Investigation of the electronic, magnetic, and thermoelectric characteristics of the transition metal-based double perovskites Ba₂XNbO₆ (X = V, Cr, Mn, and Fe) for spintronic applications

Abstract

Spin-polarized materials with high Curie temperatures are emerging aspirants for spintronics. In the present work, we conduct first-principles calculations using the WIEN2k code to examine the structural, electronic, magnetic, and thermoelectric properties of Ba₂XNbO₆ (X = V, Cr, Mn, and Fe). The thermodynamic stability is confirmed by the negative enthalpies of formation and the structural stability by the tolerance factor. The range of Curie temperatures (T_c) from 289 K to 325 K for the studied materials allows for room-temperature ferromagnetism. Inferring from their spin-polarized band structures, Ba₂VNbO₆ and Ba₂MnNbO₆ have half-metallic ferromagnetism, and Ba₂CrNbO₆ and Ba₂FeNbO₆ have ferromagnetic semiconductor behavior. The calculated exchange and crystal field energies and exchange constants reveal dominant spin–orbit coupling rather than magnetic ion clustering. The total magnetic moment analysis demonstrates that the magnetic moment ranges from 2.0µ_B to 5.00µ_B for the Fe-based compounds, reflecting the gradual addition of electrons to the 3d orbitals, and its distribution to non-magnetic sites confirms the exchange of electrons. Additionally, these compounds show great potential in thermoelectrics. Ba₂VNbO₆ reached a power factor of 8.60 W mK⁻² because of its good electrical conductivity and moderate Seebeck coefficient. These results underscore that substituting the B-site elements in Ba₂XNbO₆ (X = V, Cr, Mn, and Fe) can improve the spintronic and thermoelectric properties by modifying the electronic structure, magnetic behavior, and measurable thermoelectric power.

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1. Introduction

Spintronics is an emerging field of science that has an extensive impact on the emerging features of the electronics industry. Traditional electronic mechanisms are based on the charge of the electron. However, incorporating the spin of the electron in electronic devices gives birth to the new, quickly growing field of spintronics.¹ The benefits of spintronics over traditional electronics are their high speed, small size, low power consumption, and greater efficiency with high robustness.² Spintronic devices need spin-polarization (SP) of 100% near the Fermi level (E_F). Hence, we required magnetic materials that offer extraordinary spin-polarization around E_F.³ For this purpose, the half-metallic ferromagnetic (HMFM) materials are potential candidates showing 100% spin polarization near E_F for one spin configuration and a band gap for the other spin channel. These HMFM materials are more effective when their Curie temperature is greater than room temperature. In such compounds, asymmetry in the spin-polarized density of state (DOS) is witnessed near E_F.⁴ Spintronics has unlocked possibilities for manufacturing spin-based electronic devices, including magnetic tunnel junctions (MTJs), spin valves (SVs), and spin transistors (STs), due to the exploration of HMFM materials.⁵ Spin-based electronic devices are regulated by controlling spin-polarized currents. Spin-reliant events, including tunnel magnetoresistance (TMR) and colossal magnetoresistance (CMR), depend directly on the design and fabrication of HM materials. In spintronics, the concepts of TMR and CMR are fundamental in high-density magnetic data storage devices.⁶ In numerous types of materials, such as spinel chalcogenides, double perovskites (DPs), and transition metal oxides, scientists have used the idea of half-metallicity theoretically and experimentally. Zn-based spinel chalcogenides were predicted to be potential applicants because they respond at high temperatures. Ferromagnetic materials with frustrated ferromagnetism play a crucial role in manipulating spintronic gadgets.^7,8 Half metallic ferromagnetic materials include SnCo₂O₄, Sr₂FeMoO₆, and BaTiO₃ and have the potential to solve this puzzle.⁹ In recent years, a lot of studies have been carried out on spinel compounds owing to their superior thermoelectric (TE) and FM characteristics.¹⁰ HMFM has also been investigated in the context of the hybridization of atomic states. The HMFM response of these materials was inspected by examining the crystal field energy (CFE) and the Jahn–Teller effect (JTE).¹¹ HMFM was observed in double perovskites, which have potential for spintronics due to their stable structure, high Curie temperature (T_c), and cost-effectiveness.

Perovskite materials are categorized by the 3-D connection of their metallic cations enclosed by an octahedron of halogen anions due to their unique crystal structure.¹² The predominantly investigated single perovskites are mostly represented by the general formula ABX₃, where the A and B sides are cations and X is an anion.^13,14 A new configuration (A₂BB′X₆) has been established by doubling the atoms, where B and B′ are dissimilar cations.¹⁵ Furthermore, FM character at T_c (635 K, 725 K) is displayed by the DP with Sr₂CrYO₆ (Y = Re and Os) composition.¹⁶ To explain their TE and magnetic behaviors, an assortment of DPs has been investigated.^17,18

Several first-principles investigations have explored the magnetic and thermoelectric properties of double perovskite oxides using density functional theory (DFT)-based approaches. Fang et al. investigated Sr₂FeMO₆ (M = Mo, W, and Re), studied the role of 5d electrons in controlling their magnetoresistance characteristics, and noticed their Tc values in the range of 410 K to 419 K.¹⁹ In the stable ferromagnetic phase, the magnetic moments were computed to be 0.39/0.22/−0.86µ_B for Mo/W/Re, respectively. The change of magnetic moments (MMs) to non-magnetic (NM) nodes confirmed the spin of electrons, which allowed their customizable use in quantum gadgets.²⁰ For instance, through the use of the FP-LAPW approach, Sahnoun et al. found that Ba₂FeMoO₆ has a half-metallic ferromagnetic state where the magnetic moment on Fe sites is measured at 3.631µ_B but the Mo sites do not exhibit measurable magnetism. Moreover, it behaved like an n-type semiconductor up to 1200 K, and, at 200 K, its high ZT value was close to one.²¹ Similarly, applying DFT with exchange–correlation functionals, Khandy and Gupta studied Sr₂HoNbO₆ and found it to be a ferromagnetic semiconductor with a wide band gap of 3.6 eV and a total magnetic moment of 4.0µ_B, mostly contributed by Ho (µ_Ho = 3.95µ_B). The analysis of their thermoelectric properties indicated that these have p-type conductivity, with a peak ZT value of about 0.97 at room temperature.²² Similarly, Blaha et al. used FP-LAPW within the GGA and mBJ methods to study Sr₂TiCoO₆, showing it had magnetic behavior with 1µ_B, was both strong and brittle, and its power factor increased until 1000 K.²³ Agouri et al. carried out a study on Sr₂CeVO₆ and Ba₂CeVO₆ that identified them as stable cubic magnets and found they have half-metallic characteristics, are ductile, and have high power factors due to their n-type conductivity mostly through the different oxidation states of vanadium.²⁴ Mahmood employed the WIEN2k and BoltzTraP codes based on DFT to explore the physical characteristics of Ba₂XMoO₆ (X = Cr, Mn, Fe, and Co) which exhibited their half metallic ferromagnetism with a spin-polarization of 100%, high Curie temperatures (340–313 K), magnetic moments ranging from 3.0 to 5.0µ_B, and negative formation energies, indicating their thermodynamic stability and excellent prospective for thermoelectric and spintronic devices.²⁵ In a similar context, Caid et al. used the FP-LAPW approach in the GGA and GGA + U frameworks to examine the electronic and magnetic behavior of Ba₂ZrFeO₆. Their calculations revealed a transition from metallic to half-metallic ferrimagnetism upon including the Hubbard U term, yielding a direct band gap of 1.98 eV (up spin) and total magnetic moment of −4.0µ_B, thereby reinforcing the material's suitability for magneto-electronic applications.²⁶ Furthermore, Ishfaq et al. applied DFT with the PBE-GGA functional using WIEN2k and BoltzTraP to investigate A₂HfNiO₆ (A = Ba, Ca, and Sr), which demonstrated their half-metallic semiconducting nature with substantial band gaps of 3.70, 2.91, and 2.42 eV, respectively, along with total magnetic moments of approximately 4.0µ_B, suggesting robust magnetic ordering conducive to spintronics and thermoelectrics.²⁷ Additionally, Khandy and Gupta carried out a comprehensive study on Ba₂XNbO₆ (X = Ho and Yb) using the TB-mBJ potential and GGA and GGA + U, alongside density functional perturbation theory (DFPT) for phonon dispersion analysis. Their findings confirmed the material's half-metallicity, with spin moments of 4 and 1µ_B, dynamic stability, and high Seebeck coefficients (−347.65 µV K⁻¹ for Ho and −261.11 µV K⁻¹ for Yb), thus emphasizing their dual applicability in the spintronic and thermoelectric domains.²⁸ Cherif et al. investigated the Ba₂FeWO₆ double perovskite and reported that it exhibits an indirect band gap of 0.91 eV along the L–X direction, along with a net magnetic moment of 4µ_B. These findings, combined with its low brittleness and favorable elastic properties, suggest its potential suitability for low-temperature thermoelectric applications.²⁹ Similarly, Ullah et al. carried out a DFT study on Ba₂EuMoO₆, revealing it as a thermodynamically stable ferromagnet in the cubic phase characterized by a half-metallic nature, total magnetic moment of about 7µ_B, and a direct down-spin band gap of 2.17 eV. Importantly, its high n-type thermoelectric performance was reflected in a ZT value of 0.91, indicating promising applications in thermoelectric energy conversion.³⁰ In another study, Ali et al. explored Ba₂GdXO₆ (X = Nb and U) using both the GGA + U and HSE06 functionals and found Ba₂GdNbO₆ to be a direct band gap semiconductor (E_g ranging from 2.3 to 5.38 eV), while Ba₂GdUO₆ displayed half-metallic behavior. Both compounds exhibited ferromagnetic ground states with total magnetic moments of 7µ_B and 8µ_B, respectively. Notably, Ba₂GdUO₆ demonstrated superior thermoelectric performance due to its higher power factor and ZT.³¹ Furthermore, Nadeem et al. investigated the series Sr₂NbXO₆ (X = Mn, Fe, Co, and Ni) and reported that all compositions are half-metallic ferromagnets, with total magnetic moments of 6, 5, 4, and 2µ_B, respectively. These materials also displayed significant thermoelectric performances, indicating their potential use in both spintronic and energy conversion applications.³² A theoretical investigation of 4d-series transition metal doped graphene has been executed, in which the hybridization, magnetic moments, and Curie temperatures are explained thoroughly. The Nb-, Mo-, and T_c-doped graphene have Curie temperatures of 83, 354, and 36 K, respectively, with a doping concentration of 3.125%.³³ Similarly, Mn doping with a concentration of 5.56% has a Curie temperature of 735 K, which increases rapidly when increasing the doping to 12.5% to a value of 1120 K.³⁴

To the best of our knowledge, no theoretical investigation has been dedicated to exploring the various physical aspects of Ba₂XNb₂O₆ (X = V, Cr, Mn, and Fe) double perovskites. Therefore, the current study aims to theoretically investigate the structural, electronic, magnetic, and transport properties of Ba₂XNb₂O₆ (X = V, Cr, Mn, and Fe) DPs to examine their potential for spintronics and spintronic devices.

2. Computational methods

In the current research work, the various physical characteristics of the electronic and magnetic responses of DPs Ba₂XNb₂O₆ (X = V, Cr, Mn, and Fe) were explored using spin-polarized DFT.^35,36 The full-potential linearized augmented plane wave plus local orbital (FP-LAPW + lo) technique was used to compute the structural and energetic characteristics using the WIEN2k software.³⁷ By applying the PBEsol approximation, the structural optimization to pin the ground state energy was performed.³⁸ Since the TB-mBJ potential is an effective functional to more precisely compute the electronic band structure (BS), it has been utilized to calculate the electronic band structure.^39,40 In the present study, the Perdew–Burke–Ernzerhof (PBE) exchange–correlation function was used for structural optimization, yielding reliable estimates of lattice parameters, bulk moduli, and ground state energies. For electronic structure analysis and the evaluation of key physical properties, the modified Becke–Johnson potential (mBJ) was utilized. The mBJ approach offers a favorable balance between computational efficiency and accuracy, particularly when compared to more resource-intensive methods such as generalized gradient approximation (GGA) with onsite Coulomb correction (GGA + U) and hybrid functionals like HSE06. Notably, mBJ delivers band gap predictions and electronic state distributions that exhibit excellent agreement with experimental data, especially for semiconductor systems and transition-metal-based compounds. While GGA + U is more appropriate for systems containing localized f-electrons, such as rare earth or 5d transition elements, and HSE06 provides improved accuracy for a wide range of materials, these methods often require significantly higher computational resources and present greater complexity in practical implementation. A comparative assessment reveals that mBJ not only accelerates the convergence of electronic calculations but also simplifies the computational workflow, thereby offering a pragmatic and robust alternative for accurate electronic structure predictions, especially in studies prioritizing efficiency without compromising predictive reliability. The TB-mBJ is a versatile potential that gives accurate band gaps for transition metal-based calculations such as the GGA + U and HSE06 potentials. Zaari et al. compared the results of the magnetic moment and band gap of CdFe₂O₄ by calculating them with the TB-mBJ and GGA + U potentials. Band gaps of 1.88 eV and 1.56 eV are reported for TB-mBJ and GGA + U, respectively. The total magnetic moments of 4.17µ_B and 4.18µ_B were calculated by GGA + U and TB-mBJ and are very close to each other. The advantage of GGA + U for heavy elements is that it includes d and f states more accurately in the calculation system. The final analysis of Zaari et al. was that, after including some advantages and deficiencies of these two potentials, they are similar in accuracy.⁴¹ Furthermore, Li et al. compared the accuracy and calculation errors of the band gaps for many and varied materials, such as transition metal-based systems and mixed and doped materials. They report that the band gaps had 25% error for both the HSE06 and TB-mBJ potentials. The positive point of the TB-mBJ potential is its lower computational cost.⁴²

During optimization, as an input parameter, the values for maximum wave vector (K_max) and muffin-tin radius (R_MT) were set as R_MT × K_max = 8. The values of R_MT for V, Cr, Mn, and Fe were fixed as 1.85, 2.1, 2.5, and 2.37 a.u, respectively. The cut-off energy criteria, Gaussian factor, and angular momentum vector were set as −6.0 Ry, 18, and 10, respectively. Since the size of the k-mesh is crucial for computation, for the present calculation, 1000 k-points were selected for the finest energy convergence. Finally, for the calculation of transport parameters, the Maxwell–Boltzmann theory-based BoltzTrap code was used with a relaxation time constant of 10⁻¹⁴ s.⁴³ In our calculations, a constant relaxation time of τ = 10⁻¹⁴ s has been used based on the Boltzmann transport framework. We acknowledge that this approximation neglects the dependence of τ on carrier type, temperature, and scattering mechanisms, which reduces the accuracy of calculations. This limitation is mentioned in the revised manuscript. In reality, electrons and holes have different relaxation times, which are calculated by deformation potential theory. This is computationally very expensive and beyond the scope of the present study and our expertise. Therefore, we will clearly mention in the revised manuscript that the calculated transport coefficients give semi-quantitative trends rather than absolute values. The carrier's concentration (∼10¹⁹ cm⁻³) is not fixed as an intrinsic property; it varies with chemical potential (µ). However, for smooth calculations by the BoltzTraP code, the carrier's concentration (∼10¹⁹ cm⁻³) was adjusted at zero chemical potential, where it remains almost constant during the calculations. This is also the limitation of classical Boltzmann theory-based analysis. Finally, it is concluded that the Boltzmann theory-based BoltzTraP code only provides accurate trends of thermoelectric coefficients instead of their absolute values.

We kept the supercell size at 2 × 2 × 2, number of atoms at 80, and the timesteps at 1 fs with a total simulation time of 20 ps and used the Langevin (NPT) thermostat for the AIMD calculations. For the input of the ShengBTE code, the third-order interatomic force constants were calculated by Esfarjani et al. using their methodology.⁴⁴ In the calculations of the third-order IFCs, we replaced two atoms up to the 5th nearest neighbors. There was a minute deviation in lattice thermal conductivities at 300 K. For the 5th nearest neighbor displacement consideration, we obtained well-converged lattice thermal conductivities with a K-mesh order of 12 × 12 × 12.

3. Results and discussion

3.1. Structural properties

Fig. 1 illustrates the ball-and-stick (left) and polyhedral (right) models of the unit cell of the investigated Ba₂XNbO₆ (X = V, Cr, Mn, and Fe) in the cubic phase. In this unit cell, red, gold, green, and purple-colored balls designate the Ba, X, Nb, and O atoms, respectively. These DPs belong to the Fm3̄m (225) space group. The basic cubic structure of these DPs is a framework in which octahedra share their corners with a monovalent Ba atom placed at the center of the octahedral void. Twelve O atoms are positioned in the surroundings of each Ba atom, while six O atoms are placed in the surroundings of each X and Nb atom. It can be observed from Fig. 1 that atoms of X and Nb are placed at each octahedron's center. The positions of Wyckoff sites (0.25, 0.25, 0.25), (0, 0, 0), (0.5, 0.5, 0.5), and (x, 0, 0)⁴⁵ are engaged by the Ba, X, Nb, and O atoms, respectively.

Fig. 1 Ball-and-stick and polyhedral models of the double perovskites Ba₂XNbO₆ (X = V, Cr, Mn, and Fe).

The volume optimization procedure was performed under PBE-GGA⁴⁶ to acquire the structural parameters, including the lattice constant (a), bulk modulus (B), ground state energy (E_g), enthalpy of formation (ΔH_f), and Curie temperature (T_c). The data of volume versus energy was fixed with Birch–Murnaghan's equation of state⁴⁷ to find the ground state energy value, volume, and bulk modulus. The Curie temperature was calculated by the mean field approximation of the Heisenberg model, which has an empirical formula T_c = 2ΔE/3xK_B. where ΔE is the difference in the ground state energies of the FM and AFM states. The ‘x’ represents the concentration of magnetic ions in the formula unit, and K_B represents the Boltzmann constant. The calculated values of the Curie temperature are mentioned in Table 1. Fig. 2(a–d) shows volume optimization graphs in the ferromagnetic (FM) and anti-ferromagnetic (AFM) phases which exhibit that all these compositions are more stable. Energy differences of 0.102 eV, 0.201 eV, 0.1156 eV, and 0.129 eV were calculated for the V-, Cr-, Mn-, and Fe-doped double perovskite oxides. This is because more energy is released during the formation of the FM phase than the AFM phase, highlighting that FM states are more stable and have potential for practical uses. Furthermore, it is noted that the Fe-based composition is more stable in its FM phase because more energy is released when X is replaced by Fe instead of V, Cr, and Mn.

Table 1 Parameters a, B, T_c, and ΔH_f for Ba₂XNbO₆ (X = V, Cr, Mn, and Fe)

Compound	a (Å)	Literature a (Å)	B (GPa)	ΔHf (eV)	Tc (K)
Ba2VNbO6	8.26	8.11 (ref. 46)	151	−3.11	325
Ba2CrNbO6	8.24	—	162	−3.01	305
Ba2MnNbO6	8.16	8.12 (ref. 47)	165	−2.93	302
Ba2FeNbO6	8.14	8.11 (ref. 47)	169	−2.89	289

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Fig. 2 Relationships between the volume and total energy of Ba₂XNbO₆, where X = (a) V, (b) Cr, (c) Mn, and (d) Fe.

Table 1 lists the computed values of the lattice constant (a), bulk moduli (B), and formation energy (ΔH_f) from the optimized structures in the FM state. The Curie temperature is the key parameter that leads to ensuring the practical implementation of any material that has been computed using the Heisenberg formalism.⁴⁸ The computed values of the lattice constant are noted as 8.26 Å, 8.24 Å, 8.16 Å, and 8.14 Å with corresponding T_c values of 325 K, 305 K, 302 K, and 289 K for Ba₂VNbO₆, Ba₂CrNbO₆, Ba₂MnNbO₆, and Ba₂FeNbO₆, respectively. The value of B increases as the value of the lattice constant decreases because the expansion of the unit cell makes it less hard.⁴⁹ The computed values of B are 151 GPa for Ba₂VNbO₆, 162 GPa for Ba₂CrNbO₆, 165 GPa for Ba₂MnNbO₆, and 169 GPa for Ba₂FeNbO₆. The thermodynamic viability of newly engineered perovskite compounds can be rigorously assessed through the evaluation of their formation enthalpy (ΔH_f). A negative value of ΔH_f is indicative of intrinsic thermodynamic stability, signifying that the material is energetically favorable to form under standard conditions.⁵⁰ For the investigated double perovskites, the calculated formation energies were found to be −3.11 eV for Ba₂VNbO₆, −3.01 eV for Ba₂CrNbO₆, −2.93 for Ba₂MnNbO₆, and −2.89 eV for Ba₂FeNbO₆. These results confirm the thermodynamic stability of all four compositions, with a slight decrement in stability observed across the series from vanadium to iron. The formation enthalpy was computed using the following expression:

ΔH_f = E_total(Ba_aX_bNb_cO_d) − aE_Ba ± bE_X ± cE_Nb ± dE_O, (1)

where E_total denotes the total energy of the compound, and E_Ba, E_X, E_Nb, and E_O represent the reference energies of individual elements Ba, X, Nb, and O in their standard states, respectively. The computed ΔH_f values, summarized in Table 1, consistently exhibit negative magnitudes, reinforcing the thermodynamic favorability of the synthesized perovskites. Moreover, the observed trend in formation energy suggests a progressive decrease in thermodynamic stability as the B-site transition metal shifts from V to Fe. Furthermore, the magnetic properties of double perovskite oxides are fundamentally significant for the development of next-generation spintronic devices. A pivotal parameter in evaluating their functional applicability is the Curie temperature (T_c), which delineates the thermal boundary between the ferromagnetic and paramagnetic phases.⁵¹ For spintronic components to operate reliably at or above ambient conditions, maintaining robust spin polarization necessitates materials with a T_c exceeding room temperature. In this context, the double perovskite compounds Ba₂VNbO₆, Ba₂CrNbO₆, Ba₂MnNbO₆, and Ba₂FeNbO₆ exhibit promising characteristics, with experimentally determined Curie temperatures of approximately 325, 305, 302, and 289 K, respectively. These values, ascertained through theoretical modeling based on the Heisenberg exchange interaction framework, underscore the thermal resilience of these systems. The relatively high T_c values facilitate efficient spin injection and retention under operational conditions, thereby supporting their integration into key spintronic architectures such as magnetic tunnel junctions and spin-effect transistors. The thermal stabilities of their ferromagnetic order position these materials as viable candidates for practical spin-based electronics.⁵²

Along with thermodynamic stability, the thermal stability of the lattice is also extremely important. Here, therefore, AIMD calculations were performed to ensure thermal stability. For this purpose, the total energy (eV) was calculated again in every simulation (fs), as shown in Fig. 3(a–c). It is obvious from Fig. 3(a–c) that the energy variation is negligible for the simulation time and remains almost constant for the 300, 500, and 700 K temperatures. In addition to the above, there is no bond breaking or other phases produced, which also ensures the thermal stability of the studied materials.

Fig. 3 Ab initio molecular dynamics calculations for the simulation time and total energy of Ba₂XNbO₆, where X = (a) V, (b) Cr, (c) Mn, and (d) Fe at 300, 500, and 700 K.

3.2. Electronic properties

Electronic band structure plays a key role in optimizing the electrical and optical behaviors of any material, and Fig. 4(a–d) depicts the electronic band structures of the DP materials. Fig. 4(a–d) represents the spin-polarized (SP) band diagrams for the up- and down-spin states of the Ba₂VNbO₆, Ba₂CrNbO₆, Ba₂MnNbO₆, and Ba₂FeNbO₆ compositions, respectively.

Fig. 4 Spin-polarized electronic band structures of the double perovskite materials Ba₂XNbO₆, where X = (a) V, (b) Cr, (c) Mn, and (d) Fe.

The pink and green colors depict the electronic BS in the up- and down-spin states, respectively. The Ba₂VNbO₆ and Ba₂MnNbO₆ compositions exhibited a half-metallic ferromagnetic (HMFM) nature because the spin-up states of both compositions depict a metallic nature with bands crossing the Fermi level (E_F), and the down-spin states of both compositions show a bandgap, revealing a semiconducting nature.⁵³ It was found that all the compositions have an indirect band gap nature. The Ba₂CrNbO₆ and Ba₂FeNbO₆ compositions show a ferromagnetic semiconducting (FMSC) nature.⁵⁴ The spins of their electrons are aligned in the same direction throughout the material; therefore, these materials preserve ferromagnetic ordering. For a detailed view of the individual contributions of the states to ferromagnetism, the density of states of the studied materials are plotted in Fig. 5(a–d).

Fig. 5 TDOD and PDOS graphs for (a) Ba₂VNbO₆, (b) Ba₂CrNbO₆, (c) Ba₂MnNbO₆, and (d) Ba₂FeNbO₆.

Fig. 5(a) depicts the total (TDOS) and partial (PDOS) density of states of Ba₂VNbO₆. The TDOS plot illustrates the total dispersion of electronic states from all atoms and orbitals in the Ba₂VNbO₆ composition. The energy levels spread from −6 eV to +6 eV and enclose mutually occupied and unoccupied states compared to E_F. The vertical dotted line in the graph represents the E_F. The completely occupied states are situated below the E_F, while the unoccupied states are located above the E_F. The peaks in the graph designate areas with greater electronic state accessibility, and low DOS areas are shown by valleys. The PDOS plot for V indicates that the density of the 3d-t_2g states is a maximum at E_F. These states are responsible for the metallic nature of Ba₂VNbO₆ in the up-spin configuration, whereas in the down-spin channel, these states lie deep in the conduction band, forming the band edges for this composition, and thus are crucial for controlling the conductivity of this composition. The density of the 3d-e_g states is small compared to 3d-t_2g and lies in the conduction band in the range of 2.5 to 3.5 eV. Looking at the PDOS plots of 6s-Ba and 2p-O, it is noted that the DOS of 2p-O is larger and particularly significant in the valence band (VB) in both spin configurations in the range of −1.0 to −5.5 eV, and these states are mostly localized in energy. Furthermore, a small peak close to −4 eV due to the 2p orbitals of the O atom is also observed in the graph, indicating that the O atoms are assisting the bonding state at this energy range. The 2p-O states play an auxiliary role compared to the transition metals in determining the total electronic characteristics which is revealed by the less pronounced peak of O compared to the Ba contribution. To see the contribution of the Nb atoms to the band formation, its PDOS plot is presented in Fig. 5, where it is witnessed that the t_2g states of Nb have a high density of states in the conduction band around 4 eV, whereas a small contribution to the valence band edge formation is witnessed. In contrast, a significant contribution is observed in the VB in the energy range of −4.8 to −5.7 eV.

Fig. 5(b) is divided into four parts comprising the TDOS plots of Ba₂CrNbO₆ and the PDOS plots of Cr, Ba, O, and Nb, respectively, in both spin channels. The ferromagnetic semiconducting nature of this composition is again confirmed from this figure. In the up-spin configuration, t_2g-Cr states form the VB edge, whereas the CB edge is made by the e_g states of Cr. However, the density of states of t_2g in both spin channels is high compared to that of the e_g states. Similarly, in the case of Nb, the t_2g states are closer to E_F in the VB as compared to the t_2g states. In the case of O, the energy states are spread in the energy range of 0–5.5 eV in both spin channels. The existence of the energy states of O and transition metals like Cr and V in the same energy range affirms the formation of CrO₆ and NbO₆ octahedra in the composition Ba₂CrNbO₆.

Fig. 5(c) represents the computed TDOS plus the PDOS of the individual atoms of Ba₂MnNbO₆. The calculated TDOS indicates a half-metallic ferromagnetic character, confirmed through the calculated DOS of the up- and down-spin configurations. This is because, in up spin, the DOS crosses the E_F, revealing the metallic nature, whereas, in the down-spin configuration, the energy states appear away from the E_F, indicating an insulating nature. From the PDOS, we see the individual contributions of atoms to the DOS. The 3d state of Mn is split into 3d-t_2g and 3d-e_g states. It is clear from the plot that the 3d-e_g states of Mn cross the Fermi level and are responsible for the metallic behavior of this composition. The 2p-O states also cross the Fermi level in the same manner as the 3d-Mn states, indicating the bonding interaction between Mn and O in the formation of MnO₆ octahedra. It is noted from the graph that when Cr is replaced with Mn, the 6s state of Ba and the 2p state of O show similar behaviors in the valence band. Moreover, the DOS due to the 4d-state and its substates 4d-t_2g and 4d-e_g states of the Nb atom were also examined. In the CB, the contribution of the 4d-t_2g states of Nb is more significant than that of the e_g-Nb states.

The graph of the TDOS and PDOS for Ba₂FeNbO₆ is depicted in Fig. 5(d). It is observed that when the Cr is replaced with Fe, the DOS is moved toward the E_F in both the valence and conduction bands, confirming that the studied materials have a ferromagnetic semiconducting nature. The Fe has an extra electron compared to the Cr that is responsible for better exchange splitting in the Fe-based composition. Furthermore, the contributions to the DOS for the spin up and down states from the sub-orbitals of the Fe atom are shown, revealing that the 3d-e_g orbital of Fe has a major contribution in the valence band from 0 to −5.5 eV, but it shows a minor contribution in both spin channels in the conduction band. On the other hand, the 3d-t_2g orbital shows sharp peaks in the conduction band while showing a slight contribution in the valence band in the energy range of approximately 0 to −3 eV. Furthermore, it is observed that the 6s of Ba has a weaker contribution to the valence band compared to the O atoms, while, in the conduction band, it exhibits a smaller contribution. In addition, the contribution from the 4d-e_g and 4d-t_2g orbitals of Nb is examined, and the 4d-e_g orbital extensively participates in the valence band starting from E_F to −5 eV, as well as showing sharp peaks in both spin channels. This might be due to the strong hybridization of the 4d-e_g orbital of Nb with the 2p orbitals of O.⁵⁵ In the DPs, the odd and even numbers of electrons of transition metal d-states decide the half-metallic and magnetic semiconducting behaviors because they reveal the filling of the t_2g and e_g states due to the exchange splitting process. The odd number of electrons in V (d³) and Mn (d⁵) partially fills one channel to endow insulating characteristics, and the other channel is filled unevenly, which leaves the electrons in bands that cross the Fermi level. Therefore, it produces metallic conduction on one channel (spin up) and insulating behavior on the other channel (spin down). In contrast, Cr (d⁴) and Fe (d⁶) have an even number of electrons, which fill the t_2g and e_g states symmetrically with fully occupied bands. This fact creates band gaps in both spin channels, ensuring magnetic semiconducting behavior. Therefore, an odd number of electrons makes one channel metallic, while an even number of electrons makes it filled, which supports semiconducting behavior.

3.3. Magnetic properties

The magnetic characteristics of any material are quantified in terms of exchange energy and constants, along with their magnetic moments. Under the effects of the high spin orientation, the octahedral assembly of XO₆ (X = V, Cr, Mn, and Fe) ions split the 4d-states of V, Cr, Mn, and Fe to degenerate triplet and doublet states, where the degenerate triplet states are represented by d_xy, d_yz, and d_zx and the doublet degenerate states are designated by d_x²−y² and d_z², respectively. The high spin effect further divides this degeneracy into low and high-energy states, where low-energy states represent the triplet states and high-energy states signify the doublet states. A crystal field is generated due to this splitting of states, which is calculated through the t_2g and e_g states. The difference between the t_2g and e_g states gives the crystal field splitting energy (ΔCF) for the corresponding spin channels. The following formulas are used to calculate the crystal field energy:

Δ^↑_CF = t^↑_2g − e^↑_g and Δ^↓_CF = t^↓_2g − e^↓_g, (2)

where Δ^↑_CF and Δ^↓_CF represent the crystal field splitting energies for the up- and down-spin configurations, respectively. The ΔE_CF value is inferred from the splitting of the triplet and doublet states in the same spin presentation.⁵⁶ Table 2 shows the values of ΔE_CF for various compositions, and these values were found to be −2.76, −3.38, −1.51, and 0.00 for Ba₂VNbO₆, Ba₂CrNbO₆, Ba₂MnNbO₆, and Ba₂FeNbO₆, respectively, where the negative sign reveals the stabilizing nature of the ligand field on the collective energy of the d-orbitals. The reason behind the negative sign is that ligands give electrons to the central metal ion; due to this, the collective system's energy becomes lower, resulting in the negative value of the ΔE_CF.⁵⁷ The d-orbital of the Cr-based composition is more stable than those of V, Mn, and Fe, as shown by the corresponding values of ΔE_CF due to their strong crystal field splitting. Moreover, it is noted that the value of ΔE_CF is zero in the case of the Fe-based composition, which reveals that the effect of the ligand field on the d-orbitals of Fe is zero, resulting in no crystal field splitting. Furthermore, the values of the direct Δ_x(d) and indirect Δ_x(pd) exchange energies, which arise due to the exchange splitting of d-orbitals and exchange splitting due to pd-hybridization, are tabulated in Table 2. The higher value of the Δ_x(d) highlights exceptional magnetic performance due to stronger exchange interaction in d-orbitals, and the highest is found in the Cr-based composition with a value of −4.10 as compared to V, Mn, and Fe, whose values are −3.05, −2.79, and 0.00, respectively. The larger value of the Δ_x(pd) reveals stronger pd-hybridization, which contributes to the magnetic interactions and is observed in the Mn-based composition with a value of −1.35. The CB edge splitting ΔE_C and VB edge splitting ΔE_V help to calculate the exchange constants N_oα and N_oβ using the following expressions:where x denotes the magnetic ion concentration and 〈S〉 represents the average magnetic moments of the magnetic ions. The strength of magnetization is measured using constant exchange values.⁵⁸ Generally, positive N_oα values and negative N_oβ values show half-metallic ferromagnetism of materials due to the dominance of s–d coupling in the CB and p–d coupling in the VB. However, the quantum confinement effect makes N_oα negative due to the negative value of ΔE_C.⁵⁹ The calculated values of N_oα and N_oβ are given in Table 2. The down-spin configuration promotes a double exchange mechanism to endow ferromagnetic character,⁶⁰ which is verified by the negative N_oβ values. To determine the nature of the magnetism, the total and partial magnetic moments were calculated for each composition and are presented in Table 2. It is observed that inside the V, Cr, Mn, and Fe-based compositions, the total magnetic moments (MM) are found to be 2, 3, 4, and 5µ_B, respectively. These values align well with other reported values.^61,62 Furthermore, it is noted that the highest values of magnetism primarily come from the X atoms. The magnetic moment values that come from the Ba, Nb, and O atoms are extremely weak; this might be due to a lack of unpaired electrons.⁶³ The quantum confinement effect may be responsible for the negative MM of the O atom.

Table 2 Exchange energies and constants and magnetic moments for Ba₂XNbO₆ (X = V, Cr, Mn, and Fe) in µ_B

Parameter		Ba2VNbO6	Ba2CrNbO6	Ba2MnNbO6	Ba2FeNbO6
Exchange energy and constants	Δ(CF)	−2.76	−3.38	−1.51	0
	Δx(d)	−3.05	−4.1	−2.79	0
	Δx(pd)	−1.23	−0.29	−1.35	0
	Noα	−0.2	−0.51	−0.22	−0.42
	Noβ	−0.58	−0.14	−0.29	−0.04
Magnetic moments	MMtotal	2.0	3.0	4.0	5.0
	MMint	0.316	0.186	0.231	0.006
	MMBa	0.002	0.005	0.003	0.004
	MMNb	0.045	0.007	0.072	0.023
	MMX	1.842	2.706	3.791	4.184
	MMO	−0.033	−0.016	−0.015	−0.134

---

3.4. Thermoelectric properties

The thermoelectric features of all these compositions were computed using the BoltzTrap code, and Fig. 6(a–d) display a pictorial representation of the thermoelectric parameters in the temperature range of 100–400 K.⁶⁴ In the present investigations, we calculated the thermoelectric properties using the BoltzTrap code, with the relaxation time taken as 10⁻¹⁴ s and kept constant for the whole electronic gas system. The theory behind the computation of thermoelectric features is classical Boltzmann theory in which relaxation time (τ) is kept constant, i.e., 10⁻¹⁴ s. The carrier concentration ∼10¹⁹ was used in the calculations of the thermoelectric parameters. The computed electrical conductivity (σ) of the Ba₂XNbO₆ (X = V, Cr, Mn, and Fe) double perovskite series exhibits a monotonic, nearly linear increase with rising temperature in the range of 100–400 K, indicating enhanced carrier mobility or thermally activated transport mechanisms. To accurately evaluate the thermoelectric response, it is essential to consider the contributions from both the up- and down-spin configurations. Accordingly, key thermoelectric transport parameters, such as electrical conductivity (σ), electronic contribution to thermal conductivity (κ_e), and the Seebeck coefficient (S), were computed using spin-averaged formulations, given by eqn (4)–(6) as follows:

Fig. 6 (a) The electrical conductivity, (b) electronic part of thermal conducctivity, (c) Seebeck coefficient and (d) power factor of Ba₂XNbO₆ (X = V, Cr, Mn, Fe) in the temperature range of 100–400 K.

Fig. 6(a) shows the electrical conductivities of the studied compounds calculated against the temperature, and their range is set at 100 to 400 K. It is observed from the graph that the electrical conductivity increased with temperature for all compositions, revealing that these materials show a semiconducting character in this given range of temperature. This might be attributed to the presence of thermally activated charge carriers in all compositions.⁶⁵ At room temperature, the values of the electrical conductivities are 9.70 × 10⁵ Ωm⁻¹ for Ba₂VNbO₆, 7.90 × 10⁵ Ωm⁻¹ for Ba₂CrNbO₆, 6.20 × 10⁵ Ωm⁻¹ for Ba₂MnNbO₆, and 5.21 × 10⁵ Ωm⁻¹ for Ba₂FeNbO₆. These values indicate that the V-based composition exhibited better electronic conductivity compared to the other compositions. A graph of the thermal conductivity versus temperature is depicted in Fig. 6(b). The κ_e also increased when raising the temperature for all compounds, indicating that these materials have more thermally excited free electrons because of raised thermal vibrations.⁶⁶ The computed κ_e values are noted as 6.0, 5.30, 4.50, and 3.81 W mK⁻¹ for the V, Cr, Mn, and Fe-based compositions, respectively, at room temperature, revealing that the V-based composition shows greater κ_e than the other compositions. Furthermore, the potential gradient between the contacts of the various metals is explained by the Seebeck coefficient (S).⁶⁷ The S is measured in mV K⁻¹, as demonstrated in Fig. 6(c). At a temperature of 100 K, the values of S are 0.23 mV K⁻¹, 0.21 mV K⁻¹, 0.2 mV K⁻¹, and 0.18 mV K⁻¹ for the Fe, Mn, Cr, and V-based compositions, respectively, which reveals that the Fe-based composition possesses the highest S compared to the rest of the compositions. For all compositions, the S value decreases as temperature increases, indicating that the mobility of the charge carriers decreases due to increased electron-phonon interaction at elevated temperatures.⁶⁸ Furthermore, all the studied compositions exhibited a p-type nature, confirmed by the positive values of S. Moreover, the calculated values of S at room temperature are 0.142 mV K⁻¹, 0.120 mV K⁻¹, 0.102 mV K⁻¹, and 0.070 mV K⁻¹ for Fe, Mn, Cr, and V, respectively. In addition, the power factor (PF) was computed, as the product of σ and S², to check the performance of the studied compounds. Fig. 6(d) displays the observed values of the PF. A decreasing trend is observed from the graph of the PF. The reason behind this decrease in the PF is a decrease in S and the mobility of the charge carriers.⁶⁹ The calculated values of PF at room temperature are 8.60 W mK⁻², 7.50 W mK⁻², 6.00 W mK⁻², and 4.51 W mK⁻² for the V, Cr, Mn, and Fe-based compositions, respectively, suggesting that the V-based composition shows a higher value of PF and thus demonstrates a higher capability of conversion of heat energy to electrical energy.⁷⁰

Lattice vibrations significantly influence thermal management, operational stability, and the overall efficiency of materials employed in spintronic and thermoelectric technologies. To quantitatively evaluate this aspect, the lattice thermal conductivity κ_L of the investigated compositions was computed using the ShengBTE package, which solves the phonon Boltzmann transport equation from first principles. As shown in Fig. 7, the calculated κ_L values remain consistently low over a wide temperature range, including both ambient and elevated conditions. This pronounced suppression of phonon-mediated heat transport suggests strong anharmonic lattice interactions or phonon-scattering mechanisms intrinsic to the crystal structure. The ultralow lattice thermal conductivity underscores the thermal robustness of the materials and positions them as promising candidates for thermoelectric energy conversion and stable spintronic device applications, where efficient heat dissipation and reduced thermal conductivity are critical performance criteria.²⁸

Fig. 7 Lattice thermal conductivity (κ_L) plots of Ba₂XNbO₆ (X = V, Cr, Mn, and Fe).

4. Conclusion

In the current research, we examined the structural, electronic, magnetic, and thermoelectric properties of Ba₂XNbO₆ (X = V, Cr, Mn, and Fe) using DFT. The comparison of the energies of the AFM and FM states confirms the stability of the FM states. The formation enthalpies of the FM states were negative, confirming their thermodynamic stability. Curie temperatures decreased from 325 K for Ba₂VNbO₆ to 289 K for Ba₂FeNbO_6, which shows that magnetic interactions become weaker with more 3d-electrons present. According to their electronic structures, Ba₂VNbO₆ and Ba₂MnNbO₆ were half-metallic ferromagnets, but Ba₂CrNbO₆ and Ba₂FeNbO₆ were classified as ferromagnetic semiconductors. The total magnetic moment increased from 2.00µ_B (V) to 5.00µ_B (Fe) because more electrons were placed in the 3d orbitals. The distribution of MM to non-magnetic sites and complete spin polarizations ensures ferromagnetism by the spin of electrons instead of the accumulation of magnetic ions. The crystal field energy is overcome by the exchange of energies of 3d and 4d electrons. Furthermore, the negative values of the p–d coupling and its exchange constants lower the energy of the system to stabilize the ferromagnetism. Analysis of thermoelectric measurements revealed that Ba₂VNbO₆ has a record power factor (8.60 W mK⁻²) because of its high electrical conductivity and balanced Seebeck coefficient. Based on these findings, future studies can consider A-site or oxygen-site doping, different types of strain engineering, and heterostructure formation to boost spin polarization, achieve Curie temperatures that are higher than room temperature, and enhance thermoelectric performance, contributing to next-generation energy and spintronic devices. Ba₂VNbO₆ has the highest value of the power factor, which recommends it for thermoelectric applications.

Conflicts of interest

There is no conflict to declare.

Data availability

All data included in this paper will be made available upon request.

Acknowledgements

The authors extend their appreciation to the Deanship of Research and Graduate Studies at King Khalid University for funding this work through a Large Research Project under the grant number RGP2/355/47. The authors extend their appreciation to the Deanship of Scientific Research at Northern Border University, Arar, KSA for funding this research work through the project number “NBU-FFR-2026-674-08”.

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