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On the determination of the magnetic entropy change in materials with first-order transitions
ARTICLE IN PRESS Journal of Magnetism and Magnetic Materials 321 (2009) 3559–3566
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Current Perspectives
On the determination of the magnetic entropy change in materials with first-order transitions ¨ b, L. Caron a,b, Z.Q. Ou b,c, T.T. Nguyen b, D.T. Cam Thanh b, O. Tegus b,c, E. Bruck a
Instituto de F´ısica Gleb Wataghin, Universidade Estadual de Campinas-UNICAMP, C.P. 6165, Campinas 13 083 970, SP, Brazil Fundamental Aspects of Energy and Materials, Faculty of Applied Science, TU Delft, Mekelweg 15, 2629 JB Delft, The Netherlands c Key Laboratory for Physics and Chemistry of Functional Materials, Inner Mongolia Normal University, Hohhot 010022, PR China b
a r t i c l e in fo
abstract
Available online 9 July 2009
An accurate method to determine the magnetic entropy change in materials with hysteretic first-order transitions is presented, which is needed to estimate their potential for applications. We have investigated the effect of the maximal entropy change derived from magnetization measurements performed in different measurement processes. The results show that the isothermal entropy change can be derived from the Maxwell relations even for samples with large thermal hysteresis. In the temperature region with hysteresis, overestimating the entropy change can be avoided by measuring the isothermal magnetization of the sample after it is cooled from the paramagnetic state to the measurement temperature. In this way the so-called peak effect is not observed as shown here for a few compounds. & 2009 Elsevier B.V. All rights reserved.
Keywords: Magnetization process Magnetic entropy First-order transition Magnetic refrigeration
1. Introduction Magnetic refrigeration, based on the magnetocaloric effect (MCE), is considered as one of the most promising technologies to replace vapor-compression refrigeration, due to its low environmental impact and expected high energy efficiency [1,2]. Since the discovery of the giant MCE in Gd5(SixGe1x)4 [3,4] compounds, other giant magnetocaloric materials, MnFeP0.45As0.55 [5], MnAs1xSbx [6], La(Fe1xSix)13 [7] and La1xCaxMnO3 [8], were reported by different research groups. Recently, a colossal MCE with magnetic entropy change of 267 J/kgK for a 5 T field change was reported for MnAs [9] at hydrostatic pressure of 0.23 GPa. When partially substituting Fe (de Campos et al. [10]) or Cu (Rocco et al. [11]) for Mn in MnAs compounds, the same effects are reported even at ambient pressure. Most recently, a giant entropy change, 78 J/kgK, was reported for La0.8Ce0.2Fe11.4Si1.6 [12] at a field change of 3 T. A common feature of the alloys showing colossal magnetocaloric effects is the occurrence of considerable thermal hysteresis. This thermal hysteresis makes these materials not very suitable for applications in refrigerators that are expected to be operated cyclically at rather high cycle frequencies. However, as it is well known that hysteresis is an extrinsic property and thus may be reduced by processing, these large values of entropy change make these materials very interesting for further study.
Corresponding author. Tel.: +3115 2783158; fax: +31 20 6951500.
¨ E-mail address: [email protected] (E. Bruck). 0304-8853/$ - see front matter & 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2009.06.086
The observation of magnetic entropy changes as high as the socalled colossal ones brings two problems: the first is that the colossal values surpass the magnetic limit given by max DSM ¼ ln(2J+1), the second is that of the correct determination of the isothermal magnetic entropy change from magnetic measurements. Experimental discrepancies between the calculated and directly measured temperature change observed for Gd5Si2Ge2 has led Giguere et al. [13] to propose that the Clausius–Clapeyron relation, rather than the Maxwell relation
@S @M ¼ @B T @T B
ð1Þ
should be used. In a comment, Sun et al. [14] have shown that Clausius–Clapeyron is just a special case of the integrated Maxwell relations and thus the latter should be applicable. Nevertheless, the Clausius–Clapeyron relation has again been put forward very recently to analyze the magnetic response of (La,Pr)(Fe,Si)13 compounds [15]. Liu et al. argue that a careful study indicates that the Maxwell relation cannot be used in the vicinity of the Curie temperature because of the coexistence of paramagnetic and ferromagnetic phases, and therefore the observed huge entropy peak is a spurious result. Also Wada et al. [6] mention in their paper on Mn(As,Sb) that the peak should be considered as a spurious effect. On a more detailed derivation, de Oliveira et al. [16] showed that the so-called magnetic Clausius–Clapeyron relation does not hold for magnetic systems since it assumes there is no work done
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to take the system from the ferromagnetic to the paramagnetic state. The same work states that the Maxwell relations are not sufficient to calculate the entropy change around a first-order magnetic phase transition, and that in the region where the discontinuity itself is observed an extra term that resembles the Clausius–Clapeyron relation must be used. Since most observed first-order magnetic phase and metamagnetic transitions are rather smooth, calculations from real data using this model are seemingly impossible. Thus we reconsider the Maxwell relations, arguing they should be valid for the same reason why de Oliveira’s description fails to be of practical use: in nature, first-order transitions are far from step-like. And, as long as the first-order phase transition is properly probed, one should not incur in large errors. Here we show that the ‘‘colossal’’ peaks in the entropy change arise not from the application of the Maxwell relations but from the incorrect probing of the phase transition. In order to do that the measurement process being used to obtain such results is analyzed, keeping in mind the thermodynamical relations which describe the process. This paper, from here on, is organized in the following way: first a quick description is given of how the samples used in this paper were prepared. Then MnFeP0.8Ge0.2 is taken for a case study to explain and illustrate the various processes used to probe the phase transition, followed by an analysis of these processes. Next the results of correctly probing the phase transition are presented in contrast with the results found in literature for three other materials: La(Fe,Si)13, Mn0.99Cu0.01As and a Gd5Ge2.3Si1.7 single crystal. For Mn0.99Cu0.01As we also apply Liu’s method to show its inadequacy to this extreme case.
2. Experimental A polycrystalline sample of MnFeP0.8Ge0.2 was synthesized by ball milling and sintering process with appropriate amounts of starting materials. The sample was sintered at 1050 1C for 5 h, and then quenched immediately into water instead of slowly cooling down to ambient temperature. Also a Mn0.99Cu0.01As sample was prepared exactly as reported in Ref. [11]. La0.8Ce0.2Fe11.4Si1.6 was prepared by arc melting in a purified argon atmosphere, an excess 15% of La and Ce were added to compensate for the loss during melting. The sample was annealed for 50 h at 1100 1C and quenched into ice water. Powder X-ray diffraction (XRD) data were collected on a Philips diffractometer with Cu Ka radiation for all samples, which show that the compounds crystallize in the desired structure, with a minor second phase of MnO in the case of MnFe(P,Ge) and aFe in the case of La0.8Ce0.2Fe11.4Si1.6. The single crystal of Gd5Ge2.3Si1.7 was the same as used earlier by Tegus et al. [17]. The magnetic measurements were performed on a commercial SQUID magnetometer (Quantum Design MPMS 5XL). The magnetic entropy change is derived from the magnetization data collected at discrete equidistant temperatures using Eq. (3).
3. Results When characterizing materials presenting high magnetocaloric effects, two magnetic measurements are necessarily performed. Isofield measurements, in order to determine the critical temperature, thermal hysteresis and the width of the transition. And isothermal measurements that are used to calculate the isothermal magnetic entropy change using the Maxwell relations. Now, both isofield and isothermal measurements probe the same phenomenon, and therefore should be equivalent. This can also
be directly concluded from either the Maxwell or the Clausius–Clapeyron relations. Isofield measurements are very consistent in the sense that they always start in a temperature where the sample is either completely magnetic (ferromagnetic or otherwise) or completely paramagnetic, and without change in field the transition is probed continuously without any change in the conditions of measurement. In this manner the phase transition is always crossed in the same sense. Isothermal measurements are usually performed as follows: the sample is brought to the lowest temperature that one desires to measure, the temperature, let us say Ti, is fixed and the magnetic field is increased from 0 until Bmax in discrete steps. At each step the magnetization is measured with fixed temperature and field. Once Bmax is reached the field is brought back to zero and the temperature is increased by a certain DT to a temperature Ti+DT and the process of increasing field is repeated. This is repeated for every temperature until the maximum temperature Tf is reached. From these isothermal measurements the isothermal magnetic entropy change can be calculated integrating the Maxwell relation Z BF @MðT; BÞ DSM ðT; DBÞ ¼ dB ð2Þ @T B;p BI
Because the measurements are made at discrete temperature intervals, DSM can be numerically calculated using X MðT þ ðDT=2Þ; Bi Þ MðT ðDT=2Þ; Bi Þ DSM ðT; DBÞ ¼ DBi ð3Þ DT i where DB is the sum of DBi, M(T+DT/2,Bi) and M(TDT/2,Bi) represent the values of the magnetization in a magnetic field Bi at the temperatures T+DT/2 and TDT/2, respectively. This numerical approach is commonly employed for determining the DSM values of novel materials, because determination of the temperature derivative of the magnetization and integration as described in Eq. (2) is more complicated. For reversible processes the equivalence of these two processes is obviously achieved. However, when measuring a material with large thermal hysteresis, where a magnetization process is not fully reversible, history-dependent magnetic states are observed and the isothermal measurement just described fails to probe the phase transition correctly. In order to compare the isofield and isothermal measurements, as well as to address the thermal hysteresis issue, a sample of MnFeP0.8Ge0.2 is used for a case study. 3.1. MnFeP0.8Ge0.2: a case study MnFeP0.8Ge0.2 is known to show rather high thermal hysteresis [18]. Fig. 1 shows the temperature dependence of magnetization of this compound measured with increasing temperature, and then subsequently with decreasing temperature in applied fields of 1, 2, 3, 4 and 5 T. This sample exhibits a large thermal hysteresis of about 15 K that hardly depends on the applied field. Though the transition is clearly of first order, the strong dependence of the critical temperature on the Ge contents creates a rather broad phase transition for this compound, due to statistical variation of the Ge content [19]. In other words for this material near the critical temperature one will always find some coexistence of ferro- and paramagnetic phases. Thus, strictly speaking one should define a phase transition region; however for simplicity we assume a defined phase transition temperature. The field dependence of the phase transition temperature has been determined as the inflection point of isofield curves for both
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T (K) Fig. 1. Temperature dependence of the magnetization of MnFeP0.8Ge0.2 measured in constant fields of 1, 2, 3, 4 and 5 T with temperature increasing and decreasing in a step of 2 K, the arrows indicate the warming and cooling processes.
heating and cooling processes as shown with the marker ’ in Fig. 2. The origin of the other data points in Fig. 2 will be explained below. From this diagram we can determine the thermal evolution of the transition from the low-temperature ferromagnetic (FM) state to the high-temperature paramagnetic (PM) state. The CFM–PM line starting at about 296 K marks the isofield transition on heating, and the inverse transition follows the CPM–FM line starting at about 280 K. Since isofield and isothermal processes are equivalent, the later can also be characterized by these lines. With increasing field, the PM to FM transition follows the CPM–FM line, while the inverse transition follows the CFM–PM line. Therefore, a history-dependent magnetic state exists in the region between these two lines. When the temperature interval of this history-dependent region is small, let us say the same as the step width of the magnetization isotherms measured to determine the entropy change, we need not worry about the validity of our results to a thermodynamic analysis, since the transition is completely crossed in a temperature step. However, what happens when the hysteresis far exceeds the step size as is the case here with temperature step size of 2 K and about 15 K hysteresis? The isothermal process described earlier, to which I shall address as standard process, produces the results presented in Fig. 3. At first glance these results, other than the considerably high isothermal magnetic entropy change, seem to be reasonable [20]. In order to compare the isothermal and isofield magnetization measurements we take the temperature and field values at which the magnetization is half its maximum value, in this case around 60 Am2/kg, as it represents the critical field of the transition at that temperature. These points are represented in Fig. 2 by * marks. From this line it is clear that this isothermal process is not equivalent to the isofield process. To understand why these two measurements are not equivalent it is necessary to analyze, following the diagram on Fig. 2, what kind of process is actually being followed by the standard isothermal measurement. Let us take the first isotherm around 280 K. At this temperature the sample is ferromagnetic and the field is increased until 5 T. The magnetization saturates very quickly due to the alignment of the magnetic domains by the external magnetic field. When the field is reduced to zero the sample stays ferromagnetic until the CFM–PM line is reached. Once that happens, for a certain temperature interval around the CFM–PM line, part of the sample will be in the paramagnetic state
285
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T (K) Fig. 2. Magnetic phase diagram of the compound MnFeP0.8Ge0.2 as derived from isofield magnetization measurements shown in Fig. 1 ’ (other symbols derived from M(B) see Figs. 3(a) and 4(a)).
and part will be in the ferromagnetic state. Now we consider an isotherm around 296 K, where the sample, at zero field will be a mixture of paramagnetic and ferromagnetic phases. When the magnetic field is increased, the ferromagnetic part will again saturate very quickly due to the alignment of the magnetic domains, creating a plateau. The paramagnetic part will only respond when the CPM–FM line is reached at much higher field, in this case a little over 4 T, corresponding to the metamagnetic transition. Having this in mind, it is clear that the process used to probe the phase transition must take into account the mixed phase state and compensate for it. In order to do that we must make sure that the phase transition is always crossed in the same sense, and that is done using a similar process to that used in the isofield measurement. This consists in bringing the sample completely into the paramagnetic region, far away from the transition, in between every isotherm. This way the transition is always crossed in the same sense, and the plateaus are no longer observed. Therefore, in order to correctly probe the phase transition the following isothermal measurement was performed: the measurement is started at the lowest temperature, let us say Ti, the field is increased until its maximum value Bmax, magnetization is measured in discrete steps and then the field is brought back to zero. Next the sample is taken from Ti all the way into the paramagnetic region until a temperature far from the phase transition to make sure the sample is completely paramagnetic. Only then it is cooled down to the next measurement temperature, Ti+DT. The loop in temperature is performed before every isotherm, at zero field. The results for the so-called loop process (Fig. 4) are strikingly different from those of the standard process. In the loop process, because the sample is always cooled down from the paramagnetic state, the only region where a mixture of phases will be observed is the CPM–FM line. Because the sample always reaches a two phase region coming from a previous onephase region it gradually transforms from one phase into the other. This can be seen very clearly in Fig. 4(a) in the very nice and gradual increase of the critical field for the metamagnetic transition. In this case the material simply does not ‘‘see’’ the CFM–PM line in the measurement, but only the CPM–FM cooling line. Therefore, it will only cross the CPM–FM phase transition line once and always in the same sense. Similar results, i.e. having the material crossing a single-phase transition line during the isothermal measurements, can be obtained simply by increasing
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Fig. 3. Isothermal magnetization curves of MnFeP0.8Ge0.2 on increasing field in the vicinity of the Curie temperature (a) the magnetic entropy changes for different field changes (b).
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Fig. 4. Isothermal magnetization curves of MnFeP0.8Ge0.2 on increasing field in the vicinity of the Curie temperature (a), the magnetic entropy change for different field changes (b).
the temperature in between isotherms (with no loop) and measure with decreasing magnetic field. The plateaus are no longer observed, and the entropy change calculated from the area differences caused by the shifting metamagnetic transition is a reproducible entropy change. More than that, the entropy change falls back to values that can be accounted by the thermodynamical limit. The line obtained from the half-magnetization criterion for the loop process (+ marks on Fig. 2) is parallel to the isofield cooling and heating lines, clearly demonstrating the equivalence of isofield and isothermal pro-
cesses. Another interesting point is that the total entropy, i.e. the area under the curves in Figs. 3(b) and 4(b), are the same for both processes. This shows that no extra entropy is calculated, but that the standard process simply concentrates entropy into a very narrow temperature interval. Next, this process is verified for the three samples previously mentioned. First, La0.8Ce0.2Fe11.4Si1.6 is measured using the loop process, so that the results can be compared to those of Liu et al. [15]. Then Mn0.99Cu0.01As is studied since it shows the same size of hysteresis as MnFeP0.8Ge0.2 but with an extremely squared
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phase transition. Also the highest entropy values were observed in MnAs-based compounds. And last, this procedure is applied to a Gd5Ge2.3Si1.7 single crystal.
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unphysical results for more extreme cases like the MnAs-based compounds, as will be demonstrated next.
3.3. Mn.99Cu.01As with square thermal hysteresis 3.2. La0.8Ce0.2Fe11.4Si1.6 with moderate thermal hysteresis The La0.8Ce0.2Fe11.4Si1.6 alloy is very similar to La0.7Pr0.3Fe11.5Si1.5 that has been studied by Liu et al. [15]. For obtaining the isothermal magnetization data displayed in Fig. 5(a) that are input for Eq. (3) the loop process was used. The magnetic response recorded in this way is quite different from the response reported by Liu et al. [15]. There are no magnetization isotherms resulting from a mixture of ferro- and paramagnetic phase. As can be seen from Fig. 5(b) the isothermal magnetic entropy change as derived by applying Eq. (3) does not exceed 30 J/kgK in stark contrast to values of 99 J/kgK found by Liu et al [15] when using the same equation. Liu et al. employ an elaborate procedure based on the Clausius–Clapeyron equation to remove the undisputedly unphysical peak from their results. But, not only the Clausius–Clapeyron equation does not hold for these systems, his procedure also fails to eliminate the
Another material with large thermal hysteresis is the alloy Mn0.99Cu0.01As, as seen from the temperature dependence of the magnetization measured in 1 T (Fig. 6), the thermal hysteresis has very similar size but is squarer than in the MnFeP0.8Ge0.2 compound. In Fig. 7 we plot again the magnetic phase diagram as determined by isofield measurements ’, magnetization curves measured with the loop process E and magnetization curves measured with the standard process K. Also here only the magnetization curves measured with the loop process reflect the magnetic phase diagram as determined by isofield temperature scans. The low-field data for the loop process still show a somewhat steeper slope than expected.
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Fig. 6. Magnetization versus temperature measurement for Mn0.99Cu0.01As.
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Fig. 7. Magnetic phase diagram of Mn0.99Cu0.01As as derived from isofield measurements and as derived from M(B) measurements.
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T (K) Fig. 8. (a) Magnetization isotherms measured according to the loop process. (b) Entropy change as a function of temperature for Mn0.99Cu0.01As for the loop process with 2 and 1 K steps and the standard process with 1 K steps for a 5 T field change.
Fig. 8(a) shows the magnetization isotherms measured using the loop process. Except for the isotherm recorded at 308 K, these data exhibit nicely developed field-induced transitions over a wide temperature interval. The corresponding magnetic entropy change is displayed in Fig. 8(b). The entropy change is large over an interval of almost 20 K. We include the results from the standard process with 1 K step to emphasize that the same total entropy change is put into a very narrow temperature interval. Note that the spikes still visible in the data with the loop process are one order of magnitude smaller than for the standard process. One would expect that these spikes do not occur at all, but they also occur when estimating the entropy change from the isofield measurements [21]. Fig. 9(a) shows the magnetization isotherms as derived from the standard process and Fig. 9(b) the resulting isothermal entropy change. The results nicely reproduce earlier results of Rocco et al. [11]. Note that the peak value depends strongly on the temperature interval between the isotherms. This clearly suggests that these results are not physical, however, if one tries to correct the data in the way proposed by Liu et al. it turns
0 300
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Fig. 9. (a) Magnetization isotherms measured according to the standard process. (b) Entropy change as a function of temperature for Mn0.99Cu0.01As for 1 and 2 K steps for a 5 T field change.
out that the data point at 313 K with about 80 J/kgK should stay unchanged and only the second point with 180 J/kgK and 70 J/kgK, respectively, will be reduced.
3.4. Gd5Ge2.3Si1.7 revisited Another material exhibiting considerable thermal hysteresis is the Gd5Ge2Si2 type of compound. In an earlier paper we published results on a single crystal with composition Gd5Ge2.3Si1.7 [17]. These results were obtained by the standard process and show a very pronounced peak with a maximum value of 45 J/kgK in the isothermal entropy change. The same sample was inspected again using the loop process to measure the magnetization isotherms. The measured isotherms and calculated isothermal entropy change are displayed in Fig. 10(a) and (b), respectively. Compared with the earlier results, the peak in the magnetic entropy change is strongly reduced to 35 J/kgK. On the other hand,
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the shoulder in the isothermal entropy change that was observed for higher fields is hardly affected and the integral of the magnetic entropy is unchanged.
4. Discussion We now want to discuss what goes wrong when the magnetic entropy change is determined by the standard process. The Maxwell relation (1) tells us that the magnetic entropy changes go with the applied field at the same rate as the isofield magnetization changes with temperature. This means that when we measure magnetization isotherms at equidistant temperatures, a large change in magnetic response between adjacent isotherms reflects a large isothermal change in entropy. Equivalently a steep change of critical field with temperature results in a large entropy change. The former is only true if the adjacent isotherms reflect the temperature dependence of the isofield magnetization. In a material that shows a second-order phase transition this condition is always fulfilled independent of the field and temperature history. For a material that shows a first-order magnetic phase transition the field and temperature history plays an important
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role as indicated in the temperature dependence of the magnetization displayed in Figs. 1 and 6 and the phase-diagrams derived from different magnetic measurements compiled in Figs. 2 and 7. Here we study the different processes by considering the diagram of Mn.99Cu.01As because it is most pronounced. The black squares in Fig. 7 were constructed from isofield measurements in cooling and heating mode. This produced two approximately parallel phase lines called CFM–PM and CPM–FM. The slope of these lines is about 0.25 T/K. The arrows in Fig. 7 indicate which line is crossed when either the field or the temperature is changed. In the standard process depicted in Fig. 9(a) at low temperature the magnetization isotherms measure the ferromagnetic response. The material is soft magnetic which means very low fields are sufficient to saturate the multi-domain sample, indicated by the round marks at the bottom of Fig. 7. Because the sample has an irregular shape we do not correct for demagnetizing fields and just take half the spontaneous magnetization in the ferromagnetic state (60 Am2/kg) as criterion for ferromagnetism. At 314 K, the sample appears to be in a mixed state, part of the sample is paramagnetic and another part is ferromagnetic. A higher field (2.5 T) is therefore needed to reach 60 Am2/kg. At 316 K, the paramagnetic fraction appears to be the majority phase resulting in a field-induced ferromagnetic response only for fields above 3 T which is beyond the line CPM–FM. After the CFM–PM line is crossed we are far in the paramagnetic state and ferromagnetic response will only be found for high-applied magnetic fields that run parallel to the line CPM–FM. The maximum available field of 5 T is just sufficient to have 3 more isotherms with field-induced transitions and at 324 K we only observe a partial metamagnetic transition. Thus we find a phase line that starts off extremely steep around 314 K and then continues with a moderate slope. This phase line is in stark contrast with the results from isotherms determined from the loop process. For the loop process on the other hand the sample is always transformed to the PM state when temperature is once beyond the line CPM–FM by undergoing a zero field heating and cooling process. In this case, the magnetic response is not affected by the coexistence of the two phases. The magnetization response changes each time over the line CPM–FM from PM to FM, representing the intrinsic entropy change of the spin sub-lattice. As evidenced by the data marked in Fig. 7, the change in magnetic response probes the by temperature dependence of the isofield magnetization that is measured in a cooling mode. Thus, a reliable MCE can be determined by using the loop process. Above 315 K, the upper bound of the zero field hysteresis, the magnetic entropy change is identical for the data from both processes. Returning to Fig. 7, lets consider a sample at 312 K that has been magnetized earlier, or has been heated from low temperatures, this sample will be ferromagnetic for all fields. At 316 K independent of the history we shall see paramagnetic response and around 3.5 T field we see a field-induced transition. Therefore in a very narrow temperature interval, a large change in magnetic response is observed. Obviously, the size of the temperature interval is determined by the temperature interval between the isotherms, note the striking but unphysical difference in Fig. 9(b) between 1 and 2 K step size. This change in magnetic response depends on the extent of temperature hysteresis in the sample that is measured. If the lines CFM–PM and CPM–FM are separated by tens of degrees or the hysteresis is squarer, the change in magnetization will be more pronounced, than if they are close together or the isofield curves are more rounded. The change in magnetization is real but the magnetic entropy change calculated from magnetic measurements performed in this way, artificially concentrates the entropy change over the full width of hysteresis into the temperature step of the measurement. Therefore we cannot use the standard method to determine the maximal
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entropy change of a material with thermal hysteresis. As mentioned above analysis of the magnetization data with the process proposed by Liu et al. does not improve the results but under some conditions artificially removes the spike. The fact that the maximum value of the isothermal entropy change depends on the temperature step chosen between the isotherms, clearly indicates that the standard process does not measure the physics of the system as it does not take into account that the magnetic response depends on the sample history. The data recorded by the loop measurement record the entropy change as it occurs in a cooling mode. Alternatively, one may take isothermal magnetization measurements performed with decreasing field in combination with heating the sample in maximum field from the ferromagnetic state, in that case the data should represent the temperature dependence of the isofield magnetization measured in the heating mode. For each of the processes the area below the curves, which is generally associated with the cooling capacity, stays the same. This evidences that the entropy change originates from a magnetization process that involves not only rotation of domains. This entropy change should also be considered in the evaluation of the materials properties and not thrown away as artifact. Obviously, the standard process is putting part of this entropy change into a too narrow temperature interval resulting in the unphysical large peak values. The extent of this part is determined by the field and temperature interval with coexisting ferro- and paramagnetic phases as well as the size of the thermal hysteresis. The temperature interval is determined by the experimental conditions that probe the phase transition from single-phase paramagnetic to single-phase ferromagnetic state. Thus, when performing isothermal magnetization measurement on samples with thermal hysteresis, and the temperature step for the measurement is smaller than the thermal hysteresis or the selected temperature for the isotherm is in the two phase region, these magnetization measurements may not reflect the temperature and field dependence of the single-phase FM and PM state. In summary, the determination of the magnetic entropy change in several compounds, which show large thermal hysteresis has carefully been studied. When the magnetization measurements are performed using a loop process that takes the history dependence of the magnetization into account, it is possible to obtain the isothermal magnetic entropy change just by employing the Maxwell relations. Thus it is not necessary to perform complicated subtraction procedures as proposed by Liu et al. [15]. As was seen in our study the integrated entropy or the refrigeration capacity is independent of the method. Subtracting some part from the results will not improve them. Thus, when performing the isothermal magnetization measurement to derive
the magnetic entropy change in samples with large thermal hysteresis, one should verify that these magnetization measurements reflect the temperature and field dependence of the FM and PM state. Compared to the method proposed by Liu et al. [15] the advantage of our approach is that we determine correctly both the maximal entropy change and the cooling capacity. The authors would like to thank P.F. de Chˆatel for discussion and critical reading of the manuscript. This work was performed within the Scientific Exchange Program between the Netherlands and China, the Student Exchange Program supported by Coordenac- a~ o de Aperfeic- oamento de Pesssoal de N´ıvel Superior (Capes), and partly supported by the National Natural Science Foundation of China and the Dutch technology foundation STW. The authors would also like to thank the Fundac- a~ o de Amparo a Pesquisa do Estado de Sa~ o Paulo (FAPESP). References [1] K.A. Gschneidner, V.K. Pecharsky, A.O. Tsokol, Reports on Progress in Physics 68 (2005) 1479–1539. ¨ [2] E. Bruck, Journal of Physics D-Applied Physics 38 (2005) R381–R391. [3] V.K. Pecharsky, K.A. Gschneidner, Physical Review Letters 78 (1997) 4494–4497. [4] V.K. Pecharsky, K.A. Gschneidner, Journal of Magnetism and Magnetic Materials 167 (1997) L179–L184. ¨ [5] O. Tegus, E. Bruck, K.H.J. Buschow, F.R. de Boer, Nature 415 (2002) 150–152. [6] H. Wada, Y. Tanabe, Applied Physics Letters 79 (2001) 3302–3304. [7] F.X. Hu, B.G. Shen, J.R. Sun, Z.H. Cheng, G.H. Rao, X.X. Zhang, Applied Physics Letters 78 (2001) 3675–3677. [8] H. Terashita, J.J. Garbe, J.J. Neumeier, Physical Review B 70 (2004) 094403. [9] S. Gama, A.A. Coelho, A. de Campos, A.M.G. Carvalho, F.C.G. Gandra, P.J. von Ranke, N.A. de Oliveira, Physical Review Letters 93 (2004) 237202. [10] A. de Campos, D.L. Rocco, A.M.G. Carvalho, L. Caron, A.A. Coelho, S. Gama, L.M. Da Silva, F.C.G. Gandra, A.O. dos Santos, L.P. Cardoso, P.J. Von Ranke, N.A. de Oliveira, Nature Materials 5 (2006) 802–804. [11] D.L. Rocco, A. de Campos, A.M.G. Carvalho, L. Caron, A.A. Coelho, S. Gama, F.C.G. Gandra, A.O. dos Santos, L.P. Cardoso, P.J. von Ranke, N.A. de Oliveira, Applied Physics Letters 90 (2007) 242507. [12] C. Xu, G.D. Li, L.G. Wang, Journal of Applied Physics 99 (2006) 123913. [13] A. Giguere, M. Foldeaki, B.R. Gopal, R. Chahine, T.K. Bose, A. Frydman, J.A. Barclay, Physical Review Letters 83 (1999) 2262–2265. [14] J.R. Sun, F.X. Hu, B.G. Shen, Physical Review Letters 85 (2000) 4191. [15] G.J. Liu, R. Sun, J. Shen, B. Gao, H.W. Zhang, F.X. Hu, B.G. Shen, Applied Physics Letter 90 (2007) 032507. [16] N.A. de Oliveira, P.J. von Ranke, Physical Review B 77 (2008) 214439. ¨ [17] O. Tegus, E. Bruck, L. Zhang, Dagula, K.H.J. Buschow, F.R. de Boer, Physica B 319 (2002) 174–192. ¨ [18] D.T.C. Thanh, E. Bruck, O. Tegus, J.C.P. Klaasse, T.J. Gortenmulder, K.H.J. Buschow, Journal of Applied Physics 99 (2006) 08Q107. ¨ [19] R.P. Hermann, O. Tegus, E. Bruck, K.H.J. Buschow, F.R. de Boer, G.J. Long, F. Grandjean, Physical Review B 70 (2004) 214425. [20] Danmin Liu, Ming Yue, Jiuxing Zhang, T.M. McQueen, Jeffrey W. Lynn, Xiaolu Wang, Ying Chen, Jiying Li, R.J. Cava, Xubo Liu, Zaven Altounian, Q. Huang, Physical Review B 79 (2009) 014435. [21] H. Wada, private communication.
Contents lists available at ScienceDirect
Journal of Magnetism and Magnetic Materials journal homepage: www.elsevier.com/locate/jmmm
Current Perspectives
On the determination of the magnetic entropy change in materials with first-order transitions ¨ b, L. Caron a,b, Z.Q. Ou b,c, T.T. Nguyen b, D.T. Cam Thanh b, O. Tegus b,c, E. Bruck a
Instituto de F´ısica Gleb Wataghin, Universidade Estadual de Campinas-UNICAMP, C.P. 6165, Campinas 13 083 970, SP, Brazil Fundamental Aspects of Energy and Materials, Faculty of Applied Science, TU Delft, Mekelweg 15, 2629 JB Delft, The Netherlands c Key Laboratory for Physics and Chemistry of Functional Materials, Inner Mongolia Normal University, Hohhot 010022, PR China b
a r t i c l e in fo
abstract
Available online 9 July 2009
An accurate method to determine the magnetic entropy change in materials with hysteretic first-order transitions is presented, which is needed to estimate their potential for applications. We have investigated the effect of the maximal entropy change derived from magnetization measurements performed in different measurement processes. The results show that the isothermal entropy change can be derived from the Maxwell relations even for samples with large thermal hysteresis. In the temperature region with hysteresis, overestimating the entropy change can be avoided by measuring the isothermal magnetization of the sample after it is cooled from the paramagnetic state to the measurement temperature. In this way the so-called peak effect is not observed as shown here for a few compounds. & 2009 Elsevier B.V. All rights reserved.
Keywords: Magnetization process Magnetic entropy First-order transition Magnetic refrigeration
1. Introduction Magnetic refrigeration, based on the magnetocaloric effect (MCE), is considered as one of the most promising technologies to replace vapor-compression refrigeration, due to its low environmental impact and expected high energy efficiency [1,2]. Since the discovery of the giant MCE in Gd5(SixGe1x)4 [3,4] compounds, other giant magnetocaloric materials, MnFeP0.45As0.55 [5], MnAs1xSbx [6], La(Fe1xSix)13 [7] and La1xCaxMnO3 [8], were reported by different research groups. Recently, a colossal MCE with magnetic entropy change of 267 J/kgK for a 5 T field change was reported for MnAs [9] at hydrostatic pressure of 0.23 GPa. When partially substituting Fe (de Campos et al. [10]) or Cu (Rocco et al. [11]) for Mn in MnAs compounds, the same effects are reported even at ambient pressure. Most recently, a giant entropy change, 78 J/kgK, was reported for La0.8Ce0.2Fe11.4Si1.6 [12] at a field change of 3 T. A common feature of the alloys showing colossal magnetocaloric effects is the occurrence of considerable thermal hysteresis. This thermal hysteresis makes these materials not very suitable for applications in refrigerators that are expected to be operated cyclically at rather high cycle frequencies. However, as it is well known that hysteresis is an extrinsic property and thus may be reduced by processing, these large values of entropy change make these materials very interesting for further study.
Corresponding author. Tel.: +3115 2783158; fax: +31 20 6951500.
¨ E-mail address: [email protected] (E. Bruck). 0304-8853/$ - see front matter & 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2009.06.086
The observation of magnetic entropy changes as high as the socalled colossal ones brings two problems: the first is that the colossal values surpass the magnetic limit given by max DSM ¼ ln(2J+1), the second is that of the correct determination of the isothermal magnetic entropy change from magnetic measurements. Experimental discrepancies between the calculated and directly measured temperature change observed for Gd5Si2Ge2 has led Giguere et al. [13] to propose that the Clausius–Clapeyron relation, rather than the Maxwell relation
@S @M ¼ @B T @T B
ð1Þ
should be used. In a comment, Sun et al. [14] have shown that Clausius–Clapeyron is just a special case of the integrated Maxwell relations and thus the latter should be applicable. Nevertheless, the Clausius–Clapeyron relation has again been put forward very recently to analyze the magnetic response of (La,Pr)(Fe,Si)13 compounds [15]. Liu et al. argue that a careful study indicates that the Maxwell relation cannot be used in the vicinity of the Curie temperature because of the coexistence of paramagnetic and ferromagnetic phases, and therefore the observed huge entropy peak is a spurious result. Also Wada et al. [6] mention in their paper on Mn(As,Sb) that the peak should be considered as a spurious effect. On a more detailed derivation, de Oliveira et al. [16] showed that the so-called magnetic Clausius–Clapeyron relation does not hold for magnetic systems since it assumes there is no work done
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to take the system from the ferromagnetic to the paramagnetic state. The same work states that the Maxwell relations are not sufficient to calculate the entropy change around a first-order magnetic phase transition, and that in the region where the discontinuity itself is observed an extra term that resembles the Clausius–Clapeyron relation must be used. Since most observed first-order magnetic phase and metamagnetic transitions are rather smooth, calculations from real data using this model are seemingly impossible. Thus we reconsider the Maxwell relations, arguing they should be valid for the same reason why de Oliveira’s description fails to be of practical use: in nature, first-order transitions are far from step-like. And, as long as the first-order phase transition is properly probed, one should not incur in large errors. Here we show that the ‘‘colossal’’ peaks in the entropy change arise not from the application of the Maxwell relations but from the incorrect probing of the phase transition. In order to do that the measurement process being used to obtain such results is analyzed, keeping in mind the thermodynamical relations which describe the process. This paper, from here on, is organized in the following way: first a quick description is given of how the samples used in this paper were prepared. Then MnFeP0.8Ge0.2 is taken for a case study to explain and illustrate the various processes used to probe the phase transition, followed by an analysis of these processes. Next the results of correctly probing the phase transition are presented in contrast with the results found in literature for three other materials: La(Fe,Si)13, Mn0.99Cu0.01As and a Gd5Ge2.3Si1.7 single crystal. For Mn0.99Cu0.01As we also apply Liu’s method to show its inadequacy to this extreme case.
2. Experimental A polycrystalline sample of MnFeP0.8Ge0.2 was synthesized by ball milling and sintering process with appropriate amounts of starting materials. The sample was sintered at 1050 1C for 5 h, and then quenched immediately into water instead of slowly cooling down to ambient temperature. Also a Mn0.99Cu0.01As sample was prepared exactly as reported in Ref. [11]. La0.8Ce0.2Fe11.4Si1.6 was prepared by arc melting in a purified argon atmosphere, an excess 15% of La and Ce were added to compensate for the loss during melting. The sample was annealed for 50 h at 1100 1C and quenched into ice water. Powder X-ray diffraction (XRD) data were collected on a Philips diffractometer with Cu Ka radiation for all samples, which show that the compounds crystallize in the desired structure, with a minor second phase of MnO in the case of MnFe(P,Ge) and aFe in the case of La0.8Ce0.2Fe11.4Si1.6. The single crystal of Gd5Ge2.3Si1.7 was the same as used earlier by Tegus et al. [17]. The magnetic measurements were performed on a commercial SQUID magnetometer (Quantum Design MPMS 5XL). The magnetic entropy change is derived from the magnetization data collected at discrete equidistant temperatures using Eq. (3).
3. Results When characterizing materials presenting high magnetocaloric effects, two magnetic measurements are necessarily performed. Isofield measurements, in order to determine the critical temperature, thermal hysteresis and the width of the transition. And isothermal measurements that are used to calculate the isothermal magnetic entropy change using the Maxwell relations. Now, both isofield and isothermal measurements probe the same phenomenon, and therefore should be equivalent. This can also
be directly concluded from either the Maxwell or the Clausius–Clapeyron relations. Isofield measurements are very consistent in the sense that they always start in a temperature where the sample is either completely magnetic (ferromagnetic or otherwise) or completely paramagnetic, and without change in field the transition is probed continuously without any change in the conditions of measurement. In this manner the phase transition is always crossed in the same sense. Isothermal measurements are usually performed as follows: the sample is brought to the lowest temperature that one desires to measure, the temperature, let us say Ti, is fixed and the magnetic field is increased from 0 until Bmax in discrete steps. At each step the magnetization is measured with fixed temperature and field. Once Bmax is reached the field is brought back to zero and the temperature is increased by a certain DT to a temperature Ti+DT and the process of increasing field is repeated. This is repeated for every temperature until the maximum temperature Tf is reached. From these isothermal measurements the isothermal magnetic entropy change can be calculated integrating the Maxwell relation Z BF @MðT; BÞ DSM ðT; DBÞ ¼ dB ð2Þ @T B;p BI
Because the measurements are made at discrete temperature intervals, DSM can be numerically calculated using X MðT þ ðDT=2Þ; Bi Þ MðT ðDT=2Þ; Bi Þ DSM ðT; DBÞ ¼ DBi ð3Þ DT i where DB is the sum of DBi, M(T+DT/2,Bi) and M(TDT/2,Bi) represent the values of the magnetization in a magnetic field Bi at the temperatures T+DT/2 and TDT/2, respectively. This numerical approach is commonly employed for determining the DSM values of novel materials, because determination of the temperature derivative of the magnetization and integration as described in Eq. (2) is more complicated. For reversible processes the equivalence of these two processes is obviously achieved. However, when measuring a material with large thermal hysteresis, where a magnetization process is not fully reversible, history-dependent magnetic states are observed and the isothermal measurement just described fails to probe the phase transition correctly. In order to compare the isofield and isothermal measurements, as well as to address the thermal hysteresis issue, a sample of MnFeP0.8Ge0.2 is used for a case study. 3.1. MnFeP0.8Ge0.2: a case study MnFeP0.8Ge0.2 is known to show rather high thermal hysteresis [18]. Fig. 1 shows the temperature dependence of magnetization of this compound measured with increasing temperature, and then subsequently with decreasing temperature in applied fields of 1, 2, 3, 4 and 5 T. This sample exhibits a large thermal hysteresis of about 15 K that hardly depends on the applied field. Though the transition is clearly of first order, the strong dependence of the critical temperature on the Ge contents creates a rather broad phase transition for this compound, due to statistical variation of the Ge content [19]. In other words for this material near the critical temperature one will always find some coexistence of ferro- and paramagnetic phases. Thus, strictly speaking one should define a phase transition region; however for simplicity we assume a defined phase transition temperature. The field dependence of the phase transition temperature has been determined as the inflection point of isofield curves for both
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T (K) Fig. 1. Temperature dependence of the magnetization of MnFeP0.8Ge0.2 measured in constant fields of 1, 2, 3, 4 and 5 T with temperature increasing and decreasing in a step of 2 K, the arrows indicate the warming and cooling processes.
heating and cooling processes as shown with the marker ’ in Fig. 2. The origin of the other data points in Fig. 2 will be explained below. From this diagram we can determine the thermal evolution of the transition from the low-temperature ferromagnetic (FM) state to the high-temperature paramagnetic (PM) state. The CFM–PM line starting at about 296 K marks the isofield transition on heating, and the inverse transition follows the CPM–FM line starting at about 280 K. Since isofield and isothermal processes are equivalent, the later can also be characterized by these lines. With increasing field, the PM to FM transition follows the CPM–FM line, while the inverse transition follows the CFM–PM line. Therefore, a history-dependent magnetic state exists in the region between these two lines. When the temperature interval of this history-dependent region is small, let us say the same as the step width of the magnetization isotherms measured to determine the entropy change, we need not worry about the validity of our results to a thermodynamic analysis, since the transition is completely crossed in a temperature step. However, what happens when the hysteresis far exceeds the step size as is the case here with temperature step size of 2 K and about 15 K hysteresis? The isothermal process described earlier, to which I shall address as standard process, produces the results presented in Fig. 3. At first glance these results, other than the considerably high isothermal magnetic entropy change, seem to be reasonable [20]. In order to compare the isothermal and isofield magnetization measurements we take the temperature and field values at which the magnetization is half its maximum value, in this case around 60 Am2/kg, as it represents the critical field of the transition at that temperature. These points are represented in Fig. 2 by * marks. From this line it is clear that this isothermal process is not equivalent to the isofield process. To understand why these two measurements are not equivalent it is necessary to analyze, following the diagram on Fig. 2, what kind of process is actually being followed by the standard isothermal measurement. Let us take the first isotherm around 280 K. At this temperature the sample is ferromagnetic and the field is increased until 5 T. The magnetization saturates very quickly due to the alignment of the magnetic domains by the external magnetic field. When the field is reduced to zero the sample stays ferromagnetic until the CFM–PM line is reached. Once that happens, for a certain temperature interval around the CFM–PM line, part of the sample will be in the paramagnetic state
285
290
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305
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T (K) Fig. 2. Magnetic phase diagram of the compound MnFeP0.8Ge0.2 as derived from isofield magnetization measurements shown in Fig. 1 ’ (other symbols derived from M(B) see Figs. 3(a) and 4(a)).
and part will be in the ferromagnetic state. Now we consider an isotherm around 296 K, where the sample, at zero field will be a mixture of paramagnetic and ferromagnetic phases. When the magnetic field is increased, the ferromagnetic part will again saturate very quickly due to the alignment of the magnetic domains, creating a plateau. The paramagnetic part will only respond when the CPM–FM line is reached at much higher field, in this case a little over 4 T, corresponding to the metamagnetic transition. Having this in mind, it is clear that the process used to probe the phase transition must take into account the mixed phase state and compensate for it. In order to do that we must make sure that the phase transition is always crossed in the same sense, and that is done using a similar process to that used in the isofield measurement. This consists in bringing the sample completely into the paramagnetic region, far away from the transition, in between every isotherm. This way the transition is always crossed in the same sense, and the plateaus are no longer observed. Therefore, in order to correctly probe the phase transition the following isothermal measurement was performed: the measurement is started at the lowest temperature, let us say Ti, the field is increased until its maximum value Bmax, magnetization is measured in discrete steps and then the field is brought back to zero. Next the sample is taken from Ti all the way into the paramagnetic region until a temperature far from the phase transition to make sure the sample is completely paramagnetic. Only then it is cooled down to the next measurement temperature, Ti+DT. The loop in temperature is performed before every isotherm, at zero field. The results for the so-called loop process (Fig. 4) are strikingly different from those of the standard process. In the loop process, because the sample is always cooled down from the paramagnetic state, the only region where a mixture of phases will be observed is the CPM–FM line. Because the sample always reaches a two phase region coming from a previous onephase region it gradually transforms from one phase into the other. This can be seen very clearly in Fig. 4(a) in the very nice and gradual increase of the critical field for the metamagnetic transition. In this case the material simply does not ‘‘see’’ the CFM–PM line in the measurement, but only the CPM–FM cooling line. Therefore, it will only cross the CPM–FM phase transition line once and always in the same sense. Similar results, i.e. having the material crossing a single-phase transition line during the isothermal measurements, can be obtained simply by increasing
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loop process
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T (K)
Fig. 4. Isothermal magnetization curves of MnFeP0.8Ge0.2 on increasing field in the vicinity of the Curie temperature (a), the magnetic entropy change for different field changes (b).
the temperature in between isotherms (with no loop) and measure with decreasing magnetic field. The plateaus are no longer observed, and the entropy change calculated from the area differences caused by the shifting metamagnetic transition is a reproducible entropy change. More than that, the entropy change falls back to values that can be accounted by the thermodynamical limit. The line obtained from the half-magnetization criterion for the loop process (+ marks on Fig. 2) is parallel to the isofield cooling and heating lines, clearly demonstrating the equivalence of isofield and isothermal pro-
cesses. Another interesting point is that the total entropy, i.e. the area under the curves in Figs. 3(b) and 4(b), are the same for both processes. This shows that no extra entropy is calculated, but that the standard process simply concentrates entropy into a very narrow temperature interval. Next, this process is verified for the three samples previously mentioned. First, La0.8Ce0.2Fe11.4Si1.6 is measured using the loop process, so that the results can be compared to those of Liu et al. [15]. Then Mn0.99Cu0.01As is studied since it shows the same size of hysteresis as MnFeP0.8Ge0.2 but with an extremely squared
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phase transition. Also the highest entropy values were observed in MnAs-based compounds. And last, this procedure is applied to a Gd5Ge2.3Si1.7 single crystal.
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unphysical results for more extreme cases like the MnAs-based compounds, as will be demonstrated next.
3.3. Mn.99Cu.01As with square thermal hysteresis 3.2. La0.8Ce0.2Fe11.4Si1.6 with moderate thermal hysteresis The La0.8Ce0.2Fe11.4Si1.6 alloy is very similar to La0.7Pr0.3Fe11.5Si1.5 that has been studied by Liu et al. [15]. For obtaining the isothermal magnetization data displayed in Fig. 5(a) that are input for Eq. (3) the loop process was used. The magnetic response recorded in this way is quite different from the response reported by Liu et al. [15]. There are no magnetization isotherms resulting from a mixture of ferro- and paramagnetic phase. As can be seen from Fig. 5(b) the isothermal magnetic entropy change as derived by applying Eq. (3) does not exceed 30 J/kgK in stark contrast to values of 99 J/kgK found by Liu et al [15] when using the same equation. Liu et al. employ an elaborate procedure based on the Clausius–Clapeyron equation to remove the undisputedly unphysical peak from their results. But, not only the Clausius–Clapeyron equation does not hold for these systems, his procedure also fails to eliminate the
Another material with large thermal hysteresis is the alloy Mn0.99Cu0.01As, as seen from the temperature dependence of the magnetization measured in 1 T (Fig. 6), the thermal hysteresis has very similar size but is squarer than in the MnFeP0.8Ge0.2 compound. In Fig. 7 we plot again the magnetic phase diagram as determined by isofield measurements ’, magnetization curves measured with the loop process E and magnetization curves measured with the standard process K. Also here only the magnetization curves measured with the loop process reflect the magnetic phase diagram as determined by isofield temperature scans. The low-field data for the loop process still show a somewhat steeper slope than expected.
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Fig. 6. Magnetization versus temperature measurement for Mn0.99Cu0.01As.
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Fig. 7. Magnetic phase diagram of Mn0.99Cu0.01As as derived from isofield measurements and as derived from M(B) measurements.
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T (K) Fig. 8. (a) Magnetization isotherms measured according to the loop process. (b) Entropy change as a function of temperature for Mn0.99Cu0.01As for the loop process with 2 and 1 K steps and the standard process with 1 K steps for a 5 T field change.
Fig. 8(a) shows the magnetization isotherms measured using the loop process. Except for the isotherm recorded at 308 K, these data exhibit nicely developed field-induced transitions over a wide temperature interval. The corresponding magnetic entropy change is displayed in Fig. 8(b). The entropy change is large over an interval of almost 20 K. We include the results from the standard process with 1 K step to emphasize that the same total entropy change is put into a very narrow temperature interval. Note that the spikes still visible in the data with the loop process are one order of magnitude smaller than for the standard process. One would expect that these spikes do not occur at all, but they also occur when estimating the entropy change from the isofield measurements [21]. Fig. 9(a) shows the magnetization isotherms as derived from the standard process and Fig. 9(b) the resulting isothermal entropy change. The results nicely reproduce earlier results of Rocco et al. [11]. Note that the peak value depends strongly on the temperature interval between the isotherms. This clearly suggests that these results are not physical, however, if one tries to correct the data in the way proposed by Liu et al. it turns
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Fig. 9. (a) Magnetization isotherms measured according to the standard process. (b) Entropy change as a function of temperature for Mn0.99Cu0.01As for 1 and 2 K steps for a 5 T field change.
out that the data point at 313 K with about 80 J/kgK should stay unchanged and only the second point with 180 J/kgK and 70 J/kgK, respectively, will be reduced.
3.4. Gd5Ge2.3Si1.7 revisited Another material exhibiting considerable thermal hysteresis is the Gd5Ge2Si2 type of compound. In an earlier paper we published results on a single crystal with composition Gd5Ge2.3Si1.7 [17]. These results were obtained by the standard process and show a very pronounced peak with a maximum value of 45 J/kgK in the isothermal entropy change. The same sample was inspected again using the loop process to measure the magnetization isotherms. The measured isotherms and calculated isothermal entropy change are displayed in Fig. 10(a) and (b), respectively. Compared with the earlier results, the peak in the magnetic entropy change is strongly reduced to 35 J/kgK. On the other hand,
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Gd5Ge2.3Si1.7
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T (K) Fig. 10. (a) Magnetization isotherms on a single crystal of Gd5Ge2.3Si1.7 according to the loop process. (b) Isothermal entropy change derived from these isotherms.
the shoulder in the isothermal entropy change that was observed for higher fields is hardly affected and the integral of the magnetic entropy is unchanged.
4. Discussion We now want to discuss what goes wrong when the magnetic entropy change is determined by the standard process. The Maxwell relation (1) tells us that the magnetic entropy changes go with the applied field at the same rate as the isofield magnetization changes with temperature. This means that when we measure magnetization isotherms at equidistant temperatures, a large change in magnetic response between adjacent isotherms reflects a large isothermal change in entropy. Equivalently a steep change of critical field with temperature results in a large entropy change. The former is only true if the adjacent isotherms reflect the temperature dependence of the isofield magnetization. In a material that shows a second-order phase transition this condition is always fulfilled independent of the field and temperature history. For a material that shows a first-order magnetic phase transition the field and temperature history plays an important
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role as indicated in the temperature dependence of the magnetization displayed in Figs. 1 and 6 and the phase-diagrams derived from different magnetic measurements compiled in Figs. 2 and 7. Here we study the different processes by considering the diagram of Mn.99Cu.01As because it is most pronounced. The black squares in Fig. 7 were constructed from isofield measurements in cooling and heating mode. This produced two approximately parallel phase lines called CFM–PM and CPM–FM. The slope of these lines is about 0.25 T/K. The arrows in Fig. 7 indicate which line is crossed when either the field or the temperature is changed. In the standard process depicted in Fig. 9(a) at low temperature the magnetization isotherms measure the ferromagnetic response. The material is soft magnetic which means very low fields are sufficient to saturate the multi-domain sample, indicated by the round marks at the bottom of Fig. 7. Because the sample has an irregular shape we do not correct for demagnetizing fields and just take half the spontaneous magnetization in the ferromagnetic state (60 Am2/kg) as criterion for ferromagnetism. At 314 K, the sample appears to be in a mixed state, part of the sample is paramagnetic and another part is ferromagnetic. A higher field (2.5 T) is therefore needed to reach 60 Am2/kg. At 316 K, the paramagnetic fraction appears to be the majority phase resulting in a field-induced ferromagnetic response only for fields above 3 T which is beyond the line CPM–FM. After the CFM–PM line is crossed we are far in the paramagnetic state and ferromagnetic response will only be found for high-applied magnetic fields that run parallel to the line CPM–FM. The maximum available field of 5 T is just sufficient to have 3 more isotherms with field-induced transitions and at 324 K we only observe a partial metamagnetic transition. Thus we find a phase line that starts off extremely steep around 314 K and then continues with a moderate slope. This phase line is in stark contrast with the results from isotherms determined from the loop process. For the loop process on the other hand the sample is always transformed to the PM state when temperature is once beyond the line CPM–FM by undergoing a zero field heating and cooling process. In this case, the magnetic response is not affected by the coexistence of the two phases. The magnetization response changes each time over the line CPM–FM from PM to FM, representing the intrinsic entropy change of the spin sub-lattice. As evidenced by the data marked in Fig. 7, the change in magnetic response probes the by temperature dependence of the isofield magnetization that is measured in a cooling mode. Thus, a reliable MCE can be determined by using the loop process. Above 315 K, the upper bound of the zero field hysteresis, the magnetic entropy change is identical for the data from both processes. Returning to Fig. 7, lets consider a sample at 312 K that has been magnetized earlier, or has been heated from low temperatures, this sample will be ferromagnetic for all fields. At 316 K independent of the history we shall see paramagnetic response and around 3.5 T field we see a field-induced transition. Therefore in a very narrow temperature interval, a large change in magnetic response is observed. Obviously, the size of the temperature interval is determined by the temperature interval between the isotherms, note the striking but unphysical difference in Fig. 9(b) between 1 and 2 K step size. This change in magnetic response depends on the extent of temperature hysteresis in the sample that is measured. If the lines CFM–PM and CPM–FM are separated by tens of degrees or the hysteresis is squarer, the change in magnetization will be more pronounced, than if they are close together or the isofield curves are more rounded. The change in magnetization is real but the magnetic entropy change calculated from magnetic measurements performed in this way, artificially concentrates the entropy change over the full width of hysteresis into the temperature step of the measurement. Therefore we cannot use the standard method to determine the maximal
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entropy change of a material with thermal hysteresis. As mentioned above analysis of the magnetization data with the process proposed by Liu et al. does not improve the results but under some conditions artificially removes the spike. The fact that the maximum value of the isothermal entropy change depends on the temperature step chosen between the isotherms, clearly indicates that the standard process does not measure the physics of the system as it does not take into account that the magnetic response depends on the sample history. The data recorded by the loop measurement record the entropy change as it occurs in a cooling mode. Alternatively, one may take isothermal magnetization measurements performed with decreasing field in combination with heating the sample in maximum field from the ferromagnetic state, in that case the data should represent the temperature dependence of the isofield magnetization measured in the heating mode. For each of the processes the area below the curves, which is generally associated with the cooling capacity, stays the same. This evidences that the entropy change originates from a magnetization process that involves not only rotation of domains. This entropy change should also be considered in the evaluation of the materials properties and not thrown away as artifact. Obviously, the standard process is putting part of this entropy change into a too narrow temperature interval resulting in the unphysical large peak values. The extent of this part is determined by the field and temperature interval with coexisting ferro- and paramagnetic phases as well as the size of the thermal hysteresis. The temperature interval is determined by the experimental conditions that probe the phase transition from single-phase paramagnetic to single-phase ferromagnetic state. Thus, when performing isothermal magnetization measurement on samples with thermal hysteresis, and the temperature step for the measurement is smaller than the thermal hysteresis or the selected temperature for the isotherm is in the two phase region, these magnetization measurements may not reflect the temperature and field dependence of the single-phase FM and PM state. In summary, the determination of the magnetic entropy change in several compounds, which show large thermal hysteresis has carefully been studied. When the magnetization measurements are performed using a loop process that takes the history dependence of the magnetization into account, it is possible to obtain the isothermal magnetic entropy change just by employing the Maxwell relations. Thus it is not necessary to perform complicated subtraction procedures as proposed by Liu et al. [15]. As was seen in our study the integrated entropy or the refrigeration capacity is independent of the method. Subtracting some part from the results will not improve them. Thus, when performing the isothermal magnetization measurement to derive
the magnetic entropy change in samples with large thermal hysteresis, one should verify that these magnetization measurements reflect the temperature and field dependence of the FM and PM state. Compared to the method proposed by Liu et al. [15] the advantage of our approach is that we determine correctly both the maximal entropy change and the cooling capacity. The authors would like to thank P.F. de Chˆatel for discussion and critical reading of the manuscript. This work was performed within the Scientific Exchange Program between the Netherlands and China, the Student Exchange Program supported by Coordenac- a~ o de Aperfeic- oamento de Pesssoal de N´ıvel Superior (Capes), and partly supported by the National Natural Science Foundation of China and the Dutch technology foundation STW. The authors would also like to thank the Fundac- a~ o de Amparo a Pesquisa do Estado de Sa~ o Paulo (FAPESP). References [1] K.A. Gschneidner, V.K. Pecharsky, A.O. Tsokol, Reports on Progress in Physics 68 (2005) 1479–1539. ¨ [2] E. Bruck, Journal of Physics D-Applied Physics 38 (2005) R381–R391. [3] V.K. Pecharsky, K.A. Gschneidner, Physical Review Letters 78 (1997) 4494–4497. [4] V.K. Pecharsky, K.A. Gschneidner, Journal of Magnetism and Magnetic Materials 167 (1997) L179–L184. ¨ [5] O. Tegus, E. Bruck, K.H.J. Buschow, F.R. de Boer, Nature 415 (2002) 150–152. [6] H. Wada, Y. Tanabe, Applied Physics Letters 79 (2001) 3302–3304. [7] F.X. Hu, B.G. Shen, J.R. Sun, Z.H. Cheng, G.H. Rao, X.X. Zhang, Applied Physics Letters 78 (2001) 3675–3677. [8] H. Terashita, J.J. Garbe, J.J. Neumeier, Physical Review B 70 (2004) 094403. [9] S. Gama, A.A. Coelho, A. de Campos, A.M.G. Carvalho, F.C.G. Gandra, P.J. von Ranke, N.A. de Oliveira, Physical Review Letters 93 (2004) 237202. [10] A. de Campos, D.L. Rocco, A.M.G. Carvalho, L. Caron, A.A. Coelho, S. Gama, L.M. Da Silva, F.C.G. Gandra, A.O. dos Santos, L.P. Cardoso, P.J. Von Ranke, N.A. de Oliveira, Nature Materials 5 (2006) 802–804. [11] D.L. Rocco, A. de Campos, A.M.G. Carvalho, L. Caron, A.A. Coelho, S. Gama, F.C.G. Gandra, A.O. dos Santos, L.P. Cardoso, P.J. von Ranke, N.A. de Oliveira, Applied Physics Letters 90 (2007) 242507. [12] C. Xu, G.D. Li, L.G. Wang, Journal of Applied Physics 99 (2006) 123913. [13] A. Giguere, M. Foldeaki, B.R. Gopal, R. Chahine, T.K. Bose, A. Frydman, J.A. Barclay, Physical Review Letters 83 (1999) 2262–2265. [14] J.R. Sun, F.X. Hu, B.G. Shen, Physical Review Letters 85 (2000) 4191. [15] G.J. Liu, R. Sun, J. Shen, B. Gao, H.W. Zhang, F.X. Hu, B.G. Shen, Applied Physics Letter 90 (2007) 032507. [16] N.A. de Oliveira, P.J. von Ranke, Physical Review B 77 (2008) 214439. ¨ [17] O. Tegus, E. Bruck, L. Zhang, Dagula, K.H.J. Buschow, F.R. de Boer, Physica B 319 (2002) 174–192. ¨ [18] D.T.C. Thanh, E. Bruck, O. Tegus, J.C.P. Klaasse, T.J. Gortenmulder, K.H.J. Buschow, Journal of Applied Physics 99 (2006) 08Q107. ¨ [19] R.P. Hermann, O. Tegus, E. Bruck, K.H.J. Buschow, F.R. de Boer, G.J. Long, F. Grandjean, Physical Review B 70 (2004) 214425. [20] Danmin Liu, Ming Yue, Jiuxing Zhang, T.M. McQueen, Jeffrey W. Lynn, Xiaolu Wang, Ying Chen, Jiying Li, R.J. Cava, Xubo Liu, Zaven Altounian, Q. Huang, Physical Review B 79 (2009) 014435. [21] H. Wada, private communication.