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Z. Wang, E. Dengina, Anna Kosogor, [T. Hiroto](https://orcid.org/0000-0002-6176-5782), [Xin Tang](https://orcid.org/0000-0001-6762-6145), [N. Kulesh](https://orcid.org/0000-0001-7046-2671), [A. Bolyachkin](https://orcid.org/0000-0003-0420-1806), [T. Ohkubo](https://orcid.org/0000-0003-3548-1951), [H. Sepehri-Amin](https://orcid.org/0000-0002-7856-7897)

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[Insights into reduction of hysteresis in (Mn, Fe)2(P, Si) compounds by experimental approach and Landau theory](https://mdr.nims.go.jp/datasets/6da82c22-1e03-4a1c-8528-546182ccb64e)

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Insights into reduction of hysteresis in (Mn, Fe)2(P, Si) compounds by experimental approach and Landau theoryZ. Wang a, E. Dengina a, Anna Kosogor b,c, T. Hiroto a, Xin Tang a, *, N. Kulesh a, A. Bolyachkin a, T. Ohkubo a, H. Sepehri-Amin aa National Institute for Materials Science (NIMS), Tsukuba, Ibaraki 305-0047, Japanb Institute of Magnetism NASU and MESU, Kyiv, 03142, Ukrainec University of Vienna, Faculty of Physics, Boltzmanngasse 5, A-1090 Vienna, Austria* Corresponding author: tang.xin@nims.go.jpThe giant magnetocaloric effect in (Mn, Fe)2(P, Si) based alloys arises from a magnetoelastic ferromagnetic-paramagnetic (FM-PM) phase transition accompanied by a large thermal hysteresis. The thermal hysteresis leads to an undesirable irreversibility of the magnetocaloric effect (MCE) during cyclic operation, impeding the practical application in solid-state magnetic refrigeration. Here, we present pre-existing PM nuclei play a role in reducing hysteresis.  A combinatorial analysis, using in situ X-ray diffraction (XRD) and magneto-optical Kerr effect (MOKE) microscopy, reveals that the residual PM phases at the ferromagnetic state of the compound acts as the nuclei for the growth of the PM phases. This is kinetically favorable for the FM-PM phase transition and contributes to a smaller hysteresis from an extrinsic perspective. In addition, the smaller changes in the lattice constants during the phase transition indicates a weakened first-order phase transition. Through a combined effect of intrinsic and extrinsic contributions, a giant MCE in the magnetic entropy change of 16 J kg–1 K−1 under 2 T and a low thermal hysteresis of 3.0 K were achieved in a basic Mn–Fe–Si–P quaternary system for room temperature applications. Furthermore, based on Landau theory and experimentally obtained data, we established an H-T phase diagram and unveiled the crucial role of the magnetoelastic coupling in manipulating thermal hysteresis. Overall, the findings in this work offer a strategy to mitigate hysteresis while retaining a large MCE through extrinsic control. Keywords: (Mn, Fe)2(P, Si) magnetocaloric materials; Thermal hysteresis; Magnetoelastic phase transition; Paramagnetic nucleus; Magnetoelastic coupling 1. IntroductionIn comparison with gas compression refrigeration, the solid-state magnetic refrigeration technology based on the magnetocaloric effect (MCE) offers energy-efficient and environmentally friendly advantages [1]. The MCE can be characterized by the isothermal entropy change (ΔSm) or the adiabatic temperature change (ΔTad) upon the variation of external magnetic fields. In 1976, Pecharsky and Gschneidner et al. proposed a prototype room-temperature refrigerator using Gd as the refrigeration material [2, 3]. Gd, known as a second-order magnetic transition (SOMT) material, exhibits a small gradient of magnetization near its Curie temperature (Tc), leading to a small magnetocaloric response in ΔSm. Giant MCE can be realized by exploiting the magneto-structural transition in numerous first-order magnetic transition (FOMT) materials, such as Gd5(Si, Ge)4 [3], Fe–Rh [4], Ni–M–X (M = Mn, Co and X = Si, Ge, In, or Sn) [5-7], and La(Fe, Si)13-based [8] compounds. In these materials, the giant MCE can be attributed to the abrupt magnetization change near their transition temperatures (Ttr) due to the FOMT. Among these giant MCE materials, low-cost (Mn, Fe)2(P, Si)-based compounds with a hexagonal structure (space group P2m), exhibit a ΔSm of >20 J/kg·K at a moderate magnetic field of 2 T and tunable transition temperature from 71 K to 451 K [9, 10]. These properties make the (Mn, Fe)2(P, Si) compound a competitive candidate for the active magnetic regeneration (AMR) cycle [11]. However, the magnetic phase transition in Mn–Fe–Si–P involves a discontinuous change in lattice parameters at the transition temperature, inducing a thermal hysteresis above 10 K [12]. This not only deteriorates the reversibility of MCE due to hysteresis loss but also the durability under cyclic operation. The main approach to reduce the thermal hysteresis is to weaken FOMT by doping different elements, such as the light element B [12-14], 3d transition metal (V, Co, Ni) [15, 16], 4d transition metal (Zr, Nb, Mo, Ru) [17-20], and other (Al, Ge) [21, 22]. However, the reduction in thermal hysteresis comes at the expense of ΔSm. For instance, with V doping, the hysteresis was reduced from 1.1 to 0.7 K, and the entropy change at 2 T was deteriorated from 14.9 to 8.4 J/kg·K [18]. Among these studies, it is noteworthy that the optimum addition of B can lead to a favorable combination of small hysteresis of 1.6 K and large magnetic entropy change of 14.5 J/kg·K at 2 T [15]. However, this approach has been shown to be effective within a specific compositional range; its broader applicability remains an open question [23]. In addition, from an application perspective, relatively simple compositions are desired, which is beneficial for the upscaling process. Dung et al. found that with the increase of x above 1.4 in MnxFe1.95-xP0.50Si0.50 quaternary system, SOMT with an eliminated hysteresis can be realized with the degradation of ΔSm to below 10 J/kg·K at 2 T [26]. However, the transition temperature for such hysteresis-free compounds falls below 240 K, far from the desirable temperature range for room temperature applications. In this study, we revisited (Mn, Fe)2(P, Si)-based quaternary system with different Mn contents. Analysis of -ΔSm vs. ΔThys in Fig. 1(a) demonstrated that a well-balanced MCE and thermal hysteresis can be achieved solely by adjusting the Mn content in the Mn–Fe–Si–P quaternary system. Note that the magnetocaloric properties of these compounds can be comparable and even superior to those of widely reported B-doped systems [13, 14, 24-33]. This advantage is further highlighted when Ttr is narrowed to the range of 290~310 K (Fig. 1(b)). The developed compounds in this study display different hysteresis upon change of Mn concentration x. This allows us to investigate the fundamental origins of hysteresis within a basic quaternary (Mn, Fe)2(P, Si)-based system. Through in situ X-ray diffraction (XRD) analysis and a magneto-optical Kerr effect microscopy (MOKE), we found that the residual paramagnetic phase (PM) at the ferromagnetic state of the sample acts as the nuclei of ferromagnetic-paramagnetic (FM-PM) phase transition. This reduces the energy barrier required for FOMT, reducing a thermal hysteresis extrinsically. Furthermore, integrating experimental magnetization measurements, the H-T phase diagrams were explored within the Landau theory, which reveal the critical role of the magnetoelastic coupling in controlling the thermal hysteresis.Fig. 1. (a) Summarized entropy change -ΔSm vs. thermal hysteresis ΔThys of Fe2P-type magnetocaloric compounds with transition temperature Ttr from 241 to 320 K, including the data with (square) B-doped [13, 14, 24-34] and (circle) B-free alloys [9, 13, 16, 17, 23, 26, 35-65]. The results reported in this work are shown with the black-edged red circles. (b) Benchmark of -ΔSm vs. ΔThys within a narrower temperature interval Ttr of 300 ±10 K.2. Experimental detailsPolycrystalline MnxFe2-x(P0.48Si0.52)1.03 as-cast ingots with the nominal x of 0.98 1.18, 1.22, and 1.26 were prepared by induction melting using high-purity Mn, Fe, Mn2P, and Si as starting materials. Hereinafter, the four samples with different Mn: Fe ratios are denoted as samples Mn0.98, Mn1.18, Mn1.22, and Mn1.26, respectively. A slight off-stoichiometry in the metal/non-metal ratio of 2: 1.03 was selected to maximize the volume fraction of the Fe2P-type matrix phase following our previous work [66]. An extra 10 wt.% of Mn was added to compensate the evaporation loss of this metal during the induction melting. All the ingot slices were wrapped in Molybdenum foil and sealed in quartz ampoules under 80 kPa of Ar. Thereafter, the as-cast ingots were annealed at various temperatures from 1100 to 1200 oC for 5 hours, followed by water quenching, to obtain the largest MCE in terms of ΔSm. More detailed information on annealing conditions was described in the Figure S1 (supplementary file). The weight losses of MnxFe2-xP0.48Si0.52 samples during the annealing process range from 2 to 5 wt.%. The outermost surface layers of samples were polished to remove oxides before all measurements. The thermomagnetic properties of prepared samples were examined by a superconducting quantum interference device vibrating sample magnetometer, SQUID-VSM (Quantum Design MPMS SQUID VSM). At least three ingots for each composition were prepared and processed with their optimal annealing conditions to confirm the reproducibility of thermomagnetic curves (Figure S2, supplementary file). The scanning electron microscopy (SEM) was performed using a Carl Zeiss CrossBeam 1540EsB microscope equipped with an energy-dispersive spectroscopy (EDS) detector, Bruker XFlash 6 series. The overall alloy compositions, obtained from low-magnification SEM-EDS maps, were listed in Table S1 (supplementary file). The tendency of measured alloy composition is generally consistent with nominal compositions. In situ XRD experiments during a cooling and heating cycle, under a purged nitrogen atmosphere of 1 atm, was performed on a X-ray diffractometer (SmartLab with TTK-600 chamber, Rigaku) using a Cu Kα source at 45 kV and 200 mV. Before heating, the samples were pre-cooled down to 240 K, well below the transition temperature. The heating and cooling rates of 2 K/min rate was employed to detect phase transition around their respective transition temperature. To ensure the temperature homogeneity, we dwelled 5 min before each temperature measurements. The magneto-optical Kerr effect (MOKE) microscopy was performed using an Evico Magnetics microscope equipped with a heating system for in situ observation of the magnetostructural phase transition upon heating.3. Results and discussions3.1. Experimental resultsFig. 2. (a) Temperature dependencies of magnetization under an applied magnetic field of 1 T during heating and cooling process of MnxFe2-x(P0.48Si0.52)1.03 powdery compounds with a nominal x = 0.98, 1.18, 1.22, and 1.26. (b) Corresponding -ΔSm during heating and cooling cycle as a function of temperature under a field change of 1 and 2 T. The overlapped area represents the reversible -ΔSm for a field change of 2 T.Figure 2(a) shows the M–T curves measured under a constant magnetic field of 1 T for the optimally annealed MnxFe2-x(P0.48Si0.52)1.03 alloys. During the heating process, typical FM-PM magnetoelastic phase transitions were observed for all samples. The transition temperature for each M–T curve was determined as the temperature at which the derivative (dM/dT) was maximum. The thermal hysteresis, defined as ΔThys, represents the Ttr difference upon cooling and heating. Table 1 summarizes the magnetocaloric properties and structural parameters of the prepared alloys in more detail. With tailoring x from 0.98 to 1.18, Ttr (heating) was found to decrease from 353.7 to 302.3 K. However, with increase of x from 1.18 to 1.26, the transition temperature does not change too much as the transition temperature is sensitive to matrix phase compositions (the matrix phase composition is listed in the next section) that can be influenced by annealing condition. This can be evidenced in the literature [26]. More details are provided in supplementary note 1. The ΔThys values demonstrated a monotonous reduction from 35.7 to a minimum of 3.0 K with increase of x. Such a strong correlation between the ΔThys and Mn concentration in the alloys indicates that substituting Fe with Mn can weaken the FOMT in Mn–Fe–P–Si system. The key figure of merit of MCE, -ΔSm, was determined using the Maxwell equation via measuring isofield M–T curves upon heating and cooling near their transition temperatures, plotted in Fig. 2(b). The -Δ values for Mn0.98 were 11.4 and 21.0 J/kg·K under 1 and 2 T magnetic field, respectively, while -Δ for Mn1.18 were evaluated to be 12.1 and 22.6 J/kg·K under 1 and 2 T magnetic field, respectively. Such -Δ values are larger than those of most Mn–Fe–Si–P alloys reported in the literature (Fig. 1), but this was accompanied with a remarkable hysteresis of 35.7 and 17.0 K for Mn0.98 and Mn1.18, respectively. Increasing x to 1.22 and 1.26 resulted in a substantial reduction in the thermal hysteresis down to 8.0 and 3.0 K, respectively. Meanwhile, we found that -Δ slightly degraded to 17.9 and 16.0 J/kg·K under 2 T, respectively. Thus, a well-balanced combination of small hysteresis and large -Δwas achieved in the Mn–Fe–Si–P quaternary system without doping any other elements, as shown in Fig. 2(b). Furthermore, among all the developed compounds, the maximum overlapping area of -Δ and -Δcurves in Mn1.26 sample suggests the largest reversible -ΔSm under cycling conditions [19, 46, 67] due to smaller hysteresis in this sample. Note that this reversible magnetic entropy change is determined based on a complete phase transition scenario and may differ from the values obtained during actual cyclic operation [68]. Hereafter, Mn1.18, Mn1.22, and Mn1.26 were selected for further investigation, as they have comparable transition temperatures close to the room temperature. And -ΔSm upon heating is used as the default magnetic entropy change unless otherwise specified.Table 1. The summarized properties and structural parameters of MnxFe2-x(P0.48Si0.52)1.03 alloys with x = 0.98, 1.18, 1.22, and 1.26: Transition temperature (Ttr) on heating and cooling and thermal hysteresis (ΔThys) under a 1T field and magnetic entropy change (-Δ) measured under a magnetic field change of 1 and 2 T, the lattice parameters a and c, the c/a ratio, the unit-cell volume V and the mass fractions of the matrix phase at paramagnetic state and the impurity phase. Samples Ttr (heating), K Ttr (cooling), K ΔThys, K -Δ, J/kg·K a (Å) c (Å) c/a V (Å) Matrix phase (wt.%) 3:1 type impurity phase (wt.%) 5:3 type impurity phase (wt.%)     0–1 T 0–2T        Mn0.98 353.7 318.0 35.7 11.4 21.0 6.0575  3.4755 0.5737 110.44  95.7 4.3 0 Mn1.18 302.3 285.3 17.0 12.1 22.6 6.0829 3.4575 0.5684 110.80 92.7 6.1 1.2 Mn1.22 298.1 290.1 8.0 9.5 17.9 6.0985 3.4494 0.5656 111.10 93.0 6.2 0.8 Mn1.26 300.0 297.0 3.0 8.2 16.0 6.1022 3.4462 0.5648 111.13 94.8 5.2 0Fig. 3.  SEM-EDS maps of Mn, and Fe elements taken from the (a) Mn1.18, (b) Mn1.22, and (c) Mn1.26 alloys. (d) The table with compositions in the matrix grains of the observed three samples. It is well-known that the transition temperature and strength of FOMT is sensitive to the composition of matrix phase in Mn–Fe–Si–P system. Herein, we analyzed the composition of matrix phase using SEM-EDS technique. For all samples in this work, no compositional fluctuations can be detected in the matrix regions, indicating a uniform composition within the (Mn, Fe)2(P, Si) matrix phase. In addition, the secondary phases located at so-called “grain boundary” region are mainly comprised of major 3:1 phase (region i) and minor 5:3 phase (region ii) based on compositional analysis. The composition of matrix phase is further statistically analyzed and summarized in the table in Fig. 3(d), wherein only a slight composition change occurs upon x, such as metallic Mn: Fe and nonmetallic Si: P. The Ttr is usually controlled by the Mn: Fe and P: Si ratios of the matrix phase, namely, an increase in ratios would reduce the transition temperature [26, 60]. The ratio of Mn: Fe in our case is monotonously increased from 1.42 ± 0.08 for Mn1.18 to 1.66 ± 0.05 for Mn1.26. but its resultant negative contributions to Ttr is compensated by reduced P: Si ratios [54]. This well explains all three samples can maintain a similar Ttr around room temperature despite alloying different Mn contents. Fig. 4(a-c) shows the XRD patterns measured at 240 K, well below the transition temperature, for three samples observed above. The quantitative XRD results illustrate that the matrix phase of Mn1.18 sample entirely transforms into a low-temperature ferromagnetic (FM) phase upon cooling process, as listed in Fig. 4(d). In contrast, residual PM phases with a mass fraction of about 10 and 12 wt.% remains at 240 K for Mn1.22 and Mn1.26, respectively, which has been rarely reported in the Mn–Fe–Si–P system. Gottschall et al. [68, 69] exploited the co-existence of martensite/austenite phases in minor loops to achieve good reversibility in Heusler alloys. This indicates coexistence of these phases can be advantageous in reducing hysteresis. In this work, we can find that the presence of PM phase at the ferromagnetic state of Mn–Fe–Si–P samples has correlation to the thermal hysteresis. For instance, the Mn1.18 sample shows no PM residual phase with a hysteresis of 17.0 K. As the PM phase increases from 10 to 12 wt.%, the thermal hysteresis decreases from 8.0 to 3.0 K in Mn1.22 and Mn1.26 samples, respectively.  Fig. 4. XRD patterns measured at 240 K under a zero magnetic field for (a) Mn1.18, (b) Mn1.22, and (c) Mn1.26. (d) The mass proportion of ferromagnetic (FM) and paramagnetic (PM) phases in the (Mn, Fe)2(P, Si) phase extracted from the XRD data.Fig. 5. (a) Schematic illustration of the original and processed magnetic domains on bulk Mn1.26 sample observed with MOKE microscope. (b) Temperature dependent MOKE images of Mn1.18 (upper row) and Mn1.26 (bottom row) showing the phase nucleation and propagation during heating process. The influence of the residual paramagnetic (PM) phase in the ferromagnetic state of Mn–Fe–Si–P samples on the phase transition is investigated by visualizing the thermally driven phase transition through MOKE. Since the MOKE is observed in a zero-field state, corresponding M–T curves under a small field of 0.005 T were attached as Figure S4 (Supplementary Information). At 200 K, the Mn1.26 sample comprised the low-temperature FM phase with grey contrast and some inclusions of the high-temperature PM phase with dark contrast (Fig. 5(a)), which is consistent with the lower magnetization in M–T curves measured under a field of 0.005 T. To enhance the imaging contrast, the FM and PM phases were colored in blue and red, respectively, as schematically illustrated in Fig. 5(a). In contrast, the Mn1.18 sample with large hysteresis does not show such a mixed state at 200 K that is consistent with the XRD data at 240 K. Additionally, the Mn1.18 sample exhibits a higher areal density of cracks, indicating inferior mechanical properties to the amplitude of the discontinuities on a and c parameters. Upon heating to 260 K, the PM phase starts nucleating next to the cracks in the Mn1.18 sample. As the temperature increases to 300 K, an increase in PM nuclei is observed, but the already-nucleated PM phase does not propagate, suggesting a strong pinning effect at the PM/FM phase interface (PI 1), marked in Fig. 5 with yellow arrow. At 310 K, the cascade propagation of nucleated PM phase is observed, which is replication of a sharp phase transition in M-T curves and might be correlated to a strong FOMT in Mn1.18 sample with a large hysteresis. On the other hand, for the Mn1.26 sample, an increase in temperature from 200 K to 265 K results in the growth of the PM phase along with the observation of more PM nuclei. The weakened pinning effect allows the PM phase to easily propagate into neighboring matrix grains, as the PM/FM interface (PI 2) is marked with a yellow arrow. These results indicate that the FM-PM phase transition occurs over a wider temperature range, leading to a less sharp phase transition and weaker FOMT in the Mn1.26 sample. When the sample is in its ferromagnetic state, existence of paramagnetic nucleus would become the FM/PM transition seeds that lower the energy barrier required for phase transition as no additional energy needed for nucleation of the paramagnetic phase, thereby reducing the hysteresis.Fig. 6. In situ XRD waterfall plots of (a) Mn1.18 and (b) Mn1.26 upon heating under a zero field. (c) Lattice parameters vs. temperature extracted from XRD patterns, including a, c, c/a ratio, and V.  Figure 6 shows in situ XRD experiments to elucidate the detailed phase transition process during heating of Mn1.18 and Mn1.26 powdery samples. Note that the samples were cooled down to 240 K, well below the transition temperature of the samples prior to the XRD experiment. The evolution of 111 peak around 2θ of 39.4o indicates that a small amount of PM phase “nucleus” (marked with a red dotted line) emerges when Mn1.18 is heated to about 288 K. In contrast, strong 111 peak in low-hysteresis Mn1.26 suggests the dual phases exist after cooling down the sample and throughout the entire heating process, which is consistent with MOKE results. In addition, the change of the lattice parameters upon transition, is primarily evaluated by analyzing the magnitude of diffraction peak shifts. The peak shift during phase transition is less pronounced in the Mn1.26 sample with low hysteresis compared to the Mn1.18 sample with large hysteresis. Based on the in situ XRD data, the changes in lattice constants were extracted for Mn1.18 and Mn1.26 samples upon heating, including a, c, c/a ratio, and volume V, respectively, as shown in Fig. 6(c). When heating to around Ttr, the dramatic lattice contraction in a and expansion in c are captured. Interestingly, the impact of alloy compositions on the lattice constants of Fe2P type phases at the PM state is more pronounced than at the FM state. This leads to the different changes of lattice constants around Ttr, such as a, c, c/a ratio. For instance, the lattice parameter a change from a similar initial value of ~6.18 Å in FM state to 6.08 Å for Mn1.18 and 6.11 Å for Mn1.26 in PM state. Thus, the observed phase transition depending on Mn concentrations results into different changes in lattice constants. However, the unit cell volume V keeps a continuous change throughout the entire heating process. The smaller change in lattice parameters for Mn1.26 sample has revealed a weaker magnetoelastic coupling during the phase transition, which indicates weakened first-order character of transition.3.2. Landau-type theoretical analysisAbove, a comprehensive experimental investigation was conducted to examine the impact of extrinsic (presence of PM nucleus) and intrinsic (change in lattice parameter upon phase transition) factors on thermal hysteresis in Mn–Fe–Si–P alloys. The first-order phase transitions are always accompanied by hysteresis associated with the existence of a region of metastability of two phases. The Landau theory proves to be a powerful method for comprehensively describing various categories of phase transitions [70]. As it was shown in the XRD part (Fig. 6), Mn–Fe–Si–P alloys undergo the magnetoelastic FM to PM transition without symmetry and volume change, while only the change in lattice constants occurs. In such a case the primary order parameter is magnetization, M, while the deformation is a secondary order parameter, . The primary order parameter is a main variable in Landau theory that characterizes the phase transition, while the secondary order parameter is another property of the system that may undergo a change during a phase transition, but its change is not as significant or fundamental as that of the primary order parameter [70]. The primary order parameter is distinguished from secondary order parameter by the fact that the coefficient of the square of the primary parameter in the expression for free energy goes to zero at the transition temperature  and then changes sign. For the simplicity, we consider the one-component order parameter M, and the deformation  along z axis, in this case the Gibbs potential in the presence of magnetic field H can be expressed as: ,   (1)where  is a temperature-dependent coefficient that governs the transition temperature , the coefficients  are phenomenological constants. Depending on the sign of the coefficient , either a first-order phase transition (if ) or a second-order phase transition (if ) occurs. It should be noted, that usually purely magnetic (without changes in crystal lattice) PM-FM phase transitions are second-order phase transitions, while first-order PM-FM phase transitions are observed for magnetoelastic or magnetostructural phase transitions [71]. This suggests that exactly elastic system and magnetoelastic coupling is responsible for the appearance of first-order PM-FM phase transitions. In particular, the magnetoelastic Bean-Rodbell theory was proposed to explain the first-order PM-FM phase transition in Mn–As, which is associated with a substantial volume change [72]. They showed that if the exchange energy (or Curie temperature) depends on the lattice volume, under certain conditions the behavior of the system may be the usual second-order transition, but it can in fact become a first-order phase transition. Noteworthy that recent experiments for doped ErCo2-xFex alloys confirmed the switch from the first-order magnetostructural to second-order magnetic phase transition by reducing volume and lattice change during transition [73]. The magnetoelastic Bean-Rodbell model has been successfully applied to describe first-order magnetocaloric effect (MCE) materials, as discussed in Ref. [74] and references therein. The Landau-type expansion of the Gibbs free energy allows for the estimation of the energy landscape and, consequently, the energy barriers characterizing the MCE system. The case of Mn–Fe–Si–P alloys differs from the Bean-Rodbell approach because no volume change was detected experimentally as shown in Fig. 6; instead, changes in lattice constants occur. Therefore, here we propose an alternative method for applying the magnetoelastic model to phase transitions where no volume change occurs, but only lattice changes take place. In this case, one should analyze the elastic and magnetoelastic energy (Eq. (1)). The detailed explanation of construction of free energy in the case of secondary order parameter is presented in [70], according to it, the elastic energy can be expanded in series of :,    (2)where the polynomial coefficients are elastic constants. The magnetoelastic energy is expressed as following , where  is magnetoelastic coupling coefficient. The minimization of the function  gives:.     (3)The  value is associated with thermal expansion of the material along z axis, so with some deformation , which is not induced by phase transition at . It can be excluded from the energy by redesignation , where  is initial value of deformation  at phase transition temperature. This value is determined by the equation . This equation can be included into the background free energy . Therefore, Eq. (3) can be rewritten in the form.,     (4)where coefficient C differs from  by additives that depend on  due to the presence in Eq. (3) such terms as . Considering Eqs. (1), (2), (4) one can exclude  from the Gibbs potential: ,   (5)where renormalized coefficients are expressed as                           (6)Therefore, the interrelation between primary and secondary order parameters leads to the renormalization of energy coefficients connected with primary order parameter, M. This behavior exemplifies the typical relationship between primary and secondary order parameters (for more details see [70]). The stronger is magnetoelastic coupling the more pronounced is change in energy coefficients. The magnetoelastic coupling results in a shift of the magnetoelastic phase transition temperature, , because the phase transition temperature corresponds to condition, where the coefficient of the square of the primary parameter equals to zero. This condition for renormalized energy coefficient  is distinct from the condition , associated with the purely magnetic phase transition temperature, . Therefore, the temperature of magnetoelastic phase transition can be expressed as following:.     (7)It is seen that in the absence of magnetoelastic coupling  and  coincide. Another important point is the renormalization of the coefficient , the sign of which predetermines the order of the phase transition. Evidently, the presence of magnetoelastic coupling can lead to the change of the sign of coefficient , switching purely magnetic second-order PM-FM phase transition () to the first-order magnetoelastic phase transition (). Now we can analyze first-order magnetoelastic phase transition (), which is described by the potential Eq. (5). The equation of state can be found from the minimum condition       (8)and condition :.      (9)In the absence of magnetic field, the PM phase is stable when , the ferromagnetic phase is stable when,     (10)the region of coexistence of two phases is expressed as :.     (11)The region where two phases coexist represents a metastable state, resulting in overcooling and overheating processes, which, in turn, lead to the thermal hysteresis observed in first-order phase transitions. As so, the temperatures  (Eq. (7)) and  (Eq. (10)) correspond to the transition temperature on cooling and heating, respectively. The value  is a thermal hysteresis observed in zero magnetic field. To comprehend the hysteresis behavior under the influence of an applied magnetic field, one should examine Eq. (5). Such kind of potential (2-4-6 series in order parameter and bilinear coupling between order parameter and external field) was previously analyzed in Ref. [75], where a phase diagram for such transition was constructed. It has been shown that phase diagram has two phase stability boundaries lines, which meet at the critical point and the first-order phase transition disappears. The coordinates of this points can be found from equation of state Eq. (8) along with conditions  ,  and in our case result in                    (12)The temperatures of phase stability can be found from Eq. (8) ,    (13)where characteristic solutions, which correspond to metastable states, can be found from the condition [75]:.    (14)Substituting the solutions of Eq. (14) into Eq. (13) one can plot the stability boundaries of FM and PM phases on temperature-magnetic field phase diagram.  The energy coefficients in the potential equation (5) were determined through the fitting of theoretical magnetization versus temperature curves to experimental data (Fig. 7). The fitting was applied to all samples with x = 0.98, 1.18, 1.22, 1.26, a mass density of  is accepted. The parameters  and , which are independent of the  value, remained constant, while the renormalized parameters  and  were adjusted for each sample composition. Table 2 presents the parameters used in calculations ( and ). These values, along with Eqs. (13) and (14), enable the computation of transition temperatures (Ttr) during heating and cooling under a 1 T magnetic field, as well as the hysteresis width (ΔThys) (see Table 2). The absolute value of the coefficient  decreases with an increasing concentration of x, corresponding to the observed reduction in hysteresis as x increases. This reduction in coefficient  can be attributed to the decrease in magnetoelastic coupling, consistent with the dual decrease in the lattice parameters' jump during the phase transition with an increase in x concentration (Fig. 6). This decrease of coefficient  provides the good agreement between computed transition temperatures and hysteresis widths (Table 2) and experimental data presented in Table 1.Fig. 7. Experimental and theoretical temperature dependencies of magnetization obtained for Mn1.18 sample under a magnetic field of 1 T.Table 2. Parameters used for calculations (, ) along with the computed transition temperatures and hysteresis under a magnetic field of 1 T.  Samples , T4kg3/J3  , K Ttr (heating), K Ttr (cooling), K , K Mn0.98 -13.3 × 10-5 312 352.0 318.0 34 Mn1.18 -9.9 × 10-5 280 302.8 285.4 17.4 Mn1.22 -7.0 × 10-5 286 297.9 290.8 7.1 Mn1.26 -5.5 × 10-5 291 298.4 295.4 3Fig. 8 illustrates the experimental and theoretical H-T phase diagrams obtained for Mn1.18 and Mn1.26 samples. The experimental phase diagram depicts the dependencies of Ttr obtained during the heating and cooling of specimens in a magnetic field. Dashed lines represent linear approximations to the experimental points. The theoretical diagram, calculated from Eqs. (13) and (14), defines regions of stability for the FM and PM phases. The intervening area constitutes the metastable region, where both phases can coexist. The presence of the metastable region gives rise to overcooling and overheating processes, intrinsic to the temperature hysteresis observed in first-order phase transitions. The hysteresis decreases when approaching the critical point (), where first-order phase transition disappears. It's noteworthy that the emergence of a critical point on the phase diagram is observed under specific conditions for various potentials [70, 74]. In particular, the interrelation between the critical point and thermal hysteresis has been theoretically investigated for magnetic phase transitions in Mn–As alloy [76] and structural phase transitions in Heusler alloys [77]. The experimental confirmation of hysteresis disappearance at the critical point was observed during structural phase transitions in Ni–Fe–Co–Ga and Fe–Pd alloys [78, 79]. Moreover, the reduction of thermal hysteresis when approaching critical point was experimentally observed for the Mn1.32Fe0.71P0.5Si0.56 alloy specimen [46], the coordinates of critical point were 317 K, 16.7 T, which is quite close to our case for Mn1.18.Fig. 8. Experimental (a) and theoretical (b) phase diagrams obtained for Mn1.18 and Mn1.26. The critical point indicates the end of first-order phase transition. As the concentration of Mn increases, the phase stability lines merge, leading to a noticeable decrease in hysteresis. This reduction can be attributed to the decrease in magnetoelastic coupling, resulting in a decrease in the absolute value of the coefficient , which predetermines the hysteresis width (Eq. (11)). The coordinates of the critical point, where the first-order phase transition disappears, are also influenced by the value of . The smaller  is, the smaller the critical magnetic field where the magnetically induced first-order PM-FM phase transition is observed (Eq. (12)). It should be noted that the energy coefficient  can be influenced not only by magnetoelastic interactions but also by changes in the electronic structure of the material [80]. Noteworthy that Equation (11) describes the intrinsic hysteresis linked to the metastability region inherent in first-order phase transitions. Nevertheless, extrinsic factors such as microstructure, heterogeneous nucleation, or kinetic effects also play a role in the hysteresis. Thus, the coefficient  obtained through fitting theoretical curves to experimental data encapsulates the combined influence of factors affecting hysteresis.Discussions We have explored the potential of the (Mn, Fe)2(P, Si) quaternary system and realized well-balanced magnetocaloric properties, i.e., small thermal hysteresis of 3.0 K and large -ΔSm of 16 J kg–1 K−1 at 2 T. The obtained properties are comparable to the best properties reported to far in (Mn, Fe)2(P, Si) for room temperature refrigeration. In addition, the thermal hysteresis can be tuned depending on Mn concentration x in MnxFe2-x(P0.48Si0.52)1.03 alloys, as reported in literature [57]. This enables us to investigate the origin of the thermal hysteresis for the studied compounds. Under the framework of the Landau theory [74, 81-83], the FOMT is theoretically controlled by different energy barriers between the local and global minima in the Gibbs free energy within one cooling and heating cycle, leading to the hysteresis. The origin of the energy barrier for the phase transition can generally be classified into intrinsic and extrinsic contributions [84]. The intrinsic aspect is related to a change in crystallographic structure parameters, e.g., symmetry, lattice parameters, volume of a unit cell. This in turn alters the electronic structure and its associated magnetism. While the extrinsic influence is linked to the microstructure, such as grain size, defects, phase boundaries, etc. and kinetics under the external stimulus. For the (Mn, Fe)2(P, Si) system, lowering lattice parameter change across the phase transition has been widely known for minimization of hysteresis due to weakening of FOMT. In this work, we characterized the strength of FOMT by tracking the phase transition using in situ XRD technique. As x increases from 1.18 to 1.26 in the MnxFe2-x(P0.48Si0.52)1.03 alloys, the lattice constant changes across the transition were reduced, revealing a weaker magnetoelastic coupling. Thereby, the hysteresis is reduced from 17.0 K to 3.0 K from intrinsic aspect. Of particular interest is that we report for the first time that the residual PM phase in the ferromagnetic state of the sample also plays a role in reducing the thermal hysteresis. For a low-hysteresis Mn1.26 sample, the pre-existing PM phase acts as a nucleus for growth of PM phase during the phase transition as revealed by MOKE observation. As a result, the propagation of PM becomes kinetically easier compared to the large-hysteresis Mn1.18 sample. This phenomenon reduces the energy required for the phase transition, which could be an additional extrinsic factor for the small hysteresis in the Mn1.26 sample. In order to describe the influence of magnetoelastic coupling within the (Mn, Fe)₂(P, Si) system, an analysis of magnetic, magnetoelastic, and elastic energies was conducted using the Landau theory framework. The presence of magnetoelastic coupling was found to result in the renormalization of coefficients in the Landau expansion for the Gibbs potential (Eqs. (5), (6)). The renormalization of coefficient  by the magnetoelastic coupling coefficient  leads to the shift of phase transition temperature. Simultaneously, the renormalization of the  coefficient by magnetoelastic coupling results in the hysteresis variation. The H–T phase diagram was analyzed both theoretically and experimentally. Through fitting theoretical data to experimental points, energy coefficients were determined for different samples. It has been shown that increase of x in MnxFe2-x(P0.48Si0.52)1.03 alloys leads to the decrease of absolute value of the energy coefficient  consequently leading to a reduction in thermal hysteresis. The observed decline in  with increasing x was attributed to a corresponding decrease in the magnetoelastic coupling coefficient  . This observation aligns with the decrease in the lattice constant jump as x increases, capturing the weakening of the magnetoelastic phase transition. Fig. 9. The summarized elastic strain energy Ue at Ttr as a function of thermal hysteresis ΔThys for the (Mn, Fe)2(P, Si) alloys [9, 14, 19, 28, 33, 34, 37, 47, 50, 85, 86]. The dashed line is a guide to the eyes.The elastic energy is strongly correlated with thermal hysteresis during magnetoelastic and magnetostructural phase transitions, so let us analyze the elastic strain energy Ue caused by phase transition in (Mn, Fe)2(P, Si) based magnetocaloric alloys. The Ue is estimated by formula below [14, 19, 87]:Ue = (C11 + C12)   + 2C13e1 e3 + C33 .The elastic constants C11 = 308.4, C12 = 120.0, C13 = 144.0, and C33 = 227.8 GPa are taken from Refs. [14, 87]. As the symmetry of 2:1 type hexagonal crystals, the tensile strain within the ab plane is e1 = e2 = Δa/a and along the c axis e3 = Δc/c. The volume change across the FOMT is an almost constant, e1 = e3 [14].To confirm the extrinsic contribution from nucleus, the relationship between the elastic strain energy and thermal hysteresis for the data adopted from literature and our experimental data were plotted in Fig. 9. Compared to data reported in literature, the Mn1.22 and Mn1.26 samples exhibit larger elastic strain energy but smaller thermal hysteresis than expected. This can be attributed to the pre-existing nucleus in both samples that reduces the energy barrier required for phase transition as observed by MOKE. ConclusionIn conclusion, we have reported that the thermal hysteresis in quaternary (MnxFe2-x)(P0.48Si0.52)1.03 magnetocaloric compounds can be substantially reduced by tailoring the Mn content from x = 0.98 to 1.26. As a result, a giant MCE in magnetic entropy change of 16 J/kg·K under 2 T external magnetic field and low hysteresis of 3.0 K were simultaneously achieved in the Mn1.26Fe0.74(P0.48Si0.52)1.03 sample. To elucidate the cause of the reduced hysteresis, we performed MOKE observations and found that the pre-existing paramagnetic (PM) phase in the low-temperature ferromagnetic state of the sample acts as a nucleus for the growth of the PM phase. Therefore, the residual PM phase is kinetically favorable for driving the FM-PM phase transition in the Mn1.26 sample. In addition to the extrinsic origin mentioned above, an increase in Mn: Fe ratios in chemical compositions reduces the change in lattice constants (a, c) as reported by Dung et al. [57]. It effectively lowers the energy barrier between PM and FM phases in the energy landscape, thereby reducing the detrimental hysteresis from the intrinsic aspect, as detected by in situ XRD. The H-T phase diagram is constructed using Landau theory and is in agreement with experimental measurements, showing that the reduction in the thermal hysteresis is due to weakened magnetoelastic coupling. 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(a) Summarized entropy change -ΔSm vs. thermal hysteresis ΔThys of Fe2P-type magnetocaloric compounds with transition temperature Ttr from 241 to 320 K, including the data with (square) B-doped [13, 14, 24-34] and (circle) B-free alloys [9, 13, 16, 17, 23, 26, 35-65]. The results reported in this work are shown with the black-edged red circles. (b) Benchmark of -ΔSm vs. ΔThys within a narrower temperature interval Ttr of 300 ±10 K.Fig. 2. (a) Temperature dependencies of magnetization under an applied magnetic field of 1 T during heating and cooling process of MnxFe2-x(P0.48Si0.52)1.03 powdery compounds with a nominal x = 0.98, 1.18, 1.22, and 1.26. (b) Corresponding -ΔSm during heating and cooling cycle as a function of temperature under applied magnetic fields of 0–1 and 0–2 T. The overlapped area represents the reversible -ΔSm for a field change of 2 T.Fig. 3.  SEM-EDS maps of Mn, and Fe elements taken from the (a) Mn1.18, (b) Mn1.22, and (c) Mn1.26 alloys. (d) The table with compositions in the matrix grains of the observed three samples. Fig. 4. XRD patterns measured at 240 K under a zero magnetic field for (a) Mn1.18, (b) Mn1.22, and (c) Mn1.26. (d) The mass proportion of ferromagnetic (FM) and paramagnetic (PM) phases in the (Mn, Fe)2(P, Si) phase extracted from the XRD data.Fig. 5. (a) Schematic illustration of the original and processed magnetic domains on bulk Mn1.26 sample observed with MOKE microscope. (b) Temperature dependent MOKE images of Mn1.18 (upper row) and Mn1.26 (bottom row) showing the phase nucleation and propagation during heating process. Fig. 6. In situ XRD waterfall plots of (a) Mn1.18 and (b) Mn1.26 upon heating under a zero field. (c) Lattice parameters vs. temperature extracted from XRD patterns, including a, c, c/a ratio, and V.  Fig. 7. Experimental and theoretical temperature dependencies of magnetization obtained for Mn1.18 sample under a magnetic field of 1 T.Fig. 8. Experimental (a) and theoretical (b) phase diagrams obtained for Mn1.18 and Mn1.26. The critical point indicates the end of first-order phase transition. Fig. 9. The summarized elastic strain energy Ue at Ttr as a function of thermal hysteresis ΔTThys for the (Mn, Fe)2(P, Si) alloys [9, 14, 19, 28, 33, 34, 37, 47, 50, 85, 86]. The dashed line is a guide to the eyes.   25image1.pngimage2.jpegimage3.pngimage4.pngimage5.jpegimage6.pngimage7.wmf336.410 kg/m×oleObject1.binimage8.wmf2201.610 Tkg/JK-a=×oleObject2.binimage9.wmf9655710 Tkg/J-g=×oleObject3.binimage10.jpegimage11.jpegimage12.jpeg