Scrutinizing the stability and exploring the dependence of thermoelectric properties on band structure of 3d-3d metal-based double perovskites Ba2FeNiO6 and Ba2CoNiO6

Through the conventional DFT computation, we have designed new oxide double perovskites Ba2FeNiO6 and Ba2CoNiO6. The structural and thermodynamic stabilities are predicted by optimizing the crystal structure and evaluation of enthalpy of formation, respectively. Then by using the optimized lattice constant, we have explored the different physical properties. The GGA + mBJ electronic band-structure illustrates Ba2FeNiO6 is a half-metal with 100% spin polarization at the Fermi level. While Ba2CoNiO6 shows a ferromagnetic semiconducting nature. The change in the electronic structure when Fe is replaced by Co is explained with the help of the orbital diagram and exchange interaction. The eg-eg hybridization that happens via O-p states is strong because Fe–O–Ni and Co–O–Ni bond angles are strictly 180°. The narrow bandgaps in the semiconducting channels prompted us to analyze the applicability of these materials towards thermoelectric technology. Besides this, we have investigated the dependency of transport properties on electronic band structure. The semiconducting nature in Ba2CoNiO6 results in a significant ZT around 0.8 at room temperature makes it suitable for wasted-energy regeneration

Scrutinizing the stability and exploring the dependence of thermoelectric properties on band structure of 3d-3d metal-based double perovskites Ba 2 FeNiO 6 and Ba 2 CoNiO 6 Shabir Ahmad Mir & Dinesh C. Gupta * Through the conventional DFT computation, we have designed new oxide double perovskites Ba 2 FeNiO 6 and Ba 2 CoNiO 6 . The structural and thermodynamic stabilities are predicted by optimizing the crystal structure and evaluation of enthalpy of formation, respectively. Then by using the optimized lattice constant, we have explored the different physical properties. The GGA + mBJ electronic band-structure illustrates Ba 2 FeNiO 6 is a half-metal with 100% spin polarization at the Fermi level. While Ba 2 CoNiO 6 shows a ferromagnetic semiconducting nature. The change in the electronic structure when Fe is replaced by Co is explained with the help of the orbital diagram and exchange interaction. The e g -e g hybridization that happens via O-p states is strong because Fe-O-Ni and Co-O-Ni bond angles are strictly 180°. The narrow bandgaps in the semiconducting channels prompted us to analyze the applicability of these materials towards thermoelectric technology. Besides this, we have investigated the dependency of transport properties on electronic band structure. The semiconducting nature in Ba 2 CoNiO 6 results in a significant ZT around 0. 8

at room temperature makes it suitable for wasted-energy regeneration
With the passage of time and advancement in experimental techniques, new novel materials with promising capabilities to meet the challenges of current and futuristic technologies are being discovered. In the last year, room temperature superconductivity, and solar cells with 47.1% conversion efficiency at lab scale were successfully characterized 1,2 . Besides the experimental synthesis, density functional theory (DFT) based simulation is an efficient approximation method to find and explore new materials [3][4][5] . With the combined experimental and DFT-methods, the colossal Seebeck effect with ZT ~ 470 at around 350 K in metallic Cu 2 Se has been reported 6 . New members of series of families are being intensively designed through DFT simulations by establishing their stabilities and computation of the electronic structure along with other physical properties [7][8][9][10] . The DFT-investigation is quite helpful in predicting the nature of the materials under extreme conditions with low-cost facile methods. So, DFT examination can be considered to be the first step to define stability and predict properties of any material in a quick process before its experimental synthesis.
In the past few decades, perovskite structured materials are reported to show thought-provoking and physicsrich properties [11][12][13][14][15] . Oxide double perovskites (DPs) are the quaternary metal oxides with a chemical composition of A 2 BB′O 6 . The A-site occupants are mostly alkaline-earth metal (especially Ca, Sr and Ba) but any other metal that can exist in the + 3 or + 2-oxidation state with coordination number 12 can occupy the site. The most occurring oxidation states found for oxide double perovskites are A 2 +2 (BB′) +8 O 6 -2 and A 2 +3 (BB′) +6 O 6 -2 while A 2 +1 (BB′) +10 O 6 -2 configuration rarely occurs [16][17][18] . The B and Bʹ constituents are mostly transition/inner-transition metals. The ionic character makes these materials suitable for solid-state ionic batteries 19 . Besides this, double perovskites are known to be multifunctional showing good oxidation resistance, high Curie temperature, large spin polarization effect, high figure of merit, and other fascinating properties [20][21][22][23] . These features make double  24 . Ba 2 FeCoO 6-δ oxygen-deficient double perovskite has been reported to be a potential electrode material for super capacitive purposes 25 . Several transition metal-based double perovskites are reported to be suitable for spintronic applications. By the use of DFT computation, Ba 2 FeMnO 6 has been verified as a half-metallic ferromagnet with a high magnetic moment of 6μ B and a large thermopower 26 . A 2 MnTaO 6 (A = Sr, Ba) and Sr 2 FeCoO 6 are also ferromagnetic half-metals stable in cubic Fm-3m structure [27][28][29] . In our previous studies, we have found transition metal-based perovskites La 2 CuMnO 6 and Ba 2 CoUO 6 are ferromagnetic half-metals while Ba 2 NiUO 6 is a ferromagnetic semiconductor 30,31 .
The d-d electronic interactions in transition metal-based perovskites are strong localizing the electron orbital and spin moments 32 . The B-O-B′ bond angle in A 2 BB′O 6 features the type of exchange interaction 33 . The B-B′ interactions of double perovskites can be gradually altered by carefully manipulating the B-O-B′ bond angle to obtain the desired properties 34,35 . If both B and B′ are magnetic, the magneto-electronic structure of such configuration is governed by B-B′ interaction. However, if one of the transition-metal ions is non-magnetic (B′ say), the properties are defined by long-distance next-next-nearest neighbor B-B exchange interactions Therefore, the choice of d-atoms having different electron occupancies largely influences the magneto-electronic, transport, and mechanical properties of the DPs.

Results and discussions
The obtained results of the study and the discussion over the results are presented below under different sections. Table 1. Ground-state parameters of Ba 2 BNiO 6 double perovskites namely tolerance factor ( t ); lattice constant (a); bulk modulus (B 0 ) and its pressure derivative (B 0 ′); optimized energy of FM phase (E FM ); enthalpy of formation per atom (E Form ), cohesive energy per atom (E C ); energy difference between different magnetic phases E FM -E NM and E FM -E AFM .  17 . Ideal single perovskites are most stable in Pm-3m cubic structure given in Fig. S1 (Supplementary Information). However, the transition from single perovskite to double perovskite changes the prototype structure from Pm-3m to Fm-3m also shown in Fig. S1. The lattice constant in DPs is almost double of single perovskites. The B and B′ cations in Fm-3m structure are ordered; occupy alternative sites or even can be layer-wise ordered. However, like in the single perovskites, the size mismatch in the constituents can distort the structure. Therefore, tolerance factor (t) an empirical relation from ionic radii of constituents is widely used to predict the structure of new double perovskites 36,37 . 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, under these circumstances, BO 6 and B′O 6 octahedra distort to intensify bonding 31 . The values of the t-factor for Ba 2 BNiO 6 perovskites are presented in Table 1, the evaluated values signify both materials are stable in Fm-3m structure. To be more convinced about the stability of Ba 2 BNiO 6 (B = Fe and Co) in a cubic structure, we have carried out structural optimization. Ba 2 BNiO 6 perovskites are optimized in ferromagnetic (FM), antiferromagnetic (AFM) and non-magnetic (NM) configurations with the help of spin-polarized and non-spin polarized calculations. The Birch-Murnaghan equation is used to make fit from energy-volume data and predict the optimized parameters 38,39 .
Here, E 0 , V 0 , B 0, and B 0 ′ represent energy, volume, bulk modulus, and pressure derivative of B 0 respectively in the stress-free state. The optimization curves of the ferromagnetic phase are presented in Fig. 1. The parabolic nature of the curves authenticates the stability in a cubic structure. The optimized parameters corresponding to The spin-polarized band structure computed by employing GGA + mBJ approximation (the arrows are used to designate spin channels, ↑ for spin-up; ↓ for spin-down). www.nature.com/scientificreports/ ground-state energy along with energy differences between possible magnetic phases are presented in Table 1.
The structural parameters obtained in the present study are compatible with the already reported results of similar other double perovskites, thereby validate our results. On comparing the energy of the magnetic and non-magnetic phase mentioned in Table 1, it is evident that the ferromagnetic phase is thermodynamically most stable, it possesses the least energy. The thermodynamic stability of these compounds is further predicted by computing the formation energy (enthalpy of formation E form ). E form remarks the difference of optimized energy of the compound and energy of its constituent atoms in their standard reference states 40 . The E form for Ba 2 FeNiO 6 perovskites is obtained through the balanced chemical equation 4Ba 2 FeNiO 6 ↔ 8BaO + 2Fe 2 O 3 + 4NiO + 3O 2 , similarly for Ba 2 CoNiO 6 . The formation energy is therefore evaluated by using the following relation 41,42 .
where E Ba2BNiO6 tot is the optimized total ground state energy, and E BaO , E B2O3 , E NiO , and E O2 are total energy for BaO, Fe 2 O 3 /Co 2 O 3 , NiO, and O 2 , correspondingly. A positive E form symbolizes spontaneous and unstable character, while E form less than zero signifies material is thermodynamically stable. The cohesive energy (E C ) signifies the binding energy of the constituents of a compound. E C per atom can be determined using the relation 43,44 .
where E Ba atom , E B atom , E Ni atom , and E O atom are the bulk energies of the Ba, Fe/Co, Ni, and O-constituents, respectively. All these energies are computed by using the GGA-PBE approximation and the obtained values are given in Table 1. The obtained values of E C suggest atoms are strongly held within the materials. The negative values of E form for both compounds corroborate the thermodynamic stability 45 .
Besides the formation and cohesive energies, other thermodynamic properties like specific heat (C v ), Grüneisen parameter (γ), and Debye temperature (θ D ) are also investigated. The graphical variation of these parameters against rising temperature is specified in Fig. S2. Specific heat is the amount of energy required to raise the temperature of the material by one degree. Therefore, C v 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, C v is converted back to energy. So, the higher the heat capacity more could be energy stored, and likely the   www.nature.com/scientificreports/ material acts as an efficient regenerator. The specific heat plot of the titled materials is given in Fig. S2a. The C vvariation with temperature indicates at low-temperature C v follows T 3 law only longwave phonon are excited in this range 46 . However, towards high temperature, all the phonons are thermally excited and C v tends to Dulong limit value 3nR, R is gas constant. The discussion over variation in the γ-and θ D -parameters is presented in Supplementary Information.
Electronic and magnetic behaviour. The band structure of Ba 2 BNiO 6 double perovskites obtained by GGA is represented by Fig. S3. The GGA band structure indicates the presence of metallic character in these perovskites because the Fermi level passes through the bands. However, by GGA + mBJ the band structure changes  www.nature.com/scientificreports/ effectively. The change in band structure is obvious because GGA underestimates the exchange-correlation potential and incorporation of mBJ potential sophisticatedly improves the results 47,48 . The GGA + mBJ band profiles are presented in Fig. 2, Ba 2 CoNiO 6 is a semiconductor for both spin configurations. While Ba 2 FeNiO 6 is half-metallic, designates metallic character in the spin-up channel and displays semiconducting behavior in the spin-down channel. The half-metallic character of Ba 2 FeNiO 6 thereby suggests 100% spin polarization around the Fermi level 49 .
To further discuss the electronic properties and elucidate the band structure of the materials, we have analyzed the distribution of electrons in energy states in vicinity of Fermi level. 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. The high peaks in DOS indicate the presence of a plethora of energy states. Moreover, the atomic projected density of the states given in Fig. S4a,b, signifies energy states of interest are Fe-d, Co-d, Ni-d, and O-p. Figure 4a makes it certain that Fe-t 2g and Ni-t 2g states in the spin-up channel are filled give rise to DOS peaks beneath the Fermi level (set at 0 eV) for Ba 2 FeNiO 6 . The Fe-e g and Ni-e g states hybridize with the O-p states occupy the Fermi level. While in the spin-down channel, Ni-t 2g states are filled. But Fe-t 2g states are empty configure conduction band minima. The Fe(e g )-O(p)-Ni(e g ) states even in the spin-down channel hybridize, constitute one peak below Fermi level and one above it. Also, in the Ba 2 CoNiO 6 system, the e g -states of Co and Ni interact via O-p states. Co(e g )-O(p)-Ni(e g ) constitute two sub-bands one in the valence band and others in the conduction band. The Co-t 2g states are filled in the spin-up state and empty in the down channel. Whereas, Ni-t 2g states are filled for both channels, illustrated in Fig. 4b. None of the state's occupies the Fermi level in any of spin orientations suggesting the semi-conducting nature for Ba 2 CoNiO 6 . The bandgap values of Ba 2 BNiO 6 and similar other double perovskites are summarized in Table 2.
Origin of gap. The illustration of the driving mechanism for understanding the role of e g -states in characterizing the electronic structure is given in Fig. S5 and Figs. 5, 6. The well-known behavior of the d-states in the octahedral field is that they split into two separate degenerate sets t 2g and e g . The 3d-atoms in the Ba 2 BNiO 6 perovskites occupy alternative sites and are bonded via O-atoms. The Fe-O-Ni and Co-O-Ni bond angles are 180°. Therefore, the e g -states of Fe and Ni (similarly Co and Ni) in their respective materials are oriented linearly along the axes, thereby hybridize via O-p states symbolized in Fig. S5. However, the t 2g states of transition-metals have clover leaf type structure, don't orient directly towards the orbitals of the nearest neighbor. So, t 2g -states hybridize feebly in a lateral way with t 2g and e g -states of another transition-metal. The hybridization between and Ba 2 +2 Co +5 Ni +3 O 6 -2 . The Fe +5 , Co +5 , and Ni +3 valence electrons are most dominant in characterizing the electronic band structure. Ni 3+ (common cation in both perovskites) form a low spin state, six out of seven valence electrons fill t 2g -states. The last electron enters into hybridized Fe(e g )-Ni(e g ) and Co(e g )-Ni(e g ) bonding states in their respective materials. Concerning the electrons of other transition-metal, in Ba 2 FeNiO 6 the three unpaired electrons in Fe +5 half fill t 2g . Implying t 2g -states are filled in the spin-up channel, thus localized in the valence band. However, the single electron in hybridized bonding states of (Fe)e g -(Ni)e g partially fills them. Therefore, (Fe)e g -(Ni)e g occupy the Fermi level of the spin-up channel. The Co 5+ in Ba 2 CoNiO 6 forms a high spin state, three valence electrons out of four fill spin-up t 2g -states. While the fourth electron enters to Co(e g )-Ni(e g ) bonding states. Therefore, one electron from Co 5+ and the other from Ni 3+ half fill the hybridized (Co)e g -(Ni)e g bonding states, bring them down to the valence band and open the gap at the Fermi level of the spin-up channel. In the spin-down channel of both materials, only Ni-t 2g states are filled. While all other d-states are empty, reside in the conduction band with a gap at the Fermi level. The overall number of unpaired electrons in FeNi-and CoNiperovskites is 4 and 5, respectively. Due to which the total magnetic moment is 4μ B and 5μ B in Ba 2 FeNiO 6 and Ba 2 CoNiO 6 , respectively, following the Slater-Pauling rule 50 .
Thermoelectric properties. The thermoelectric response of Ba 2 BNiO 6 materials has been analyzed by studying the behavioral variation in transport parameters with chemical potential at different temperatures. The total conductivity (σ) and Seebeck coefficient (S) are defined with the help of two current model. According to which; σ tot = σ ↑ + σ ↓ and S tot = (S ↑ σ ↑ + S ↓ σ ↓ )/σ tot ; where arrows designate respective spin-channels [51][52][53] . The variation in total Seebeck coefficient, total conductivities along with the figure of merit (ZT) is presented in Figs. 7, 8, 9, 10. 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. So, it is natural for titled materials that spin-up and spin-down electrons have dissimilar transport behavior, which is discussed in Supplementary Information. The total Seebeck coefficient of FeNi-based double perovskite has several kinks, www.nature.com/scientificreports/ presented in Fig. 7a, nevertheless, the peak values remain low. However, CoNi-based perovskite shows a high Seebeck coefficient with main peaks centered on either side of the Fermi level, see Fig. 7b. On comparing the magnitude of S, Ba 2 CoNiO 6 shows higher thermopower than Ba 2 FeNiO6. This is because of the semiconducting nature in both spin channels of Ba 2 CoNiO 6 compared to the metallic nature in one spin channel of Ba 2 FeNiO 6 . With the increase in temperature the peak values of |S| decreases gradually. The decreasing behavior can be credited to the smearing of energy bands. The variation in the total electronic conductivity coefficient over relaxation time (σ/τ) is presented in Fig. 8a,b. The electronic conductivity variation with temperature is gentle, but with the chemical potential, it is steep. Corresponding to the forbidden gap in band structure conductivity has to be zero, therefore vanishing conductivity at 0 eV for CoNi-indicate the semiconducting nature. However, with the rise in temperature conductivity increases slightly at the Fermi level it is because of energy band smearing. On the other side, although FeNi-based perovskite is a half-metal, total conductivity depicts the metallic-like behavior.
The conductivity peaks at the Fermi level should be wholly contributed by the spin-up channel electrons. To confirm this, we have plotted variation in conductivity against the chemical potential of spin up and down channels separately, shown in Fig. S6 and Fig. S7. The chemical potential (μ) predominantly decides which carriers (electrons and holes) take part in transport phenomenon 54 . So, the transport properties, both conductivities, and Seebeck coefficient, extensively depend on the value of chemical potential. Also, from Figs. 7 and 8 it is conclusive that at a fixed temperature, a rapid variation in the electrical conductivity and Seebeck coefficient occurs with a change in chemical potential. The electronic band profile has parabolic degenerate bands in the vicinity of the Fermi level with different k-dispersion. To correlate the influence of electronic properties on the transport properties, we have plotted the Seebeck coefficient, electrical conductivity at 300 K together with volumetric DOS for Ba 2 CoNiO 6 . The comparative variation in these parameters within the vicinity of the Fermi level is presented in Fig. S8. The large Seebeck coefficient around the Fermi level can be ascribed to the vanishing density of states 55 . As we move away from the Fermi level, the DOS peaks increase result in a decrease in the Seebeck coefficient. Corresponding to every peak in DOS there is a peak in the conductivity. While zero conductivity can be seen against bandgap/pseudo-gap. The highly populated DOS regions show a low Seebeck coefficient and high conductivity. Thereby, confirming that magnitude of transport properties is directly related to the behavior of energy levels close to the Fermi level. With the increase in temperature, energy band smear and some states that were empty at T = 0 K are now filled because the electrons changeover states due to gain in thermal energy. As the band smearing happens, the Seebeck www.nature.com/scientificreports/ coefficient at high DOS regions increases while conductivity decreases. The reverse is the case for the low/zero DOS populated energies. The effect of carrier concentration on the magnitude of S-and σ/τ-is described in Supplementary Information. The electronic thermal conductivity over relaxation time (κ e /τ) presented in Fig. 9a,b also demonstrates a similar kind of behavior against chemical potential variation as shown by σ/τ. 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 a prime factor that characterizes the desirability of materials towards thermoelectric applicability. The variation in ZT with chemical potential at different temperatures is illustrated via Fig. 10a,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. Although Ba 2 FeNiO 6 has 100% spin polarization at the Fermi level, however, the thermoelectric results are poor mostly due to high thermal conductivity and low Seebeck coefficient. The overall transport properties of Ba 2 FeNiO 6 have effectively dominant character of the metallic channel. The spin split ZT provided 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 in ZT on either side of the Fermi level with prominent peaks having ZT ~ 0.8, slightly lower than the reported value for Ba 2 FeMoO 6 ~ 0.99 21 . The high value of ZT in Ba 2 CoNiO 6 can be attributed to the semiconducting band profile. But with temperature rise, the peak values drop down because of the combined effects of the decrease in Seebeck value and increase in thermal conductivity.
Computation methodology. All the calculations in the present study have been carried out with the help of the Wien2k simulation code in its full potential formalism 56 . The ground-state electron density for the perovskites is obtained with help of the Kohn-Sham (K-S) equation; wherein the exchange and correlation interaction have been estimated by well-known generalized gradient approximation under Perdew, Burke, and Ernzerhof parameterization 57 . Moreover, we have facilitated GGA by the modified Becke-Johnson (mBJ) potential to be more accurate 58 . The more detailed information regarding the parameters fitted for the present study is mentioned in Supplementary Information. The transport applicability is explored with the help of Boltztrap code 59  www.nature.com/scientificreports/ works efficiently 60,61 . The thermodynamic properties are obtained with the help of Gibbs2 package integrated with Wien2k code 62 .

Conclusion
In the presented work, the structural stability along with electronic and transport properties are investigated using the FP-LAPW scheme with GGA and GGA + mBJ approximations. The GGA + mBJ functional employed is found to produce the half-metallic charter for Ba 2 FeNiO 6 and semiconducting nature for Ba 2 CoNiO 6 . The possible strong exchange interaction occurs between the e g -states of Fe (Co) and e g -states of Ni because Fe-O-Ni and Co-O-Ni bond angles are 180°. The t 2g -orbitals of transition metals are less involved in the exchange interaction. However, the finite overlap between these orbitals can be introduced by tuning the bond angles and lattice parameters as a consequence of mismatching the cations, thereby distorting the octahedra. The resultant transport properties suggest Ba 2 FeNiO 6 conduct like that of metal while Ba 2 CoNiO 6 conductivity happens only after carries gain sufficient thermal energy. The ZT for Ba 2 CoNiO 6 turns out to be ~ 0.8 at room temperature. The 100% spin polarization at the Fermi level indicates FeNi-based perovskite can be used for spintronic applications and high ZT supports Ba 2 CoNiO 6 to be used as a thermoelectric material.

Data availability
The datasets generated by the computation and thereafter analyzed would be available from Mr. Shabir Ahmad Mir on reasonable request. www.nature.com/scientificreports/