|
Ba2CoNiO6
|
Ba2FeMnO6[21]
|
↑
|
↓
|
↑
|
↓
|
GGA
|
--
|
--
|
--
|
--
|
4.59
|
5.72
|
7.0
|
GGA+mBJ
|
--
|
1.66
|
0.11
|
1.22
|
4.0
|
5.0
|
|
Origin of semiconducting gap
The illustration of the driving mechanism for the understanding role of eg-states in the origin of such a band profile is given in Fig 5(a, b, c). The well-known behavior of the d-states in the octahedra field is that they split into two separate degenerate sets t2g and eg sub-sets. The 3d-atoms in Ba2BNiO6 configuration occupy alternative sites that are bonded via O-atoms. The B-O-Ni bond angle for the present set of materials is 1800. Therefore, the eg-states of Fe and Ni (similarly Co and Ni) in their respective materials are linearly along the axes. However, the t2g states are in the plane don’t orient directly towards the orbitals of the nearest neighbor so mostly don’t take part in the hybridization, hybridize feebly in a lateral way with t2g and eg-states of another atom. The hybridization between d-states of transition atom with O-p can also be visualized from charge density distribution shown in Fig. 5(b). The B and Ni-atoms are covalently bonded with O-atom. While Ba-O bonds show ionic bonding.
The orbital splitting and formation of bonding and anti-bonding eg-states are represented via Fig. 5 (c). The nominal valance states of the present set of double perovskites are Ba2+2B+5Ni+3O-2, maintaining the charge stability. The B+5 and Ni+3 electrons are most dominant in characterizing the electronic band structure. The six electrons out of seven in Ni3+ for both perovskites fill t2g-states forming a low spin state. The last electron of Ni3+ enters hybridized eg-eg bonding states. Now in Ba2FeNiO6, the three unpaired electrons in Fe+5 half fill t2g, i.e., t2g states are filled in spin-up channel localize them in the valance band. However, a single electron in hybridized bonding states of (Fe)eg-(Ni)eg bonding states fills them partially and make them reside at the Fermi level of the spin-up channel as can be seen from the Fig. 5(c). Likewise, Co5+ in Ba2CoNiO6 also forms a high spin state, three electrons out of four fill spin-up t2g state. While the last (fourth) electron and one from Ni3+, half fill the hybridized bonding (Co)eg-(Ni)egstates, bring them down to the valance band in the spin-up channel and open the gap at Fermi level. In the spin-down channel of both materials, only Ni-t2gstates are filled lie in the valence band, while all other d-states that are empty reside in the CB with a gap at the Fermi level. The overall number of unpaired electrons in FeNi- and CoNi-perovskites is 4 and 5, due to which the magnetic respective materials are 4μB and 5μB, following the Slater-Pauling rule [36].
Thermoelectric Properties
The transport character of the materials is directly linked to the electronic band structure. Ideally, the perfect thermoelectric material should possess a large thermopower like an insulator and low resistivity like a metal, which is almost impossible to attain practically. However, low bandgap degenerate semiconductors enjoy the sweet spot among the thermoelectric applicable materials. Therefore, such materials are on the hunt for thermoelectric technology and are expected to show a good thermoelectric response. The thermoelectric response of Ba2MNiO6 materials has been recorded by analyzing the variation of various transport parameters with chemical potential at different temperatures. The dissimilar electronic filling in spin channels of magnetic materials suggests electrons in these spin channels experience different driving forces thereby exhibit variant scattering rates. The variation in the transport parameters viz. Seebeck coefficient, electronic conductivities is directly linked with the nature of the energy bands around the Fermi level. The variation in total Seebeck coefficient, total conductivities along with the figure of merit (ZT) is presented in Fig. 6 to Fig. 9. However, the electronic band profile of spin up and spin down channels are entirely different. So, it is naturals for titled materials to exhibit dissimilar variational behavior of transport parameters in spin-up and spin-down channels discussed in Supplementary Information. The resultant conduction and Seebeck coefficient are defined with the help of two current model. According to which: and ; where arrows designate up and down channels [37,38,39]. The total Seebeck coefficient of FeNi-based double perovskite has several kinks, presented in Fig. 6(a), nevertheless, the peaks values remain low. While on the other side CoNi- shows a high Seebeck coefficient with main peaks occurring on either side of the Fermi level at 0.00 Ry refer to Fig. 6(b). With temperature rise, the peak values decline rapidly, nevertheless, the peak location remains the same. On comparing the magnitude of S, Ba2CoNiO6 shows higher thermopower than Ba2FeNiO6. The is mostly because of the semiconducting nature in both spin channels of Ba2CoNiO6 compared to the metallic nature in one spin channel of Ba2FeNiO6. Thereby, confirming that magnitude of Seebeck is directly related to the behavior of energy levels close to the Fermi level. With the increase in temperature the peak values of |S| decreases greatly. The decreasing behavior of |S| can be credited to the smearing of energy bands.
The variation in the total electronic conductivity is presented in Fig. 7 (a, b). The variation in conductivity with temperature is gentle but with the chemical potential, it is steep. Corresponding to the forbidden gap in band structure conductivity would be zero, therefore vanishing conductivity at 0 Ry of CoNi-reflect the semiconducting nature. However, with the rise in temperature conductivity increases slightly at the Fermi level it is because of the smearing of energy bands. On the other side, although the band structure of FeNi-based perovskite reflects half-metallic nature, total conductivity depicts the behavior of the conductor. The conductivity peaks at the Fermi level are wholly contributed by spin up channel, which is demonstrated by plotting conductivity against the chemical potential of spin up and down channels separately shown in Supplementary Information. As the value of chemical potential changes, the Seebeck and conductivity coefficients at some places increase with temperature, and somewhere shows a decreasing trend. The behavior is expected because of the presence of pseudo gaps and fewer states at some energy values of the band structure. Howsoever, as the temperature increases, bands smear, and some states that were empty at T = 0K are now filled because electrons make the transition due to a gain in thermal energy. The highly populated DOS regions have low effective mass, thereby low Seebeck coefficient and high conductivity. As the band smearing happens the effective mass increase, therefore, Seebeck coefficient increase while conductivity decrease. The reverse is the case for the low DOS populated regions. The electronic thermal conductivity presented in Fig. 8(a, b) also demonstrates a similar kind of behavior against chemical potential variation. However, the thermal conductivity increases abruptly with temperature compared to electric conductivity. Also, FeNi- shows higher conducting capacity compared to CoNi-perovskite. It because of the half-metallic electronic profile.
The figure of merit (ZT) is the ultimate factor that characterizes the desirability of materials towards thermoelectric applicability. The variation in ZT with chemical potential at different temperatures is illustrated via Fig. 9 (a, b). The ZT-value of FeNi-based perovskite is very low, the highest value goes around 0.013 that too away from the Fermi level at -0.12Ry. Although Ba2FeNiO6 has 100% spin polarization at the Fermi level however the thermoelectric results are poor mostly because of high thermal conductivity and low Seebeck coefficient. The overall transport properties of Ba2FeNiO6 have the effectively dominant character of the metallic channel. The spin split ZT is shown in the Supplementary Information, wherefrom it can be seen spin-down channel of FeNi-based having semiconducting nature has ZT~1. However, CoNi-based perovskite offers significant peaks on either side of the Fermi level with prominent peaks having ZT~0.8 in closeness to the Fermi level. the high value of ZT in Ba2CoNiO6 can be credited to the semiconducting band profile. However, with the temperature rise, the peak values drop down because of the combined effects of the decrease in Seebeck value and an increase in thermal conductivity.