On the magnetocaloric effect in biphasic FeCrSiB amorphous composites

Abstract

Room-temperature magnetic refrigeration using the Ericsson cycle relies on materials with a large, nearly constant magnetocaloric (MC) effect over a wide temperature range. Here, we examine the MC properties of biphasic composite systems made from Fe–Cr–Si–B amorphous alloys. The precursor compositions, Fe_68.8Cr_11.2Si₆B₁₄ and Fe_70.4Cr_9.6Si₆B₁₄, show a second-order magnetic phase transition with values of the Curie temperature (T_C) of 290 K and 340 K, respectively. Analysis of critical exponents reveals that both alloys display intermediate critical behavior, between mean-field and three-dimensional Heisenberg universality classes. Guided by the individual isothermal magnetic entropy change versus temperature curves [ΔS(T)], we have designed a composite by combining the two alloys in proportions that yield a nearly constant |ΔS_comp|, achieving a value ≈1.7 J kg⁻¹ K⁻¹ for a magnetic field change of µ₀ΔH = 5 T. This table-like behavior persists over a temperature range of about 50 K, as determined by the T_C values of the parent alloys. We also assessed the effective refrigerant capacity (RC_eff) and temperature-averaged entropy change (TEC) for the composite. The customized MC response delivers a competitive RC_eff (87 J kg⁻¹ at µ₀ΔH = 5 T) and a TEC that is almost independent of the operating temperature range. This shows that rational design of Fe-based amorphous composites provides an adaptable solution for Ericsson-cycle magnetic refrigeration by balancing performance, operating temperature, and material cost.

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Introduction

Metallic amorphous materials, or metallic glasses, are solids lacking long-range atomic order, meaning that the atomic positions do not follow a periodic pattern like in crystalline materials. This structural disorder greatly influences physical properties. Most of ferromagnetic metallic glasses exhibit isotropic magnetic behavior, mainly due to the absence of magnetocrystalline anisotropy (directional dependence of magnetic properties in crystals) and grain-boundary effects (coming from the interfaces between different crystalline regions). This usually leads to reduced hysteresis losses (less energy loss during magnetic cycling), enhanced corrosion resistance, and improved mechanical resilience.¹ These features have made amorphous metallic alloys key functional materials in soft magnetic technologies and, more recently, in energy-related applications.²

Moreover, metallic glasses have attracted particular attention for solid-state refrigeration via the magnetocaloric effect (MCE),³ which refers to the heating or cooling of a material when it is exposed to a change in magnetic field. Magnetic refrigeration is a sustainable alternative to vapor-compression cooling, offering higher efficiency and eliminating the use of greenhouse refrigerant gases.⁴ The MCE originates from the isothermal magnetic entropy change (ΔS) associated with field-induced spin ordering and depends sensitively on magnetic exchange interactions and thermodynamic response.

The conventional magnetocaloric (MC) materials with promising functionality are primarily crystalline rare-earth intermetallics and compounds exhibiting a first-order magnetostructural phase transition (FOPT). These materials frequently exhibit substantial ΔS values, but they also present drawbacks such as thermal and magnetic hysteresis, mechanical brittleness, and limited cycling stability.⁵ In contrast, metallic glasses usually exhibit second-order magnetic phase transitions (SOMPT) with negligible hysteresis favored by the as-mentioned lack of long-range structural coupling.⁵ Their MCE is frequently lower than that of FOPT materials. However, their broad magnetic transitions, excellent soft magnetic properties, mechanical stability, and compositional tunability can yield competitive refrigerant capacities and reliable operation features.⁶

Among amorphous MC systems, rare-earth- and transition-metal-based alloys constitute the two main families.⁷ Rare-earth-based glasses can display large MCE values, but their Curie temperature (T_C) values are habitually confined to the cryogenic regime, limiting their application to magnetic refrigeration for gas liquefaction.⁷ However, Fe-based amorphous alloys represent a versatile alternative for room temperature (RT) refrigeration. They combine second-order transitions with a T_C close to RT and low raw material cost.⁶ These characteristics make them particularly attractive for near-RT magnetic refrigeration.^8–10 Nevertheless, despite the favorable properties exhibited by second-order materials, they rarely display the “table-like” entropy change profile (i.e., almost constant temperature-dependence of ΔS) required for an Ericsson-type refrigeration cycle.¹¹

One well-established and effective strategy to overcome this limitation is to design composite materials that combine phases with slightly different values of their T_C to extend and flatten the ΔS(T) curve.^12–18 Fe-based amorphous alloys are especially suitable as parent phases for such composites, thanks to their large thermal stability (with crystallization temperatures typically above 750 K^19–21) and their relatively low cost compared with the rare-earth-based metallic glasses. In particular, interest in Fe–Cr–Si–B amorphous alloys has recently increased, both from a fundamental perspective focused on magnetic relaxation processes and crystallization kinetics,²² and also for MC applications.^23–25

In this work, we investigate the MC response of Fe–Cr–Si–B melt-spun ribbons with Curie temperatures close to room temperature and their composites. In particular, we have selected Fe_70.4Cr_9.6Si₆B₁₄ and Fe_68.8Cr_11.2Si₆B₁₄ ribbons from our previous systematic study in the (Fe_1−xCr_x)₈₀Si₆B₁₄ system,²⁵ given that their T_C separation of approximately 50 K is particularly suitable for designing a composite with an almost flat isothermal magnetic entropy change. Particular emphasis is placed on the functional design of a broad table-like ΔS(T) response for Ericsson-cycle refrigeration, the magnetic-field dependence of the composite magnetocaloric behavior, and its impact on application-relevant figures of merit such as the effective refrigerant capacity (RC_eff) and the temperature-averaged entropy change (TEC). Furthermore, through critical exponent analysis, we establish a direct correlation between scaling behavior and the extended ΔS(T) curves observed in these amorphous systems. Our results demonstrate that rare-earth-free Fe-based amorphous composites can achieve competitive MC performance while combining low magnetic losses, scalability, and cost-effective processing, making them promising candidates for RT magnetic cooling.

Experimental methods and numerical calculations

Two amorphous ribbons, 25 µm thick and 1 mm wide, with nominal compositions Fe_70.4Cr_9.6Si₆B₁₄ (ribbon A) and Fe_68.8Cr_11.2Si₆B₁₄ (ribbon B) were fabricated by melt-spinning as described elsewhere.²⁵ X-ray diffraction (XRD) analysis was performed in a high-resolution Rigaku Smartlab diffractometer with Cu-Kα₁ radiation to evaluate the amorphous nature of the samples. Fig. S1 depicts the XRD patterns of both samples. The absence of any sharp Bragg reflections, and the appearance of the characteristic broad halo of amorphous materials evidence the lack of long range crystalline order.

Isothermal magnetization curves, M(µ₀H), as well as magnetization vs. temperature curves, M(T), were collected under applied magnetic field values up to 8 T in the 150–400 K range using a Quantum Design PPMS EverCool-9T system equipped with a vibrating sample magnetometer (VSM). The ribbon length was aligned with the direction of the magnetic field to minimize demagnetizing-field effects. The isothermal ΔS(T) curves were determined by numerically integrating a set of isothermal M(µ₀H) curves using the Maxwell relation:²⁶

The magnetic and magnetocaloric measurements were performed on multiple pieces cut from the same melt-spun ribbons, ensuring representative sampling of each composition. The reported data reflect consistent behavior across these pieces, indicating good batch homogeneity typical of rapidly quenched Fe-based amorphous alloys.

For each value of the applied magnetic field the isothermal magnetic entropy change of the composite was calculated as:²⁷

where, ΔS_A(T) and ΔS_B(T) refer to the isothermal magnetic entropy change measured for each individual ribbon, while the symbol α denotes the weight fraction (mass proportion) of the A-type ribbons.

Results and discussion

Magnetic behavior of individual alloys

The inset in Fig. 1 shows the M(T) curves (normalized to the value at T = 150 K) for ribbons A and B under a low 5 mT applied magnetic field. Magnetization decreases gradually with temperature, vanishing at the second-order ferromagnetic (FM) to paramagnetic (PM) phase transition (T = T_C). Further support for the ferromagnetic character of the samples is provided by the hysteresis loop (depicted in Fig. S2), which exhibits a characteristic low coercive field of 19 A m⁻¹ (0.24 Oe). There is not any evidence of secondary magnetic phases in the M(T) data within the measured temperature range. The values of the Curie temperature, estimated from the minima of the dM/dT versus T curves (not shown), are T^A_C = 340(2) K for ribbon A and T^B_C = 290(2) K for ribbon B.²⁵ These values fall within the temperature range relevant for household magnetic refrigeration, including RT cooling and air-conditioning. The difference in T_C between the two compositions is attributed to variations in the Fe–Fe exchange interactions, which are influenced by the Cr content and the local atomic environment.²⁸

Fig. 1 Isothermal magnetization curves, M(µ₀H), measured at temperatures ranging from 150 to 400 K in steps of 5 and 10 K for (a) ribbons A, and (b) ribbons B. Inset: Temperature dependence of the reduced magnetization M/M_150 K, where M_150 K is the magnetization measured at 150 K under an applied magnetic field of 5 mT. The inflection point of the curve provides an estimate of the Curie temperatures of both samples (see the text).

Isothermal magnetization curves measured over the temperature interval 150–400 K for both ribbons are shown in Fig. 1. In all cases, magnetization increases rapidly in the low magnetic field region, thus indicating a low magnetic anisotropy.

Under higher values of the applied magnetic field, the M(µ₀H) curves exhibit a tendency toward saturation; however, full saturation is not achieved at temperatures close to T_C even for µ₀H = 8 T. Therefore, a fit of the M(µ₀H) curves to a classical approach-to-saturation law was used to estimate the value of saturation magnetization (M_S) at zero magnetic field:

where a is a coefficient related to magnetic hardness and χ₀ is the field-independent magnetic susceptibility.²⁹ As shown in Fig. S3, the resulting M_S(T) values for ribbon B are consistently lower than those of ribbon A. There are two main reasons for this observation: the reduced T^B_C, due to the enhancement of the antiferromagnetic Fe–Cr exchange coupling with increasing Cr content,²⁸ and the reduced Fe content of the B ribbon.

The concavity of the Arrott's plots evolves progressively with increasing temperature, as shown in Fig. S4, changing from convex to concave behavior. According to Banerjee's criterion,³⁰ this evolution is characteristic of an FM–PM SOMPT. Moreover, T_C can be estimated by extrapolating the high-magnetic field linear region of each isotherm toward the zero-field region; the isotherm whose extrapolation passes through the origin corresponds to T_C.³¹ For ribbons A and B, these estimated values are approximately 370 K and 330 K, respectively, higher than those estimated from the minimum in the dM/dT vs. T curves.

Furthermore, the convex curvature persists even at temperatures above T_C. These observations suggest that the conventional analysis of the Arrott's plots is inadequate to accurately describe the critical behavior of this system. Therefore, modified Arrott plots (MAPs), which incorporate the critical exponents β and γ, should be used to determine the magnetic transition parameters more reliably [see Fig. 2 for M^1/β vs. (μ₀H/M)^1/γ plots].³² By means of the Kouvel–Fisher (KF) method,³³ the critical exponents β = 0.423(5), γ = 1.6(1), and δ = 4.8(3) (δ values were derived from the Widom relation,³⁴ δ = 1 + γ/β) were estimated for ribbon A. For ribbon B, β = 0.423(3), γ = 1.62(2), and δ = 4.83(7) were obtained. These critical exponents differ from those expected in the mean-field (MF) model (i.e., β = 0.5, γ = 1.0, δ = 3.0).³⁵ In fact, the experimental value of the exponent β lies between those of the MF and 3D-Heisenberg predictions (β = 0.365).³⁵ This finding points to a magnetic regime governed by the coexistence of long-range interactions, typical of MF behavior, and short-range correlations consistent with a 3D-Heisenberg model.^36,37

Fig. 2 Modified Arrott plots (M^1/β vs. (μ₀H/M)^1/γ) measured between 150 K and 400 K for (a) ribbons A, and (b) ribbons B. The critical exponents β and γ were determined from the Kouvel–Fisher method (see below).

However, the relatively large values of γ obtained for both alloys are significantly higher than those predicted by conventional universality classes such as the 3D Heisenberg (γ ≈ 1.386) or 3D Ising (γ ≈ 1.24) models,³⁵ indicating that the magnetic phase transition cannot be fully described within the framework of an ideal homogeneous ferromagnet.^38,39 This anomalous behavior it is primarily associated with the intrinsic structural disorder characteristic of the amorphous state. In Fe-based amorphous alloys, the absence of long-range crystallographic order produces a broad distribution of Fe–Fe interatomic distances and local atomic environments, which in turn generates a wide distribution of exchange interaction strengths. As a consequence, the magnetic coupling becomes spatially inhomogeneous, leading to enhanced spin fluctuations and strong local variations in magnetic correlations in the vicinity of T_C.

The MAPs resulting from these critical exponents are shown in Fig. 2. Clearly, the plots exhibit the expected concavity in the low-magnetic-field region, which evolves toward a linear trend at moderate and high values of the applied magnetic field. Furthermore, the extrapolation of the isothermal plot at T_C passes through the origin, thereby confirming the proper selection of critical exponents. The temperature dependence of the spontaneous magnetization, M_s, was determined by linearly extrapolating the high-field MAPs to the y-axis intercept for T < T_C.³⁷ Similarly, the inverse of the magnetic susceptibility (χ₀⁻¹) was derived from the MAPs at T > T_C (both are depicted in Fig. 3).

Fig. 3 Temperature dependence of the spontaneous magnetization M_S and the inverse magnetic susceptibility χ₀⁻¹ for (a) ribbon A, and (b) ribbon B. The solid points are the experimental data extracted from modified Arrot plots, while the lines represent the fits to the Kouvel–Fisher equations, M_s = M₀(1 − T/T_C)^β and χ₀⁻¹ = Γ(T/T_C − 1)^γ (see text for further details).

The M_S(T) and χ₀⁻¹(T) curves for both A and B ribbons vanish at T = T_C within the experimental uncertainty. A noticeable deviation from linearity in χ₀⁻¹(T) is observed in the temperature region immediately above T_C, indicating the presence of persistent short-range magnetic correlations within the paramagnetic range. This behavior is consistent with the analysis of the critical exponents.

Magnetocaloric properties of individual ribbons

The temperature dependence of the isothermal magnetic entropy change for ribbons A and B under different magnetic field values (1–8 T) is shown in Fig. S5a and b. The curves exhibit wide peaks with caret-like shape characteristic of SOMPT materials. For ribbon A, the maximum of |ΔS| reaches approximately 2.1(4) J kg⁻¹ K⁻¹ [1.1(2) J kg⁻¹ K⁻¹] under µ₀ΔH = 5 T [2 T] around 340 K, while for ribbon B, |ΔS|^max = 1.8(4) J kg⁻¹ K⁻¹ [0.93(19) J kg⁻¹ K⁻¹] near 290 K. The width of the ΔS(T) curves for µ₀ΔH = 2 T reaches 138 K and 128 K for the ribbons A and B, respectively. This broadening is characteristic of Fe-based metallic glasses and originates from their inherent structural disorder, which ultimately leads to the relatively wide magnetocaloric response.^40,41

MCE offers a powerful framework for systematically probing critical behavior by incorporating the complete temperature dependence of the isothermal entropy change. In this context, the method proposed by Franco et al.^42,43 provides a robust tool for determining whether different materials belong to the same universality class. This approach relies on constructing a normalized entropy change curve by (i) scaling each |ΔS|(T) dataset to |ΔS|^max and (ii) rescaling the temperature axis as:

where T_r represents a reference temperature below T_C corresponding to |ΔS|/|ΔS|^max = a_S, typically taken as 0.5.⁴² The rescaled ΔS(T) datasets in Fig. 4a and b collapse onto a single curve, a hallmark of a single-phase magnetic transition.⁴⁴ This scaling behavior reflects the universality class of both materials. Furthermore, the renormalized curves for both ribbons overlap, indicating that the two compositions belong to the same universality class,^43,44 as expected from the similar critical exponents β and γ.

Fig. 4 Normalized magnetic entropy change curves |ΔS|/|ΔS|^max as a function of the rescaled temperature θ (see eqn (4)) for (a) ribbon A, and (b) ribbon B. The collapse of all curves onto a single universal curve confirms the scaling behavior and the second-order nature of the transition. (c) Comparison of the universal curves of ribbon A (red circles) and ribbon B (green squares). The nearly identical shape of both master curves indicates that both alloys belong to the same universality class, consistent with their similar critical exponents (see text for further details).

Magnetocaloric properties of biphasic composites

After the individual characterization of ribbons A and B, we evaluated the MC response of the biphasic composites made by the two ribbons with different relative weight ratios. To this end, the isothermal magnetic entropy change of the composites was calculated by linearly combining the ΔS(T) curves of the ribbons A and B (see eqn (2)). This strategy has been previously proved to be effective in predicting the behavior of biphasic composites designed from Fe-based amorphous ribbons.^8,45

Fig. S6 represents the temperature dependence of |ΔS_comp| for a weight fraction of 0.5 under µ₀ΔH up to 8 T. Also, for the magnetic field change (μ₀ΔH ≤ 1 T) a double-peak profile is still evident in the |ΔS_comp(T)| curves, reflecting the magnetic transitions of the individual alloys. However, upon increasing the magnetic field (μ₀H ≥ 1 T), the broadening of the single ΔS(T) curves yields a more uniform superposition, ultimately leading to a flat region usually known as table-like behavior.

Fig. 5a displays the temperature dependence of |ΔS_comp(T)| for composites with different weight fractions (α ranging from 0 to 1) under a magnetic field change of 5 T. Due to the combination of two alloys with different values of the Curie temperature, the maximum of |ΔS_comp(T)| is reduced compared with the |ΔS|^max of the individual ribbons. In the case of the composite with α = 0.5, |ΔS_comp|^max = 1.7 J kg⁻¹ K⁻¹ for µ₀ΔH = 5 T, smaller than |ΔS_A|^max and |ΔS_B|^max (see Fig. S5 and S6). Notably, a table-like shape of the curve emerges for specific weight fractions. Among the simulated composites, the most pronounced table-like behavior for µ₀ΔH = 5 T (highlighted as black line in Fig. 5a) is obtained for α = 0.5. The resulting plateau extends approximately between T^A_C (340 K) and T^B_C (300 K). For lower or higher values of α the plateau becomes less uniform and eventually a single-peak profile appears, with the maximum shifting toward the magnetic transition temperature of the dominant phase. For example, at α = 0.8, a well-defined peak appears near 300 K, with a narrower temperature span than in the table-like profile.

Fig. 5 (a) Isothermal magnetic entropy change of the composite (ΔS_comp) as a function of temperature and weight fraction A-type ribbon (α) for the biphasic composite under an applied magnetic field change of 5 T. The composite exhibits a table-like ΔS_comp profile over ≈50 K, delimited by the Curie temperatures of the two ribbons. (b) |ΔS_comp|(T) curves calculated from the ΔS(T) curves measured for A and B amorphous ribbons under μ₀ΔH = 2 T, 5 T, and 8 T. For each value of the magnetic field, the curve corresponding to the table-like variation has been highlighted. The cut-off point |ΔS_θ| is indicated by a dashed line. (c) Applied magnetic field dependence of the magnetic entropy change for ribbons A and B for characteristic temperatures T^A_C, T^B_C, and T_θ. Lines correspond to the theoretical fits of ΔS vs. μ₀H obtained from eqn (7) (see text for details).

Fig. 5b shows the simulated |ΔS_comp|(T) curves for composites with different weight fractions under magnetic field changes of 2, 5, and 8 T. The optimum values of α evolve between 0.4 for 2 T and 0.6 for 8 T, due to the distinct magnetic field dependence of the individual ΔS(T) curves (see Fig. 5c). As previously noted,⁴⁵ all |ΔS_comp|(T) curves intersect at a common point, ΔS_θ, at T_θ for a given applied magnetic field. The ΔS_θ value provides a practical constraint for engineering table-like composites, since it limits the maximum |ΔS_comp| achievable in the plateau region. Notably, T_θ is found to be field-independent (T_θ = 315 K up to µ₀ΔH = 8 T), suggesting that ΔS_A(T_θ) and ΔS_B(T_θ) exhibit a similar magnetic field dependence. Fig. 5c depicts the magnetic field dependence of ΔS_θ together with ΔS of the individual ribbons at T^A_C, T^B_C, and T_θ.

Interestingly, ΔS_A(T_θ) and ΔS_B(T_θ) remain essentially identical, as predicted above. Across the entire magnetic field range, |ΔS_θ| exhibits lower values than |ΔS_A(T^A_C)| and |ΔS_B(T^B_C)|, but higher than |ΔS_A(T^B_C)| and |ΔS_B(T^A_C)|, with |ΔS_θ| ≈ 0.82, 1.8 and 2.6 J kg⁻¹ K⁻¹ for µ₀ΔH = 2, 5 and 8 T, respectively. Noteworthy, the magnetic-field dependence of both the individual and biphasic |ΔS(T)| curves follows a nonlinear power-law behavior, |ΔS| ∝ µ₀ΔHⁿ, typical of materials experiencing a SOMPT.⁴³

The exponent n depends on both temperature and magnetic field in FM materials and it can be estimated as:⁴⁶

Moreover, the value of n at T_C is linked to the critical exponents β and δ governing the SOMPTs:⁴⁶

n(T_C) = 1 + (1/δ)(1 − 1/β) (6)

which makes n a valuable parameter for probing the universality class of magnetic phase transitions using magnetocaloric data. For materials following an MF approach, this critical exponent is expected to be n = 2/3 at T = T_C.⁴³ In our case, the values of n obtained from the critical exponents are 0.716(7) and 0.718(5) for the ribbons A and B, respectively, close to the value given from the MF prediction and consistent with the analysis presented in the previous section.

Furthermore, we have analyzed the magnetic field dependence of the maximum of the isothermal magnetic entropy change, which can be described using Landau's theory for SOMPT as:^47,48

|ΔS|^max = A(H + H₀)^2/3 − AH₀^2/3 + BH^4/3 (7)

where A and B are intrinsic composition-dependent parameters, and H₀ is an extrinsic parameter that primarily reflects sample quality and microstructural effects. The isothermal magnetic entropy change depends primarily on the spontaneous magnetization (i.e., related to β) and its temperature variation, while deviations in the critical susceptibility do not affect significantly the validity of the fit.⁴⁹ The lines in Fig. 5c represent the fit of the |ΔS|^max vs. µ₀ΔH data. In this work, we restricted the analysis to the 2/3-power law. The obtained values for A and H₀ are compiled in Table 1. The intrinsic parameter A slightly varies between ribbons A and B, consistent with the similar magnetic exchange coupling expected for both samples. In contrast, the extrinsic H₀ parameter exhibits a more pronounced difference between the two ribbons. This behavior is less straightforward to interpret because H₀ is highly sensitive to microstructural imperfections. The observed larger H₀ value for ribbon B can be attributed to a higher atomic disorder. The values obtained from the fit fall within the ranges reported for other magnetic systems undergoing a SOMPT, such as the 2 : 17 alloys Pr_1.64Sm_0.36Fe₁₇⁵⁰ and Tb₂Fe₁₇,⁵¹ as well as other families of compounds, including Mn₅Ge_3−xZn_x,⁵² Gd_1−xSm_xCo₂,⁵³ and Ni-doped MnCoGe.⁵⁴

Table 1 Fitted parameters A and H₀ obtained from the refinement of the experimental data shown in Fig. 5(c) using eqn (7). The reduced chi-squared (χ²) values are also provided as a measure of the goodness of the fit. The uncertainties correspond to one standard deviation

	Composite α = 0.5	Ribbon A TC = T^AC	Ribbon A TC = T^BC	Ribbon A TC = Tθ	Ribbon B TC = T^AC	Ribbon B TC = T^BC	Ribbon B TC = Tθ
A µ0^−2/3 (J kg^−1 K^−1 T^−2/3)	0.758 (3)	0.756 (2)	0.7082 (8)	0.7237(9)	1.7 (4)	0.662 (2)	0.739 (4)
µ0H0 (T)	1.14 (3)	0.210 (7)	1.74 (1)	0.776 (8)	5 (4)	0.178 (6)	0.79 (3)
χ^2	1.23	1.07	1.06	1.06	1.36	1.06	1.17

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To evaluate the suitability of a magnetic material for refrigeration applications, two primary figures of merit are commonly considered. The first is the refrigerant capacity (RC), that quantifies the heat absorbed from the cold reservoir and released to the hot reservoir during an Ericsson cycle of refrigeration, and can be estimated using three different criteria:^55–57 (i) RC-1, defined as the area of a rectangle with height |ΔS|^max and base equal to the full width at half maximum of the |ΔS(T)| curve (δT_FWHM); (ii) RC-2, given by the area under |ΔS(T)| within δT_FWHM; and (iii) RC-3, obtained via the Wood and Potter method, which identifies the rectangle with maximum area under the |ΔS(T)| curve. Fig. S7 schematically illustrates the procedure used to determine RC-1 and RC-2. These definitions are consistent with a reversible Ericsson cycle operating between a cold reservoir at T_cold and a hot reservoir at T_hot, in which the maximum energy that can be extracted from the cold reservoir is T_hot |ΔS(T_cold)|. Furthermore, by the second law of thermodynamics, the energy released at the hot reservoir must be equal to T_cold |ΔS(T_cold)|.

Consequently, any ΔS values exceeding |ΔS(T_cold)| do not contribute to the refrigeration process; in other words, the peak of the |ΔS(T)| curve does not fully reflect the material thermodynamic utility. Although the definitions of RC commonly used in the literature provide a reference for comparing materials, they fail to ascertain the practical performance of systems with broad or table-like entropy changes. For such materials, a figure of merit that better denotes the practical heat transfer across the plateau is the effective refrigerant capacity, RC_eff, defined as:⁸

RC_eff = |ΔS_θ|·ΔT (8)

where ΔT represents the width of the plateau and |ΔS_θ| the value of ΔS_comp within the plateau (sketched in Fig. S7). Fig. 6 depicts the |ΔS_comp|(T) curve for the composite with α = 0.5 under µ₀ΔH = 5 T, showing a broad plateau (≈300–340 K) with |ΔS_θ| ≈ 1.7 J kg⁻¹ K⁻¹, which yields an RC_eff of 87 J kg⁻¹.

Fig. 6 Comparison of the isothermal magnetic entropy change for the individual ribbons and the composite with A-ribbon weight fraction α = 0.5 for µ₀ΔH = 5 T. The shaded blue area highlights the table-like region.

Over the same temperature interval, the composite extracts more internal energy from the cold reservoir during a refrigeration cycle than either alloy alone, by efficiently combining the contributions of both phases. Moreover, although the individual alloys exhibit higher ΔS peaks (see Fig. 6), their narrower temperature span limits the usable range in an hypothetical Ericsson cycle. In contrast, the composite provides a functional plateau between 300 K and 340 K, which aligns well with the typical operating range of conventional refrigeration systems. To place the performance of the biphasic composite with α = 0.5 in the context of the current state of the art, data for several table-like MC materials reported in the literature are given in Table 2, with values of RC_eff as a function of the central temperature (T*) under 2 and 5 T applied magnetic field values.

Table 2 Comparison of magnetocaloric parameters for various composites reported in the literature and for the current Fe–Cr–Si–B composite with weight fraction α = 0.5, measured under magnetic field changes of 2 and/or 5 T as indicated. The listed quantities are: the application temperature T*, the table-like temperature range ΔT_table, the isothermal entropy change of the composite ΔS_comp, and the effective refrigerant capacity RC_eff. Where available, values for both field changes are provided

Family of alloys	µ0H (T)	T* (K)	ΔTtable (K)	|ΔScomp| (J kg^−1 K^−1)	RCeff (J kg^−1)	Ref.
Fe(80−x)CrxSi6B14	2/5	315	50/50	0.8/1.7	37.5/87	This work
Fe87La(7−x)Ce5Co1	1.5/5	310	58/58	1.5/3.1	85/181	62
Fe(88−x)NdxCr8B4	2/5	335	40/40	1.5/3.2	60/128	63
Fe(78−x)CexSi4Nb5B12Cu1	—/5	310	—/40	—/2.1	40/84	64
Fe(88–x)CoxZr8B4	—/5	305	—/20	—/3.0	25/60	65
La(1−x)PrxFe10.7Co0.8Si1.5	2/5	285	10/10	5/7.5	30/75	66
(La0.6Ba0.2Sr0.2MnO3)1−x/(Co2O3)x	2/4.5	320	40/40	1/2.0	40/80	67
La2/3Ba1/3MnO(3−x)	1/–	325	35/–	1.3/–	46/–	68
Gd50Co48(Fe,Mn)2	2/5	250	40/40	1.7/4.3	85/172	69
Ho5Pd2/Ho3Pd2	2/5	25	19/25	3/10	—/429	16
(Er2Cr2C3)x/(Ho2Cr2C3)(1−x)	2/–	10	15/–	6/–	60/–	70
ErZn2/ErZn	2/5	12	10/10	8/16	96/232	71
Fe87Pr10−xB3Cex	—/5	316	—/30	—/3.8	—/114.9	72

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It is worth noting that rare-earth-based composites generally exhibit higher RC_eff due to their larger isothermal magnetic entropy change peak values. However, these systems typically operate at cryogenic temperatures or require critical raw materials. In contrast, the Fe-based composite presented here operates near RT and offers a notably broad temperature span (ΔT_table ≈ 50 K). The obtained RC_eff value of 87 J kg⁻¹ at 5 T is competitive among rare-earth free materials, surpassing other Fe-based metallic glasses and thin films. Under an applied magnetic field of 2 T, the magnetic entropy change reaches 0.8 J kg⁻¹ K⁻¹. Nevertheless, the broad operational temperature span (∼50 K) results in an RC_eff of 37.5 J kg⁻¹, which is among the highest values reported for rare-earth-free compositions operating near RT.

The second figure of merit derives from the critical assessment conducted by Griffith et al.,⁵⁸ which underscores the inherent complexity of magnetocaloric materials. It emphasizes that reducing their performance to a single figure of merit inevitably leads to oversimplification. Consequently, meaningful comparisons must consider temperature and magnetic field-dependent properties, hysteresis effects, and practical factors such as thermal conductivity, corrosion resistance, and mechanical stability.

Within this framework, the temperature-averaged entropy change has emerged as a complementary metric for material screening and for comparing performance across different caloric families (magnetocaloric, electrocaloric, and elastocaloric) over application-relevant temperature spans.^59–61 Our results for Fe–Cr–Si–B amorphous composites are consistent with this perspective. Our work prioritizes achieving a broad, table-like ΔS profile over approximately 50 K. Furthermore, the SOMPT characteristic of these amorphous alloys ensure intrinsically low thermal and magnetic hysteresis, thereby avoiding the cyclability issues that often complicate the interpretation of TEC in first-order systems (Fig. 7).

Fig. 7 Temperature dependence of |ΔS_comp| measured under µ₀ΔH = 5 T. Shaded rectangles illustrate the different regions around the midpoint temperature (T_mid) with the same temperature span (ΔT_lift) used for determining TEC (see eqn 9). The height of each region corresponds to the nearly constant table-like value of |ΔS_comp| within the selected interval. Inset: TEC of the composite with α = 0.5 as a function of the applied magnetic field for selected ΔT_lift values within the magnetic-field range where the table-like MCE behavior is preserved. The procedure for calculating error bars is detailed in the SI.

TEC is therefore proposed as a simple, material-based figure of merit and is defined as:⁵⁸

where ΔT_lift and T_mid denote the width and midpoint of the temperature interval over which the integral is evaluated, respectively. In essence, TEC provides the maximum average entropy change within all temperature intervals of width ΔT_lift. Fig. 7 and its inset reveal that the TEC in the table-like region exhibits no appreciable dependence on T_mid and ΔT_lift.

These observations underscore that in Fe-based amorphous composites exhibiting a table-like MCE, TEC remains essentially independent of the operating temperature range, providing reliable and versatile performance for Ericsson-cycle magnetic refrigeration processes. This makes such composites especially advantageous for practical RT applications, where stable and field-independent performance is critical. Consequently, this material provides a suitable compromise between performance, operational range, and cost, reinforcing the suitability of Fe-based amorphous alloys for domestic refrigeration applications.

The wide table-like magnetocaloric effect originates from the dual-phase amorphous structure. Specifically, when two amorphous phases are combined with an appropriate fraction ratio and a suitable difference in their Curie temperatures (owing to different Fe–Fe distances and exchange coupling interactions), the working temperature span is significantly broadened. This effect, together with the comparable magnetic entropy changes of the individual phases, results in a high overall magnetocaloric response.

Conclusions

The magnetic critical behavior and magnetocaloric performance of Fe_70.4Cr_9.6Si₆B₁₄ and Fe_68.8Cr_11.2Si₆B₁₄ amorphous ribbons have been systematically investigated. Both alloys exhibit a second-order magnetic phase transition, with critical exponents that deviate from the conventional mean-field predictions and instead reflect a coexistence of long- and short-range magnetic interactions, consistent with an intermediate universality class. The close correspondence between the exponent sets for both compositions demonstrates that they indeed share the same universality class, a conclusion further reinforced by the trend followed by their magnetocaloric behavior. The ribbons achieve maximum isothermal magnetic entropy changes of 2.1 J kg⁻¹ K⁻¹ (1.1 J kg⁻¹ K⁻¹) and 1.8 J kg⁻¹ K⁻¹ (0.93 J kg⁻¹ K⁻¹), respectively, under a change of the applied magnetic field of 5 T (2 T) underscoring the intrinsic efficiency of these amorphous systems.

For evaluating materials intended for Ericsson-cycle refrigeration, where a broad and nearly constant ΔS response is preferred over a sharp magnetic entropy change peak, RC_eff and TEC are particularly relevant figures of merit. Building on the findings obtained for the Fe–Cr–Si–B amorphous ribbons, the optimally designed and engineered composite that integrates both amorphous ribbons delivers a broad table-like ΔS profile with a nearly constant magnitude of ΔS ≈ 1.7 J kg⁻¹ K⁻¹ under an applied magnetic field change of 5 T across a temperature span of about 50 K. This behavior results in a highly competitive RC_eff among rare-earth-free materials operating near room temperature and a TEC that remains nearly independent of ΔT_lift. In addition, the SOMPT of these amorphous alloys minimizes hysteresis effects, which is advantageous for cyclic operation. Crucially, this performance achieved in low-magnetic-loss Fe-based amorphous systems without reliance on critical raw chemical elements establishes Fe–Cr–Si–B metallic glasses and their composites not merely as model systems for probing magnetic phase transitions, but as genuinely viable candidates for scalable, sustainable, next-generation magnetocaloric refrigeration technologies.

Author contributions

A. S.-P.: data curation, formal analysis, visualization, writing original draft; J. L.-G.: formal analysis, validation, visualization, writing original draft.; K. P.-A.: visualization, data curation; J.-M. P.: formal analysis, visualization, funding acquisition; J.-L. S.: investigation; J.-A. B.: formal analysis, validation, visualization, writing final version; P. G. conceptualization, supervision, writing final version. P. A.-A.: conceptualization, data curation, investigation, funding acquisition, methodology, project administration, software, visualization, writing original draft. All authors participated in the writing, review and editing of the final version of the manuscript.

Conflicts of interest

The authors declare that there are no conflicts to declare.

Data availability

The raw data supporting the findings of this study, including magnetization isotherms, critical exponent analysis, and entropy change calculations, are available from the corresponding author upon reasonable request. Processed data and the analysis scripts used for the calculation of ΔS, RC_eff, and TEC have been deposited in https://zenodo.org/communities/magcoolest/. Supplementary information (SI) is available. See DOI: https://doi.org/10.1039/d6ma00481d.

Acknowledgements

This work was financially supported by SEK-25-GRU-GIC-24-113 from SEKUENS, Principality of Asturias and the Spanish MCIN/AEI/10.13039/501100011033 and ERDF, EU PID2022-138256NB-C21 and PID2022-138256NA-C22. A. S.-P. acknowledges funding received through the Avenir 2025/2026 fellowship awarded by the Embassy of France in Spain. J.-M. P. acknowledges the Ramón y Cajal grant RYC2023–044841-I funded by MICIU/AEI/10.13039/501100011033. J. L. S. L. ackowledges the support received from Laboratorio Nacional de Nanociencias y Nanotecnología (LINAN, IPICyT) and M.Sc. Beatriz Rivera Escoto for her technical support.

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