Studying Magnetization Distribution in Magnetic Thin Films under Transversal Application of Magnetic Fields

Author(s)
Aleksey Anatolievich Zakharenko

ABSTRACT

The problem of magnetization change across the direction of magnetic field for a magnetic layer with non-symmetric boundary conditions was treated. The exact solution of the problem for the magnetization components mx and my was written in the form of complex combination of Jacobian elliptic functions and elliptic integrals. This allows one to demonstrate both the static mode and all dynamic modes for the mag-netization distribution across the layer thickness. The static mode and several dynamic modes, as well as the first and second derivatives of the magnetization components, were calculated. Also, average values of the magnetization components ?mx? and ámy? for the static mode and three dynamic modes were calculated in dependence on the magnetic field. The obtained results can represent an interest in the large amount of ap-plications of magnetic devices such as recording media, memory chips, and computer disks. The results are also useful for checking different numerical methods recently applied to study the problem, because it is thought that any numerical method cannot demonstrate solutions for the dynamic modes.

The problem of magnetization change across the direction of magnetic field for a magnetic layer with non-symmetric boundary conditions was treated. The exact solution of the problem for the magnetization components mx and my was written in the form of complex combination of Jacobian elliptic functions and elliptic integrals. This allows one to demonstrate both the static mode and all dynamic modes for the mag-netization distribution across the layer thickness. The static mode and several dynamic modes, as well as the first and second derivatives of the magnetization components, were calculated. Also, average values of the magnetization components ?mx? and ámy? for the static mode and three dynamic modes were calculated in dependence on the magnetic field. The obtained results can represent an interest in the large amount of ap-plications of magnetic devices such as recording media, memory chips, and computer disks. The results are also useful for checking different numerical methods recently applied to study the problem, because it is thought that any numerical method cannot demonstrate solutions for the dynamic modes.

Cite this paper

nullA. Zakharenko, "Studying Magnetization Distribution in Magnetic Thin Films under Transversal Application of Magnetic Fields,"*Journal of Modern Physics*, Vol. 1 No. 1, 2010, pp. 33-43. doi: 10.4236/jmp.2010.11004.

nullA. Zakharenko, "Studying Magnetization Distribution in Magnetic Thin Films under Transversal Application of Magnetic Fields,"

References

[1] L. Landau and E. Lifshits, “On the Theory of the Disper-sion of Magnetic Permeability in Ferromagnetic Bodies,” Physikalische Zeitschrift der Sowjetunion, Vol. 8, 1935, pp. 153-169.

[2] L. Landau and E. Lifshits, “On the Theory of the Disper-sion of Magnetic Permeability in Ferromagnetic Bodies,” Ukrainian Journal of Physics, Special Issue, Vol. 53, 2008, pp. 14-22.

[3] J. Li, “A Two-Dimensional Landau-Lifshitz Model in Studying Thin Film Micromagnetics,” Abstract and Ap-plied Analysis, Vol. 13, 2009.

[4] J.-N. Li, X.-F. Wang and Z.-A. Yao, “An Extension Landau-Lifshitz Model in Studying Soft Ferromagnetic Films,” Acta Mathematicae Applicatae Sinica, English Series, Vol. 23, No. 3, 2007, pp. 421-432.

[5] L. G. Korzunin, B. N. Filippov and F. A. Kassan-Ogly, “Dynamics of Vortex-like Domain Walls in Triaxial Magnetic Films with the Goss Orientation of the Sur-face,” Technical Physics, Vol. 52, No. 11, 2007, pp. 1453-1457.

[6] B. N. Filippov and L. G. Korzunin, “Non-linear Dynam-ics of Domain Walls with Vortex-like Inner Structure in Magnetically-single-axis Films with Plane Anisotropy,” Journal of Experimental and Technical Physics, Moscow, Vol. 121, No. 2, 2002, pp. 372-387.

[7] P. L. Sulem, C. Sulem and C. Bardos, “On the Continu-ous Limit for a System of Classical Spins,” Communica-tions in Mathematical Physics, Vol. 107, No. 3, 1986, pp. 431-454.

[8] W. E and X.-P. Wang, “Numerical Methods for the Lan-dau-Lifshitz Equation” SIAM Journal on Numerical Analysis, Vol. 38, No. 5, 2000, pp. 1647-1665.

[9] A. DeSimone, R. V. Kohn, S. Müller and F. Otto, “A Reduced Theory for Thin-Film Micromagnetics,” Com-munications on Pure and Applied Mathematics, Vol. 55, No. 11, 2002, pp. 1408-1460.

[10] A. DeSimone, R. V. Kohn, S. Müller, F. Otto and R. Schäfer, “Two-Dimensional Modelling of Soft Ferro-magnetic Films,” Proceedings of the Royal Society of London, Series A, Vol. 457, No. 2016, 2001, pp. 2983- 2991.

[11] A. Aharoni, E. H. Frei and S. Shtrikman, “Theoretical Approach to the Asymmetrical Magnetization Curve,” Journal of Applied Physics, Vol. 30, No. 12, 1959, pp. 1956-1961.

[12] N. M. Salansky and M. S. Eruchimov, “The Peculiarities of Spin-Wave Resonance in Films with Ferro-Antiferro-magnetic Interaction,” Thin Solid Films, Vol. 6, No. 2, 1970, pp. 129-140.

[13] Y. V. Zakharov and E. A. Khlebopros, “Magnetization Curves and Frequencies of Magnetic Resonance in Films with Domain Structures on Anti-Ferromagnetic Substrate,” Solid State Physics, Moscow, Vol. 22, No. 12, 1980, pp. 3651-3657.

[14] H. Bateman and A. Erd´elyi, “Higher Transcendental Functions,” McGraw-Hill, New York, Vol. 3, 1955.

[15] M. Abramowitz and I. A. Stegun, Eds., “Handbook of Mathematical Functions with Formulas,” Graphs and Mathematical Tables, National Bureau of Standards in Applied Mathematics, Washington, 1964, p. 1058.

[16] L. Collatz, “Eigenvalue Problems with Engineering Ap-plications,” Fizmatgiz, Moscow, 1968.

[17] Y. V. Zakharov, “Static and Dynamic Instability of a Ferromagnetic Layer under Magnetic Reversal,” Doklady of Russian Academy of Science, Vol. 344, No. 3, 1995, pp. 328-332.

[18] M. A. Lavrent’ev and A. Y. Ishlinskii, “Dynamic Buck-ling Modes of Elastic Systems,” Doklady Akademii Nauk USSR (Moscow), Vol. 64, No. 6, 1949, pp. 779-782.

[19] Y. V. Zakharov and A. A. Zakharenko, “Dynamic Insta-bility in the Nonlinear Problem of a Cantilever,” Compu-tational Technologies, Vol. 4, No. 1, 1999, pp. 48-54.

[20] M. Johnson, “Bipolar Spin Switch,” Science, Vol. 260 No. 5106, 1993, pp. 320-323.

[21] D. J. Monsma, J. C. Lodder, T. J. A. Popma and B. Dieny, “Perpendicular Hot Electron Spin-Valve Effect in a New Magnetic Field Sensor: The Spin-Valve Transistor,” Phy- sical Review Letters, Vol. 74, No. 26, 1995, pp. 5260- 5263.

[22] M. Johnson, “Spin Injector in Metal Films: The Bipolar Spin Transistor,” Materials Science and Engineering B, Vol. 31, 1995, pp. 199-205.

[23] V. Y. Shur and E. L. Rumyantsev, “Kinetics of Ferro-electric Domain Structure during Switching: Theory and Experiment,” Ferroelectrics, Vol. 151, 1994, pp. 171-189.

[1] L. Landau and E. Lifshits, “On the Theory of the Disper-sion of Magnetic Permeability in Ferromagnetic Bodies,” Physikalische Zeitschrift der Sowjetunion, Vol. 8, 1935, pp. 153-169.

[2] L. Landau and E. Lifshits, “On the Theory of the Disper-sion of Magnetic Permeability in Ferromagnetic Bodies,” Ukrainian Journal of Physics, Special Issue, Vol. 53, 2008, pp. 14-22.

[3] J. Li, “A Two-Dimensional Landau-Lifshitz Model in Studying Thin Film Micromagnetics,” Abstract and Ap-plied Analysis, Vol. 13, 2009.

[4] J.-N. Li, X.-F. Wang and Z.-A. Yao, “An Extension Landau-Lifshitz Model in Studying Soft Ferromagnetic Films,” Acta Mathematicae Applicatae Sinica, English Series, Vol. 23, No. 3, 2007, pp. 421-432.

[5] L. G. Korzunin, B. N. Filippov and F. A. Kassan-Ogly, “Dynamics of Vortex-like Domain Walls in Triaxial Magnetic Films with the Goss Orientation of the Sur-face,” Technical Physics, Vol. 52, No. 11, 2007, pp. 1453-1457.

[6] B. N. Filippov and L. G. Korzunin, “Non-linear Dynam-ics of Domain Walls with Vortex-like Inner Structure in Magnetically-single-axis Films with Plane Anisotropy,” Journal of Experimental and Technical Physics, Moscow, Vol. 121, No. 2, 2002, pp. 372-387.

[7] P. L. Sulem, C. Sulem and C. Bardos, “On the Continu-ous Limit for a System of Classical Spins,” Communica-tions in Mathematical Physics, Vol. 107, No. 3, 1986, pp. 431-454.

[8] W. E and X.-P. Wang, “Numerical Methods for the Lan-dau-Lifshitz Equation” SIAM Journal on Numerical Analysis, Vol. 38, No. 5, 2000, pp. 1647-1665.

[9] A. DeSimone, R. V. Kohn, S. Müller and F. Otto, “A Reduced Theory for Thin-Film Micromagnetics,” Com-munications on Pure and Applied Mathematics, Vol. 55, No. 11, 2002, pp. 1408-1460.

[10] A. DeSimone, R. V. Kohn, S. Müller, F. Otto and R. Schäfer, “Two-Dimensional Modelling of Soft Ferro-magnetic Films,” Proceedings of the Royal Society of London, Series A, Vol. 457, No. 2016, 2001, pp. 2983- 2991.

[11] A. Aharoni, E. H. Frei and S. Shtrikman, “Theoretical Approach to the Asymmetrical Magnetization Curve,” Journal of Applied Physics, Vol. 30, No. 12, 1959, pp. 1956-1961.

[12] N. M. Salansky and M. S. Eruchimov, “The Peculiarities of Spin-Wave Resonance in Films with Ferro-Antiferro-magnetic Interaction,” Thin Solid Films, Vol. 6, No. 2, 1970, pp. 129-140.

[13] Y. V. Zakharov and E. A. Khlebopros, “Magnetization Curves and Frequencies of Magnetic Resonance in Films with Domain Structures on Anti-Ferromagnetic Substrate,” Solid State Physics, Moscow, Vol. 22, No. 12, 1980, pp. 3651-3657.

[14] H. Bateman and A. Erd´elyi, “Higher Transcendental Functions,” McGraw-Hill, New York, Vol. 3, 1955.

[15] M. Abramowitz and I. A. Stegun, Eds., “Handbook of Mathematical Functions with Formulas,” Graphs and Mathematical Tables, National Bureau of Standards in Applied Mathematics, Washington, 1964, p. 1058.

[16] L. Collatz, “Eigenvalue Problems with Engineering Ap-plications,” Fizmatgiz, Moscow, 1968.

[17] Y. V. Zakharov, “Static and Dynamic Instability of a Ferromagnetic Layer under Magnetic Reversal,” Doklady of Russian Academy of Science, Vol. 344, No. 3, 1995, pp. 328-332.

[18] M. A. Lavrent’ev and A. Y. Ishlinskii, “Dynamic Buck-ling Modes of Elastic Systems,” Doklady Akademii Nauk USSR (Moscow), Vol. 64, No. 6, 1949, pp. 779-782.

[19] Y. V. Zakharov and A. A. Zakharenko, “Dynamic Insta-bility in the Nonlinear Problem of a Cantilever,” Compu-tational Technologies, Vol. 4, No. 1, 1999, pp. 48-54.

[20] M. Johnson, “Bipolar Spin Switch,” Science, Vol. 260 No. 5106, 1993, pp. 320-323.

[21] D. J. Monsma, J. C. Lodder, T. J. A. Popma and B. Dieny, “Perpendicular Hot Electron Spin-Valve Effect in a New Magnetic Field Sensor: The Spin-Valve Transistor,” Phy- sical Review Letters, Vol. 74, No. 26, 1995, pp. 5260- 5263.

[22] M. Johnson, “Spin Injector in Metal Films: The Bipolar Spin Transistor,” Materials Science and Engineering B, Vol. 31, 1995, pp. 199-205.

[23] V. Y. Shur and E. L. Rumyantsev, “Kinetics of Ferro-electric Domain Structure during Switching: Theory and Experiment,” Ferroelectrics, Vol. 151, 1994, pp. 171-189.