Softening of Hard Magnetic Behavior and the Maximum Coercive Field in Zr₂RhTl as Revealed by Effective Field Theory

Abstract

The present study investigated the magnetic hysteresis properties (coercivity and remanent magnetization) of the Zr₂RhTl Heusler alloy using effective field theory (EFT). The study found that the coercive field of Zr₂RhTl reaches a maximum at a specific critical temperature, T_ch, at which the hardness of magnetic materials increases with the coercive field. This behavior is called the “critical hardness temperature (T_ch)”. The hardness of the Zr₂RhTl Heusler alloy increases with temperature until T_ch, reaching a maximum at T_ch. In contrast, it exhibits soft magnetic behavior at T < T_ch and T > T_ch. We suggest that this maximum hardness behavior can enable a new class of thermo-hardness sensors (THSs) and actuators (THAs).

1. Introduction

Half-metallic ferromagnets (HMFs) in spintronics have been an appealing subject of study in recent decades. Half metals, owing to their 100% spin polarization, electronic band structure, and magnetic properties, behave like semiconductors, and are widely used in spintronics, MR, and MRAM components [1,2,3]. Exploration of half-metallic Heusler alloys is widely anticipated both from a fundamental and an application perspective. In 1983, de Groot et al. [4] first presented the half-Heusler compound NiMnSb as an HMF material. Heusler alloys have attracted considerable attention due to their high Curie temperature (T_C) and tunable electronic structure [5,6,7]. Generally, Heusler compounds can be classified into ternary and equiatomic quaternary Heusler compounds. Ternary Heusler compounds consist of half-Heusler and full-Heusler families. The chemical formulas of these compounds are XYZ and X₂YZ, respectively. X and Y elements are represented by various transition metals (B group), and Z elements stand for main group elements (3A–8A) in the periodic table. Novel Heusler compounds can also be achieved with the XX’YZ composition. In this case, one of the X atoms is replaced with a new transition element, X′. Half-metallic properties can be observed in different examinations. Among these, a key indicator is the band structure, which shows a band gap for each spin state. While one of the spin band structures behaves like a metal, the other acts like a semiconductor. In addition, to classify the investigated alloy as a half metal, the alloy’s total magnetization must match the Slater–Pauling rule (SP rule), Mt = Zt–24 [8,9,10,11,12,13,14,15,16,17]. However, a modified SP rule, Mt = Zt–28, has also been suggested for these alloys [8]. In conductors, when an external magnetization is applied while current is flowing in the circuit, the voltage difference created by the Hall effect can be measured more precisely and effectively using the high spin polarization of Heusler alloys. This precision is also important in actuators for converting electromagnetic effects into mechanical responses. Therefore, better performance can be achieved with Heusler compounds in applications such as sensors and actuators.

The unique characteristics of Heusler alloys confer multifunctional properties arising from magnetoelastic coupling, including shape memory, magnetic super elasticity, and magnetocaloric and barocaloric effects. These properties of Heusler alloys have attracted researchers for their use as sensors, actuators, magnetic storage media, energy harvesters, and magnetic cooling devices [18,19,20,21].

The present study investigated the magnetic hysteresis properties (coercivity and remanence magnetization) of the Zr₂RhTl Heusler alloy using the effective field theory (EFT) developed by Kaneyoshi [22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40]. We found that Zr₂RhTl and its components behave as both soft and hard magnets, depending on temperature. Zr₂RhTl and its components act as a hard magnet at a specific critical temperature (T_ch, critical hardness temperature), whereas they behave as a soft magnet at T < T_ch and T > T_ch. This maximum-hardness behavior can open the door to modeling and produce a new class of thermo-hardness sensors (THSs) and actuators (THAs).

2. Materials and Methods

2.1. Model and EFT Formulations

In our previous study [41], we modeled and investigated the temperature dependence of the magnetization of the Zr₂RhTl Heusler alloy using EFT [22]. The necessary DFT data for the materials and methods were obtained using the Wien2k program with GGA and GGA+U approaches, as described in our previously published article [41]. Volume–energy optimization was performed using the Murnaghan equation of state, and the lattice constant, equilibrium energy, band structure, and density of states were calculated.

This paper examines the external-field dependence of the magnetization of that alloy. Since the crystal structure of Zr₂RhTl and its EFT equations are the same, we follow the same procedures as given below.

Figure 1 illustrates the crystal structure of the Zr₂RhTl Heusler alloy. In Figure 1, Zr1 is Zirconium atoms in the surface, Zr2 is Zirconium atoms in the core, Rh is Rhodium atoms, and Tl is thallium atoms in the core. Zr1 and Zr2 occupy two distinct crystallographic positions. In full Heusler compounds, element X occupies two separate positions in the unit cell in the form X₂YZ. For cubic (α = β = γ = 90) space group 216 (f-43 m), elements X1 and X2 should be in the 0/0.25 Wyckoff positions, respectively.

2.2. Hamiltonian

The Hamiltonian and magnetizations of Zr₂RhTl are given as follows [41]:

$$\begin{array}{ll}\hfill {\mathcal{H}}_{\mathrm{Zr}2\mathrm{RhTl}}=& -{\mathrm{J}}_{\mathrm{s}}{\displaystyle \sum _{〈\mathrm{Zr}1,\mathrm{Rh}〉}{\mathrm{S}}_{\mathrm{Zr}1}^{\mathrm{z}}{\mathrm{S}}_{\mathrm{Rh}}^{\mathrm{z}}-{\mathrm{J}}_{\mathrm{c}}{\displaystyle \sum _{〈\mathrm{Zr}2,\mathrm{Tl}〉}{\mathrm{S}}_{\mathrm{Zr}2}^{\mathrm{z}}{\mathrm{S}}_{\mathrm{Tl}}^{\mathrm{z}}}}-{\mathrm{J}}_{\mathrm{i}1}{\displaystyle \sum _{〈\mathrm{Zr}1,\mathrm{Zr}2〉}{\mathrm{S}}_{\mathrm{Zr}1}^{\mathrm{z}}{\mathrm{S}}_{\mathrm{Zr}2}^{\mathrm{z}}}-{\mathrm{J}}_{\mathrm{i}2}{\displaystyle \sum _{〈\mathrm{Zr}1,\mathrm{Tl}〉}{\mathrm{S}}_{\mathrm{Zr}1}^{\mathrm{z}}{\mathrm{S}}_{\mathrm{Tl}}^{\mathrm{z}}}-{\mathrm{J}}_{\mathrm{i}3}{\displaystyle \sum _{〈\mathrm{Zr}2,\mathrm{Rh}〉}{\mathrm{S}}_{\mathrm{Zr}2}^{\mathrm{z}}{\mathrm{S}}_{\mathrm{Rh}}^{\mathrm{z}}}\hfill \\ & -{\mathrm{J}}_{\mathrm{i}4}{\displaystyle \sum _{〈\mathrm{Tl},\mathrm{Rh}〉}{\mathrm{S}}_{\mathrm{Tl}}^{\mathrm{z}}{\mathrm{S}}_{\mathrm{Rh}}^{\mathrm{z}}}-{\Omega }_{\mathrm{Zr}1}{\displaystyle \sum _{\mathrm{Zr}1}{\mathrm{S}}_{\mathrm{Zr}1}^{\mathrm{r}}-{\Omega }_{\mathrm{Zr}2}{\displaystyle \sum _{\mathrm{Zr}2}{\mathrm{S}}_{\mathrm{Zr}2}^{\mathrm{r}}}-{\Omega }_{\mathrm{Rh}}{\displaystyle \sum _{\mathrm{Rh}}{\mathrm{S}}_{\mathrm{Rh}}^{\mathrm{r}}-{\Omega }_{\mathrm{Tl}}{\displaystyle \sum _{\mathrm{Tl}}{\mathrm{S}}_{\mathrm{Tl}}^{\mathrm{r}}}}}\hfill \\ & -\mathrm{h}\left({\displaystyle \sum _{\mathrm{Zr}1}{\mathrm{S}}_{\mathrm{Zr}1}^{\mathrm{z}}+{\displaystyle \sum _{\mathrm{Zr}2}{\mathrm{S}}_{\mathrm{Zr}2}^{\mathrm{z}}}+{\displaystyle \sum _{\mathrm{Rh}}{\mathrm{S}}_{\mathrm{Rh}}^{\mathrm{z}}+{\displaystyle \sum _{\mathrm{Tl}}{\mathrm{S}}_{\mathrm{Tl}}^{\mathrm{z}}}}}\right)\hfill \end{array}$$

(1)

In the Hamiltonian, S^r = ±1 µB (r = z or x) denotes the Pauli spin operator, and the external magnetic field is denoted by h. The transverse field is Ω i (i = Zr1, Zr2, Rh, and Tl). J_i (i = s, c, i1, i2, i3, and i4) represents the exchange interaction between two nearest-and next-nearest-neighbor atoms, and the values of the J are calculated from the relationship J_i = k_i/d_i [29,30,31,32,33,34,35,36,37,38,39,40,41,42]. The parameters d_i (i = s, c, i1, i2, i3, and i4) are the distances between two magnetic atoms on the unit lattice, and they are obtained from the lattice constant (d_s = d_c = 6.37 Å, d_i1 = d_i2 = d_i3 = d_i4 = 2.758 Å). k is the exchange interaction constant, J is ferromagnetic if k > 0, and J is antiferromagnetic if k < 0. In this work, the partial exchange interaction constants are obtained as k_s = 100, k_i1 = 1390, k_i2 = 100, k_i3 = 0.001, k_i4 = 0.001, and k_c = 0.001 to fit with the results of the DFT calculation for Zr₂RhTl [41].

2.3. Magnetizations

The magnetizations of Zr1 (mZr1), Zr2 (mZr2), Rh (mRh), and Tl (mTl) are given by Equation (2) within EFT [22,41]. The numbers (6, 4, 4) represent the nearest- and next-nearest-neighbor atoms. For example, Zr1 has six Rh, four Zr2, and four Tl.

$$\begin{array}{ll}\hfill {\mathrm{m}}_{\mathrm{Zr}1}=& {\left[{\cos \mathrm{h}(\mathrm{J}}_{\mathrm{s}}\nabla {)+\mathrm{m}}_{\mathrm{Rh}}{\sin \mathrm{h}(\mathrm{J}}_{\mathrm{s}}\nabla )\right]}^6{\left[{\cos \mathrm{h}(\mathrm{J}}_{\mathrm{i}1}\nabla {)+\mathrm{m}}_{\mathrm{Zr}2}{\sin \mathrm{h}(\mathrm{J}}_{\mathrm{i}1}\nabla )\right]}^4\hfill \\ & {\left[{\cos \mathrm{h}(\mathrm{J}}_{\mathrm{i}2}\nabla {)+\mathrm{m}}_{\mathrm{Tl}}{\sin \mathrm{h}(\mathrm{J}}_{\mathrm{i}2}\nabla )\right]}^4{\left.{\mathrm{F}}_{\mathrm{s}-1/2}(\mathrm{x})\right|}_{\mathrm{x}=0},\hfill \\ \hfill {\mathrm{m}}_{\mathrm{Zr}2}=& {\left[{\cos \mathrm{h}(\mathrm{J}}_{\mathrm{c}}\nabla {)+\mathrm{m}}_{\mathrm{Tl}}{\sin \mathrm{h}(\mathrm{J}}_{\mathrm{c}}\nabla )\right]}^6{\left[{\cos \mathrm{h}(\mathrm{J}}_{\mathrm{i}1}\nabla {)+\mathrm{m}}_{\mathrm{Zr}1}{\sin \mathrm{h}(\mathrm{J}}_{\mathrm{i}1}\nabla )\right]}^4\hfill \\ & {\left[{\cos \mathrm{h}(\mathrm{J}}_{\mathrm{i}3}\nabla {)+\mathrm{m}}_{\mathrm{Rh}}{\sin \mathrm{h}(\mathrm{J}}_{\mathrm{i}3}\nabla )\right]}^4{\left.{\mathrm{F}}_{\mathrm{s}-1/2}(\mathrm{x})\right|}_{\mathrm{x}=0},\hfill \\ \hfill {\mathrm{m}}_{\mathrm{Rh}}=& {\left[{\cos \mathrm{h}(\mathrm{J}}_{\mathrm{s}}\nabla {)+\mathrm{m}}_{\mathrm{Zr}1}{\sin \mathrm{h}(\mathrm{J}}_{\mathrm{s}}\nabla )\right]}^6{\left[{\cos \mathrm{h}(\mathrm{J}}_{\mathrm{i}3}\nabla {)+\mathrm{m}}_{\mathrm{Zr}2}{\sin \mathrm{h}(\mathrm{J}}_{\mathrm{i}3}\nabla )\right]}^4\hfill \\ & {\left[{\cos \mathrm{h}(\mathrm{J}}_{\mathrm{i}4}\nabla {)+\mathrm{m}}_{\mathrm{Tl}}{\sin \mathrm{h}(\mathrm{J}}_{\mathrm{i}4}\nabla )\right]}^4{\left.{\mathrm{F}}_{\mathrm{s}-1/2}(\mathrm{x})\right|}_{\mathrm{x}=0},\hfill \\ \hfill {\mathrm{m}}_{\mathrm{Tl}}=& {\left[{\cos \mathrm{h}(\mathrm{J}}_{\mathrm{c}}\nabla {)+\mathrm{m}}_{\mathrm{Zr}2}{\sin \mathrm{h}(\mathrm{J}}_{\mathrm{c}}\nabla )\right]}^6{\left[{\cos \mathrm{h}(\mathrm{J}}_{\mathrm{i}2}\nabla {)+\mathrm{m}}_{\mathrm{Zr}1}{\sin \mathrm{h}(\mathrm{J}}_{\mathrm{i}2}\nabla )\right]}^4\hfill \\ & {\left[{\cos \mathrm{h}(\mathrm{J}}_{\mathrm{i}4}\nabla {)+\mathrm{m}}_{\mathrm{Rh}}{\sin \mathrm{h}(\mathrm{J}}_{\mathrm{i}4}\nabla )\right]}^4{\left.{\mathrm{F}}_{\mathrm{s}-1/2}(\mathrm{x})\right|}_{\mathrm{x}=0},\hfill \end{array}$$

(2)

The terms in Equation (2) include the differential operator, and F_1/2(x) is given for spin-1/2 particles, as described in [22],

$${\mathrm{F}}_{1/2}(\mathrm{x})={\displaystyle \frac{\left(\mathrm{x}+\mathrm{h}\right)}{\sqrt{{\left(\mathrm{x}+\mathrm{h}\right)}^2+{\Omega }_{\mathrm{i}}^2}}}\tan \mathrm{h}\left[\beta \sqrt{{\left(\mathrm{x}+\mathrm{h}\right)}^2+{\Omega }_{\mathrm{i}}^2}\right];(\mathrm{i}=\mathrm{Zr}1,\mathrm{Zr}2,\mathrm{Rh}\mathrm{and}\mathrm{Tl}),$$

(3)

where k_B is Boltzmann’s constant, β = 1/k_BT_A, and the absolute temperature is given by T_A. H is the external applied field. Finally, the equations for the total magnetization (MT_Zr2RhTl) and Slater–Pauling (MT_SP) magnetization (m_I is the interstitial magnetization) of Zr₂RhTl are written as follows,

$${\mathrm{MT}}_{\mathrm{Zr}2\mathrm{RhTl}}=\left({\mathrm{m}}_{\mathrm{Zr}1}{+\mathrm{m}}_{\mathrm{Zr}2}{+\mathrm{m}}_{\mathrm{Rh}}{+\mathrm{m}}_{\mathrm{Tl}}\right)$$

(4)

$${\mathrm{MT}}_{\mathrm{SP}}=\left(\left({\mathrm{m}}_{\mathrm{Zr}1}{+\mathrm{m}}_{\mathrm{Zr}2}{+\mathrm{m}}_{\mathrm{Rh}}{+\mathrm{m}}_{\mathrm{Tl}}\right)+{\mathrm{m}}_{\mathrm{I}}\right)$$

(5)

In addition to the partial magnetization of atoms in the unit cell, there is another type of magnetization that arises from the interaction of the density of states (DOS) of the atoms in the unit cell. This interaction, which we call interstitial magnetization, arises from the interaction of the d orbitals of the elements in the regions between the atoms within the cell. This magnetization is calculated using the Generalized Gradient Approximation—Perdew Burke Ernzerhof (GGA-PBE) method in Density Functional Theory (DFT) in two regions: (1) muffin-tin spheres (MTs), which determine the magnetization of the atoms in the unit cell, and (2) internal interactions in the unit cell outside the MT spheres. The magnetizations obtained from the regions outside the MT spheres are interstitial magnetizations. When calculating the total magnetic moment, the partial magnetic moments and interstitial magnetizations of the elements are added together, thus determining the magnetization of the entire unit cell.

3. Results

Numerical Results

Figure 2 shows the dependence of magnetization on the external magnetic field (magnetic hysteresis curves) for Zr₂RhTl and its components (Zr1, Zr2, Rh, and Tl) at isothermal temperatures of T = 50, 150, 300, 450, 600, 750, 900, and 1000 K in Figure 2a–h, respectively. As shown, the areas and coercive field points of the hysteresis loops increase as the temperature increases until a certain critical temperature, T < T_ch = 350 K (see Table S1), and then reach a maximum at T = Tch = 350 K, after which they decrease. However, they decrease as the temperature increases at T > T_ch = 350 K. Since the hardness and softness of magnetic materials are characterized by their coercivity [17], we suggest a new critical temperature for Zr₂RhTl and its components as “critical hardness temperature (T_ch = 350 K)”.

Zr₂RhTl and its components exhibit hard magnetic behavior at T = T_ch but are soft magnets at T < T_ch and T > T_ch. Similar hysteresis behavior is observed in the Co₂FeSn Heusler alloy synthesized using a simple low-cost electrodeposition method by Duan and Kou [43]. Duan and Kou found that higher defect concentrations result in higher H_C values in the samples. When the current density exceeds 2.0 A/dm², the Co content in the samples decreases with increasing current density, which in turn decreases the H_C of the Co₂FeSn samples. Tian et al. found that the saturation magnetization (M_s) and coercivity (H_c) increase from 144.84 kAm⁻¹ and 15.27 kAm⁻¹ to 175.13 kAm⁻¹ and 125.20 kAm⁻¹ as the Co content increases in Ni–Co coatings synthesized by electrodeposition [44]. Mizusaki et al. found that coercivity appears at about x = 0.4, decreasing after reaching a maximum of 4.13 kOe (0.41 T) at x = 0.45. This indicates that the ferromagnetic behavior changes from hard magnetic (Ru₂Mn_0.6Fe_0.4Ge) to soft magnetic (Ru₂FeGe) as the Fe content increases in the Ru₂Mn_1−xFe_xGe system [45].

Figure 3 shows the temperature dependence of the coercivity and remanent magnetizations of the Zr₂RhTl. In Figure 3a, coercive field points are obtained at H_c = 9, 19, 22, 23, 25, 26, 28, 30, 31, 33, 34, 35, 36, 37, 38, 37, 36, 35, 34, 32, 31, 30, 28, 26, 24, 21, 19, 16, 14, 12, 9, 7, 5, 3, 2, 1, 0, 0, 0, 0 and 0 T for T = 1, 25, 50, 75, 100, 125, 150, 175, 200, 225, 250, 275, 300, 325, 350, 375, 400, 425, 450, 475, 500, 525, 550, 575, 600, 625, 650, 675, 700, 725, 750, 775, 800, 825, 850, 875, 900, 925, 950, 975, and 1000 K, respectively.

One notes that the coercive field points of Zr₂RhTl increase as the temperature increases until T_ch = 350 K. Above Tch = 350 K, they decrease as the temperature rises. These results are significant because the hardness and softness of magnetic materials are characterized by their coercivity. Namely, magnetic materials are hard when their coercivity is high, or they are soft [17,46] when their coercivity is lower. Soft magnetic materials have low coercive field points and narrow hysteresis loop areas, whereas hard magnetic materials have high coercive field points and wide hysteresis loop areas [17,46]. However, Zr₂RhTl has high coercive field points at a certain temperature (Tch = 350 K) but low coercive field points at other temperatures (Tch ≠ 350 K). Thus, Zr₂RhTl exhibits both soft and hard behaviors. At this point, the Zr₂RhTl Heusler alloy has a maximum coercive field point (H_c^max = 38 T) at T_ch = 350 K. In contrast, the coercive field points decrease at T < T_ch = 350 K and at T > T_ch = 350 K. Thus, Zr₂RhTl exhibits a maximum hardness behavior at T = T_ch and shows soft magnetic behavior at T < T_ch and T > T_ch (see Table S1). Therefore, the hardness of Zr₂RhTl depends on temperature. Similar coercivity maximum behavior versus temperature is observed by Sanchez Llamazares et al. [47]. They found that as the temperature increases, H_C rises and peaks at 1017 Oe around 50 K. Above 50 K, H_C progressively decreases, approaching zero close to 195 K. Still, above this temperature, H_C increases again to 20 Oe, becoming zero close to 308 K. The very different coercivity values measured in both temperature intervals suggest a notable difference in the magnetocrystalline anisotropy of both martensites (i.e., the low-temperature MST shows higher magnetocrystalline anisotropy than the one existing above 195 K) in Ni_51.2Mn_36.8Sn_12.0 ribbons. In Figure 3b, remanence magnetizations (Mr) of the total Zr₂RhTl and its components (Zr1, Zr2, Rh, and Tl) are obtained at 1–1000 K with 25 K steps (see Table S1). The remanent magnetization of Zr1 has the highest value among the other components. On the other hand, the remanent magnetization of Tl has the lowest value. Additionally, Alrahamneh et al. [48] reported that Tc = 536.28 K for Zr₂RhGa, 972 K for Zr₂RhIn, and 607 K for Zr₂RhAl [49]. We find that Tc = 897.43 K for Zr2RhTl [41]. Therefore, Ga and Al have a softness effect, whereas In and Tl have a hardness effect on the Zr2Rh Heusler alloys. Therefore, we suggest that the Zr atoms dominate the magnetic properties of Zr₂RhTl. Using the theoretical and experimental results for the maximum coercivity (maximum hardness) behavior at a certain temperature, we suggest that Zr₂RhTl Heusler alloy is a potential candidate for producing a new class of thermo-hardness sensors (THSs) and actuators (THAs).

4. Conclusions

Using the effective field theory (EFT) developed by Kaneyoshi [22], the magnetic hysteresis properties of the Zr₂RhTl Heusler alloy were analyzed, revealing a peak in the coercive field and maximum hardness at a critical hardness temperature Tch = 350 K. While the alloy and its individual components behave as soft magnets both below and above this threshold, the Zr atoms (Zr1 and Zr2) exhibit higher remanence than Rh and Tl, making them the dominant influence on the system’s magnetic properties. This observation of maximum coercivity at a specific temperature aligns with experimental results for similar systems, such as Co₂FeSn [43], Ni–Co coatings [44], the Ru₂Mn_1−xFe_xGe system [45], and Ni_51.2Mn_36.8Sn_12.0 ribbons [47], suggesting that the unique transition between soft and hard magnetic behaviors in Zr₂RhTl could be leveraged to develop a new class of thermo-hardness sensors (THSs) and actuators (THAs).