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Multiferroics, can simultaneously exhibit multiple ferroic orders, including magnetic order, electric order and elastic order. Among these orders there exist intimately coupling effects. Multiferroics is significant for technological applications and fundamental research. The interplay between ferroelectricity and magnetism allows a magnetic control of ferroelectric properties and an electric control of magnetic properties, which can yield new device concepts. Recent experimental research shows that the Fe/BaTiO3 compound exhibits a prominent magnetoelectric effect, which originates from a change in bonding at the ferroelectric-ferromagnet interface that changes the interface magnetization when the electric polarization reverses, and thus offering a new route to controlling the magnetic properties of multilayer compound heterostructures by the electric field. Motivated by recent discoveries, in this paper we investigate theoretically the thermodynamics of a finite ferroelectric-ferromagnetic chain. A microscopic Heisenberg spin model is constructed to describe magnetoelectric properties of this composite chain, in which electric and magnetic subsystem are coupled through interfacial coupling. However, this vector model is not integrable in general. Therefore, one has to resort to numerical calculations for the thermodynamic properties of such a system. A uniform discrete spin vector is adopted here to approximate the original continuous one, and then the transfer-matrix method is employed to derive the analytical expression. To verify its rationality and effectiveness, the zero-field specific heat of a classical spin chain is solved based on this simplified model, and compared with the exact solution. It demonstrates that the main characteristics obtained by previous research are well reproduced here, and the whole variant trend is also identical. And then the quantities concerned in this paper are calculated, including the magnetization, polarization, magnetoelectric susceptibility, and specific heat. The influence of interfacial coupling, external field, and single-ion anisotropy on the magnetoelectric effect of the composite chain are examined in detail. The results reveal that the interfacial coupling enhances the magnetization and polarization. And in the magnetic field driven magnetoelectric susceptibility, the large magnetoelectric correlation effects are observed, indicating that the magnetic behaviors can be effectively controlled by an external electric field. Meanwhile, it is also found that the external field and single-ion anisotropy both suppress the magnetoelectric susceptibility. In addition, interestingly, the specific heat of system presents a three-peak structure under high electric field, which stems from the thermal excitation of spin states as well as dipole moment caused jointly by electric field and temperature.
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Keywords:
- composite multiferroic chain /
- Heisenberg model /
- transfer-matrix technique /
- magnetoelectric coupling
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[3] Wei L, Hu Z, Du G, Yuan Y, Wang J, Tu H, You B, Zhou S, Qu J, Liu H, Zheng R, Hu Y, Du J 2018 Adv. Mater. 30 1801885
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[13] Ding L J, Yao K L, Fu H H 2011 J. Mater. Chem. 21 449
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[16] Odkhuu D, Kioussis N 2018 Phys. Rev. B 97 094404
[17] Wang Z, Grimson M J 2015 J. Appl. Phys. 118 124109
[18] Gao R, Xu Z, Bai L, Zhang Q, Wang Z, Cai W, Chen G, Deng X, Cao X, Luo X, Fu C 2018 Adv. Electron. Mater. 4 1800030
[19] Liu X T, Chen W J, Jiang G L, Wang B, Zheng Y 2016 Phys. Chem. Chem. Phys. 18 2850
[20] Tokunaga Y, Taguchi Y, Arima T, Tokura Y 2012 Nat. Phys. 8 838
[21] Gao X S, Liu J M, Chen X Y, Liu Z G 2000 J. Appl. Phys. 88 4250
[22] Fisher M E 1964 Am. J. Phys. 32 343
[23] Juhász Junger I, Ihle D 2005 Phys. Rev. B 72 064454
[24] Härtel M, Richter J 2011 Phys. Rev. E 83 214412
[25] Gong S J, Jiang Q 2004 Phys. Lett. A 333 124
[26] Zhai L J, Wang H Y 2015 J. Magn. Magn. Mater. 377 121
[27] Thakur P, Durganandini P 2018 Phys. Rev. B 97 064413
[28] Tokura Y, Seki S, Nagaosa N 2014 Rep. Prog. Phys. 77 076501
[29] Liu M W, Chen Y, Song C C, Wu Y, Ding H L 2011 Solid State Commun. 151 503
[30] Song C C, Chen Y, Liu M W 2010 Physica B 405 439
[31] Juhász Junger I, Ihle D, Bogacz L, Janke W 2008 Phys. Rev. B 77 174411
[32] Venkataiah G, Shirahata Y, Itoh M, Taniyama T 2011 Appl. Phys. Lett. 99 102506
[33] Blöte H W J 1975 Physica B+C 79 427
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[1] Eerenstein W, Mathur N D, Scott J F 2006 Nature 442 759
[2] Fiebig M, Lottermoser T, Meier D, Trassin M 2016 Nat. Rev. Mater. 1 16046
[3] Wei L, Hu Z, Du G, Yuan Y, Wang J, Tu H, You B, Zhou S, Qu J, Liu H, Zheng R, Hu Y, Du J 2018 Adv. Mater. 30 1801885
[4] Nozaki T, Sahashi M 2018 Jpn. J. Appl. Phys. 57 0902A2
[5] Brivio S, Petti D, Bertacco R, Cezar J C 2011 Appl. Phys. Lett. 98 092505
[6] Duan C G, Jaswal S S, Tsymbal E Y 2006 Phys. Rev. Lett. 97 047201
[7] Sahoo S, Polisetty S, Duan C G, Jaswal S S, Tsymbal E Y, Binek C 2007 Phys. Rev. B 76 092108
[8] Horley P P, Sukhov A, Jia C, Martinez E, Berakdar J 2012 Phys. Rev. B 85 054401
[9] Chotorlishvili L, Khomeriki R, Sukhov A, Ruffo S, Berakdar J 2013 Phys. Rev. Lett. 111 117202
[10] Rondinelli J M, Stengel M, Spaldin N A 2008 Nat. Nanotechnol. 3 46
[11] Cai T, Ju S, Lee J, Sai N, Demkov A A, Niu Q, Li Z, Shi J, Wang E 2009 Phys. Rev. B 80 140415
[12] Sirker J 2010 Phys. Rev. B 81 014419
[13] Ding L J, Yao K L, Fu H H 2011 J. Mater. Chem. 21 449
[14] Paglan P A, Nguenang J P, Ruffo S 2018 Europhys. Lett. 122 68001
[15] Sukhov A, Jia C, Horley P P, Berakdar J 2010 J. Phys.: Condens. Matter 22 352201
[16] Odkhuu D, Kioussis N 2018 Phys. Rev. B 97 094404
[17] Wang Z, Grimson M J 2015 J. Appl. Phys. 118 124109
[18] Gao R, Xu Z, Bai L, Zhang Q, Wang Z, Cai W, Chen G, Deng X, Cao X, Luo X, Fu C 2018 Adv. Electron. Mater. 4 1800030
[19] Liu X T, Chen W J, Jiang G L, Wang B, Zheng Y 2016 Phys. Chem. Chem. Phys. 18 2850
[20] Tokunaga Y, Taguchi Y, Arima T, Tokura Y 2012 Nat. Phys. 8 838
[21] Gao X S, Liu J M, Chen X Y, Liu Z G 2000 J. Appl. Phys. 88 4250
[22] Fisher M E 1964 Am. J. Phys. 32 343
[23] Juhász Junger I, Ihle D 2005 Phys. Rev. B 72 064454
[24] Härtel M, Richter J 2011 Phys. Rev. E 83 214412
[25] Gong S J, Jiang Q 2004 Phys. Lett. A 333 124
[26] Zhai L J, Wang H Y 2015 J. Magn. Magn. Mater. 377 121
[27] Thakur P, Durganandini P 2018 Phys. Rev. B 97 064413
[28] Tokura Y, Seki S, Nagaosa N 2014 Rep. Prog. Phys. 77 076501
[29] Liu M W, Chen Y, Song C C, Wu Y, Ding H L 2011 Solid State Commun. 151 503
[30] Song C C, Chen Y, Liu M W 2010 Physica B 405 439
[31] Juhász Junger I, Ihle D, Bogacz L, Janke W 2008 Phys. Rev. B 77 174411
[32] Venkataiah G, Shirahata Y, Itoh M, Taniyama T 2011 Appl. Phys. Lett. 99 102506
[33] Blöte H W J 1975 Physica B+C 79 427
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