Multifunctional antiperovskites driven by strong magnetostructural coupling

Abstract

Based on density functional theory calculations, we elucidated the origin of multifunctional properties for cubic antiperovskites with noncollinear magnetic ground states, which can be attributed to strong isotropic and anisotropic magnetostructural coupling. Of 54 stable magnetic antiperovskites M₃XZ (M = Cr, Mn, Fe, Co, and Ni; X = selected elements from Li to Bi except for noble gases and 4f rare-earth metals; and Z = C and N), 14 are found to exhibit the Γ_4g/Γ_5g (i.e., characterized by irreducible representations) antiferromagnetic magnetic configurations driven by frustrated exchange coupling and strong magnetocrystalline anisotropy. Using the magnetic deformation as an effective proxy, the isotropic magnetostructural coupling is characterized, and it is observed that the paramagnetic state is critical to understand the experimentally observed negative thermal expansion and to predict the magnetocaloric performance. Moreover, the piezomagnetic and piezospintronic effects induced by biaxial strain are investigated. It is revealed that there is not a strong correlation between the induced magnetization and anomalous Hall conductivities by the imposed strain. Interestingly, the anomalous Hall/Nernst conductivities can be significantly tailored by the applied strain due to the fine-tuning of the Weyl points energies, leading to promising spintronic applications.

Introduction

Smart materials like multiferroic materials with enhanced coupling between different degrees of freedom (e.g., mechanical, electronic, and magnetic) are promising for engineering devices for future applications such as sensors, transducers, memories, and spintronics^1,2,3. Cubic antiperovskite (APV) compounds host the two most appealing aspects of multiferroics, e.g., magnetoelectric coupling and piezomagnetic effect (PME)^3,4. In APV materials, the strong magnetoelectric coupling is achieved by combining piezoelectric and piezomagnetic heterostructure composites^5,6,7. A significant PME is reported for Mn-based nitrides like Mn₃SnN, making such compounds a suitable component for fabricating magnetoelectric composite^8,9. The PME in APVs can be attributed to the strong magnetostructural coupling, which manifests itself also as giant negative thermal expansion (NTE)^10,11,12 and magnetocaloric/barocaloric effect^{9,13,14,15,16,17}. From the materials perspective, many Mn-based APV carbides go through a first-order magnetic phase transition and possess a large magnetocaloric effect^14,18. For instance, Mn₃GaC exhibits a huge magnetic entropy change (ΔS_M) of 15 J/kgK under an applied magnetic field of 2T¹⁶. The strong magnetostructural coupling in APVs is driven by the cubic-to-cubic first-order transition wherein a change in the crystal volume brings about a change in the frustrated magnetic states. Last but not least, APVs have been investigated recently due to the presence of a treasury of multifunctionality such as superconductivity¹⁹, thermoelectric²⁰, magnetostriction²¹.

Particularly, from the topological transport properties point of view, the existence of finite anomalous Hall conductivity (AHC) in noncollinear antiferromagnets has attracted noticeable attention due to possible applications in AFM spintronics for information storage and data processing^22,23,24,25. The spin-dependent transport phenomena can provide spin-polarized charge current and large pure spin current, which could be achieved premised on two fundamental properties, i.e., AHC and spin Hall conductivity. The kagome lattice turns out to be an elementary model to host giant AHC^26,27,28. Recently, Mn-based APV nitrides have been proposed to exhibit large AHC in frustrated AFM kagome lattice^29,30,31,32. It is observed that Mn₃GaN exhibits vanishing and non-vanishing AHC for two different magnetic ordering Γ_5g and Γ_4g, respectively^29,30. In this regard, for magnetic materials with noncollinear AFM ground states, the non-vanishing AHC is only feasible with specific magnetic space group symmetry, i.e., the AHC tensor depends on the magnetic group symmetry³³. For instance, Mn₃X (X = Ga, Ge, and Sn) and Mn₃Z (Z = Ir, Pt, and Rh) have been reported to have a different form of AHC tensor as a result of different magnetic ordering^22,34,35,36. A possible phase transition (Γ_5g ↔ Γ_4g) attained by strain or chemical modification could make these materials suitable for AFM spintronic applications. The spin-polarized current could also be generated by temperature gradient instead of the applied electric fields, resulting in anomalous Nernst conductivity (ANC), also termed as spin caloritronics^37,38. A large ANC in noncollinear AFMs could be useful for establishing spin caloritronics devices that exhibit useful prospects in energy conversion and information processing. Noticeably, a large ANC of 1.80 AK⁻¹m⁻¹ has been reported for APV Mn₃NiN at 200 K³⁹, which is slightly less than half of the highest reported ANC of 4.0 AK⁻¹m⁻¹ in Co₂MnGa^40,41. Enhancing the AHC and ANC by applying strain could be crucial for realizing AFM spintronic devices.

In this work, we carried out a systematic analysis of 54 cubic APV systems (Pm$\bar{3}$m) with chemical formula M₃XZ (see Supplementary Fig. 1) to determine their magnetic ground states, tunability via biaxial strain, and the resulting spintronic properties (Fig. 1). The selected 54 cubic APVs are stable, i.e., fulfilling all three stability criteria such as thermodynamical, mechanical, and dynamical stabilities, which were recently evaluated in our high-throughput screening⁴². Explicit calculations were performed to obtain the energies of eight phases, i.e., Γ_4g, Γ_5g, non-magnetic (NM), ferromagnetic (FM), collinear AFM-1 (cAFM-1), collinear AFM-2 (cAFM-2), collinear AFM-3 (cAFM-3), and M-1 configurations. Moreover, the magnetic anisotropy energy (MAE) defined as the energy difference between Γ_5g and Γ_4g is examined to understand the origin of the noncollinear magnetic states with the help of spin-orbit coupling (SOC) energy. Moreover, a detailed analysis of the lattice constant variation with respect to the magnetic states reveals that the paramagnetic (PM) state is critical in the magnetic phase transition, enabling us to predict potential NTE and magnetocaloric materials. Last but not least, the PME was studied by introducing biaxial strains (compressive and tensile), which causes possible phase transitions between Γ_5g ↔ Γ_4g, and leads to a significant modification in the AHC and ANC, dubbed as a piezospintronic effect^30,43. In-depth analysis on symmetry analysis and electronic structure suggests that the piezospintronic effect is originated from the existence of Weyl nodes whose position can be tailored by strain, resulting in promising applications for future spintronic devices.

Results

Validation and prediction of magnetic ground state

As the magnetic transition metal atoms are located at the face centers of the cubic cell for magnetic APVs, it leads to a frustrated kagome lattice in the (111)-plane. Correspondingly, noncollinear magnetic structures are expected when the interatomic exchange interaction is AFM for the nearest neighbors. As shown in Fig. 2, Γ_4g and Γ_5g are the two most common magnetic configurations reported for APVs, resulting in 120° magnetic angle configurations within the (111)-plane between three magnetic moments. The Γ_4g state can be obtained from Γ_5g by simultaneously rotating the moments of three metal atoms within the (111)-plane by 90°. The APVs with the noncollinear magnetic ground state are listed in Table 1, in comparison to the available experimentally measured results. Interestingly, all the APVs with noncollinear magnetic ground states are nitrides including Cr and Mn, while all carbides end up with the FM and cAFM-2 ground state except Mn₃SnC, which exhibits the Γ_5g state (see Supplementary Table 1).

Fig. 1: Abbreviations.

The list of abbreviations used in the manuscript.

Fig. 2: Magnetic configurations.

Possible magnetic structures of antiperovskites: a Γ_5g, b Γ_4g, c FM, d PM, e cAFM-1, f cAFM-2, g cAFM-3, and h M-1. The spin chirality for Γ_4g and Γ_5g is assumed to be +1.

Table 1 Compilation of the APVs (M₃XZ) with noncollinear magnetic ground state.

Mn₃GaC⁴⁴ and Mn₃Ga_0.95N_0.94¹⁵ are reported to adopt collinear AFM and M-1 magnetic configurations, respectively. Thus, we considered additionally cAFM-1, cAFM-2, cAFM-3, and M-1 configurations (see Fig. 2e–h). It is confirmed that Mn₃GaC exhibit the cAFM-2 magnetic ground state, the same as Mn₃AlC and Mn₃InC. This is in contrast to the previous experimental reports on both Mn₃AlC⁴⁵ and Mn₃InC⁴⁶ with a FM configuration. Moreover, Cr₃SnN and Mn₃SnN exhibit the cAFM-3 magnetic ground state, where it is demonstrated that Mn₃SnN shows four different magnetic and crystallographic phases (see Supplementary Note 1)^47,48. Such discrepancies might be attributed to the temperature effect, indicating a strong interplay between magnetism and crystal structures at finite temperatures that deserves further detailed investigations. Interestingly, none of the APV is stabilized in the M-1 phase (see Supplementary Table 1). It is noted that there are also other magnetic configurations such as ferrimagnetic and canted states for Mn₃XC (X = Sn and Zn), which will be saved for future investigations.

The magnetic ground states of noncollinear APVs are in good agreement with the experimental measurements. For instance, both Mn₃GaN¹³ and Mn₃ZnN^47,49 have the Γ_5g magnetic ground state, which are consistent with our density functional theory (DFT) calculations. Interestingly, many Mn-based APVs exhibit a mixed Γ_4g + Γ_5g magnetic ordering, with a possible meta-magnetic transition to the other magnetic phases⁴⁷. For instance, Mn₃AgN is characterized by two distinct magnetic phase transitions. A mixed Γ_4g + Γ_5g phase exists below 55 K, whereas pure Γ_5g state persists at intermediate temperature range (55 K < T < 290 K), and lastly, it undergoes the magnetic transition to PM state at 290 K⁴⁷. Likewise, Mn₃NiN displays such mixed magnetic phases⁴⁷.

The mixed magnetic ordering can be attributed to the MAE between the Γ_5g and Γ_4g states, which can be expressed as:

$${\rm{MAE}}={{\rm{E}}}_{{{{\Gamma }}}_{5g}}-{{\rm{E}}}_{{{{\Gamma }}}_{4g}},$$

(1)

where E indicates the total energy of the corresponding magnetic configuration. For Mn₃AgN and Mn₃ NiN with experimentally observed mixed magnetic states, the corresponding MAE is 0.063 and 0.144 meV, respectively. Nevertheless, it is still unclear why Mn₃AgN is stabilized in the Γ_5g state between 55 and 290 K. We suspect that those compounds with MAE greater than 0.2 meV between Γ_4g and Γ_5g should have pure noncollinear magnetic states. In our study, the Mn-based APVs Mn₃XN with X = Ag, Co, Ir, Ni, Pd, and Rh exhibit Γ_4g while those with X = Au, Ga, Hg, In, Pt, and Zn display the Γ_5g as the magnetic ground state (see Table 1 and Supplementary Table 1). This is consistent with the recent DFT calculations for Mn₃XN (X = Ni, Zn, Ga, and Pt)⁵⁰, except for Mn₃XN (X = In, Pd, and Ir). In ref. ⁵⁰, Mn₃InN exhibit Γ_4g state while Mn₃XN (X = Ir and Pd) display Γ_5g state. The MAE for Mn₃InN calculated by Huyen et al. is 74.6 meV, which is much larger than our MAE (–1.84 meV). As detailed below, a large MAE could be expected for the compound with strong SOC; thus, it is questionable for Mn₃InN with such a large MAE of 74.6 meV⁵⁰. To clarify the discrepancy, we performed calculations of the MAE using the Quantum Espresso (QE) code⁵¹ with the same parameters consider in Huyen et al. study. The QE and VASP calculations provide the same magnetic ground state and MAE for Mn₃XN (X = Ir and Pt) systems, which is in contradiction to the Huyen study performed using QE⁵⁰.

Lastly, based on our high-throughput calculations⁴², two unreported Cr-based APVs are stable, our calculations reveal that Cr₃IrN and Cr₃PtN have Γ_4g as the lowest energy magnetic configuration (see Table 1 and Supplementary Table 1). While among APV nitrides, Mn₃AuN and Mn₃HgN are the other two unreported noncollinear systems with Γ_5g magnetic ground state. To the best of our knowledge, there is no experimental neutron diffraction measurement available till now for such APVs, making them interesting for future studies. To shed more light on the origin of MAE, we performed detailed calculations of the SOC energy in order to obtain atomically resolved contributions to the MAE. Like MAE, the change in the total SOC energy (ΔE_SOC) of M₃XN can be defined as the difference between the SOC energy (E_SOC) of the Γ_5g and Γ_4g magnetic states⁵².

$${{\Delta }}{{\rm{E}}}_{{\rm{SOC}}}={{\rm{E}}}_{{\rm{SOC}}}({{{\Gamma }}}_{5g})-{{\rm{E}}}_{{\rm{SOC}}}({{{\Gamma }}}_{4g}),$$

(2)

In the same way, the atomically resolved change of SOC energy (δE_SOC) can also be defined for each atomic species, as shown in Table 1 and Supplementary Table 3. The positive (negative) values of MAE and ΔE_SOC indicate the Γ_4g (Γ_5g) magnetic ground state. On the one hand, it is found that the sign of MAE is the same as that of ΔE_SOC (see Table 1). That is, both the ΔE_SOC and MAE are equally valid to characterize the magnetic ground states, with a linear scaling behavior observed (see Supplementary Fig. 2). For example, Cr₃PtN has Γ_4g magnetic ground state with an MAE of 2.868 meV and ΔE_SOC of 4.70 meV. The atomic resolved δE_SOC of Pt and Cr are 5.639 and –0.938 meV. Evidently, the δE_SOC of Pt is larger than Cr δE_SOC originating in the strong SOC of Pt. As a result, the δE_SOC of Pt is a determining factor for the MAE and ΔE_SOC sign. On the other hand, for most APVs, the contribution of the X elements to MAE is more significant than that of the magnetic elements M, as indicated by the atomic resolved δE_SOC (see Table 1). Taking the Mn₃XN with X = Co, Rh, and Ir as examples, the MAE are 1.22, 1.70, and 10.55 meV, corresponding to the dominant δE_SOC of X as 1.72, 3.46, and 20.31 meV, respectively. It is noted that δE_SOC of Mn is smaller than 3% of ΔE_SOC for such compounds. This can be attributed to the enhanced strength of atomic SOC of the X elements, e.g., the strength of atomic SOC for Co, Rh, and Ir atoms is about 0.065, 0.152, and 0.452 eV, respectively⁵³. Therefore, the enhanced atomic SOC strength of the X elements is favorable for the strong MAE of the APV compounds.

NTE and barocaloric

Turning now to the magnetostructural coupling, which can be best represented for APVs by the NTE, magnetocaloric, and PME. NTE materials exhibit contraction in the lattice parameters with respect to temperature in contrast to most materials.⁴⁹ To study NTE of APVs, we evaluated the relative change in the lattice constants Δa/a₀ on the transition from the noncollinear magnetic ground state to PM state (where a₀ is the lattice constant of Γ_4g or Γ_5g noncollinear state and Δa is lattice constant difference between PM and Γ_4g or Γ_5g state) and compared with the Δa/a₀ values obtained from the experimental measurement at their transition temperature (see Fig. 3a)^11,12,49. The average of the evaluated Δa/a₀ is in good agreement with the experimental observation except for Mn₃GaN (see 3a). Together with the mismatch for the other cases, we suspect it may be due to the finite temperature effect, as our DFT calculations are done at zero Kelvin.

Fig. 3: Negative thermal expansion and barocaloric.

a The relative change in the lattice constant (Δa/a₀) and b the magnetic deformation ∑_M for the Cr₃X N and Mn₃XN antiperovskites (X and X are the elements present on the x-axis in violet and black, respectively). The lattice constants of noncollinear, FM, PM, and NM states are used to determine Δa/a₀ and ∑_M.

Importantly, it is found that the PM state is critical and cannot be approximated as the NM state, e.g., the Δa/a₀ between the noncollinear and NM states is about five times as large as the experimental value (see Fig. 3a). An ameliorated estimation could be achieved by defining the PM phase as a collinear AFM arrangement generated by using the disordered local moment (DLM) theory⁵⁴. The Δa/a₀ of Cr₃IrN and Cr₃PtN on the transition from Γ_4g magnetic ground state to PM are –0.0053 and –0.0067, respectively. That is, Cr-based APVs undergo the lattice contraction with Δa/a₀ comparable to the reported NTE in Mn-based APV materials such as Mn₃GaN and Mn₃ZnN (see Fig. 3a). Furthermore, the equilibrium lattice constants decrease and increase for the Mn₃XN (X = Ga, Hg, In, Ni, Pd, Pt, and Zn) and Mn₃XN APVs (X = Au, Co, Ir, and Rh) on the transition from a noncollinear magnetic ground state to PM state, respectively (see Fig. 3a).

Recently, Bocarsly et al. proposed a magnetic deformation proxy ∑_M to predict the magnetocaloric materials as the theoretical evaluation of (ΔS_M) is very challenging⁵⁵. This proxy measures the volume deformation of the NM and magnetic structure, which correlates well with the experimentally measured ΔS_M values. In their study, they screened 167 FM materials, including 7 APVs. For the APVs, it is noted that ∑_M scales linearly with the percentage volume difference ΔV_M between the FM and NM unit cell (see Supplementary Fig. 3). Inspired by their work, we evaluated ∑_M for the APVs with FM and noncollinear magnetic ground state in order to estimate the potential magnetocaloric effect. For FM APVs, it is found that the ∑_M is most significant for the Fe-based APVs within the 1.2–1.7% range (see Supplementary Table 5). For example, the ∑_M of Fe₃RhN, Fe₃PtN, and Fe₃NiN are 1.67%, 1.59%, and 1.54%, respectively, whereas Co-, Mn-, and Ni-based APVs have ∑_M < 1.0% except for Mn₃ZnC (1.33%) (see Supplementary Table 5). Thus, we expect a significant magnetocaloric effect in Fe₃RhN, Fe₃PtN, and Fe₃NiN, based on the argument that the ∑_M approximate to 1.5% could exhibit a considerable magnetocaloric effect⁵⁵.

As to the noncollinear APVs, we evaluated the ∑_M by considering four ΔV_M combinations, namely, the noncollinear magnetic ground state with PM and NM states (NC-PM and NC-NM), the FM with PM and NM states (FM-PM and FM-NM) (see Fig. 3b and Supplementary Table 6). It is observed that the ∑_M value is small by using the ΔV_M of the NC-PM and FM-PM states, in comparison to the other two combinations (see Fig. 3b). Nevertheless, a small value of ∑_M does not mean a low ΔS_M⁵⁵. For instance, the experimentally measured ΔS_M of Mn₃GaN is very significant (22.3 Jkg⁻¹K⁻¹), and it also exhibits giant barocaloric effect¹³. Overall, the ∑_M of Mn₃GaN should be largest among all the APVs. The calculated ∑_M of Mn₃GaN is largest 0.805% for the Γ_5g and PM state combination, which verifies the experimental observation, whereas the ∑_M of Mn₃GaN for the other three combinations is not the largest among the APVs (see Fig. 3b). Therefore, it can be concluded for the noncollinear systems that the approach of Bocarsly et al. proxy does not provide reasonable ∑_M obtained from the ΔV_M combination of NC-NM, FM-NM, and FM-PM states. The implicit correlation of ΔS_M with ∑_M indicates the strong magnetostructural coupling in APVs with significant ΔV_M (NC-PM). This conduces to the prediction of possible NTE and magnetocaloric materials with sizable ∑_M by virtue of significant ΔV_M (see Supplementary Table 6). For example, the Cr₃XN APVs (X = Ir and Pt) with appreciable ∑_M could exhibit potential applications as NTE and magnetocaloric materials.

Piezomagnetism

PME provides another effective characterization of the magnetostructural coupling⁶, which manifests itself as the response of magnetization to strain. The origin of PME is explained based on symmetry together with an illustration of the magnetic spin directions (see Supplementary Fig. 4 and Supplementary Note 2). For APVs with noncollinear magnetic ground states, the total bulk magnetization is vanishing, but a net magnetization can be induced under finite compressive/tensile strain, as reported for Mn-based APVs^6,8. Our calculations confirm their results on the Mn-based APVs except for the Mn₃CoN and Mn₃RhN (see Supplementary Fig. 5), where the resulting net magnetization from our (Zemen et al.)⁸ calculations are 0.646 (0.305) and 0.214 (−0.143) μ_B/f.u., respectively. Interestingly, for Mn₃CoN, a net magnetic moment of 0.362 μ_B/f.u. is induced at Co atoms under 1% tensile strain, whereas the local magnetic moment of Co is zero in the cubic unstrained state. The magnetic moment direction of Co is aligned in (111) plane similar to Mn atom direction in the Γ_4g state. As a result, Mn₃CoN could be an interesting material for elastocaloric and magneto-elastic applications. Surprisingly, the PME effect is asymmetric with respect to the applied compressive and tensile strains (see Supplementary Figs. 6 and 7). Most APVs exhibit more significant PME with tensile strain except Cr₃PtN (see Supplementary Fig. 6), e.g., the net magnetization for Cr₃PtN at 1% compressive strain is as large as 0.21 μ_B/f.u. in comparison to 0.15 μ_B/f.u. at 1.0% tensile strain.

The magnitude of the magnetoelectric effect is minimal in the intrinsic bulk AFM materials, such as Cr₂O₃⁵⁶. The two-phase heterostructure materials consolidate the magnetoelectric effect. A recent study corroborates APV as a potential material for magnetoelectric composite^6,57. The piezoelectric perovskite could be one suitable substrate for such composite heterostructure as they also have comparable lattice parameters with APVs^5,58. The heterostructure amalgamate the piezoelectric and piezomagnetic properties, which are coupled by an interfacial strain. Based on our PME analysis, we propose Cr- and Mn-based noncollinear systems with significant PME as potentials candidates for magnetoelectric composite.

Interestingly, the biaxial strain can induce phase transition between different magnetic configurations, i.e., Γ_4g and Γ_5g, which can further lead to a significant change in the transport properties as discussed below. It is found that a few APV materials undergo a magnetic phase transition due to the imposed biaxial strain (see Supplementary Tables 4 and 7). For instance, cubic Mn₃AuN has Γ_5g as the magnetic ground state, which preserves under tensile strain. However, a transition into a distorted Γ_4g state is obtained by applying a compressive strain of 0.5 and 1.0% (see Supplementary Table 4). Such transitions can be attributed to significant changes in the MAE and with respect to the biaxial strain (see Supplementary Table 4), but there is no consistent MAE trend as the absolute value of MAE is mostly determined by the atomic SOC strengths.

Anomalous Hall conductivity (unstrained)

It is well known that FM compounds commonly exhibit AHC due to the presence of a net magnetization⁵⁹, where the broken time-reversal symmetry (${\mathcal{T}}$) and SOC are two essential prerequisites. Noncollinear AFMs also display non-zero AHC, as observed in Mn₃X (X = Ir, Ge, and Sn)^22,34. As a linear response property, the occurrence of AHC can be elucidated based on the symmetry analysis. For cubic APVs, the magnetic space group of Γ_5g and Γ_4g are $R\bar{3}m$ (166.97) and $R\bar{3}m^{\prime}$ (166.101), respectively. For Γ_5g, the local magnetic moment spin directions of M atoms are invariant under the mirror symmetry transformation M${}_{(0\bar{1}1)}$, M${}_{(10\bar{1})}$, and M${}_{(\bar{1}10)}$, leading to vanishing AHC (see Supplementary Table 8)^29,30, whereas in the Γ_4g state, the above-mentioned mirror symmetries are broken, but the product of mirror (M) and time-reversal symmetry ${\mathcal{T}}$ retain the Γ_4g configuration. This results in finite AHCs with all three off-diagonal components non-zero but of the same amplitude (see Supplementary Table 8)²⁹.

Explicit calculations confirm the above symmetry arguments. For instance, the AHC is zero for Mn₃XN (X = Au, Hg, In, Pt, and Zn) with the Γ_5g ground state (see Supplementary Figs. 8b and 9), whereas the significant AHC is observed for the APVs with the Γ_4g ground state, such as the AHC of Cr₃IrN and Cr₃PtN are 414.6 and 278.6 S/cm, respectively. The AHC of APVs is summarized in the Supplementary information (see Supplementary Figs. 8–10 and 16). In comparison to the previous studies on an individual or a few compounds^29,32, our results are in good agreement (see Supplementary Table 9). For instance, the AHC of Mn₃SnN from our and calculations by Gurung et al. are 106.5 and 133 S/cm²⁹, respectively, whereas the value (−73.9 S/cm) obtained by Huyen et al.⁵⁰ is of opposite sign in addition to the difference in the absolute values. In this regard, for Mn₃NiN, the sign of experimentally observed AHC is confirmed by our DFT calculations³¹, consistent with that obtained by Zhou et al^32,39. This might be due to opposite magnetization direction while maintaining the noncollinear configurations or chirality of noncollinear spin configurations specified in the calculations, subject to further investigations. Moreover, it is observed that the absolute values of AHC for the same compound in the same magnetic ground state are scattered as well (see Supplementary Table 9). We suspect that this is due to the different lattice constants and numerical parameters used in different calculations. For instance, Zhou et al. performed calculations using the experimental lattice constants on Mn₃NiN and other APV compounds, which are on average about 1.0–1.3% smaller than the fully relaxed lattice constants in this work.

Strain-induced AHC (piezospintronics)

As discussed above, all cubic APVs with noncollinear magnetic ground states display significant PMEs where a net magnetization can be induced by applying biaxial strain; an interesting question is whether the AHC can be tailored as well. From the symmetry point of view, for the Γ_5g state, the magnetic space group changes from $R\bar{3}m$ (166.97) to C2/m (12.58) with biaxial strain, resulting in finite σ_x and σ_y with the same amplitude but opposite sign, while the σ_z component remains zero³⁰. This is confirmed by our calculations on those compounds with the Γ_5g ground state. For instance, the AHC induced by 1% biaxial strain is as large as −71.2 S/cm for Mn₃PtN (see Supplementary Fig. 8b), comparable to that in Mn₃HgN (see Supplementary Fig. 9). The Mn₃InN and Mn₃ZnN have quite low AHC due to weak SOC strength (see Supplementary Fig. 9).

Similarly, the AHC of APVs in the Γ_4g state can also be tailored by the biaxial strain. In this case, the magnetic space group is reduced from $R\bar{3}m^{\prime}$ (166.101) to $C2^{\prime} /m^{\prime}$ (12.62), the resulting σ_x and σ_y components are the same while the σ_z component possesses a distinct value (see Supplementary Fig. 8a and Supplementary Table 8). Explicit evaluations of the AHC for APVs in distorted Γ_4g states verify the symmetry arguments, where the σ_x and σ_y components behave the same with respect to strain (see Supplementary Fig. 8a). Interestingly, it is observed that more significant changes occur in AHC for the Γ_4g compounds than the Γ_5g cases. For instance, the σ_z component of AHC in Cr₃IrN attains the largest AHC of 693.1 S/cm at 1% tensile strain, with an increase of 278 S/cm compared to the value in the cubic geometry (see Supplementary Fig. 8a). Moreover, the moderate strain can even lead to a sign change of the AHC, as observed on the σ_z component of Mn₃AgN and Mn₃CoN with 0.5% compressive and 1% tensile strain, respectively (see Supplementary Fig. 10).

One interesting question is whether the piezospintronic effect (i.e., the variation of AHC induced by biaxial strain) correlates with the PME, as summarized in Fig. 4 for the variation of such quantities comparing the cases with 1% tensile strain and cubic geometries. Obviously, ∣ΔAHC∣ cannot be directly inferred from the induced net magnetization, particularly exemplified by the APVs in the Γ_4g magnetic ground state. For example, the ∣ΔAHC∣ and net magnetization of Cr₃IrN are 278.5 S/cm (σ_z component) and 0.082 μ_B/f.u., respectively, while for Mn₃SnN, the ∣ΔAHC∣ and net magnetization are 35.5 S/cm (σ_z component) and 0.47 μ_B/f.u., respectively. Moreover, for Mn₃CoN, the net magnetization and ∣ΔAHC∣ of σ_x are 250.7 S/cm and 0.646 μ_B/f.u. It signifies that the ∣ΔAHC∣ could be large for the small net magnetization and vice versa (see Fig. 4). Interestingly, the ∣ΔAHC∣ and net magnetization are smaller for those compounds in the Γ_5g magnetic state, in comparison to those of such materials with the Γ_4g magnetic state, e.g., the ∣ΔAHC∣ and net magnetization of Mn₃ZnN are 9.17 S/cm and 0.029 μ_B/f.u., respectively.

Fig. 4: Piezospintronic.

The piezospintronic effect is illustrated by the change in AHC (∣ΔAHC∣) at 1% tensile strain and cubic phase. The ∣ΔAHC∣ and net magnetization is shown for the Cr₃X N and Mn₃XN antiperovskites (X and X are the elements present on the x-axis, respectively). The left, middle, and right panels correspond to the Γ_4g, phase transition between Γ_4g and Γ_5g, and Γ_5g state, respectively.

Particularly, the APVs undergoing a phase transition between Γ_4g and Γ_5g states driven by the epitaxial strain exhibit strong piezospintronic effect while the magnitude of the PME is marginal. For instance, Mn₃ZnN has a Γ_5g magnetic ground state in the cubic phase with vanishing AHC. A phase transition into the Γ_4g state by 1% tensile strain results in a considerable ∣ΔAHC∣ of 152.2 S/cm for the σ_y component (see Fig. 4). On the other hand, the phase transition from the Γ_4g to Γ_5g states for Mn₃PdN leads to a ∣ΔAHC∣ as large as 58.6 S/cm (Fig. 4). Therefore, we suspect that those compounds with Γ_4g states are more promising to host a strong piezospintronic effect.

Anomalous Nernst effect

The intriguing behavior of AHC under biaxial strain can be attributed to the fine-tuning of the electronic structure. To further illustrate such sensitivity, we investigated the ANC, which shares the same symmetry as AHC but is proportional to the derivative of AHC with respect to energy following the Mott formula⁶⁰. Taking Mn₃NiN as an example, the corresponding ANC is as large as 16,569 S/cm.eV. This is consistent with 1.80 AK⁻¹m⁻¹ at 200 K obtained by Zhou et al.³⁹, up to a factor of two but with alike dependence with respect to the energy around the Fermi energy (see Supplementary Fig. 15). The numerical discrepancy might be due to the different lattice constants used in the calculations, as discussed above. Such a large ANC of Mn₃NiN is comparable to the largest values observed experimentally in Co₂FeGe (3.16 AK⁻¹m⁻¹), Co₂MnGa (4.0 AK⁻¹m⁻¹), and Fe₃Ga (3.0 AK⁻¹m⁻¹)^40,41,61,62. Interestingly, the strain also has a stronger influence on the ANC. For the tetragonally distorted Mn₃NiN in the Γ_4g state, the ANC becomes as large as 20,035 S/cm.eV at –0.5% compressive strain for the α_x and α_y components, i.e., enhanced by 21% compared to that of the cubic case. Last but not least, a sign change can be induced in the ANC by the biaxial strain. For instance, the ANC corresponding to the α_x and α_y components of Cr₃IrN changes to –3196.9 S/cm.eV at 1% compressive strain starting from 6823.4 S/cm.eV for the cubic case in the Γ_4g state. This is also observed for Mn₃AgN and Mn₃CoN (see Supplementary Fig. 11). The ANC of APV compounds as a function of biaxial strain is summarized in Supplementary Figs. 11–14.

To explicate the origin of the piezospintronic effects on both AHC and ANC, our detailed analysis of the electronic structure reveals that the tunability of AHC and ANC by strain can be attributed to the presence of Weyl points close to the Fermi energy. Taking Mn₃PdN as an example, as shown in Fig. 5a, the ANC can be tuned between 7037.3 S/cm.eV at 1% compressive strain to −14,599 S/cm.eV at 1.5% compressive strain, with a sign change at 1.3% strain. Such an ANC is comparable to that observed in Mn₃NiN, awaiting further experimental validation. The band structure shows no obvious changes due to the applied strain (see Fig. 5b). However, it is found that there are 12 Weyl points at (0.43, −0.35, and −0.49) and equivalent k-points with an energy of 20 meV below the Fermi energy in the cubic phase. Such Weyl points will be shifted across the Fermi energy upon applying compressive strain, e.g., reaches to −1.75 meV for −1.3% biaxial strain, and 1.87 meV above the Fermi energy for 1.4% strain (see above inset of Fig. 5a). The AHC is mostly enhanced when the Weyl points are located at the Fermi energy, due to the singular behavior of the Berry curvature at the Weyl nodes (see below inset of Fig. 5a). This is consistent with ref. ⁵⁰ and our observation in Mn₃GaN^30,50. For magnetic materials, the Weyl nodes with opposite chiralities (i.e., Berry curvatures with opposite sign) are located at the same energy, but the total contributions to the AHC can be significant, particularly if the Weyl nodes are within “several” or “a few” meV around the Fermi energy. Correspondingly, the sign of ANC, which is determined by the derivative of AHC can be tuned as the Weyl points are shifted across the Fermi level. We suspect such tunability of AHC and concomitant ANC by manipulating the Weyl points with the biaxial strain or other possible stimuli is promising for future applications, such as Fe₃Al with giant transversal thermoelectric effects⁶².

Fig. 5: Anomalous Nernst effect.

a The calculated ANC of Mn₃PdN in the Γ_4g magnetic state. The Weyl points (0.43, −0.35, −0.49, and equivalent coordinates) shift across the Fermi level with strain (inset above) and the Berry curvature for the 1.4% compressive biaxial strain (inset below). b The band structure of Mn₃PdN for unstrained (black) and 1.4% compressive biaxial strained (blue). The presence of Weyl point near the Fermi energy is indicated by black (cubic) and blue dashed line (−1.4% biaxial strain).

Discussion

We carried out a systematic analysis of 14 cubic APV M₃XZ compounds with noncollinear magnetic ground states, focusing on the magnetic properties driven by isotropic and anisotropic magnetostructural coupling. It is found that there exists a strong competition between different noncollinear magnetic configurations where a large MAE clearly defines the noncollinear magnetic ground state, while small MAE leads to the mixed Γ_4g and Γ_5g configurations. For such materials with mixed magnetic ions, the MAE cannot be understood based on the perturbation theory, whereas the SOC energy can be used to get a reliable atomic resolved contribution, which is mostly determined by the strength of atomic SOC. The magnetic ground state analysis resulted in the prediction of four unreported APVs with noncollinear ground state, especially Cr₃XN APVs (X = Ir and Pt) with the Γ_4g magnetic configuration. The isotropic magnetostructural coupling indicated by the magnetic deformation Δa/a₀ can be considered as an effective descriptor for the magnetocaloric effect. However, we observed that the magnetic deformation is better measured comparing the PM and magnetically ordered states, resulting in better agreement with the experimental NTE results, rather than comparing the NM and ordered state. Therefore, we suggest that the recently proposed proxy for predicting magnetocaloric effects based on the magnetic deformation could be improved by using quasi-random approximation of the PM state. More interestingly, biaxial strain not only causes a significant PME in such materials but also leads to a strong influence on MAE, AHC, and ANC after considering SOC. Based on detailed symmetry analysis, we performed an explicit evaluation of the AHC and found that those compounds with the Γ_4g magnetic ground state have large AHC, which is susceptible to the epitaxial strain. Nevertheless, there is no strong correlation between the net magnetization and induced AHC when finite strain is applied. Detailed analysis reveals that the sensitivity of AHC and derived ANC with respect to strain can be attributed to the fine-tuning of energies for the Weyl nodes, opening up further possibilities for engineering spintronic devices in the future.

Methods

Density functional theory calculation

Our DFT calculations are performed using the projector augmented wave method as effectuated in the VASP package⁶³. The exchange-correlation functional is approximated using the generalized gradient approximation as parameterized by Perdew–Burke–Ernzerhof⁶⁴. We used an energy cutoff of 500 eV for the plane-wave basis set, and a uniform k-points grid of 13 × 13 × 13 within the Monkhorst-pack scheme for the Brillouin zone integrations. The Methfessel–Paxton scheme is used to determine the partial occupancies of orbitals with a smearing width of 0.06 eV. The SOC is considered in all the calculations. In order to verify the magnetic ground state of various compounds, the more accurate total energy calculations are performed using a higher energy cutoff of 600 eV and a dense k-mesh of 25 × 25 × 25 (see Supplementary Note 3).

The magnetic ground states are obtained by comparing the total energies (see Supplementary Table 1) of eight magnetic configurations, i.e., Γ_4g, Γ_5g, NM, FM, cAFM-1, cAFM-2, cAFM-3, and M-1 states (see Fig. 2), where the lattice constants are fully optimized for Γ_4g, Γ_5g, FM, PM, and NM state (see Supplementary Table 2). To verify the effects of the Coulomb interaction, we performed a series of DFT+U (U = 2–10 eV for the Mn-3d orbitals) calculations on Mn₃GaN. It is observed that the Γ_5g magnetic ground state does not change with respect to the applied U values (see Supplementary Table 10). Therefore, we suspect that the Coulomb interaction is likely to have a marginal influence on the magnetic ground state of such a class of compounds. Moreover, to investigate the piezomagnetic and piezospintronic effects, biaxial in-plane strain is applied, which reduces the crystalline symmetry from cubic to tetragonal. The optimal lattice constants along c-direction are evaluated by the polynomial fitting of the energies from a series of calculations.

Modeling of paramagnetic state

The PM state is modeled based on the DLM picture (see Fig. 2d), where a special quasi-random structure modeled using a 2 × 2 × 2 supercell by imposing zero total magnetization⁶⁵. That is, a supercell with the same number of up and down moments is considered with a pair correlation function same as A₅₀B₅₀ random alloys, generated using the alloy theoretic automated toolkit code⁶⁶. It is noted that our results reveal that systematic treatment of the PM state is needed in order to provide a quantitative description of the magnetostructural coupling in such compounds, where the recently developed spin-lattice dynamics approach is promising for future detailed investigations⁶⁷.

Anomalous Hall and Nernst conductivity calculation

The AHC is evaluated using the WannierTools code⁶⁸, where the required accurate tight-binding models are obtained by the maximally localized Wannier functions (MLWFs) using the Wannier90 code⁶⁹. The s, p, and d orbitals of M and X atoms and the s and p orbitals for N atom are considered, resulting in 80 MLWFs in total for every noncollinear APVs. The AHC is computed by integrating the Berry curvature on a uniform 401 × 401 × 401 k-points mesh to guarantee good accuracy, which can be expressed as⁷⁰:

$${\sigma }_{xy}=-\frac{{e}^{2}}{\hslash }\int \frac{d{\bf{k}}}{{\left(2\pi \right)}^{3}}\mathop{\sum}\limits_{n}f\left[\right.\epsilon \left({\bf{k}}\right)-\mu \left]\right.{{{\Omega }}}_{n,xy}\left({\bf{k}}\right)$$

(3)

$${{{\Omega }}}_{n,xy}\left({\bf{k}}\right)=-2{\rm{Im}}\mathop{\sum}\limits_{m\ne n}\frac{\langle {\psi }_{{\bf{k}}n}| {v}_{x}| {\psi }_{{\bf{k}}m}\rangle \langle {\psi }_{{\bf{k}}m}| {v}_{y}| {\psi }_{{\bf{k}}n}\rangle }{\left[\right.{\epsilon }_{m}\left({\bf{k}}\right)-{\epsilon }_{n}\left({\bf{k}}\right){\left]\right.}^{2}}$$

(4)

where e is elementary charge, μ is the chemical potential, ψ_n/m denotes the Bloch wave function with energy eigenvalue ϵ_n/m, v_x/y is the velocity operator along Cartesian x/y direction, and $f[\epsilon \left({\bf{k}}\right)-\mu ]$ is Fermi-Dirac distribution function. The ANC α_xy is evaluated based on the Mott relation, which yields:

$${\alpha }_{xy}=-\frac{{\pi }^{2}{k}_{B}^{2}T}{3e}\frac{d{\sigma }_{xy}}{d\epsilon }{\left|\right.}_{\epsilon = \mu }$$

(5)

In this work, we evaluated only the derivative of AHC at the Fermi level dσ_xy/dϵ, which provides a quantitative measurements of the ANC.