Effect of Magnetic Field and Hydrostatic Pressure on Metamagnetic Isostructural Phase Transition and Multicaloric Response of Fe₄₉Rh₅₁ Alloy

Abstract

The effect of a high magnetic field up to 12 T and a high hydrostatic pressure up to 12 kbar on the stability of the metamagnetic isostructural phase transition and the multicaloric effect of Fe₄₉Rh₅₁ alloy has been studied. The phase transition temperature shifts under the magnetic field and the hydrostatic pressure on with the rates of dT_m/μ₀dH = −9.2 K/T and dT_m/dP = 3.4 K/kbar, respectively. The magnetocaloric and multicaloric (under two external fields) effects were studied via indirect method using Maxwell relations. The maximum of the entropy change is increasing toward the high temperature region from ∆S~2.5 J/(kg K) at 305 K to ∆S~2.7 J/(kg K) at 344 K under simultaneously applied magnetic field of 0.97 T and hydrostatic pressure of 12 kbar. The obtained results were explained using the first-principle calculations of Gibbs energies and the phonon spectra of the ferromagnetic and the antiferromagnetic phases. Taking into account the low concentration of antisite defects in the calculation cells allows us to reproduce the experimental dT_m/dP coefficient.

1. Introduction

The FeRh alloys with the near-equiatomic composition are characterized by a CsCl-type crystal structure with a first-order magnetic phase transition from the antiferromagnetic (AFM) state to the ferromagnetic (FM) at the same average temperature T_m, which depends strongly on the chemical composition [1,2,3]. These alloys exhibit significant changes in the magnetization and lattice parameter without the change in the crystal symmetry in the vicinity of the T_m [4], which can be called the magnetovolume effect at the metamagnetic isostructural phase transition (MIPT). The FeRh alloys are convenient model objects for studying the nature of the MIPT, due to their simple crystal structure and the bright effects observed near T_m. There are several methods to control the MIPT temperature in FeRh alloys such as a magnetic field [5], a hydrostatic pressure [6], an electric field-induced strain [7,8,9], a heat treatment [10,11], a chemical substitution [12], and an ion irradiation [13]. The first studies of the temperature dependences of magnetization, Young’s modulus, and the crystal lattice parameter were carried out via dilatometric method on the equiatomic alloy Fe₅₀Rh₅₀ [2], and the rates of MIPT temperature shift dT_m/dP and dT_m/μ₀dH in the hydrostatic pressure p and the magnetic field μ₀H were estimated. The FeRh alloys exhibit high magnetization saturation up to 130 Am²/kg in the FM phase [14], and the MIPT is accompanied by a strong anomaly of the specific heat capacity [15] and an isotropic volume increase of ~1% [2]. The magnetostriction of the Fe₅₀Rh₅₀ alloy was firstly measured in the temperature range 290–400 K in magnetic fields up to 15 T [16]. The published phase diagrams for near-equatomic FeRh alloys as a function of a magnetic field and a pressure present the collected data of the theoretical calculations and the experimental results obtained from the samples with different compositions and fabrication protocols [17,18,19]. For example, the P–T and the μ₀H–T diagrams obtained from experimental data for Fe₄₉Rh₅₁ alloy are limited by the hydrostatic pressure of 5 kbar [19] and the magnetic fields of 6 T [20].

The FeRh alloys have a number of unique functional properties in the external magnetic and mechanical fields known as “giant” caloric and multicaloric effects [17,19,20,21,22,23] due to the phenomena near T_m described above. The large magnetocaloric effect (MCE) [21], the elastocaloric effect [22] and the barocaloric effect (BCE) [20] were observed in FeRh alloys near the MIPT temperatures. For the first time, the MCE in the Fe₄₈Rh₅₂ alloy was reported in [24], where the dependence of the critical magnetic field on the MIPT temperature was established and the observed inverse MCE (the decreasing of the sample temperature under increasing of the external magnetic field) was described as a negative factor for constructing the critical magnetic field diagram. The near-equiatomic FeRh alloys were proposed as promising materials for the magnetic cooling at the room temperatures due to record values of MCE, where the values of the adiabatic temperature changes were as follows: ΔT_ad = −12.9 K at μ₀∆H = 1.95 T in the Fe₄₉Rh₅₁ [21,25]; ΔT_ad = −20.2 K at μ₀∆H = 8 T in the Fe₄₈Rh₅₂ [26]. In addition, the MCE was studied via the direct method in pulsed fields μ₀∆H = 50 T in the Fe₄₉Rh₅₁ alloy [27]. The BCE in Fe₄₉Rh₅₁ alloys was studied under hydrostatic pressure up to 2.5 kbar [20,28]. The effect of the magnetic field and the hydrostatic pressure on the MIPT trough the electrical resistance measurements were demonstrated in (Fe_1−xNi_x)₄₉Rh₅₁ alloys at the hydrostatic pressure up to 13 kbar [29], as well as in the Fe₄₈Rh₅₂ alloy at the applied magnetic field up to 8 T [30].

One of the promising approaches of the solid-state cooling based on the caloric effects is the multicaloric approach, which is based on the combination of the different external fields (magnetic, mechanical, electric) [31,32]. The multicaloric approach based on the combination of the MCE and the BCE under the external magnetic field up to 5 T and the hydrostatic pressure up to 5 kbar was experimentally used for the Fe₄₉Rh₅₁ alloy [19]. Moreover, the uniaxial mechanical force can be used to improve the efficiency of systems operating on MCEs in magnetic materials with the first-order phase transitions [23,33]. The uniaxial compression in combination with the magnetic field can be used to control hysteresis effects that negatively contribute to the efficiency of magnetic cooling systems [34].

The present work is aimed to investigate the two interrelated phenomena in FeRh alloys within the framework of a single study: (1) the effect of the combination of the magnetic field and the hydrostatic pressure on the temperature of the phase transition; and (2) the measurements of the multicaloric effect in magnetic field combined with the hydrostatical pressure. The FeRh samples with the same Fe₄₉Rh₅₁ composition but with two different heat treatment protocols were used as an object. The ab initio approach, which includes the total energy and the phonons calculations, was employed in order to describe the experimental P–T diagram.

2. Materials and Methods

The ingot of Fe₄₉Rh₅₁ alloy was obtained via induction melting in an argon atmosphere. The samples in shape of the disks with a diameter of 5 mm and thickness of 1 mm were cut out of the ingot. The samples were annealed in the pre-pumped quartz ampoules (~1 Pa) at a temperature of 1273 K. The first series of Fe₄₉Rh₅₁ samples was annealed for 72 h and quenched in ice water following this—the FR72h samples—and the second series was annealed for 48 h and cooled in the furnace—the FR48h samples. The structure and the elemental composition of the samples were confirmed via energy dispersion spectroscopy (EDS) and X-ray diffraction analysis (XRD) and presented in earlier work [35]. The MIPT temperatures of FR72h and FR48h samples were determined using two different methods: (1) the differential scanning calorimetry (DSC) on the calorimeter Netzsch DSC 204F1 Phoenix in zero magnetic field; and (2) the temperature magnetometry on the vibration magnetometer Versa Lab Quantum Design in a low magnetic field of 1 mT. The rate of the temperature change in these experiments was 10 K/min.

The magnetic field dependences of the magnetization of FR72h sample at the different temperatures in vicinity of the MIPT (300–328 K) were measured using the vibration magnetometer Quantum Design PPMS-14T in the magnetic field up to 8 T. The temperature dependences of the electrical resistance of FR72h sample at the different magnetic fields (0–12 T) were measured using the four-point method on Quantum Design PPMS-14T with the insert for the transport properties measurements. The experiments for studies of the multicaloric effects near the MIPT of FR48h sample at simultaneously applied magnetic field (0.155 T or 0.97 T) and hydrostatic pressure (up to 12 kbar) were carried out using the Domenicali-type pendulum magnetometer [36]. The mass of the FR48h sample was approximately 12.5 mg. The non-magnetic high-pressure cells with the sample were mounted at the end of the pendulum. This cell was made of the beryllium bronze, and pure indium (In) was used as the pressure transmitting medium. The pressure in the cell was determined based on the calibration of this system on chromium telluride (CrTe), which has the well-known phase transition temperatures shift on the applied pressure of −6 K/kbar. The MCE and the multicaloric effect were estimated via indirect method with the help of the well-known Maxwell relations.

The theoretical study of the ground-state properties of FeRh alloy was conducted using the density functional theory (DFT) calculations within the projector augmented wave method (PAW) method as implemented in the VASP package [37,38]. To describe the exchange correlation effects, both the generalized-gradient approximation (GGA) in the form of Perdew–Burke–Ernzerhof (PBE) [39] and the recently developed strongly constrained and appropriately normed (SCAN) meta-GGA [40] were used for the exchange correlation functional. The PAW potentials were selected with the following set of valence electrons: 3p⁶3d⁷4s¹ for Fe and 4p⁶4d⁸5s¹ for Rh. The plane wave cut-off parameter was taken as 700 eV and the Brillouin zone sampling is confined to 8 × 8 × 8 k-mesh generated by the Monkhorst-Pack scheme. The 16-atom cubic supercell with the space group Fm-3m and FM, AFM orders of Fe atoms was taken into account in the geometry optimization calculations. The zero temperature phonon spectrum calculations was performed in the harmonic approximation considering the small displacement method (0.01 Å) as implemented in the Phonopy package [41]. The 2 × 2 × 2 (32 atoms) supercell with four basis atoms in a fcc unit cell for AFM phase and 3 × 3 × 3 (54 atoms) supercell with two basis atoms in a bcc unit cell for FM phase were chosen. In order to investigate the thermodynamic properties of the FeRh under temperature and pressure, the quasi-harmonic Debye model implemented in the GIBBS code [42,43] was applied.

3. Results and Discussion

3.1. Experimental Studies

The DSC thermograms at the heating and cooling protocols for FR72h and FR48h samples are shown in Figure 1 (left axes). The temperatures of the start (AF_S and F_S) and the finish (AF_F and F_F) of the MIPT for the FR72h sample were determined from the DSC curves (Figure 1a) as AF_S = 317 K, AF_F = 308 K at cooling (from FM to AFM state) and F_S = 321 K, F_F = 330 K at heating (from AFM to FM state). The latent heat of the MIPT for FR72h sample was calculated as the area under the DSC peaks: λ_C = 4300 J/kg at cooling, and λ_H = 4400 J/kg at heating (Figure 1a). The corresponding MIPT temperatures for the FR48h sample were determined to be AF_S = 316 K, AF_F = 295 K at cooling and F_S = 306 K, F_F = 328 K at heating (Figure 1b). The latent heat of the MIPT for FR48h sample was as follows: λ_C = 4300 J/kg at cooling, and λ_H = 3600 J/kg at heating (Figure 1b).

The temperature dependences of the magnetization for FR72h and FR48h samples at heating and cooling in low magnetic field of 1 mT are shown in Figure 1 (right axes). These curves show the temperature hysteresis of the first order MIPT between two phases of Fe₄₉Rh₅₁: AFM at lower temperatures and FM at higher temperatures. The MIPT temperatures obtained from the DSC analysis are consistent with the temperatures from magnetometry data with an accuracy of 1 K (Figure 1a,b). In general, the differences in the heat treatment of the FR72h and FR48h samples greatly affected on AF_F and F_S temperatures (lower in FR48h).

Figure 2a demonstrates the magnetic field dependences of the magnetization M(μ₀H) for FR72h sample at different temperatures from the range 300–328 K in magnetic field of 8 T. The magnetic field-induced MIPT is observed in Figure 2a, similar to that were observed in Fe₄₈Rh₅₂ alloy in [44]. The magnetization hysteresis at the lowest temperature (300 K, the black curve in Figure 2a) is wide because the sample initially is completely in the AFM phase (lower than AF_F = 308 K). The magnetization hysteresis at highest temperature (328 K, the yellow curve in Figure 2a) is narrow because, initially, the sample is close to complete transition to the FM phase (F_F = 330 K).

Figure 2b demonstrates the temperature dependences of the electrical resistivity for the FR72h sample at heating and cooling at different magnetic fields up to 12 T. These measurements show the shifts in the MIPT temperatures under the high magnetic fields. The temperature hysteresis of the MIPT in Fe₄₉Rh₅₁ alloy according to the measurements results shifts to the lower temperature region with the increasing in magnetic fields, which is explained by the presence of the nuclei of the high-temperature FM phase at the low temperatures. Additionally, as a result, some broadening of the temperature hysteresis is observed with the increasing of the external magnetic field (Figure 3b), which is in good agreement with the results obtained for the Fe₄₈Rh₅₂ alloy [30].

Figure 3 demonstrates the temperature dependences of the magnetization of the FR48h sample at applied hydrostatic pressure from 0 to 12 kbar in magnetic fields of 0.155 T (Figure 3a) and 0.97 T (Figure 3b). The temperature hysteresis of the MIPT in the Fe₄₉Rh₅₁ alloy shifts to the higher temperatures region with the increase in hydrostatic pressure, and there is a certain narrowing of the temperature hysteresis at the same time. This is explained by the strong magneto-elastic coupling between the magnetic and structural subsystems in the solid. The external hydrostatic pressure compresses the crystal lattice and thus blocks the origination of the high-temperature FM phase with the larger unit cell volume. However, studies in such a range of magnetic fields and pressures for FeRh alloys are not known in the literature, and this would certainly expand our knowledge about the nature of the MIPT of these alloys.

If both the magnetization and the entropy are continuous functions of the temperature and the magnetic field, then an infinitesimal change in isobaric-isothermal entropy can be associated with the magnetization M, the magnetic field strength H and the absolute temperature T using one of Maxwell’s relations in integral form [45,46,47]:

$$\Delta S_{mag}={\mu }_0{\displaystyle \underset{0}{\overset{H}{\int }}\left(\frac{\partial M}{\partial T}\right)}dH$$

(1)

It is possible to calculate the magnetic entropy change ∆S in FR72h sample of Fe₄₉Rh₅₁ alloy connected with the external magnetic field change via Equation (1) using the data of the isothermal magnetization in Figure 2a. The calculation results for μ₀∆H = 1 and 2 T are shown in Figure 4a, while the maximum value was ∆S = 5.6 J/(kg K) at μ₀∆H = 1 T. This result correlates well with the calculated data from the heat capacity measurement in [48]: ∆S = 5.5 J/(kg K) at μ₀∆H = 1 T for Fe₄₉Rh₅₁ alloy.

Additionally, it is possible to calculate the magnetic entropy change ∆S in FR48h sample of Fe₄₉Rh₅₁ alloy connected with the external magnetic field change via Equation (1) using the data of the magnetization under different external pressures in Figure 3b. The calculation results for μ₀∆H = 0.97 T at cooling protocol are shown in Figure 4b, while the maximum obtained value was ∆S = 2.7 J/(kg K) under the external pressure p = 12 kbar. Thus, there is the difference in ∆S more than 2 times for FR72h and FR48h samples at the magnetic field changes of 1 T, which is explained by the difference in the heat treatment of the samples.

The magnetic phase μ₀H–T diagram based on the data of the temperature dependence of the electrical resistance for FR72h sample of Fe₄₉Rh₅₁ alloy in high magnetic fields was presented in Figure 5. The MIPT temperatures in different magnetic fields up to 12 T from Figure 2b were plotted on phase μ₀H–T diagram. It was established that the magnetic field shifts the MIPT to the region of the lower temperatures with the average rate dT_m/μ₀dH = −9.2 K/T. The obtained value generally corresponds with known data in the literature for the Fe₄₉Rh₅₁ alloy [20]: dT_m/μ₀dH = −9.6 K/T in magnetic field up to 6 T.

The phase P–T diagram based on the data of the temperature dependence of the magnetization (at 0.97 T) for FR48h sample of Fe₄₉Rh₅₁ alloy under the high hydrostatic pressure is presented in Figure 5. The MIPT temperatures in different external hydrostatic pressures up to 12 kbar from Figure 3b were plotted in the P–T. diagram. The restores stability of the AFM order and shifts the MIPT to the region of higher temperatures with the average rate dT_m/dP = 3.4 K/kbar. The obtained value is lower than the data presented in the literature for the Fe₄₉Rh₅₁ alloy [20]: dT_m/dP = 6.4 K/kbar in hydrostatic pressure up to 2.5 kbar—this discrepancy is explained by the difference in the heat treatment of the samples.

3.2. DFT Calculations

Since Fe-based alloys belong to strongly correlated electronic systems, there is an interest in studying the exchange correlation effects on the ground state properties of the AFM and FM phases of FeRh compound. Figure 6a displays the total energy difference as a function of lattice constant with respect to the AFM ground state calculated with PBE and SCAN. To find an optimized lattice parameter, the energy curves are fitted to the Birch-Murnaghan equation of state. As can be seen that both functionals predict the AFM order as energetically favorable. However, for both types of magnetic ordering, the SCAN lattice parameters (2.981 Å and 2.962 Å for FM and AFM) are found to be slightly smaller than those of the PBE one (3.003 Å and 2.988 Å for FM and AFM). It is worth noting that additional electron correlation effects beyond the GGA yield a significantly smaller energy difference ΔE (by approximately an order of magnitude) in comparison to that of PBE. Thus, the calculated SCAN value is ΔE = 2.3 meV/atom. In regard to the magnetic moments, the SCAN slightly overestimates them compared to those for the PBE. This is a known feature of the SCAN functional due to self-interaction corrections, which can be measured using the Coulomb energy U [49,50,51,52,53].

Next, let us discuss the phonon spectra for the AFM phase, which are calculated in the harmonic approximation using the SCAN and PBE functionals. It should be noted that we omit the results for the FM case here, since they show similar behavior without any sign of instability, as reported in earlier DFT calculations (e.g., Refs. [54,55,56]). As evident from Figure 6b, the PBE calculations demonstrate the phonon instability observed at the X point that contradicts the experimental data. On the other side, the meta-GGA calculations lead to the degeneracy of the anomaly of the optical phonon spectrum at the X point, thereby making the AFM phase dynamically stable at 0 K as in a case of FM phase. In general, a similar phonon spectra can be observed for both FM and AFM phases. The calculated Debye temperatures ${\Theta }_D$ from the phonon dispersion relations are found to be approximately 370 K and 394 K for FM and AFM phase, respectively.

We now turn to a discussion of the calculated metamagnetic temperature between the AFM and FM phases as a function of applied pressure and compare these with the experimental data shown in Figure 7. To calculate the Gibbs energy for both FM and AFM phases, we consider only a lattice contribution through the quasi-harmonic Debye model [42,43], which allows us to calculate thermodynamic properties as a function of temperature and pressure from the DFT energy-volume E(V) points and Debye temperatures taken from the phonon spectra calculations. It should be noted that the free-energy curves for FM and AFM phases calculated within the PBE E(V) data do not intersect due to a large energy difference between the two phases that is in agreement with earlier theoretical works [54,56]. However, the consideration of SCAN E(V) curves in combination with ${\Theta }_D$ as input in the Gibbs energy calculations predict well the transition temperature T_m~310 K under ambient pressure, which is in good agreement with the experiment. This is mainly due to the smallest difference in the total energies of FM and AFM phases. However, an application of hydrostatic pressure up to 12 kbar leads to a sufficient increase in T_m up to 550 K, which contradicts the experimental data. The theoretical rate of change in T_m with pressure is dT_m/dP~20 K/kbar, while the experiment gives ~3.4 K/kbar. It should be noted a similar effect of pressure has been observed in earlier theoretical calculations [17,18].

In order to resolve this contradiction, we considered the antisite defects in a high-temperature FM phase, which can occur in an ordered alloy when atoms of different types exchange their positions. For the investigation of point defects, we considered a periodically repeated supercell with 32, 64, 128, and 256 atoms, in which only one antisite pair (Fe–Rh) is embedded. Obviously, the larger the supercell, the lower the concentration of the antisite defect. As an example, the FM composition (Fe₁₅Rh₁)–(Rh₁₅Fe₁) formed on a 32-atom defect supercell has the antisite defect fraction of 1/16. We denote this defect composition as [email protected]%. In accordance with our calculations, there is no metamagnetic transition for this composition due to a large energy difference between both phases ΔE ≈ 47 meV/atom, whereas it takes place for compositions with a smaller defect concentration. As is evident from the P–T diagram, the transition temperature is only slightly affected by an applied pressure for [email protected]% (the defect supercell with 64 atoms, ΔE ≈ 23 meV/atom). However, the further reduction in defect fraction in the FM phase by half reveals a good agreement between the theory and the experiment for [email protected]% (the defect supercell with 128 atoms, ΔE ≈ 16 meV/atom). We would like to note that the subsequent change in the defect concentration results in an increase in the dT_m/dP slope, which becomes close to the case of pure AFM and FM phases, as exemplified by [email protected]% (the defect supercell with 256 atoms, ΔE ≈ 3.7 meV/atom). We are reminded that ΔE for compound with a pure FM and AFM phase is calculated to be 2.3 meV/atom, as shown in Figure 6a. Thus, we can conclude that the theoretical slope of dT_m/dP strongly depends on the defect concentration due to a change in ΔE and ${\Theta }_D$ between a pure AFM phase and a defect FM one.

4. Conclusions

The effects of the high hydrostatic pressure up to 12 kbar and the high magnetic field up to 12 T on the stability of the MIPT in Fe₄₉Rh₅₁ alloy samples were studied. As expected, the strategy of the combination of the external fields leads to opposite contributions: the magnetic field expands the FM region and shifts the MIPT to the lower temperatures with the rate dT_m/μ₀dH = −9.2 K/T, and the hydrostatic pressure restores the stability of the AFM order and shifts the MIPT to the higher temperatures with the rate dT_m/dP = 3.4 K/kbar. The study of the multicaloric effect demonstrates that the hydrostatic pressure can be used for the tuning of the maximum of the magnetic entropy change, which has prospective applications for the upgrading of the magnetic cooling systems. The obtained experimental P–T diagram is explained by using the DFT calculations within the exchange correlation effects beyond the well-known GGA. The experimental dT_m/dP slope is reproduced theoretically when the antisite defects were considered in the computational supercell.