Inverse magnetocaloric effect in Heusler Ni44.4Mn36.2Sn14.9Cu4.5 alloy at low temperatures

1. Introduction

The promising idea of solid-state magnetic cooling (SMC) by means of adiabatic demagnetization of paramagnetic salts was used in the early 1930s to develop new methods to obtain the low temperatures below 1 K [1]. The SMC is based on the magnetocaloric effect (MCE), which is a reversible change in an entropy ΔS_iso under isothermal conditions or in a temperature ΔT_ad under adiabatic conditions of a magnetic material due to the external magnetic field changes [2,3]. The MCE is an efficient tool in the study of magnetic phase transitions (PTs), which is among the most urgent problems in modern solid-state physics. The maximum of the MCE is reached in the vicinity of magnetic PTs [3,4]. The magnetic materials with magnetic PTs in the required range of working temperatures are chosen for SMC. The promising applications of the SMC at low temperatures in particular for the liquefaction of gases such as N₂, He, H₂ or natural gases have been mentioned in [5,6]. The advantages of such materials for SMC at low temperatures are related to the fact that the magnetic heat capacity increases in the region of the magnetic PT and becomes comparable (and sometimes exceeds) to the heat capacity of the crystal lattice of a solid. This circumstance makes SMC more promising at low temperatures, where the lattice heat capacity of metals is much higher than that at room temperature. It is commonly accepted at present that low (cryogenic) temperatures are temperatures below 120 K [7].

The MCE can be either direct or inverse under a magnetization of a magnetic material [2,3,4]. The direct MCE is a decrease in entropy at isothermal conditions or an increase in temperature at adiabatic conditions. Against, the inverse MCE is a positive isothermal change in entropy or a negative adiabatic change in temperature under magnetization. Both direct and inverse MCE can be used for SMC at low temperatures. The inverse MCE can be useful when it is necessary to rapidly cool an area in the presence of magnetic field., It is sufficed to exert magnetic field on the refrigerant for this purpose. Thus, the refrigerant will cool down, and it will also cool the area of magnetic field. The effect can be used to cool a superconducting magnet and the surrounding liquefied gas [8], which will make it possible to reduce the consumption of liquefied gas.

The rare-earth metals [4] and their intermetallic compounds [9] are the most promising materials for SMC at low temperatures. The magnetostructural PTs in such alloys are normally 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) [9,10], Gd₂In [11]. The inverse MCE can also be observed as a result of magnetization rotation upon magnetization of a highly anisotropic single-crystal sample along the hard magnetization axis in RCo₅ and Tb–Gd [12,13]. However, the rare earth metals are expensive, and their compounds are breakable and rapidly oxidize on the air, so they are difficult to use in the proposed scheme of the SMC.

Inverse MCE at low temperatures can also be exhibited by Mn-based compounds, for example in Mn₅Si₃ [14]. Significant inverse MCE can be observed in several ferromagnetic (FM) Heusler alloys based on Ni–Mn–Z (Z = In, Sb, Sn), but usually they show it closer to the room temperatures [15,16,17,18]. Some compositions of the Heusler alloys show the inverse MCE at low temperatures, for example the Ni(-Co)-Mn-Ti Heusler alloys [19]. We investigated the Ni_44.4Mn_36.2Sn_14.9Cu_4.5 Heusler alloy with the inverse MCE at low temperature perspective for the SMC and tried to explain its properties theoretically in this paper.

2. Materials and Methods

2.1. Samples Characterization

Initially, the composition of the Heusler alloy was chosen according to [20]. Polycrystalline ingot of the Ni_44.4Mn_36.2Sn_14.9Cu_4.5 Heusler alloy was synthesized by the argon arc-melting method from high purity elements. The ingot was remelted seven times for the elimination the chemical segregation of the composition. The obtained ingot had the mass of about 80 g. The samples in the form of the 1 mm thick plates were cut and sealed into evacuated quartz ampoules. As is shown earlier the melting point this alloy is 1313 K [21] due to the temperature of the homogenization annealing was selected as 1133 K. The sealed samples were subjected to homogenization annealing for 24 hours, after that they were quenched in water.

The X-ray diffraction and the X-ray phase analysis was performed on a Rigaku Ultima V X-ray diffractometer using Cu-Kα radiation. The crystal structure of the studied sample at room temperature is cubic (space group Fm-3m) with lattice parameter a₀ = 5.991 A.The analysis of the microstructure and elemental composition were carried out on a Vega 3-SBH Tescan scanning electron microscope (SEM) equipped with sensors for the backscattered electrons and the energy-dispersive X-ray (EDX) analysis (Oxford Instruments). The specimens were prepared by electropolishing in an electrolyte with 90 % n-butyl alcohol (C₄H₉OH) and 10 % HCl. The microstructure of the alloy by SEM is represented by a single-phase state and is shown in Figure 1a. The dark points on the thin section are etching pits. The grains have size in order of 100 μm according to the orientational contrast. The microstructure can be characterized as an equiaxed one in general. The not blurred contrast means that the grain boundaries are high-angle ones. The chemical elements maps of the section area are obtained by EDX analysis and presented in Figure 1b. The distribution of the Ni, Mn, Sn and Cu is uniform, and the areas with localization of the any elements are absent. It can be concluded, that annealing at 1133 K for 24 hours followed by quenching makes possibly to obtain a single-phase state of the sample so within the possibility of EDX analysis.

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

Magnetic field dependence of the magnetization of the Ni_45.3Mn_35.9Sn_14.3Cu_4.5 Heusler alloy for different temperatures in the vicinity of the magnetostructural PT from 50 to 130 K with increasing and decreasing of the magnetic fields are shown in Figure 2b. The results were obtained in Bitter magnet’s magnetic fields up to 10 T by using an original technique detail described in [22].

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 in 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 VASP package [23,24]. The generalized gradient approximation in the formulation of Perdue, Burke, and Ernzerhof [25] was chosen to calculate the exchange-correlation energy. Electron-ion interactions are described by 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 by a 5×5×4 mesh of k points centered at Gamma point. The convergence criteria for the total energy is 10^-6 eV/at. A 2×2×2 supercell of 64 atoms with 8 atom tetragonal unit cell (L1₀, Im3 ̅m, #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 и 4.69 at.%), where the latter is close to the experimental Ni_44.4Cu_4.5Mn_36.2Sn_14.9. Two magnetic configurations were considered: FM, where the Mn₁, Mn₂ and Ni magnetic moments are parallelly aligned, and FiM, where magnetic moments of Mn₂ atoms are 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 code implemented in the SPR-KKR package [26] was applied for calculating of the magnetic exchange coupling constants in austenitic and martensitic phase 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 [27]. The model lattice with periodic boundary conditions is consisted of 5488 atoms (for Ni₂MnSn: 2744 Ni atoms, 1372 Mn and Ga atoms) and is obtained by multiplying 7×7×7 times the 16-atom unit cell. To form Ni_50-xCu_xMn_35.94Sn_14.06 (x = 1.56, 3.12 и 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) by using the magnetization curves with increasing of field from 0 to 10 T and presented in Figure 2b.

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

(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 hysteresis of the magnetostructural PT are irreversible, because reversible MCE values in this region can be obtained by demagnetizing of the sample or by turning on the magnetic field again.

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

∆Q = –T∆S_iso,

(2)

The maximum value was ΔQ = 1.06 kJ/kg at T = 113 K in magnetic field of 10 T, which is an order of magnitude lower than the maximum known value for the MnAs compound at room temperature in the same magnetic field [28].

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 by using the Equation (3)

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

(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 ΔQ maximal value, which is made by the Heusler Ni_44.4Mn_36.2Sn_14.9Cu_4.5 alloy is interesting for using as a magnetocaloric working body in the temperature range of natural gas liquefaction [8], despite the presence of field hysteresis of the magnetostructural PT.

3.2. Direct MCE Measurements

The extraction method in Bitter magnet’s field up to 10 T was used for direct measurements of the MCE in the Heusler Ni_44.4Mn_36.2Sn_14.9Cu_4.5 alloy under adiabatic conditions ∆T_ad. This method in details was described in [29]. 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 down 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 ∆Т_ad value in magnetic fields of 1.8 T and 10 T: closed circles – sequential heating, open circles – thermal cycling.

The inverse MCE is observed in the region of low temperatures in a field of 1.8 T in both regimes of measuring (Figure 4). The maximum value of the inverse MCE in a field of 1.8 T was observed in the vicinity of the 1^st order PT is ∆T_ad = –0.5 K at an initial temperature T₀ = 117 K (Figure 4). The inverse MCE increases with increasing of the magnetic field up to 10 T, reaching the value ∆T_ad = –2.7 K at T₀ = 117 K (Figure 4). It is interesting that direct MCE in a magnetic field of 10 T is observed in the FiM martensite phase and reaching the value ∆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 proceed to discuss the results of geometric optimization of the crystal structures of the austenitic and martensitic phases as well as the calculation of magnetic properties for 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 with response to volume-conserving elongations and compressions of the cubic L2₁ structure along c axis. The results are shown for both FM and FiM solutions (Figure 5a). One can see that the FM solution exhibits only one global cubic minimum at c/a = 1 being 11 meV/atom higher in energy than the FiM one and indicating an instability of FM tetragonal phase. For all compounds, the FiM ordering is energetically preferable compared to the FM one for both cubic austenitic and tetragonal martensitic phases. The global minimum for the FiM tetragonal phase takes place 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 tetragonal phase varies slightly from 1.25 to 1.27 with increasing Cu content due to a reduce (an increase) of tetragonal lattice constant a_t (c), respectively (see Table 1). In addition, it is found that the optimized cubic lattice constant increases slightly with the addition of Cu. This is explained by the slightly larger atomic radius of Cu (r = 1.28 Å) compared to Ni (r = 1.24 Å). It should be noted that partial substitution of Ni by Cu reveals an almost linear decrease in the energy difference ($∆E_{{L2}_1-{L1}_0}$) between the FIM cubic and tetragonal structure, as evident from Figure 5b. For the parent compound with x = 0, $∆E_{{L2}_1-{L1}_0}$ is calculated to be 7.145 meV/atom, whereas in the case of x = 4.69 at.%, it reduces to 3.89 meV/atom. The decrease in the energy barrier with the Cu content indicates indirectly the reduce of the martensitic transition temperature T_m. The T_m temperature can be estimated from a rough approximation: $∆E_{{L2}_1-{L1}_0}\approx {k_{B}T}_m$ 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 calculations, we considered the lattice contribution to the free energy only, 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, F_lat(T,V) is the lattice contribution calculated within the Debye model [30].

In Figure 6, we illustrate the free energies of L2₁ cubic and L1₀ tetragonal phases as well as the free energy difference $∆F=F_{{L2}_1}-F_{{L1}_0}$ for Ni_45.31Cu_4.69Mn_35.94Sn_14.06 alloy as an example. For $∆F$> 0, the L1₀ phase (matensite) is more preferable and vica verse for $∆F$< 0, the L2₁ phase (austenite) is stable. As evident from the figure, both free energy curves reveal a non-linear behavior with a temperature and intersect with each other at low temperature. The martensitic transition temperature extracted from $∆F$ = 0 is about 88 K, which agrees well with the experimental one (85.75 K) for Ni_44.4Mn_36.2Sn_14.9Cu_4.5, where T_m is 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 $∆E_{{L2}_1-{L1}_0}$ at T = 0 K and ${∆\Theta }_D^{{L2}_1-{L1}_0}$ at T > 0 K. For Ni_45.31Cu_4.69Mn_35.94Sn_14.06, ${∆\Theta }_D^{{L2}_1-{L1}_0}$ is 13 K in absolute value. We would like to note that T_m is sensitive to ${∆\Theta }_D^{{L2}_1-{L1}_0}$, and an increase in ${∆\Theta }_D^{{L2}_1-{L1}_0}$ (${\Theta }_D^{{L2}_1}$<${\Theta }_D^{{L1}_0}$) leads to a reduce in T_m, the austenitic phase becomes stable in the whole temperature range at ${∆\Theta }_D^{{L2}_1-{L1}_0}$ is about 30 K in absolute value. On the hand, an increase in ${∆\Theta }_D^{{L2}_1-{L1}_0}$ (${\Theta }_D^{{L2}_1}$>${\Theta }_D^{{L1}_0}$) shifts T_m to higher temperatures.

Figure 7 shows the pairwise exchange coupling constants J_ij as function of 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. We would like to note that magnetic interactions reveal a similar behavior for all compounds under study since 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, except that J_ij (Mn₂-Mn₂) within the 1^st and the 2^nd coordination shell of the cubic structure exhibits the 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 =$\sqrt[]{2}/2$) are characterized by a strong AFM interaction, which gradually decreases with increasing d/a. In contrast to cubic structure, the strongest AFM exchange (≈ - 20 meV) between the four nearest atoms Mn₁-Mn₂ located in the (110) plane is found for tetragonal structure due to the smallest distance (d/a = 0.5), whereas two next nearest atoms Mn₁-Mn₂ (d/a ≈ 0.627) strongly interact ferromagnetically with J_ij ≈ 11.5 meV. As for the Mn₁₍₂₎-Ni interactions, they show similar behavior for both phases, demonstrating the FM exchange only between the nearest neighbors. Generally, the behavior of J_ij coupling constants is a similar to those of Ni-Mn-(Ga, In, Sb) alloys reported previously [31,32,33,34].

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 by Cu influences a small change in the parameters of crystal structures for both phases, it will also affect weakly the exchange constants between the nearest pairs of Ni Mn₁, Mn₂ (See Table 2). For cubic structure, a slight enhancement of the FM interactions J_{Mn1(2)-Mn1(2)} and J_Mn1(2)-Ni, as well as 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 tetragonal structure, a weakening of FM and AFM interactions between Ni, Mn₁, Mn₂ atoms is observed. However, the Curie temperature of martensite T_C^M increases in a similar way as 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, which includes the predicted T_m and T_C temperatures and experimentally measured values of T_m and T_C^A. With increasing the 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 $∆E_{{L2}_1-{L1}_0}$ and ${∆\Theta }_D^{{L2}_1-{L1}_0}$, which have a maximum 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 Curie temperature, MC simulations and mean-field approximation (MFA) show a similar trend of T_C^A 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. Discussions

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 experimental one 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 FiM one, where all magnetic moments of ${\mathrm{M}\mathrm{n}}_1^{\uparrow }$ and ${\mathrm{M}\mathrm{n}}_2^{\downarrow }$ 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 yields twice larger ${\mu }_{tot}$. 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 is larger than experimental one. The dashed line in Figure 2a are calculated curve (approximation) of the temperature dependence of the magnetization in μ₀H = 3T for the sample in austenite phase without a magnetostructural PT. This curve was calculated by using the approach proposed in [35]. The obtained magnetization value is 110 emu/g by using this approach (Figure 2a).

To explain the observed discrepancy, we suggest that not all Mn₂ atoms located at the Sn sublattice have antiparallel magnetic moment with respect to Mn₁ atoms located at the regular Mn sublattice. To prove this, we additionally preformed CPA calculations of ${\mu }_{tot}$ for L2₁-Ni_45.31Cu_4.69Mn_35.94Sn_14.06 (or ${\mathrm{N}\mathrm{i}}_{45.31}{\mathrm{C}\mathrm{u}}_{4.69}{\mathrm{M}\mathrm{n}}_{25}^{\uparrow }{\mathrm{M}\mathrm{n}}_{10.94-x}^{\downarrow }{\mathrm{M}\mathrm{n}}_x^{\uparrow }{\mathrm{S}\mathrm{n}}_{14.06}$) using SPR-KKR code as 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 a fraction of ${\mathrm{M}\mathrm{n}}_2^{\uparrow }$ atoms and reaches 5.94 ${\mu }_B$ or 125.6 emu/g for 100 % amount of ${\mathrm{M}\mathrm{n}}_2^{\uparrow }$ spins aligned parallel to ${\mathrm{M}\mathrm{n}}_1^{\uparrow }$ ones (FM order). We find that the experimental value of ${\mu }_{tot}$ can be reached theoretically if only ≈40 % of Mn₂ atoms interact antiferromagnetically with Mn₁ atoms. This finding allows to conclude that the experimental sample has some kind of FiM order with the predominant amount of FM ordered Mn atoms.

The inverse MCE at low temperatures in the Heusler Ni_33.7Co_14.8Mn_35.4Ti_16.1 alloy show the similar values at low temperatures [19]: maximal ∆T_ad = –2.7 K at T₀ = 90 K in magnetic field of 20 T. Perhaps this is a physical limit on the value of the inverse MCE for the Heusler alloys with PTs at low temperatures, which is significantly lower than in known alloys with inverse MCE at room temperatures [36].

5. Conclusions

The inverse MCE in the Heusler Ni_33.7Co_14.8Mn_35.4Ti_16.1 alloy in the region of low temperatures is observed in magnetic field of 1.8 T in both regimes of measuring: at sequential heating and at thermal cycling. The maximum value of the inverse MCE in the vicinity of the 1^st order PT in magnetic field of 1.8 T was observed ∆T_ad = –0.5 K at an initial temperature T₀ = 117 K. The inverse MCE is reaching up to ∆T_ad = –2.7 K at T₀ = 117 K with increasing of the magnetic field up to 10 T. The direct MCE in the FiM martensite phase is observed in magnetic field of 10 T and reaching the mximal value ∆T_ad = 0.7 K at the sequential heating regime in the temperature range of 20–60 K.

First principles and MC approaches applied to determine the phase diagram for Ni_50-xCu_xMn_35.94Sn_14.06 (x = 0, 1.56, 3.12, and 4.69) compounds FM ordered, which are similar in composition with experimental sample. It is shown that the Cu doping leads to decrease in T_m temperature and an increase in T_C^A temperature. This is 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 exhibits some type of FiM order in the austenite, with a predominance of ferromagnetically ordered Mn atoms. The theory is well confirmed by experiment and will make it possible to predict the transition temperatures, the magnetizations and the MCE properties of Heusler Ni-Mn-Sn family when the part of Sn atoms replaced by Cu.