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 M_{3}XZ (M = Cr, Mn, Fe, Co, and Ni; X = selected elements from Li to Bi except for noble gases and 4f rareearth 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 finetuning 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 Mnbased nitrides like Mn_{3}SnN, 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 Mnbased APV carbides go through a firstorder magnetic phase transition and possess a large magnetocaloric effect^{14,18}. For instance, Mn_{3}GaC exhibits a huge magnetic entropy change (ΔS_{M}) of 15 J/kgK under an applied magnetic field of 2T^{16}. The strong magnetostructural coupling in APVs is driven by the cubictocubic firstorder 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 superconductivity^{19}, thermoelectric^{20}, magnetostriction^{21}.
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 spindependent transport phenomena can provide spinpolarized 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, Mnbased APV nitrides have been proposed to exhibit large AHC in frustrated AFM kagome lattice^{29,30,31,32}. It is observed that Mn_{3}GaN exhibits vanishing and nonvanishing 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 nonvanishing AHC is only feasible with specific magnetic space group symmetry, i.e., the AHC tensor depends on the magnetic group symmetry^{33}. For instance, Mn_{3}X (X = Ga, Ge, and Sn) and Mn_{3}Z (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 spinpolarized 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^{−1}m^{−1} has been reported for APV Mn_{3}NiN at 200 K^{39}, which is slightly less than half of the highest reported ANC of 4.0 AK^{−1}m^{−1} in Co_{2}MnGa^{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 M_{3}XZ (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 highthroughput screening^{42}. Explicit calculations were performed to obtain the energies of eight phases, i.e., Γ_{4g}, Γ_{5g}, nonmagnetic (NM), ferromagnetic (FM), collinear AFM1 (cAFM1), collinear AFM2 (cAFM2), collinear AFM3 (cAFM3), and M1 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 spinorbit 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}. Indepth 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 cAFM2 ground state except Mn_{3}SnC, which exhibits the Γ_{5}_{g} state (see Supplementary Table 1).
Mn_{3}GaC^{44} and Mn_{3}Ga_{0.95}N_{0.94}^{15} are reported to adopt collinear AFM and M1 magnetic configurations, respectively. Thus, we considered additionally cAFM1, cAFM2, cAFM3, and M1 configurations (see Fig. 2e–h). It is confirmed that Mn_{3}GaC exhibit the cAFM2 magnetic ground state, the same as Mn_{3}AlC and Mn_{3}InC. This is in contrast to the previous experimental reports on both Mn_{3}AlC^{45} and Mn_{3}InC^{46} with a FM configuration. Moreover, Cr_{3}SnN and Mn_{3}SnN exhibit the cAFM3 magnetic ground state, where it is demonstrated that Mn_{3}SnN 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 M1 phase (see Supplementary Table 1). It is noted that there are also other magnetic configurations such as ferrimagnetic and canted states for Mn_{3}XC (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 Mn_{3}GaN^{13} and Mn_{3}ZnN^{47,49} have the Γ_{5g} magnetic ground state, which are consistent with our density functional theory (DFT) calculations. Interestingly, many Mnbased APVs exhibit a mixed Γ_{4g} + Γ_{5g} magnetic ordering, with a possible metamagnetic transition to the other magnetic phases^{47}. For instance, Mn_{3}AgN 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 K^{47}. Likewise, Mn_{3}NiN displays such mixed magnetic phases^{47}.
The mixed magnetic ordering can be attributed to the MAE between the Γ_{5g} and Γ_{4g} states, which can be expressed as:
where E indicates the total energy of the corresponding magnetic configuration. For Mn_{3}AgN and Mn_{3} NiN with experimentally observed mixed magnetic states, the corresponding MAE is 0.063 and 0.144 meV, respectively. Nevertheless, it is still unclear why Mn_{3}AgN 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 Mnbased APVs Mn_{3}XN 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 Mn_{3}XN (X = Ni, Zn, Ga, and Pt)^{50}, except for Mn_{3}XN (X = In, Pd, and Ir). In ref. ^{50}, Mn_{3}InN exhibit Γ_{4g} state while Mn_{3}XN (X = Ir and Pd) display Γ_{5g} state. The MAE for Mn_{3}InN 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 Mn_{3}InN with such a large MAE of 74.6 meV^{50}. To clarify the discrepancy, we performed calculations of the MAE using the Quantum Espresso (QE) code^{51} with the same parameters consider in Huyen et al. study. The QE and VASP calculations provide the same magnetic ground state and MAE for Mn_{3}XN (X = Ir and Pt) systems, which is in contradiction to the Huyen study performed using QE^{50}.
Lastly, based on our highthroughput calculations^{42}, two unreported Crbased APVs are stable, our calculations reveal that Cr_{3}IrN and Cr_{3}PtN have Γ_{4g} as the lowest energy magnetic configuration (see Table 1 and Supplementary Table 1). While among APV nitrides, Mn_{3}AuN and Mn_{3}HgN 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 M_{3}XN can be defined as the difference between the SOC energy (E_{SOC}) of the Γ_{5g} and Γ_{4g} magnetic states^{52}.
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, Cr_{3}PtN 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 Mn_{3}XN 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, respectively^{53}. 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/a_{0} on the transition from the noncollinear magnetic ground state to PM state (where a_{0} 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/a_{0} values obtained from the experimental measurement at their transition temperature (see Fig. 3a)^{11,12,49}. The average of the evaluated Δa/a_{0} is in good agreement with the experimental observation except for Mn_{3}GaN (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.
Importantly, it is found that the PM state is critical and cannot be approximated as the NM state, e.g., the Δa/a_{0} 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) theory^{54}. The Δa/a_{0} of Cr_{3}IrN and Cr_{3}PtN on the transition from Γ_{4g} magnetic ground state to PM are –0.0053 and –0.0067, respectively. That is, Crbased APVs undergo the lattice contraction with Δa/a_{0} comparable to the reported NTE in Mnbased APV materials such as Mn_{3}GaN and Mn_{3}ZnN (see Fig. 3a). Furthermore, the equilibrium lattice constants decrease and increase for the Mn_{3}XN (X = Ga, Hg, In, Ni, Pd, Pt, and Zn) and Mn_{3}XN 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 challenging^{55}. 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 Febased APVs within the 1.2–1.7% range (see Supplementary Table 5). For example, the ∑_{M} of Fe_{3}RhN, Fe_{3}PtN, and Fe_{3}NiN are 1.67%, 1.59%, and 1.54%, respectively, whereas Co, Mn, and Nibased APVs have ∑_{M} < 1.0% except for Mn_{3}ZnC (1.33%) (see Supplementary Table 5). Thus, we expect a significant magnetocaloric effect in Fe_{3}RhN, Fe_{3}PtN, and Fe_{3}NiN, based on the argument that the ∑_{M} approximate to 1.5% could exhibit a considerable magnetocaloric effect^{55}.
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 (NCPM and NCNM), the FM with PM and NM states (FMPM and FMNM) (see Fig. 3b and Supplementary Table 6). It is observed that the ∑_{M} value is small by using the ΔV_{M} of the NCPM and FMPM states, in comparison to the other two combinations (see Fig. 3b). Nevertheless, a small value of ∑_{M} does not mean a low ΔS_{M}^{55}. For instance, the experimentally measured ΔS_{M} of Mn_{3}GaN is very significant (22.3 Jkg^{−1}K^{−1}), and it also exhibits giant barocaloric effect^{13}. Overall, the ∑_{M} of Mn_{3}GaN should be largest among all the APVs. The calculated ∑_{M} of Mn_{3}GaN is largest 0.805% for the Γ_{5g} and PM state combination, which verifies the experimental observation, whereas the ∑_{M} of Mn_{3}GaN 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 NCNM, FMNM, and FMPM states. The implicit correlation of ΔS_{M} with ∑_{M} indicates the strong magnetostructural coupling in APVs with significant ΔV_{M} (NCPM). 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 Cr_{3}XN 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 coupling^{6}, 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 Mnbased APVs^{6,8}. Our calculations confirm their results on the Mnbased APVs except for the Mn_{3}CoN and Mn_{3}RhN (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 Mn_{3}CoN, 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, Mn_{3}CoN could be an interesting material for elastocaloric and magnetoelastic 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 Cr_{3}PtN (see Supplementary Fig. 6), e.g., the net magnetization for Cr_{3}PtN 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 Cr_{2}O_{3}^{56}. The twophase 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 Mnbased 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 Mn_{3}AuN 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 magnetization^{59}, where the broken timereversal symmetry (\({\mathcal{T}}\)) and SOC are two essential prerequisites. Noncollinear AFMs also display nonzero AHC, as observed in Mn_{3}X (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 abovementioned mirror symmetries are broken, but the product of mirror (M) and timereversal symmetry \({\mathcal{T}}\) retain the Γ_{4g} configuration. This results in finite AHCs with all three offdiagonal components nonzero but of the same amplitude (see Supplementary Table 8)^{29}.
Explicit calculations confirm the above symmetry arguments. For instance, the AHC is zero for Mn_{3}XN (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 Cr_{3}IrN and Cr_{3}PtN 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 Mn_{3}SnN from our and calculations by Gurung et al. are 106.5 and 133 S/cm^{29}, 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 Mn_{3}NiN, the sign of experimentally observed AHC is confirmed by our DFT calculations^{31}, 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 Mn_{3}NiN and other APV compounds, which are on average about 1.0–1.3% smaller than the fully relaxed lattice constants in this work.
Straininduced 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 zero^{30}. 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 Mn_{3}PtN (see Supplementary Fig. 8b), comparable to that in Mn_{3}HgN (see Supplementary Fig. 9). The Mn_{3}InN and Mn_{3}ZnN 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 Cr_{3}IrN 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 Mn_{3}AgN and Mn_{3}CoN 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 Cr_{3}IrN are 278.5 S/cm (σ_{z} component) and 0.082 μ_{B}/f.u., respectively, while for Mn_{3}SnN, the ∣ΔAHC∣ and net magnetization are 35.5 S/cm (σ_{z} component) and 0.47 μ_{B}/f.u., respectively. Moreover, for Mn_{3}CoN, 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 Mn_{3}ZnN are 9.17 S/cm and 0.029 μ_{B}/f.u., 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, Mn_{3}ZnN 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 Mn_{3}PdN 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 finetuning 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 formula^{60}. Taking Mn_{3}NiN as an example, the corresponding ANC is as large as 16,569 S/cm.eV. This is consistent with 1.80 AK^{−1}m^{−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 Mn_{3}NiN is comparable to the largest values observed experimentally in Co_{2}FeGe (3.16 AK^{−1}m^{−1}), Co_{2}MnGa (4.0 AK^{−1}m^{−1}), and Fe_{3}Ga (3.0 AK^{−1}m^{−1})^{40,41,61,62}. Interestingly, the strain also has a stronger influence on the ANC. For the tetragonally distorted Mn_{3}NiN 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 Cr_{3}IrN 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 Mn_{3}AgN and Mn_{3}CoN (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 Mn_{3}PdN 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 Mn_{3}NiN, 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 kpoints 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 Mn_{3}GaN^{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 Fe_{3}Al with giant transversal thermoelectric effects^{62}.
Discussion
We carried out a systematic analysis of 14 cubic APV M_{3}XZ 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 Cr_{3}XN APVs (X = Ir and Pt) with the Γ_{4g} magnetic configuration. The isotropic magnetostructural coupling indicated by the magnetic deformation Δa/a_{0} 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 quasirandom 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 finetuning 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 package^{63}. The exchangecorrelation functional is approximated using the generalized gradient approximation as parameterized by Perdew–Burke–Ernzerhof^{64}. We used an energy cutoff of 500 eV for the planewave basis set, and a uniform kpoints grid of 13 × 13 × 13 within the Monkhorstpack 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 kmesh 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, cAFM1, cAFM2, cAFM3, and M1 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 Mn3d orbitals) calculations on Mn_{3}GaN. It is observed that the Γ_{5}_{g} 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 inplane strain is applied, which reduces the crystalline symmetry from cubic to tetragonal. The optimal lattice constants along cdirection 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 quasirandom structure modeled using a 2 × 2 × 2 supercell by imposing zero total magnetization^{65}. That is, a supercell with the same number of up and down moments is considered with a pair correlation function same as A_{50}B_{50} random alloys, generated using the alloy theoretic automated toolkit code^{66}. 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 spinlattice dynamics approach is promising for future detailed investigations^{67}.
Anomalous Hall and Nernst conductivity calculation
The AHC is evaluated using the WannierTools code^{68}, where the required accurate tightbinding models are obtained by the maximally localized Wannier functions (MLWFs) using the Wannier90 code^{69}. 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 kpoints mesh to guarantee good accuracy, which can be expressed as^{70}:
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 FermiDirac distribution function. The ANC α_{xy} is evaluated based on the Mott relation, which yields:
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.
Data availability
All relevant computational results generated or analyzed during this study are included in the manuscript and Supplementary information files in the form of tables and figures.
Code availability
The AHC was calculated using open source code Wannier90 and WannierTools. Postprocessing scripts used to calculate the derivative of AHC can be provided on request from the corresponding authors.
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Acknowledgements
The authors are grateful and acknowledge TU Darmstadt Lichtenberg highperformance computer support for the computational resources where the calculations were conducted for this project. The authors thank Prof. Manuel Richter of IFW Dresden for providing the SOC strength data and discussion. This project was supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) ProjectID 405553726TRR 270. N.M.F. acknowledges European Research Council (ERC) funding for financial support under the European Union’s Horizon 2020 research and innovation programme (Grant No. 743116 project Cool Innov). The work of J.Z. was supported by the Ministry of Education, Youth and Sports of the Czech Republic from the OP RDE program under the project International Mobility of Researchers MSCAIF at CTU No. CZ.02.2.69/0.0/0.0/18_{−}070/0010457.
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Singh, H.K., Samathrakis, I., Fortunato, N.M. et al. Multifunctional antiperovskites driven by strong magnetostructural coupling. npj Comput Mater 7, 98 (2021). https://doi.org/10.1038/s4152402100566w
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DOI: https://doi.org/10.1038/s4152402100566w
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