najoNanosystems: Physics, Chemistry, MathematicsНаносистемы: физика, химия, математика2220-80542305-7971Университет ИТМО10.17586/2220-8054-2020-11-1-50-64najo-467Research ArticlePHYSICSФИЗИКАMagnetoelectric effects theory by Heisenberg method based on permutation group symmetry of nanoparticlesLebleS.

Kaliningrad 236000

lebleu@mail.ruImmanuel Kant Baltic Federal University, Institute of Physics, Mathematics and Informational TechnologyRussian Federation2020310720251115064Copyright © Leble S., 20252025Leble S.Leble S.Данная работа распространяется под лицензией Creative Commons Attribution 4.0.This work is licensed under a Creative Commons Attribution 4.0 License.https://nanojournal.ifmo.ru/jour/article/view/467

The Heisenberg theory of ferromagnetism is widened to include external electric field action. The material relations are derived by means of differentiation of logarithm of partition function with respect to the magnetic and electric fields. The mean energy coefficients as the exchange integrals combinations are expressed via characters of irreducible representations of corresponding permutation groups by the Heitler method. The thermodynamic equations of state for polarization and magnetization, as functions of the electric and magnetic fields, are derived and illustrated by figures. The magnetization and hysteresis curves in magnetization – magnetic field components plane are built. The theory is applied to nanoparticles, the particle partition function is modeled as the product of the surface and bulk parts. The statistical sum is constructed having explicit expressions for the mean energy in terms of exchange integrals and number of closest neighbors for surface and bulk atoms. The relative contribution of the surface and bulk terms is evaluated.

multielectron statespermutation group symmetrymean energyGauss distributionHeisenberg chainelectric fieldnanoparticlesmultiferroicsThe author acknowledges discussions with V. RodionovaReferencesHeisenberg W. Zur Theorie des Ferromagnetismus. Zs. Phys., 1928, 49, P. 619–636. Zur Quantentheorie des Ferromagnetismus. Probleme der Modernen Physik, A. Sommerfeld Festschrift, Leipzig, 1928, P. 114–122.Heisenberg W. Zur Theorie des Ferromagnetismus. Zs. Phys., 1928, 49, P. 619–636. Zur Quantentheorie des Ferromagnetismus. Probleme der Modernen Physik, A. Sommerfeld Festschrift, Leipzig, 1928, P. 114–122.Vazquez M.´ Magnetic Nano- and Microwires: Design, Synthesis, Properties and Applications. Woodhead Publishing Series in Electronic and Optical Materials. Elsevier Science, 2015, 314 p.Vazquez M.´ Magnetic Nano- and Microwires: Design, Synthesis, Properties and Applications. Woodhead Publishing Series in Electronic and Optical Materials. Elsevier Science, 2015, 314 p.Leble S. Heisenberg chain equations in terms of Fockian covariance with electric field account and multiferroics in nanoscale. Nanosystems: Physics, Chemistry, Mathematics, 2019, 10(1), P. 18–30.Leble S. Heisenberg chain equations in terms of Fockian covariance with electric field account and multiferroics in nanoscale. Nanosystems: Physics, Chemistry, Mathematics, 2019, 10(1), P. 18–30.Heitler W. Staorungsenergie und Austausch beim Mehrk¨ aorperproblem.¨ Zs. Phys. 1927, 46, P. 47. Zur Gruppentheorie der hom´’oopolaren chemischen Bindung, 1928, 835 p.Heitler W. Staorungsenergie und Austausch beim Mehrk¨ aorperproblem.¨ Zs. Phys. 1927, 46, P. 47. Zur Gruppentheorie der hom´’oopolaren chemischen Bindung, 1928, 835 p.Inglis D.R. The Heisenberg Theory of Ferromagnetism. Phys. Rev., 1932, 42, P. 442.Inglis D.R. The Heisenberg Theory of Ferromagnetism. Phys. Rev., 1932, 42, P. 442.Bitter F. On the Interpretation of Some Ferromagnetic Phenomena, Phys. Rev, 1932, 39, P. 337–345.Bitter F. On the Interpretation of Some Ferromagnetic Phenomena, Phys. Rev, 1932, 39, P. 337–345.Tyler F. The magnetization-temperature curves of iron, cobalt and nickel. Phil. Mag., 1931, 11, P. 596.Tyler F. The magnetization-temperature curves of iron, cobalt and nickel. Phil. Mag., 1931, 11, P. 596.Coey J.M.D. Magnetism and magnetic materials. Cambridge University Press, Cambridge, 2009.Coey J.M.D. Magnetism and magnetic materials. Cambridge University Press, Cambridge, 2009.Aharoni A. Introduction to the Theory of Ferromagnetism, Clarendon Press, Oxford, 1996, 192 p.Aharoni A. Introduction to the Theory of Ferromagnetism, Clarendon Press, Oxford, 1996, 192 p.Nakano F. The Heisenberg Theory of Ferromagnetism of the Crystal constituted by the Atoms with Spin One. Progress of Theoretical Physics, 1953, 9(4), P. 403–413.Nakano F. The Heisenberg Theory of Ferromagnetism of the Crystal constituted by the Atoms with Spin One. Progress of Theoretical Physics, 1953, 9(4), P. 403–413.Heisenberg W. Zur Theorie des Magnetpstriktin und der Magnetisierungkurve. Zs. Phys., 1931, 69, P. 287–297.Heisenberg W. Zur Theorie des Magnetpstriktin und der Magnetisierungkurve. Zs. Phys., 1931, 69, P. 287–297.Singh N.The story of magnetism: from Heisenberg, Slater, and Stoner to Van Vleck, and the issues of exchange and correlation. arXiv:1807.11291 [cond-mat.str-el], 2018.Singh N.The story of magnetism: from Heisenberg, Slater, and Stoner to Van Vleck, and the issues of exchange and correlation. arXiv:1807.11291 [cond-mat.str-el], 2018.Borovik E.S., Eremenko V.V., Milner A.S. Lectures on magnetism. 3-rd ed. Revised. and add. M.: Fizmatlit, 2005. (In Russian)Borovik E.S., Eremenko V.V., Milner A.S. Lectures on magnetism. 3-rd ed. Revised. and add. M.: Fizmatlit, 2005. (In Russian)Lakshmanan M. The fascinating world of the Landau–Lifshitz–Gilbert equation: an overview. Phil. Trans. R. Soc. A, 2011, 369, P. 1280–1300.Lakshmanan M. The fascinating world of the Landau–Lifshitz–Gilbert equation: an overview. Phil. Trans. R. Soc. A, 2011, 369, P. 1280–1300.Schollwock U., Richter J., Farnell D.J.J., Bishop R.F.¨ Quantum Magnetism, Lecture Notes in Physics, 645, Springer-Verlag, Berlin, 2004, 381 p.Schollwock U., Richter J., Farnell D.J.J., Bishop R.F.¨ Quantum Magnetism, Lecture Notes in Physics, 645, Springer-Verlag, Berlin, 2004, 381 p.Martinez-Garcia J.C., Rivas M., Lago-Cachon D., Garcia J.A. First-order reversal curves analysis in nanocrystalline ribbons.´ J. Phys. D: Appl. Phys., 2014, 47, P. 015001.Martinez-Garcia J.C., Rivas M., Lago-Cachon D., Garcia J.A. First-order reversal curves analysis in nanocrystalline ribbons.´ J. Phys. D: Appl. Phys., 2014, 47, P. 015001.Martinez-Garcia J.C., Rivas M., Garcia J.A. Induced ferro-ferromagnetic exchange bias in nanocrystalline systems. Journal of Magnetism and Magnetic Materials, 2015, 377, P. 424.Martinez-Garcia J.C., Rivas M., Garcia J.A. Induced ferro-ferromagnetic exchange bias in nanocrystalline systems. Journal of Magnetism and Magnetic Materials, 2015, 377, P. 424.Klaus D. Sattler-Ed. Handbook of Nanophysics: Nanoparticles and Quantum Dots, CRC press, 2011.Klaus D. Sattler-Ed. Handbook of Nanophysics: Nanoparticles and Quantum Dots, CRC press, 2011.Gaikwad A.S., More S.S., et al. Int. Res. J. of Science and Engineering, 2018, A5, P. 41–44.Gaikwad A.S., More S.S., et al. Int. Res. J. of Science and Engineering, 2018, A5, P. 41–44.Valasek J. Piezo-electric and allied phenomena in Rochelle salt. Phys. Rev., 1921, 17, P. 475–481.Valasek J. Piezo-electric and allied phenomena in Rochelle salt. Phys. Rev., 1921, 17, P. 475–481.Spaldin, Nicola A., Fiebig, Manfred. The renaissance of magnetoelectric multiferroics. Science, 2005, 309(5733), P. 391–392.Spaldin, Nicola A., Fiebig, Manfred. The renaissance of magnetoelectric multiferroics. Science, 2005, 309(5733), P. 391–392.Birola T., Benedek N.A., et al. The magnetoelectric effect in transition metal oxides: Insights and the rational design of new materials from first principles Current Opinion. Solid State and Materials Science, 2012, 16(5), P. 227–242.Birola T., Benedek N.A., et al. The magnetoelectric effect in transition metal oxides: Insights and the rational design of new materials from first principles Current Opinion. Solid State and Materials Science, 2012, 16(5), P. 227–242.Reichl L.E. A Modern Course in Statistical Physics, 2-nd Edition. A Wiley-interscience publication, John Wiley and Sons, inc., New York, Chichester, Weinheim, Brisbane, Singapore, Toronto, 1998, 822 p.Reichl L.E. A Modern Course in Statistical Physics, 2-nd Edition. A Wiley-interscience publication, John Wiley and Sons, inc., New York, Chichester, Weinheim, Brisbane, Singapore, Toronto, 1998, 822 p.Heisenberg W. Zur Quantentheorie des Ferromagnetismus. Probleme der Modernen Physik, A. Sommerfeld Festschrift, Leipzig, 1928, P. 114– 122.Heisenberg W. Zur Quantentheorie des Ferromagnetismus. Probleme der Modernen Physik, A. Sommerfeld Festschrift, Leipzig, 1928, P. 114– 122.Jossang P., J¨ ossang A.C., Pouillet S.M. member of the French Academy of Sciences, discovered the “Curie” point in ... 1832.¨ Science Tribune, January 1997, URL: http://www.tribunes.com/tribune/art97/jos2e.htm.Jossang P., J¨ ossang A.C., Pouillet S.M. member of the French Academy of Sciences, discovered the “Curie” point in ... 1832.¨ Science Tribune, January 1997, URL: http://www.tribunes.com/tribune/art97/jos2e.htm.Popov I.Yu. Hydrodynamic stability and perturbation of the Schrodinger operator. Lett. Math. Phys., 1995, 35(2), P. 155–161.Popov I.Yu. Hydrodynamic stability and perturbation of the Schrodinger operator. Lett. Math. Phys., 1995, 35(2), P. 155–161.Fock V. An Approximate Method for Solving the Quantum Many-Body Problem. Zs. Phys., 1930, 61, P. 126–148; TOI 1931, 5, 51, 1; UFN 1967, 93, 2, 342.Fock V. An Approximate Method for Solving the Quantum Many-Body Problem. Zs. Phys., 1930, 61, P. 126–148; TOI 1931, 5, 51, 1; UFN 1967, 93, 2, 342.Dzyaloshinskii I.E. On the magneto-electrical effect in antiferromagnets. J. Exp. Theor. Phys., 1959, 37, P. 881–882.Dzyaloshinskii I.E. On the magneto-electrical effect in antiferromagnets. J. Exp. Theor. Phys., 1959, 37, P. 881–882.Nasirpouri F., Barzegar S., et al. Mesophase micelle-assisted electrodeposition and magnetisation behavior of meso-porous nickel films for efficient electrochemical energy and magnetic device applications. Applied Surface Science, 2019, 471, P. 776–785.Nasirpouri F., Barzegar S., et al. Mesophase micelle-assisted electrodeposition and magnetisation behavior of meso-porous nickel films for efficient electrochemical energy and magnetic device applications. Applied Surface Science, 2019, 471, P. 776–785.

The authors declare that there are no conflicts of interest present.