Introduction

Since the theoretical prediction of topological insulators1,2,3,4 and topological semi-metals5,6,7, researchers in the field of condensed matter and materials science have extensively endeavored to design, characterize, and synthesize topological materials3,4,6,8 considering their potential applicability in the field of quantum information9, coherent spin transport10, and high-efficiency catalysis11. Recent advances in elementary band representation theory have accelerated the prediction of insulating and (semi-)metallic non-magnetic and magnetic topological phases of matter in real materials12,13,14,15,16,17,18. However, the prediction of stable magnetic topological materials with large transport characteristics such as large anomalous Hall conductivity (AHC) and Nernst conductivity (ANC) in strongly correlated magnetic oxides is more challenging, and systematic calculations have been conducted only for a few selected strongly correlated oxides19,20,21,22,23,24,25,26,27,28.

In correlated magnetic systems, the interplay between magnetism, crystal symmetry, and spin-orbit coupling (SOC) underlies the various topological transport phenomena and is highly intriguing from an experimental perspective29,30,31. In three-dimensional (3D) ferromagnets, spin-polarized bands often display topologically non-trivial nodal points and nodal lines (NLs)32,33,34,35,36,37,38. The lifting of band degeneracy because of the SOC around a nodal structure gives rise to increased Berry curvature around it, leading to a variety of topological transport phenomena, including the anomalous Hall effect (AHE) as well as the anomalous Nernst effect (ANE)35,39,40,41.

In the case of magnetic topological materials with large AHC and ANC conductivities, it is important to have a magnetic site with a localized magnetic moment, which can provide a high energy scale for magnetism, as well as a magnetic site occupied by a heavy element with strong SOC. From the perspective of designing magnetic topological materials, it is also important to have materials with physically separated ions that host magnetism and ions that host strong SOC, to avoid the problem resulting from the interplay between the correlation effect and SOC at the same site. A promising class of materials in this context is the rocksalt-ordered42 double perovskite (DP) compounds with the general formula A2BB’O6. These compounds present the possibility of hosting a 3d transition metal (TM) ion at the B site and 4d/5d TM ions at the B’ site, offering a suitable playground for exploring exotic properties that depend on the competing energy scales associated with the bandwidth, Coulomb correlation, and SOC for the 3d and 4d/5d orbitals at these two sites. From an experimental point of view, these compounds are easy to produce, are robustly stable against oxidation in ambient air, and possess a high magnetic ordering temperature. However, because most conventional topological materials have low transition temperatures, their application to potential technologies is complicated. Several prominent 3d-4d/5d DP compounds with remarkable properties, such as half-metallicity at room temperature43 and colossal magnetoresistance44 are known. These compounds are excellent candidates for real-life applications, such as spintronics and low-dissipation devices45. In addition, in many of these 3d-4d/5d DP materials, the low-energy physics is primarily governed by the t2g-orbital bands of the 4d/5d TM ions with strong SOC, with the possibility of a topological band crossing near the Fermi energy with highly efficient transport characteristics, such as AHC and ANC. In spite of the extensive studies of the electronic and magnetic properties of these materials46,47,48,49, the topological characteristics of these materials have not yet been adequately investigated. This motivated our study, in which we considered selected cubic and tetragonal 3d-4d/5d-based experimentally reported DP with the aim to explore and understand stable topological materials with large AHC and ANC. Our results show the possible existence of a clean topological band crossing near the Fermi energy, primarily contributed by the 5d-t2g orbitals with large AHC and ANC. In certain DP compounds, particularly in Ba2NiReO6, a mirror symmetry-protected topological nodal line and their drumhead surface state are observed very close to the Fermi energy. In addition, the AHC and ANC of these compounds are almost three times larger than those of the extensively studied prototype metallic ferromagnetic oxide material SrRuO319,41.

Results and discussion

Electronic structures and magnetic ground state

In the present study, we systematically investigated 3d-4d/5d-based cubic and tetragonal stable DP compounds in an attempt to identify and further our understanding of stable magnetic topological quantum materials with large AHC. The primary structural building blocks of the considered compounds are the corner-sharing B(3d)-O6 and B’(4d/5d)-O6 octahedra (cf. Fig. 1a, b). As a result of the tetragonal distortion, the octahedra in a tetragonal DP are slightly distorted, whereas they are perfect octahedra in a cubic DP. In the cubic crystal field of regular octahedra, the d-orbitals split into a higher energy level of two-fold (four-fold including spin) degenerate eg orbitals and a lower energy level of three-fold (sixfold including spin) degenerate t2g orbitals, as schematically shown in Fig. 1c. The electron-electron correlation is expected to decrease from 3d to 4d to 5d transition metal (TM) elements, whereas the bandwidth and spin-orbit coupling strength increases from 3d to 4d to 5d TM elements owing to the enhanced delocalization of electronic wave functions and the higher atomic number (Z), respectively. Consequently, exchange splitting at the 3d-B site is likely to be quite large. However, the crystal field splitting at the 4d/5d-B’ site is expected to be very large relative to the exchange splitting owing to the strong delocalization of the electronic wave function. Large crystal field splitting at the 4d/5d-B’ site results in the stabilization of the low spin state, and when the 4d/5d-B’ orbital is less than or equal to half filled, the empty eg orbitals would lie far above the Fermi level, with a negligible impact on the electronic structure and transport properties. The main contributing orbitals around the Fermi energy would be the three 4d/5d-t2g orbitals with strong SOC. In general, the energetically filled 3d orbitals of the B sites, lie below the Fermi energy, which has a finite degree of hybridization with the occupied 4d/5d-t2g orbitals through the O-p orbitals.

Fig. 1: Crystal structures and schematic energy level diagram.
figure 1

Crystal structure of a cubic space group Fm-3m (No. 225) and b tetragonal space group I4/m (No. 87) double perovskites. c Schematic energy level diagram of the d-levels in the cubic octahedron crystal field at the 3d B and 4d/5d B' sites. The arrows indicate the direction of magnetic moment at the B and B' magnetic sites.

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From both an experimental and theoretical perspective, it is important to have a topological material with a clean band structure with a topological crossing near the Fermi energy. Depending on the electron occupation of the 3d orbitals of the B site (thereby effectively controlling the exchange splitting), which lie below the Fermi energy, we can effectively control the mixing of the occupied 3d orbitals with the partially occupied 4d/5d-t2g states. In the present study, we considered 3d-4d/5d-based cubic and tetragonal DPs with 0, 1, 2, or 3 electrons in the 4d/5d-t2g state near the Fermi energy, which can provide strong SOC and many electrons in the 3d B sites. Our calculations indicated that whenever B-3d magnetic sites have fully half-filled d-shells or half-filled eg states, because of the enhanced exchange splitting, the filled 3d orbitals are pushed below the Fermi energy, significantly reducing mixing with the occupied 4d/5d-t2g states. As a result, a clean band structure with predominant 4d/5d-t2g orbitals character is observed around the Fermi energy. For example, in Fig. 2a, b, we present the band structure of Sr2CoReO6 and Ba2NiReO6 with electron occupancy at the B and B’-sites of Co-d7, Ni-d8, and Re-d1, respectively. Owing to the completely half-filled Ni-eg states in the compound, we observed a clean band structure with predominantly partially filled Re-t2g states near the Fermi energy compared to the Sr2CoReO6 compound.

Fig. 2: Effect of 3d orbital filling on the electronic structure.
figure 2

Schematic energy level diagram, orbital filling of the d-levels in the octahedron crystal field at the Co/Ni-3d and Re-5d sites and band structure with SOC for a tetragonal double perovskites, Sr2CoReO6, and b cubic double perovskites, Ba2NiReO6.

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In the following section, we describe the magnetic ground state, charge state, electron occupancies, and filling of d-orbitals for the considered DP compounds. In order to understand the magnetic ground state for the DP compounds, we first compared the total energy of their non-magnetic, ferromagnetic, and ferri-magnetic structures. In Supplementary Table 1, we summarize the total energy calculations for the different magnetic structures. The bold entries indicate magnetic ground states for each DP compound. In addition, we calculated magnetic exchange interactions (J) between magnetic elements B-3d and B’-4d/5d to further confirm and understand the magnetic ground state. The calculated exchange coupling constants (J) as a function of the distance between B-3d and B’-4d/5d elements for all the considered DP compounds are summarized in Supplementary Figs. 2, 3. A negative sign in the exchange interactions between B-3d and B’-4d/5d in Supplementary Figs. 2, 3 indicates a ferri-magnetic ground state, and a positive sign indicates a ferromagnetic alignment. From the estimated exchange coupling, we further calculated the magnetic transition temperature Tc using the mean field formula50 for all the DP compounds. In Supplementary Table 2, we summarize the mean field estimated Tc for all the considered DP compounds. We also compared the mean field estimate of the magnetic transition temperature of the DP compounds with the experimentally reported magnetic transition temperature (cf. Supplementary Table 2) and for most of the DP compounds, the mean field50 estimated Tc is found to be underestimated compared to that of the reported measured magnetic transition temperature.

Considering the magnetic ground state structure, we did electronic structure calculations for all the considered DP compounds. In order to understand the detailed electronic structure, B-3d and B’-4d/5d orbital occupancies and orbital filling in more detail, the density of states with GGA+U for all the considered DP compounds are shown in Supplementary Figs. 4, 5. In the Fe-based cubic DP, we find a magnetic moment of ~4.0 μB, indicating a nominal valency of Fe3+ with a high-spin state d5 occupancy [t2g(3↑), eg(2↑)] consistent with the experimental result reported for the Fe based compounds51,52,53. On the other hand, the calculated density of states (DOS) and magnetic moment indicated a nominal valency of 2+ at the Mn, Co, and Ni with the following occupancies for d5 [t2g(3↑), eg(2↑)], d7 [t2g(3↑, 2↓), eg(2↑)] and d8 [t2g(3↑, 3↓), eg(2↑)], respectively. The d-level occupancy at the 4d/5d-B’ site of Mo, W, and Re varies from 0 to 2 (cf. Table 1). The low-energy state is primarily governed by the 4d/5d-t2g states (cf. orbital-resolved band structure of the cubic DP in the Supplementary Information). So, it is expected to have a negligible contribution from the 4d/5d-t2g orbital near the Fermi energy for the completely unoccupied 4d/5d-t2g state. Therefore, compounds with half-filled 3d orbitals and completely unoccupied 4d/5d-t2g orbitals usually lead to an insulating solution as found in Ba2MnMoO6 and Ba2NiWO6.

Table 1 Summary of the results of selected materials considered in this study.
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In the tetragonal group of DP compounds, from the calculated DOS and magnetic moment, the sites occupied by Co, Ni and Fe, Cr are found to be in the charge state 2+ and 3+, respectively, with occupancies of d7 [t2g(3↑,2↓), eg(2↑)], d8 [t2g(3↑, 3↓), eg(2↑)], d5 [t2g(3↑), eg(2↑)], and d3 [t2g(3↑)], respectively. In the group of tetragonal DP compounds, Sr2NiMoO6 and Sr2CrOsO6 are found to be insulating. In the Sr2CrOsO6 compound, both Cr and Os are in the d3 configuration and aligned anti-parallel to each other. Both completely filled Cr-t2g up the spin channel and Os-t2g down-spin channel lie below the Fermi energy and the empty eg states of Cr and Os lie above the Fermi energy (cf. orbital-resolved band structure of tetragonal DP in the Supplementary Information). This leads to an insulating solution whereas in the Sr2NiMoO6 compound Mo is the d0 state, thus a negligible contribution around the Fermi energy. The half-filled Ni-eg states lie below the Fermi energy and give rise to an insulating solution. In Supplementary Table 2, we summarize the detailed charge state of each individual element, electron occupancies, and orbital fillings of each B-3d and B’-4d/5d element of the considered DP compounds.

AHC and ANC of considered DP compounds

Based on the knowledge we gained of the electronic structure and magnetic ground state for all the considered DP compounds, we subsequently investigated the topological transport characteristics, AHC and ANC. Table 1 summarizes the calculated AHC and ANC for all the compounds at the Fermi energy and Supplementary Fig. 1 shows a graphic representation. It is clearly evident that the value of AHC and ANC around the Fermi energy is more or less of the same order of magnitude for the cubic and tetragonal group of DP compounds, for example, the cubic DP Ba2NiReO6 and tetragonal DP Sr2NiReO6. To correlate the similar trend that was observed for the AHC and ANC in the cubic and tetragonal group of DP compounds, we investigated their magnetic space group in detail using the FINDSYM54,55 software package. Considering the SOC with magnetization along 001 direction, the magnetic space groups (MSG) of the cubic and tetragonal DP compounds are determined to be I4/mm’m’ and I4/m, respectively. Both MSG have a mirror mz along the z-axis, and lack mx and my, which is why both cubic and tetragonal DPs present a gapless nodal line at kz = 0 but gapped nodal lines at kx = 0 and ky = 0. However, these gapped nodal lines have a different origin, in the case of cubic DPs mx and my are broken because of the coupling between SOC and magnetism, whereas in the case of tetragonal DPs they were never present. Hence, for the cubic DP compounds, the main source of the Berry curvature around the Fermi energy is found to be the gapped topological nodal line in the presence of SOC and is discussed in more detail in the selected material with a large AHC and ANC section.

Selected material with large AHC and ANC

The calculated value of the AHC of the Ba2NiReO6 is almost three times larger than that of the ferromagnetic SrRuO3, which is a prototype of ferromagnetic oxide materials and has been studied extensively19,41. In addition, the determination of the energy dependence of the AHC and ANC is also important to account for the energy shifts away from the Fermi energy. Therefore, we additionally show the largest possible value in a range of 200 meV around the Fermi energy in Table 1. This occurs because minor oxygen nonstoichiometry is commonly found in perovskite oxides56,57,58 and has the effect of a small degree of doping and, consequently, a shift in the Fermi level. The largest values of AHC and ANC we found are −571 S cm−1 and −2.83 A m−1 K−1 in Ba2NiReO6. Results show that changes in the Fermi energy can have a large influence on the anomalous transport coefficients, AHC, and ANC (cf. Supplementary Figs. 14–27). To understand the large anomalous transport coefficients of Ba2NiReO6, we investigated its electronic and magnetic structure and topological features in more detail, and present the results in the following section.

First, using the GGA+U method, we compared the total energies of the different magnetic structures—non-magnetic, ferromagnetic, and ferri-magnetic—to determine the magnetic ground state of DP compounds. The calculations revealed the ferromagnetic ground state to be energetically favorable. The calculated spin-polarized DOS, considering the ferromagnetic ground state with the GGA+U calculation, is shown in Fig. 3. Within the crystal field of a regular octahedron, the d-levels split into t2g and eg states, and the d-states of Ni and the Re ions are exchanged and the crystal field splits. The exchange splitting at the 3d-Ni site is expected to be high compared to the 5d-Re site, whereas the large crystal field splitting at the 5d-Re site stabilizes in the low spin state. The calculations with the GGA+U showed that the localized magnetic moment at the Ni-3d and Re-5d sites is found to be 1.70 and 0.7 μB, respectively, which indicates that Ni2+(3d8) is in the high spin ([t2g(3↑, 3↓) eg(2↑)]) and Re6+(5d1) in the low spin [t2g(1↑)] states. In the DP Ba2NiReO6 compound, as Ni is in the high spin d8 configuration, both the majority and minority spin channels of t2g and the majority spin channel of eg are completely filled, whereas the minority spin channel of eg remains completely empty and lies above the Fermi energy (cf. Fig. 3). On the other hand, low spin Re6+(5d1) has only one electron and, from Fig. 3, it is evident that the states close to the Fermi level (EF) are mainly contributed by the partially filled Re-[t2g(1↑)] states in the up spin channel. The partially filled Re-t2g states cross the Fermi level in the majority spin channel, which causes this spin channel to become conducting. In contrast, a clear gap arises in the minority spin channel, leading to a half-metallic ferromagnetic ground state with full spin polarization. From the orbital-resolved partial DOS (cf. Fig. 3), it is clearly evident that a trivial band gap of 1.2 eV exists in the minority spin channel of Ba2NiReO6, whereas that in the majority spin channel contributes significant states to form a peak-like DOS near the Fermi level, which consist mainly of Re-t2g states hybridized with O-2p states.

Fig. 3: Site projected DOS.
figure 3

Spin-polarized GGA+U density of states for the DP Ba2NiReO6, projected onto Ni-d (blue), Re-d, (cyan), and O-p (orange) states.

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As the states close to the Fermi energy are dominated by the partially filled Re-t2g states, the effect of the SOC in the electronic structure is expected to be high. The electronic band structure, with the GGA+U+SOC calculations considering the ferromagnetic ground state, is shown in Fig. 4a. With the constructed tight-binding Hamiltonian considering the atomic orbital-like MLWFs of the Ni-d, Re-d, Sr-d, and O-p states, the calculated intrinsic contribution of the AHC and the ANC as a function of the Fermi energy is shown in Fig. 4b, c, respectively. From the computed value of the AHC as a function of Fermi energy, we observed a large peak of the AHC around the Fermi energy (cf. Fig. 4b). We also crosschecked our electronic structure, AHC and ANC calculations for a different choice of U values at the Ni-3d and Re-5d site, but no significant changes were observed in the basic electronic structure and in the peak of AHC and ANC near the Fermi energy (cf. Supplementary Fig. 6). This demonstrates the robustness of our calculation. In the next section, we investigated the peak in the AHC in detail, taking into account the values of U = 4.0 and 1.6 eV at Ni-3d and Re-5d sites, respectively.

Fig. 4: Electronic structure, AHC, and ANC.
figure 4

a Band structure in the presence of spin-orbit coupling (SOC) and Hubbard U. b anomalous Hall conductivity (AHC), σ, and c anomalous Nernst conductivity (ANC), α, for the DP Ba2NiReO6.

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Detailed topological analysis of selected material

The band structure of the DP Ba2NiReO6 undergoes a very distinct change because of the strong SOC of the Re−5d states. Near the Fermi level, most of the band touching points are gapped. Along the K−Γ direction, the valence and conduction bands of Re-t2g intersect with each other. To understand the topological nature of the band crossing along K−Γ, we examined the electronic structure in more detail. In the absence of magnetism and SOC, the cubic DP Ba2NiReO6 preserves mx, my, mz, C4x, C4y, and C4z symmetries. Without SOC, the internal spin space, and the lattice space are decoupled from each other and spins can be separated into spin-up and spin-down sectors. Thus, in the absence of the SOC, cubic Ba2NiReO6 possesses three mirror planes at kz = 0, ky = 0, kx = 0, and symmetry-protected nodal lines can appear in any of the three planes37,59,60. Considering the magnetic ground state with GGA+U, we observe the symmetry-protected nodal line in each of the three planes (kx = 0, ky = 0, kz = 0) along the K−Γ direction, as shown with green lines in Fig. 5a–c.

Fig. 5: Symmetry-protected nodal line.
figure 5

Band structure calculated along the K - Γ direction in the planes kz = 0 (a), ky = 0 (b), and kx = 0 (c) for the DP Ba2NiReO6. Green lines: band structure with GGA+U, blue lines: band structure with GGA+U+SOC. In the kz = 0 plane, the crossing indicated by the red circle is intact due to the protected mirror plane while in the kx = 0 or ky = 0 planes, nodal lines gapped out in the presence of SOC (magnetization, M ∥ (001).

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The situation is different when we take into account the SOC, in the presence of which the spin space (coupled up and down-spin channels) and the lattice space are coupled with each other, thus breaking the spin-rotational symmetry. Particularly, when considering the moments aligned along the z-direction, i.e., the magnetization along the 001-direction, the mx and my symmetries are broken, thus breaking the symmetry protection of nodal lines in the kx = 0 and ky = 0 planes, leading to a gapped nodal line along the K−Γ high-symmetry line (cf. Fig. 5c, b). Instead, for the moment aligned along the z-direction, the mz symmetry is still preserved, which leads to the gapless nodal line in the kz = 0 plane (cf. Fig. 5a). As a result of the gapped nodal lines, a large Berry curvature could be induced in the Brillouin zone (BZ), consequently enhancing the AHC37,59,60. Following the above discussion, we expect the nodal lines to be gaped away from the kz = 0 plane, which can lead to the formation of Weyl points. Thus, we searched the Weyl points in the whole Brillouin zone considering the two bands that form the nodal line and found Weyl points near the Fermi energy (35.5 meV above the Fermi energy). In Supplementary Table 3 and Supplementary Fig. 12, we show the coordinates, topological charge, bulk band structure, and surface spectrum showcasing the topologically protected Fermi arc connecting the projection of the Weyl nodes.

We calculated the Berry curvature contribution in the kx and ky = 0 planes to further investigate the contribution of these gapped nodal lines to the AHC. As expected, due to the presence of the mirror symmetry at the kz = 0 plane with SOC (M ∥ 001), a closed nodal line would be visible and, consequently, the closed nodal line results in an almost zero Berry curvature contribution in the kz = 0 plane. In Fig. 6a, we show the band gap in the kx = 0 plane by considering the two bands that form the nodal line for the DP Ba2NiReO6. In the kx = 0 plane, in which the mirror symmetry is broken due to the magnetization along the z-direction (001), the nodal lines are separated by a gap, as shown in Fig. 6a. Consequently, the gapped nodal lines make a large contribution to the Berry curvature in the system, as shown in Fig. 6b. To quantify the contributions that the gapped nodal lines make to the total AHC, we restricted the integration of Eq. (2) to slices around the kx = 0 and ky = 0 planes, as these are the planes in which the gapped lines reside. Considering the thickness of the slices at 20% of the full BZ, we observe that 80 % of the AHC value can be attributed to the gapped nodal line structure of the DP Ba2NiReO6, which suggests that the gapped nodal lines are the primary source of AHC. We further studied the topological crossing and the transport characteristics (AHC and ANC) considering the SOC with magnetization along the 100 and 010 directions. As the cubic DP compounds possess mx and my mirror planes at the kx and ky=0 planes, we find a gapless nodal line in the kx = 0 plane for the SOC with magnetization along the 100 while a gapless nodal line is found in the ky = 0 plane for the SOC with magnetization along the 010 (cf. Supplementary Fig. 10). Since the nodal line features remain unchanged for the SOC with magnetization along 001, 100, or 010 directions, the values of the AHC and ANC are expected to be the same, as shown explicitly in Supplementary Fig. 11.

Fig. 6: Gaped nodal line and Berry curvature.
figure 6

a Band gap in the kx = 0 plane with magnetization M∥ (001). The gap structure resembles that of the original nodal line. b Berry curvature in the kx = 0 plane with M∥ (001). Strong Berry curvature contribution is visible along the gaped nodal line.

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One of the hallmark features of the nodal line compounds are the protected drumhead surface states, consisting of surface states covering the projection of the nodal line on the surface. Therefore, to find the topologically protected surface state of DP Ba2NiReO6, the surface states were calculated using the WannierTools software package61,62, which employs tight-binding Hamiltonians constructed by maximally localized Wannier functions63,64. In the topological nodal line compounds, the topological invariants65 and, thus, the surface spectrum are protected by crystalline symmetries. Therefore only surfaces that preserve them will show topologically protected surface states66,67,68. In the case of the DP Ba2NiReO6, the mirror symmetry-protected nodal line is observed in the plane kz = 0, thus one could expect a surface state to prevail on the (001) surface. In Fig. 7a we show the projection of the band dispersion onto the (001) surface, following the high-symmetry k-paths shown in Fig. 7b. On the projected (001) surface, topological non-trivial surface states are clearly visible, connecting the two gapless points within Γ1−X1 and M1−Γ1 to form a two-dimensional drumhead surface. Moreover, the dispersion of the drumhead surface states is observed to be small compared to other material classes68,69 and located very close to the Fermi energy, contributing a large surface DOS at the Fermi energy, which is greatly beneficial for experimental realization as well as in spintronics applications70.

Fig. 7: Topological surface state.
figure 7

a Band dispersion of Ba2NiReO6 on the projected (001) surface. b BZs of bulk and (001) surfaces with high-symmetry points of the fcc structure.

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Summary

In summary, using the first-principles density functional calculations, Wannier interpolation, and symmetry analysis, we have systematically investigated the electronic structure and topological transport characteristics, AHC, and ANC of a list of stable cubic and tetragonal rocksalt ordered DP compounds (A2BB’O6), with the aim to search and understand stable magnetic topological materials with large AHC and ANC. We presented here a comprehensive study of an intrinsic anomalous transport for the list of representative DP materials considering the 3d TM magnetic ions at the B site and 4d/5d TM magnetic ions at the B’ site which provide a high energy scale for magnetism as well as strong spin-orbit coupling and explored the possibility of realizing a clean electronic band structure with a large Berry curvature, hence large AHC and ANC. Our systematic calculations and analysis of the electronic structure for the list of representative DPs materials showed that when DP compounds have a half-filled B site 3d or eg states and partially filled 4d/5d-t2g states, due to the enhanced exchange splitting at the B sites, mixing with the partially filled 4d/5d-t2g states reduce significantly and leads to the possibility of a clean topological band crossing. This is primarily attributed to strong SOC in the 5d-t2g orbitals near the Fermi energy with large AHC.

From the symmetry analysis, the magnetic space group of the cubic and tetragonal DP compounds considering the SOC with magnetization along z direction, is found to be in the I4/mm’m’ and I4/m, respectively, with a mirror symmetry mz which leads to a symmetry-protected nodal line in the kz = 0 plane. In the presence of magnetism and SOC with magnetization along z direction, mx and my mirror symmetries of the cubic structure are broken, resulting in a gapped topological nodal line in the kx = 0 and ky = 0 planes which are found to be the primary source of the Berry curvature for the cubic DP compounds around the Fermi energy. The presence of mirror symmetries and a clean topological band crossing primarily resulting from the 5d-t2g state with strong SOC, were found to be crucial for large AHC and ANC, which, in combination with the magnetism and SOC, produces a large BC in the band structure.

In certain DP compounds, particularly in Ba2NiReO6, we found a symmetry-protected nodal line near the Fermi energy with very large AHC and/or ANC values that are three times larger than the extensively studied prototype metallic ferromagnetic oxide material SrRuO319,41. In the DP Ba2NiReO6, slightly dispersing mirror symmetry-protected drumhead surface states and Weyl points are observed around the Fermi energy making it a good candidate for using the topological properties in the spintronics applications. Furthermore, the electronic structure, topological transport characteristics, AHC, and ANC of its optimized structure are found to remain unchanged, demonstrating the robustness of our calculations. We additionally found that both AHC and ANC are strongly dependent on the position of the Fermi level and, therefore, on the doping level of the investigated material.

This work demonstrates the versatility and usability of cubic and tetragonal DP based on 3d and 4d/5d TM ions to realize a clean topological band crossing near the Fermi energy, primarily resulting from the 4d/5d-t2g state with a large AHC. Our computational study pave the way for high-performance stable magnetic topological materials based on oxide DP.

Methods

First-principles DFT calculations

For the calculations, we considered stable cubic and tetragonal 3d-4d/5d-based DP compounds whose structural data was taken from the ICSD database71 with a 3d magnetic element at the B site and 4d/5d magnetic element at the B’ site to enable us to investigate topological transport characteristics such as the anomalous Hall and Nernst effects. We considered the experimentally measured lattice constant, atomic position, and space group as inputs for the density functional theory (DFT) calculations. We also checked the calculations with the optimized structures for selected compounds, and found that the electronic structure and transport characteristics, AHC, and ANC almost remain unchanged around Fermi energy (cf. Supplementary Figs. 28, 29 and Supplementary Table 4). The DFT-based plane-wave projected augmented wave (PAW) method, as implemented in the Vienna ab initio simulation package, was employed72,73,74. We used a 12 × 12 × 12 (12 × 12 × 10) k-points mesh and a plane-wave cutoff of 520 eV, for the self-consistent calculations of cubic (tetragonal) compounds. These selections of the k-mesh and the plane-wave cut-off were found to result in good convergence of the total energy. We used the Perdew-Burke-Ernzerhof (PBE)75 exchange-correlation functional within the generalized gradient approximation (GGA). The electron-electron correlation effects beyond GGA at the strongly correlated 3d and 4d/5d magnetic sites were taken into account using the supplemented on-site Hubbard U correction via the GGA+U method76. Considering that the B site is occupied by a 3d transition metal and the B’ site by a 4d/5d transition metal with wider bands compared to the B site, the correlation effect would be expected to be stronger at the B site compared to the B’ site. Thus, we chose U values of 4.0 eV and 1.6 eV for the B and B’ sites of the considered 3d-4d/5d-based DPs, respectively. The selection of these values was motivated by the U values used for 3d-4d/5d-based DP compounds in previous studies47,48,49,77, in which the electronic structure and magnetic ground state were found to be in good agreement with the experimental results. We initially checked our calculations by varying the U values of a few compounds, but overall, the electronic structure and magnetic ground state remained unaltered. The effect of the SOC is included in the fully relativistic schemes. Magnetic anisotropy calculation (cf. Supplementary Table 5) shows that the z axis (i.e., magnetization along 001) is an easy axis for tetragonal DP compounds, while cubic DP compounds are isotropic in agreement with experimental observation28. Thus, for all calculations, magnetization is chosen to be parallel to the (001) direction.

Transport characteristics AHC and ANC calculation

We calculated the contribution of the intrinsic Berry curvature to the anomalous Hall conductivity by employing the Wannier interpolation technique63. To compute the Berry curvature, we first constructed maximally localized Wannier functions (MLWFs) Hamiltonian projected from the GGA+U+SOC Bloch wave functions using the Wannier90 tool64,78,79. Atomic orbital-like MLWFs of the A-d, B-3d, B’-4d/5d, and O-p states were considered to construct the tight-binding Hamiltonian, which reproduced the spectrum of the system accurately in the energy window of ±2.0 eV around the Fermi energy. The wannierization was repeated with different inner and outer energy windows until it reproduced the spectrum of the system quite accurately in the energy window of ±2.0 eV around the Fermi energy (cf. Supplementary Figs. 44–47).

With the tight-binding model Hamiltonian, we calculated the intrinsic AHC using the linear response Kubo formula approach as follows80:

$${\Omega }_{n}^{z}({{{\bf{k}}}})=-{\hslash }^{2}\mathop{\sum}\limits_{n\ne m}\frac{2Im\langle {u}_{n{{{\bf{k}}}}}| {\hat{v}}_{x}| {u}_{m{{{\bf{k}}}}}\rangle \langle {u}_{m{{{\bf{k}}}}}| {\hat{v}}_{y}| {u}_{n{{{\bf{k}}}}}\rangle }{{({\epsilon }_{m{{{\bf{k}}}}}-{\epsilon }_{n{{{\bf{k}}}}})}^{2}},$$
(1)

where \({\Omega }_{n}^{z}({{{\bf{k}}}})\) is the Berry curvature of band \(n,\hslash {\hat{v}}_{i}=\partial \hat{H}({{{\bf{k}}}})/\partial {k}_{i}\) is the i’th velocity operator, unk and ϵnk are the eigenstates and eigenvalues of the Hamiltonian \(\hat{H}({{{\bf{k}}}})\), respectively.

Subsequently, we calculated the AHC, given by:

$${\sigma }_{H}^{A}=-\frac{{{{{\rm{e}}}}}^{2}}{\hslash }\mathop{\sum}\limits_{n}{\int}_{BZ}\frac{d{{{\bf{k}}}}}{{\left(2\pi \right)}^{3}}{f}_{n}{\Omega }_{n}^{z}({{{\bf{k}}}})$$
(2)

and the ANC, \({\alpha }_{H}^{A}\), as proposed by Xiao et al.81,82

$${\alpha }_{H}^{A}=-\frac{1}{T}\frac{{{{{\rm{e}}}}}^{2}}{\hslash }\mathop{\sum}\limits_{n}{\int}_{{{{\rm{BZ}}}}}\frac{d{{{\bf{k}}}}}{{\left(2\pi \right)}^{3}}{\Omega }_{n}^{z}({{{\bf{k}}}})\left[({E}_{n}-{E}_{{{{\rm{F}}}}}){f}_{n}+{k}_{B}T{{{{\ln }}}}\left(1+\exp \left(\frac{{E}_{n}-{E}_{{{{\rm{F}}}}}}{-{k}_{B}T}\right)\right)\right],$$
(3)

where T is the actual temperature, fn is the Fermi distribution, and EF is the Fermi level. We used a k-point mesh of 300 × 300 × 300 for the calculation of the AHC and ANC using Eqs. (2) and (3), respectively, and T = 300 K for the calculation of the ANC.

Magnetic exchange and surface state calculation

In order to compute magnetic exchange interactions (J), we used Green’s function approach83,84, implemented from the combination of the Wannier9064,78,79 and Tb2j software packages85. In this approach, energy variations are utilized due to infinitesimal local spin rotations83. Surface states were calculated using tight-binding methods by the combination of the Wannier9064,78,79 and WannierTools software packages61,62. We used the VESTA86, sumo87, and PyProcar88 tools for the lattice structure and projected band structure plots.