Magnetic structures and correlated physical properties in antiperovskites

Collinear magnetic structure

Both collinear AFM and collinear FM structures were determined by neutron diffraction in Mn₃GaC as early as the 1970s. Upon warming, Mn₃GaC displays several magnetic phase transitions: an AFM-intermediate (AFM-IM) phase transition at 160.1 K, an intermediate-FM (IM-FM) phase transition at 163.9 K, and a FM-paramagnetic (FM-PM) transition at 248 K^[52]. As shown in Figure 2A, the determined magnetic moments m of AFM Mn₃GaC alternates along the [111] direction with a propagation vector k = (1/2, 1/2, 1/2), corresponding to m = 1.8 ± 0.1 μ_B/Mn at 4.2 K reported by Fruchart et al. and m = 1.54 μ_B/Mn at 150 K revealed by Çakır et al.^[53-55]. As seen from Figure 2B, the propagation vector for FM Mn₃GaC is k = (0, 0, 0), displaying m = 1.3 ± 0.1 μ_B/Mn at 193 K^[53,54]. The direction [111] of easy magnetization is also confirmed by the Mössbauer effect^[56].

Figure 2. The collinear (A) AFM structure and (B) FM structure of Mn₃GaC.

Mn₃ZnN is characterized by two first-order magnetic transitions: PM-AFM at 183 K and AFM-AFM at 140 K. As shown in Figure 3A, the low temperature AFM phase is collinear within a $$ \sqrt{2} a, \sqrt{2} a \text {, } 2a $$ sub-lattice of manganese, exhibiting two inequivalent magnetic atoms with different ordered magnetic moments^[57]. The values of moments 0.61 μ_B on Mn1 (000) and 1.03 μ_B on Mn2 (1/4, 1/4, 1/4) were clarified by Fruchart et al.^[58]. In 2012, we determined the ordered moments of 1.40 μ_B on Mn1 and 2.44 μ_B on Mn2 at 80 K for Mn₃Zn_0.99N by neutron diffraction^[20]. Moreover, for Mn_3.19Zn_0.77N_0.94, we observed a collinear FIM structure with a propagation vector k = (0, 0, 0) below 200 K [Figure 3B]. The magnetic moments alternate along the [111] direction with different values, corresponding to 0.5(1) (face-centered atom) and 1.3(7) (corner atom) μ_B/Mn for Mn atoms at 120 K^[21].

Figure 3. The collinear (A) AFM structure of Mn₃ZnN^[57] and (B) the FIM structure of Mn_3.19Zn_0.77N_0.94^[21].

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. In Γ^4g and Γ^5g magnetic structures, two Mn atoms form an angle of 120° with each other in the (111) plane, and the magnetic moments of the three atoms cancel each other out to generate a zero net magnetic moment.

Figure 4. The non-collinear (A) Γ^5g and (B) Γ^4g AFM structures of antiperovskites.

So far, five typical undoped antiperovskites displaying the non-collinear magnetic structures of Γ^4g and Γ^5g types have been reported experimentally, including Mn₃NiN, Mn₃ZnN, Mn₃GaN, Mn₃AgN, and Mn₃SnN. The magnetic structure of Mn₃NiN below 163 K, Mn₃ZnN between 140 K and 183 K, Mn₃GaN below 298 K, and Mn₃AgN 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 Mn₃NiN between 163 K and 266 K, Mn₃AgN below 55 K and Mn₃SnN 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 Mn₃Ni_0.9N_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 Mn₃Zn₉₆N, the sample fully transforms to the Γ^5g phase upon cooling below 185 K, and the low-temperature collinear magnetic structure existing in Mn₃ZnN disappears^[20].

The non-collinear Γ^5g AFM structure is also examined by both neutron diffraction and the correlated thermal expansion behavior in doped antiperovskites, such as Mn₃Cu_1-x(Ge, Sn)_xN, Mn₃Zn_1-x(Ge, Sn)_xN, and Mn₃Ga_1-xSn_xN, etc.^[5-22]. Particularly, the undoped Mn₃CuN does not display the Γ^5g magnetic configuration. As shown in Figure 5, a cubic structure with the Γ^5g magnetic configuration was determined by neutron study in Mn₃Cu_1-xGe_xN (x ≥ 0.15), which is suggested to be a key ingredient of a large magnetovolume effect in antiperovskites^[18]. Meanwhile, the NTE behavior determined by Γ^5g magnetic order was also observed in Mn₃Zn_1-xGe_xN^[11,20]. In Mn₃Cu_1-xSn_xN, 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.

Figure 5. Phase diagram of Mn₃Cu_1-xGe_xN. T_C and T_N denote the Curie and Néel temperatures, respectively^[18].

Non-coplanar magnetic structures

A non-coplanar FIM structure with the propagation vector k = (1/2, 1/2, 0) has been reported in Mn₃CuN^[54,60]. The magnetic ordering temperature of the Mn₃CuN 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 Mn₃SnC with a magnetic ordering temperature of 294 K has the same type of magnetic structure as Mn₃CuN. Moreover, as shown in Figure 6A, an orthorhombic magnetic structure model with P₁ symmetry was determined in Mn₃Cu_0.89N_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 Mn₃Ga_0.95N_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 $$ \sqrt{2} a, \sqrt{2} a \text {, } a $$, 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.

Figure 6. The non-coplanar FIM structures of (A) Mn₃Cu_0.89N_0.96^[39], (B) Mn₃Ga_0.95N_0.94^[35], (C) Mn₃SbN^[61], and (D) Mn_3.39Co_0.61N^[62].

Recently, the non-collinear FIM structures were determined in the (1-x)Mn₃GaN- xMn₃SbN (0.2 ≤ x ≤ 0.8) heterogeneous system^[61][Figure 6C]. Upon cooling, Mn₃SbN undergoes a PM-FIM phase transition at ~353 K and FIM-M2 phase transition at ~250 K. Neutron diffraction pattern at 300 K reveals the non-collinear 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, 0.5, 0.75), corresponding to (−2.47, 2.47, −2.16) μ_B/Mn, (2.47, −2.47, −2.16) μ_B/Mn, (2.47, −2.47, −2.16) μ_B/Mn, and (−2.47, 2.47, −2.16) μ_B/Mn, respectively. At 5 K, another non-coplanar magnetic structure M2 with $$ \sqrt{2} a, \sqrt{2} a \text {, } a $$ was revealed in Mn₃SbN. 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 Mn₃SnN 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+xNi_1-xN and Mn_3.39Co_0.61N 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.

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.^[63-69]. 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 ZrW₂O₈^[67] and ScF₃^[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 LaCu₃Fe₄O₁₂^[69], BiNiO₃^[70], and YbGaGe^[63] and the ferroelectric characteristics in PbTiO₃-BiFeO₃^[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 Ca₂Ru_1−xCr_xO₄^[73] and near ZTE in FeNi Invar^[74] and SrRuO₃^[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^[75-80].

Some manganese nitrogen compounds (such as Mn₃ZnN at 185 K, Mn₃GaN 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 Mn₃Cu_1-xGe_xN^[5]. For Mn₃CuN, 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 Mn₃Cu_1-xGe_xN 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 Mn₃Cu_0.53Ge_0.47N and Mn₃Cu_0.5Ge_0.5N 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].

Figure 7. Linear thermal expansion of antiperovskites (A, B) Mn₃Cu_1-xGe_xN^[5], (C) Mn₃Ga_0.5Ge_0.4Mn_0.1N_1-xC_x^[7], (D) Mn₃Cu_0.6Si_xGe_0.4-xN^[8], (E) Mn_3+xAg_1-xN^[9], and (F) Mn₃Zn_1-xGe_xN^[11].

The thermal expansion behavior of Mn₃Ga_0.5Ge_0.4Mn_0.1N_1-xC_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 Mn₃Ga_0.5Ge_0.4Mn_0.1N_1-xC_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 Mn₃Cu_0.5Sn_0.5N_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 Mn₃Cu_0.6Si_xGe_0.4−xN and obtained a low-temperature 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, Mn₃Cu_0.6Si_0.15Ge_0.25N 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 Mn₃Cu_0.6Si_xGe_0.4-xN and Mn₃Cu_1-xGe_xN, it can be seen that the single doping of Ge has a narrow volume mutation temperature range in the low temperature region (such as Mn₃Cu_0.8Ge_0.2N 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.

Lin et al. reported the thermal expansion and magnetic properties of antiperovskite manganese nitrides Mn_3+xAg_1-xN^[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_3.6Ag_0.4N compound shows ZTE with α = -0.48 × 10^-6 K^-1 (temperature range 5 - 87 K). Moreover, Lin et al. revealed a giant NTE covering room temperature in nanocrystalline Mn₃GaN_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 Mn₃Zn_1-xGe(Sn)_xN has been investigated by us^[11,12]. Figure 7F shows the variation of lattice constant with temperature in Mn₃Zn_1-xGe_xN. The doping of Ge broadens the magnetovolume effect of Mn₃Zn_1-xGe_xN 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 Mn₃Zn_1-xSn_xN compounds^[12]. On the other hand, the regulation of the thermal expansion behavior of Mn₃NiN-based compounds has also been reported^[13,14]. Antiperovskite Mn₃Ni_0.5Ag_0.5N shows NTE behavior in a wide temperature range (260-320 K) near room temperature with α = -12 × 10^-6 K^-1. The Mn₃Ni_0.5Cu_0.5N 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+xNi_1-xN^[15].

Song et al. revealed the ZTE behavior of Mn₃Cu_0.5Ge_0.5N due to the size effect^[16]. When Mn₃Cu_0.5Ge_0.5N 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 ultra-nanocrystalline sample exhibits ZTE behavior in a wide temperature range ΔT = 218 K (12-230 K) with α = 1.18 × 10^-7 K^-1.

Figure 8. (A) ZTE behavior of Mn₃Cu_0.5Ge_0.5N^[16]; (B) local lattice distortion of Mn₃Cu_1-xGe_xN^[17]; (C) the relationship between the anomalous change of the lattice and the atomic magnetic moments for Γ^5g phase of Mn₃Zn_xN^[20].

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 Mn₃Cu_1-xGe_xN, 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 Mn₃Cu_1-xGe_xN^[17] [Figure 8B]. As suggested by the pair distribution function (PDF) analysis, Mn₃Cu_1-xGe_xN maintains a cubic structure within a certain doping range, while the Mn₆N octahedrons in Mn₃Cu_1-xGe_xN 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 Mn₃Cu_1-xSn_xN 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 Mn₃(Zn, M)_xN(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 Mn₃Zn_xN. 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 Mn₃Zn_xN, 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 Mn₃Zn_xN 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 spin-lattice coupling between the lattice constant and the atomic magnetic moment in Mn₃Zn_xN. In addition, the strong spin-lattice coupling that can be tuned to achieve ZTE behavior was further confirmed in antiperovskite Mn_3+xNi_1−xN and Mn_3.19Zn_0.77N_0.94 within Γ^5g phase^[15,21]. Meanwhile, in Mn₃Ga_1−xSn_xN 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 Mn₃GaN 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 Mn₃AN. 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 Mn₃GaC. 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, Mn₃GaC generates a 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 Mn₃GaC 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 Mn₃GaC, 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_3.338Ni_0.651N 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_3.338Ni_0.651N.

Figure 9. (A) Giant magnetoresistance effect of Mn₃GaC at selected temperatures^[26]; (B) magnetoresistance of Mn_3.338Ni_0.651N after cooling in zero field and in ±9 T^[27]; (C) anomalous Hall conductivity versus field measured in single crystalline Mn₃Ni_0.35Cu_0.65N film on MgO (111) substrate^[30].

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 non-collinear 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 Mn₃A(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 Mn₃GaN 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 Mn₃GaN, 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 Mn₃Ni_1-xCu_xN, 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 Mn₃GaN can be controlled by a small applied biaxial strain [Figure 10A]^[23]. Figure 10B shows the net magnetic moment of Mn₃GaN 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 Mn₃XN (X = Rh, Pd, Ag, Co, Ni, Zn, Ga, In, Sn), revealing an extraordinarily large piezomagnetic effect in Mn₃SnN at room temperature^[33]. Boldrin et al. demonstrate experimentally that a giant piezomagnetic effect is indeed manifest in the AFM antiperovskite Mn₃NiN^[34].

Figure 10. (A) Variation of magnetic moment of Mn Atoms in the (111) Plane of Γ^5g AFM Mn₃GaN with axial strain^[23]; (B) variation of net magnetic moment and rotation angle of Mn atomic magnetic moment with axial strain for Mn₃GaN^[23]; (C) piezomagnetic effect determined by magnetization curve in Mn₃Ga_0.95N_0.94^[35]; (D) piezomagnetic effect of in Mn₃Ga_0.95N_0.94 at 130 K and 170 K^[35].

In 2016, the baromagnetic effect of Mn₃Ga_0.95N_0.94 was determined by both neutron diffraction analysis and magnetic measurements^[35]. Interestingly, Mn₃Ga_0.95N_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 Mn₃GaC^[36]. The collinear AFM- intermediate magnetic transition of Mn₃GaC 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 Mn₃GaC were changed, thereby affecting the magnetocaloric effect^[37]. The magnetic entropy of Mn₃GaC_0.78 decreased to 3.7 J/kg·K under a 5 T magnetic field. In Mn_3-xCo_xGaC, 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].

Figure 11. Magnetocaloric effect of (A) Mn₃GaC^[36] and (B) Mn₃Cu_0.89N_0.96^[39].

A large magnetic entropy change was observed in Mn₃Cu_0.89N_0.96^[39] [Figure 11B]. By introducing vacancies, the onset of the FIM-PM transition is slightly reduced from 150 K of Mn₃CuN to 147.7 K of Mn₃Cu_0.89N_0.96, and a new non-coplanar FIM structure with an orthorhombic symmetry was determined. The total entropy change of Mn₃Cu_0.89N_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 Mn₃Cu_0.89N_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.

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 Mn₃GaN^[40]. As shown in Figure 12A, when a hydrostatic pressure change of 139 MPa is applied, Mn₃GaN 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. Matsunami et al. further suggests that the magnitude of the barocaloric effect of Mn₃GaN 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 Mn₃NiN [Figure 12B]^[41]. It is worth noting that a large barocaloric entropy change, which is a factor of 1.6 than that of Mn₃GaN, is observed. Boldrin et al. proposed that the barocaloric effect of Mn₃NiN originates from multisite exchange interactions amongst the local Mn magnetic moments and their coupling with itinerant electron spins^[41].

Figure 12. (A) The entropy change of Mn₃GaN under different hydrostatic pressures as a function of temperature^[40]; (B) isothermal entropy and adiabatic temperature changes^[41].