The field of antiferromagnetic (AFM) spintronics has recently garnered attention due to its ultrafast dynamics, zero net magnetization, and stability against external magnetic fields, which hold promise for future computing power^[1–4]. In these materials, information can be encoded by manipulating the antiferromagnetic order via domain wall motion^[5], optical excitation^[6], spin-orbit torques^[3], and staggered local relativistic fields^[7]. Among the different categories of antiferromagnet, metallic antiferromagnet are particularly unique as they provide an ideal platform for fundamental studies and applications due to the strong interplay between electrons, spin, phonons, and photons^[8–9]. However, metallic antiferromagnets are rare, and the known materials display AFM order only well below room temperature (RT), which limits their practical applications. For instance, the transition-metal oxides Nb₁₂O₂₉^[10], Ca₃Ru₂O₇^[11], CaCrO₃^[12], and LaNiO₃^[13] exhibit Néel transition temperature T_N of 12, 56, 90, and 157 K, respectively. Although RuO₂ exhibits AFM order at around RT (T_N ~ 300 K), the magnetic moment is as small as 0.05µ_B (µ_B represents Bohr magneton) at about 300 K. Therefore, exploring materials with RT magnetic order is of utmost importance for practical magnetic applications^[1].

Recently, research on the magnetic properties of double-perovskite oxides A₂BB'O₆, where B and B' are ordered 3d and 4d/5d transition metal ions, has demonstrated the emergence of homogeneous and long-range magnetic order due to periodic intrinsic magnetic ions and magnetic exchange interactions^[2–9]. These materials exhibit multiple functions resulting from multiple physical interactions, such as spin, orbit, dipole, and lattice. For example, such interactions give rise to magnetoelectric coupling based multiferroics^[14–15], half-metallicity^[16], colossal magnetoresistance^[16], and ferromagnetic (FM) semiconductor and insulator^[17] properties in RT perovskite-oxides magnets.

Theoretical analysis of A₂BB'O₆ materials with divalent A-site alkali earth cations reveals that the maximum number of unpaired 3d and 4d/5d electrons can reach 3d⁵ (${{t}}_{2\mathrm{g}}^{3}{{e}}_{\mathrm{g}}^{2}$ in high spin) or 4d³/5d³ (${{t}}_{2\mathrm{g}}^{3}{{e}}_{\mathrm{g}}^{0}$) combination over the B- and B'-sites, such as in A₂Fe³⁺B'⁵⁺O₆ (B' = Ru, Os) and A₂Mn²⁺B'⁶⁺O₆ (B' = Rh, Ir) under charge balance. For example, the Ca₂FeOsO₆ material exhibits electrically insulated behavior and undergoes a FM transition at about 320 K driven by lattice distortion and chemical pressure^{[6, 18–19]}. In contrast, spin frustration arising from the disordered alignment of Fe³⁺/Ru⁵⁺ in A₂FeRuO₆ (Pbnm symmetry)^[20–21] may result in spin-glass or AFM behavior^[22]. Notably, the Fe/Ru ordered double perovskite Sr₂FeRuO₆ is stabilized in film form by hetero-interface strain (chemical pressure) on SrTiO₃ substrate, achieving a Curie temperature (T_C) of 390 K^[23]. To date, there are no reports on A₂Mn²⁺Rh⁶⁺O₆ or A₂Mn²⁺Ir⁶⁺O₆.

The emerging research on exotic perovskites under high-pressure and high-temperature (HPHT) conditions has led to the investigation of transition metal cations at the A-site, resulting in enhanced magnetic interactions when compared to conventional analogs^{[14–15, 17, 24–34]}. The occupancy of the A-site by high-spin d⁵-Mn²⁺ cations can significantly increase the number of unpaired d electrons, thereby promoting strong magnetic interactions in Mn₂BB'O₆^[35]. Theoretically, there are 411 possible combinations of Mn₂BB'O₆ in Mn₂B¹⁺B'⁷⁺O₆, Mn₂B²⁺B'⁶⁺O₆, and Mn₂B³⁺B'⁵⁺O₆, as shown in Fig. 1. Mn₂B⁴⁺B'⁴⁺O₆ is unlikely to form ordered double perovskites and is ignored here. Up to now, only 20 compounds have been experimentally validated to crystallize in Mg₃TeO₆-type (R$\overline{3}$)^[36–37], Ni₃TeO₆-type (R3)^[38–39], β-Li₃VF₆-type (C2/c)^[40–42], and GdFeO₃-type (P2₁/n)-type^[43–44] structures, depending on the chemical and synthesis parameters (see Fig. 1).

Figure  1.  (a) Possible combinations of Mn₂BB′O₆ in the chemical space of B and B′ cations under charge balance (The possible, reported, and inexistent compounds are displayed in green, red, and blank, respectively. The numbers marked for the 20 known compounds denote the space group types listed on the right side.); (b)–(e) the crystal structures of some of the known compounds, including Mg₃TeO₆-type (R$\overline{3}$)^[36–37], Ni₃TeO₆-type (R3)^[38–39], β-Li₃VF₆-type (C2/c)^[40–42], and GdFeO₃-type (P2₁/n)^[43–44](To ensure that our calculations are compatible with different magnetic configurations and atomic distortions, we have useda 40-atom super-cell, which can be considered as a (2, 2, 2) multiple of the ideal single perovskite structure alongthe three axes of Cartesian coordinates^[50–51]. (b)–(e) also show the examples of collinear magnetic structure thatwe considered structure optimization with (b) FM, (c) Type Ⅰ AFM, (d) Type Ⅱ AFM and (e) Type Ⅲ AFM.)

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Taking the number of unpaired d electrons (n_d), chemical valences and ionic radius into account, it is noteworthy that eight transition-metal-only (TMO) double perovskites with n_d between 14 and 18 exhibit robust magnetoelectric properties, which provides motivation for further exploration in this category. More than 70 TMO double perovskites can be formed in Mn₂BB′O₆ by ordered combination of 3d and 4d/5d ions over B- and B′-sites under charge balance, offering a vast space to explore new multifunctional materials. However, the normally arranged atoms will have a certain displacement and deformation due to the influence of strong magnetism^[45], making TMO double perovskites’ symmetry further reduced than that of conventional nonmagnetic perovskites. As a result, tolerance factor (t)^[46–47] and octahedral factor (µ)^[48–49] as geometric descriptors for conventional perovskites with large A-cations are not longer suitable for the TMO double perovskites. Therefore, discovering these materials mainly relies on empirical intuition and costly trial-and-error in HPHT synthesis.

Recently, the successful prediction and precise synthesis of A₃TeO₆ have provided insight into the screening of new materials oriented towards structure and function, which can help avoid the costly trial-and-error process involved in preparing magnetically exotic perovskites under HPHT^[52–53]. These studies have used first-principles density functional theory (DFT) calculations based on the Murnaghan equation of state for solids^[54] to show that the phase stability between competing polymorphs can be estimated using the phase diagram of the relative enthalpy (∆H), the total energy (E) and the pressure (p). The reported DFT-based workflow, which adopts a restricted collinear magnetic structure, has been shown to provide considerable accuracy in predicting the polymorph and corresponding synthesis parameters (such as pressure). However, for TMO double perovskites with 4d/5d ions at the B′-site, the variation in the energy scale of possible interactions, such as magnetic interactions, spin-orbit coupling (SOC), Dzyaloshinskii-Moriya (DM) interaction, Hubbard U and Hund’s interactions, cause a challenge for calculating these materials reliably and efficiently using ab initio many-body methods^[45]. This remains an open challenge in providing reliable and timely predictions for this strongly correlated perovskites family^[55–56]. Conversely, validation calculations based on experimentally reported TMO double perovskite-related compounds are expected to facilitate the understanding of dominant factors in inverse prediction and rational design.

In this article, we will firstly conduct test calculations on the known transition metal oxide Mn₂BB'O₆, establish calculation models, and then these models will be used to predict the pressure-dependent polymorph evolution and synthesis parameters of the TMO double perovskite Mn₂FeOsO₆, which contains a maximum n_d of 18. Additionally, we will propose several magnetoelectric properties of Mn₂FeOsO₆. Due to the high cationic disordering in corundum-type, Mn₂FeRuO₆ is beyond the scope of this work, shown experimentally^[25].

1. Symmetry Analysis

For double perovskite materials with the chemical formula A₂BB'O₆, the ideal crystal structure is usually doubled in all three crystallographic directions compared to a single perovskite ABO₃, resulting in a space group of Fm$\overline{3}$m^[32]. However, this only occurs when the size of the A-site cation is much larger than that of the B-site cation, i. e. with tolerance factor t around 1^[57]. When the size of the A-site cation is comparable to that of the B-site cation, i. e. with tolerance factor t less than or equal to 0.85, the BO₆ octahedra will tilt to optimize the A-cation bonding^[58]. According to Glazer’s tilt rule, there can be 12 tilt systems in A₂BB'O₆ perovskites, as determined by Howard et al^[59–60]. As mentioned in our previous article^[53], exotic double perovskites with transition metals and t≤0.85 may adopt one of six possible structures identified by database screening, including distorted GdFeO₃-type double perovskite (P2₁/n), B-site ordered LiSbO₃ derivatives (Pnn2), Mg₃TeO₆ (R$\overline{3}$)^[36–37], corundum derivatives (LiNbO₃-type R3c, ilmenite R$\overline{3}$, and ordered ilmenite or Ni₃TeO₆-type R3)^[38–39], bixbyite (Ia$\overline{3}$), and β-Li₃VF₆ (C2/c)^[40–42]. Based on chemical and geometrical considerations, the size and the charge differences between the A-site and B-site cations studied in this paper do not energetically favor the formation of bixbyite (Ia$\overline{3}$) and LiSbO₃ derivatives (Pnn2). Therefore, we only consider the remaining four structures (R3, R$\overline{3}$, C2/c and P2₁/n).

Stretching the ideal face-centered cubic structure Fm$\overline{3}$m along the <111> direction results in the R$\overline{3}$ structure, where the BO₆ octahedra tilts and the space group changes to a^–a^–a^– in Glazer tilt systems. However, in practice, the orders of B and B' coordination can destroy the space-reversal symmetry, leading to the R3 polymorphs. The reported R3c structure is unlikely to happen for rock-salt double perovskite structures.

Meanwhile, the C2/c and P2₁/n structures can be obtained by sliding the ideal Fm$\overline{3}$m structure along the <110> direction and considering the tilt of the BO₆ octahedra. In the case of small A-cations, the A cation is unlikely to maintain the ideal position, leading to the distortion of the lattice and the formation of the P2₁/n structure with a⁺b^–b^– in Glazer tilt systems. Both C2/c and P2₁/n configurations are commonly observed in double perovskite structures.

The single perovskite structure is also the most abundant in Pnma (a⁺b^–b^–) and R$\overline{3}$c (a⁰b^–b^–) configurations^{[48, 61]} since the tilt direction of BO₆ octahedra in the adjacent two layers of perovskite structure tends to be opposite. In the Glazer symbol, therefore, compared with in-phase BO₆ rotations, such as Im3 (a⁺a⁺a⁺) and Pmmn (a^–b⁺b⁺), the anti-phase BO₆ rotations are more common. When it comes to double perovskites A₂BB'O₆, two typical progress of octahedral tilts with decreasing t has been found as^[62–63]

We note that in material calculations, A-site cations are always repelled or attracted by B-site cations due to the influence of magnetism and Jahn-Teller effect, making it difficult to maintain a square arrangement in a plane. Therefore, the common tetragonal lattice (I4/m (a⁰a⁰c^–) or P4/mnc (a⁰a⁰c⁺)) in perovskite is hard to appear under the condition that both A- and B-sites of double-perovskite are transition metal ions. In fact, experimental research on this type of materials has also confirmed our conclusion, i. e. there is no report on tetragonal lattice double-perovskite oxides with A-site alkali earth cations, B and B′ ordered transition metal ions so far, as shown in Fig. 1(a).

In this work, we investigated the effect of magnetism on the lattice distortion of double perovskite structures, in addition to the BO₆ octahedral tilt induced by the small A-cation. Previous studies have also reported the influence of magnetism on the lattice distortion phenomenon^{[22–23, 45]}. It is important to note that the nearest-neighbor (NN) and next nearest-neighbor (NNN) interactions can have similar strengths. In the ideal double perovskite structure, there are many possible magnetic configurations. However, experimental observations have shown that only four magnetic configurations are mostly observed. There are three types of AFM order, namely, type-Ⅰ, type-Ⅱ, and type-Ⅲ, depending on the relative magnitudes of NN and NNN interactions. For type-Ⅰ order, the spins are anti-parallel between the [100] planes and parallel within the [100] plane. Type Ⅱ order occurs when the NN interactions are weaker than the NNN interactions, with the spins arranged anti-parallel between the [111] planes and parallel within the [111] plane. Type-Ⅲ order is an intermediate form in which the NN interactions are relatively strong, while the NNN interactions are still significant.

In addition to these four configurations, there are other disordered magnetic phases such as magnetic frustration and spin-glass-type behavior. However, these phases are less relevant to the present study, therefore, we limit our DFT calculation and analysis of structural optimization to the four aforementioned configurations. When analyzing the magnetic properties of materials, we consider as many magnetic configurations as possible to obtain more reliable magnetic properties.

Since most experimental synthesis of transition-metal double perovskites require HPHT conditions^[64–66], we considered the synthesis pressure as a crucial parameter. To investigate the effect of pressure on the stability of double perovskite structures, we calculated the energies of all four lattice configurations and four magnetic configurations for each material under varying pressure conditions.

2. Model and Methods

To simulate the application of relatively large pressure, we started with a relatively small initial volume for each polymorph with a specific set of ingredients. We then gradually increased the simulating pressure by reducing the lattice to determine the most stable structure type at each given pressure. After that, we obtained the total energy (E) and volume (V) curve of each compound, and fit it with the Brich-Murnaghan function

where B₀, $B'_0$, V₀ and E₀ are fitting parameters. We finally obtained the relative energy-pressure (E-p) relation to determine the phase transition of each compound

Our DFT calculations were performed using the Vienna ab initio simulation package (VASP)^[67–70] with the projector augmented wave (PAW) method. We used the spin-polarized generalized gradient approximation (GGA) and the Perdew Burke-Ernzerhof (PBE) function to treat the electron exchange correlation potential^[71]. The plane wave kinetic energy cutoff was set to 400 eV.

To ensure our calculations are compatible with different magnetic configurations and atomic distortions, we used a 40-atom supercell, which can be considered as a (2, 2, 2) multiple of the ideal single perovskite structure along the three axes of Cartesian coordinate^[50–51]. The Γ point only in VASP was used to accelerate the crystal structure optimization until the force on each atom is less than 0.001 eV/Å. We built initial structure models for calculations by elemental substitutions of the crystallographic information files of known compounds with elemental substitutions.

To investigate the effect of pressure on the stability of double perovskite structures, we applied isotropic pressure during the calculations. In DFT calculations, we have considered various factors, such as collinear magnetic structure, SOC, DFT+U (U_Fe = 4.6 eV, U_Mn = 4 eV, U_Os = 2.0 eV), and non-collinear magnetic structure to determine which factors would have undeniable impact on the structure.

For Mn₂FeOsO₆, its magnetic properties can be described by the Heisenberg Hamiltonian

where the symbol S_i (S_i^z) represents the spin operator at the lattice site i (along the z direction), J_ij refers to the Heisenberg exchange between S_i and S_j, D refers to the DM interaction, and $\varDelta$_s represents the single-ion Ising anisotropy.

There are 8 magnetic atoms in a supercell, and it is needed to consider 64 types of magnetic exchange interactions between these magnetic ions. If we ignore magnetic exchange interaction J between identical magnetic ions (i. e. J_ii = 0) and the directional dependence between different magnetic ions (i. e. J_ij = J_ji), there are 28 distinct values left. Therefore, we need to calculate these 28 different magnetic exchange interaction values individually.

We calculated 64 different magnetic configurations among the total 2⁸ = 256 configurations in the magnetic unit cell containing 8 magnetic ions. Because the magnetic anisotropy is basically negligible, we used collinear magnetic DFT calculations to save the resources requirement. These 64 magnetic configurations are enough to obtain 28 unknown J. In order to simplify the calculation, we used the linear matrix method to calculate J one by one using the minimum dichotomy method

where $\boldsymbol{\boldsymbol{\tilde{J}}}$ is a 1×28 matrix contained 28 different magnetic exchange interaction values, $\boldsymbol{\tilde{E}}$ is a 64×1 matrix contained 64 different energies obtained by DFT calculations, and $\boldsymbol{\tilde{S}}\otimes\boldsymbol{\tilde{S}}$ represents the direct product between magnetic moments of the magnetic ions.

To study the spin dynamic behaviours, we applied a linear spin wave theory by using SpinW package to compute the excitation spectrum^[72]

where $\mathit{\mathit{\boldsymbol{k}}}$ is normalized wave vector, ω is frequency of the excitation spectrum, $S^{\alpha,\beta}$ (α, β = x, y, z) is the dynamic structure factor, ${\tilde k_\alpha }{\text{, }}{\tilde k_\beta }$ are components of the normalized wave vector $\mathit{\boldsymbol{k}}$, ${\delta_{\alpha ,\beta }}$ is the Dirac delta function which takes the value of 1 only when α=β, otherwise it takes 0.

3. Validation Calculations on Known Two Double Perovskites

Table 1 provides the details of the synthesis conditions, structural symmetry, electron configurations, and physical properties of TMO double perovskites A₂BB'O₆ where B- and B′-site cations is transition-metal atoms. The Re-based compounds crystallize in a distorted GdFeO₃-type monoclinic structure (P2₁/n), while Mo-based and W-based compounds are more likely to adopt rhombohedral symmetry (R3). Previous studies have shown that DFT caculations in consideration of only collinear magnetic structures can yield reasonable and acceptable overall trends of synthesis condition and E-p phase diagrams for TMO double perovskites^[52–53].

Table  1.  Structural and electronic properties of the 7 experimentally realized TMO double perovskite-related compounds, along with the predicted Mn₂FeOsO₆. Here $n_{\mathrm{d}}$≥14 provide certain displacement and deformation due to the influence of strong magnetism, resulting in low symmetry (R3 and P2₁/n)

Compound	Space group	nd	Synthesis conditions	Physical properties	Magnetic structure	Ref.
Mn2FeMoO6	R3	16	1350 ℃, 8 GPa	Multiferroic, TC = 337 K	Type-Ⅱ AFM	[17]
Mn2FeWO6	R3	14	1400 ℃, 8 GPa	Second-Harmonic generation, TN = 70 K	Type-Ⅰ AFM	[24]
Mn2MnWO6	R3	15	1400 ℃, 8 GPa	TN = 58 K	Type-Ⅰ AFM	[15]
Mn2CoReO6	P21/n	14	1300 ℃, 8 GPa	Semiconductor, TN = 94 K	Type-Ⅱ AFM	[26]
Mn2FeReO6	P21/n	17	1350 ℃, 5/11 GPa	Half-metal, TC = 520 K	Type-Ⅲ AFM	[16, 29–30]
Mn2MnReO6	P21/n	17	1400 ℃, 5/8 GPa	Semiconductor, TN = 50 K	Type-Ⅱ AFM	[29, 31]
Mn2NiReO6	P21/n	17	1573 ℃, 8 GPa	Semiconductor, TN = 80 K	Type-Ⅱ AFM	[73]
Mn2FeOsO6	P21/n	18	5 GPa	Metal, TN = 680 K	Type-Ⅲ AFM	This work

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We investigated the effect of other possible properties, such as DFT+U (U_Fe = 4.6 eV, U_Mn = 4 eV, U_Os = 2.0 eV), and the role of non-collinear magnetic structure, on the energy-volume (E-V) results, but found that these interactions do not impact the results significantly. Therefore, we concluded that these additional interactions could be ignored in structural optimization, and we only adopted collinear magnetic structures in the structural optimization calculations to save computational resources.

To be clear, we started with Mn₂MnReO₆ with the known P2₁/n structure synthesized in previous studies^{[16, 29–30]}. To evaluate the effect of magnetism on the crystal structure, we calculated the E-V curve using the Brich-Murnaghan function. It turns out that for collinear magnetic structures, the magnetism contribution to the energy difference is on the order of 0.1 eV/atom, accordingly, having an effect on the lattice structure. However, the E-p curve calculated using Eq. (3) shows that, the R3 structure is the most stable one when p≤10 GPa, with a minimum energy difference of less than 0.1 eV per cell without considering the magnetic structure. This small energy difference results in inconsistency between the calculated R3 output and the experimental P2₁/n result. Therefore, we must consider the magnetic structure in TMO double perovskites. We conjectured that the magnetic exchange between magnetic ions in Mn₂MnReO₆ would affect the crystal structure, and the magnetic contribution of atoms should be considered in the DFT calculations of structure optimization. After considering magnetic structure, the type-Ⅱ AFM configuration has the lowest energy of the four collinear magnetic structures considered, which is a good validation of previous experiment results^[29–31].

In the present study, we investigated the polymorphic phase transitions of five double perovskite compounds, namely Mn₂MnReO₆, Mn₂CoReO₆, Mn₂FeReO₆, Mn₂FeMoO₆, and Mn₂FeWO₆. In our calculations, we have considered the four different magnetic configurations and obtained their respective E-V relations (see Fig. 2(a)–Fig.2(e)). However, due to the existence of metastable magnetic structures, the total magnetic moment after DFT convergence is found to be inconsistent with the calculations of a single magnetic configuration. The E-V curves also exhibit abnormal points and protrusions, rendering the fitting of these curves with the Brich-Murnaghan function inaccurate. To address this issue, we selected the magnetic structure with the lowest energy under each V point of the four magnetic configurations as the source of E-V curve. By uniformly fitting these E-V curves with the lowest energy, we obtained appropriate parameters to produce a sufficiently reliable E-p curve.

Figure  2.  E-p curves of Mn₂FeWO₆ and Mn₂FeMoO₆, as well as Mn₂FeOsO₆ with the magnetic structures in Table 1. (All these results are fitted by Brich-Murnaghan function Eq. (3) in consideration of the lowest energy of four collinear magnetic configurations.)

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Upon adding magnetism, we observed that although the influence of magnetism on each structure is different, the total energy of the system is reduced by about 1 eV per cell. For Mn₂MnReO₆, such a small influence of magnetic exchange energy is able to reverse the two energy levels of R3 and P2₁/n structures under the external pressure around 7 GPa, beyond which the P2₁/n structure is more stable and in good agreement with the experimental results^{[29, 31]}. Notably, 5 GPa is needed to prepare P2₁/n Mn₂MnReO₆ in the experiment^[29], and the gap of 2 GPa between theory and experiment may be caused by the 1400 ℃ treatment and can be considered as within the theoretical error range.

We then applied this method to the related double perovskite compounds that had been experimentally prepared. In our calculations, the lowest synthesis energy configuration of Mn₂CoReO₆ is always the P2₁/n structure, which roughly echoes the experimental results^[26]. For Mn₂FeReO₆, the synthesis energies of the four structures exhibit little differences in the pressure range considered. The energy of R$\overline{3}$ overlaps that of P2₁/n at 5 GPa, being consistent with the synthetic conditions in the experiment^[16]. The situations of Mn₂FeMoO₆ and Mn₂FeWO₆ are similar. Both compounds show the R3 structure under high pressure, and the synthesis conditions are about 6 GPa. We also checked the calculations without considering magnetic conditions and found that they do not match experiments. Validation calculations on these five synthesized double perovskite materials prove that our proposed method for computing polymorph-dependent phase diagrams is reliable.

4. Magnetism of Mn₂FeOsO₆

In this section, we present predictions on the structure and the properties of Mn₂FeOsO₆, a compound that has not yet been experimentally synthesized. Our DFT calculations demonstrate that the C2/c structure is relatively stable under atmospheric pressure, as illustrated in Fig. 2(f). However, under a pressure greater than 5 GPa, the P2₁/n structure with type-Ⅲ AFM configuration becomes the most stable.

In addition, our calculations on the possible magnetic structures of Mn₂FeOsO₆ revealed a standard AFM configuration, where the magnetism of each magnetic ion is reversed. Specifically, in a 40-atom supercell, four spins of the eight Mn atoms are opposite to the other four, and two spins of the four Fe atoms are also opposite to the other two. We have also confirmed that the magnetic anisotropy is relatively small, as the energy difference per cell caused by magnetism in the lattice vectors a, b, and c (Fig. 3(a)) directions is less than 0.1 meV.

Figure  3.  DFT results of Mn₂FeOsO₆ in consideration of SOC, DFT+U (U_Fe = 4.6 eV, U_Mn = 4 eV, U_Os = 2.0 eV) and non-collinear magnetism: (a) the predicted atomic and magnetic structures, with a distorted GdFeO₃-type monoclinic structure (P2₁/n) as for Mn₂FeReO₆^{[16, 29–30]} (The arrow on each atom represents the spin orientation and the size of spin value. Every spin is almostlaying at <001> direction, forming a relatively ideal AFM configuration.); (b) energy band structure near Fermi level andspin-resolved density of states (DOS) (Mn₂FeOsO₆ has no bandgap, forming as conductor. There is a suspectedformation of Dirac points between wavevectors B-D, marked by red rectangle and arrow. The DOS of spin-upand spin-down are basically the same, verifying the AFM configuration of Mn₂FeOsO₆.)

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We then proceeded to approximate the magnetic properties of Mn₂FeOsO₆. Our calculations showed that the total magnetic anisotropy energy (MAE) in a single magnetic unit cell is negligible ($\varDelta$_s<0.1 meV) in consideration of SOC, and the spins of magnetic atoms exhibit almost collinear configuration with only a degree difference of less than 5°, as depicted in Fig. 3(a). Therefore, in the estimation of magnetic properties, we neglected the MAE, as well as DM interaction, and only considered Heisenberg exchange in Eq. (4).

As illustrated in Fig. 3, Mn₂FeOsO₆ contains 8 magnetic ions in a single magnetic unit cell. Among them, the four Mn ions possess a local magnetic moment of approximately 4.5µ_B, and the two Fe ions possess a local magnetic moment of approximately 4µ_B, while the two Os ions possess a local magnetic moment of approximately 1µ_B. Although basically Os ions is not magnetic, they exhibit magnetism induced by the surrounding strong magnetic Mn and Fe ions.

In our magnetic exchange model for Mn₂FeOsO₆, we calculated 28 different J due to the irregular configuration of the magnetic ions in the P2₁/n structure. Here, we present a few examples to illustrate the numerical values of these J . The AFM exchanges (J > 0) between the NN Mn ions (J₁₃≈J₂₄) and NNN ones are approximately equal to the values reported in [74]. However, due to the low symmetry of P2₁/n, there are two different distances between NN Mn ions, resulting in slightly different FM exchanges (J₂₃<J₁₄) induced by indirect Ruderman-Kittel-Kasuya-Yosida (RKKY) exchanges. Thus, our group of magnetic exchange models, which contains 8 different magnetic ions, is considered to be reliable.

According to the aforementioned magnetic structure model, we constructed the corresponding exchange interactions between magnetic ions, as shown in Table 2. The result of spin dynamic behaviours of Mn₂FeOsO₆ is shown in Fig. 4(b). Note that we have ignored the DM interaction and single-ion Ising MAE in the Hamiltonian. An obvious value of intensity is shown at AFM order wave vector Q_AFM = (0, 0, 1). As the NN interactions are on the order of 10 meV, the maximum of the magnon dispersion reaches 600 meV. Interestingly, the 8 bands along the Z−D momentum path all show characteristics of flat band.

Table  2.  Magnetic exchange $J_{i j}$ between magnetic ions within a magnetic unit cell of Mn₂FeOsO₆

i	j	Jij/meV		i	j	Jij/meV		i	j	Jij/meV		i	j	Jij/meV
Mn1	Mn2	0.97		Mn2	Mn3	−19.20		Mn3	Fe1	1.01		Mn4	Os2	−15.77
Mn1	Mn3	40.48		Mn2	Mn4	39.88		Mn3	Fe2	13.49		Fe1	Fe2	−12.01
Mn1	Mn4	−20.07		Mn2	Fe1	15.06		Mn3	Os1	−16.74		Fe1	Os1	35.37
Mn1	Fe1	−14.33		Mn2	Fe2	18.32		Mn3	Os2	15.65		Fe1	Os2	35.37
Mn1	Fe2	−16.58		Mn2	Os1	12.26		Mn4	Fe1	−1.26		Fe2	Os1	9.14
Mn1	Os1	−12.72		Mn2	Os2	25.43		Mn4	Fe2	−13.27		Fe2	Os2	−33.49
Mn1	Os2	−27.35		Mn3	Mn4	28.61		Mn4	Os1	17.05		Os1	Os2	−6.61

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Figure  4.  Magnetism of Mn₂FeOsO₆: (a) calculated magnetization and magnetic susceptibility via classical Monte Carlo simulation (The red line represents the theoretical total magnetic moment M divided by saturation magnetic moment M_max, while the blue line represents the relative magnetic susceptibility χ, both indicating the phase transition positions around temparature T about 480 K.); (b) simulated excitation spectrum using SpinW package (The vertical axis represents the energy of the excitation spectrum,and the colors correspond to the intensity of the excitation spectrum.)

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In Fig. 4(a), the magnetization and magnetic susceptibility of Mn₂FeOsO₆ were calculated using classical Monte Carlo simulations with Metropolis algorithm. It was observed that the magnetic susceptibility curve has two peaks, indicating the possible existence of two different AFM T_N in this system, consistent with previous experiments^[29]. We propose that this phenomenon is due to two sets of AFM atoms at A and B sub-lattices. The higher AFM T_N is directly dependent on the Mn ion at A-sites. The estimated T_N is above 680 K, which is higher than the T_N (or T_C ) observed in related experiments. It is noteworthy that the estimated transition temperature is relatively conservative. The actual transition temperature would be further increased if the MAE resulting from shape and DM interaction is considered.

Additionally, the electronic band structure of Mn₂FeOsO₆ was computed, shown in Fig. 3(b). Due to the low lattice symmetry, the energy bands are complex and disordered. However, in contrast to other TMO double perovskites, Mn₂FeOsO₆ exhibits metallic properties, with a high DOS near the Fermi surface. The DOS indicates that the energy band near the Fermi level is primarily contributed by the 3d electrons of the magnetic ions. Furthermore, the system presents an AFM configuration, where the spin up and down DOS are consistent, and the total magnetization equals zero. Thus, Mn₂FeOsO₆ is an ideal AFM metallic material.

Finally, the excitation spectrum mode of the above spin model was computed using the spin-wave method. The excitation mode, shown in Fig. 4(b), reveal clear AFM imprints in the <100> direction, consisting of two sets of different AFM excitation spectra, supporting the previously estimated AFM configuration obtained by our Monte Carlo method.

5. Conclusions

In conclusion, we extend a method that can determine the complex magnetism configuration and synthesis condition of double perovskite materials at the same time consuming relatively low computational resources. Our method is shown to provide satisfactory results for several known double perovskite compounds.

We apply this method to investigate the promising material Mn₂FeOsO₆, which has not yet been experimentally synthesized. Our analysis reveals that this material has a P2₁/n structural configuration with a high density of states near the Fermi level, making it an ideal AFM metal with a total magnetic moment of zero. Using a set of matrix operations, we are able to obtain the complex magnetic exchange interaction of the P2₁/n structure, and identify two sets of AFM T_N. Our findings are further verified by classical Monte Carlo simulations and spin wave excitation spectrum analysis.

Overall, we theoretically identify an ideal AFM metal with extremely high AFM T_N, which has promising applications in the field of spintronics. Our method is an effective approach to study complex magnetism in double perovskite materials and can be extended to investigate other related materials.