Magnetic and magnetocaloric properties of Dy₅Pd₂: role of magnetic irreversibility

Abstract

In this report the magnetic behavior and magnetocaloric response of the intermetallic compound Dy₅Pd₂ has been investigated from detailed magnetization measurements. Magnetic cluster glass behavior has been observed in the crystalline Dy₅Pd₂ compound below 38 K. Magnetocaloric entropy changes evaluated using Maxwell’s relation (MXR) from zero-field-cooled magnetization isotherms, which is used mostly in the literature, shows a giant inverse magnetocaloric effect (MCE) below 20 K and a large conventional MCE around 60 K in this compound. The observed value of inverse MCE is ∼29 J kg⁻¹ K⁻¹ for 50 kOe external magnetic field change at T = 3.5 K. In this context, the origin of the giant inverse MCE and its temperature dependence has been studied. The applicability of MXR and procedure to obtain reversible MCE using MXR which is important for practical application has been discussed. Dy₅Pd₂ exhibits a reversible MCE with a peak value of magnetic entropy change (ΔS^max_M) 8.3 J kg⁻¹ K⁻¹ at 60 K under a magnetic field of 50 kOe and an irreversible inverse MCE below 20 K.

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1 Introduction

On application of an external magnetic field the magnetic moments of paramagnetic (PM) or ferromagnetic (FM) materials have the tendency to align along the direction of the magnetic field. As a result in the presence of an external magnetic field under adiabatic conditions an increase of temperature occurs to compensate for the decrease of magnetic entropy, which is known as the magnetocaloric effect¹ (MCE). The MCE is quantified¹ by the adiabatic temperature change ΔT_ad, or as the isothermal magnetic entropy change −ΔS_M. In contrast to PM or FM materials, in some magnetic materials on application of an external magnetic field the configurational entropy of the spin structure increases. As a result, the application of a magnetic field adiabatically generates cooling (−ΔS_M negative) of the material due to the decrease of lattice entropy, which is known as an inverse magnetocaloric effect² (IMCE). The irreversibility due to metastable states also generates positive entropy.³

In most of the literature, authors use Maxwell’s relation (MXR) to evaluate magnetocaloric (MC) entropy change using zero-field-cooled (ZFC) isothermal magnetization data. The use of Maxwell’s relation is only valid if the system is in thermodynamic equilibrium.⁴ On the other hand, zero-field-cooled states do not yield information about the equilibrium states. The positive entropy due to the irreversible process^5–8 evaluated using MXR from ZFC magnetization may be confused with reversible IMCE. This raises some important questions; (i) can one calculate reversible MC entropy changes accurately from magnetization data of the materials having metastable states and irreversibilities; (ii) can one calculate the amount of irreversibility and the dependencies involved in these irreversible processes. The origin of the giant irreversible IMCE and the procedure to discriminate the irreversible part of MCE from the reversible one has not yet been studied in detail and therefore a detailed methodology needs to be worked out.

Intermetallic R₅Pd₂ (R = rare-earth) compounds have drawn much attention because the R³⁺ ions are arranged in a triangular configuration having geometrical frustration.^9–12 Dy₅Pd₂ is likely to show glassy magnetic behavior associated with the frustration due to the atomic disorder and layered crystal structure, which are inherent features of R₅Pd₂-type compounds.^12,13 We have investigated the magnetic behavior of Dy₅Pd₂ at different magnetic field strengths under zero-field-cooled warming (ZFCW), field cooled cooling (FCC), and field cooled warming (FCW) modes. Our dc and ac magnetic measurements reveal that Dy₅Pd₂ exhibits a non-equilibrium cluster glass (CG) magnetic state consistent with the triangular lattice configuration favoring spin frustration¹⁴ and glass dynamics. The magnetic entropy change (−ΔS_M) evaluated from the measured M(H) curves under ZFC protocol shows large irreversible negative to giant reversible positive values with increasing temperature. Magnetic entropy change due to the application of external magnetic field has also been calculated from isothermal M(H) curves extracted from M(T) curves measured under different applied magnetic fields in ZFC, FCC and FCW protocols. In contrast, the giant IMCE has not been observed in the MC entropy change calculated from FCC and FCW magnetization data. In this aspect, the validity of MXR and applicability of the procedure to calculate MCE by using a magnetization method has been described. An estimation of internal entropy production and its temperature dependence has been presented.

2 Experimental details

The polycrystalline Dy₅Pd₂ compound was prepared by arc melting of constituent elements in an argon atmosphere. Dy of purity 99.9% was bought from Alfa Aesar (a Johnson Matthey company) and Pd of purity higher than 99.99% was bought from Arora-Matthey Limited. The sample was re-melted several times to ensure homogeneity. The X-ray diffraction (XRD) study was carried out around room temperature by using a Rigaku diffractometer using a Cu-Kα source of wavelength 1.54 Å. The dc magnetization measurements were carried out in the temperature range 2–300 K and for applied magnetic field up to 7 T using a superconducting quantum interference device (SQUID VSM, Quantum Design) magnetometer. In ZFC mode at first the sample was cooled in the absence of an external magnetic field from the temperature which is well above the transition temperature to the lowest possible temperature and the required magnetic field was applied, followed by the measurement during the warming cycle. After ZFC measurement, without switching off the applied magnetic field, magnetization was measured during cooling (FCC mode) and also during warming (FCW mode). AC susceptibility measurements were taken in the temperature range 4–90 K using a commercial susceptometer (CryoBIND model) at frequencies in the range 37–1737 Hz. Specific heat studies were carried out using the semi-adiabatic heat pulse method.

3 Results and discussion

The room temperature XRD spectrum of the prepared sample is shown in Fig. 1. The lattice structure of the bulk crystalline sample has been analyzed by a standard profile fitting method using the FULLPROF 2009 program. The XRD pattern confirms the single-phase nature of the compound, which crystallizes in a cubic Dy₅Pd₂ type (space group Fd3m) structure. The evaluated value of the lattice parameter of Dy₅Pd₂ from the XRD pattern using Rietveld profile fitting is found to be 13.517(1) Å. The Bragg R-factor and RF-factor are 2.31% and 3.26% respectively. The lattice parameter of the sample is in agreement with the previously reported value.¹⁵

Fig. 1 Powder X-ray diffraction pattern and profile fitting of the XRD spectrum of Dy₅Pd₂.

The temperature dependence of ZFC susceptibility (χ) data for an applied magnetic field of 50 Oe is shown in Fig. 2(a). χ(T) exhibits a peak at about T_f = 38 K. The high temperature region of the inverse susceptibility curve has been fitted using the Curie–Weiss law. With decreasing temperature χ⁻¹(T) deviates from the Curie–Weiss straight line with cooling below 80 K. This deviation from the Curie–Weiss law indicates the beginning of the development of the short-range magnetic correlations. The estimated value of the effective magnetic moment (μ_eff) and the paramagnetic Curie temperature (θ_P) using the slope of the temperature dependence of 1/χ in the PM region comes out to be 11.43 μ_B per Dy-atom and +50 K, respectively. These values are in agreement with previously reported values.¹⁶ The estimated value of the frustration parameter⁹ (f = θ_P/T_f) is 1.3. The temperature dependence of ZFC, FCC and FCW dc susceptibility (χ) at different values of the externally applied magnetic field are shown in Fig. 2(b). From the figure it is evident that the magnetic susceptibility of Dy₅Pd₂ exhibits irreversible behavior while measuring under ZFC and FC protocols. The extent of irreversibility and irreversibility temperature T_irr(H) decreases with increasing applied magnetic field. This is generally observed in glass-like systems.¹⁷ Though the value of the frustration parameter is not too high, the bifurcation in M_ZFC and M_FC has been observed for a maximum value of externally applied magnetic field as high as 50 kOe.

Fig. 2 (a) The temperature dependence of susceptibility and inverse susceptibility measured in the presence of 50 Oe external magnetic field. The solid line is a linear fit to the data. (b) Zero-field-cooled warming, field-cooled warming and field-cooled cooling branches of magnetization measured as a function of temperature under different applied magnetic fields for Dy₅Pd₂ indicating large irreversibility in low temperature region.

To investigate the nature of the underlying magnetic state, ac susceptibility measurements have been taken. The ac susceptibility data at ac field H_ac = 1.5 Oe with frequency (ω) 37–1737 Hz is shown in Fig. 3(a) and (b). Both the real and the imaginary part of ac susceptibility data for 37 Hz frequency show a maximum at T_f = 41 K. The susceptibility maximum at T_f shifts with ω (Fig. 4). A criterion that is often used to compare the frequency dependence of T_f in different glassy systems is to compare the relative shift in freezing temperature per decade of frequency: δT_f = ΔT_f/[T_fΔ(log ω)]. The obtained value of δT_f is 0.017, which is intermediate between the canonical spin glass (SG) systems (e.g. δT_f for CuMn ∼0.005) and non-interacting ideal superparamagnetic systems¹⁸ (e.g. δT_f for a-[Ho₂O₃(B₂O₃)] ∼0.28). The value of δT_f is more consistent with those of CG.¹⁹

Fig. 3 (a) The real part of ac susceptibility [χ′(T,ω)] data. (b) χ′(T,ω) in the temperature region ranging from 40 to 48 K.

Fig. 4 The T_f dependence and fit of the experimental data of χ′(T) with the Vogel–Fulcher law.

The frequency dependence of freezing temperature T_f (Fig. 4) obtained from the real part of the ac susceptibility (χ′) was fitted with the empirical Vogel–Fulcher (VF) law: τ = τ₀ exp[E_a/k_B(T_f − T₀)], where τ is relaxation time, E_a is activation energy, and T₀ is VF temperature. The best fitting gives E_a/k_B = 193.5 K, τ₀ = 10⁻¹² s and T₀ = 35.205 K. The VF temperature T₀ is slightly smaller than T_f obtained from ac measurements. The non-zero value of T₀, which is the measure of intercluster strength,²⁰ indicates the presence of finite interaction between the clusters.²¹ Altogether, the frequency dependence of the freezing temperature T_f provides clear evidence for the formation of a CG state in Dy₅Pd₂.

Zero-field heat capacity [C₀(T)] data of the Dy₅Pd₂ are shown in Fig. 5. Previously R₅Pd₂ compounds were suggested to exhibit long range AFM ordering¹⁰ with the possibility of temperature driven spin-reorientation transitions at lower temperatures. However no distinct feature was observed in our measured C₀(T) data around T_f which confirms the absence of long range magnetic ordering.^22,23 The peak observed in the M_ZFC(T) curve and the upturn of the M_FC(T) curve below T_f are characteristic of cluster glass systems with high concentrations of magnetic atoms.^13,24

Fig. 5 The heat capacity C_p as a function of temperature for the cluster glass Dy₅Pd₂ both in the presence and absence of an external magnetic field.

The isothermal magnetization curves as a function of applied magnetic field are displayed in Fig. 6. Before carrying out isothermal ZFC M(H) measurement at different temperatures, the temperature of the sample was raised to well above T_f each time and then cooled down slowly (1 K min⁻¹) to the measuring temperature in the absence of a magnetic field. Isothermal magnetization curves as a function of applied magnetic field were extracted by transposing the isofield FCC and FCW temperature dependence of magnetization data measured at different magnetic field values. With decreasing temperature the ZFC isotherm steadily drops for T < 20 K giving S-shaped M(H) curves and magnetization is far from saturation under magnetic fields as high as 70 kOe. Magnetization isotherms obtained under FCC and FCW protocols steadily increase up to the lowest measured temperature and are very similar to each other but different from those measured under ZFC protocol in the low temperature region (T < 20 K).

Fig. 6 The isothermal magnetization curves with increasing field at three different temperatures measured under a ZFC protocol and extracted from the M(T) data under FCC and FCW protocols.

The change in magnetic entropy of a system is usually derived by integrating M_ZFC(H) data using MXR:^1,25,26

Here H_min and H_max represent the initial and final values of the magnetic field. Eqn (1) is integrated numerically in the desired range of magnetic fields on the basis of the set of M_ZFC(H) curves at different temperatures. We have calculated the area between magnetic isotherms at neighboring temperatures, and then evaluated the entropy change by dividing the temperature difference. The temperature dependence of −ΔS_M for 70 kOe magnetic field change (ΔH) is shown in Fig. 7. −ΔS_M changes continuously from giant negative values (IMCE) at the lowest temperature to giant positive values (MCE) as temperature increases and after showing a broad maximum around 60 K it tends to zero. The maximum observed value of −ΔS_M for 70 kOe magnetic field change is (−ΔS^max_M=) 8.3 J kg⁻¹ K⁻¹ at T = 57.5 K [relative cooling power (RCP) = ΔS^max_M × ΔT_FWHM = 5.51 J cm⁻³], which is comparable to other giant MC materials.^27–29 The maxima in the temperature dependence of magnetic entropy change have been observed at higher temperatures than the glass transition temperature, which is consistent with earlier reports³⁰ on the magnetocaloric effect in glassy materials. However the giant IMCE observed in the low temperature region is −30.5 J kg⁻¹ K⁻¹ for 7 T magnetic field change at 3.5 K, which is surprisingly large.

Fig. 7 Magnetic entropy changes under a magnetic field change of 7 T obtained from the heat capacity and magnetization (under ZFCW, FCW and FCC protocols) measurements.

The temperature dependence of the entropy²⁵ (S) curve can be obtained from C_H(T) data at the corresponding magnetic field using the relationship³¹ . The plot of −ΔS_M(T) shown in Fig. 7 follows a positive caret-like shape with a maximum around 60 K. The values of −ΔS_M(T) and temperature dependence for different ΔH evaluated from both ZFC magnetization and specific heat data are similar to each other around 60 K and above. Below 60 K gradual deviation from each other was observed, while the magnetic entropy change (−ΔS_M) obtained from the MXR changes continuously from giant negative values to giant positive values with increasing temperature. In contrast, the −ΔS_M determined from the heat capacity data is always greater than zero.

The SG state is a non-equilibrium one in the thermodynamical sense and the ZFC magnetization does not yield information of the thermal equilibrium behavior.⁴ According to mean-field theory there exist a large number of degenerate thermodynamic states with the same macroscopic properties but with different microscopic configurations separated by free-energy barriers in phase space exhibiting many valleys.³² Multiple metastable states due to random magnetic anisotropy (RMA) give rise to irreversibility. Varying the external field can make the system jump to the nearest local minima.³³ The results of measurement then depend on the kinetics of the phase transition and on the experimental procedure.

Here entropy change in Dy₅Pd₂ has been calculated following the method adopted in the reversible region, i.e., from isothermal ZFC magnetization data. As a result, there is an additional contribution introduced in the estimation of −ΔS_M from ZFC magnetization data due to the metastable nature of the measured state which is not taken into consideration. The irreversible positive internal entropy production due to RMA should correlate with the irreversibility in ZFC and field cooled (FC) magnetization which was also observed in rare-earth based bulk metallic glasses.³³ On the other hand the FC magnetization is reversible.³⁴ In FC measurement the system goes to the near equilibrium state without further “minima hop”.⁴ We have calculated magnetic entropy change due to the application of an external magnetic field in Dy₅Pd₂ by using MXR also from the magnetization isotherms obtained by FCC and FCW protocols. The temperature dependence of −ΔS_M, shown in Fig. 7, around and above the freezing temperature is very similar to those observed under the ZFC protocol. However, the temperature dependence in a lower temperature regime is distinctly different. The large IMCE observed at low temperature is absent in the −ΔS_M measured by FCC and FCW protocols. The magnetic entropy change (−ΔS_M) determined from the FCC and FCW magnetization data using MXR is always larger than zero.

The above results indicate that the irreversible entropy in a lower temperature region may be related to the irreversibility observed in ZFC and FC magnetization data. The ZFC susceptibility is the FC susceptibility modified by the random anisotropy field.³⁵ The temperature at which a maximum is observed in the ZFC susceptibility and the irreversibility starts is related to the magnitude and the temperature variation of the coercivity (H_c), which is a measure of the magnetic anisotropy.

The temperature dependence of the coercive field H_c in Dy₅Pd₂ has been shown in Fig. 8(a). It can be very well described by the exponential function^20,36 H_c = H_c0 exp(−αT). As a result eqn (2) can be expressed as

Fig. 8 (a) Temperature dependence of coercive field. (b) Temperature dependence of irreversible magnetization normalized with respect to ZFC magnetization and irreversible entropy changes calculated both from irreversible magnetization and taking the difference between FC and ZFC entropy changes. The irreversible entropy calculated in both ways gives similar results. The solid lines are the fitted curves.

The temperature dependence of (M_FC − M_ZFC)/M_ZFC (=H_c/H) below the freezing temperature for 10 kOe applied magnetic field in Dy₅Pd₂ is shown in Fig. 8(b) which follows exp(−αT) dependence with the value of fitting parameter (α=) 0.153(1) K⁻¹.

To investigate the origin of the giant irreversible IMCE measured under the ZFC protocol using MXR we have analyzed the difference in magnetic entropy change measured in the ZFC and FC protocols. The irreversible magnetic entropy change^37,38 (δS_irr) from difference (irreversible) magnetization [(M_FCW − M_ZFC) and (M_FCC − M_ZFC)] by using MXR has been calculated and is shown in Fig. 8(b). Fig. 8(b) shows that the temperature dependence of δS_irr in the low temperature region can be fitted well by an exponential dependence [exp(−αT)/T] with α-value 0.156(4). The value of the fitting parameter α obtained from the irreversible magnetization and irreversible entropy changes agree well. The difference of magnetic entropy changes between ZFC and FC protocols for 5 T magnetic field change are also shown in Fig. 8(b), which follow a similar dependence. These results show the additional contribution of positive entropy production on the estimation of MCE using MXR from ZFC magnetization in systems having hysteresis related to non-equilibrium behavior. The metastable states increase with decreasing temperature and entropy also increases exponentially.

4 Conclusions

In the intermetallic compound Dy₅Pd₂, the most commonly used procedure of calculating MCE from ZFC magnetization measurement using MXR results in a giant IMCE as well as a large MCE. Further detailed analysis using FC magnetization and heat capacity data confirms that the giant IMCE observed in Dy₅Pd₂ is due to magnetic irreversibility and the MC entropy change measured by using MXR from ZFC magnetization data does not give a correct estimation of the reversible magnetic entropy change. The difference between the entropy changes measured using the ZFC and FC protocols reflects the irreversibility in the non-equilibrium magnetic behavior. The present paper highlights the cluster glass magnetic behavior in Dy₅Pd₂ and the methodology for calculating the reversible magnetic entropy change which is the usable part for practical application using MXR from the FCW or FCC magnetization isotherms extracted from isofield magnetization curves.

Acknowledgements

Tapas Paramanik would like to acknowledge DAE-India for the fellowship.

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