New insight into magneto-structural phase transitions in layered TbMn₂Ge₂-based compounds

Abstract

The Tb_1−xY_xMn₂Ge₂ series (x = 0, 0.1, 0.2) compounds are found to exhibit two magnetic phase transitions with decreasing temperature: from the paramagnetic state to the antiferromagnetic interlayer state at T_N^inter and from an antiferromagnetic interlayer structure to a collinear ferrimagnetic interlayer structure at T_C^inter. Compared with the slight change of T_N^inter (409 K, 410 K and 417 K for x = 0, 0.1 and 0.2 respectively), the replacement of Y for Tb leads to a significant decrease in T_C^inter from 97.5 K for x = 0 to 74.6 K for x = 0.2. The variation in T_C^inter can be ascribed to the combination of two effects: (1) chemical pressure and (2) magnetic dilution effect by Y substitution for Tb. Besides, a strong anisotropic magnet-volume effect has been detected around T_C^inter in all compounds with Δa/a = 0.125%, 0.124% and 0.130% for x = 0, 0.1 and 0.2, respectively while no obvious effect is detected along the c-axis. The maximum magnetic entropy change were found to be −ΔS_max = 9.1 J kg⁻¹ K⁻¹, 11.9 J kg⁻¹ K⁻¹ and 6.3 J kg⁻¹ K⁻¹ with a field change from 0 T to 5 T for x = 0, 0.1, 0.2 respectively.

Introduction

Since the discovery in 1997 of a giant magnetocaloric effect (GMCE) originating from a discontinuous first order magnetic transition in Gd₅Si₂Ge₂¹, room-temperature magnetic refrigeration based on the magnetocaloric effect (MCE) has attracted significant attention due to its energy efficiency and environment friendly in comparison with conventional gas compression-expansion refrigeration². A number of materials which exhibit giant magnetic entropy changes at magnetic transitions have been investigated, including MnFeP_0.45As_0.55³, MnAs_1−xSb_x⁴, Ni-Mn-Sn-based alloys⁵, Ni-Mn-Ga^6,7, and La(Fe,Si)₁₃⁸. The key features of these systems are the temperature- and magnetic field-induced first-order magneto-structural or magneto-elastic phase transitions. Given these promising developments, magnetic materials which exhibit a large magnetocaloric effect have been studied extensively, both experimentally and theoretically, over the past two decades with the overall aim of increasing the efficiency of magnetic refrigeration techniques^9,10. While a key focus is exploration of materials that exhibit a pronounced magnetocaloric effect at room temperature, materials that operate in the low temperature region are also useful in meeting the cooling requirements for fields such as gas liquefaction or attaining millikelvin for experimental research facilities. However, so far only a few materials such as GdLiF₄, GdF₃ and Gd₃Ga₅O₁₂ are used commercially¹¹. As reflected by the increase in exploration of materials which exhibit a large MCE below room temperature^10,11,12,13, the search for materials which exhibit large magnetocaloric effects over temperature ranges relevant for hydrogen and natural gas liquefaction are also important for exploring potential applications.

Some RT₂X₂ compounds (R = rare earth, T = transition metal, and X = Si or Ge) have been found to exhibit large MCE values with small hysteresis losses near their low magnetic transition temperatures^{11,13,14,15,16}. For example, the magnetic entropy values of RNi₂Si₂ (R = Dy, Ho, Er) compounds are 21.3 J kg⁻¹ K⁻¹, 21.7 J kg⁻¹ K⁻¹ and 22.9 J kg⁻¹ K⁻¹ around 6.5 K, 4.5 K and 3.5 K respectively during a change of magnetic induction intensity from 0–5 T¹⁶, while the magnetic entropy of ErCr₂Si₂ attained 29.7 J kg⁻¹ K⁻¹ near the magnetic ordering temperature 4.5 K¹⁷. The crystal structure of the RT₂X₂ series is body centred tetragonal ThCr₂Si₂-type (with space group I4/mmm)^15,18,19, with the sequence -R-X-T-X-R- atomic layers stacked along the c-axis. The rare earth elements typically exhibit large magnetic moment (for example μ_Tb = 8.8 μ_B in TbMn₂Si₂ at 5 K)²⁰ and correspondingly make a large contribution to the magnetocaloric effect^14,15,17. Given the sensitivity of the magnetic state in RMn₂X₂ to the intra-planar Mn-Mn spacing^{15,19,21,22,23,24}, compounds in this series are found to display a rich variety of interesting phenomena, including superconductivity, magnetism, mixed valence, heavy fermions, and Kondo behaviour^25,26,27. This diversity enables control of the interplay between the R-Mn and Mn-Mn exchange interactions in RMn₂X₂ through external factors such as pressure²⁸, temperature and magnetic field²⁹ meaning that such compounds have the potential for competitive performance^15,19,24. The notations used in this paper to describe the magnetic structure type and critical transition temperatures are defined by Venturini et al.²² Using standard magnetic methods^19,30, TbMn₂Ge₂ was reported to be antiferromagnetic below Néel temperature T_N = 410 K with the AFil antiferromagnetic interlayer structure (i.e. a collinear antiferromagnetic structure between adjacent Mn planes in a + − + − sequence along the c-axis²².) Below T_C = 100 K, TbMn₂Ge₂ exhibits a collinear ferrimagnetic structure in which the Tb moments order ferromagnetically and couple antiferromagnetically with the Mn moment²³. Furthermore, in a later study for the Tb_1−xY_xMn₂Ge₂ series (x = 0–0.4), it was reported that the replacement of Y for Tb leads to significant modifications of both the Curie temperature (from 76 K for TbMn₂Ge₂ to almost 0 K for Tb_0.4Y_0.6Mn₂Ge₂) and magnetovolume effect (the volume effect is ∆V/V = 3.2 × 10^–3 and 2.7 × 10⁻³ for x = 0 and 0.1 respectively)³⁰. The magnetic phase transitions around T_C in the Tb-rich Tb_1−xY_xMn₂Ge₂ compounds were shown to be first order³⁰, offering scope for large magnetocaloric effects around the region of their Curie temperatures.

Here we present a systematic study of the magnetic transition from antiferromagnetism to ferromagnetism in a series of Tb_1−xY_xMn₂Ge₂ samples (x = 0, 0.1, 0.2) using a combination of methods including variable temperature x-ray diffraction (XRD), specific heat, differential scanning calorimetry (DSC) and magnetization measurements. The overall aim is to understand fully the influence of Y substitution for Tb on magnetocaloric effects and search for novel magnetocaloric materials that may be suitable for operation over the hydrogen and natural gas liquefaction temperature ranges.

Method

The polycrystalline Tb_1−xY_xMn₂Ge₂ samples with x = 0, 0.1, 0.2 were prepared by arc melting constituent elements of 99.9% purity under argon atmosphere. For improved crystallization and chemical homogeneity, the samples were annealed in vacuum-sealed quartz tube at 850 °C for 7 days after arc melting. The dc magnetic measurements were performed using a Quantum Design 9 T physical properties measurement system (PPMS). The magnetic behaviour was investigated over the range from 5 K to 340 K in a magnetic field 0.01 T. Differential scanning calorimetry measurements were performed on differential scanning calorimetry equipment (DSC 204 F1 Phoenix^®) from 340 K to 500 K. Magnetization-field loops were obtained at temperatures close to the Curie temperature of samples with magnetic fields over the range 0–5 T. The heat capacity measurements were performed on a Quantum Design 14 T physical properties measurement system scanning from 2 K to 250 K. The samples were characterized and the structures determined by variable temperatures XRD measurements over the temperature range (12–300 K) using a PANAlytical diffractometer with Cu-Kα radiation.

Results and Discussion

Structural behaviour

The room temperature x-ray diffraction study shows that all samples are single phase and that patterns can be indexed with a space group of I4/mmm as expected. The Rietveld refinements have been carried out using the FullProf package³¹ with the main results shown in Fig. 1(a,b and c) for x = 0, 0.1 and 0.2 respectively. It can be seen from Fig. 1. that the variations of lattice parameters of a and c with temperature display strong anisotropy: the lattice parameter c (red solid circle) increases monotonically with increasing temperature while a pronounced discontinuity is observed in the a lattice parameter (black solid square) around the Curie temperature T_c for each sample (the transition temperatures were determined as the point where the value of dM/dT is minimum). Similar behaviours for TbMn₂Ge₂ were also determined by Morellon et al.²³ for which an anomaly in the thermal expansion along the a-axis was found near T_c. The discontinuity in the a lattice parameter around T_c leads to the associated decrease in the unit cell volume for all samples as also evident in Fig. 1. These behaviours are very similar to the behaviour reported for Pr_0.5Y_0.5Mn₂Ge₂¹⁵ (i.e. PrMn₂Ge₂ diluted by Y), but different from NdMn₂Ge_0.4Si_1.6³², NdMn_1.9Ti_0.1Si₂³³ and NdMn_1.7Cr_0.3Si₂^34,35 (where Mn diluted with transition metal Ti or Ge diluted by Si) for which the lattice parameter a decreases with increasing temperature around T_C while the lattice parameter c expands. In order to derive the magneto-volume effect below T_C, we have calculated the contribution from lattice vibration using the Debye model:

Figure 1

Temperature dependence of the lattice constants a, c and unit cell volume: (a) TbMn₂Ge₂, (b) Tb_0.9Y_0.1Mn₂Ge₂ and (c) Tb_0.8Y_0.2Mn₂Ge₂. The dashed lines show the phonon contribution to the lattice expansion as evaluated from the Gruneisen relation.

where β is the volume thermal expansion coefficient of the parameter state, k is the compressibility, γ is the Gruneisen constant and C_v is the specific heat at constant volume caused by lattice vibrations. C_v was derived from the Debye theory of the specific heat using the value of the Debye temperature ^θ_D (as derived from our specific heat measurements for each of the samples as described below for the three compositions):

where k_B is the Boltzmann constant and N is the number of the atoms. The thermal expansion for the hypothetical paramagnetic state is derived on integrating Eq. (2) with respect to temperature. The parameter was adjusted to obtain the best least-squares fitting to the successive data points of the observed thermal expansion curve well above the magnetic ordering temperature (based on the fact that the magnetic contribution in the antiferromagnetic region to total thermal expansion can be ignored for these types of compounds)³².

The temperature dependence of the unit cell volumes based on Debye theory for the TbMn₂Ge₂, Tb_0.9Y_0.1Mn₂Ge₂ and Tb_0.8Y_0.2Mn₂Ge₂ samples are shown by the dashed lines in Fig. 1(a,b and c) with pronounced magneto-volume effects evident below their magnetic transition temperatures T_C = 94 K, T_C = 83 K and T_C = 70 K respectively. The discontinuous nature of the changes in a lattice parameter and unit cell volume V at the Curie temperatures as shown in Fig. 1, are consistent with the first order nature of these transitions as discussed fully below. The changes in the lattice parameter a are Δa/a = 0.125%, Δa/a = 0.124% and Δa/a = 0.130% for x = 0, 0.1 and 0.2 respectively with spontaneous volume magnetostriction ω_s (=ΔV_m/V) at 5 K determined as: TbMn₂Ge₂ - ω_s = 4.1 × 10⁻³; Tb_0.9Y_0.1Mn₂Ge₂ - ω_s = 3.2 × 10⁻³ and Tb_0.8Y_0.2Mn₂Ge₂ - ω_s = 5.8 × 10⁻³.

Magnetic phase transition

The magnetisation of the three samples have been measured in a field of B = 0.01 T over the temperature range 5–340 K. As in Fig. 2(a,b and c) the TbMn₂Ge₂, Tb_0.9Y_0.1Mn₂Ge₂ and Tb_0.8Y_0.2Mn₂Ge₂ samples were respectively measured on warming from 5 K in three states: after cooling in zero field (ZFC heating) and after cooling and heating in a field of B = 0.01 T (FC cooling and FC heating). As is evident from the magnetization versus temperature curves of Fig. 2(a,b and c), there is an abrupt change in magnetisation at the Curie temperature T_C^inter that marks the magnetic phase transition from a collinear antiferromagnetism (AFil)²² at higher temperature to a collinear ferrimagnetic structure along the c axis at lower temperature according to the neutron diffraction study on TbMn₂Ge₂²³. Of the three samples, TbMn₂Ge₂ has the highest T_C^inter(warm) = 97.5 K and T_C^inter(cool) = 93.0 K transitions respectively as determined from the FC heating and cooling M-T curves, while the values for Tb_0.9Y_0.1Mn₂Ge₂ are derived to be T_C^inter(warm) = 87.5 K and T_C^inter(cool) = 81.8 K with the values for Tb_0.8Y_0.2Mn₂Ge₂ being T_C^inter(warm) = 74.6 K and T_C^inter cool) = 66.0 K (normally the transition temperature during the FC process is chosen as the Curie temperature T_c). As expected, the higher the level of doping of non-magnetic Y atoms in Tb_1−xY_xMn₂Ge₂, the lower the magnetic phase transition temperature³⁰.

Figure 2

Temperature dependence of magnetization on ZFC heating, FC cooling and FC heating processes under a field of B = 0.01 T: (a) TbMn₂Ge₂, (b) Tb_0.9Y_0.1Mn₂Ge₂ and (c) Tb_0.8Y_0.2Mn₂Ge₂. (d) the differential scanning calorimetry curves for the three samples over the range ~300–500 K.

Differential scanning calorimetry measurements have been carried out on the Tb_1−xY_xMn₂Ge₂ samples over the temperature range 300–500 K (Fig. 2(d)) in order to investigate the paramagnetic to antiferromagnetic transition^10,23 at T_N^inter. As revealed by the DSC results in Fig. 2(d), the T_N^inter transition temperatures are found to increase slightly with increasing Y concentration - T_N^inter = 409 K, T_N^inter = 410 K and T_N^inter = 417 K for x = 0.0, 0.1 and 0.2 respectively. Compared with the reduction in ferromagnetic transition temperature on replacement of Tb atoms by Y atoms, the paramagnetic to antiferromagnetic transition temperatures are found to exhibit a slight increase (Fig. 2). The increase in T_N^inter values is due to enhancement of the Mn-Mn exchange interaction as a result of the slight reduction of Mn-Mn distance. This behaviour is similar to the PrMn₂Ge_2−xSi_x system³⁶ in which the paramagnetic to antiferromagnetic transition temperatures are found to increase slightly while the antiferromagnetic to ferromagnetic transition temperatures decrease on replacing Ge with Si.

The temperature dependences of magnetization for TbMn₂Ge₂, Tb_0.9Y_0.1Mn₂Ge₂ and Tb_0.8Y_0.2Mn₂Ge₂ under various external magnetic fields are presented in Fig 3(a,b and c) respectively. As expected the ferromagnetic transition temperature T_C^inter is shifted to higher temperature with increase in applied magnetic field. For example, the transition temperatures are T_C^inter = 101.8 K, T_C^inter = 92 K and T_C^inter = 76 K for TbMn₂Ge₂, Tb_0.9Y_0.1Mn₂Ge₂ and Tb_0.8Y_0.2Mn₂Ge₂ respectively in an external magnetic field of B = 1 T, while the transition temperatures are shifted to T_C^inter = 107 K, T_C^inter = 98.2 K and T_C^inter = 81 K respectively in a field of B = 5 T. The field dependence of the magnetic transition temperatures are summarized in Fig. 3(d). The values of dT_c/dB (obtained on linear fitting of the experimental data in Fig. 3(d) along with a summary of experimental data determined for Tb_1−xY_xMn₂Ge₂ (x = 0, 0.1, 0.2) in this investigation, are provided in Table 1.

Figure 3: The magnetisation versus temperature curves during cooling under various magnetic field (B = 1–5 T).

(a) TbMn₂Ge₂, (b) Tb_0.9Y_0.1Mn₂Ge₂, (c) Tb_0.8Y_0.2Mn₂Ge₂. (d) The variation of ferromagnetic transition temperature T_c with magnetic field for the three samples. The dashed lines represent linear fits to the T_c-B curves leading to dT_c/dB values for each sample.

Table 1 Experimental data for the three Tb_1−xY_xMn₂Ge₂ samples (x = 0, 0.1 and 0.2).

Y doping in Tb_1−xY_xMn₂Ge₂ - Chemical pressure effect

As noted above, the effect of replacing the magnetic rare earth Tb with the nonmagnetic ion Y in Tb_1−xY_xMn₂Ge₂ is to weaken the exchange interaction between magnetic ions due to the dilution effect. The magnetic behaviour of Y-doped Tb_1−xY_xMn₂Ge₂ will also be modified as a result of chemical pressure due to differences in the atomic radii of the Tb(1.80 Å) and Y(1.78 Å) ions and resultant changes in lattice parameters. In order to separate these two contributions - dilution effect and pressure effect - and their influence on the variation in magnetic transition temperature, the decrease of T_C by chemical pressure was calculated as follows. The chemical pressure Δp was calculated^20,22,33 according to the Murnaghan equation below:

where V₀, B₀ and are the volume, the bulk modulus and its first derivative of TbMn₂Ge₂ and V is the volume of the unit cell at room temperature of the Y doped samples. Here, due to the similarity of crystal structure for the RMn₂Ge₂ system, we assume that the values of B₀ and for PrMn₂Ge₂ (B₀ = 38.0 Gpa,  = 19.5 as derived from our synchrotron data under external pressure³⁷) can be applied to TbMn₂Ge₂ at room temperature. Given that the doped materials Tb_1−xY_xMn₂Ge₂ (x = 0.1, 0.2) retain the ThCr₂Si₂-type tetragonal structure, the chemical pressure Δp caused by doping can be assumed to have the same effect as mechanical pressure. According to previous findings that describe the pressure effect on the magnetic properties of TbMn₂Ge₂ (dT_c/dP = −2.9 K/kbar)³⁸, the values of ΔT_C can be deduced by the relationship:

where Δp is the calculated chemical pressure. The calculated values of ΔT_C for Tb_0.9Y_0.1Mn₂Ge₂ and Tb_0.8Y_0.2Mn₂Ge₂ are ΔT_C =1.94 K and ΔT_C = 8.7 K respectively. As noted above (see also Fig. 1, Figs 2 and 3.) the effect of replacing Tb atoms with Y atoms in Tb_1−xY_xMn₂Ge₂ also contributes to the decrease in the Curie temperature. It can therefore be concluded that chemical pressure accounts for ~17.3% and ~32.2% of the decrease in transition temperatures for x = 0.1 and 0.2 respectively. In addition, the value of dT_c/dp can be derived using the Clausius-Clapeyron thermodynamic relation as follows:¹⁵

The values of the change in the unit cell volume ΔV_m he change in moment Δμ during magnetic phase transition around T_C and dT_c/dB for each sample were taken from the present experimental results listed in Table.1. The derived results are dT/dp = −3.03 K/kbar, −2.84 K/kbar and −3.07 K/kbar for TbMn₂Ge₂, Tb_0.9Y_0.1Mn₂Ge₂ and Tb_0.8Y_0.2Mn₂Ge₂, respectively. These calculated values are in general accord with the value of dT_c/dp = −2.9 K/kbar for TbMn₂Ge₂³⁸, deviating by ~4.5%, ~2.1% and ~5.9% for the x = 0.0. 0.1, 0.2 samples respectively.

Magnetocaloric effect

Graphs of the magnetization as a function of applied field are shown for TbMn₂Ge₂, Tb_0.9Y_0.1Mn₂Ge₂, and Tb_0.8Y_0.2Mn₂Ge₂ at temperatures around T_C^inter in Fig. 4(a,b and c) respectively. It can be seen that with increasing temperature beyond T_C^inter, a field-induced metamagnetic phase transition from the antiferromagnetic state to the ferromagnetic state at certain temperatures has been detected. The region of the metamagnetic phase transition for TbMn₂Ge₂ is indicated by arrows in Fig. 4(a) as a typical example. This behaviour indicates that the region of ferromagnetic ordering in Tb_1−xY_xMn₂Ge₂ can be shifted to higher temperatures by a stronger applied magnetic field.

Figure 4: Curves of isothermal magnetization versus magnetic field at temperatures around T_c.

(a) TbMn₂Ge₂, (b) Tb_0.9Y_0.1Mn₂Ge₂ and (c) Tb_0.8Y_0.2Mn₂Ge₂.

The nature of the magnetic transitions (first order or second order) was analysed using Arrott plots with the magnetisation expressed in the usual way as graphs of M² versus B/M (Fig. 5). As can be seen in Fig. 5(a,b and c), negative slopes are detected in the M² versus B/M graphs for the TbMn₂Ge₂ and Tb_0.9Y_0.1Mn₂Ge₂ samples thus indicating that the antiferromagnetic to ferromagnetic processes are first order³⁹. However, Some papers^40,41 reported that for compounds near the critical point (from first order to second order magnetic phase transition) such as DyCo₂, this criterion of Arrott plots do not always work properly. It is also noted that the negative slopes for Tb_0.8Y_0.2Mn₂Ge₂ around the antiferromagnetic to ferromagnetic transition was reduced compared with those for the TbMn₂Ge₂ and Tb_0.9Y_0.1Mn₂Ge₂ samples. However, the first order transition characters of all the three samples can be confirmed from our variable temperatures crystal structure analyses above, where strong magneto-elastic coupling around T_C^inter has been detected (Fig. 1).

Figure 5: Arrott plots: isotherms graphs of M² versus B/M for decreasing magnetic fields at temperatures around T_c.

(a) TbMn₂Ge₂, (b) Tb_0.9Y_0.1Mn₂Ge₂ and (c) Tb_0.8Y_0.2Mn₂Ge₂.

The magnetic entropy changes ΔS_M for all samples have been determined from the isothermal magnetization curves of Fig. 4(a,b and c), by using the standard Maxwell relationship:

The calculated temperature dependent magnetic entropy changes for the Tb_1−xY_xMn₂Ge₂ samples with x = 0, 0.1 and 0.2 for both increasing field and decreasing field processes between field changes of ΔB = 0–1 T and ΔB = 0–5 T are shown in Fig. 6(a,b and c) respectively with the maximum values ΔS_max shown as a function of applied field in the insets of Fig. 6. With a field change of ΔB = 0–5 T, the value of −ΔS_max are 9.1 J/kgK, 11.9 J/kgK and 6.3 J/kgK for TbMn₂Ge₂, Tb_0.9Y_0.1Mn₂Ge₂ and Tb_0.8Y_0.2Mn₂Ge₂ respectively, demonstrating that the entropy change for Tb_0.9Y_0.1Mn₂Ge₂ is the largest of the three samples. As it is clear from Fig. 4, while TbMn₂Ge₂ has the highest fraction of magnetic rare earth element and largest saturation magnetization (42.5 Am²/kg at 84 K), its large hysteresis loss (7.40 J/kg) leads to reduction in the magnetic entropy change. By comparison, with the lowest concentration of magnetic rare earth Tb, the Tb_0.8Y_0.2Mn₂Ge₂ sample displays the lowest saturation magnetization (only 32.5 Am²/kg even at 55 K) and the smallest hysteresis loss 5.21 J/kg), while as shown in Fig. 4(b), Tb_0.9Y_0.1Mn₂Ge₂ with medium concentration of Tb has a relatively large saturation magnetization of 38.0 Am²/Kg at 84 K and small hysteresis loss (5.36 J/kg). The refrigerant capacity (RCP), defined as the product of −ΔS_max and the full width at half maximum of the −ΔS_max curve, for the three samples are: 93.3 J/kg, 102.9 J/kg, 62.4 J/kg for TbMn₂Ge₂, Tb_0.9Y_0.1Mn₂Ge₂ and Tb_0.8Y_0.2Mn₂Ge₂ respectively, with a field change of ΔB = 0–5 T. The MCE value of Tb_0.9Y_0.1Mn₂Ge₂ is comparable to those of other materials for a field change of ΔB = 0–5 T including: GdCoAl, −ΔS_max(T) = 10.4 J/kgK at 100 K³⁴, TbCoAl, −ΔS_max(T) = 10.5 J/kgK at 70 K³⁴ and GdMn₂Ge₂, −ΔS_max(T) = 1.2 J/kgK at 95 K²⁸, all of which, in common with Tb_0.9Y_0.1Mn₂Ge₂, importantly exhibit negligible field and thermal hysteresis losses.

Figure 6: The magnetic entropy changes around the ferromagnetic transition temperatures for applied magnetic fields from 1–5T.

(a) TbMn₂Ge₂, (b) Tb_0.9Y_0.1Mn₂Ge₂ and (c) Tb_0.8Y_0.2Mn₂Ge₂ (black full symbols ◾ ★ ▴ ▾ ♦ for 1–5 T respectively during increasing field and the red empty symbols correspond to 1–5 T for the decreasing field). The insets show the variation of the maximum values of magnetic entropy changes for the decreasing field values.

Moreover, it is well accepted that first-order phase transitions are accompanied by a latent heat and the barocaloric effect can be expected. In fact giant barocaloric effect has been found in several systems recently including Mn₃GaN (∆S_bar = 22.3 J/K kg)⁴² and Ni–Mn–In magnetic superelastic alloys (∆S_bar = 27.7 J/K kg)⁴³. Based on the fact that all these three Tb_1−xY_xMn₂Ge₂ samples exhibit strong magnetovolume effect around magnetic phase transition, we have calculated the barocaloric effect using the Clausius_Clapeyron relation⁴³. The barocaloric effects entropy change ∆S_bar, have been derived to be ∆S_bar = 9.6 J/kgK, 13.5 J/kgK and 13.2 J/kgK for TbMn₂Ge₂,Tb_0.9Y_0.1Mn₂Ge₂ and Tb_0.8Y_0.2Mn₂Ge₂ respectively. These barocaloric values indicate that these materials can be considered as a potential candidate for mechanocaloric effects over the hydrogen and natural gas liquefaction temperature ranges.

Heat Capacity

The heat capacity of TbMn₂Ge₂ over the temperature range 2–250 K is shown in Fig. 7(a). The sharp peak in the heat capacity near the Curie temperature of TbMn₂Ge₂ on both zero magnetic field and a field of 2 T reflects the first order character of the magnetic phase transition. The peak in specific heat shifts from ~98 K to 102.6 K for magnetic fields of 0 T and 2 T respectively; this behaviour corresponds well to the values of the Curie temperature of TbMn₂Ge₂ - 97.5 K (B = 0 T) to 103 K (B = 2 T) - obtained for the magnetization measurements (Fig. 3(d)).

Figure 7: The specific heat capacity relative parameters for TbMn₂Ge₂.

(a) The specific heat capacity C_P of TbMn₂Ge₂ over the temperature range 2–250 K in zero magnetic field (red solid square) and a field of 2 T (blue solid triangle). (b) A graph of C_p/T versus T² for TbMn₂Ge₂ at temperatures below 10 K. (c) Magnetic entropy change −ΔS as a function of temperature derived from the specific heat data of Fig. 7(a) for ΔB = 0–2 T. The inset shows the corresponding adiabatic temperature change, ΔT_ad.

The heat capacity C(T) of a metallic magnetic material includes contributions from phonons, electrons and magnons and can be described as follows:

where C_ph, C_el and C_m are the lattice, electronic, and magnetic contributions respectively⁴⁴. In the absence of a magnetic phase transition, the heat capacity can be described as:

where γ and β are the electronic and phonon heat capacity coefficients, respectively. For the specific heat of TbMn₂Ge₂ at low temperatures T ≤ 1 K, well away from the magnetic transition, as shown in Fig. 7(b), a fit to the graph of C_p/T versus T² leads to γ = (65.2 ± 0.95)mJ/molK², β = (4.53 ± 0.156) × 10⁻⁴ J/molK⁴. The electronic density of states N(E_F) at the Fermi surface can be calculated by the formula:⁴⁴

where k_B is the Boltzmann constant. For the TbMn₂Ge₂ compound, the value of N(E_F) is derived to be (5.54 ± 0.08) state/eV atom. Likewise, the Debye temperature θ_D can also be obtained by:

where R is the universal gas constant and the number of atoms n = 5⁴⁵. The Debye temperature for TbMn₂Ge₂ was determined as θ_D = 278 ± 3 K.

The magnetic entropy change, −ΔS_M (T, B) can also be derived from measurements of the in-field heat capacity using the expression thermodynamic relations below:⁴⁶

where C(T, B) and C(T, 0) are the values of the heat capacity measured in field B and zero field, respectively. The maximum of magnetic entropy change has been derived to be −ΔS_M =2.6 J/kgK for TbMn₂Ge₂ (field change of ΔB = 2 T), which is smaller than the value (−ΔS_M = 5.9 J/kgK) deduced from the isothermal magnetization curves. This behaviour may be due to the fact that a straightforward numerical integration using Maxwell equation based on magnetization curves is not applicable in the phase-separated state as described in ref. 47,48. The corresponding adiabatic temperature change, −ΔT_ad (shown as inset of Fig. 7(c) can be evaluated from −ΔS_M (T, B) and the heat capacity data.

The equivalent heat capacity parameters for Tb_0.9Y_0.1Mn₂Ge₂ and Tb_0.8Y_0.2Mn₂Ge₂ are shown in Figs 8 and 9, respectively. The Debye temperatures were found to increase from 281 K for TbMn₂Ge₂ to 344 K for Tb_0.9Y_0.1Mn₂Ge₂ and 354 K for Tb_0.8Y_0.2Mn₂Ge₂; with the increases understood in terms of the differences in their molecular mass³¹. The adiabatic temperature changes near the Curie temperature are found to decrease from −ΔT_ad = 2.6 K for TbMn₂Ge₂, to −ΔT_ad = 2.3 K for Tb_0.9Y_0.1Mn₂Ge₂ and 1.8 K for Tb_0.8Y_0.2Mn₂Ge₂.

Figure 8: The specific heat capacity relative parameters for Tb_0.9Y_0.1Mn₂Ge₂.

(a) The specific heat capacity C_P of Tb_0.9Y_0.1Mn₂Ge₂ over the temperature range 2–250 K in zero magnetic field (red solid square) and a field of 2 T (blue solid triangle). (b) A graph of C_p/T versus T² for Tb_0.9Y_0.1Mn₂Ge₂ at temperatures below 10 K. (c) Magnetic entropy change −ΔS as a function of temperature derived from the specific heat data of Fig. 8a for ΔB = 0–2 T. The inset shows the corresponding adiabatic temperature change, ΔT_ad.

Figure 9: The specific heat capacity relative parameters for Tb_0.8Y_0.2Mn₂Ge₂.

(a) The specific heat capacity C_P of Tb_0.8Y_0.2Mn₂Ge₂ over the temperature range 2–250 K in zero magnetic field (red solid square) and a field of 2 T (blue solid triangle). (b) A graph of C_p/T versus T² for Tb_0.8Y_0.2Mn₂Ge₂ at temperatures below 10 K. (c) Magnetic entropy change −ΔS as a function of temperature derived from the specific heat data of Fig. 9(a) for ΔB = 0–2 T. The inset shows the corresponding adiabatic temperature change, ΔT_ad.

The electron density at the Fermi surface is found to decrease from 5.54 state/eV atom for TbMn₂Ge₂ to 2.18 state/eV atom and 3.06 state/eV atom for Tb_0.9Y_0.1Mn₂Ge₂ and Tb_0.8Y_0.2Mn₂Ge₂ respectively. All the fitting results including electronic heat capacity coefficient γ, phonon heat capacity coefficient β, electronic density of states N(EF) and Debye temperature θ_D are summarized in Table 2. The modification of the electron density at the Fermi surface may be related to the difference of electronic configuration of Y and Tb as well as the unit cell size variation. The latter may lead to the variation in the degree of hybridization of Mn 3d states with p states of Ge with decreasing interatomic distances for Y doped samples. Similar behaviour has been found in the La_1−xY_xMn₂Si₂ system where the electron density is derived to be 2.83 states/eV atom for x = 0, 2.51 states/eV atom for x = 0.25, 2.54 states/eV atom for x = 0.3 and 1.47 states/eV atom for x = 1.0⁴⁹. Moreover, it is also noted that the electron density at the Fermi level for TbMn₂Si₂²⁹ was reported to be 2.38 states/eV atom, which is close to the values reported here for Tb_1−xY_xMn₂Ge₂ samples.

Table 2 Calculated heat capacity parameters for Tb_1−xY_xMn₂Ge₂ (x = 0, 0.1, 0.2).

Conclusions

In conclusion, we have carried out a detailed investigation around the region of the magnetic transitions of compounds in the Tb_1−xY_xMn₂Ge₂ series (x = 0, 0.1, 0.2) by variable temperature x-ray diffraction, heat capacity, differential scanning calorimetry and magnetic measurements. Two magnetic phase transitions occur at T_N^inter and T_C^inter for each of the three samples. The antiferromagnetic transition at T_N^inter is shown to increase slightly with increase in the Y concentration, while the ferromagnetic transition at T_C^inter drops significantly. The mechanism of reduction of T_C due to the substitution of Y for Tb has been analysed and chemical pressure is found to play a significant role. Moreover, the entropy change of Tb_0.9Y_0.1Mn₂Ge₂ is found to exhibit very good magnetocaloric performance around T_C^inter (−ΔS = 11.9 J kg⁻¹ K⁻¹ and RCP = 102.9 J kg⁻¹ for a field change of ΔB = 0–5 T) with a small hysteresis loss of 5.36 J/kg. This behaviour reflects the potential suitability of Tb_0.9Y_0.1Mn₂Ge₂ for operation as a magnetic refrigerant below the nature gas liquefaction temperature. The Debye temperature and the density of states N(E_F) at the Fermi level have been determined and analyzed from the heat capacity.