In the last decade, intensive research of the topology of electronic structures has deepened our understanding of the exotic physical properties of novel condensed matters and established a whole new classification. After the discovery of topological insulators,^[1,2] topological semimetals (TSMs) have attracted considerable attention in recent years. The TSMs are characterized by the symmetry protected bulk band crossings near the Fermi energy E_F.^[3–6] According to the degeneracy of nodes, they can be classified into Dirac semimetals (DSMs), Weyl semimetals (WSMs), multi-degenerate semimetals, and so on.^[7] Among these TSMs, DSMs have Dirac nodes that can be viewed as two Weyl nodes with opposite chirality degenerate at the same k-point. Such a degeneracy can be lifted by breaking either inversion symmetry (IS) or time-reversal symmetry (TRS), resulting in two types of WSMs; i.e., nonmagnetic noncentrosymmetric WSMs and magnetic WSMs.^[7] The first class of WSMs has been theoretically predicted and experimentally confirmed.^[8–10] Magnetic WSMs have been predicted in several materials, such as pyrochlore iridates, Heusler compounds, and the related materials such as Heusler alloys Co₂TiX (X = Si, Ge, or Sn), Co₂MnAl, and half-Heusler compound GdPtBi, etc.^[4,11–14]

Recently, Heusler compound Co₂ZrSn has also been predicted to host Weyl fermions.^[15] Moreover, it belongs to the family of 26 valence electron Heusler systems, which are known to exhibit half-metallic ferromagnetism (HMFM) with saturated magnetic moments of per formula unit.^[16,17] According to the classification proposed by Coey,^[18] Co₂ZrSn is a type of I_A HMFM, in which the spin-majority band is crossing E_F (metallic) and there is a band gap for the spin-minority band. Because it is an HMFM compound with Curie temperature (T_c) near/higher than room temperature,^[19] besides the unique feature of harboring Weyl fermions, it is vitally important to the spintronics applications due to the nearly 100% spin polarization of conduction electrons at E_F.^[20] Previous studies on Co₂ZrSn single crystals show that there are some vacancies in the position of Co with the content of Co varying from 1.59 to 1.84. With higher Co content, the sample exhibits higher T_c and the T_c is above room temperature when the content of Co is larger than 1.70.^[19] Meanwhile, when Cu is doped into the Co site, the T_c of Co_2−yCu_xZrSn decreases linearly and the low-temperature spin glass behavior appears when y is close to 1.^[21] This trend can be explained by the weakening of half-metallicity of Co₂ZrSn with the Cu substitutions, leading to the breakdown of the ferromagnetism.^[21] To understand the itinerant ferromagnetism in Co₂ZrSn further, a comprehensive investigation of phase transition and critical exponents is urgently needed, which will not only provide an insight into the range of ferromagnetic interaction but also the symmetry of the order parameter within a given universality class.^[22,23] Among the various properties of ferromagnets, magnetocaloric effect (MCE) is another important feature that can be used for magnetic refrigeration, which is an environmentally friendly energy-conversion technology.

In this paper, the magnetic critical behavior and magnetocaloric effect of Co_2−xZrSn in the vicinity of the paramagnetic–ferromagnetic (PM–FM) phase transition region are studied in detail. It is found that Co_2−xZrSn undergoes a second-order phase transition with a long-range order in the nature of magnetic interaction. Moreover, the obtained critical exponents are found to follow the theoretically predicted values for mean-field model. Further analysis of the field-dependent magnetic entropy change indicates that the maximum magnetic entropy change of Co_2−xZrSn is about at 50 kOe.

Polycrystal Co_2−xZrSn was synthesized using arc-melting method under an Ar atmosphere. High purity Zr particles and Co pieces were melted first to achieve a homogeneous mixture of the compound metal. Then, the obtained alloy was brought close to the Sn shots and remelted. The sample was flipped three times to avoid breaking up over a long time of arc-melting. Finally the arc-melted sample was cooling down naturally. The crystal structure and phase purity were identified by powder x-ray diffraction (pXRD) using the Cu–Kα radiation of Bruker D8. Rietveld refinements of the data were performed with the TOPAS package.^[24] The magnetization measurements were conducted using a Quantum Design MPMS3.

The XRD patterns of Co_2−xZrSn can be readily indexed to the crystal structures of Heusler L21 phase (Fm3m) (Fig. 1(a)). No obvious characteristic peaks of other impurities are detected. The intense diffraction peaks suggest that the as-synthesized products are well crystallized. The fitted lattice parameter a is 6.21266(8) Å. When compared to previous single crystal results, it suggests that the value of x in Co_2−xZrSn is about 0.33.^[19] The temperature dependence of magnetization M(T) at H = 1 kOe with zero-field-cooling (ZFC) and field-cooling (FC) models are plotted in Fig. 1(b). The M(T) curves show a typical PM–FM phase transition. Inset of Fig. 1(b) shows the T_c is ∼280 K, which is defined as the minimum value of the . This is consistent with previous results for Co_2−xZrSn single crystals.^[19] The overlap of ZFC and FC M(T) curves suggests the absence of spin-glass behavior in Co_2−xZrSn. Figure 1(c) shows the magnetization hysteresis loop of Co_2−xZrSn measured at 2 K. The saturated moment M_s is about /f.u., in agreement with that of Co_2−xZrSn single crystals.^[19] Meanwhile, the magnetic coercivity (H_c) of Co_2−xZrSn is smaller than 50 Oe, which indicates the Co_2−xZrSn is a super soft magnet.

Fig. 1.

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	Fig. 1. (a) The powder XRD pattern and Rietveld refinement of Co2−xZrSn. Vertical tick marks is Bragg reflections in the Fm3m space group. (b) Temperature dependence of magnetization at H = 1 kOe with ZFC and FC modes. Inset: as a function of temperature near Tc. (c) Isothermal M(H) loop at 2 K. (d) Isothermal magnetization curves from 260 K to 300 K with 2-K temperature interval.

To further understand the fundamental feature of the FM phase transition in Co_2−xZrSn, we measured the isothermal magnetization curves M(H) from 260 K to 300 K with temperature step of 2 K as shown in Fig. 1(d). According to the mean-field theory, the ferromagnet undergoes a second order phase transition with long range order interactions, the Landau free energy G(T,M) could be expressed as:^[25]

Based on this equation, we can plot a Arrott plot of M ²versus H/M, which should be a series of straight lines, and the positive intercepts of these straight lines on the H/M axis determine the second-order magnetic transition. But we should analyze the M(H) results for the short-range FM phase transition systems by using the Arrot–Noaks equation of state , where is the reduced temperature, and a and b are constants.^[26] We constructed the Arrott plots, using 3D XY model (β = 0.345, γ = 1.316), 3D Heisenberg model (β = 0.365, γ = 1.386), 3D Ising model (β = 0.325, γ = 1.24), and mean field model (β = 0.5, γ = 1.0), as given in Fig. 2. All these models yield quasi-straight lines in the high-field region, the positive intercepts reveals a second-order PM–FM transition in Co_2−xZrSn. However, none of the models yield a series of parallel straight lines in the high-field range and the line at T_c passes through the origin. Therefore, it is difficult to distinguish which model is the best for the determination of critical exponents.

Fig. 2.

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	Fig. 2. The MAP of (a) 3D XY model, (b) 3D Heisenberg model, (c) 3D Ising model, and (d) mean-field model.

To determine better parameters, we calculated their normalized slopes (NS) which are defined at the critical point as , where S(T) is the slope of versus at high field. Because the most appropriate model should be the one close to 1.0 in the vicinity of T_c,^[27] the mean-field is the best model for Co_2−xZrSn system, as shown in Fig. 3. However, the Arrott plot with the critical exponents β = 0.5 and γ = 1 could not lead to a set of parallel lines in the high magnetic field region. According to the scaling hypothesis, critical exponents β and γ can be briefly described by the following equation,^[25,28]

Fig. 3.

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	Fig. 3. Temperature dependence of the normalized slopes .

Fig. 4.

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	Fig. 4. (a) The MAP plot for Msversus T (left) and versus T (right). (b) The KF plot for versus T (left) and versus T. (c) The plot of the isotherm M versus H around T = 280 K. Insert: log(M) versus log(H). (d) Scaling plots of M versus H below and above Tc = 280 K. Isert: log (M) versus log (H).

Figure 4(b) shows that the critical exponents obtained from the KF method are β = 0.415(2) with T_c = 280.1(7) K and γ = 1.08(5) with T_c = 280.4(1) K. These are consistent with the values derived from the modified Arrott plots. The critical exponent δ can be determined by the critical isotherm analysis from M(H) at T_c following Eq. (5). The isothermal magnetization M(H) at T_c = 280 K is shown in fig 4(c), where the inset plots with the logarithmic scale. The inset shows that the log(M)–log(H) relation yields straight line at higher field range with the slope 1/δ, where δ = 3.53 is obtained. According to the statistical theory, the critical exponents should agree with the Widom scaling relation:^[30]

Using the β and γ values determined from the modified Arrott plot and Kouvel–Fisher (KF) plot, we obtain δ = 3.66 and δ = 3.61, respectively, which close to the above value of δ. Furthermore, there is an important criterion for the critical regime based on the prediction of scaling hypothesis,^[25]

The universality class of the magnetic phase transition depends on range of the exchange distance J(r). It decays as , where σ is the range of the interaction, r is the distance, d is the dimensionality, and b is the spatial scaling factor. The the parameter of σ can be calculated by the equation,^[31] 10 where n is the spin dimensionality, , . According to the renormalization group analysis, the spin interactions can be classified by the value of σ. The implies long-range spin interaction dominated in the system, while implies the short-range spin interaction. In the Co_2−xZrSn system, the experimental value of γ is 1.26, for any combination of {d:n} the values of σ are always smaller than 2. This suggests that the long-range spin interaction plays an important role in the Co_2−xZrSn system.

To gain more information on critical properties of Co_2−xZrSn, we have studied the magnetic entropy change . According to thermal dynamics, the magnetic entropy change is defined as,^[32]

For the magnetization measured at small discrete fields and temperature intervals, can be approximately expressed as,^[33]

Fig. 5.

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	Fig. 5. (a) The trend plots of with temperature variation at the different magnetic fields. (b) Field dependence of . The solid line represents a fit of to the data. (c) Field dependence of RCP.

The relative cooling power (RCP) is also an important indicator for characterizing the magnetothermal material, its mathematical definition is,^[34]

In summary, the magnetic properties of Co_2−xZrSn polycrystal are studied in detail. It is found that the PM–FM transition in Co_2−xZrSn polycrystal is the second-order phase transition. The critical exponents β, γ, and δ obtained from the MAP and KF methods are consistent each other. These parameters conform to the scaling rate very well. In addition, these critical exponents suggest that the magnetic transition of Co_2−xZrSn polycrystal can be described by the mean-field model. Moreover, the and RCP are about and at H = 50 kOe, respectively.