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Properties of magnetocaloric materials with a distribution of Curie temperatures
Journal of Magnetism and Magnetic Materials 324 (2012) 564–568
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Journal of Magnetism and Magnetic Materials journal homepage: www.elsevier.com/locate/jmmm
Properties of magnetocaloric materials with a distribution of Curie temperatures C.R.H. Bahl n, R. Bjørk, A. Smith, K.K. Nielsen Fuel Cells and Solid State Chemistry Division, Risø National Laboratory for Sustainable Energy, Technical University of Denmark, Frederiksborgvej 399, 4000 Roskilde, Denmark
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a b s t r a c t
Article history: Received 13 July 2011 Received in revised form 18 August 2011 Available online 3 September 2011
The magnetocaloric properties of inhomogeneous ferromagnets that contain distributions of Curie temperatures are considered as a function of the width of such a distribution. Assuming a normal distribution of the Curie temperature, the average adiabatic temperature change, DTad , the isothermal magnetic entropy change, Ds, and the heat capacity, cp , in zero magnetic field and an applied magnetic field of m0 H ¼ 1 T, have been calculated using the mean field model of ferromagnetism. Interestingly, both the peak position and amplitude of each of these parameters vary differently with the width of the distribution, explaining the observed mismatch of peak temperatures reported in experiments. Also, the field dependence of DTad and Ds is found to depend on the width of the distribution. & 2011 Elsevier B.V. All rights reserved.
Keywords: Magnetocaloric effect Curie temperature Inhomogeneous ferromagnet
1. Introduction Magnetic refrigeration relies on the heating and cooling of a magnetocaloric material due to the application and removal of a magnetic field, see, e.g., Ref. [1]. This so-called magnetocaloric effect has a maximum value close to, but not generally coinciding with, the magnetic phase transition temperature of the magnetocaloric material. Hence, for applications where high values of the magnetocaloric effect are desired, it is important to know the transition temperature (Curie temperature) with some degree of accuracy. In some magnetocaloric materials the Curie temperature may be controlled by, e.g., chemical doping. Here materials may be tailored to specific applications. In order to achieve high temperature spans in magnetic refrigeration devices layered regenerators consisting of a number of different magnetocaloric materials may be used. Such graded regenerators are often built from a number of similar materials with a relatively small spacing of the transition temperatures [2,3]. Examples of such materials that may be chemically doped in order to tune the Curie temperature include LaðFe,Co,SiÞ13 [4], La0:67 Ca0:33x Srx MnO3 [5] and Gd1x Tbx or Gd1x Erx [2]. The required spacing between the transition temperatures will depend on the application, but will in general be of the order of the adiabatic temperature change, i.e. a few degrees. To achieve a tuning of the Curie temperature on such a small scale requires an accurate control of the composition and of any spatial inhomogeneities. Thus a consistent method for analyzing and reporting the
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Corresponding author. E-mail address: [email protected] (C.R.H. Bahl).
0304-8853/$ - see front matter & 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2011.08.044
temperature dependence of the magnetocaloric effect in these materials is needed. In general three physical properties are needed to characterize the magnetocaloric effect fully [6]. The first is the isothermal magnetic entropy change, Ds, linked to the magnetization, m, in a magnetic field change from H1 to H2 through Z H2 @m dH, ð1Þ Ds ¼ m0 H1 @T where m0 is the vacuum permeability and T is the temperature. The other two properties are the specific heat capacity, cp ðT,HÞ, and the adiabatic temperature change, DTad , given by Z H2 T @m dH: ð2Þ DTad ¼ m0 H1 cp @T At the lowest fields the derivative @m=@T is large very close to the Curie temperature in materials displaying a second order phase transition. As the field is increased the inflection point of m will gradually move to higher temperatures, while @m=@T rapidly decreases. Thus, if the initial field H1 is close to zero and H2 is a modest field up to about 1 T both integrals in Eqs. (1) and (2) will give rise to peaks that will be almost coincident with the Curie temperature, as found in, e.g, Refs. [7,8]. Using readily available analysis lab equipment, cp , at least in zero applied field, can be fairly easily measured, whereas calculating Ds from Eq. (1) requires an extensive set of magnetization data if high accuracy is required. DTad may be calculated from Eq. (2), requiring the aforementioned magnetization data as well as knowledge of the field dependence of cp . Alternatively it may be directly measured, but this requires non-standard, often custom built lab equipment [8]. Finally, Ds and DTad may be
C.R.H. Bahl et al. / Journal of Magnetism and Magnetic Materials 324 (2012) 564–568
1
800 T = 302.9 K 750 cp [J kg−1 K−1]
Δ Tad [K]
0.8 0.6 0.4 0.2
μ0Happl = 0 T μ0Happl = 1 T
565
T = 298.5 K
700 650 600
0
550 280
290 T [K]
300
280
310
290 T [K]
300
310
Fig. 1. Experimentally measured values of (A) DTad in an applied field of m0 H ¼ 1 T and (B) cp in zero and 1 T applied field for a sample of La0:66 Ca0:24 Sr0:09 Mn1:05 O3 . The vertical lines indicate the peak temperatures. A difference in the peak temperature of 4.4 K is observed. The data is corrected for demagnetisation.
2. Numerical simulation results The MFM results have been calculated using the parameters of gadolinium as given in Ref. [18]. Gadolinium was chosen as the model system due to its role as benchmark material for magnetic refrigeration and a fairly good agreement between MFM results and experimental data, see, e.g., Ref. [19]. The entropy change and adiabatic temperature change are calculated from the magnetization and heat capacity using Eqs. (1) and (2).
0.4
0.3
p
derived from heat capacity data using the functional relationship sðT,H1 Þ ¼ sðT þ DTad ,H2 Þ and sðT,H1 Þ ¼ sðT,H2 Þ þ DsðT,H1 -H2 Þ, valid for materials with a second order phase transition. For the reasons outlined above the magnetocaloric parameters most often reported in the literature are zero field cp and Ds, and the peak temperature of either of these is often referred to as the Curie temperature of the material. Differences between the peak temperatures found in the three parameters when measuring the same sample have been observed for, e.g., La0:66 Ca0:24 Sr0:09 Mn1:05 O3 measured at Risø DTU using the equipment discussed in Ref. [8]. Here the zero field cp peaks at about 4.4 K below the value where DTad peaks, as shown in Fig. 1. Similar discrepancies have been observed in other manganite systems, where the peak temperature of cp has been found to be less than the Curie temperature determined as the inflection point of m in an applied field of 10 mT [9], the peak of Ds [9–11] and the metal-insulator transition [12]. Reductions of the peak temperatures of cp have also been observed in other classes of magnetocaloric materials, e.g. Tb5 Si2 Ge2 [13] and LaðFe,Co,SiÞ13 [14]. A local distribution in Curie temperatures within a material may arise due to differences of the exchange environments of the magnetic ions because of, e.g., defects, strain or impurities in the structure. In the following we will show how such a distribution of Curie temperatures within a material will influence measurements of each of the magnetocaloric properties. It has previously been shown, by numerical modeling, how a normal distribution of Curie temperatures can shift the peak of Ds to lower temperatures [15]. Here, we confirm this result and expand the study to include the influence on cp and DTad of the width of the distribution. The distribution is introduced through the mean field model of ferromagnetism to calculate the temperature and field dependence of the magnetization (see, e.g., Ref. [16]) as this is a robust model known to give correct trends for a large number of materials. This model is combined with the Debye and Sommerfeld models to include the full heat capacity in the calculations [17]. In the following this combined model will simply be referred to as MFM.
0.2
0.1
0 280
290 T [K]
300
310
Fig. 2. The distributions of Curie temperatures used in the present study. The range of standard deviations shown is s ¼ 1 K–10 K and p is the probability.
In a sample having a distribution of Curie temperatures, the measured values of Ds, cp and DTad will be spatial thermal averages over the entire sample. The basic premise of the present study is that the spatial average may be replaced by an average over an ensemble of spatially uniform samples having a randomly distributed Curie temperature. This will be the case if the variation of the Curie temperature occurs on a scale which is much less than the size of the sample. Then each small part of the sample only feels the average magnetic properties of the rest of the sample, since the internal field in the sample only varies on the same scale as the sample itself. In the opposite case, when the spatial variation of the Curie temperature occurs on a scale comparable to the sample size (as in the case, e.g., of a graded material), it is essential to include demagnetization effects [20]. Normal distributions of 10,000 random Curie temperatures have been used as input for the MFM to simulate having a distribution of Curie temperatures. All other input parameters have been kept constant in these calculations. Each of the normal distributions of Curie temperatures is centered around 293 K and characterized by its standard deviation, s. Values in the range s ¼ 0 K–10 K are modeled in order to simulate different degrees of impurities or defects. The distributions used in this study are shown in Fig. 2. Using MFM with the distributions of Curie temperatures outlined above, the final values of the magnetocaloric parameters when the sample is in thermal equilibrium have been calculated. For the values of the specific entropy and heat capacity this simply corresponds to an average of the values for the random Curie
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temperatures. The final adiabatic temperature change of a sample in thermodynamic equilibrium has been calculated by assuming that the heat generated inside the sample is conserved. For each initial temperature, Tstart , the DTad,i of each of the N individual Curie temperatures of the sample is calculated using MFT in the applied field H and setting the Curie temperature to the ith Curie temperature. The energy balance is then calculated according to N Z X i
Tstart þ DTad,i Tstart
cp,i dT ¼
N Z X
Tstart þ DTad,final
cp,i dT,
Tstart
i
ð3Þ
where DTad,final is the resulting adiabatic temperature change, which is the same for all N Curie temperatures. This equation is solved numerically for DTad,final . The calculated results are presented in Fig. 3. It is observed that with a modest distribution of Curie temperatures (standard deviation, s ¼ 3 K), the only result is a rounding of the sharp
5
σ=0K σ=3K σ = 10 K
3 2 1
3 2 1
0
0 260
280
300 T [K]
350
320
340
280
300 T [K]
300
σ=0K σ=3K σ = 10 K
300
260
cp (1 T) [J kg−1 K−1]
cp (0 T) [J kg−1 K−1]
σ=0K σ=3K σ = 10 K
4 ΔTad [K]
−Δs [J kg−1 K−1]
4
peaks in Ds and DTad . A larger value of s ¼ 10 K results in a severe broadening, reduction of peak amplitude and shift of peak temperature, consistent with the trend observed for Ds in Ref. [15]. For the zero applied field heat capacity the introduction of a distribution immediately rounds the otherwise sharp transition and shifts the beginning of the transition to lower values. In the heat capacity calculated under an applied field of 1 T this effect is much less pronounced. Fig. 4(a) shows the dependence of the temperatures at which the peak values are found on the width of the Curie temperature distribution. As discussed above even small values of s lead to a significant change of the zero field heat capacity peak temperature, but much less change in the heat capacity in an applied field of 1 T. Therefore, the heat capacity in an applied field of 0.1 T has been included in Fig. 4(a). This initially has the same weak dependence on the distribution as the 1 T one but at modest values of s starts to follow the curve for zero applied field. In
250 200 150
320
340
σ=0K σ=3K σ = 10 K
280 260 240 220 200 180
260
280
300 T [K]
320
340
260
280
300 T [K]
320
340
100
290 Peak value [%]
Peak temperature [K]
Fig. 3. The effect of a distribution of Curie temperatures on each of the magnetocaloric parameters. (A) The isothermal magnetic entropy change, Ds, (B) the adiabatic temperature change, DTad , (C) the heat capacity, cp , in zero magnetic field and (D) the heat capacity in an applied magnetic field of m0 H ¼ 1 T.
280 270
cp (0 T) cp (0.1 T) cp (1 T) ΔTad (1 T) Δs (1 T)
260 250 0
2
90 cp (0 T) cp (0.1 T) cp (1 T) ΔTad (1 T) Δs (1 T)
80 70 60
4
6 σ [K]
8
10
0
2
4
6
8
σ [K]
Fig. 4. Effect of a distribution of Curie temperatures, (A) the peak temperatures and (B) the peak values.
10
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0.9 −Δs ΔTad
0.85
β [−]
0.8
0.75
0.7
0.65 0
2
4
σ [K]
6
8
10
Fig. 5. Effect of a distribution of Curie temperatures on the field dependence, aHb , of Ds and DTad at the temperature of the peak value.
Fig. 4(b) the peak amplitudes are plotted as a function of the standard deviation. The values have all been normalized to 100% in the zero distribution case. Here, it is observed how while the amplitudes of the three calculated heat capacities are only weakly affected, the situation for Ds and DTad is different. Even a modest width of the distribution will reduce the peak amplitudes by more than 10%, lowering the magnetocaloric performance of such a material. Close to the Curie temperature both DTad and Ds have been reported to scale approximately as H2=3 (see Ref. [21]). This is in accordance with the MFM. We have also studied the influence of a distribution of Curie temperatures on the scaling of these properties. For both DTad and Ds the value at the peak temperature has been found at a range of applied fields, with a spacing of 0.05 T. The data is fitted in the range 0.5 T to 1.5 T by a single exponential expression, aHb , giving the results shown in Fig. 5. As s increases it is seen how the value of b steadily increases. The small error bars show that a single exponential expression provides an acceptable fit to the data. The results shown depend slightly on the range over which the fits are made. Here, the range relevant for devices with permanent magnet assemblies has been used. However, the trend that b increases with s has also been verified for other fitting ranges.
3. Discussion The results in Figs. 3–5 show the effects of a distribution of Curie temperatures on the magnetocaloric properties of ferromagnetic materials. Using ferromagnetic resonance (FMR) measurements the presence of such distributions have been observed in manganites, with widths of about 3 K being reported [22,23]. The value of the Curie temperature of a material is given by the strength of the exchange coupling between magnetic ions. That is, the size of the exchange interaction, the coordination of ions and the value of the spin of these. Any local variation in one or more of these will induce a change in the Curie temperature. As discussed in Ref. [24] such variations are on a macroscopic scale in real ferromagnets. Thus, individual grains or domains each have distinct Curie temperatures, albeit these volumes are very small compared to the sizes of the samples being measured. The properties measured will therefore be an average of these making this approach valid. It has been found that the method of sample preparation of perovskite oxides can have an effect on the spatial homogeneity of the sample [25]. This may be due to different amounts or distributions of defects or inhomogeneities depending on the method of preparation. Also, doping the structure may broaden
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the distribution of Curie temperatures [9,12]. The presence of dopants in the structure alters the chemical environments of the magnetic ions introducing a spread in the exchange energy and thus Curie temperature. The distribution of dopants in manganites has been found to be inhomogeneous [26,27], potentially leading to a distribution of local Curie temperatures. Using an old method to study inhomogeneities by Arrott plots [28] Bebenin et al. [29] recently studied the effects of different dopants in La1x Ax MnO3 . They found that doping different divalent ions (A¼Sr, Ba, Ca) results in different distributions, from s ¼ 0:8 K for Sr to s ¼ 5–8 K for Ca. The above discussion is mainly focused on the presence of inhomogeneities in perovskite oxides, but the same considerations may also be made for other types of magnetocaloric materials that may contain locally inhomogeneous exchange environments. Indeed, assuming a rectangular distribution of Curie temperatures, widths of about 5 K for EuO and as much as 11 K for polycrystalline Gd were recently reported [30]. These values were found by studying the offset of the field dependence o ðDsÞmax from the expected H2=3 scaling, where ðDsÞmax is the maximum value Ds attains. In a recent paper the scaling was extended with a H4=3 term and this expression was used to fit ðDsÞmax [31]. This was used to fit data measured on Gd samples of various purities. Assuming a rectangular distribution of Curie temperatures, widths of between 0.3 K for a single crystal sample and 11 K for a less pure polycrystalline sample were found. Here, we also show how even a modest value of s can influence the field dependence of ðDsÞmax and also ðDTad Þmax , see Fig. 5. For simplicity only a single exponent term, b, is used. This is found to give good fits to the data. The increase of b with increasing s indicates a stronger field dependence of both properties, which actually is an advantage from the point of view of applications, as it will compensate some of the reduced amplitude of these as indicated in Fig. 4(b). The evaluation of the magnetocaloric properties in the presence of a distribution of Curie temperatures can, except for DTad , be considered to be like a convolution. Therefore the asymmetry of a peak determines the temperature shift of this in the presence of a distribution of Curie temperatures. Thus the significant shift of cp observed in Fig. 4(A) is due to the highly asymmetric nature of the peak, something that for second order materials is observed both in MFM (see Fig. 3(C)) and in experiments (see Fig. 1(B)). As the symmetry increases for the Ds and DTad peaks (see Figs. 3(A) and (B)) the shift of the peak due to the distribution decreases (see Fig. 4(A)). The ability of MFM to accurately describe the magnetic properties of materials may be questioned. Indeed, the qualitative resemblance of MFM predictions and experimental data is clearly better for some materials than others. However, in the present paper MFM is primarily used as a convenient way of introducing and modeling the effect of a distribution of Curie temperatures. The lower temperature of the peak of cp in an applied field compared to the zero field value, apparent in Figs. 3(C) and (D) and Fig. 4(A), is due to the characteristics of MFM. This is contrary to the experimental situation where the applied field peak is in general found at a higher temperature than the zero field peak (see Fig. 1(B)); however, the shape of the applied field peak is quite well described by MFM. The results given here are obtained assuming a normal distribution of Curie temperatures. However, the choice of distribution is not critical; other distributions, such as a Lorentzian or rectangular distribution, will result in trends similar to those found here. A further possible reason for a discrepancy between the temperature at which Ds peaks and the Curie temperature has recently been discussed by Franco et al. [32]. When using the
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critical exponents given by the three-dimensional Heisenberg model a shift of the Ds peak towards higher temperatures is observed, whereas the two peaks coincide when using the critical exponents from MFM. However, it is not clear whether there will be a relative shift between the peak values as no similar analysis of the heat capacity or adiabatic temperature change is given. Such a relative shift is observed experimentally as well as in the MFM using normal distributions of Curie temperatures.
4. Conclusion It has been shown that the introduction of a distribution in the Curie temperatures of a ferromagnet will shift the peak temperature and amplitude of the adiabatic temperature change, DTad , the isothermal magnetic entropy change, Ds, and the heat capacity, cp . Such a distribution will exist in many real ferromagnets. The variation of these magnetocaloric properties with the width of the distribution depends on the shape and symmetry of the single Curie temperature peak, with asymmetric peaks, as often found in the zero-field heat capacity of second order phase transitions, being affected most. A standard deviation of the distribution of 10 K will in gadolinium lead to a reduction of the DTad and Ds peaks by about 30% and a shift of the peaks of 2 K and 5 K, respectively. Such a distribution is consistent with experiments, and will be important to take into account when constructing multi-material magnetocaloric devices. However, the decrease of the magnetocaloric effect is partially compensated by an increased field dependence of both DTad and Ds in the presence of a distribution of Curie temperatures.
Acknowledgment The authors would like to thank Dr. C. Ancona-Torres for supplying the experimental data and Dr. L. Theil Kuhn for a critical reading of the manuscript. Also, the authors would like to acknowledge the support of the Programme Commission on Energy and Environment (EnMi) (Contract no. 2104-06-0032) which is part of the Danish Council for Strategic Research. References [1] V.K. Pecharsky, K.A. Gschneidner Jr, Magnetocaloric affect and magnetic refrigeration, Journal of Magnetism and Magnetic Materials 200 (1999) 44. [2] A. Rowe, A. Tura, Experimental investigation of a three-material layered active magnetic regenerator, International Journal of Refrigeration 29 (2006) 1286–1293. [3] S. Russek, J. Auringer, A. Boeder, J. Chell, S. Jacobs, C. Zimm, The performance of a rotary magnet magnetic refrigerator with layered beds, in: Proceedings of the Fourth International Conference on Magnetic Refrigeration at Room Temperature, Baotou, Inner Mongolia, China, 2010, pp. 339–349. [4] M. Katter, V. Zellmann, G.W. Reppel, K. Uestuener, Magnetocaloric properties of LaðFe,Co,SiÞ13 bulk material prepared by powder metallurgy, IEEE Transactions on Magnetics 44 (2008) 3044–3047. [5] A.R. Dinesen, S. Linderoth, S. Mørup, Direct and indirect measurement of the magnetocaloric effect in La0:67 Ca0:33x Srx MnO3 7 d (xA[0; 0.33]), Journal of Physics: Condensed Matter 17 (2005) 6257. [6] A.M. Tishin, Y.I. Spichkin, The Magnetocaloric Effect and Its Applications, Institute of Physics Bristol, 2003. [7] A. Szewczyk, H. Szymczak, A. Wisniewski, K. Piotrowski, R. Kartaszynski, B. Dabrowski, S. Kolesnik, Z. Bukowski, Magnetocaloric effect in La1x Srx MnO3 for x¼ 0.13 and 0.16, Applied Physics Letters 77 (2000) 1026. [8] R. Bjørk, C.R.H. Bahl, M. Katter, Magnetocaloric properties of LaFe13xy Cox Siy and commercial grade Gd, Journal of Magnetism and Magnetic Materials 322 (2010) 3882–3888.
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Properties of magnetocaloric materials with a distribution of Curie temperatures C.R.H. Bahl n, R. Bjørk, A. Smith, K.K. Nielsen Fuel Cells and Solid State Chemistry Division, Risø National Laboratory for Sustainable Energy, Technical University of Denmark, Frederiksborgvej 399, 4000 Roskilde, Denmark
a r t i c l e i n f o
a b s t r a c t
Article history: Received 13 July 2011 Received in revised form 18 August 2011 Available online 3 September 2011
The magnetocaloric properties of inhomogeneous ferromagnets that contain distributions of Curie temperatures are considered as a function of the width of such a distribution. Assuming a normal distribution of the Curie temperature, the average adiabatic temperature change, DTad , the isothermal magnetic entropy change, Ds, and the heat capacity, cp , in zero magnetic field and an applied magnetic field of m0 H ¼ 1 T, have been calculated using the mean field model of ferromagnetism. Interestingly, both the peak position and amplitude of each of these parameters vary differently with the width of the distribution, explaining the observed mismatch of peak temperatures reported in experiments. Also, the field dependence of DTad and Ds is found to depend on the width of the distribution. & 2011 Elsevier B.V. All rights reserved.
Keywords: Magnetocaloric effect Curie temperature Inhomogeneous ferromagnet
1. Introduction Magnetic refrigeration relies on the heating and cooling of a magnetocaloric material due to the application and removal of a magnetic field, see, e.g., Ref. [1]. This so-called magnetocaloric effect has a maximum value close to, but not generally coinciding with, the magnetic phase transition temperature of the magnetocaloric material. Hence, for applications where high values of the magnetocaloric effect are desired, it is important to know the transition temperature (Curie temperature) with some degree of accuracy. In some magnetocaloric materials the Curie temperature may be controlled by, e.g., chemical doping. Here materials may be tailored to specific applications. In order to achieve high temperature spans in magnetic refrigeration devices layered regenerators consisting of a number of different magnetocaloric materials may be used. Such graded regenerators are often built from a number of similar materials with a relatively small spacing of the transition temperatures [2,3]. Examples of such materials that may be chemically doped in order to tune the Curie temperature include LaðFe,Co,SiÞ13 [4], La0:67 Ca0:33x Srx MnO3 [5] and Gd1x Tbx or Gd1x Erx [2]. The required spacing between the transition temperatures will depend on the application, but will in general be of the order of the adiabatic temperature change, i.e. a few degrees. To achieve a tuning of the Curie temperature on such a small scale requires an accurate control of the composition and of any spatial inhomogeneities. Thus a consistent method for analyzing and reporting the
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Corresponding author. E-mail address: [email protected] (C.R.H. Bahl).
0304-8853/$ - see front matter & 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2011.08.044
temperature dependence of the magnetocaloric effect in these materials is needed. In general three physical properties are needed to characterize the magnetocaloric effect fully [6]. The first is the isothermal magnetic entropy change, Ds, linked to the magnetization, m, in a magnetic field change from H1 to H2 through Z H2 @m dH, ð1Þ Ds ¼ m0 H1 @T where m0 is the vacuum permeability and T is the temperature. The other two properties are the specific heat capacity, cp ðT,HÞ, and the adiabatic temperature change, DTad , given by Z H2 T @m dH: ð2Þ DTad ¼ m0 H1 cp @T At the lowest fields the derivative @m=@T is large very close to the Curie temperature in materials displaying a second order phase transition. As the field is increased the inflection point of m will gradually move to higher temperatures, while @m=@T rapidly decreases. Thus, if the initial field H1 is close to zero and H2 is a modest field up to about 1 T both integrals in Eqs. (1) and (2) will give rise to peaks that will be almost coincident with the Curie temperature, as found in, e.g, Refs. [7,8]. Using readily available analysis lab equipment, cp , at least in zero applied field, can be fairly easily measured, whereas calculating Ds from Eq. (1) requires an extensive set of magnetization data if high accuracy is required. DTad may be calculated from Eq. (2), requiring the aforementioned magnetization data as well as knowledge of the field dependence of cp . Alternatively it may be directly measured, but this requires non-standard, often custom built lab equipment [8]. Finally, Ds and DTad may be
C.R.H. Bahl et al. / Journal of Magnetism and Magnetic Materials 324 (2012) 564–568
1
800 T = 302.9 K 750 cp [J kg−1 K−1]
Δ Tad [K]
0.8 0.6 0.4 0.2
μ0Happl = 0 T μ0Happl = 1 T
565
T = 298.5 K
700 650 600
0
550 280
290 T [K]
300
280
310
290 T [K]
300
310
Fig. 1. Experimentally measured values of (A) DTad in an applied field of m0 H ¼ 1 T and (B) cp in zero and 1 T applied field for a sample of La0:66 Ca0:24 Sr0:09 Mn1:05 O3 . The vertical lines indicate the peak temperatures. A difference in the peak temperature of 4.4 K is observed. The data is corrected for demagnetisation.
2. Numerical simulation results The MFM results have been calculated using the parameters of gadolinium as given in Ref. [18]. Gadolinium was chosen as the model system due to its role as benchmark material for magnetic refrigeration and a fairly good agreement between MFM results and experimental data, see, e.g., Ref. [19]. The entropy change and adiabatic temperature change are calculated from the magnetization and heat capacity using Eqs. (1) and (2).
0.4
0.3
p
derived from heat capacity data using the functional relationship sðT,H1 Þ ¼ sðT þ DTad ,H2 Þ and sðT,H1 Þ ¼ sðT,H2 Þ þ DsðT,H1 -H2 Þ, valid for materials with a second order phase transition. For the reasons outlined above the magnetocaloric parameters most often reported in the literature are zero field cp and Ds, and the peak temperature of either of these is often referred to as the Curie temperature of the material. Differences between the peak temperatures found in the three parameters when measuring the same sample have been observed for, e.g., La0:66 Ca0:24 Sr0:09 Mn1:05 O3 measured at Risø DTU using the equipment discussed in Ref. [8]. Here the zero field cp peaks at about 4.4 K below the value where DTad peaks, as shown in Fig. 1. Similar discrepancies have been observed in other manganite systems, where the peak temperature of cp has been found to be less than the Curie temperature determined as the inflection point of m in an applied field of 10 mT [9], the peak of Ds [9–11] and the metal-insulator transition [12]. Reductions of the peak temperatures of cp have also been observed in other classes of magnetocaloric materials, e.g. Tb5 Si2 Ge2 [13] and LaðFe,Co,SiÞ13 [14]. A local distribution in Curie temperatures within a material may arise due to differences of the exchange environments of the magnetic ions because of, e.g., defects, strain or impurities in the structure. In the following we will show how such a distribution of Curie temperatures within a material will influence measurements of each of the magnetocaloric properties. It has previously been shown, by numerical modeling, how a normal distribution of Curie temperatures can shift the peak of Ds to lower temperatures [15]. Here, we confirm this result and expand the study to include the influence on cp and DTad of the width of the distribution. The distribution is introduced through the mean field model of ferromagnetism to calculate the temperature and field dependence of the magnetization (see, e.g., Ref. [16]) as this is a robust model known to give correct trends for a large number of materials. This model is combined with the Debye and Sommerfeld models to include the full heat capacity in the calculations [17]. In the following this combined model will simply be referred to as MFM.
0.2
0.1
0 280
290 T [K]
300
310
Fig. 2. The distributions of Curie temperatures used in the present study. The range of standard deviations shown is s ¼ 1 K–10 K and p is the probability.
In a sample having a distribution of Curie temperatures, the measured values of Ds, cp and DTad will be spatial thermal averages over the entire sample. The basic premise of the present study is that the spatial average may be replaced by an average over an ensemble of spatially uniform samples having a randomly distributed Curie temperature. This will be the case if the variation of the Curie temperature occurs on a scale which is much less than the size of the sample. Then each small part of the sample only feels the average magnetic properties of the rest of the sample, since the internal field in the sample only varies on the same scale as the sample itself. In the opposite case, when the spatial variation of the Curie temperature occurs on a scale comparable to the sample size (as in the case, e.g., of a graded material), it is essential to include demagnetization effects [20]. Normal distributions of 10,000 random Curie temperatures have been used as input for the MFM to simulate having a distribution of Curie temperatures. All other input parameters have been kept constant in these calculations. Each of the normal distributions of Curie temperatures is centered around 293 K and characterized by its standard deviation, s. Values in the range s ¼ 0 K–10 K are modeled in order to simulate different degrees of impurities or defects. The distributions used in this study are shown in Fig. 2. Using MFM with the distributions of Curie temperatures outlined above, the final values of the magnetocaloric parameters when the sample is in thermal equilibrium have been calculated. For the values of the specific entropy and heat capacity this simply corresponds to an average of the values for the random Curie
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temperatures. The final adiabatic temperature change of a sample in thermodynamic equilibrium has been calculated by assuming that the heat generated inside the sample is conserved. For each initial temperature, Tstart , the DTad,i of each of the N individual Curie temperatures of the sample is calculated using MFT in the applied field H and setting the Curie temperature to the ith Curie temperature. The energy balance is then calculated according to N Z X i
Tstart þ DTad,i Tstart
cp,i dT ¼
N Z X
Tstart þ DTad,final
cp,i dT,
Tstart
i
ð3Þ
where DTad,final is the resulting adiabatic temperature change, which is the same for all N Curie temperatures. This equation is solved numerically for DTad,final . The calculated results are presented in Fig. 3. It is observed that with a modest distribution of Curie temperatures (standard deviation, s ¼ 3 K), the only result is a rounding of the sharp
5
σ=0K σ=3K σ = 10 K
3 2 1
3 2 1
0
0 260
280
300 T [K]
350
320
340
280
300 T [K]
300
σ=0K σ=3K σ = 10 K
300
260
cp (1 T) [J kg−1 K−1]
cp (0 T) [J kg−1 K−1]
σ=0K σ=3K σ = 10 K
4 ΔTad [K]
−Δs [J kg−1 K−1]
4
peaks in Ds and DTad . A larger value of s ¼ 10 K results in a severe broadening, reduction of peak amplitude and shift of peak temperature, consistent with the trend observed for Ds in Ref. [15]. For the zero applied field heat capacity the introduction of a distribution immediately rounds the otherwise sharp transition and shifts the beginning of the transition to lower values. In the heat capacity calculated under an applied field of 1 T this effect is much less pronounced. Fig. 4(a) shows the dependence of the temperatures at which the peak values are found on the width of the Curie temperature distribution. As discussed above even small values of s lead to a significant change of the zero field heat capacity peak temperature, but much less change in the heat capacity in an applied field of 1 T. Therefore, the heat capacity in an applied field of 0.1 T has been included in Fig. 4(a). This initially has the same weak dependence on the distribution as the 1 T one but at modest values of s starts to follow the curve for zero applied field. In
250 200 150
320
340
σ=0K σ=3K σ = 10 K
280 260 240 220 200 180
260
280
300 T [K]
320
340
260
280
300 T [K]
320
340
100
290 Peak value [%]
Peak temperature [K]
Fig. 3. The effect of a distribution of Curie temperatures on each of the magnetocaloric parameters. (A) The isothermal magnetic entropy change, Ds, (B) the adiabatic temperature change, DTad , (C) the heat capacity, cp , in zero magnetic field and (D) the heat capacity in an applied magnetic field of m0 H ¼ 1 T.
280 270
cp (0 T) cp (0.1 T) cp (1 T) ΔTad (1 T) Δs (1 T)
260 250 0
2
90 cp (0 T) cp (0.1 T) cp (1 T) ΔTad (1 T) Δs (1 T)
80 70 60
4
6 σ [K]
8
10
0
2
4
6
8
σ [K]
Fig. 4. Effect of a distribution of Curie temperatures, (A) the peak temperatures and (B) the peak values.
10
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0.9 −Δs ΔTad
0.85
β [−]
0.8
0.75
0.7
0.65 0
2
4
σ [K]
6
8
10
Fig. 5. Effect of a distribution of Curie temperatures on the field dependence, aHb , of Ds and DTad at the temperature of the peak value.
Fig. 4(b) the peak amplitudes are plotted as a function of the standard deviation. The values have all been normalized to 100% in the zero distribution case. Here, it is observed how while the amplitudes of the three calculated heat capacities are only weakly affected, the situation for Ds and DTad is different. Even a modest width of the distribution will reduce the peak amplitudes by more than 10%, lowering the magnetocaloric performance of such a material. Close to the Curie temperature both DTad and Ds have been reported to scale approximately as H2=3 (see Ref. [21]). This is in accordance with the MFM. We have also studied the influence of a distribution of Curie temperatures on the scaling of these properties. For both DTad and Ds the value at the peak temperature has been found at a range of applied fields, with a spacing of 0.05 T. The data is fitted in the range 0.5 T to 1.5 T by a single exponential expression, aHb , giving the results shown in Fig. 5. As s increases it is seen how the value of b steadily increases. The small error bars show that a single exponential expression provides an acceptable fit to the data. The results shown depend slightly on the range over which the fits are made. Here, the range relevant for devices with permanent magnet assemblies has been used. However, the trend that b increases with s has also been verified for other fitting ranges.
3. Discussion The results in Figs. 3–5 show the effects of a distribution of Curie temperatures on the magnetocaloric properties of ferromagnetic materials. Using ferromagnetic resonance (FMR) measurements the presence of such distributions have been observed in manganites, with widths of about 3 K being reported [22,23]. The value of the Curie temperature of a material is given by the strength of the exchange coupling between magnetic ions. That is, the size of the exchange interaction, the coordination of ions and the value of the spin of these. Any local variation in one or more of these will induce a change in the Curie temperature. As discussed in Ref. [24] such variations are on a macroscopic scale in real ferromagnets. Thus, individual grains or domains each have distinct Curie temperatures, albeit these volumes are very small compared to the sizes of the samples being measured. The properties measured will therefore be an average of these making this approach valid. It has been found that the method of sample preparation of perovskite oxides can have an effect on the spatial homogeneity of the sample [25]. This may be due to different amounts or distributions of defects or inhomogeneities depending on the method of preparation. Also, doping the structure may broaden
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the distribution of Curie temperatures [9,12]. The presence of dopants in the structure alters the chemical environments of the magnetic ions introducing a spread in the exchange energy and thus Curie temperature. The distribution of dopants in manganites has been found to be inhomogeneous [26,27], potentially leading to a distribution of local Curie temperatures. Using an old method to study inhomogeneities by Arrott plots [28] Bebenin et al. [29] recently studied the effects of different dopants in La1x Ax MnO3 . They found that doping different divalent ions (A¼Sr, Ba, Ca) results in different distributions, from s ¼ 0:8 K for Sr to s ¼ 5–8 K for Ca. The above discussion is mainly focused on the presence of inhomogeneities in perovskite oxides, but the same considerations may also be made for other types of magnetocaloric materials that may contain locally inhomogeneous exchange environments. Indeed, assuming a rectangular distribution of Curie temperatures, widths of about 5 K for EuO and as much as 11 K for polycrystalline Gd were recently reported [30]. These values were found by studying the offset of the field dependence o ðDsÞmax from the expected H2=3 scaling, where ðDsÞmax is the maximum value Ds attains. In a recent paper the scaling was extended with a H4=3 term and this expression was used to fit ðDsÞmax [31]. This was used to fit data measured on Gd samples of various purities. Assuming a rectangular distribution of Curie temperatures, widths of between 0.3 K for a single crystal sample and 11 K for a less pure polycrystalline sample were found. Here, we also show how even a modest value of s can influence the field dependence of ðDsÞmax and also ðDTad Þmax , see Fig. 5. For simplicity only a single exponent term, b, is used. This is found to give good fits to the data. The increase of b with increasing s indicates a stronger field dependence of both properties, which actually is an advantage from the point of view of applications, as it will compensate some of the reduced amplitude of these as indicated in Fig. 4(b). The evaluation of the magnetocaloric properties in the presence of a distribution of Curie temperatures can, except for DTad , be considered to be like a convolution. Therefore the asymmetry of a peak determines the temperature shift of this in the presence of a distribution of Curie temperatures. Thus the significant shift of cp observed in Fig. 4(A) is due to the highly asymmetric nature of the peak, something that for second order materials is observed both in MFM (see Fig. 3(C)) and in experiments (see Fig. 1(B)). As the symmetry increases for the Ds and DTad peaks (see Figs. 3(A) and (B)) the shift of the peak due to the distribution decreases (see Fig. 4(A)). The ability of MFM to accurately describe the magnetic properties of materials may be questioned. Indeed, the qualitative resemblance of MFM predictions and experimental data is clearly better for some materials than others. However, in the present paper MFM is primarily used as a convenient way of introducing and modeling the effect of a distribution of Curie temperatures. The lower temperature of the peak of cp in an applied field compared to the zero field value, apparent in Figs. 3(C) and (D) and Fig. 4(A), is due to the characteristics of MFM. This is contrary to the experimental situation where the applied field peak is in general found at a higher temperature than the zero field peak (see Fig. 1(B)); however, the shape of the applied field peak is quite well described by MFM. The results given here are obtained assuming a normal distribution of Curie temperatures. However, the choice of distribution is not critical; other distributions, such as a Lorentzian or rectangular distribution, will result in trends similar to those found here. A further possible reason for a discrepancy between the temperature at which Ds peaks and the Curie temperature has recently been discussed by Franco et al. [32]. When using the
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critical exponents given by the three-dimensional Heisenberg model a shift of the Ds peak towards higher temperatures is observed, whereas the two peaks coincide when using the critical exponents from MFM. However, it is not clear whether there will be a relative shift between the peak values as no similar analysis of the heat capacity or adiabatic temperature change is given. Such a relative shift is observed experimentally as well as in the MFM using normal distributions of Curie temperatures.
4. Conclusion It has been shown that the introduction of a distribution in the Curie temperatures of a ferromagnet will shift the peak temperature and amplitude of the adiabatic temperature change, DTad , the isothermal magnetic entropy change, Ds, and the heat capacity, cp . Such a distribution will exist in many real ferromagnets. The variation of these magnetocaloric properties with the width of the distribution depends on the shape and symmetry of the single Curie temperature peak, with asymmetric peaks, as often found in the zero-field heat capacity of second order phase transitions, being affected most. A standard deviation of the distribution of 10 K will in gadolinium lead to a reduction of the DTad and Ds peaks by about 30% and a shift of the peaks of 2 K and 5 K, respectively. Such a distribution is consistent with experiments, and will be important to take into account when constructing multi-material magnetocaloric devices. However, the decrease of the magnetocaloric effect is partially compensated by an increased field dependence of both DTad and Ds in the presence of a distribution of Curie temperatures.
Acknowledgment The authors would like to thank Dr. C. Ancona-Torres for supplying the experimental data and Dr. L. Theil Kuhn for a critical reading of the manuscript. Also, the authors would like to acknowledge the support of the Programme Commission on Energy and Environment (EnMi) (Contract no. 2104-06-0032) which is part of the Danish Council for Strategic Research. References [1] V.K. Pecharsky, K.A. Gschneidner Jr, Magnetocaloric affect and magnetic refrigeration, Journal of Magnetism and Magnetic Materials 200 (1999) 44. [2] A. Rowe, A. Tura, Experimental investigation of a three-material layered active magnetic regenerator, International Journal of Refrigeration 29 (2006) 1286–1293. [3] S. Russek, J. Auringer, A. Boeder, J. Chell, S. Jacobs, C. Zimm, The performance of a rotary magnet magnetic refrigerator with layered beds, in: Proceedings of the Fourth International Conference on Magnetic Refrigeration at Room Temperature, Baotou, Inner Mongolia, China, 2010, pp. 339–349. [4] M. Katter, V. Zellmann, G.W. Reppel, K. Uestuener, Magnetocaloric properties of LaðFe,Co,SiÞ13 bulk material prepared by powder metallurgy, IEEE Transactions on Magnetics 44 (2008) 3044–3047. [5] A.R. Dinesen, S. Linderoth, S. Mørup, Direct and indirect measurement of the magnetocaloric effect in La0:67 Ca0:33x Srx MnO3 7 d (xA[0; 0.33]), Journal of Physics: Condensed Matter 17 (2005) 6257. [6] A.M. Tishin, Y.I. Spichkin, The Magnetocaloric Effect and Its Applications, Institute of Physics Bristol, 2003. [7] A. Szewczyk, H. Szymczak, A. Wisniewski, K. Piotrowski, R. Kartaszynski, B. Dabrowski, S. Kolesnik, Z. Bukowski, Magnetocaloric effect in La1x Srx MnO3 for x¼ 0.13 and 0.16, Applied Physics Letters 77 (2000) 1026. [8] R. Bjørk, C.R.H. Bahl, M. Katter, Magnetocaloric properties of LaFe13xy Cox Siy and commercial grade Gd, Journal of Magnetism and Magnetic Materials 322 (2010) 3882–3888.
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