# Correlation between deformation and total entropy change at the first-order magnetic transition of Fe2P-based magnetocaloric materials

F. Guillou, H. Yibole, B. Narsu, V. Hardy

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F. Guillou, H. Yibole, B. Narsu, V. Hardy. Correlation between deformation and total entropy change at the first-order magnetic transition of Fe2P-based magnetocaloric materials. Results in Physics, 2023, 44, pp.106203. 10.1016/j.rinp.2022.106203. hal-04305446

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1 Correlation between deformation and total entropy change at the first-order magnetic
2 transition of Fe₂P-based magnetocaloric materials

3 F. Guillou¹, H. Yibole¹,*, B. Narsu¹, and V. Hardy²

4 1- College of Physics and Electronic Information, Inner Mongolia Key Laboratory for Physics
5 and Chemistry of Functional Materials, Inner Mongolia Normal University, 81 Zhaowuda Road,
6 Hohhot 010022, Inner Mongolia, People's Republic of China

7 2- Normandie University, ENSICAEN, UNICAEN, CNRS, CRISMAT, 14000 Caen, France

8 *hyibole@imnu.edu.cn

9 Highlights :

10 - latent heat or transition entropy are key parameters in giant magnetocaloric materials

11 - a general scaling of ΔS<sub>tr</sub> with Δ(c/a) is observed in Fe<sub>2</sub>P-related materials family

12 - deformation has a crucial role even in volume preserving FOMTs

13 Abstract:

14 While there is a growing consensus that lattice is an important source of entropy change at
15 structural or elastic first-order magnetic transitions (FOMT) with finite volume change, the case
16 of volume preserving transitions (ΔV = 0) remains more controversial. Here, the evolution of the
17 total transition entropy change within the Fe2P family is presented as a function of the
18 discontinuity on the ratio of the cell parameters. A striking correlation between ΔStr and Δ(c/a) is
19 observed. This relationship is a first step towards a systematization of the FOMT properties in
20 the Fe2P materials system. It also demonstrates that deformation can play a leading role in
21 volume preserving FOMT.

22 Keywords:

23 Latent heat; First-order Magnetic Transition; Magnetism; Magnetocaloric effect;

24 Thermomagnetic materials
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Magnetic transitions often involve simultaneous changes in the electronic ($S_{el}$), lattice ($S_{latt}$), and magnetic ($S_{mag}$) properties. The expression $S = S_{latt}+S_{el}+S_{mag}$ is commonly employed to express how these degrees of freedoms and their entropy are expected to sum up. By extension, at first-order magnetic transition (FOMT) with transition temperature ($T_{tr}$), the transition entropy $\Delta S_{tr} = L/T_{tr}$ related to the latent heat ($L$) is expected to originate from one or several of these contributions. Unfortunately, estimating individually each term is challenging. On top of that, the above summation relies on a strong and disputable assumption that the degrees of freedom are independent, i.e. there is no cooperativity between them.

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Understanding the driving mechanisms for the latent heat of giant magnetocaloric materials is of outmost importance for applications, as the optimization primarily consists in triggering a large latent heat for an external magnetic field change as small as possible. Following the discovery of a giant magnetocaloric effect with magneto-structural FOMT in $Gd_{5}Si_{2}Ge_{2}$, a structural entropy change representing about half of the magnetocaloric effect was proposed as primary reason for overcoming the performances of elemental Gd [1,2]. This led to a general scaling that the larger the relative volume change $\Delta V/V$, the stronger the lattice contribution and therefore the higher the magnetocaloric effect [3]. This approach may however be confusing as it considers the isothermal entropy change induced by a magnetic field change (referred to as $\Delta S$), whose relationship with the transition entropy discontinuity in zero field (referred to as $\Delta S_{tr}$) is indirect. In addition, an accurate determination of $\Delta S$ can be challenging at FOMT due to the spike artefact. A $\Delta V/V$ scaling of the magnetocaloric effect appears also applicable to isotropic magneto-elastic FOMT such as in $La(Fe,Si)_{13}$. For this system, after some early controversies on whether the lattice entropy change counteracts the magnetic one [4], direct insights using inelastic resonant x-ray scattering techniques have led to a consensus that the lattice entropy change adds up to the magnetic contribution and may be responsible for nearly half the isothermal entropy change [5-8]. However, $\Delta V$ or $\Delta V/V$ is not a convenient parameter for characterizing magneto-elastic anisotropic FOMT as in hexagonal $Fe_{2}P$ compounds where the contraction on the $a$ axis is compensated by an expansion along the $c$ axis leading to nearly negligible volume changes.

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In view of the strong interest in $Fe_{2}P$ compounds [9-28], insights into the mechanism of their FOMT are highly desirable. $Fe_{2}P$-based compounds can exhibit ferromagnetic transitions with
extreme behaviors. Depending on chemical compositions, it is possible to achieve: i) second-order transitions (typically Mn₂ₓFeₓP₀.₅Si₀.₅ with x < 0.7) [9]; ii) FOMT with particularly small hysteresis and latent heat (case of the Fe₂P binary parent) [10,11]; iii) FOMT with small hysteresis but significant latent heat (best compositions for magnetocaloric or thermomagnetic applications, e.g. MnFe₀.₉₅P₀.₅₈₅Si₀.₃₄B₀.₀₇₅) [12]; and iv) up to giant hysteresis and giant latent heat as in MnFe₀.₉₅P₂/₃Si₁/₃ [13]. In all these materials the volume discontinuity is either limited (ΔV/V < 0.2 %) or down to virtually zero. For the Fe₂P-based family, no consensus has ever emerged about the dominant(s) contribution(s) to ΔSₜr and even the most recent reports still led to puzzling results. First, there is a one order of magnitude difference on the integrated magnetic entropy between Fe₂P and MnFe₀.₉₅P₀.₅₈₅Si₀.₃₄B₀.₀₇₅ [11,16]. This is hardly understandable in view of their comparable saturation magnetizations (Mₛₜₐt approximately 2.9 and 4.0 µB/f.u., respectively). Second, in contrast to early theoretical studies pointing out that the latent heat has mostly a magnetic origin [17], recent phonon calculations and inelastic scattering experiments suggest that nearly all the transition entropy (ΔSₜr) has a structural origin [18,19]. Finally, former attempts at scaling the FOMT in Fe₂P-based materials (either via *L*, hysteresis or even ΔS) failed to encompass the broad variety of properties observed in this system. Dung *et al.* pointed out that isothermal entropy changes (ΔS) are correlated with |Δ*a*/a| and |Δ*c*/c| in a couple of (Mn,Fe)₂(P,Si) samples [25]. In MnFe(P,Si,B) compounds, Maschek *et al.* found that the thermal hysteresis varies like Δ(c/a), but no specific correlation could be established for ΔS [22]. In (Mn,Fe)₂(P,Si) compounds, You *et al.* reported a linear relation between hysteresis and c/a (after a given threshold), while they pointed an absence of correlation between *L* and (c/a) [29]. Miao *et al.* claimed the existence of a striking relationship between thermal hysteresis and elastic strain energy for a large panel of Fe₂P compounds [20]. It must be emphasized, however, that all these previous observations carry some ambiguities. First, even though ΔS is correlated with ΔSₜr, it also involves other parameters such as the sensitivity of the transition to the magnetic field Tₜr(B) or even extrinsic ones like the transition width. Second, thermal hysteresis can also not be regarded as an intrinsic quantity for a FOMT. Indeed hysteresis is known to be impacted by external features like the microstructure and is even not a necessity for FOMTs, as materials with nearly negligible hysteresis but sizable latent heat can be observed like in (Mn,V,Fe)₂(P,Si,B) [23]. Actually, ΔSₜr (or *L*=TₜrΔSₜr) are more genuine intrinsic quantities, directly connected to the Erhenfest definition of first-order transitions.
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Seeking to bring back together the diversity in FOMTs observed in Fe₂P materials, **Figure 1** presents the transition entropy ΔS<sub>tr</sub> as a function of the elastic deformation quantified as the discontinuity in the cell parameters ratio Δ(c/a). The MnFeP₀.₆₆-xSi₀.₃₄Bₓ series is particularly important as it covers a wide ΔS<sub>tr</sub> range while keeping chemical substitutions limited [13]. Results for Fe₂P, MnFe(P,Si) and MnFe(P,Ge) were included depending upon availability of ΔS<sub>tr</sub> and temperature dependent structural data. The criteria used to determine Δ(c/a) and ΔS<sub>tr</sub> as well as the list of the eighteen considered samples are presented in Supplementary Materials.

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A striking correlation is observed between the transition entropy ΔS<sub>tr</sub> and the deformation Δ(c/a). We emphasize, however, that this correlation does not imply a predominant lattice component in the entropy change; it neither addresses the respective weights of the magnetic, lattice or electronic contributions within the net ΔS<sub>tr</sub> value. First of all, one must keep in mind that the experimental determinations of ΔS<sub>tr</sub> can only yield a global quantity incorporating all the degrees of freedom. Second, we stress that even when using additional experimental data, the aim at separating the various contributions in ΔS<sub>tr</sub> is basically impossible owing to the intrication between degrees of freedom that characterizes FOMTs. Recently, it has been proposed that the elastic deformation energy could be expressed as ΔU<sub>elastic</sub>/V<sub>mol</sub> ≈ (C<sub>11</sub>+C<sub>12</sub>)ε<sub>1</sub><sup>2</sup>+2C<sub>13</sub>ε<sub>1</sub>ε<sub>3</sub>+(C<sub>33</sub>ε<sub>3</sub><sup>2</sup>)/2 where ε<sub>ij</sub> denotes the strains and c<sub>ij</sub> the elastic constants [20]. Considering the theoretical elastic constants for MnFeSi<sub>0.33</sub>P<sub>0.58</sub>B<sub>0.08</sub> [21] and the experimental strains for MnFe<sub>0.95</sub>P<sub>0.595</sub>Si<sub>0.33</sub>B<sub>0.075</sub> [13], the above equation leads to a ΔS<sub>elastic</sub> = ΔU<sub>elastic</sub>/T<sub>tr</sub> estimated in the range 8 – 18 J kg<sup>-1</sup> K<sup>-1</sup>, which would be compatible with the experimental ΔS<sub>tr</sub> = 16 J kg<sup>-1</sup> K<sup>-1</sup>. Nevertheless, the above equation involves squared strains, whereas ΔS<sub>tr</sub> appears to linearly scale with Δ(c/a). Even though large uncertainties make it difficult to distinguish the two possibilities, a ΔS<sub>tr</sub>-Δ(c/a) scaling was found to be the most reliable.

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The main result of Figure 1 is that the deformation appears as the source of the *total* transition entropy change ΔS<sub>tr</sub>. But no complementary information about the various contributions to this overall result is accessible. Let us also specify that the correlation appears so far limited to Fe<sub>2</sub>P materials. Considering iso-structural FOMTs in other hexagonal materials, (Sc,Ti)Fe<sub>2</sub> (Δ(c/a) ≈ 0.004 and ΔS<sub>tr</sub> ≈ 7 J kg<sup>-1</sup> K<sup>-1</sup> [31]) would be close to fit in, but (Hf,Ta)Fe<sub>2</sub> (Δ(c/a) ≈ 0.014 and ΔS<sub>tr</sub> ≈ 4 J kg<sup>-1</sup> K<sup>-1</sup> [32]) strongly departs from the trend. This might reflect that each materials family has its own underlying FOMT mechanism, leading to distinct ΔS<sub>tr</sub>-deformation relations.
To conclude, the scaling shown in Fig. 1 is a first step towards a systematization of the FOMT properties in the Fe₂P materials family. From an applicative viewpoint, this correlation between ΔS<sub>tr</sub> and Δ(c/a) also illustrates the challenges met when optimizing Fe₂P-related magnetocaloric materials. In order to reach large isothermal entropy changes, one needs sizable ΔS<sub>tr</sub>, but the latter necessarily goes along with sizable deformations having negative consequences such as hysteresis or mechanical embrittlement. Besides tailoring the microstructure, fine tuning of ΔS<sub>tr</sub> is therefore a crucial issue to target magnetocaloric materials compatible with applications.

<figure><img src="image_2.png" /><figcaption>**Figure 1:** Transition entropy of the FOMT as a function of deformation Δ(c/a) of the lattice parameters for Fe₂P-based materials. The dashed line marks out the linear regression with intercept fixed at 0 (R² = 98.7%).</figcaption></figure>
127 Acknowledgments

128 This work was supported by the Natural Science Foundation of China (grant Nos. 52150610486,
129 11904188, 51961033, and 11864030), the Inner Mongolia autonomous region (grant Nos. NJYT-
130 20-A17 and NJZY20025) and Inner Mongolia Normal University (grant Nos. 2018YJRC002 and
131 2018YJRC003).

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