Influence of the demagnetizing field on the determination of the magnetocaloric effect from magnetization curves

Influence of the demagnetizing field on the determination of the magnetocaloric effect from magnetization curves R. Caballero-Flores, V. Franco, A. Conde, and L. F. Kiss Citation: Journal of Applied Physics 105, 07A919 (2009); doi: 10.1063/1.3067463 View online: http://dx.doi.org/10.1063/1.3067463 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/105/7?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Effect of hydrostatic pressure on magnetic entropy change and critical behavior of the perovskite manganite La0.4Bi0.3Sr0.3MnO3 J. Appl. Phys. 113, 023904 (2013); 10.1063/1.4774107 Influence of Co and Ni addition on the magnetocaloric effect in Fe 88 − 2 x Co x Ni x Zr 7 B 4 Cu 1 soft magnetic amorphous alloys Appl. Phys. Lett. 96, 182506 (2010); 10.1063/1.3427439 Magnetocaloric response of FeCrB amorphous alloys: Predicting the magnetic entropy change from the Arrott–Noakes equation of state J. Appl. 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Box 49, 1525 Budapest, Hungary 共Presented 14 November 2008; received 20 August 2008; accepted 2 November 2008; published online 12 February 2009兲 The influence of the demagnetizing factor 共N兲on the magnetic entropy change 共⌬SM兲curves is studied for materials with a second order phase transition. For this purpose, a soft magnetic amorphous ribbon is measured for different orientations of the magnetic field with respect to the plane of the sample. For temperatures below the Curie temperature 共TC兲, the increase in Ncauses a decrease in ⌬SM, while for temperatures above TCno change in the shape of the curves has been found for the different orientations, as expected. In order to eliminate this influence of Nand compare the ⌬SM共T兲curves for samples with different shapes, the recently proposed universal curve for the magnetocaloric effect can be used. © 2009 American Institute of Physics. 关DOI: 10.1063/1.3067463兴 The recent increase in the research on magnetocaloric effect 共MCE兲is due to its promising application for magnetic refrigeration at room temperature. It could allow more efficient and environmental friendly systems than those based on the conventional gas compression-expansion cycle. Although the larger MCE response corresponds to materials with a first order magneto-structural phase transition 共the socalled giant magnetocaloric effect1兲, most of the current refrigerator prototypes still use materials with a second order phase transition due to the lack of thermal hysteresis,2being Gd a paradigmatic case. In the literature, it is usual to disregard the value of the demagnetizing factor to calculate the magnetic entropy change from magnetization curves. In the case of ribbon shaped samples measured with the applied field contained in the plane of the ribbon, the influence of the demagnetizing factor Ncan be neglected due to the large aspect ratio. However, this is not the typical case in the study of magnetocaloric materials. The aim of this work is twofold: on the one hand, to analyze the influence of Non the temperature dependence of ⌬SM, and on the other hand, to find a method to compare results obtained from samples with different shapes. An amorphous ribbon of Fe77Cr8B15 composition 共1mm wide and ⬃20 ␮ m thick兲was obtained by melt spinning. The amorphous character of the as-quenched alloy was checked by x-ray diffraction. The field 共up to 15 kOe兲and temperature 共300–500 K兲dependence of magnetization 共M兲 of 3 mm long samples have been measured in a vibrating sample magnetometer with different orientations of the applied magnetic field with respect to the plane of the sample. More details about the microstructural and magnetic characterization of the sample are given elsewhere.3Rotation of the sample around an axis perpendicular to the ribbon plane gives no evidence in the magnetization curves of neither a preferential magnetization axis contained in the plane of the ribbon, nor of an out of plane easy axis. Taking into account the amorphous character of the sample, together with its extreme magnetic softness, we can assume that the main contribution to the magnetic anisotropy is determined by shape. The magnetic entropy change due to the application of a magnetic field Hhas been evaluated using a numerical approximation to the equation ⌬SM= 冕 0 H 冉 ⳵ M ⳵ T 冊 H dH,共1兲 where the partial derivative is replaced by finite differences and the integration is performed numerically. Recently, a phenomenological universal curve for the magnetic entropy change has been proposed4as a method to compare the properties of different materials and to make extrapolations to fields and/or temperatures outside the available experimental range. It is constructed by normalizing the magnetic entropy change curves with respect to their peak ⌬SM pk and rescaling the temperature axis. Initially two different reference temperatures were used for scaling: ␪ 共2 ref兲= 再 −共T−TC兲/共Tr1−TC兲TⱕTC, 共T−TC兲/共Tr2−TC兲T⬎TC, 冎 共2兲 where Tr1and Tr2are the temperatures of the two reference points of each curve and TCis the Curie temperature. However, it has been recently shown theoretically5and confirmed experimentally6that it should be enough to use a single reference temperature Tr, ␪ 共1 ref兲=共T−TC兲/共Tr−TC兲.共3兲 The reference temperatures have been selected as the temperatures corresponding to ⌬SM共Tr兲/⌬SM pk=0.5. For the a兲Electronic mail: [email protected]. JOURNAL OF APPLIED PHYSICS 105, 07A919 共2009兲 0021-8979/2009/105共7兲/07A919/3/$25.00 © 2009 American Institute of Physics105, 07A919-1 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 150.214.182.17 On: Thu, 21 Jan 2016 14:57:24 present study the Curie temperature has been identified as the temperature of the peak entropy change Tpk. Figure 1shows the field dependence of the room temperature magnetization of samples with different orientations with respect to the applied magnetic field. While there is no detectable influence of the orientation of the field when it is contained in the plane of the sample, the application of Hat different angles shows a tilting of the magnetization curve due to shape anisotropy. The small differences between the continuous line 共sample on a bottom mount sample holder rotated around an axis perpendicular to its plane兲and the 0° curve 共sample on a side mount sample holder with the field contained in plane兲are due to the experimental uncertainties. In order to make an accurate comparison between the magnetic entropy change curves, it should be taken into account that there could be a minor influence of small off-axis rotation of the sample, which alters the saddle point, slightly affects the moment value and, consequently, the calculated value of 兩⌬SM兩. Therefore, it is preferable to normalize the curves to analyze the influence of the orientation of H共Fig. 2兲. While above TCthe differences between curves are inside the experimental error, at low temperatures there is a systematic decrease in 兩⌬SM兩as the demagnetizing factor Nincreases. The construction of the universal curve is a more adequate method for analyzing the influence of N. Figure 3 shows the rescaled magnetic entropy change curves for different maximum applied fields using a single reference temperature. Two different orientations of the field are presented: in plane 共N⬇0兲and perpendicular to the plane of the sample 共N⬇1兲. While in the first case the collapse of the experimental data onto a single rescaled curve can be achieved, this is not the case for the perpendicular orientation. Increasing the maximum applied field in this second case produces an increase in the reduced magnetic entropy change for the same value of the reduced temperature. However, this increase is not homogeneous: as field is increased, the separation between the different curves is smaller. The explanation of the low temperature evolution of ⌬SM/⌬SM pk should be related to the fact that for higher demagnetizing factors, the field required to saturate the sample is larger, as seen in Fig. 1. Let us consider that the magnetic equation of state of a ferromagnetic material in the proximity of the transition temperature can be approximately described by the Arrott–Noakes equation of state,7which can be written as H1/ ␥ =a共T−TC兲M1/ ␥ +bM1/ ␤ +1/ ␥ ,共4兲 where ␤ and ␥ are the critical exponents. After proper manipulation of the equation of state, Eq. 共1兲can be rewritten as ⌬SM=− 冕 Ms M a ␥ M共a共T−TC兲+bM1/ ␤ 兲 ␥ −1dM,共5兲 where Msstands for the spontaneous magnetization and the upper limit of the integral is the magnetization corresponding to the maximum applied field at the given temperature. For a general case this expression cannot be simplified. In the case of a mean field model the values of the critical exponents 共 ␤ =0.5; ␥ =1兲reduce it to FIG. 1. 共Color online兲Field dependence of the room temperature magnetization of the sample with the magnetic field applied at different orientations with respect to the plane of the ribbon, rotating the sample around an axis contained in its plane 共0° corresponds to the field in plane; 90° to the field perpendicular to the plane兲. Continuous line corresponds to the rotation of the sample around an axis perpendicular to the plane of the sample. FIG. 2. 共Color online兲Influence of the orientation of the field on the normalized magnetic entropy change calculated from the magnetization data for a maximum applied field of 15 kOe. Symbols correspond to the rotation of the sample around an axis contained in the plane of the sample 共field forming an angle with respect to the ribbon plane兲; continuous line corresponds to the rotation around an axis perpendicular to the plane of the ribbon 共field in plane兲. FIG. 3. 共Color online兲Rescaled magnetic entropy change data using a single reference temperature for different maximum field values applied in the plane of the ribbon and perpendicular to this plane. 07A919-2 Caballero-Flores et al. J. Appl. Phys. 105, 07A919 共2009兲 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 150.214.182.17 On: Thu, 21 Jan 2016 14:57:24 ⌬SM共mean field兲=−1 2a共M2−Ms 2兲.共6兲 The equation of state should be applicable to materials in technical saturation. However, for low applied fields and temperatures below TC, both domain wall movement and magnetization rotation should be taken into account, and the experimental value of magnetization Mexp is smaller than M predicted by Eq. 共4兲. Therefore, Eq. 共6兲can be used to justify the decrease in ⌬SM/⌬SM pk with increasing Hangle: for field perpendicular to the plane, the value of Mexp 2for moderate field values is much smaller than when the field is applied in plane, making ⌬SM/⌬SM pk smaller. As Hincreases, the difference between Mexp and Mdecreases, leading to the finding that the separation between reduced magnetic entropy change curves is smaller for larger field values. For large enough field values 共or for smaller angles of the applied field兲, the rescaled curve tends to that of the sample measured with the field in plane, in agreement with the trend observed in Fig. 2for the angular dependence of the curve. Although rescaling ⌬SM/⌬SM pk curves with the use of a single reference temperature is not able to eliminate the influence of the demagnetizing factor, Fig. 4shows that when the temperature axis is rescaled with the use of Eq. 共2兲, all the experimental data points for the different orientations of the field collapse onto the same curve. In this case, to make more evident the quality of the collapse at the tails of the curves, the reference temperatures have been selected as ⌬SM共Tr兲/⌬SM pk=0.6, with Tr1the one fulfilling that condition for each curve below TC, and Tr2above it 共this is a more demanding test than a factor of 0.5, as the possible deviations would be magnified兲. The reason for this collapse is that although the field dependence of the magnetic entropy change is distorted from what is expected from the equation of state, this change does not have a strong temperature dependence except for temperatures close to the peak, where N does not play an important role. Therefore, the second reference temperature can approximately account for this distortion below TC, allowing the collapse of the experimental data. In conclusion, the demagnetizing factor has an influence in the low temperature tails of the magnetic entropy change, narrowing the peak and affecting the calculated value of the refrigerant capacity for samples with different shapes 共RC decreases with increasing N兲. In order to compare experimental results from samples with different shapes, the universal curve constructed with two reference temperatures can be used, eliminating the influence of Non the experimental data. This work was supported by the Spanish Ministry of Science and Innovation and EU FEDER 共Project Nos. MAT 2007-65227 and CIT-420000-2008-9兲, the PAI of the Regional Government of Andalucía 共Project No. P06-FQM- 01823兲, the Hungarian-Spanish Academic Exchange Program for 2007-2008 共MTA-CSIC, Project No. 04; 2006HU0015兲, and the Hungarian Research Fund 共Contract No. OTKA K 68612兲. R.C.F. acknowledges a research fellowship from the Regional Government of Andalucía. 1V. K. Pecharsky and K. A. Gschneidner, Jr., Phys. Rev. Lett. 78,4494 共1997兲. 2A. M. Tishin and Y. I. Spichkin, The Magnetocaloric Effect and its Applications 共Institute of Physics, Bristol, 2003兲. 3V. Franco, A. Conde, and L. F. Kiss, J. Appl. Phys. 104, 033903 共2008兲. 4V. Franco, J. S. Blázquez, and A. Conde, Appl. Phys. Lett. 89, 222512 共2006兲. 5V. Franco, A. Conde, J. M. Romero-Enrique, and J. S. Blázquez, J. Phys.: Condens. Matter 20, 285207 共2008兲. 6V. Franco, J. S. Blázquez, and A. Conde, J. Appl. Phys. 103, 07B316 共2008兲. 7A. Arrott and J. E. Noakes, Phys. Rev. Lett. 19,786共1967兲. FIG. 4. 共Color online兲Rescaled magnetic entropy change data using two reference temperatures for different maximum field values applied in the plane of the ribbon and perpendicular to this plane. 07A919-3 Caballero-Flores et al. J. Appl. Phys. 105, 07A919 共2009兲 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 150.214.182.17 On: Thu, 21 Jan 2016 14:57:24