Critical exponents and magnetocaloric response in Zintl phase EuIn₂ X₂ (X = As, P)

Abstract

In the present studies, we have explored the nature of magnetic transition, critical magnetization scaling, and magnetocaloric effect of Zintl phase single crystalline EuIn₂ X₂ (X = P, As) compounds synthesized by flux-method. The magnetization susceptibilities of the two compounds revealed the ferromagnetically ordered Eu²⁺ moment below 24 K in EuIn₂P₂ while in EuIn₂As₂ Eu²⁺ moment couple antiferromagnetically below 16.3 K. A comprehensive study of the magnetization data is performed in the critical regime utilizing modified Arrott plots (MAP), the Kouvel-Fisher (KF) method, universal curve scaling, and the magnetocaloric effect. For EuIn₂P₂, the magnetization isotherms follow MAP with critical exponents β = 0.257, γ = 0.86, and δ = 4.45 for the magnetic field applied along the c-axis (H||c) and β = 0.226, γ = 1.10, and δ = 5.87 for the magnetic field applied along the ab-plane (H||ab). For EuIn₂As₂, the values are β = 0.245, γ = 0.81, and δ = 4.31 and β = 0.244, γ = 0.89, and δ = 4.65 for H||c and H||ab plane, respectively. The Kouvel-Fisher and universal scaling plots of the magnetization isotherms further corroborated the reliability of our analysis and the exponents’ values. A reasonable magnetocaloric effect -ΔS_m = 9.55 Jkg⁻¹K⁻¹ and 10.80 Jkg⁻¹K⁻¹ for 5T magnetic field change is observed for H||c and H||ab plane for EuIn₂P₂, respectively. The values of -ΔS_m for EuIn₂As₂ are 10.40 Jkg⁻¹K⁻¹ and 12.75 Jkg⁻¹K⁻¹ for H||c and H||ab plane. The critical exponents derived from the magnetocaloric effect (MCE) are n = 0.49(5), 0.45(3) and δ = 5.17(4), 5.74(1) for EuIn₂P₂ and n = 0.33(1), 0.34(2) and δ = 5.08(5), 6.54(6), respectively for H||c and H||ab plane for EuIn₂P₂ and EuIn₂As₂. The exponents derived from the MCE are marginally elevated, but they remain in substantial concordance with the values inferred from the magnetic isotherm study.

Introduction

The study of magnetic phase transitions in Zintl phase compounds has garnered significant attention due to the interplay between localized 4f-electron magnetism and the structural complexity inherent in these materials. In this context, the isostructural Zintl phases EuIn₂As₂ and EuIn₂P₂ (crystal structure: hexagonal, space group: P6₃/mmc), demonstrating topological properties, are intriguing. EuIn₂As₂ exhibits antiferromagnetic (AFM) A-type ordering of Eu²⁺ magnetic layers at a Néel temperature T_N ≈ 18 K¹. In this configuration, Eu magnetic moments align ferromagnetically within each ab layer and stack along the c-axis antiferromagnetically. Theoretical predictions suggest that the A-type order can lead to an axion insulator (AXI) state. This state may appear either as a topological crystalline insulator (TCI) with certain gapless surfaces or as a higher-order topological insulator (HOTI) with chiral hinge states. The specific manifestation depends on the orientation of Eu magnetic moments^1,2. However, the neutron diffraction reveals that the compounds exhibit low-symmetry helical AFM order³. For H||c, in addition to the long-range AFM order, magnetization indicates the presence of short-range ferromagnetic (FM) orders and correlations, leading to magnetic polarons formation^1,4. These FM clusters may linger even above the AFM transition, resulting in the emergence of unconventional transport properties¹. Multiple magnetic orders and interactions in EuIn₂As₂ are crucial for understanding its topological nature or tuning its topological states^1,2,3. To understand how magnetism modulates topological behaviors, it is essential to clarify the magnetic interactions. Therefore, an in-depth study of critical phenomena across the magnetic transition is essential for EuIn₂As₂. At criticality, the critical behavior can reveal key features, for instance, the magnetic interaction mechanisms, the spatial decay of correlations, the correlation length, and the spin dimensionality. However, as indicated by the magnetic phase diagram for EuIn₂As₂, near the critical temperature, upon the application of a magnetic field, the weak AFM order at T_N rapidly transforms to FM ordering, known as the field-induced FM (FFM) state. EuIn₂P₂, on the other hand, exhibits a canted FM order below T_C = 24 K⁴ with ab initio simulations prediction to be a FM Weyl semimetal, where the emergence of Weyl points or Weyl nodal lines is contingent on the alignment of magnetic moments along the a- or c-axis, respectively⁵. Electronic structure calculations suggest that EuIn₂P₂ behaves as a semiconductor with a band gap of 0.4 eV⁶, which agrees well with our previous experimental observation of 1.3 eV for H = 0 and 0.5 eV for H >0⁷, confirming that the narrow gap of ~ 3.2 meV, observed by Jiang et al⁴. was probably from contributions of diverse interband transitions. The semiconducting behavior was a coincidental phenomenon. EuIn₂P₂ shows variable-range hopping transport above ~ 40 K, with negative magnetoresistance scaling quadratically with field strength and magnetization⁷. The distinct magnetic ground states of EuIn₂P₂ and EuIn₂As₂ make them ideal for comparative studies of phase transition behavior. Critical exponents associated with magnetic transitions are fundamental parameters that describe the divergence of physical quantities near the critical temperature. Their values depend on the universality class, which reflects the symmetry of the order parameter (n) and the dimensionality of the system (d)⁸. Determining the critical exponents of EuIn₂P₂ and EuIn₂As₂ is crucial for understanding the nature of magnetic interactions in these Zintl phases and assessing their adherence to established theoretical models (e.g., mean-field theoretical model⁹, Heisenberg model¹⁰, Ising model¹¹, and tricritical mean field model¹². Another important property of these systems is the magnetocaloric effect (MCE), which refers to the reversible thermal response of a material when exposed to an external magnetic field. The MCE is significant for potential applications in magnetic refrigeration and offers a unique way to investigate critical behavior. The dependence of the maximum entropy change (ΔS_m^max ∝ µ₀Hⁿ) on the magnetic field directly relates to the critical exponents, providing a comprehensive understanding of magnetic phase transitions^13,14.

This study aims to thoroughly investigate the critical exponents and MCE in EuIn₂P₂ and EuIn₂As₂ single crystals through detailed magnetic measurements. By comparing these systems’ FM transition in EuIn₂P₂ with the FFM transitions in EuIn₂As₂, we seek to uncover the underlying mechanisms driving their critical behaviors and elucidate their universality classes. Our findings will contribute to the broader field of critical phenomena in correlated electron systems.

Fig. 1

(a-b) Magnetic field dependence of magnetization for EuIn₂P₂. (c-d) Arrott plots (M² vs. H/M). The left panels correspond to H||c, and the right panels show H||ab.

Results and discussion

Figure 1(a-b) represents the magnetic isotherms M(H) for EuIn₂P₂, respectively, along the H||c axis and H||ab plane in the vicinity of the Curie temperature T_C = 24 K, the point of inflection on the susceptibility curve (see Fig. S1). Arrott plots (M² vs. H/M, Fig. 1(c-d)) reflect the positive curvature, indicating the phase transition from FM to paramagnetic (PM) state is of second order¹⁵. To comprehensively analyze the second-order phase transition, a critical analysis of magnetization data characterized by the set of critical exponents β, γ, and δ was carried out^16,17. The critical exponents can be determined by analysing the spontaneous magnetization (M_S), zero field susceptibility ($\:{\chi\:}_{0}^{-1}$), and magnetization isotherm at the T_C, by following the set of asymptotic relations referred as power laws:

$$\:{M}_{S}\left(0,T\right)=\left[{M}_{S}\left(0\right)\right]{\left(-\epsilon\:\right)}^{\beta\:},\:\epsilon\:<0$$

(1)

$$\:{\chi\:}_{0}^{-1}\left(0,T\right)=\left[\frac{{H}_{0}}{{M}_{S}\left(0\right)}\right]{\left(\epsilon\:\right)}^{\gamma\:},\:\epsilon\:>0$$

(2)

$$\:\:\:\:\:\:\:M\left(H,\:{T}_{C}\right)=D{H}^{1/\delta\:},\:\epsilon\:=0\:$$

(3)

where $\:\epsilon\:=\:\frac{T-{T}_{C}}{{T}_{C}}$ is the reduced temperature, and 𝑀_𝑆(0), 𝐻₀∕𝑀_𝑆(0), and 𝐷 are the critical amplitudes^18,19,20. The critical exponents (β, γ, and δ) are connected by critical wisdom relation as²¹:

$$\:\delta\:=1+\frac{\gamma\:}{\beta\:}$$

(4)

Fig. 2

(a-b) Modified Arrott plots (M^1/β vs. (H/M)^1/γ) for EuIn₂P₂. (c-d) series of M_S(T) and $\:{\chi\:}_{0}^{-1}\left(T\right)$ curves from the MAP method showing the convergence of β and γ along the H||c and H||ab plane, respectively. The arrows point out the initial to final values of β and γ.

Arrott plot is the conventional method to deduce the critical exponents and the critical temperature with the assumption to follow the mean-field theory for which β = 0.5 and γ = 1. Then, the isotherms are plotted to yield a set of parallel straight lines around T_C in the high-field region, indicating the existence of mean-field interactions¹⁵. The technique gives precise T_C as the critical isotherm passes through the origin, it also provides M_S(T) as the intercept on the positive M² axis and $\:{\chi\:}_{0}^{-1}\left(T\right)$ as an intercept on the µ₀H/M axis²². However, all the curves in Fig. 1(c-d) exhibit nonlinear trend with downward curvature, even at high fields, indicating a deviation from the mean-field theory. Accounting for the demagnetization field effect (µ₀H_int = µ₀H - NM, N = 1/3), as shown in Fig. S2, the resulted correction in magnetization isotherm is negligible for both the orientations^23,24. To more accurately characterize the magnetic isotherms, we use a modified Arrott plot (MAP) based on the Arrott-Noaks Eqs. 5^18,19,20:

$$\:{\left(\frac{H}{M}\right)}^{1/\gamma\:\:}=A+B{M}^{1/\beta\:}$$

(5)

where 𝜀 represents the reduced temperature, while A and B are constants¹⁸. Fig S3 and S4 render the Heisenberg model (β = 0.365 and γ = 1.336), Ising model (β = 0.325 and γ = 1.24), and tricritical mean field model (β = 0.25 and γ = 1) plotted for EuIn₂P₂ along H||c and H||ab, respectively. All of them in the high field region show straight lines having considerable curvature near T_C. The critical behavior of EuIn₂P₂ does not seem to align with any universal class of models. Consequently, we employed a self-consistent iterative approach to create an optimized modified Arrott plot with appropriate critical exponents^25,26,27, achieving the convergence in fitting for β = 0.257 and γ = 0.86 for H||c and β = 0.226 and γ = 1.1 for H||ab as shown in Fig. 2(a-b). For both directions, the isotherm at T = 24 K passes through the origin, indicating a critical temperature of T_C = 24 K. Figures 2(c-d) depict the series of M_S (intercepts on M^1/β) and $\:{\chi\:}_{0}^{-1}\left(T\right)$ (intercepts on (µ₀H/M)^1/γ) curves obtained from MAP method indicating the convergence of β and γ values. The power-laws (Eqs. (1-2)) fit to M_S and χ₀⁻¹ (Fig. 3(a-b)) yielding β = 0.255(4), T_C = 24.6(1) K, γ = 0.856(1), T_C = 24.1(1) K for H||c and β = 0.225(1), T_C = 24.4(3) K, γ = 1.067(1), T_C = 24.0(1) K for H||ab. Straight line fit to plot of log(M_S) and log(χ₀⁻¹) against log(-𝜀) and log(𝜀) (Fig. 3(c-d)) gives more accurate values of β and γ, namely 0.249(2) and 0.855(3) for H||c, and 0.224(3) and 1.068(2) for H||ab.

Fig. 3

(a-b) Zero field susceptibility ($\:{\chi\:}_{0}^{-1}$), and spontaneous magnetization (M_S) as a function of temperature on the right and left y-axes for EuIn₂P₂, respectively, for H||c and H||ab. The red solid lines are the fits with Eqs. 1-2. (c-d) log(𝑀_𝑆) vs. log(-𝜀) and log(χ₀ ⁻¹) vs. log(ԑ) with T_C = 24 K as obtained from MAP, the solid lines are straight line fits. (e-f) The Kouvel-Fisher (KF) plot of χ₀ ⁻¹ and M_S as a function of temperature on the right and left y-axes, obtained from the (µ₀H/M)^1/γ and M^1/β intercepts of the modified Arrott plots (MAP) near T_C.

Kouvel-Fisher (KF) method is another method to acquire the critical exponents²⁸. This method involves dividing Eqs. (1) and (2) by their respective derivatives:

$$\:\frac{T-{T}_{C}}{\beta\:}=\frac{{M}_{S}}{{dM}_{S}/dT}$$

(6)

$$\:\frac{{T}_{C}-T}{\gamma\:}=\frac{{\chi\:}_{0}^{-1}}{d{\chi\:}_{0}^{-1}/dT}$$

(7)

Fits of Eqs. (6) and (7) to temperature-dependent $\:\frac{{M}_{S}}{{dM}_{S}/dT}$ and $\:\frac{{\chi\:}_{0}^{-1}}{d{\chi\:}_{0}^{-1}/dT}$ (Fig. 3(e, f)) deduced β = 0.257(5) with T_C = 24.5(2) K and γ = 0.864(1) with T_C = 24.0(1) K for H||c and β = 0.227(1) with T_C = 24.1(4) K and γ = 1.09(2) with T_C = 23.9(2) K for H||ab.

Fig. 4

(a-b) Critical isotherms for EuIn₂P₂, inset: log-log plot of critical isotherm along H||c and H||ab plane, respectively. The red solid lines are fits of Eq. 3.

Using Eq. (3), the critical exponent δ can be determined from the M(H) isotherm at T = T_C. The fitting on the logarithmic scale (insets of Fig. 4(a, b))gives δ = 4.42(1) for H||c and 5.85(3) for H||ab, which agrees well with δ value estimated from Eq. (4) (critical wisdom relation) using β and γ obtained from various techniques and tabulated in Table 1. The values of β, γ, and δ obtained from different methods agree. The theoretically expected values for the mean-field model, Ising model, and Heisenberg model are also listed in Table 1 for comparison.

In EuIn₂As₂, AFM order is delineated in the temperature derivative of susceptibility at Néel temperature of T_N = 16.3 K for a field of 0.01 T both along H||c and H||ab (see Fig. S5 (a, b))^1,29. The magnetic isotherms in Fig. 5(a-b) reveal distinct temperature-dependent characteristics. The M(H) curves show a linear paramagnetic behavior in the high-temperature regime. In contrast to this, at low temperatures, the system exhibits a two-stage response: a linear magnetization increase at low fields, followed by the saturation at higher fields, indicating a field-induced transition to a ferromagnetic state below T_C (determined from Arrott plot analysis). This behavior is consistent with critical exponent analysis for a second-order FM phase transition. Furthermore, the observed universality in the magnetic entropy change ΔS_m(T, H) strictly applies to the thermodynamic FM state. The successful scaling collapse of ΔS_m(T, H) data under high fields provides conclusive evidence for stabilizing the FFM phase in EuIn₂As₂ at high magnetic fields, as will be discussed later.

Fig. 5

(a-b) Magnetic field dependence of magnetization for EuIn₂As₂, inset: temperature dependence of magnetization. (c-d) Arrott plots (M² vs. H/M) for H||ab and H||c, respectively.

Table 1 Critical exponents (β, γ, δ, and n) and critical temperatures (T_C) obtained for EuIn₂P₂ and EuIn₂As₂using various methods in this work, including the modified Arrott plot (MAP), Kouvel-Fisher (KF) plots, critical isotherm analysis, Widom scaling, and magnetocaloric effect (MCE)/relative cooling power (RCP) analysis. The theoretically predicted values of critical exponents for different universality classes adapted from ref.⁴² are also included for comparison.

Fig. 6

(a-b) Modified Arrott plots (M^1/β vs. (H/M)^1/γ) for EuIn₂As₂. (c-d) series of M_S(T) and $\:{\chi\:}_{0}^{-1}\left(T\right)$ curves from the MAP method delineating the convergence of β and γ along the H||c and H||ab plane, respectively.

The Arrott plots (Fig. 5(c-d)) revealed the non-mean-field-type interactions. Like EuIn₂P₂, EuIn₂As₂ does not follow any established theoretical model following the modified Arrott plot method. For H||ab (Fig. 6a), the iterative method led to convergence at β and γ values of 0.244 and 0.89, respectively, with T_C = 18 K (we use T_C as we assume here the FFM state) passing through the origin and for H||c (Fig. 6b) β and γ values are 0.254 and 0.81 respectively with T_C = 17 K. M_S and χ₀⁻¹ were obtained for various β and γ until their convergence (Fig. 6(c-d)). Fitting with Eqs. (1) and (2), as shown in Fig. 7(a, b), provided β = 0.245(3) with 𝑇_C = 18.2(1) and γ = 0.899(2) with 𝑇_C = 17.2(2) K for H||ab, and β = 0.244(3) with 𝑇_C = 17.2(1) and γ = 0.807(3) with 𝑇_C = 16.7(3) K for H||c. The β and γ determined from various methods, including straight line fit of log(M_S) and log(χ₀ ⁻¹) with respect to log(|𝜀|) (Fig. 7(c- d), Kouvel Fisher (KF) method (Fig. 7(e-f)), and δ value from critical isotherm analysis and critical wisdom relation (Fig. S6(a-b)) are gathered in Table 1. No difference in magnetization isotherms was observed after considering the effect of demagnetizing field, as shown in Fig. S7.

Fig. 7

(a-b) Spontaneous magnetization M_S and χ₀⁻¹ as a function of temperature on the left and right y-axes for EuIn₂As₂, respectively, for H||ab and H||c. The solid lines are the fits with Eqs. 1–2. (c-d) log(𝑀_𝑆) and log(χ₀⁻¹) vs. log(|𝜀|) with T_C = 18 K for H||ab and T_C = 17 K H||c as obtained from MAP, the solid lines are straight line fits. (e, f) The Kouvel-Fisher plot of M_S and χ₀ ⁻¹, obtained from the intercepts of the modified Arrott plots (MAP) near T_C.

In the critical regime, the field and temperature dependence of magnetization follow a universal scaling equation^17,20:

$$\:M\left(H,\epsilon\:\right)=\:{\epsilon\:}^{\beta\:}{f}_{\pm\:}\left(\frac{H}{{\epsilon\:}^{\beta\:+\gamma\:}}\right)$$

(8)

where f₊ applies for T > 𝑇_C, and f₋ applies for T < 𝑇_C. A notable feature of this equation is that,, rather than detailing the interplay among the three variables M, H, and 𝜀, it reduces the relationship to two scaled variables, M∣𝜀∣^−β and H∣𝜀∣^−(γ+β). By plotting M∣𝜀∣^−β versus H∣𝜀∣^−(γ+β), it is expected that all data points for T > 𝑇_C collapse onto a single curve represented by the function f₊, while data points for T < 𝑇_C fall onto another curve described by f₋. Using the critical exponents β and γ and the critical temperature 𝑇_C estimated from the method for both EuIn₂P₂ and EuIn₂As₂ along H||c and H||ab, a plot of M∣𝜀∣^−β versus H∣𝜀∣^−(γ+β) is shown in Fig. 8(a-b) for EuIn₂P₂ and Fig. 8(c-d) for EuIn₂As₂. The insets of Fig. 8 present the same results on a log-log scale. The figure illustrates that the scaling equation of state, as stated in the above form, is perfectly satisfied in the present systems.

Fig. 8

(a-b), Scaling plots above and below T_C according to Eq. (4). The different symbols represent various temperatures. Inset shows the same plot using the log-log scale, indicating two universal curves below and above T_C, respectively, for H||c and H||ab plane for EuIn₂P₂. (c-d) The same plots for EuIn₂As₂ above and below T_C, for H||ab plane and H||c axis, respectively.

Although the estimated critical exponents in this study do not align strictly with standard universality classes, critical exponents are often significantly influenced by factors such as disorder and/or competing interactions. Therefore, it is essential to investigate whether they converge toward any known universality classes as they reach their asymptotic values. To address this, the effective critical exponents β_eff and γ_eff for both compounds were calculated and plotted in Fig. 9 as functions of 𝜀, following the relations^15,30:

$$\:{\beta\:}_{\text{e}\text{f}\text{f}}=\frac{{d(lnM}_{S}\left({\upepsilon\:}\right))}{d\left({ln}\left({\upepsilon\:}\right)\right)}\text{for}\:T<{T_C};\:{\gamma\:}_{\text{e}\text{f}\text{f}}=\frac{d\left(ln{\chi\:}_{0}^{-1}\right({\upepsilon\:}\left)\right)}{d\left({ln}\left({\upepsilon\:}\right)\right)}\text{for}\:T>{T_C}\:$$

(9)

Fig. 9

The effective critical exponents (a) β_eff and (b) γ_eff for EuIn₂P_2, and (c) β_eff and (d) γ_eff for EuIn₂As₂ are plotted as a function of reduced temperature ε, above and below T_C.

For both EuIn₂P₂ and EuIn₂As₂, β_eff (Fig. 9(a-c)) and γ_eff (Fig. 9(b-d)) reflect a nonmonotonic change with 𝜀 approaching a value of 0.241 and 0.855 for H||c and 0.217 and 1.04 for H||ab in case of EuIn₂P₂ at lowest of 𝜀 = 𝜀_min = 9.8 × 10⁻². For EuIn₂As₂, β_eff and γ_eff for H||c at 𝜀_min = 6.3 × 10⁻² are 0.245, 0.77, and for H||ab, the values are 0.247, 0.91, respectively. These values closely align with the critical exponents β and γ in Table 1, obtained through various analysis methods. They appear to converge to the values in the asymptotic regime, indicating that the compounds under investigation do not belong to known universality classes. However, the estimated critical exponents can be somehow explained with the tricritical mean-field model, but relatively large deviations exist for EuIn₂P₂ and EuIn₂As₂, similar to the related systems EuSn₂P₂³¹ and EuCuP³². This proximity to tricriticality may stem from the complexity of their magnetic phase diagrams. For EuIn₂As₂^1,29, multiple ordered phases, including AFM and metamagnetic states coexist near T_C, with the possibility of a tricritical point emerging at low fields where second-order and first-order transitions meet. In our case, the primary second-order transition is accompanied by first-order metamagnetic transitions, which could drive the system toward tricritical behavior without fully reaching it. A full magnetic phase diagram of EuIn₂P₂ is not provided, but it is dominated just by the increase of T_C with magnetic field. Nevertheless, in small magnetic fields the magnetic components due to the small tilting of the magnetization out of the ab plane will also provide a competition of the magnetic phases supporting the tricritical behavior. Moreover, the clear anisotropy in the critical exponents especially for EuIn₂P₂ (Table 1) underscores the role of magnetic anisotropy, the layered structure (P6₃/mmc) promotes quasi-two dimensional magnetic interactions within ab-plane. The critical fluctuations are therefore constrained by this structural dimensionality, and the applied field direction probes different effective dimensionalities of the spin system, leading to the observed anisotropic exponent values. The effective exponents (Fig. 9) further confirm that the asymptotic critical regime is reached and these values are intrinsic, not a crossover artifact. Further complicating this picture is the interplay between magnetic fluctuations and topological properties. For instance, EuIn₂As₂ exhibits low-energy collective modes, such as magnons, which are highly sensitive to temperature changes around the magnetic transition. This strong coupling between magnetic order and topological surface states and the emergence of a narrow surface magnetic gap at the AFM transition temperature may contribute to the observed deviations from conventional models used to describe critical behavior³³.

Fig. 10

(a, b) Temperature-dependent magnetic entropy change (ΔS_m) for EuIn₂P₂ at the magnetic field change in the range from 0.2 T to 5 T, for H||c and H||ab. (c) The rotational magnetic entropy change (ΔS_R) was calculated for different values of the magnetic field change for EuIn₂P₂. (d, e) ΔS_m and (f) ΔS_R for EuIn₂As₂ in similar conditions to EuIn₂P₂.

In addition to analyzing the magnetization scaling, we have also determined critical exponents from the magnetic field dependence of the magnetic entropy change (ΔS_m), associated with the magnetocaloric effect (MCE). Initially, Oesterreicher and Parker proposed a universal relationship to describe the magnetic field dependence of ΔS_m as -ΔS_m^max ∝ µ₀ΔH^2/3, where ΔS_m^max is the maximum of -ΔS_m³⁴. However, subsequent experimental studies have shown that the power-law exponent often deviates significantly from the proposed value of 2/3 (representing the mean-field universality class). Using the Arrott–Noakes equation, as outlined in Eq. (11), Franco demonstrated that -ΔS_m (T_C, H) can generally be expressed as³⁵:

$$-\Delta{S}_m^{max}=\Delta{S}_m(T_C,H)\propto\mu_0\Delta{H}^n$$

(10)

where

$$\:n=1+\frac{\beta\:-1}{\beta\:+\gamma\:}$$

(11)

.

The magnetic entropy change ΔS_m (T, H) was calculated and presented in Fig. 10 based on Maxwell’s equation of the form^34,36:

$$\:\varDelta\:{S}_{\text{m}}\left(T,\:H\right)={\int\:}_{0}^{H}\left(\delta\:M/\delta\:T\right)dH$$

(12)

For EuIn₂P₂, Fig. 10(a-b) shows the maximum of ΔS_m (T, H) at ~ 24 K, both for H||ab and H||c. For H||ab, ΔS_m^max = −10.80 Jkg⁻¹K⁻¹ and for H||c, ΔS_m^max = −9.55 Jkg⁻¹K⁻¹ for a maximum magnetic field change (µ₀ΔH) of 5 T. For EuIn₂As₂ and magnetic field change of 5 T, Fig. 10(d-e) indicates a maximum entropy change at ~ 17.5 K with values ΔS_m^max = −12.75 Jkg⁻¹K⁻¹ for H||ab and ΔS_m^max = −10.40 Jkg⁻¹K⁻¹ for H||c. The values of the T at which the maximum value of entropy change appears for EuIn₂P₂ and EuIn₂As₂ are compatible with our Arrott-Noakes results. For EuIn₂P₂, the peaks are symmetric, while for EuIn₂As₂, the peaks are broader and more asymmetric. This is due to the change in the magnetic structure of EuIn₂As₂, i.e., from AFM to FFM type, which is further supported by the observation of negative magnetic entropy changes (-ΔS_m)^26,37,38.

To further investigate the effect of the magnetic anisotropy, we estimated the rotating magnetic entropy change (ΔS_R), also known as the rotating magnetocaloric effect (RMCE), defined as:

$$\Delta{S}_R(T,H)=\Delta{S}_m(T,H_ab)-\Delta{S}_m(T,H_c)$$

(13)

where H_ab and H_c indicate the magnetic field applied along the ab plane and the c axis, respectively³⁹. The maximum value of ΔS_R was recorded to be 1.8 Jkg⁻¹K⁻¹ and 3.2 Jkg⁻¹K⁻¹ for EuIn₂P₂ and EuIn₂As₂, respectively, for the magnetic field of 1.5 T and 5 T, as displayed in Figs. 10 (c) and (f). Larger values of ΔS_R suggest more substantial magnetic anisotropy, driven by the crystal field environment and the anisotropic magnetic exchange interactions. The more assertive anisotropic behavior in EuIn₂As₂ may be attributed to its electronic structure and enhanced spin-lattice coupling compared to EuIn₂P₂, as seen in the exchange interaction analysis²⁹. These anisotropic features can be more clearly visualized in the contour plots (Fig. 11) of magnetic entropy change ΔS_m and ΔS_R for studied compounds under different magnetic field orientations. The contours of ΔS_R nicely illustrate one of the advantages of RMCE over the standard MCE. Namely, it turns out that the best performance of the effect can occur in moderate magnetic fields, as shown in Fig. 11(c-f), whereas the standard MCE is stronger with the field.

Fig. 11

(a-b) Contour plots of magnetic entropy change (-ΔS_m) for EuIn₂P₂ for H||c and H||ab plane. (c) Contour plots of the rotational magnetic entropy change (ΔS_R) for EuIn₂P₂. (d-e) Contour plots of -ΔS_m for EuIn₂As₂ for H||c and H||ab. (f) Contour plots of ΔS_R for EuIn₂As₂.

Another crucial parameter that assesses the performance of magnetocaloric materials is relative cooling power (RCP), which is determined as RCP = |ΔS_m^max| ×ΔT_FWHM, where ΔT_FWHM is the full width at half maximum of ΔS_m^33,35. RCP represents the amount of heat transferred between the hot and cold reservoirs. At 5 T, RCP reaches 262.86 Jkg⁻¹ and 212.03 Jkg⁻¹ for H||ab and H||c in the case of EuIn₂P₂ (right axis of Fig. 12 (a-b)), while for EuIn₂As₂ the RCP values reach 210.97 Jkg⁻¹ and 141.49 Jkg⁻¹ for H||ab and H||c (right axis of Fig. 12 (c-d)).

The magnetocaloric parameters obtained are not exceptional, but they are comparable to those of other similar materials^43,44,45. Table 2 summarizes the parameters for the studied materials and compares them with selected Eu-based materials in the form of single crystals.

Table 2 Comparison of magnetocaloric effects in selected Eu-based materials in form of single crystals. FM - ferromagnetic, AFM - antiferromagnetic, T_ord - magnetic ordering temperature, |ΔS_m^max| - absolute maximum value of the entropy change, RCP - relative cooling power. Magnetocaloric parameters are evaluated for the magnetic field change µ₀ΔH = 5 T.

Fitting Eq. (11) on -ΔS_m^max as a function of changing magnetic field µ₀ΔH (Fig. S8) in the region of FFM transition gives the values of n equal to 0.57(3) and 0.75(1) for EuIn₂P₂ and 0.79(2) and 0.64(3) for EuIn₂As₂, for H||ab and H||c, respectively. These values are relatively higher than those estimated from Eq. (11) for β and γ acquired by different methods, as summarized in Table 1. However, following the technique of Huang et al.⁴⁰, in which they proposed that the entropy change can be defined as:

$$-\Delta{S}_m^{max}=C+D\times\mu_0\Delta{H}^n$$

(14)

where C and D are fitting parameters. However, in Ref.⁴⁰, no physical basis for introducing the parameter C is given. The value of n determined by this method is 0.45(3) and 0.49(5) for EuIn₂P₂ and 0.34(2) and 0.33(0) for EuIn₂As₂, for H||ab and H||c, respectively (see left axes of Fig. 12). Using Eq. (15) yields different values for n, which are more similar to those determined from MAP.

Similar to -ΔS_m^max, RCP also follows a power law in the region of FFM transition³⁴:

$$RCP\propto\mu_0\Delta{H}^{(1+1/\delta)}$$

(15)

fitting Eq. (16) allowed us to estimate the value of δ, which agrees well with the MAP estimations. The values of δ are also presented in Table 1.

Fig. 12

Magnitude of the maximum peak value of ΔS_m (ΔS_m^max) and RCP represented as a function of the magnetic field displayed on the left and right y-axes, respectively, for H||c and H||ab. Panels (a-b) correspond to EuIn₂P₂, and panels (c-d) correspond to EuIn₂As₂. The corresponding scaling power laws fit both quantities for magnetic fields above the saturation field. Red solid lines are fits of Eq. (15) for the left panel and Eq. (16) for the right panel.

Based on the universality principle, the magnetic entropy change (−ΔS_m(T, H)) in the FFM region can be normalized into a universal curve. Specifically, −ΔS_m(T, H) can be scaled as ΔS΄_m = ΔS_m/ΔS_m^max. The temperature is renormalized into a scaled temperature (θ) defined as follows⁴¹:

$$\:{\uptheta\:}=\left\{\begin{array}{c}{{\uptheta\:}}_{-}=-\frac{T-{T}_{C}}{{T}_{r1}-{T}_{C}},\:\:T\le\:{T}_{C}\\\:{{\uptheta\:}}_{+}=\frac{T-{T}_{C}}{{T}_{r2}-{T}_{C}},\:\:T>{T}_{C}\end{array}\right.$$

(16)

Fig. 13

Normalized entropy change ΔS_m/ΔS_m^max vs. reduced temperature θ: (a, b) for EuIn₂P₂ and (c, d) for EuIn₂As₂.

Fig. 14

Normalized entropy change ΔS_m/ΔS_m^max scaled by the deduced exponents in MAP: (a-b) for EuIn₂P₂ and (c-d) for EuIn₂As₂.

where T_r1and T_r2 are temperatures at which -ΔS_m = -ΔS_m^max/2. As illustrated in Fig. 13(a-d), all the scaled curves collapse into a universal curve under a high magnetic field region. This collapse confirms that the high-field PM to FM transition is second-order in EuIn₂P₂ and EuIn₂As₂. Further, the magnetic entropy change − ΔS_m(T, H) can also be expressed using an alternative scaling equation:

$$\:{-\varDelta\:{S}_{m}=H}^{n}g\left(\frac{\epsilon\:}{{H}^{1/({\upbeta\:}+{\upgamma\:})}}\right)$$

(17)

where β, γ, and n represent the critical exponents, and g denotes a regular scaling function³⁰. The values of β and γ estimated from MAP were used while n obtained from power law fitting (Eq. 15) of -ΔS_m^max was used. All the plots of -ΔS_m/Hⁿ vs. $\:\frac{\epsilon\:}{{H}^{1/(\beta\:+\gamma\:)}}$ collapse onto a universal curve in the high-field FM region, as shown in Figs. 14(a-d) for both the studied compounds. This observation further confirms the reliability of the determined critical exponents. Since the above scaling procedure for -ΔS_m(T, H) applies only to materials undergoing a second-order FM phase transition, the well-scaled curves in the high-field region provide strong Evidence that the PM-FFM phase transition in this region is of second-order nature. In contrast, the deviation from scaling in the low-field region suggests the presence of a first-order phase transition under low-field conditions due to the metamagnetic transitions³².

In conclusion, in this study, we performed a detailed investigation of the critical behavior and magnetocaloric response in the Zintl phases EuIn₂P₂ and EuIn₂As₂, both of which exhibit second-order magnetic transitions, a direct ferromagnetic (FM) transition in EuIn₂P₂ (T_C ≈ 24 K), and a field-induced ferromagnetic transition (FFM) in EuIn₂As₂ (T_C ≈ 17–18 K). Critical exponents (β, γ, δ, n), extracted through multiple self-consistent methods, deviate significantly from standard 3D universality classes, reflecting the influence of three key factors: (i) proximity to tricriticality and competing interactions, evidenced by exponent values approaching tricritical mean-field limits and by the coexistence of AFM and metamagnetic transitions in EuIn₂As₂; (ii) magnetic anisotropy and quasi-two-dimensionality, arising from the layered P6₃/mmc structure, which produces distinct exponents for H∥c and H∥ab; and (iii) coupling between localized 4f moments and itinerant topological states, which can renormalize spin correlations and suppress conventional critical scaling. Magnetocaloric measurements, including significant magnetic and rotational entropy changes, corroborate the anisotropic critical behavior and validate the second-order nature of the high-field transitions with EuIn₂As₂ demonstrating superior magnetocaloric performance, with maximum entropy changes of 12.75 Jkg⁻¹K⁻¹ (H||ab) and 10.40 Jkg⁻¹K⁻¹ (H||c) at 5 T. These results demonstrate that magnetic criticality in EuIn₂P₂ and EuIn₂As₂ is governed by a non-conventional universality class shaped by the interplay of low-dimensional structure, competing magnetic orders, and topological electronic states.

Methods

Single crystals of EuIn₂As₂ were grown using the In-As flux. High-purity elements (Eu, In, and As) in a molar ratio of 1:12:3 were sealed in an alumina crucible and heated to 1000 °C under an inert atmosphere. The mixture was held at this temperature for 24 h to ensure homogeneity, and then cooled to 700 °C at 2 °C/h. The resulting crystals were separated from flux via centrifugation yielding air-stable, silver-shiny crystals alongside InAs crystals as byproducts. For EuIn₂P₂ crystals, pure elements of Eu, In, P (molar ratio; 3:110:6) were reacted using In as flux. The mixture was heated to 1100 °C for 30 h, followed by slow cooling to 600 °C at 2 °C/h. Crystals stoichiometry was determined using a scanning electron microscope (SEM) (FEI Technologies) coupled with an energy-dispersive x-ray spectrometer (EDS) (Genesis XM4). A detailed experimental description of the samples can be found in prior studies published^7,29. The QD PPMS-9T, a Quantum Design Physical Property Measurement System equipped with a vibrating sample magnetometer (VSM), was used for the field-dependent and temperature dependent magnetic (DC) measurements.