Overview of the Characteristic Features of the Magnetic Phase Transition with Regards to the Magnetocaloric Effect: the Hidden Relationship Between Hysteresis and Latent Heat
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Abstract
The magnetocaloric effect has seen a resurgence in interest over the last 20 years as a means towards an alternative energy efficient cooling method. This has resulted in a concerted effort to develop the socalled “giant” magnetocaloric materials with large entropy changes that often come at the expense of hysteretic behavior. But do the gains offset the disadvantages? In this paper, we review the relationship between the latent heat of several giant magnetocaloric systems and the associated magnetic field hysteresis. We quantify this relationship by the parameter Δμ _{0} H/ΔS _{L}, which describes the linear relationship between field hysteresis, Δμ _{0} H, and entropy change due to latent heat, ΔS _{L}. The general trends observed in these systems suggest that itinerant magnets appear to consistently show large ΔS _{L} accompanied by small Δμ _{0} H (Δμ _{0} H/ΔS _{L} = 0.02 ± 0.01 T/(J K^{−1} kg^{−1})), compared to local moment systems, which show significantly larger Δμ _{0} H as ΔS _{L} increases (Δμ _{0} H/ΔS _{L} = 0.14 ± 0.06 T/(J K^{−1} kg^{−1})).
Keywords
Latent Heat Phase Transition Heat Capacity Magnetocaloric Effect Magnetic Entropy ChangeIntroduction
In magnetic materials the application of a magnetic field under adiabatic conditions can result in heating due to the magnetocaloric effect, where a reduction in magnetic entropy results in an increase of lattice entropy. This suggests the possibility of a cyclic process to achieve cooling, a technology that has the advantages of an absence of greenhouse gases and a potential increase in engine system efficiency compared to conventional vapour compression systems.[1] While utilisation of the magnetocaloric effect was proposed by Debye[2] and Giauque and MacDougall[3] for cooling below 1 K (−272 °C) in the early twentieth century, it was not until the seminal papers by Brown,[4] and Pecharsky and Gschneidner[5] that room temperature magnetic refrigeration started to be considered a viable application.
Almost two decades on, there are now a handful of materials that have been identified as suitable candidates for solidstate refrigeration. For the most part these material systems are described as exhibiting a giant magnetocaloric effect (GMCE); typically defined as an entropy change that exceeds that of the standard benchmark material, Gd. A common feature of such GMCE materials is magnetovolume or magnetostructural coupling that results in field and thermal hysteresis, an example of which can be found in Gd_{5}(Ge_{1−x }Si_{ x })_{4}. In this material system, an appropriate choice of Si content (x = 0.5) produces an orthorhombic to monoclinic structural transition that coincides with a ferromagnetic (FM) to paramagnetic (PM) transition.[6] While the entropy change at this field driven phase transition is promising (ΔS ~ 15 J K^{−1} kg^{−1}), this magnitude of entropy change requires large operating fields (~2 to 5 T) and the system shows significant field hysteresis (Δμ _{0} H ~ 1 T). In order to develop future attractive materials for use in a commercial solidstate refrigeration device, a compromise needs to be reached between the magnitude of ΔS, the required magnetic field, material fatigue and hysteretic losses.
In this paper, the general characteristics of the phase transition in a collection of material systems considered interesting with regards to applications of the magnetocaloric effect will be discussed. The magnetovolume, structural and elastic coupling in these material systems that leads to a moderate or giant magnetocaloric effect will also be considered in light of the associated detrimental hysteresis and its relationship with the development of latent heat.
Experimental
The full details of preparation routes for each of the samples discussed in this paper are reported elsewhere. Single crystal Gd_{5}Ge_{2}Si_{2} was prepared by the Bridgman method;[7] DyCo_{2} was prepared by arc melting;[8] the La(Fe,Si)_{13} bulk ingots were prepared by arc melting followed by annealing at 1323 K (1596 °C) for 7 days; the Co(Mn_{1−x }Fe_{ x })(Si_{1−y }Ge_{ y }) alloys were prepared by induction melting followed by annealing at 1223 K (1496 °C);[9] Mn_{1.95}SbCr_{0.05} was prepared by arc melting;[10] La_{0.67}Ca_{0.33}MnO_{3} and RMnO_{3} were prepared by standard solidstate methods.[11,12]
Magnetization measurements were carried out using a Quantum Design vibrating sample magnetometer (VSM) for temperatures ranging from 77 K to 300 K (−196 °C to 27 °C) and at a field sweep rate of 0.5 T/min.
Microcalorimetry measurements were obtained for 100 μm fragments using a commercial Xensor (TCG3880) SiN membrane gauge that has been adapted to work as an ac calorimeter[13] or an adiabatic temperature probe[14] in a cryostat capable of 0 to 8 T and 4.2 K to 295 K (−268.9 °C to 22 °C). (The sample size is typically limited to the size of the heater on the SiN chip: 50 × 100 μm.) As an ac calorimeter the heat capacity, C _{p}, is measured by the application of an ac temperature modulation to the sample while held in He exchange gas. The solution to the heat transfer equation yields C _{p} of the sample as a function of the phase and amplitude of the resultant thermal modulation (with respect to the source signal). Due to the nature of the ac technique, it measures C _{p} alone, and not the latent heat, ΔQ _{L}. While any latent heat that occurs on first driving the phase transition may be registered, as it is neither reversible nor necessarily in phase with the temperature modulation, it will not yield a repeatable measurement.[15] In order to fully sample the latent heat, a separate measurement is required: the adiabatic temperature probe.[14] When operated in this mode, the helium exchange gas is evacuated and the temperature change due to a change in applied magnetic field is registered. The noise floor of this measurement is of the order of 1 μV, which is equivalent to 1 nJ.
Characteristics of the Continuous Phase Transition
 (a)
A continuous change in magnetic moment,
 (b)
Good agreement with the Curie–Weiss relationship for inverse susceptibility,
 (c)
Continuous changes in the heat capacity as a function of field.
It should be noted that while heat capacity measurements are often presented as a function of temperature, we instead display them here with respect to field. This is in order to complement the latent heat data presented later. Also, for measurements close to room temperature, small changes in the heat capacity might be observed due to sample movement (Figure 1(b) at 296 K (23 °C)) as the adhesion of the thermal grease used to fix the sample onto the membrane starts to decrease. This results in the small hysteresis observed in Figure 1(b) where it would otherwise not be expected.
Mechanisms for the Onset of the FirstOrder Phase Transition
 (a)
A coupled structural transition (e.g., Gd_{5}Ge_{2}Si_{2} changes from an orthorhombic FM to monoclinic PM,[6])
 (b)
Strong magnetoelastic coupling (e.g., competing exchange interactions in CoMnSi,[19] or the Jahn–Teller distortion in manganites[20]),
 (c)
An associated change in volume, i.e., magnetovolume coupling (e.g., La(Fe,Si)_{13} exhibits a volume change, ΔV/V, of ~1 pct[21]).
In each case, the energy barrier that results in a firstorder phase transition is due to some change in the crystal lattice that occurs alongside a magnetic change of state.[18]
The major advantage of the microcalorimetry technique presented here is that the first order characteristics of the phase transition (i.e., latent heat, ΔQ _{L}) can be separated from continuous changes in heat capacity. While other techniques such as differential scanning calorimetry can be used to estimate ΔQ _{L}, this often also includes disorder broadening of the phase transition that makes it difficult to distinguish between the actual latent heat and gradual changes in the heat capacity for complex phase transitions. The size of the sample measured using this technique is one factor that limits the averaging effect of disorder broadening, another is the thermal modulation technique itself. Overall, this results in an independent measure of how much the total entropy change, ΔS _{tot}, increases as latent heat is introduced, while enabling correlation of ΔQ _{L}, with the observed hysteresis of the phase transition, Δμ _{0} H. When studying the material system in this way it will generally fall under one of two cases (or a combination of the two), as outlined in the following sections.
Case 1: StepLike Changes in Heat Capacity
For the Gd_{5}Ge_{2}Si_{2} single crystal this results in a value of ΔS _{L} = 6.39 ± 0.3 J K^{−1} kg^{−1} at 285 K (12 °C), compared to the total entropy change at this temperature of ΔS _{tot} = 12.5 ± 0.6 J K^{−1} kg^{−1}.[23] This value agrees well with the structural contribution determined by Liu et al.[24] and Pecharsky et al.[25] to be between 40 and 60 pct of ΔS _{tot}.
 (a)
A continuous change in bulk magnetic moment that is comprised of multiple sharp changes in the local magnetic moment (nucleation), with slightly different critical fields (Figure 2(a)).
 (b)
A linear trend in the Curie–Weiss plot (inverse susceptibility) away from T _{c}, and then a departure from linearity with a sharp drop to zero at T _{c} (Figure 2(b)),
 (c)
A steplike change in the heat capacity accompanied by latent heat.
Typically, with Case 1 phase transitions, the latent heat and change in heat capacity are only weakly dependant on temperature and where there are changes in the magnitude of the latent heat it is not directly reflected in the heat capacity.
Case 2: Coupled Heat Capacity and Latent Heat Behavior
 (a)
A continuous change in magnetic moment (Figure 3(a)),
 (b)
Divergence of the inverse susceptibility from the linear behavior expected by the Curie–Weiss law (Figure 3(b)), indicating two different Curie temperatures: T _{o} (in the absence of a volume change) and T _{c} (the observed Curie temperature),
 (c)
A large change in the heat capacity (ΔC > 100 pct) accompanied by latent heat, both of which change dramatically with the increasing temperature (Figure 3(c) and (d)).
The shape of the heat capacity curve is also significantly different to both continuous and Case 1 phase transitions. For example, in Figure 3(c) a peak in C _{p} can be observed (that in a measurement that does not separate the contributions might be mistakenly attributed to latent heat) that moves to higher magnetic fields as the temperature is increased. Another important point to note is that the magnitude of the change in heat capacity is significantly larger for Case 2 phase transitions. In addition, these changes can exceed observations for continuous phase transitions (such as Gd in Figure 1(b)).
Exceptions
Disorder broadening[28] (of the phase transition) may also make it difficult to identify whether it is first order (or not). While it is common to use the Banerjee criterion to determine whether a phase transition is first order, as this criterion is based on the mean field approximation it is likely to break down in the case of itinerant magnets where spin fluctuations may play a larger role.[8,12]
Hysteresis as a Function of Entropy Gain
In Figure 5(a), some results for the CoMn_{1−x }Fe_{ x }Si_{1−y }Ge_{ y } alloy (which is sensitive to strain and has strong magnetocrystalline anisotropy), are shown. In this case, while each individual measurement followed a linear trend (aside from the Fe doped sample, which will be discussed later), it appears that the gradient of this line, Δμ _{0} H/ΔS _{L}, was also sensitive to strain and field orientation. For example, by quenching the CoMnSi ingot after melting, strain is introduced that inhibits the phase transition. This results in an increase in Δμ _{0} H/ΔS _{L} of the quenched CoMnSi (quenched).[30] Additionally, as the size of the fragment measured in microcalorimetry is of the order of a single crystallite it is possible that this could be aligned according to the easy axis of magnetization resulting in a decrease of Δμ _{0} H/ΔS _{L} (Ge doped).[31] Lastly, it was also shown that for the Fe doped CoMnSi at low temperatures the structural contribution to entropy change continues to increase, while the magnetic entropy change has saturated. As these two contributions compete in this material system, this results in a decrease of the total entropy change, and indeed the latent heat, thus Δμ _{0} H/ΔS _{L} is no longer constant (Fe doped).[32] Overall, while these additional factors can influence the absolute value of Δμ _{0} H/ΔS _{L}, each of these examples still followed a general linear trend.

Case 1—Where magnetostructural or magnetoexchange effects typically dominate, Δμ _{0} H/ΔS _{L} = 0.14 ± 0.06 T K kg J^{−1}.

Case 2—Where magnetovolume or magnetoelastic effects typically dominate, Δμ _{0} H/ΔS _{L} = 0.02 ± 0.01 T K kg J^{−1}.
Note that the large error on the value of Δμ _{0} H/ΔS _{L} here is to encompass the selection of materials shown. This result suggests that although magnetostructural coupling could, in principle, lead to larger entropy changes, the associated hysteresis is significantly large. In particular, magnetostructural coupling appears to be less attractive in comparison to other routes of introducing firstorder behavior, such as magnetovolume or magnetoelastic coupling: there is a fivefold increase of the value of Δμ _{0} H/ΔS _{L} for Case 1 phase transitions compared to Case 2 phase transitions.
Conclusions
We have shown that for firstorder magnetic phase transitions the general characteristics can fall into one of two categories: Case 1, where magnetostructural coupling probably plays a large role and the latent heat is largely independent of temperature; and Case 2 where an itinerant metamagnetic phase transition occurs and the heat capacity and latent heat both change dramatically with temperature. Once identified, it appears that Case 1 phase transitions exhibit larger increases of Δμ _{0} H with respect to ΔS _{L}, compared to Case 2 phase transitions (Δμ _{0} H/ΔS _{L} = 0.14 ± 0.06 and 0.02 ± 0.01 T K kg J^{−1}, respectively).
The results of this work indicate that the largest gain is to be had in itinerant magnets (that typically exhibit Case 2 characteristics), where spin fluctuations may play a role in lowering the energy barrier to the phase transition. An additional benefit is the ease with which they can be tuned with respect to field, temperature, or composition in order to approach the tricritical point.[1] Overall, this suggests that material systems which exhibit an easily tunable critical point (i.e., where the phase transition moves from continuous to first order) are desirable not only because they enable better control of the desired properties (ΔS, ΔT _{ad}, and T _{c}), but also because of the lower hysteresis (Δμ _{0} H) associated with them.
While the last 20 years have seen an increase in known material systems that exhibit favourable MC properties, the detrimental impact of hysteresis, durability and poor thermal conductivity are starting to become apparent.[33, 34, 35, 36] This has resulted in a shift of focus towards material systems that lie closer to the critical point: on the cusp of firstorder and continuous phase transitions. As this technology approaches maturity, it is reasonable to speculate that focus will continue to shift towards tricritical magnets that can be easily shaped, react well with thermal engineering, and respond rapidly to changing magnetic field.
Notes
Acknowledgments
The authors would like to express their gratitude to the various individuals who provided us with good quality samples, without which this paper would not have been possible. In particular: T. A. Lograsso and Y. Mudryk of Ames Laboratory (Gd_{5}Ge_{2}Si_{2}, DyCo_{2}), K. G. Sandeman (CoMnSi based alloys), J. Lyubina and O. Gutfleisch (La(Fe,Si)_{13}), L. Caron (Mn_{1.95}SbCr_{0.05}), J. Turcaud (La_{0.67}Ca_{0.33}MnO_{3}) and A. Berenov (RMnO_{3}). L. F. C. acknowledges funding for this work from EPSRC EP/G060940/1.
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