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The study on the field dependence of magnetocaloric effect (MCE) is considered to be of fundamental and practical importance, since it not only guides us in understanding and optimizing the MCE, but also helps us estimate the MCE for higher magnetic field which is not available in some laboratories. The magnetic field (0H) dependence of magnetic entropy change (△SM) has been studied extensively in many materials with second-order magnetic transition. However, the field dependence of MCE for first-order magnetic transition (FOMT) materials has not been sufficiently studied due to their complexity and diversity. In the present work, polycrystalline Mn0.6Fe0.4NiSi0.5Ge0.5, Ni50Mn34Co2Sn14, and LaFe11.7Si1.3 compounds with FOMT are prepared, and the magnetic and magnetocaloric properties are investigated systematically. In order to avoid a spurious △SM, the M-0H curves are measured in a loop process. The M-0H curves are corrected by taking into account the demagnetization effect, i.e. Hint=Hext-NdM. It is found that the -△SM follows a linear relationship -△SM=-△S0 +0H with the variation of magnetic field in Mn0.6Fe0.4NiSi0.5Ge0.5 compound when 0H 1 T. In addition, it is also noted that the △SM is approximately proportional to the square of 0H at low field. The origin of this linear relationship between △SM and 0H at high field and the deviation at low field are discussed by numerically analyzing the Maxwell relation. In addition to the △SM peak value, it is found that other △SM values at different temperatures also follow the linear relation at high field by performing the same numerical analysis. Moreover, it is found that the fitted △SM curve matches the experimental data very well. This result indicates that the linear relationship between △SM and 0H could be utilized to predict the △SM for higher magnetic field change when the field is lower than the saturation field. The applicability of this linear relationship is also verified in other systems with first-order magnetostructural transformation, such as Ni50Mn34Co2Sn14. However, it fails to describe the field dependence of △SM in LaFe11.7Si1.3, which exhibits a strong field dependence of transition temperature. Consequently, our study reveals that a linear dependence of △SM on 0H could occur in magnetostructural transition materials, which show the field independence of transition temperature.
[1] Smith A, Bahl C R H, Bjrk R, Engelbrecht K, Nielsen K K, Pryds N 2012 Adv. Energy Mater. 2 1288
[2] Moya X, Kar-Narayan S, Mathur N D 2014 Nat. Mater. 13 439
[3] Shen B G, Hu F X, Dong Q Y, Sun J R 2013 Chin. Phys. B 22 017502
[4] Zheng X Q, Shen J, Hu F X, Sun J R, Shen B G 2016 Acta Phys. Sin. 65 217502 (in Chinese)[郑新奇，沈俊，胡凤霞，孙继荣，沈保根 2016 65 217502]
[5] Zhang H, Shen B G 2015 Chin. Phys. B 24 127504
[6] Franco V, Conde A 2010 Int. J. Refrig. 33 465
[7] Zhang D K, Zhao J L, Zhang H G, Yue M 2014 Acta Phys. Sin. 63 197501 (in Chinese)[张登魁，赵金良，张红国，岳明 2014 63 197501]
[8] Tegus O, Brck E, Buschow K H J, de Boer F R 2002 Nature 415 150
[9] Liu J, Gottschall T, Skokov K P, Moore J D, Gutfleisch O 2012 Nat. Mater. 11 620
[10] Pecharsky V K, Gschneidner Jr K A 1997 Phys. Rev. Lett. 78 4494
[11] Liu E K, Wang W H, Feng L, Zhu W, Li G J, Chen J L, Zhang H W, Wu G H, Jiang C B, Xu H B, de Boer F 2012 Nat. Commun. 3 873
[12] Shen J, Li Y X, Sun J R, Shen B G 2009 Chin. Phys. B 18 2058
[13] Zhang H, Shen B G, Xu Z Y, Zheng X Q, Shen J, Hu F X, Sun J R, Long Y 2012 J. Appl. Phys. 111 07A909
[14] Zhang H, Shen B G, Xu Z Y, Shen J, Hu F X, Sun J R, Long Y 2013 Appl. Phys. Lett. 102 092401
[15] Franco V, Blzquez J S, Ingale B, Conde A 2012 Annu. Rev. Mater. Res. 42 305
[16] Zheng X Q, Shen B G 2017 Chin. Phys. B 26 027501
[17] Wang Y X, Zhang H, Wu M L, Tao K, Li Y W, Yan T, Long K W, Long T, Pang Z, Long Y 2016 Chin. Phys. B 25 127104
[18] Oesterreicher H, Parker F T 1984 J. Appl. Phys. 55 4334
[19] Franco V, Blzquez J S, Conde A 2006 Appl. Phys. Lett. 89 222512
[20] Patra M, Majumdar S, Giri S, Iles G N, Chatterji T 2010 J. Appl. Phys. 107 076101
[21] Franco V, Blzquez J S, Conde A 2006 J. Appl. Phys. 100 064307
[22] Franco V, Blzquez J S, Milln M, Borrego J M, Conde C F, Conde A 2007 J. Appl. Phys. 101 09C503
[23] Bonilla C M, Herrero-Albillos J, Bartolom F, Garca L M, Parra-Borderas M, Franco V 2010 Phys. Rev. B 81 224424
[24] Casanova F, Batlle X, Labarta A, Marcos J, Maosa L, Planes A 2002 Phys. Rev. B 66 212402
[25] Wei Z Y, Liu E K, Li Y, Xu G Z, Zhang X M, Liu G D, Xi X K, Zhang H W, Wang W H, Wu G H, Zhang X X 2015 Adv. Electron. Mater. 1 1500076
[26] Tao K, Zhang H, Long K W, Wang Y X, Wu M L, Xiao Y N, Xing C F, Wang L C, Long Y 2017 Intermetallics 91 45
[27] Liu G J, Sun J R, Shen J, Gao B, Zhang H W, Hu F X, Shen B G 2007 Appl. Phys. Lett. 90 032507
[28] Giguere A, Foldeaki M, Gopal B R, Chahine R, Bose T K, Frydman A, Barclay J A 1999 Phys. Rev. Lett. 83 2262
[29] Caron L, Ou Z Q, Nguyen T T, Cam Thanh D T, Tegus O, Brck E 2009 J. Magn. Magn. Mater. 321 3559
[30] Li Y W, Zhang H, Tao K, Wang Y X, Wu M L, Long Y 2017 Mater. Des. 114 410
[31] Pecharsky V K, Gschneidner Jr K A 1999 J. Appl. Phys. 86 565
[32] Fldeki M, Chahine R, Bose T K, Barclay J A 2000 Phys. Rev. Lett. 85 4192
[33] Sun J R, Hu F X, Shen B G 2000 Phys. Rev. Lett. 85 4191
[34] Zou J D, Shen B G, Gao B, Shen J, Sun J R 2009 Adv. Mater. 21 693
[35] Amaral J S, Amaral V S 2009 Appl. Phys. Lett. 94 042506
[36] Amaral J S, Amaral V S 2010 J. Magn. Magn. Mater. 322 1552
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[1] Smith A, Bahl C R H, Bjrk R, Engelbrecht K, Nielsen K K, Pryds N 2012 Adv. Energy Mater. 2 1288
[2] Moya X, Kar-Narayan S, Mathur N D 2014 Nat. Mater. 13 439
[3] Shen B G, Hu F X, Dong Q Y, Sun J R 2013 Chin. Phys. B 22 017502
[4] Zheng X Q, Shen J, Hu F X, Sun J R, Shen B G 2016 Acta Phys. Sin. 65 217502 (in Chinese)[郑新奇，沈俊，胡凤霞，孙继荣，沈保根 2016 65 217502]
[5] Zhang H, Shen B G 2015 Chin. Phys. B 24 127504
[6] Franco V, Conde A 2010 Int. J. Refrig. 33 465
[7] Zhang D K, Zhao J L, Zhang H G, Yue M 2014 Acta Phys. Sin. 63 197501 (in Chinese)[张登魁，赵金良，张红国，岳明 2014 63 197501]
[8] Tegus O, Brck E, Buschow K H J, de Boer F R 2002 Nature 415 150
[9] Liu J, Gottschall T, Skokov K P, Moore J D, Gutfleisch O 2012 Nat. Mater. 11 620
[10] Pecharsky V K, Gschneidner Jr K A 1997 Phys. Rev. Lett. 78 4494
[11] Liu E K, Wang W H, Feng L, Zhu W, Li G J, Chen J L, Zhang H W, Wu G H, Jiang C B, Xu H B, de Boer F 2012 Nat. Commun. 3 873
[12] Shen J, Li Y X, Sun J R, Shen B G 2009 Chin. Phys. B 18 2058
[13] Zhang H, Shen B G, Xu Z Y, Zheng X Q, Shen J, Hu F X, Sun J R, Long Y 2012 J. Appl. Phys. 111 07A909
[14] Zhang H, Shen B G, Xu Z Y, Shen J, Hu F X, Sun J R, Long Y 2013 Appl. Phys. Lett. 102 092401
[15] Franco V, Blzquez J S, Ingale B, Conde A 2012 Annu. Rev. Mater. Res. 42 305
[16] Zheng X Q, Shen B G 2017 Chin. Phys. B 26 027501
[17] Wang Y X, Zhang H, Wu M L, Tao K, Li Y W, Yan T, Long K W, Long T, Pang Z, Long Y 2016 Chin. Phys. B 25 127104
[18] Oesterreicher H, Parker F T 1984 J. Appl. Phys. 55 4334
[19] Franco V, Blzquez J S, Conde A 2006 Appl. Phys. Lett. 89 222512
[20] Patra M, Majumdar S, Giri S, Iles G N, Chatterji T 2010 J. Appl. Phys. 107 076101
[21] Franco V, Blzquez J S, Conde A 2006 J. Appl. Phys. 100 064307
[22] Franco V, Blzquez J S, Milln M, Borrego J M, Conde C F, Conde A 2007 J. Appl. Phys. 101 09C503
[23] Bonilla C M, Herrero-Albillos J, Bartolom F, Garca L M, Parra-Borderas M, Franco V 2010 Phys. Rev. B 81 224424
[24] Casanova F, Batlle X, Labarta A, Marcos J, Maosa L, Planes A 2002 Phys. Rev. B 66 212402
[25] Wei Z Y, Liu E K, Li Y, Xu G Z, Zhang X M, Liu G D, Xi X K, Zhang H W, Wang W H, Wu G H, Zhang X X 2015 Adv. Electron. Mater. 1 1500076
[26] Tao K, Zhang H, Long K W, Wang Y X, Wu M L, Xiao Y N, Xing C F, Wang L C, Long Y 2017 Intermetallics 91 45
[27] Liu G J, Sun J R, Shen J, Gao B, Zhang H W, Hu F X, Shen B G 2007 Appl. Phys. Lett. 90 032507
[28] Giguere A, Foldeaki M, Gopal B R, Chahine R, Bose T K, Frydman A, Barclay J A 1999 Phys. Rev. Lett. 83 2262
[29] Caron L, Ou Z Q, Nguyen T T, Cam Thanh D T, Tegus O, Brck E 2009 J. Magn. Magn. Mater. 321 3559
[30] Li Y W, Zhang H, Tao K, Wang Y X, Wu M L, Long Y 2017 Mater. Des. 114 410
[31] Pecharsky V K, Gschneidner Jr K A 1999 J. Appl. Phys. 86 565
[32] Fldeki M, Chahine R, Bose T K, Barclay J A 2000 Phys. Rev. Lett. 85 4192
[33] Sun J R, Hu F X, Shen B G 2000 Phys. Rev. Lett. 85 4191
[34] Zou J D, Shen B G, Gao B, Shen J, Sun J R 2009 Adv. Mater. 21 693
[35] Amaral J S, Amaral V S 2009 Appl. Phys. Lett. 94 042506
[36] Amaral J S, Amaral V S 2010 J. Magn. Magn. Mater. 322 1552
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