Magnetic structures and correlated physical properties in antiperovskites

Compounds with perovskite structures have become one of the focuses in both materials science and condensed matter physics because of their fascinating physical properties and potential functionalities correlated to magnetic structures. However, the understanding of the intriguing physical properties is still at an exploratory stage. Herein, owing to the magnetic frustration prompted by Mn 6 N or Mn 6 C octahedra, the abounding magnetic structures of antiperovskites, including collinear antiferromagnetic, collinear ferromagnetic, collinear ferrimagnetic, non-collinear magnetic, and non-coplanar magnetic spin configurations, are systematically introduced through the updated coverage. In addition, owing to the “spin-lattice-charge” coupling of antiperovskites, a large number of physical properties, such as anomalous thermal expansion, giant magnetoresistance, anomalous Hall effect, piezomagnetic/baromagnetic effects, magnetocaloric effect, barocaloric effect, etc ., are summarized by combining the discussions of the determined magnetic structures. This review aims to clarify the current research progress in this field, focusing on the relationship between the magnetic structures and the correlated physical properties, and provides the conclusion and outlook on further performance optimization and mechanism exploration in antiperovskites.

The so-called antiperovskite structure refers to a structure that is similar to perovskite.As shown in Figure 1, the face-centered position occupied by non-metallic elements, such as oxygen, in the original perovskite structure is occupied by transition group element atoms M, especially the magnetic element M = Mn, Fe, Ni, etc.The body center position originally occupied by metal elements is occupied by nonmetallic elements N or C, and the original vertex position is occupied by metal element X, thus forming a lattice belonging to a cubic unit cell with chemical formula M 3 XN(C) (M = Mn, Fe, Ni; X = Zn, Ga, Cu, Al, In, Sn).Among them, face-centered magnetic atoms (such as Mn) and body-centered N (C) atoms can form NMn 6 or CMn 6 octahedra, and six magnetic atoms Mn are located at the six corners of the octahedron, which is prone to magnetic frustration.Thereby it generates the abounding magnetic structures, including collinear antiferromagnetic (AFM), collinear ferromagnetic (FM), collinear ferrimagnetic (FIM), noncollinear magnetic, and non-coplanar magnetic spin configurations [49][50][51] .On the other hand, the abundant magnetic structures in antiperovskite Mn 3 XN(C) compounds are very sensitive to changes in temperature, magnetic field, pressure, composition, and grain size.Its abnormal lattice change, magnetic phase transition, and electronic transport properties are interrelated and affect each other, showing its rich physical properties.
In this paper, we will summarize the magnetic structures and correlated physical properties in antiperovskites.We present the potential application of antiperovskites as novel materials in various emerging fields.In order to further optimize performance and explore mechanisms, the issues such as exploration of new magnetic structures, synthesis of single crystal samples, and practical application research for the in-depth research are deserved in the part of outlook.

MAGNETIC STRUCTURES IN MN-BASED ANTIPEROVSKITES
The research on the magnetic structures of antiperovskites mainly focuses on Mn-based compounds.Herein, the collinear, non-collinear, and non-coplanar magnetic structures in Mn-based antiperovskites will be introduced in this review.

Non-collinear magnetic structures
It is worth noting that two non-collinear AFM phases belonging to Γ 4g and Γ 5g types, respectively, have been studied extensively in antiperovskites.In this case, the compounds remain cubic with the propagation vector k = (0, 0, 0).For Γ 5g type shown in Figure 4A, the magnetic moments of Mn atoms are located in the (111) plane with a triangular arrangement.As seen from Figure 4B, the magnetic moments of Γ 4g are triangularly located in the (111) plane achieved by rotating 90° coherently from Γ 5g type.and (B) the FIM structure of Mn 3.19 Zn 0.77 N 0.94 [21] .So far, five typical undoped antiperovskites displaying the non-collinear magnetic structures of Γ 4g and Γ 5g types have been reported experimentally, including Mn 3 NiN, Mn 3 ZnN, Mn 3 GaN, Mn 3 AgN, and Mn 3 SnN.The magnetic structure of Mn 3 NiN below 163 K, Mn 3 ZnN between 140 K and 183 K, Mn 3 GaN below 298 K, and Mn 3 AgN between 55 and 290 K belongs single-phase Γ 5g type, corresponding to the magnetic moments 0.98 μ B /Mn (T = 77 K), 1.21 μ B /Mn (T = 159 K), 1.17 μ B /Mn (T = 4.2 K), and 3.1 μ B /Mn (T = 4.2 K), respectively [54] .This type of magnetic structure has had an important impact on the mechanism exploration of the NTE behavior, piezomagnetic effect, and barocaloric effect of antiperovskites.In addition, the magnetic structures of Mn 3 NiN between 163 K and 266 K, Mn 3 AgN below 55 K and Mn 3 SnN between 237 K and 357 K are composed of Γ 4g and Γ 5g types, showing the magnetic moments 0.8 μ B /Mn at 250 K, 3.1 μ B /Mn and 2.5 μ B /Mn at 250 K at 4.2 K, respectively [54] .It was suggested that the magnetic behavior of antiperovskites is very sensitive to the differences of sample composition.In Mn 3 Ni 0.9 N 0.96 , the magnetic structure is a combination of Γ 4g and Γ 5g symmetries below T N = 264 K, and the moments undergo a rotation in the (111) plane upon warming [59] .Moreover, for Mn 3 Zn 96 N, the sample fully transforms to the Γ 5g phase upon cooling below 185 K, and the low-temperature collinear magnetic structure existing in Mn 3 ZnN disappears [20] .
Mn 3 Zn 1-x Ge x N [11,20] .In Mn 3 Cu 1-x Sn x N, the AFM transition closely coupled with the volume change is broadened upon Sn doping, producing the NTE behavior [19] .The characterization of magnetic structures in doped systems still requires careful study by neutron scattering.

Non-coplanar magnetic structures
A non-coplanar FIM structure with the propagation vector k = (1/2, 1/2, 0) has been reported in Mn 3 CuN [54,60] .The magnetic ordering temperature of the Mn 3 CuN compound is 143 K [54] .With cooling, the compound shows a transition from a high-temperature cubic phase to a low-temperature tetragonal phase.Herein, the magnetic moment direction of Mn atoms in the z = 0 plane is along the [001] direction, while the atoms in the z = 0.5 plane have two magnetic components, namely the FM arrangement in the [001] direction and the "square" AFM arrangement in the z = 0.5 plane [54] .It is worth noting that the antiperovskite Mn 3 SnC with a magnetic ordering temperature of 294 K has the same type of magnetic structure as Mn 3 CuN.Moreover, as shown in figure 6A, an orthorhombic magnetic structure model with P 1 symmetry was determined in Mn 3 Cu 0.89 N 0.96 .The sub-lattice of a magnetic structure is 2c × 2a × b, where a, b, and c are nuclear lattice parameters.At 6 K, neutron diffraction revealed that the Mn moments show an AFM component of 3.65 μ B /Mn on the z = 0.5 plane and a FM component of 0.91 μ B /Mn parallel to the y-axis on the x = 0.25 and 0.75 planes [39] .
Figure 6B gives a non-collinear magnetic structure M-1 of Mn 3 Ga 0.95 N 0.94 .The M-1 phase remains 79% in coexistence with Γ 5g magnetic configuration between 6 K and 50 K [35] .It can be seen that the sub-lattice of the M-1 phase is a, where a is the lattice parameter of the nuclear structure.For the M-1 phase, the results of neutron diffraction indicate that Mn atoms comprise three different locations, including Mn1 1a (0, 0, 0), Mn2 2b (0.5, 0.5, 0), and Mn3 4d (0.25, 0.25, 0.5).Mn1 and Mn2 display the AFM components along the z axis.Mn3 consists of two magnetic components; one is the "square" AFM component on the plane z = 0.5, and the other one is the FM component along a z axis direction.At 6 K, the AFM moment is 0.89 μ B /Mn for Mn1 and Mn2, while Mn3 includes a FM moment of 2.18 μ B /Mn and an AFM moment of 0.7 μ B /Mn.
~353 K and FIM-M2 phase transition at ~250 K. Neutron diffraction pattern at 300 K reveals the noncollinear FIM phase with a sub-lattice a, a, 2c where a and c are the lattice parameters of the tetragonal nuclear structure.The Mn atoms are located at six different types of sites with a P4 space group.Mn1 (0.5, 0.5, 0) and Mn2 (0.5, 0.5, 0.5) display the FM component 2.5 μ B /Mn along the z axis.Moreover, the tiled magnetic moments were uncovered for Mn3 (0.5, 0, 0.25), Mn4 (0, 0.5, 0.25), Mn5 (0.5, 0, 0.75), and Mn6 (0, 0. ) μ B /Mn, respectively.At 5 K, another non-coplanar magnetic structure M2 with a was revealed in Mn 3 SbN.Herein, Mn atoms on the plane z = 0.5 show a "square" AFM arrangement with the moment 2.3 μ B /Mn, while the other Mn atoms display the AFM component along the z axis with the moment 2.5 μ B /Mn.The minor differences between the presented magnetic structure and the previously reported one may arise from the tiny elemental components [54] .Even more interesting in antiperovskites is that the propagation vector k = (0, 0, k z ) of Mn 3 SnN varies with temperature from k z = 0.25 at 50 K to k z = 0.125 at 237 K [54] .
The effect of magnetic element doping on the magnetic structure was also investigated in antiperovskites.For Mn-doped Mn 3+x Ni 1-x N and Mn 3.39 Co 0.61 N compounds [Figure 6D], a FM component along the [111] direction coexisting with canted Γ 5g AFM component was resolved by neutron diffraction technique [15,62] .Table 1 summarizes the magnetic structures and corresponding temperature ranges of typical antiperovskites.

PHYSICAL PROPERTIES OF ANTIPEROVSKITES
The research on antiperovskite structure compounds can be traced back to the 1930s when there were not many studies on physical properties.Since the 1980s, this type of compound has been paid attention by scientists, and the basic physical properties of antiperovskites have been studied by means of neutron diffraction, X-ray diffraction (XRD), Mössbauer spectroscopy, and nuclear magnetic resonance.Extensive research of these basic physical properties mainly includes crystal structures, magnetic properties (magnetic structures), phase diagrams, etc.At the beginning of the 21st century, superconductivity, giant magnetoresistance, magnetocaloric effect, abnormal thermal expansion, and near-zero temperature coefficient of resistance behaviors were successively reported in antiperovskites.The discovery of these physical properties prompted more and more researchers to pay attention to antiperovskites and their Ferrimagnetic Non-coplanar T < 143 K applications.In the past decade or so, a large number of physical properties correlated to magnetic structures have been reported.

Anomalous thermal expansion in manganese-based antiperovskites
Materials with zero thermal expansion (ZTE) and NTE behaviors have attracted widespread attention because of their broad applications in modern technology, such as high-precision optical instruments, microelectronics, aerospace devices, etc. [63][64][65][66][67][68][69] .A great deal of work has focused on the discovery of new materials and the improvement of thermal expansion properties.Nevertheless, the investigations for the mechanism of anomalous thermal expansion (ATE) (mainly including ZTE and NTE) are still needed.For ZrW 2 O 8 [67] and ScF 3 [68] , the mechanism associated with the soft phonon mode of the frame structure is adopted; moreover, the ATE behavior of the material has a strong coupling effect with other physical properties, such as the valence state change in LaCu 3 Fe 4 O 12 [69] , BiNiO 3 [70] , and YbGaGe [63] and the ferroelectric characteristics in PbTiO 3 -BiFeO 3 [71] ; in addition, the ATE behavior emerges with magnetic transitions in various materials, such as the NTE in La(Fe,Si,Co) 13 [72] and Ca 2 Ru 1−x Cr x O 4 [73] and near ZTE in FeNi Invar [74] and SrRuO 3 [75] .It is worth noting that although a large number of studies have shown that the Invar effect is related to the magnetic properties of materials, an adequate understanding of this property is still required.Therefore, the exploration of new materials with ATE will contribute to the clarification of mechanisms [75][76][77][78][79][80] .
Some manganese nitrogen compounds (such as Mn 3 ZnN at 185 K, Mn 3 GaN at 298 K, etc.) are accompanied by a sudden change in volume during the magnetic transition, that is, the so-called magnetovolume effect.
In 2005, Takenaka et al. reported the NTE behavior in the Ge-doped antiperovskite structure compound Mn 3 Cu 1-x Ge x N [5] .For Mn 3 CuN, the compound itself has no magnetovolume effect.Through the doping of Ge, the discontinuous volume change caused by the magnetic volume effect is broadened, thereby realizing the regulation of the thermal expansion coefficient and temperature range of the NTE behavior.With increasing the doping amount of Ge, the magnetovolume effect of Mn 3 Cu 1-x Ge x N was broadened and moved to the high temperature region, resulting in NTE behavior near room temperature.As shown in Figure 7A and 7B, near room temperature, the linear expansion coefficient α of Mn 3 Cu 0.53 Ge 0.47 N and Mn 3 Cu 0.5 Ge 0.5 N are -16 × 10 -6 K -1 and -12 × 10 -6 K -1 , respectively.In order to further reduce the material cost, Takenaka et al. used Sn as the dopant, which is cheaper than Ge.The doping of Sn can also broaden the NTE behavior of antiperovskites [6] .
The thermal expansion behavior of Mn 3 Ga 0.5 Ge 0.4 Mn 0.1 N 1-x C x was also reported, and a single-phase ZTE material with a wider temperature range has been obtained [7] .As shown in Figure 7C, the thermal expansion behavior of Mn 3 Ga 0.5 Ge 0.4 Mn 0.1 N 1-x C x changes with the doping of C. When x = 0.1, the compound exhibits low thermal expansion in the temperature range of 190-272 K with |α| < 0.5 × 10 -6 K -1 .In addition, a very close correlation between N content and NTE behavior was found in Mn 3 Cu 0.5 Sn 0.5 N 1-δ .The N content in the compound decreases with the increase of the sintering temperature.When the sintering temperature is 950 degrees, the linear expansion coefficient of the compound with the N content of about 0.8 in the temperature range of 307-355 K with |α| < 0.5 × 10 -6 K -1 .
Huang et al. carried out research on Ge and Si co-doped Mn 3 Cu 0.6 Si x Ge 0.4−x N and obtained a lowtemperature NTE material [8] .As shown in Figure 7D, with the co-doping of Si, the NTE temperature range of the compound moves to a lower temperature.When x = 0.15, Mn 3 Cu 0.6 Si 0.15 Ge 0.25 N shows NTE behavior in a wide temperature range in the temperature range of 120-184 K, and its linear expansion coefficient is α = -16 × 10 -6 K -1 .Comparing Mn 3 Cu 0.6 Si x Ge 0.4-x N and Mn 3 Cu 1-x Ge x N, it can be seen that the single doping of Ge has a narrow volume mutation temperature range in the low temperature region (such as Mn 3 Cu 0.8 Ge 0.2 N around 155 K), while the co-doping of Si can make that the temperature range of the volume change of the compound is broadened and the behavior of NTE appears.The co-doping method provides a way to regulate the thermal expansion behavior of single-phase materials.
Linet al. reported the thermal expansion and magnetic properties of antiperovskite manganese nitrides Mn 3+x Ag 1-x N [9] .The substitution of Mn for Ag effectively broadens the temperature range of NTE and moves it to low temperatures [Figure 7E].When x = 0.6, the Mn 3.6 Ag 0.4 N compound shows ZTE with α = -0.48× 10 -6 K -1 (temperature range 5 -87 K).Moreover, Sun et al. revealed a giant NTE covering room temperature in nanocrystalline Mn 3 GaN x [10] .By reducing the average grain size to ~10 nm, the temperature window ΔT for NTE exceeds 100 K, and α remains as large as -30 ppm/K (-21 ppm/K) for x = 1.0 (x = 0.9).
The influence of Ge and Sn doping on the thermal expansion behavior of Mn 3 Zn 1-x Ge(Sn) x N has been investigated by us [11,12] .Figure 7F shows the variation of lattice constant with temperature in Mn 3 Zn 1-x Ge x N. The doping of Ge broadens the magnetovolume effect of Mn 3 Zn 1-x Ge x N and moves the temperature zone to the higher one, thereby realizing the regulation of NTE behavior.A similar behavior was also observed in Sn-doped Mn 3 Zn 1-x Sn x N compounds [12] .On the other hand, the regulation of the thermal expansion behavior of Mn 3 NiN-based compounds has also been reported [13,14] .Antiperovskite Mn 3 Ni 0.5 Ag 0.5 N shows NTE behavior in a wide temperature range (260-320 K) near room temperature with α = -12 × 10 -6 K -1 .The Mn 3 Ni 0.5 Cu 0.5 N exhibits NTE in the temperature range of 160-240 K (ΔT = 80 K) with α = -22.3× 10 -6 K -1 .Interestingly, a new type of Invar-like material exhibiting ZTE has been revealed in Mn 3+x Ni 1-x N [15] .Song et al. revealed the ZTE behavior of Mn 3 Cu 0.5 Ge 0.5 N due to the size effect [16] .When Mn 3 Cu 0.5 Ge 0.5 N was prepared from polycrystalline samples (average size of 2.0 μm) to ultra-nanocrystals (average size of 12 nm), the occupancy rate of Mn in the sample changed from 100% to 78.7% [Figure 8A].Meanwhile, the ultrananocrystalline sample exhibits ZTE behavior in a wide temperature range ΔT = 218 K (12-230 K) with α = 1.18 × 10 -7 K -1 .
The mechanism for the NTE of antiperovskites was investigated by Iikub et al.The neutron diffraction results indicate that the non-collinear Γ 5g AFM structure plays a key role in the magnetovolume effect of Mn 3 Cu 1-x Ge x N, which leads to the appearance of NTE behavior.Moreover, Iikub et al. further revealed that the local lattice distortion plays a very important role in the NTE of Mn 3 Cu 1-x Ge x N [17] [Figure 8B].As suggested by the pair distribution function (PDF) analysis, Mn 3 Cu 1-x Ge x N maintains a cubic structure within a certain doping range, while the Mn 6 N octahedrons in Mn 3 Cu 1-x Ge x N rotate along the z-axis with Ge doping to form a local lattice distortion.This structural instability displays a strong correlation with the broadness of the growth of the ordered magnetic moment, which is considered as a trigger for broadening the volume change [18] .Moreover, Tong et al. studied the magnetic transition broadening and local lattice distortion in Mn 3 Cu 1-x Sn x N with NTE [19] .The PDF results indicate that the distribution of Cu/Sn-Mn bonds is linked to the fluctuations of the AFM integral.This may account for the broadening of the volume change in antiperovskites.
Through the study of Mn 3 (Zn, M) x N(M = Ag, Ge), we revealed the quantitative relationship between thermal expansion and atomic magnetic moments in antiperovskites and realized the regulation of thermal expansion [20] .A collinear AFM structure M PTE and a non-collinear AFM structure Γ 5g are observed in Mn 3 Zn x N. Herein, the M PTE phase displays PTE behavior, while the Γ 5g configuration shows NTE behavior.The NTE of Γ 5g phase can balance the contributions from PTE generated by the anharmonic vibration in the sample, producing the ZTE of antiperovskites.By introducing vacancies into Mn 3 Zn x N, the existence of a temperature range for Γ 5g configuration can be effectively regulated, thereby obtaining a ZTE material with a wider temperature range.In addition, we also discussed the quantitative relationship between the anomalous change of the lattice and the atomic magnetic moments for the Γ 5g phase.As shown in Figure 8C, both the lattice change a NTE -a T and the atomic magnetic moment m in Mn 3 Zn x N gradually decrease with the increase of temperature, and the change trends for both factors are consistent.By defining r(T) = (a NTE -a T )/m, it is obtained that r(T) hardly changes with temperature where a NTE , a T and m are the lattice constants and magnitude of the ordered magnetic moment, which confirms that there is a strong  [16] ; (B) local lattice distortion of Mn 3 Cu 1-x Ge x N [17] ; (C) the relationship between the anomalous change of the lattice and the atomic magnetic moments for Γ 5g phase of Mn 3 Zn x N [20] .
spin-lattice coupling between the lattice constant and the atomic magnetic moment in Mn 3 Zn x N. In addition, the strong spin-lattice coupling that can be tuned to achieve ZTE behavior was further confirmed in antiperovskite Mn 3+x Ni 1−x N and Mn 3.19 Zn 0.77 N 0.94 within Γ 5g phase [15,21] .Meanwhile, in Mn 3 Ga 1−x Sn x N with Γ 5g phase, the increase of the phonon contribution to the thermal expansion induced by Sn doping and the corresponding decrease of dm/dT are revealed to be the key parameters for tuning the magnetovolume effect [22] .
The first-principle calculations have been adopted for understanding the NTE behavior of antiperovskites.The primary theoretical works focus on the comparison of differences in equilibrium volumes of antiperovskites with different magnetic structures.Lukashev et al. found that the equilibrium volume of the Γ 5g AFM state in Mn 3 GaN is larger than that of the PM state, which confirms that the magnetic transition in the material can lead to volume change (magnetovolume effect) [23] .Qu et al. calculated the energy-lattice curves of various magnetic configurations, and the results show that the Γ 5g AFM state has the largest volume.This work also confirms that the Γ 5g AFM state has the largest volume compared to other magnetic configurations [24] .In addition, Mochizuki et al. constructed a classical spin model with frustrated exchange interactions and magnetic anisotropy to study the nontrivial magnetic orders in the antiperovskite Mn 3 AN.With a replica-exchange Monte Carlo technique, the Γ 5g and Γ 4g spin configurations, known to trigger the NTE, have been reproduced [25] .

Electronic transport properties in antiperovskites
There is a strong correlation between the lattice, spin, and charge of Mn 3 GaC.Therefore, a giant magnetoresistance effect was found near its collinear AFM -collinear FM magnetic transition [26] .The size of magnetoresistance can be expressed by [ρ(H)ρ(0)]/ρ(0), and ρ(H) and ρ(0) represent the resistivity when the external magnetic field is finite and 0, respectively.As shown in Figure 9A, Mn 3 GaC generates a (A) Giant magnetoresistance effect of Mn 3 GaC at selected temperatures [26] ; (B) magnetoresistance of Mn 3.338 Ni 0.651 N after cooling in zero field and in ±9 T [27] ; (C) anomalous Hall conductivity versus field measured in single crystalline Mn 3 Ni 0.35 Cu 0.65 N film on MgO (111) substrate [30] .
magnetoresistance of about 50% under an external magnetic field of 3 kOe.With the further increase of the external magnetic field, the magnetoresistance value is almost unchanged, but its peak width is broadened and reaches 20 K at 50 kOe.Kamishima et al. suggested that the magnetoresistance effect in Mn 3 GaC is aroused by the difference of resistivity between AFM and FM states, and the external magnetic field can induce the temperature shift of AFM-FM phase transition [26] .In addition, an electroresistance-like behavior of the antiperovskite Mn 3 GaC, revealed by a resistivity change of 50% due to the local Joule heating, is reported around the collinear AFM-intermediate phase transition.The currents significantly reduce the proportion of the higher resistivity AFM phase relative to the lower resistivity interphase with warming, showing a change in resistivity.On the other hand, for a non-coplanar magnet Mn 3.338 Ni 0.651 N with triangular lattice, a high-resistivity state can be frozen along the direction of the cooling field while a low-resistivity state is determined in the reversed field direction, indicating an asymmetry with respect to H [Figure 9B].This characteristic further demonstrates a switchable scalar spin chirality of Mn 3.338 Ni 0.651 N.
Recently, the anomalous Hall effect, originating from the nonvanishing momentum space Berry curvature, has been reported in the non-collinear AFM antiperovskites.Among the magnetic orders, a typical noncollinear AFM configuration is Γ 4g , whose atomic magnetic moments point to the triangle "inside" or "outside" in the triangular lattice of the antiperovskite (111) surface, forming a phase similar to that of Mn 3 A(X = Sn, Ge, Pt) non-collinear magnets.Another typical AFM phase Γ 5g can be obtained by rotating the atomic magnetic moments in Γ 4g by 90 degrees in the (111) plane.Both of these two magnetic phases have zero scalar chirality, and theoretical studies show that the former and the latter magnetic order display a finite and zero anomalous Hall resistivity, respectively.In 2019, Gurung et al. used symmetry analysis and density functional theory to study the anomalous Hall conductance in non-collinear magnetic antiperovskites, revealing that the Γ 4g magnetic phase in Mn 3 GaN shows a finite value of anomalous Hall conductivity, while the Γ 5g magnetic phase displays zero anomalous Hall conductivity [28] .In 2020, Samathrakis et al. theoretically calculated the tailoring of the anomalous Hall effect in the non-collinear antiperovskite Mn 3 GaN, revealing the large intrinsic anomalous Hall effect caused by the strain in the Γ 5g and Γ 4g magnetic phases [29] .In 2019, Zhao et al. experimentally observed the anomalous Hall effect in the non-collinear AFM Mn 3 Ni 1-x Cu x N, which is attributed to the nonzero Berry curvature of the Γ 4g magnetic phase in momentum space [Figure 9C] [30] .The research on the anomalous Hall effect of antiperovskites is attracting widespread attention for novel spintronic applications.

Piezomagnetic/baromagnetic effects in antiperovskites
The piezomagnetic effect has been reported in non-collinear AFM antiperovskites [23,33] .In 2008, Lukashev et al. predicted that the non-collinear magnetic structure of Mn 3 GaN can be controlled by a small applied biaxial strain [Figure 10A] [23] .Figure 10B shows the net magnetic moment of Mn 3 GaN and the rotational angle of the magnetic moment of Mn atoms as a function of axial strain.It can be seen that the atomic magnetic moment rotates when the strain is applied.This piezomagnetic effect is linear and displays magnetization reversal with the applied strain.As the compressive strain is 1%, the magnetization is about 0.04 μ B /f.u.Therefore, this property can be utilized for the application of the magnetoelectric effect, such as a combination of piezomagnetic and piezoelectric phases or a combination of magnetostrictive and piezoelectric phases [31,32] .In addition, Zemen et al. theoretically performed a systematic study of the piezomagnetic effect in nine cubic antiperovskites Mn 3 XN (X = Rh, Pd, Ag, Co, Ni, Zn, Ga, In, Sn), revealing an extraordinarily large piezomagnetic effect in Mn 3 SnN at room temperature [33] .Boldrin et al. demonstrate experimentally that a giant piezomagnetic effect is indeed manifest in the AFM antiperovskite Mn 3 NiN [34] .
In 2016, the baromagnetic effect of Mn 3 Ga 0.95 N 0.94 was determined by both neutron diffraction analysis and magnetic measurements [35] .Interestingly, Mn 3 Ga 0.95 N 0.94 displays a new tetragonal non-coplanar magnetic structure M-1 below 50 K, which is in coexistence with Γ 5g spin configuration under atmospheric pressure.As shown in Figure 10C and D, the sample exhibits the piezomagnetic effect.When the applied pressure is 750 MPa at 130 K, the magnetic phase transition from M-1 to Γ 5g AFM appears, generating the piezomagnetic characteristic of 0.63 μ B /f.u.Combined with the refined results of neutron diffraction, the change of Mn-Mn distance and spin rearrangement caused by pressure is considered to be the trigger of the observed baromagnetic effect.

Magnetocaloric effect
The magnetocaloric effect of antiperovskites was primarily reported in Mn 3 GaC [36] .The collinear AFMintermediate magnetic transition of Mn 3 GaC showing a first-order characteristic can be controlled by an external magnetic field, generating the magnetocaloric effect.Figure 11A shows the temperature dependence of the maximum value of magnetic entropy change ΔS mag .The peak of ΔS mag reaches 17 J/(kg•K) when the external magnetic field is 10 kOe, and the peak value broadens to a "platform" shape with further increase of the magnetic field.In addition, by introducing C vacancies, the magnetic properties of Mn 3 GaC were changed, thereby affecting the magnetocaloric effect [37] .The magnetic entropy of Mn 3 GaC 0.78 decreased to 3.7 J/kg•K under a 5 T magnetic field.In Mn 3-x Co x GaC, Co doping can reduce the first-order phase transition temperature from 164 K to 100 K without a significant decrease of magnetic entropy and realize the magnetocaloric effect covering a wider temperature range (50-160 K) [38] .
A large magnetic entropy change was observed in Mn 3 Cu 0.89 N 0.96 [39] [Figure 11B].By introducing vacancies, the onset of the FIM-PM transition is slightly reduced from 150 K of Mn 3 CuN to 147.7 K of Mn 3 Cu 0.89 N 0.96 , and a new non-coplanar FIM structure with an orthorhombic symmetry was determined.The total entropy change of Mn 3 Cu 0.89 N 0.96 obtained by DSC is about 60 J/kg•K, while the maximum magnetic entropy change ΔS mag is 13.52 J/kg•K under a magnetic field of 50 kOe near the temperature of FIM-PM transition.Neutron diffraction results indicate that the magnetic entropy change of Mn 3 Cu 0.89 N 0.96 is caused by the magnetic transition from the AFM to the FM component in the tetragonal phase and the phase transition from cubic to tetragonal under a magnetic field.; (B) variation of net magnetic moment and rotation angle of Mn atomic magnetic moment with axial strain for Mn 3 GaN [23]   ; (C) piezomagnetic effect determined by magnetization curve in Mn 3 Ga 0.95 N 0.94 [35]   ; (D) piezomagnetic effect of in Mn 3 Ga 0.95 N 0.94 at 130 K and 170 K [35]   .

Barocaloric effect
A significant barocaloric effect is expected when strong cross-correlations between the volume and magnetic order appear in materials.In 2015, Matsunami et al. reported the giant barocaloric effect enhanced by the frustration of the AFM phase in Mn 3 GaN [40] .As shown in Figure 12A, when a hydrostatic pressure change of 139 MPa is applied, Mn 3 GaN exhibits an entropy change of 22.3 J kg -1 K -1 .By applying a depressurization of 93 MPa, the change of adiabatic temperature is determined to be about 5 K.The entropy change of Mn 3 GaN under different hydrostatic pressures as a function of temperature [40] ; (B) isothermal entropy and adiabatic temperature changes [41] .
Matsunami et al. further suggests that the magnitude of the barocaloric effect of Mn 3 GaN is determined by the volume change at the transition and stability of the AFM phase against the pressure [40] .In 2018, Boldrin et al. further investigated the barocaloric effect in the geometrically frustrated antiferromagnet Mn 3 NiN [Figure 12B] [41] .It is worth noting that a large barocaloric entropy change, which is a factor of 1.6 than that of Mn 3 GaN, is observed.Boldrin et al. proposed that the barocaloric effect of Mn 3 NiN originates from multisite exchange interactions amongst the local Mn magnetic moments and their coupling with itinerant electron spins [41] .

CONCLUSION AND OUTLOOK
As reviewed in this article, owing to the magnetic frustration prompted by Mn 6 N or Mn 6 C octahedra, antiperovskites display the abounding magnetic structures, including collinear AFM, collinear FM, collinear FIM, non-collinear magnetic and non-coplanar magnetic spin configurations.In antiperovskites, the magnetic phase transition (magnetic structures), abnormal lattice change, and electronic transport properties are interrelated and affect each other, showing a large number of physical properties such as ATE, electronic transport properties, piezomagnetic/baromagnetic effects, magnetocaloric effect, barocaloric effect, etc.Therefore, antiperovskites will be an excellent candidate for exploring new smart materials.In order to further optimize performance and explore mechanisms, the following issues for indepth research deserve attention and solutions: Exploration of new magnetic structures.The examination of new physical properties is one of the important directions of the development of modern smart materials.Due to the strong correlation of "lattice-spincharge", antiperovskites show a series of rich and unique physical properties within some specific magnetic structures.Although the determination of the magnetic structures is a central issue in antiperovskites, there is still a lack of systematic and in-depth research, especially on how the magnetic structures and correlated physical properties evolve in the case of elemental doping, variated temperatures, varied magnetic fields, and pressurization.
Synthesis of single crystal samples.The current research work on antiperovskites is mainly focused on polycrystalline.From the perspective of mechanism research and application, single crystal research has greater advantages.However, it is difficult to precisely control the nitrogen/carbon contents of antiperovskites in preparation, and the change of contents has a great influence on its physical properties.Therefore, the synthesis of three-dimensional single crystal materials with excellent physical properties is challenging.
Practical application research.Selecting some typical antiperovskites with fascinating physical properties, the practical applications can be explored in the fields of optics, microelectronics, refrigeration, aerospace, etc.

Figure 2 .
Figure 2. The collinear (A) AFM structure and (B) FM structure of Mn 3 GaC.

Figure 5 .
Figure 5. Phase diagram of Mn 3 Cu 1-x Ge x N. T C and T N denote the Curie and Néel temperatures, respectively[18] .

Figure 8 .
Figure 8. (A) ZTE behavior of Mn 3 Cu 0.5 Ge 0.5 N[16]  ; (B) local lattice distortion of Mn 3 Cu 1-x Ge x N[17] ; (C) the relationship between the anomalous change of the lattice and the atomic magnetic moments for Γ 5g phase of Mn 3 Zn x N[20] .

Figure 9 .
Figure 9.(A) Giant magnetoresistance effect of Mn 3 GaC at selected temperatures[26] ; (B) magnetoresistance of Mn 3.338 Ni 0.651 N after cooling in zero field and in ±9 T[27] ; (C) anomalous Hall conductivity versus field measured in single crystalline Mn 3 Ni 0.35 Cu 0.65 N film on

Figure 10 .
Figure 10.(A) Variation of magnetic moment of Mn Atoms in the (111) Plane of Γ5g AFM Mn 3 GaN with axial strain[23]

Figure 12 .
Figure 12. (A)The entropy change of Mn 3 GaN under different hydrostatic pressures as a function of temperature[40] ; (B) isothermal entropy and adiabatic temperature changes[41] .