Magnetocaloric Effect in R₆Fe₂₃: R = Dy, Ho, Er, and Tm

Abstract

We present a mean field study on the R₆Fe₂₃ system, where R = Dy, Ho, Er, and Tm, to calculate the magnetization, magnetic heat capacity, and the magnetocaloric effect (MCE) (isothermal entropy change (ΔS_m) and the adiabatic temperature change (ΔT_ad)) for different field changes up to 5 T and at temperatures ranging from 0 to 600 K. The maximum ΔS_m, using the trapezoidal method, for the R₆Fe₂₃ system is in the range 4.9–9.8 J/K mol, and the maximum ΔT_ad is in the range 9.56–15.17 K for a field change ΔH = 5 T. The largest ΔS_m and largest ΔT_ad are found for Tm₆Fe₂₃ to be 9.8 J/K mol and 15.17 K at Curie temperature T_c = 489 K, for ΔH = 5 T. The relative cooling power RCP(S) is in the range 148–560 J/mol for ΔH = 5 T, which is comparable to that of bench-mark materials, e.g., Gd. Also, the RCP based on the adiabatic temperature change, RCP(T) is in the range 449–1092 K² for ΔH = 5 T, which is comparable also to that of bench-mark materials, e.g., Gd. We investigated the type of phase transition in the light of universal curves, Arrott plots, and the behavior of the magnetic moment, magnetic heat capacity, and MCE (ΔS_m, ΔT_ad), which confirm that the type of phase transition at T_c of this system is second-order phase transition (SOPT). A calculation of some critical exponents adds more evidence that the MFT is fairly suitable to handle the aforementioned properties in the studied systems.

1 Introduction

The magnetocaloric effect (MCE) has been discussed before for magnetic and rare-earth intermetallic compounds [1,2,3]. There are different functional materials useful for technological applications such as magnetic refrigeration technology, which is hoped to be environmentally safer and a more efficient alternative to traditional refrigeration technology [4,5,6]. Many studies, on MCE, were done by using the mean field theory MFT, e.g., TmFe₂ [7], R₂Fe₁₄B [8], Gd-Co [9], LaMnO [10], and R₃Co₁₁B₄ [11]. Several studies have been carried out on the R₆Fe₂₃ compounds, e.g., magnetic properties, magnetostriction, electronic, and transport properties, as well as on crystal structure, lattice vibrations, and x-ray photoemission [12,13,14,15,16,17,18,19]. It is known that R₆Fe₂₃ compounds crystallize in Th₆Mn₂₃ type structure. Rare earth 4f-transition metals 3d intermetallic compounds show interesting magnetic properties. The two sublattice molecular field theory proved to be fairly suitable for calculating the magnetization of the compounds [12]. There are, however, few studies on the MCE of R₆Fe₂₃; therefore, our motivation to study in detail the magnetothermal properties and MCE (∆S_m, ∆T_ad) in the R₆Fe₂₃ system is justified. In addition, we calculated the relative cooling powers RCP(S) and RSP(T) as figures of merit. Investigating the order of the magnetic transition, via the temperature and field dependences of the magnetothermal and MCE properties, the universal curve, and the Arrott plots is also done.

2 Model and Analysis

By using the MFT, the exchange fields of rare earth elements and Fe sublattices can be expressed as follows [12, 20]:

$${\mathrm{H}}_{\mathrm{R}}\left(\mathrm{T}\right) =\mathrm{H}+\mathrm{d}\left[{ 6\mathrm{n}}_{\mathrm{RR}}{\mathrm{M}}_{\mathrm{R}}\left(\mathrm{T}\right)+ {23\mathrm{n}}_{\mathrm{RFe}}{\mathrm{M}}_{\mathrm{Fe}}\left(\mathrm{T}\right)\right]$$

(1)

$${\mathrm{H}}_{\mathrm{Fe}}\left(\mathrm{T}\right)=\mathrm{H}+\mathrm{d}\left[{6\mathrm{ n}}_{\mathrm{RFe}}{\mathrm{M}}_{\mathrm{R}}\left(\mathrm{T}\right)+{23\mathrm{n}}_{\mathrm{FeFe}}{\mathrm{M}}_{\mathrm{Fe}}\left(\mathrm{T}\right)\right]$$

(2)

The symbols in Eqs. (1) and (2) have their usual meaning [12]. The molecular field coefficients n_RR, n_FeFe, and n_RFe are dimensionless.

The magnetic moments of rare-earth M_R(T) and iron M_Fe(T) at temperature T.

$${\mathrm{M}}_{\mathrm{R }}\left(\mathrm{T}\right)={\mathrm{M}}_{\mathrm{R}}\left(0\right){\mathrm{B}}_{{\mathrm{J}}_{\mathrm{R}}} [\frac{{\mathrm{M}}_{\mathrm{R}}\left(0\right){\mathrm{H}}_{\mathrm{R}}\left(\mathrm{T}\right)}{{\mathrm{k}}_{\mathrm{B}}\mathrm{T}} ]$$

(3)

$${\mathrm{M}}_{\mathrm{Fe }}\left(\mathrm{T}\right)={\mathrm{M}}_{\mathrm{Fe}}\left(0\right){\mathrm{B}}_{{\mathrm{J}}_{\mathrm{Fe}}}\left[\frac{{\mathrm{M}}_{\mathrm{Fe}}\left(0\right){\mathrm{H}}_{\mathrm{Fe}}\left(\mathrm{T}\right)}{{\mathrm{k}}_{\mathrm{B}}\mathrm{T}}\right]$$

(4)

${B}_{J}\left(x\right)$ is the well-known Brillouin function and $x=\frac{{M}_{J}H}{{k}_{B}T}$

The total magnetic moment can be calculated from

$${\mathrm{M}}_{\mathrm{Total}}\left(\mathrm{T}\right)={ 6\mathrm{M}}_{\mathrm{R}}\left(\mathrm{T}\right)\pm 23{\mathrm{M}}_{\mathrm{Fe}}\left(\mathrm{T}\right)$$

(5)

From the following Maxwell relation, the magnetic entropy change is given by

$${\left(\frac{\partial \mathrm{S}(\mathrm{T})}{\partial \mathrm{H}}\right)}_{\mathrm{T}}={\left(\frac{ \partial \mathrm{M}(\mathrm{T})}{\partial \mathrm{T}}\right)}_{\mathrm{H}}$$

$$\Delta {\mathrm{S}}_{\mathrm{M}}(\mathrm{T})={\int }_{{\mathrm{H}}_{0}}^{{\mathrm{H}}_{\mathrm{f}}}\frac{\partial \mathrm{M}\left(\mathrm{T}\right)}{\partial \mathrm{T}}\mathrm{ dH}$$

(6)

The above integral could be cast into a summation by using the well-known trapezoidal rule [21].

A universal curve [22] is the relation between △S_m/△${\mathrm{S}}_{\mathrm{m}}^{\mathrm{peak}}$ vs. Ѳ. Where Ѳ is defined from the following:

$$\theta=\left(T-T_C\right)/\left(Tr-T_C\right),$$

where T_r is the reference temperature, it can be chosen such that [22]

$$\Delta {S}_{m}\left(Tr\right)=0.7\Delta {\mathrm{S}}_{\mathrm{m}}^{\mathrm{peak}}$$

(7)

The total heat capacity C_tot is calculated from the sum of the magnetic C_m, the electronic C_e, and the lattice C_l heat capacities [23, 24].

First, from the temperature-first derivative of the magnetic energy, we can calculate the magnetic contribution to heat capacity as the following equation:

$$\mathrm{U}=-\frac{1}{2}[{\mathrm{n}}_{\mathrm{RR}}{\mathrm{M}}_{\mathrm{R}}^{2}\left(\mathrm{T}\right)+{\mathrm{n}}_{\mathrm{FeFe}}{\mathrm{M}}_{\mathrm{Fe}}^{2}\left(\mathrm{T}\right)+2{\mathrm{n}}_{\mathrm{RFe}}{\mathrm{M}}_{\mathrm{R}}\left(\mathrm{T}\right){\mathrm{M}}_{\mathrm{Fe}}\left(\mathrm{T}\right) ]$$

(8)

$${\mathrm{C}}_{\mathrm{M}}\left(\mathrm{T}\right)= \frac{\partial \mathrm{U}}{\partial \mathrm{T}}$$

(9)

Second, the electronic heat capacity is calculated by [25]

$${C}_{e}\left(T\right)={\gamma }_{e}T=\left({\pi }^{2}/3\right){N}_{a}{k}^{2}N\left({E}_{f}\right)T$$

(10)

γ_e is the electronic heat capacity coefficient, and N (E_f) is the density of states at Fermi energy.

Third, the lattice heat capacity is calculated as follows:

$${\mathrm{C}}_{\mathrm{l}}(\mathrm{T})=9\mathrm{ Na }{K}_{B}{\left(\frac{\mathrm{T}}{{\theta}_{D}}\right)}^{3}{\int }_{0}^{{{\theta }_{\mathrm{D}}}/{_\mathrm{T}}}\frac{{e}^{x }{\mathrm{x}}^{4}}{{ (\mathrm{e}}^{\mathrm{x}}-{1)}^{2}}\mathrm{ dx}$$

(11)

where ϴ_D is Debye temperature and x = ϴ_D / T.

The adiabatic temperature change [23] is given by

$$\Delta {\mathrm{T}}_{\mathrm{ad}}={\int }_{{\mathrm{H}}_{0}}^{{\mathrm{H}}_{\mathrm{f}}} \frac{\mathrm{T}}{{\mathrm{C}}_{\mathrm{tot}}(T,H)} \frac{\partial \mathrm{M}\left(\mathrm{T}\right)}{\partial \mathrm{T}}\mathrm{ dH}$$

(12)

where C_tot is the total heat capacity.

The Arrott plots (M² vs. ∆S_m) and (H/M vs. M²) are used to investigate the order of the phase transition from the sign of the plot’s slopes. Namely, positive slopes indicate second-order phase transition, according to Arrot-Belov-Kouvel (ABK) [26, 27].

Ginsburg theory is expressed as follows [28]:

$$F=\frac{1}{2}\mathrm{ A}\left(\mathrm{T}\right){\mathrm{M}}^{2}+\frac{1}{4}\mathrm{B}\left(\mathrm{T}\right){\mathrm{M}}^{4}+\frac{1}{6}\mathrm{C}\left(\mathrm{T}\right){\mathrm{M}}^{6}-\mathrm{M}.\mathrm{H}$$

(13)

From the equilibrium condition at $T_c$, the magnetic equation of state is given by

$$\frac{\mathrm{H}}{\mathrm{M}} =\mathrm{ A}\left(\mathrm{T}\right)+\mathrm{B}\left(\mathrm{T}\right){\mathrm{M}}^{2}+\dots$$

(14)

where A(T) and B(T) are Landau’s coefficients.

The RCP [29] is considered as figure of merit for the magnetocaloric materials and is defined from magnetic entropy change as:

$$RCP\left(S\right)=\Delta {S}_{max}\left(T\right)*\delta {T}_{FWHM}$$

(15)

And also from the adiabatic temperature change,

$$RCP\left(T\right)=\Delta {T}_{max}\left(T\right)*\delta {T}_{FWHM}$$

(16)

3 Results and Discussion

3.1 Magnetization

Figure 1a–d exhibits the calculated magnetic moment of the two sublattices of rare earth R, Fe, and the total magnetic moment where R = Dy, Ho, Er, and Tm, respectively. Ferrimagnetic coupling is present in these compounds, with compensation points except for Tm₆Fe₂₃. We can show that the magnetic moment for Dy atom at 0 K is M_Dy (0) = g_Dy J_Dy = (4/3) × (15/2) = 10 μ_B/atom. The magnetic moments for R = Ho, Er, and Tm are calculated by Herbst and Croat [12]. As known, the magnetic moment for rare earth R is localized whereas that of the 3d-transition elements Fe in rare-earth intermetallic compounds. So, the magnetic moment of Fe sublattice is obtained from the experimental data of the total magnetic moment and the calculated magnetic moments of the rare earth R, for example, in Dy system M_Fe (0) = 48.8/23 = 2.12 μ_B/atom. Table 1 shows the experimental [12] and theoretical data of both the total magnetic moments and the Curie temperatures; for the R₆Fe₂₃ system, also, the percentage difference between the experimentally determined moments and calculated magnetic moment at T = 1 K [12] is shown in Table 1. This difference is only ≤ 2.4%, and the difference in the T_c data is ≤ 2.86%. As shown, the mean field theory succeeded in studying the magnetization for R₆Fe₂₃ and the rare earth intermetallic compounds such as R₃Co₁₁B₄ [11]. Magnetization calculations showed that, for example, both Dy₆Fe₂₃ and Dy₃Co₁₁B₄ are ferrimagnetic compounds with the total magnetic moments 11.6 and 16.6 μ_B/f.u, respectively.

Fig. 1

Total and sublattice magnetizations vs. temperature in zero magnetic field for a Dy₆Fe₂₃, b Ho₆Fe₂₃, c Er₆Fe₂₃, and d Tm₆Fe₂₃

Table 1 The calculated and experimental net magnetic moment, in zero field and 0 K, and the Curie temperatures for R₆Fe₂₃ system

3.2 Total Heat Capacity

Figure 2 shows the field dependence of the magnetic heat capacity as function of temperature for Ho₆Fe₂₃ at different magnetic fields up to 5 T. The maximum magnetic heat capacity decreases by increasing the applied field around Curie temperature, which is typical for compounds with second-order phase transition, for example, Ho₃Co₁₁B₄ [11] and TmFe₂ [7]. The electronic heat capacity is obtained from Eq. (10), and the coefficient γ_e is given from the materials project [30] as shown in Table 3. The lattice heat capacity is calculated from Debye temperature by Eq. (11). The Debye temperatures for most crystals are around room temperature.

Fig. 2

Temperature dependence of magnetic specific heat for Ho₆Fe₂₃ in external fields of 0, 1.5, 3, and 5 T

3.3 The Isothermal Entropy Change (△S_m)

△S_m has been calculated using the Maxwell relation from Eq. (6) and also using the trapezoidal rule. Figure 3a–d shows (△S_m vs. T) at applied fields up to 5 T, for R = Dy, Ho, Er, and Tm, respectively. Both direct and inverse MCE, i.e., two peaks are present: the first peak at T_c and the second at a temperature below the compensation temperature. The data of magnetic entropy change △S_m using both Maxwell’s relation and the trapezoidal rule showed agreement between the two methods, at low field changes, as shown in Table 2. For the sake of comparison with bench-mark materials and other R₆Fe₂₃ compounds, we compare our results of ΔS_m, which is in the range of 3.9 to 9.8 J/mol K for a field change ∆H = 5 T, with that of Gd metal, i.e., 1.48 J/mol K at ΔH = 5 T as reported by Wang et al. [31], and also with Jemmali et al. [32], where ΔS_m of Er₆Fe₂₃ is 3.64 J/mol K at ΔH = 1.4 T.

Fig. 3

Temperature dependence of the magnetic entropy change for a Dy₆Fe₂₃, b Ho₆Fe₂₃, c Er₆Fe₂₃, and d Tm₆Fe₂₃, for field changes of 1.5, 3, and 5 T

Table 2 The isothermal magnetic entropy change, for different magnetic field changes, in R₆Fe₂₃ system using the trapezoidal and Maxwell methods

3.4 Adiabatic Temperature Change (ΔT_ad)

We report in this part ΔT_ad. Because of the weak dependence of the total heat capacity on the applied field, around Curie temperature for the R₆Fe₂₃ compounds, as shown in Fig. 4 for example Dy₆Fe₂₃ system, the term T/C is taken out of the integral in Eq. (12). Figure 5a–c shows the adiabatic temperature change ΔT_ad for R = Dy, Ho, and Tm using Eq. (12) for applied fields up to 5 T. The maximum value for ΔT is 15.17 K, as shown in Table 3 for applied field 5 T, and in the case of Tm₆Fe₂₃, the temperature is decreasing by a rate of 3.03 K/T.

Fig. 4

The dependence of T/C on temperature in H = 1.5, 3, and 5 T for Dy₆Fe₂₃

Fig. 5

Adiabatic temperature change vs. T for a Dy₆Fe₂₃, b Ho₆Fe₂₃, c Tm₆Fe₂₃, for field changes of 1.5, 3, and 5 T

Table 3 The adiabatic temperature change ΔT_ad (K), for different magnetic field changes, in R₆Fe₂₃ system

3.5 Relative Cooling Power (RCP)

RCP is based on the isothermal process RCP(S), which is calculated by Eq. (15), for different field changes. The RCP(S) of Er₆Fe₂₃ at a field of 1.4 T is 95.76 J/mol, as reported by Jemmali et al. [32], and also RCP(S) of Gd metal is 150.72 J/mol at field change 5 T as reported by Wang et al. [31]. Also, RCP is based on the adiabatic temperature change RCP(T) by Eq. (16). It has no physical meaning, but it is used for numerical comparison of other MC compounds. A large RCP(T) generally indicates a better magnetocaloric material, as shown in Table 5. The relative cooling power RCP(T) is in the range 449–1092 K² for a field change ΔH = 5 T, which is compared with 967 K² of Gd at field 6 T, as reported by Gschneidner et al. [3]. The calculations show that the RCP(S) and RCP(T) increase with increasing the applied magnetic field, as shown in Tables 4 and 5.

Table 4 Relative cooling power RCP(S) in J/mol for different field changes, in R₆Fe₂₃ system

Table 5 Relative cooling power RCP(T) in K² for different field changes, in R₆Fe₂₃ system

3.6 The Universal Curve and Arrott Plot

Figure 6 displays the universal curves (ΔS_m vs. Ѳ) for Dy₆Fe₂₃ compound in applied fields of 1.5, 3, and 5 T. It can be clearly shown in Fig. 6 that the data of different applied fields collapse into a single universal curve, which shows the phase transition in the R₆Fe₂₃ system is a second-order phase transition.

Fig. 6

Universal curves of the Dy₆Fe₂₃ for field changes of 1.5, 3, and 5 T

According to Arrot-Belov-Kouvel (ABK) [26, 27], the order of the phase transition involved the second-order SOPT or the first-order FOPT, from the sign of the plots slopes. Namely, positive slopes indicate SOPT, whereas negative slopes or s-shaped slopes indicate FOPT. Figure 7a, b shows (M²) vs. (H/M) and ΔS_m vs. M² plots for the Dy₆Fe₂₃ system; the positive slopes with a minimum around T_C indicate that the type of phase transition is SOPT.

Fig. 7

a H/M vs. M² and b ΔS_m vs. M² for Dy₆Fe₂₃ compound in applied fields of 0.1, 1, 2, 3,4, and 5 T. T_c = 540 K

3.7 The Field Dependence of ΔS_m and the Critical Exponents

Figure 8 shows the ΔS_m vs. (H/T_C)^2/3 for R₆Fe₂₃ compounds. According to the mean field theory, the relation ΔS_m vs. (H/T_C)^2/3 is a criterion for the existence of the SOPT [33, 34]. It would be instructive to evaluate some of the critical exponents [35,36,37,38] and compare them with those of the mean-field theory. We have calculated the parameters n, β, δ, and ɤ, where n = 1 + (β − 1) / (β + ɤ) [37]. The parameter δ has been evaluated from the isothermal magnetization curve where M ~ H ^1/δ [38].

Fig. 8

ΔS_m vs. (Tesla/K)^2/3 for R₆Fe₂₃ compounds in applied fields from 0.1 to 5 T

The mean-field parameters are β = 0.5, ɤ = 1, and δ = 3. Our calculation showed that β and ɤ are, at most, 8% and 16% off the mean-field values, respectively. The exponent δ is at most 30% off the mean-field value.

4 Conclusion

We calculated the magnetothermal properties and magnetocaloric effect, △S_m and △T_ad, for R₆Fe₂₃ compounds using the MFT. The magnetization calculation exhibited that the R₆Fe₂₃ when R = Dy, Ho, Er, and Tm are ferrimagnetic compounds. The compensation point of the system is in the range of 86–230 K, and a Curie temperature is in the range of 489–540 K. For example, for R = Dy, the compensation point close to T_comp = 240 K and T_c = 540 K. The magnetic entropy change △S_m is calculated by Maxwell relation and by using the trapezoidal method. The highest ordinary MCE △S_m and △T_ad are 9.8 J/mol K and 15.17 K for R = Tm at applied field change 5 T. The RCP(S) is fairly comparable to those of Gd and Er₆Fe₂₃. The RCP(T) is in the range 449–1092 K² for a field change ΔH = 5 T, which is comparable to that of bench-mark materials, e.g., Gd. The field and temperature dependencies of the magnetic moment, △S_m, △Ta_d, the universal curves, and Arrott plots are investigated that the type of phase transition in the R₆Fe₂₃ system is SOPT. The MFT is appropriate for studying the magnetic properties and MCE of the R₆Fe₂₃ system.