Scrutinizing the Stability and Exploring the Dependence of Thermoelectric Properties on Band Structure of 3d Metal-Based Double Perovskites: An ab-initio Analysis

Through the conventional DFT computation, we have designed new oxide double perovskites Ba2BNiO6 (B = Fe and Co). The structural and thermodynamic stabilities are dened by optimizing the crystal energy and determination of tolerance factor and cohesive energies. Thereafter, at the optimized lattice constant, we have explored the different physical properties. The GGA+mBJ electronic band-structure depicts the semiconducting nature for Ba2CoNiO6 while half-metallic with 100% spin polarization for Ba2FeNiO6. The origin of such a diverse band prole upon changing Fe to Co is explained with the help of the orbital diagram and exchange interaction. The eg-eg interaction is strong in these perovskites compared to eg-t2g and t2g-t2g hybridization. The strong exchange interaction among eg states via O-p states happens because the B-O-Ni angle is strictly 180°. Furthermore, due to the narrow bandgaps, we have explored the transport properties to express the applicability of these materials towards thermoelectric technology. Also, herein we have investigated the dependency of transport properties on band prole. The semiconducting nature in Ba2CoNiO6 results in a high ZT~0.8 at room temperature makes it suitable for energy restoration.


Introduction
With the passage of time and advancement in experimental techniques new materials that are multifunctional and could be used in current and futuristic technologies are being discovered. The discovery of the new novel materials retaining the promising capabilities to meet the challenges for new technological applications is at new heights. In the last year, room temperature superconductivity and solar cells with 47.1% conversion e ciency at lab scale were successfully characterized [1,2]. Besides the experimental synthesis of the novel materials, density functional theory (DFT) based simulations have been veri ed to be one of the most accurate and e cient methods to nd and explore new materials [3,4,5]. With the combined experimental and DFT-methods, unusual temperature dependence with ZT~470 at around 350K has been reported [6]. New members of a 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 economic-facile methods. So, DFT examination can be considered to be the rst step to de ne stability and explore properties of any material in a quick process before its experimental synthesis.
In the present era, perovskites represent one of the noble families of materials being prestigious for the scienti c community. 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 alkali-earth metal 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 double perovskites are A 2 +2 (BB ) +8 O 6 -2 and A 2 +3 (BB ) +6 O 6 -2 but rarely occur in A 2 +1 (BB ) +10 O 6 -2 [11,12,13]. The B and B constituents are mostly transition/inner transition metals. The ionic character makes these materials suitable for solid-state ionic batteries [14]. Besides this double perovskite materials are known to be multifunctional showing good oxidation resistance, high curie temperature, large spin polarization effect, a high gure of merit, and other fascinating properties [15,16,17,18]. These features make double perovskites apposite for many advanced technologies. In this paper, we have designed two new DPs, Ba 2 BNiO 6 (M = Fe and Co), having a 3d-3d combination of transition metals Fe +5 (3d 3 , t 2g 3 e g 0 )/Co +5 (3d 4 , t 2g 3 eg 1 ) and Ni 3+ (3d 7 , t 2g 6 eg 1 ).  [19]. Ba 2 FeCoO 6δ oxygen-de cient double perovskite has been reported to be a potential electrode material for super capacitive purposes [20]. However, by the computation means, Ba 2 FeMnO 6 has been reported to show half-metallic nature with a high magnetic moment of 6μ B with a large thermopower [21]. A 2 MnTaO 6 (A = Sr, Ba) are found to be cubic half-metallic ferromagnetic materials [22,23]. 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 [24,25].
In the transition metal-based perovskites, the d-d electronic interaction is so strong leading to localize the electron orbital and spin moments [26]. The B-O-B′ bond angle features the type of exchange interaction [27]. The B-B′ interactions of double perovskites can be are gradually altered by carefully manipulating the B-O-B′ bond angle to obtain the desired properties [28,29]. If both constituents (B and B′) are magnetic, the magneto-electronic structure of such con guration is governed by B-O-B′ next neighbor B-B′ interaction. However, if one of the transition-site ions is non-magnetic (B′ say), the properties are de ned by long-distance next-next-nearest neighbor (B-B) exchange interactions Therefore, the choice of d-atoms having different electron occupancies largely affects the magneto-electronic thereby transport and mechanical properties of the DP's.

Results And Discussions
The obtained results of the study and the discussion over the results are presented below under different sections Structural and Thermodynamic Stabilities Double perovskites A 2 BB O 6 are a modi ed form of simple perovskites ABO 3 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 BO 6 and B O 6 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 Ba 2 BNiO 6 (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 t from energy-volume data and predict the optimized parameters [30,31].
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 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. The thermodynamic stability of these compounds is predicted by computing the cohesive energy (E C ).
The cohesive energy of a compound is de ned by the difference between the total cell energy calculated at the equilibrium lattice constant, and the atomic energy calculated for the fundamental state con guration 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. Speci c heat (C v ) is the amount of energy required to raise the temperature of that material by one degree. Therefore, speci c 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 speci c 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 e cient regenerator. The speci c heat plot of the titled materials is given in Fig. S2(a). The C vvariation with temperature indicates at low-temperature C v follows T 3 law only longwave phonon are excited in this range [33]. However, towards high temperature, all the phonons are thermally excited and C v tends to Dulong limit value 3nR, R is gas constant [34]. Moreover, the plot of C v , re ects there is no structural phase transition in the entire temperature range. The other thermodynamic parameters like Grüneisen parameter (γ) and Debye temperature (θ D ) are discussed in the 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 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 pro le represented in Fig. 2 Table 2. Origin of semiconducting gap The illustration of the driving mechanism for the understanding role of e g -states in the origin of such a band pro le is given in Fig 5(a, b, c). The well-known behavior of the d-states in the octahedra eld is that they split into two separate degenerate sets t 2g and e g sub-sets. The 3d-atoms in Ba 2 BNiO 6 con guration In the spin-down channel of both materials, only Ni-t 2g states are lled 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 4μ B and 5μ B , 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 Ba 2 MNiO 6 materials has been recorded by analyzing the variation of various transport parameters with chemical potential at different temperatures. The dissimilar electronic lling 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 coe cient, electronic conductivities is directly linked with the nature of the energy bands around the Fermi level. The variation in total Seebeck coe cient, total conductivities along with the gure of merit (ZT) is presented in Fig. 6 to Fig. 9. However, the electronic band pro le 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 coe cient are de ned with the help of two current model. According to which: and ; where arrows designate up and down channels [37,38,39].
The total Seebeck coe cient 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 coe cient 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, Ba 2 CoNiO 6 shows higher thermopower than Ba 2 FeNiO6. The is mostly 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 . Thereby, con rming 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-re ect 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 re ects 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 coe cients 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 lled because electrons make the transition due to a gain in thermal energy. The highly populated DOS regions have low effective mass, thereby low Seebeck coe cient and high conductivity. As the band smearing happens the effective mass increase, therefore, Seebeck coe cient 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 pro le.
The gure 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)

Computation Methodology
All the calculations in the present study have been carried out with the help of the Wein2k simulation code in its full potential formalism [40]. 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 parameterized of Perdew, Burke, and Ernzerhof [41]. The more detailed information regarding the parameters tted for the present study is mentioned in Supplementary Information. Moreover, we have facilitated GGA by the modi ed Becke-Johnson (mBJ) potential to be more accurate [42]. The transport e ciency is computed with the help of Boltztrap code [43], wherein the Boltzmann equation is solved under the approximation of constant relaxation time (τ). The relaxation time is a variable parameter, the magnitude of 'τ' undoubtedly affects the transport features. However, if the variations in 'τ' are gentle on the energy scale, then constant relaxation time approximation works e ciently [44,45]. The thermodynamic properties are obtained with the help of Gibbs2 package integrated with Wien2k code [46].

Conclusion
In the presented work, the structural stability along with electronic and transported properties are investigated using the FP-LAPW scheme with GGA and GGA+mBJ approximations. The mBJ functional employed is found to produce to half-metallic charter for Ba 2 FeNiO 6 and semiconducting nature for Ba 2 CoNiO 6 . The possible strong exchange interaction happens between the e g of B (Fe and Ni) and e gorbitals of Ni because the B-O-Ni angle is strictly 180°. The t 2g -orbitals of transition metals are less involved in the exchange interaction. However, the nite 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 heavily conduct like that of metal while Ba 2 CoNiO 6 conductivity happens only after carries gain su cient 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 support Ba 2 CoNiO 6 to be used as a thermoelectric material.

Declarations
Con ict of Interest Statement: