najoNanosystems: Physics, Chemistry, MathematicsНаносистемы: физика, химия, математика2220-80542305-7971Университет ИТМО10.17586/22208054201785572578najo-602Research ArticlePHYSICSФИЗИКАThermal stability of magnetic states in submicron magnetic islandsLiashkoS. Y.

Kronverkskiy, 49, St. Petersburg, 197101

107 Reykjav´ık

sergei.liashko@gmail.comLobanovI. S.

49, St. Petersburg, 197101

lobanov.igor@gmail.comUzdinV. M.

Kronverkskiy, 49, St. Petersburg, 197101

St. Petersburg, 198504

v_uzdin@mail.ruJ´onssonH.

107 Reykjav´ık

Los Alamos, NM 87545

hj@hi.isITMO University; ScienceScience Institute and Faculty of Physical Sciences, University of IcelandRussian FederationITMO University, KronverkskiyRussian FederationITMO University; St. Petersburg State UniversityRussian FederationScience Institute and Faculty of Physical Sciences, University of Iceland; Center for Nonlinear StudiesRussian Federation20171208202585572–578Copyright © Liashko S.Y., Lobanov I.S., Uzdin V.M., J´onsson H., 20252025Liashko S.Y., Lobanov I.S., Uzdin V.M., J´onsson H.Liashko S.Y., Lobanov I.S., Uzdin V.M., J´onsson H.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/602

The lifetime of magnetic states in single domain micromagnetic islands is calculated within the harmonic approximation to transition state theory. Stable magnetic states, minimum energy paths between them and first order saddle points determining the activation energy are analyzed and visualized on twodimensional energy surfaces. An analytical expression is derived for the preexponential factor in the Arrhenius rate expression for the reversal of the magnetic moment when the external field is directed either along the anisotropy axis or perpendicular to it.

preexponential factormagnetic islandsactivation energyrate theoryspin iceThis work was supported by the Icelandic Research Fund, the Academy of Finland (grant 278260) and the Government of the Russian Federation (grant 074U01) and by grant 161110330 of Russian Science Foundation.ReferencesCoffey W.T., Garanin D.A. and McCarthy D.J. Crossover formulas in the Kramers theory of thermally activated escape rates Application to spin systems.Advances in Chemical Physics, 2001, 117, P. 483.Coffey W.T., Garanin D.A. and McCarthy D.J. Crossover formulas in the Kramers theory of thermally activated escape rates Application to spin systems.Advances in Chemical Physics, 2001, 117, P. 483.Braun H.B. Topological effects in nanomagnetism: from superparamagnetism to chiral quantum solitons. Advances in Physics, 2012, 61, P. 1.Braun H.B. Topological effects in nanomagnetism: from superparamagnetism to chiral quantum solitons. Advances in Physics, 2012, 61, P. 1.Wigner E. The transition state method. Trans. Faraday Soc., 1938, 34, P. 29–41.Wigner E. The transition state method. Trans. Faraday Soc., 1938, 34, P. 29–41.Kramers H.A. Brownian motion in a field of force and the diffusion model of chemical reactions. Physica, 1940, 7, P. 284–304.Kramers H.A. Brownian motion in a field of force and the diffusion model of chemical reactions. Physica, 1940, 7, P. 284–304.Vineyard G.H. Frequency factors and isotope effects in solid state rate processes. J. Phys. Chem. Solids, 1957, 3, P. 121.Vineyard G.H. Frequency factors and isotope effects in solid state rate processes. J. Phys. Chem. Solids, 1957, 3, P. 121.Brown Jr.W.F. Thermal fluctuations of a singledomain particle. Phys. Rev., 1963, 130, P. 1677.Brown Jr.W.F. Thermal fluctuations of a singledomain particle. Phys. Rev., 1963, 130, P. 1677.Brown Jr.W.F. Thermal Fluctuations of Fine Ferromagnetic Particles. IEEE Trans. Magn., 1979, MAG15, P. 1196–1208.Brown Jr.W.F. Thermal Fluctuations of Fine Ferromagnetic Particles. IEEE Trans. Magn., 1979, MAG15, P. 1196–1208.Braun H.B. Kramers’s rate theory, broken symmetries and magnetization reversal. J. Appl. Physics, 1994, 76, P. 6310–6315.Braun H.B. Kramers’s rate theory, broken symmetries and magnetization reversal. J. Appl. Physics, 1994, 76, P. 6310–6315.Visscher P.B., Zhu R. Lowdimensionality energy landscapes: Magnetic switching mechanisms and rates. Physica B, 2012, 407, P. 1340– 1344.Visscher P.B., Zhu R. Lowdimensionality energy landscapes: Magnetic switching mechanisms and rates. Physica B, 2012, 407, P. 1340– 1344.Fiedler G., Fidler J., Lee J., Schrefl T., Stamps R.L., Braun H.B. and Suess D., Direct calculation of the attempt frequency of magnetic structures using the finite element method. J. Appl. Phys., 2012, 111, P. 093917(7).Fiedler G., Fidler J., Lee J., Schrefl T., Stamps R.L., Braun H.B. and Suess D., Direct calculation of the attempt frequency of magnetic structures using the finite element method. J. Appl. Phys., 2012, 111, P. 093917(7).Bessarab P.F., Uzdin V.M. and J´onsson H. Harmonic Transition State Theory of Thermal Spin Transitions. Phys. Rev. B, 2012, 85, P. 184409(4).Bessarab P.F., Uzdin V.M. and J´onsson H. Harmonic Transition State Theory of Thermal Spin Transitions. Phys. Rev. B, 2012, 85, P. 184409(4).Bessarab P.F., Uzdin V.M. and J´onsson H. Potential Energy Surfaces and Rates of Spin Transitions. Z. Phys. Chem., 2013, 227, P. 1543– 1557.Bessarab P.F., Uzdin V.M. and J´onsson H. Potential Energy Surfaces and Rates of Spin Transitions. Z. Phys. Chem., 2013, 227, P. 1543– 1557.Bessarab P.F., Uzdin V.M. and J´onsson H. Calculations of magnetic states and minimum energy paths of transitions using a noncollinear extension of the AlexanderAnderson model and a magnetic force theorem. Phys. Rev. B, 2014, 89, P. 214424(12).Bessarab P.F., Uzdin V.M. and J´onsson H. Calculations of magnetic states and minimum energy paths of transitions using a noncollinear extension of the AlexanderAnderson model and a magnetic force theorem. Phys. Rev. B, 2014, 89, P. 214424(12).W. Wernsdorfer, E. Bonet Orozco, K. Hasselbach, A. Benoit, B. Barbara, N. Demoncy, A. Loiseau, H. Pascard and D. Mailly. Experimental evidence of the N´eelBrown model of magnetization reversal. Phys. Rev. Lett., 1997, 78, P. 1791.W. Wernsdorfer, E. Bonet Orozco, K. Hasselbach, A. Benoit, B. Barbara, N. Demoncy, A. Loiseau, H. Pascard and D. Mailly. Experimental evidence of the N´eelBrown model of magnetization reversal. Phys. Rev. Lett., 1997, 78, P. 1791.Bessarab P.F., Uzdin V.M. and J´onsson H. Size and Shape Dependence of Thermal Spin Transitions in Nanoislands. Phys. Rev. Lett., 2013, 110, P. 020604(5).Bessarab P.F., Uzdin V.M. and J´onsson H. Size and Shape Dependence of Thermal Spin Transitions in Nanoislands. Phys. Rev. Lett., 2013, 110, P. 020604(5).Krause S., Herzog G., Stapelfeldt T., BerbilBautista L., Bode M., Vedmedenko E. Y., Wiesendanger R. Magnetization reversal of nanoscale islands: How size and shape affect the Arrhenius Pprefactor. Phys.Rev. Lett., 2009, 103, P. 127202(4).Krause S., Herzog G., Stapelfeldt T., BerbilBautista L., Bode M., Vedmedenko E. Y., Wiesendanger R. Magnetization reversal of nanoscale islands: How size and shape affect the Arrhenius Pprefactor. Phys.Rev. Lett., 2009, 103, P. 127202(4).Farhan A., Derlet P.M., Kleibert A., Balan A., Chopdekar R.V., Wyss M., Anghinolfi L., Nolting F., Heyderman L.J. Exploring hypercubic energy landscapes in thermally active finite artificial spinice systems. Nature Physics, 2013, 9, P. 375(8).Farhan A., Derlet P.M., Kleibert A., Balan A., Chopdekar R.V., Wyss M., Anghinolfi L., Nolting F., Heyderman L.J. Exploring hypercubic energy landscapes in thermally active finite artificial spinice systems. Nature Physics, 2013, 9, P. 375(8).Liashko S.Y., Uzdin V.M. and J´onsson H. The effect of temperature and external field on transitions in elements of kagome spin ice. New J. Phys., 2017, (accepted) DOI: 10.1088/13672630/aa8b96.Liashko S.Y., Uzdin V.M. and J´onsson H. The effect of temperature and external field on transitions in elements of kagome spin ice. New J. Phys., 2017, (accepted) DOI: 10.1088/13672630/aa8b96.Ivanov A., Bessarab P.F., Uzdin V.M. and J´onsson H. Magnetic exchange force microscopy: Theoretical analysis of induced magnetization reversals. Nanoscale, 2017, 9, P. 13320–13325.Ivanov A., Bessarab P.F., Uzdin V.M. and J´onsson H. Magnetic exchange force microscopy: Theoretical analysis of induced magnetization reversals. Nanoscale, 2017, 9, P. 13320–13325.Schmidt R., Schwarz A., Wiesendanger R. Magnetization switching utilizing the magnetic exchange interaction. Phys. Rev. B, 2012, 86, P. 174402(6).Schmidt R., Schwarz A., Wiesendanger R. Magnetization switching utilizing the magnetic exchange interaction. Phys. Rev. B, 2012, 86, P. 174402(6).Lobanov I.S., J´onsson H. and Uzdin V. M. Mechanism and activation energy of magnetic skyrmion annihilation obtained from minimum energy path calculations. Phys. Rev. B , 2016, 94, P. 174418(7).Lobanov I.S., J´onsson H. and Uzdin V. M. Mechanism and activation energy of magnetic skyrmion annihilation obtained from minimum energy path calculations. Phys. Rev. B , 2016, 94, P. 174418(7).Uzdin V.M., Potkina M.N., Lobanov I.S., Bessarab P.F., J´onsson H. The effect of confinement and defects on the thermal stability of skyrmions. Physica B, 2017, (in press), DOI; 10.1016/j.physb.2017.09.040.Uzdin V.M., Potkina M.N., Lobanov I.S., Bessarab P.F., J´onsson H. The effect of confinement and defects on the thermal stability of skyrmions. Physica B, 2017, (in press), DOI; 10.1016/j.physb.2017.09.040.Uzdin V.M., Potkina M.N., Lobanov I.S., Bessarab P.F., J´onsson H. Energy surface and lifetime of magnetic skyrmions. J. Magn. Magn. Mat., (accepted). Manuscript available at https://arxiv.org/pdf/1707.00124.pdf.Uzdin V.M., Potkina M.N., Lobanov I.S., Bessarab P.F., J´onsson H. Energy surface and lifetime of magnetic skyrmions. J. Magn. Magn. Mat., (accepted). Manuscript available at https://arxiv.org/pdf/1707.00124.pdf.Bessarab P.F., Uzdin V.M. and J´onsson H. Method for finding mechanism and activation energy of magnetic transitions, applied to skyrmion and antivortex annihilation. Comp. Phys. Commun., 2015, 196, P. 335–347.Bessarab P.F., Uzdin V.M. and J´onsson H. Method for finding mechanism and activation energy of magnetic transitions, applied to skyrmion and antivortex annihilation. Comp. Phys. Commun., 2015, 196, P. 335–347.Bessarab P. F. Comment on “Path to collapse for an isolated N´eel skyrmion”. Phys. Rev. B , 2017, 95, P. 136401(2).Bessarab P. F. Comment on “Path to collapse for an isolated N´eel skyrmion”. Phys. Rev. B , 2017, 95, P. 136401(2).Moskalenko M. , Bessarab P.F., Uzdin V.M. and J´onsson H. Qualitative Insight and Quantitative Analysis of the Effect of Temperature on the Coercivity of a Magnetic System. AIP Advances, 2016, 6, P. 025213(8).Moskalenko M. , Bessarab P.F., Uzdin V.M. and J´onsson H. Qualitative Insight and Quantitative Analysis of the Effect of Temperature on the Coercivity of a Magnetic System. AIP Advances, 2016, 6, P. 025213(8).Stoner E.C., Wohlfarth E.P. A mechanism of magnetic hysteresis in heterogeneous alloys. Phil. Trans. Roy. Soc. A, 1948, 240, P. 599–642.Stoner E.C., Wohlfarth E.P. A mechanism of magnetic hysteresis in heterogeneous alloys. Phil. Trans. Roy. Soc. A, 1948, 240, P. 599–642.Coffey W.T, Crothers D.S.F., Dormann J.L., Kalmykov Y.P., Kennedy E.C., Thermally activated relaxation time of a single domain ferromagnetic particle subjected to a uniform field at an oblique angle to the easy axis: Comparison with experimental observations. Phys. Rev. Lett., 1998, 80, P. 5655–5568.Coffey W.T, Crothers D.S.F., Dormann J.L., Kalmykov Y.P., Kennedy E.C., Thermally activated relaxation time of a single domain ferromagnetic particle subjected to a uniform field at an oblique angle to the easy axis: Comparison with experimental observations. Phys. Rev. Lett., 1998, 80, P. 5655–5568.Poperechny I.S., Raikher Yu.L., Stepanov V.I. Dynamic magnetic hysteresis in singledomain particles with niaxial anisotropy. Phys.Rev. B, 2010, 82, P. 174423(14).Poperechny I.S., Raikher Yu.L., Stepanov V.I. Dynamic magnetic hysteresis in singledomain particles with niaxial anisotropy. Phys.Rev. B, 2010, 82, P. 174423(14).Franco V., Conde A. Thermal effects in a StonerWohlfarth model and their influence on magnetic anisotropy determination. J. Magn. Magn. Mat., 2004, 278, P. 28–38.Franco V., Conde A. Thermal effects in a StonerWohlfarth model and their influence on magnetic anisotropy determination. J. Magn. Magn. Mat., 2004, 278, P. 28–38.De Vries J., Bolhuis T., Abelmann L. Temperature dependence of the energy barrier and switching field of submicron magnetic islands with perpendicular anisotropy. New J. Phys., 2017, (accepted) DOI:10.1088/13672630/aa8082.De Vries J., Bolhuis T., Abelmann L. Temperature dependence of the energy barrier and switching field of submicron magnetic islands with perpendicular anisotropy. New J. Phys., 2017, (accepted) DOI:10.1088/13672630/aa8082.Engelen J.B.C., Delalande M., le Febre A.J., Bolhuis T., Shimatsu T., Kikuchi N., Abelmann L., Lodder J.C. Thermally induced switching field distribution of a single CoPt dot in a large array. Nanotechnology, 2010, 21, P. 035703(7).Engelen J.B.C., Delalande M., le Febre A.J., Bolhuis T., Shimatsu T., Kikuchi N., Abelmann L., Lodder J.C. Thermally induced switching field distribution of a single CoPt dot in a large array. Nanotechnology, 2010, 21, P. 035703(7).J´onsson H., Mills G., Jacobsen K.W. Classical and Quantum Dynamics in Condensed Phase Simulations, edited by Berne B.J., Ciccotti G., Coker D.F. (World Scientific, Singapore, 1998), P. 385–404.J´onsson H., Mills G., Jacobsen K.W. Classical and Quantum Dynamics in Condensed Phase Simulations, edited by Berne B.J., Ciccotti G., Coker D.F. (World Scientific, Singapore, 1998), P. 385–404.Henkelman G., Uberuaga B.P., J´onsson H. A climbing image nudged elastic band method for finding saddle points and minimum energy paths. J. Chem. Phys., 2000, 113, P. 9901–9904.Henkelman G., Uberuaga B.P., J´onsson H. A climbing image nudged elastic band method for finding saddle points and minimum energy paths. J. Chem. Phys., 2000, 113, P. 9901–9904.Vlasov S., Bessarab P.F., Uzdin V.M. and J´onsson H. Classical to quantum mechanical tunneling mechanism crossover in thermal transitions between magnetic states. Faraday Discuss., 2016, 195, P. 93–109.Vlasov S., Bessarab P.F., Uzdin V.M. and J´onsson H. Classical to quantum mechanical tunneling mechanism crossover in thermal transitions between magnetic states. Faraday Discuss., 2016, 195, P. 93–109.Vlasov S., Bessarab P.F., Uzdin V.M. and J´onsson H. Calculations of the onset temperature for tunneling in multispin systems. Nanosystems: Physics, Chemistry, Mathematics, 2017, 8, P. 454–461.Vlasov S., Bessarab P.F., Uzdin V.M. and J´onsson H. Calculations of the onset temperature for tunneling in multispin systems. Nanosystems: Physics, Chemistry, Mathematics, 2017, 8, P. 454–461.

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