Inverse Magnetocaloric Effect in Heusler Ni_44.4Mn_36.2Sn_14.9Cu_4.5 Alloy at Low Temperatures

Abstract

Direct measurements of the magnetocaloric effect were performed in a Heusler Ni_44.4Mn_36.2Sn_14.9Cu_4.5 alloy at cryogenic temperatures in magnetic fields up to 10 T. The maximum value of the inverse magnetocaloric effect in a 10 T field was ∆T_ad = –2.7 K in the vicinity of the first-order magnetostructural phase transition at T₀ = 117 K. Ab initio and Monte Carlo calculations were performed to discuss the effect of Cu doping into a Ni-Mn-Sn compound on the ground-state structural and magnetic properties. It is shown that with increasing Cu content the martensitic transition temperature decreases and the Curie temperature of austenite slightly increases. In general, the calculated transition temperatures and magnetization values correlated well with the experimental ones.

1. Introduction

The hopeful idea of solid-state magnetic cooling (SMC) through the adiabatic demagnetization of paramagnetic Gd salts was used in 1933 to develop novel methods to achieve low temperatures below 1 K [1]. SMC is based on the magnetocaloric effect (MCE), which is a reversible change in the temperature ΔT_ad (under adiabatic conditions) or entropy ΔS_iso (under isothermal conditions) of a magnetic material due to external magnetic field changes [2,3]. The MCE is an efficient apparatus for the investigation of magnetic phase transitions (PTs), which are among the most exigent problems in modern solid-state physics. The maximum MCE is reached in the vicinity of magnetic PTs [3,4]. Magnetic materials with magnetic PTs in the required working temperature range are chosen for SMC. The promising applications of SMC at low temperatures, particularly for the liquefaction of gases such as N₂, He, H₂ or natural gases, have been mentioned in [5,6,7,8]. The advantage of such materials for SMC at low temperatures is connected with the increase in their magnetic heat capacity in the magnetic PT region. It becomes comparable to (and sometimes exceeds) the crystal lattice heat capacity. This fact makes SMC more promising at low temperatures, where the lattice heat capacity of metals is much higher than that at room temperature. It is currently accepted that low (cryogenic) temperatures are those below 120 K [9].

The MCE can be either direct or inverse under the magnetization of a magnetic material [2,3,4]. The direct MCE is an increase in temperature at adiabatic conditions or a decrease in entropy at isothermal conditions. Conversely, the inverse MCE is a positive isothermal change in entropy or a negative adiabatic change in temperature under magnetization. Both the direct and inverse MCE can be applied for SMC at low temperatures. The inverse MCE can be applicable when it is necessary to quickly cool a magnetic field area. For this purpose, a magnetic field is simply applied to the working body. Thus, the working body will cool down, and it will also cool the magnetic field area. This effect can be applied to the cooling of a superconducting magnet and the surrounding liquefied gas, making it possible to reduce the expenditure of liquefied gas.

The rare-earth metals [4] and their intermetallic compounds [10] are the most promising materials for SMC at low temperatures. The magnetostructural PTs in such alloys are commonly observed at temperatures above 120 K [2]. Most often, they show a direct MCE in the vicinity of the PTs. The inverse MCE is observed in their intermetallic compounds with antiferromagnetic (AFM) ordering: RCu₂, R₂Fe₁₇, RFe₃ (R—heavy rare-earth metal) [10,11,12], Gd-based alloys (Gd₂In [13], GdRuSi [14]). The inverse MCE can also be observed as a result of magnetization rotation upon the magnetization of a highly anisotropic single-crystal sample along the hard magnetization axis in, for example, RCo₅ [15] and Tb₂CoMnO₆ [16]. However, the rare-earth metals are expensive, and their compounds are breakable and rapidly oxidize in air, so they are difficult to use in the proposed SMC scheme.

Significant inverse MCE can be observed in several ferromagnetic (FM) Heusler alloys based on Ni–Mn–Z (Z = In, Sn, Sb), but they usually show the effect closer to room temperature [17,18,19,20,21,22]. Some Heusler alloy compositions show the inverse MCE at low temperatures, for example, the Ni(-Co)-Mn-Ti Heusler alloys [23]. In general, the Heusler alloys are cheap to produce, as they consist of inexpensive chemical elements and do not require long-term heat treatment, unlike alloys based on expensive rare-earth elements. Therefore, the goal of our work was to study the inverse MCE at low temperatures in a Heusler composition, which was chosen according to [24]. We investigated a Ni_44.4Mn_36.2Sn_14.9Cu_4.5 Heusler alloy with the inverse MCE at low temperature for SMC and try to explain its properties theoretically in this paper.

2. Materials and Methods

2.1. Samples Characterization

A polycrystalline ingot of the Ni_44.4Mn_36.2Sn_14.9Cu_4.5 Heusler alloy was synthesized via the argon arc-melting method from high-purity elements. The ingot was remelted seven times to eliminate the chemical segregation of the composition. The obtained ingot had a mass of about 80 g. The samples, in the form of 1 mm thick plates, were placed in quartz ampoules, in which a vacuum of about 10 Pa was created. The ampoules were then sealed. Previous research found that the melting point of this alloy is 1313 K [25]; therefore, the temperature of the homogenization annealing was selected as 1133 K. The sealed samples were subjected to homogenization annealing for 24 h, and then they were quenched in water.

The quenched plates were prepared for structure analysis by electropolishing in an electrolyte with 90% n-butyl alcohol (C₄H₉OH) and 10% HCl. X-ray diffraction and X-ray phase analyses were performed on a “Rigaku” Ultima V X-ray diffractometer using Cu-Kα radiation. The crystal structure of the studied sample at room temperature was cubic (space group Fm-3m), with lattice parameter a₀ = 5.991 Å. The analyses of the microstructure and elemental composition were carried out on a “Tescan” Vega 3-SBH scanning electron microscope (SEM) equipped with sensors for backscattered electrons and energy-dispersive X-ray (EDX) analysis on a “Oxford Instruments” X-Act. The specimens were prepared by electropolishing in an electrolyte with 90% n-butyl alcohol (C₄H₉OH) and 10% HCl. The microstructure of the alloy as observed via SEM is represented by a single-phase state in Figure 1a. The dark points on the thin section are etching pits. The grains have a size on the order of 100 μm, according to the orientational contrast. The microstructure can generally be characterized as equiaxed. The unblurred contrast means that the grain boundaries are high-angle. The chemical element maps of the section area obtained by EDX analysis are presented in Figure 1b. The distribution of the Ni, Mn, Sn and Cu is uniform, and there are no areas with the localization of any elements. It can be concluded that annealing at 1133 K for 24 h followed by quenching made it possible to obtain a single-phase-state sample, although this determination is constrained by the limitations of EDX analysis.

The magnetic properties of the Ni_44.4Mn_36.2Sn_14.9Cu_4.5 Heusler alloy samples were studied using standard magnetometry methods (ZFC-FC-FH protocols) using a “Quantum Design” MPMS-7T SQUID magnetometer in a low magnetic field of 10 mT (inset in Figure 2a) and in fields of 1 T and 3 T in the wide temperature range of 2–400 K (Figure 2a). The alloy demonstrated two magnetic PTs of first and second order. The 1st-order magnetostructural PT from a ferrimagnetic (FiM) martensitic phase to an FM austenitic phase was observed at the cryogenic temperatures. The characteristic start and finish temperatures of the martensite and austenite states obtained via the tangential method are denoted as M_S = 101 K, M_F = 54 K, A_S = 70 K, and A_F = 118 K, respectively. A temperature hysteresis of about 20 K was observed during the heating/cooling process (inset in Figure 2a). The application of high magnetic fields shifted the region of the martensitic phase to lower temperatures at the rate of –2.2 K/T (Figure 2a), which made it possible to realize the inverse MCE in this alloy in relatively low magnetic fields at cryogenic temperatures. The 2nd order magnetic PT was at the Curie temperature, T_C ≈ 350 K.

Figure 2b shows the magnetic field dependence of the magnetization of the Ni_45.3Mn_35.9Sn_14.3Cu_4.5 Heusler alloy at different temperatures in the vicinity of the magnetostructural PT from 50 to 130 K, with increasing and decreasing magnetic field strengths. Measurements of the field dependences of magnetization were carried out using a self-build vibrating magnetometer with a stepper motor [26] in a Bitter magnet field up to 10 T. The sample temperature was established and maintained during each measurement. Before each measurement, the sample was cooled down to 4.2 K without a magnetic field, i.e., below the M_F temperature. This was carried out to remove residual austenite phase that may have formed as a result of the previous magnetization/demagnetization processes.

2.2. Computational Methods

In order to study the role of Cu addition on the structural and magnetic properties of Ni-Mn-Sn alloy, we also carried out theoretical studies within the framework of ab initio calculations and Monte Carlo (MC) simulations.

First-principles calculations of the ground-state properties of the austenitic and martensitic phases of Cu-doped Ni-Mn-Sn alloys were performed using the projector augmented wave (PAW) method implemented in the Vienna Ab initio Simulation Package (VASP ver. 6.2) [27,28]. The generalized gradient approximation in the formulation of Perdew, Burke, and Ernzerhof [29] was chosen to calculate the exchange-correlation energy. Electron–ion interactions are described using PAW potentials with the electronic configurations as follows: 3p⁶3d⁸4s² for Ni, 3p⁶3d⁶4s¹ for Mn, 3d¹⁰4s²4p² for Sn, and 3p⁶3d¹⁰4s¹ for Cu. The cutoff energy of plane waves was 450 eV. The Brillouin zone was sampled using a 5 × 5 × 4 mesh of k points centered at the Gamma point. The convergence criterion for the total energy was 10⁻⁶ eV/at. A 2 × 2 × 2 supercell of 64 atoms with an 8-atom tetragonal unit cell (L1₀, I4/mmm, #139) was considered.

The geometric optimization procedure was carried out for the compounds Ni_32−xCu_xMn₂₃Sn₉ (x = 1, 2, and 3), which correspond to Ni_50−xCu_xMn_35.94Sn_14.06 (x = 1.56, 3.12 and 4.69 at.%), where the latter is close to the experimental Ni_44.4Cu_4.5Mn_36.2Sn_14.9. To form Ni_50−xCu_xMn_35.94Sn_14.06 (x = 1.56, 3.12 and 4.69 at.%), the Mn₂ and Cu atoms were randomly distributed at Sn and Ni sites. Two magnetic configurations were considered: FM, where the Mn₁, Mn₂ and Ni magnetic moments were aligned in parallel, and FiM, where the magnetic moments of Mn₂ atoms were reversed with respect to the Mn₁ and Ni magnetic moments. Here Mn₁ and Mn₂ are Mn atoms located in the regular Mn sublattice and Sn sublattice, respectively.

The spin-polarized relativistic Korringa–Kohn–Rostoker package (SPR-KKR ver. 7.7) implemented in [30] was applied for calculation of the magnetic exchange coupling constants in the austenitic and martensitic phases of Ni-Cu-Mn-Sn. The formation of non-stoichiometric compositions was performed in the coherent potential approximation. Brillouin zone integration was performed using a special point method on a 57 × 57 × 57 k-grid (4495 k-points) to calculate self-consistent potentials and exchange interaction integrals, respectively. The energy convergence threshold was set to 0.01 mRy.

The temperature dependencies of the magnetization were modeled using the MC method, the Heisenberg Hamiltonian and Metropolis algorithm [31]. The model lattice with periodic boundary conditions consisted of 5488 atoms (for Ni₂MnSn: 2744 Ni atoms, 1372 Mn and Ga atoms) and was obtained by multiplying the 16-atom unit cell 7 × 7 × 7 times. To form Ni_50−xCu_xMn_35.94Sn_14.06 (x = 1.56, 3.12 and 4.69 at.%), the Mn₂ and Cu atoms were randomly distributed at Sn and Ni sites. The number of MC steps per temperature value was 1 × 10⁵. To achieve thermal equilibrium in the system, the first 10⁴ steps of the MC were discarded. Magnetization averaging was performed for 225 configurations for every 400 MC steps.

3. Results

3.1. Indirect MCE Estimation

The isothermal change of magnetic entropy ΔS_iso was obtained from Maxwell’s thermodynamic relations (1) using magnetization curves with increasing field from 0 to 10 T, and is presented in Figure 2b.

$$\mathsf{\Delta }{S}_{iso}={{\displaystyle \int }}_0^{\mathrm{H}}{\left(\frac{\partial M\left(T, H\right)}{\partial T}\right)}_{H}dH.$$

(1)

The maximum value of the magnetic entropy change was ΔS_iso = 9.5 J/(kg K) at T = 113 K in μ₀H = 10 T (Figure 3a). The ΔS_iso values obtained in the vicinity of the hysteresis of the magnetostructural PT are irreversible because reversible MCE values in this region can be obtained by demagnetizing the sample or by turning the magnetic field on again.

The isothermal heat ∆Q is also an important parameter of the working body of a magnetic refrigerator; it allows one to estimate the amount of heat that can be taken from a cooled bath as a result of one ideal cycle of isothermal magnetization/demagnetization processes. Figure 3b shows the ΔQ values calculated from the ΔS_iso values using Equation (2):

∆Q = −T∆S_iso.

(2)

The maximum value was ΔQ = 1.06 kJ/kg at T = 113 K in a magnetic field of 10 T, which is an order of magnitude lower than the maximum known value for MnAs at room temperature in the same magnetic field [32].

The presence of field hysteresis (Figure 2a) in the vicinity of the magnetostructural PT in the sample’s magnetization/demagnetization cycle leads to irreversible heat release δQ, which was calculated using Equation (3):

$$\partial Q={\mu }_0{{\displaystyle \oint }}^{ }HdM.$$

(3)

Figure 3b shows the δQ values calculated for different temperatures with the maximal value δQ = 0.09 kJ/kg at T = 110 K. This value is only 8.5% of the maximal ΔQ value, which makes the Heusler Ni_44.4Mn_36.2Sn_14.9Cu_4.5 alloy interesting for use as a magnetocaloric working body in the temperature range of natural gas liquefaction [2], despite the presence of field hysteresis in the magnetostructural PT.

3.2. Direct MCE Measurements

The extraction method in a Bitter magnet field of up to 10 T was used for direct measurement of the MCE in the Heusler Ni_44.4Mn_36.2Sn_14.9Cu_4.5 alloy under adiabatic conditions ∆T_ad. This method was described in detail in [33]. The Bitter magnet produced a steady magnetic field. The temperature change of the sample due to MCE was measured using a differential chromel–gold thermocouple with an accuracy of 0.05 K. The temperature was controlled using a Lake Shore thermo-controller. The sample was placed within a thermal screen to minimize heat exchange with the environment. Movement of the sample to and from the region of the maximum magnetic field was forced with the help of a special actuator. The sample was moved for a duration of 1 s from the center of the Bitter magnet to the outside (or vice versa) at a distance of 35 cm. It should be noted that the magnetic field values before magnetization and after demagnetization were less than 2% of the maximum field in the center of the Bitter coil.

The studies were carried out in two different regimes: (1) sequential heating of the sample, (2) thermal cycling of the sample. In the first case, the sample temperature was successively raised and ∆T_ad was measured at selected temperatures. In the second case, the sample was pre-cooled to 4.2 K, and then its temperature was raised to the required temperature for measurement. Figure 4 shows the results of measurements of the ∆T_ad value in magnetic fields of 1.8 T and 10 T: closed circles indicate sequential heating and open circles indicate thermal cycling. The inverse MCE was observed in the low-temperature region in a field of 1.8 T in both measurement regimes (Figure 4). The maximum value of the inverse MCE in a field of 1.8 T was observed in the vicinity of the 1st order PT, with a value of ∆T_ad = –0.5 K at an initial temperature T₀ = 117 K (Figure 4). The inverse MCE increased with increasing magnetic field up to 10 T, reaching ∆T_ad = –2.7 K at T₀ = 117 K (Figure 4). It is interesting that direct MCE in a magnetic field of 10 T was observed in the FiM martensite phase and reached ∆T_ad = 0.7 K (Figure 4) in the sequential heating regime in the temperature range of 20–60 K.

3.3. Computational Results

We will now discuss the results of the geometric optimization of the crystal structures of the austenitic and martensitic phases as well as the calculation of the magnetic properties of Heusler Ni_50−xCu_xMn_35.94Sn_14.06 alloys.

Figure 5 shows the energy landscape E(c/a) of Ni_50−xCu_xMn_35.94Sn_14.06, including the response to volume-conserving elongations and compressions of the cubic L2₁ structure along the c axis. The results are shown for both FM and FiM solutions (Figure 5a). It can be seen that the FM solution exhibits only one global cubic minimum at c/a = 1, which is 11 meV/atom higher in energy than observed for FiM, indicating an instability of the FM tetragonal phase. For all compounds, the FiM ordering was energetically preferable to the FM ordering for both cubic austenitic and tetragonal martensitic phases. The global minimum for the FiM tetragonal phase took place at around c/a = 1.25.

A close look at the FiM E(c/a) curves shown in Figure 5b reveals that the c/a ratio of the tetragonal phase varied slightly from 1.25 to 1.27 with increasing Cu content due to a reduction (increase) in the tetragonal lattice constant a_t (c), respectively (see

Table 1. Optimized lattice parameters and Debye temperatures of cubic and tetragonal structures of Ni_50−xCu_xMn_35.94Sn_14.06. The Debye temperatures were calculated from elastic moduli.

Phase	Parameters	x = 0	x = 1.56	x = 3.12	x = 4.69
FiM cubic (c/a = 1)	a0 (Å)	5.953	5.955	5.959	5.963
	${\Theta }_D$ (K)	316	308	293	282
FiM tetragonal (c/a = 1.27)	at (Å)	5.526	5.513	5.502	5.500
	c (Å)	6.908	6.947	6.988	6.994
	c/a	1.25	1.26	1.27	1.27
	${\Theta }_D$ (K)	333	325	308	295

). In addition, the optimized cubic lattice constant increased slightly with the addition of Cu. This is explained by the slightly larger atomic radius of Cu (r = 1.28 Å) compared to that of Ni (r = 1.24 Å). It should be noted that partial substitution of Ni with Cu revealed an almost linear decrease in the energy difference ($\Delta E_{L{2}_1-L{1}_0}$) between the FIM cubic and tetragonal structure, as is evident from Figure 5b. For the parent compound with x = 0, $\Delta E_{L{2}_1-L{1}_0}$ was calculated to be 7.145 meV/atom, whereas in the case of x = 4.69 at.%, it reduced to 3.89 meV/atom. The decrease in the energy barrier with the Cu content indirectly indicates the reduction in the martensitic transition temperature T_m. The T_m temperature can be estimated from a rough approximation: $\Delta E_{L{2}_1-L{1}_0}\approx k_B{T}_m$, where k_B is the Boltzmann constant. According to this expression, T_m ≈ 83 K for Ni₅₀Mn_35.94Sn_14.06 and 45 K for Ni_45.31Cu_4.69Mn_35.94Sn_14.06.

A more correct way to estimate the martensitic transition temperature is to calculate the free energies of the austenitic and martensitic phases. For simplicity of calculation, we only considered the lattice contribution to the free energy, which at low temperatures plays a predominant role compared to the electronic and magnetic contributions:

$$F\left(T,V\right)=E\left(V\right)+F_{lat}\left(T,V\right),$$

(4)

where E(V) is the ground-state energy calculated at T = 0 K, and F_lat(T,V) is the lattice contribution calculated within the Debye model [34].

In Figure 6, we illustrate the free energies of L2₁ cubic and L1₀ tetragonal phases as well as the free-energy difference $\Delta F=F_{L{2}_1}-F_{L{1}_0}$ for the Ni_45.31Cu_4.69Mn_35.94Sn_14.06 alloy as an example. For $\Delta F$> 0, the L1₀ phase (martensite) is preferable and, vice versa, the L2₁ phase (austenite) is stable for $\Delta F$< 0. As is evident from the figure, both free-energy curves reveal a non-linear behavior with temperature and intersect with each other at low temperatures. The martensitic transition temperature extracted from $\Delta F$ = 0 is about 88 K, which agrees well with the experimental value (85.75 K) for Ni_44.4Mn_36.2Sn_14.9Cu_4.5, where T_m was computed via T_m = (M_s + M_f + A_f + A_s)/4. The low T_m temperature suggests that the zero-point vibrational energy ($\frac{9}{8}{k}_B{\Theta }_D$) plays a predominant role as compared to a vibrational entropy in the F_lat term. Thus, T_m is mainly contributed by $\Delta E_{L{2}_1-L{1}_0}$ at T = 0 K and $\Delta {\Theta }_D^{L{2}_1-L{1}_0}$ at T > 0 K. For Ni_45.31Cu_4.69Mn_35.94Sn_14.06, $\Delta {\Theta }_D^{L{2}_1-L{1}_0}$ is 13 K in absolute value. We would like to note that T_m is sensitive to $\Delta {\Theta }_D^{L{2}_1-L{1}_0}$, and an increase in $\Delta {\Theta }_D^{L{2}_1-L{1}_0}$ (${\Theta }_D^{L{2}_1}$ < ${\Theta }_D^{L{1}_0}$) leads to a reduction in T_m; the austenitic phase becomes stable in the whole temperature range at a $\Delta {\Theta }_D^{L{2}_1-L{1}_0}$ of about 30 K in absolute value. On the other hand, an increase in $\Delta {\Theta }_D^{L{2}_1-L{1}_0}$ (${\Theta }_D^{L{2}_1}$ > ${\Theta }_D^{L{1}_0}$) shifts T_m to higher temperatures.

Figure 7 shows the pairwise exchange coupling constants J_ij as function of the distance between i and j atoms for FiM-L2₁ and FiM-L1₀ Ni_45.31Cu_4.69Mn_35.94Sn_14.06 alloy as an example. The magnetic interactions reveal a similar behavior for all compounds under study because Cu atoms are nonmagnetic. For both phases, intra-sublattice J_ij constants between Mn atoms (Mn₁-Mn₁ and Mn₂-Mn₂) show similar damped oscillatory behavior as a function of the distance between atoms up to d/a ≈ 2, with the exception of the interactions for the three coordination shells of cubic structure. In this case, J_ij (Mn₁-Mn₁) demonstrates a reducing FM character, whereas J_ij (Mn₂-Mn₂) within the first and second coordination shells exhibits a reducing AFM character.

The cubic inter-sublattice interactions between the nearest Mn₁-Mn₂ atoms located at a smaller distance (d/a = 0.5) compared to the nearest pairs Mn₁₍₂₎-Mn₁₍₂₎ (d/a =$\surd 2/2$) are characterized by a strong AFM interaction, which gradually decreases with increasing d/a. In contrast to the cubic structure, the strongest AFM exchange (≈–20 meV) between the four nearest atoms Mn₁-Mn₂ located in the (110) plane is found for the tetragonal structure due to the smallest distance (d/a = 0.5), whereas the two next nearest atoms Mn₁-Mn₂ (d/a ≈ 0.627) exhibit a strong ferromagnetic interaction with J_ij ≈ 11.5 meV. The Mn₁₍₂₎-Ni interactions show a similar behavior for both phases, demonstrating the FM exchange only between the nearest neighbors. Generally, the behavior of J_ij coupling constants is similar to those that were previously reported for Ni-Mn-Sn alloys [35].

Let us make a general remark concerning the behavior of the exchange interaction parameters for the remaining compounds in the cubic and tetragonal phases. Since the substitution of Ni with Cu influences a small change in the crystal structure parameters for both phases, it will also weakly affect the exchange constants between the nearest pairs of Mn₁–Ni and Mn₂–Ni (see

Table 2. The nearest exchange coupling constants and Curie temperatures calculated within the MC method and MFA for cubic and tetragonal structures of Heusler Ni_50−xCu_xMn_35.94Sn_14.06 alloys.

x	Mn1-Ni	Mn2-Ni	Mn1-Mn1	Mn2-Mn2	Mn1-Mn2	TC (MC)	TC (MFA)
FiM cubic structure
0	2.961	2.341	1.082	−0.936	−6.065	425	423.4
1.56	2.977	2.364	1.201	−0.837	−5.925	430	433.5
3.12	2.988	2.384	1.492	−0.772	−5.859	452	444.8
4.69	2.999	2.402	1.702	−0.678	−5.722	464	453.1
FiM tetragonal structure
0	3.514	2.308	2.035	1.859	−21.902	484	544.9
1.56	3.502	2.299	1.985	1.826	−21.513	506	547.5
3.12	3.495	2.294	1.942	1.798	−21.071	514	549.6
4.69	3.489	2.287	1.906	1.779	−20.598	523	551.4

). For the cubic structure, a slight enhancement of the FM interactions J_{Mn1(2)-Mn1(2)} and J_Mn1(2)-Ni and a weakening of the AFM interaction J_Mn1-Mn2 with increasing Cu content is observed, which also affects the Curie temperature of austenite (T_C^A). In the case of the tetragonal structure, a weakening of FM and AFM interactions between Ni, Mn₁ and Mn₂ atoms is observed. However, the Curie temperature of martensite, T_C^M, increases similarly to T_C^A. The increase in T_C^M is caused by the weakening of the strong AFM interaction between the nearest Mn₁-Mn₂, the change of which has a larger contribution to the magnetic energy compared to the change of J_{Mn1(2)-Mn1(2)} and J_Mn1(2)-Ni interactions as a function of Cu content.

Figure 8a shows the T–x phase diagram for Ni_50−xCu_xMn_35.94Sn_14.06, including the predicted T_m and T_C temperatures and the experimentally measured values of T_m and T_C^A. With increasing Cu doping level, T_m reduces nonlinearly from 133 K (x = 0 at.%) to 88 K (x = 4.69 at.%). This is mainly due to a decrease in the energy barrier $\Delta E_{L{2}_1-L{1}_0}$ and $\Delta {\Theta }_D^{L{2}_1-L{1}_0}$, both of which have maxima at x = 0 (see Figure 5 and Table 1). The predicted value of T_m for Ni_45.31Cu_4.69Mn_35.94Sn_14.06 is close to the experimental one for Ni_44.4Cu_4.5Mn_36.2Sn_14.9. As for the Curie temperature, MC simulations and mean-field approximation (MFA) show a similar T_C^A trend with the increase in Cu content. Nevertheless, the experimental value of T_C^A for Ni_44.4Cu_4.5Mn_36.2Sn_14.9 is less than the theoretical one by about 100 K.

4. Discussion

Let us next discuss the calculated total magnetic moment ${\mu }_{tot}$ for Ni_45.31Cu_4.69Mn_35.94Sn_14.06 and its relation to the experimentally obtained moment for Ni_44.4Cu_4.5Mn_36.2Sn_14.9. It follows from Figure 6 that the magnetic reference state for both austenite and martensite is an FiM one, where all magnetic moments of ${\mathrm{Mn}}_1^{\uparrow }$ and ${\mathrm{Mn}}_2^{\downarrow }$ are oppositely aligned. As a consequence, the ${\mu }_{tot}$ is calculated to be almost equal 2.16 and 2.12 ${\mu }_B$ or 45.6 and 44.8 emu/g, respectively. The latter value is close to the experimental magnetization of martensite (≈53 emu/g). However, there is some discrepancy between the calculated and measured values of ${\mu }_{tot}$ for austenite. The experiment yielded a ${\mu }_{tot}$ value that was double the calculated value (90 emu/g—dashed line in Figure 8b). On the other hand, the calculated value of ${\mu }_{tot}$ for FM austenite, which is energetically unfavorable, is about 5.94${\mu }_B$ or 125.6 emu/g larger than the experimental one. The dashed line in Figure 2a is the calculated curve (approximation) of the temperature dependence of the magnetization in μ₀H = 3 T for the sample in the austenite phase without a magnetostructural PT. This curve was calculated using the approach proposed in [36]. The magnetization value obtained using this approach is 110 emu/g (Figure 2a). This is the estimation of the maximal magnetization of the austenitic phase “without martensitic transition”, and it is very close to the calculated value.

To explain the observed discrepancy, we suggest that not all Mn₂ atoms located at the Sn sublattice have an antiparallel magnetic moment with respect to Mn₁ atoms located at the regular Mn sublattice. To prove this, we additionally performed CPA calculations of ${\mu }_{tot}$ for L2₁-Ni_45.31Cu_4.69Mn_35.94Sn_14.06 (or ${\mathrm{Ni}}_{45.31}{\mathrm{Cu}}_{4.69}{\mathrm{Mn}}_{25}^{\uparrow }{\mathrm{Mn}}_{10.94-x}^{\downarrow }{\mathrm{Mn}}_x^{\uparrow }{\mathrm{Sn}}_{14.06}$) using SPR-KKR code as a function of Mn₂ atoms at Sn sites with a parallel magnetic moment to that of Mn₁ atoms, as shown in Figure 8b. As can be seen from the figure, ${\mu }_{tot}$ increases linearly with the fraction of ${\mathrm{Mn}}_2^{\uparrow }$ atoms and reaches 5.94 ${\mu }_B$ or 125.6 emu/g for 100% of the ${\mathrm{Mn}}_2^{\uparrow }$ spins aligned parallel to the ${\mathrm{Mn}}_1^{\uparrow }$ ones (FM order). We found that the experimental value of ${\mu }_{tot}$ can be reached theoretically if only ≈40% of the Mn₂ atoms interact antiferromagnetically with the Mn₁ atoms. This finding allows us to conclude that the experimental sample had some kind of FiM order, with a predominance of FM-ordered Mn atoms.

The inverse MCE at low temperatures in the Heusler Ni_33.7Co_14.8Mn_35.4Ti_16.1 alloy showed similar values at low temperatures [23]: maximal ∆T_ad = –2.7 K at T₀ = 90 K in a magnetic field of 20 T. This may be a physical limit on the value of the inverse MCE for Heusler alloys with PTs at low temperatures, which is significantly lower than in known Heusler alloys with the inverse MCE at room temperature, for example [37]. The further study of medium-entropy [38,39,40] and high-entropy [41,42,43] alloys that do not contain rare-earth elements will reveal compositions with high inverse MCE for application in low-temperature SMC.

5. Conclusions

The inverse MCE in a Heusler Ni_44.4Mn_36.2Sn_14.9Cu_4.5 alloy in the low-temperature region was observed in a magnetic field of 1.8 T in both measurement regimes: sequential heating and thermal cycling. The maximum value of the inverse MCE in the vicinity of the first-order PT in a magnetic field of 1.8 T was observed to be ∆T_ad = –0.5 K at an initial temperature T₀ = 117 K. The inverse MCE reached up to ∆T_ad = –2.7 K at T₀ = 117 K with the increase in the magnetic field up to 10 T. The direct MCE in the FiM martensite phase was observed in a magnetic field of 10 T and reached the maximal value of ∆T_ad = 0.7 K in the sequential heating regime in the temperature range of 20–60 K.

First-principles and MC approaches were applied to determine the phase diagram for FM-ordered Ni_50−xCu_xMn_35.94Sn_14.06 (x = 0, 1.56, 3.12 and 4.69) compounds, which were similar in composition to the experimental sample. Cu doping led to a decrease in the T_m temperature and an increase in the T_C^A temperature. This was mainly due to the slight change in the optimized lattice constants, magnetic exchange interactions and Debye temperatures for austenite and martensite. From the analysis of our computational results, we suggest that the experimental sample exhibited some type of FiM order in the austenite, with a predominance of ferromagnetically ordered Mn atoms. This theory was satisfactorily confirmed by our experiment, which will make it possible to predict the transition temperatures, magnetizations and MCE properties of Heusler Ni-Mn-Sn family alloys when a portion of the Sn atoms are replaced with Cu.