najoNanosystems: Physics, Chemistry, MathematicsНаносистемы: физика, химия, математика2220-80542305-7971Университет ИТМО10.17586/2220-8054-2023-14-2-216-222najo-143Research ArticlePHYSICSФИЗИКАNucleation and collapse of magnetic topological solitons in external magnetic fieldНуклеация и коллапс магнитных топологических солитонов во внешнем магнитном полеhttps://orcid.org/0000-0002-1380-2454ПоткинаМ. Н.PotkinaM. N.

Maria N. Potkina – Faculty of Physics

St. Petersburg, 197101

potkina.maria@yandex.ruhttps://orcid.org/0000-0001-8789-3267ЛобановИ. С.LobanovI. S.

Igor S. Lobanov,– Faculty of Physics

St. Petersburg, 197101

lobanov.igor@gmail.comhttps://orcid.org/0000-0002-9505-0996УздинВ. М.UzdinV. M.

Valery M. Uzdin– Faculty of Physics

St. Petersburg, 197101

v_uzdin@mail.ruITMO UniversityRussian Federation202303062025142216222Copyright © Potkina M.N., Lobanov I.S., Uzdin V.M., 20252025Поткина М.Н., Лобанов И.С., Уздин В.М.Potkina M.N., Lobanov I.S., Uzdin V.M.This work is licensed under a Creative Commons Attribution 4.0 License.https://nanojournal.ifmo.ru/jour/article/view/143

The dependence of the lifetimes and rates of spontaneous nucleation of topological magnetic soli tons on the external magnetic field is calculated within the framework of the harmonic transition state theory for magnetic degrees of freedom. For two-dimensional magnetic skyrmions, the influence of the magnetic field on the collapse rate was found to be greater than on the nucleation rate. This is explained by the weaker dependence of the energy of the transition state on the external field compared to the energy of the metastable skyrmion. The balance of the nucleation and collapse of skyrmion rates makes it possible to determine the average equilibrium concentration of skyrmions in a thin film as a function of the external field and temperature. It is shown that skyrmion and antiskyrmion states can exist simultaneously in quasi-two-dimensional thin films in tilted external magnetic field. The minimum energy paths for the collapse of these topological solitons and magnetic configurations in the vicinity of saddle point have been found and compared.

В рамках гармонического приближения теории переходного состояния для магнитные степени свободы рассчитана зависимость времен жизни и частот спонтанного зарождения топологических магнитных солитонов от внешнего магнитного поля. Для двумерных магнитных скирмионов влияние магнитного поля на частоту распадов оказалась больше, чем на частоту процессов зарождения. Это объясняется более слабой зависимостью от магнитного поля энергии переходного состояния  по сравнению с энергией метастабильного состояния скирмиона. Баланс частот нуклеации и коллапса скирмионов позволяет определить среднюю равновесную концентрацию скирмионов в тонкой пленке в зависимости от внешнего поля и температуры. Показано, что скирмионное и антискирмионное состояния могут существовать одновременно в квазидвухмерных  тонких пленках в наклонном внешнем магнитном поле. Найдены магнитные конфигурации вблизи седловой точки и проведено сравнение путей с минимальным перепадом энергии  для коллапса этих топологических солитонов.

теория переходного состояниятопологические магнитные солитонынуклеацияколлапсвремя жизниtransition state theorytopological magnetic solitonsnucleationcollapselifetimeThe study was supported by the Russian Science Foundation grant No. 22-72-00059, https://rscf.ru/en/project/22-72-00059/.ReferencesParkin S.S.P., Hayashi M., Thomas L. Magnetic Domain-Wall Racetrack Memory. Science, 2008, 320, 5837, P. 190–194.Parkin S.S.P., Hayashi M., Thomas L. Magnetic Domain-Wall Racetrack Memory. Science, 2008, 320, 5837, P. 190–194.Wiesendanger R. Nanoscale magnetic skyrmions in metallic films and multilayers: A new twist for spintronics. Nat. Rev. Mater., 2016, 1, P. 16044.Wiesendanger R. Nanoscale magnetic skyrmions in metallic films and multilayers: A new twist for spintronics. Nat. Rev. Mater., 2016, 1, P. 16044.Finocchio G., B¨ uttner F., Tomasello R., Carpentieri M., Kl¨au M. Magnetic skyrmions: from fundamental to applications. J Phys. D: Appl. Phys., 2016, 49, P. 423001.Finocchio G., B¨ uttner F., Tomasello R., Carpentieri M., Kl¨au M. Magnetic skyrmions: from fundamental to applications. J Phys. D: Appl. Phys., 2016, 49, P. 423001.Fert A., Reyren N., Cros V. Magnetic skyrmions: Advances in physics and potential applications. Nat. Rev. Mater., 2017, 2, P. 17031.Fert A., Reyren N., Cros V. Magnetic skyrmions: Advances in physics and potential applications. Nat. Rev. Mater., 2017, 2, P. 17031.Bessarab P.F., Uzdin V.M., J´ onsson H. Harmonic transition-state theory of thermal spin transitions. Phys. Rev. B, 2012, 85, P. 184409.Bessarab P.F., Uzdin V.M., J´ onsson H. Harmonic transition-state theory of thermal spin transitions. Phys. Rev. B, 2012, 85, P. 184409.Coffey W.T., Garanin D.A., Mccarthy D.J. Crossover formulas in the Kramers theory of thermally activated escape rates—application to spin systems. Adv. Chem. Phys., 2001, 117, P. 483–765.Coffey W.T., Garanin D.A., Mccarthy D.J. Crossover formulas in the Kramers theory of thermally activated escape rates—application to spin systems. Adv. Chem. Phys., 2001, 117, P. 483–765.Lobanov I.S., Potkina M.N., Uzdin V.M. Stability and lifetimes of magnetic states of nano- and microstructures (Brief review) JETP Lett., 2021 113, P. 833–847.Lobanov I.S., Potkina M.N., Uzdin V.M. Stability and lifetimes of magnetic states of nano- and microstructures (Brief review) JETP Lett., 2021 113, P. 833–847.Bessarab P.F., Uzdin V.M., J´ onsson H. Size and Shape Dependence of Thermal Spin Transitions in Nanoislands. Phys. Rev. Lett., 2013, 110, P. 020604.Bessarab P.F., Uzdin V.M., J´ onsson H. Size and Shape Dependence of Thermal Spin Transitions in Nanoislands. Phys. Rev. Lett., 2013, 110, P. 020604.Suess D., Vogler C., Bruckner F., Heistracher P., Slanovc F., Abert, C. Spin torque efficiency and analytic error rate estimates of skyrmion racetrack memory. Sci. Rep., 2019, 9, P. 4827.Suess D., Vogler C., Bruckner F., Heistracher P., Slanovc F., Abert, C. Spin torque efficiency and analytic error rate estimates of skyrmion racetrack memory. Sci. Rep., 2019, 9, P. 4827.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: Condens. Matter., 2018, 549, 15, P. 6–9.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: Condens. Matter., 2018, 549, 15, P. 6–9.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. Mater., 2018, 459, P. 236–240.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. Mater., 2018, 459, P. 236–240.Potkina M.N., Lobanov I.S., Uzdin V.M. Nonmagnetic impurities in skyrmion racetrack memory Nanosystems: phys., chem., math., 2020, 11 (6), P. 628–635.Potkina M.N., Lobanov I.S., Uzdin V.M. Nonmagnetic impurities in skyrmion racetrack memory Nanosystems: phys., chem., math., 2020, 11 (6), P. 628–635.Lobanov I.S., Uzdin V.M. Lifetime, collapse, and escape paths for hopfions in bulk magnets with competing exchange interactions. Phys. Rev. B, 2023, 107, P. 104405.Lobanov I.S., Uzdin V.M. Lifetime, collapse, and escape paths for hopfions in bulk magnets with competing exchange interactions. Phys. Rev. B, 2023, 107, P. 104405.Potkina M.N., Lobanov I.S., J´onsson H., Uzdin V.M. Skyrmions in antiferromagnets: Thermal stability and the effect of external field and impuri ties. J. Appl. Phys., 2020, 127, P. 213906.Potkina M.N., Lobanov I.S., J´onsson H., Uzdin V.M. Skyrmions in antiferromagnets: Thermal stability and the effect of external field and impuri ties. J. Appl. Phys., 2020, 127, P. 213906.Voronin K.V. Lobanov I.S., Uzdin V.M. Activation Energy and Mechanisms for Skyrmion Collapse in Synthetic Antiferromagnets JETP Lett., 2022 116, P. 242–248.Voronin K.V. Lobanov I.S., Uzdin V.M. Activation Energy and Mechanisms for Skyrmion Collapse in Synthetic Antiferromagnets JETP Lett., 2022 116, P. 242–248.Sampaio J., Cros V., Rohart S., Thiaville A., Fert A., Nucleation, stability and current-induced motion of isolated magnetic skyrmions in nanos tructures. Nat. Nanotech., 2013. 8, P. 839Sampaio J., Cros V., Rohart S., Thiaville A., Fert A., Nucleation, stability and current-induced motion of isolated magnetic skyrmions in nanos tructures. Nat. Nanotech., 2013. 8, P. 839Romming N., Hanneken C., Menzel M., Bickel J.E., Wolter B., von Bergmann K., Kubetzka A., Wiesendanger R. Writing and Deleting Single Magnetic Skyrmions. Science, 2013, 341, 6146, P. 636–639.Romming N., Hanneken C., Menzel M., Bickel J.E., Wolter B., von Bergmann K., Kubetzka A., Wiesendanger R. Writing and Deleting Single Magnetic Skyrmions. Science, 2013, 341, 6146, P. 636–639.Iwasaki J., Mochizuki M., Nagaosa N. Current-induced skyrmion dynamics in constricted geometries. Nat. nanotech., 2013, 8, P. 742–747.Iwasaki J., Mochizuki M., Nagaosa N. Current-induced skyrmion dynamics in constricted geometries. Nat. nanotech., 2013, 8, P. 742–747.Desplat, L., Meyer, S., Bouaziz, J., Buhl, P.M., Lounis, S., Dup´e, B. and Hervieux, P.A. Mechanism for ultrafast electric-field driven skyrmion nucleation. Phys. Rev. B, 2021, 104, P. L060409.Desplat, L., Meyer, S., Bouaziz, J., Buhl, P.M., Lounis, S., Dup´e, B. and Hervieux, P.A. Mechanism for ultrafast electric-field driven skyrmion nucleation. Phys. Rev. B, 2021, 104, P. L060409.Zhang X., Zhou Y., Song K.M., Park T.E., Xia J., Ezawa M., Liu X., Zhao W., Zhao G., Woo S. Skyrmion-electronics: writing, deleting, reading and processing magnetic skyrmions toward spintronic applications. J. Phys.: Cond. Mat., 2020, 14, P. 143001.Zhang X., Zhou Y., Song K.M., Park T.E., Xia J., Ezawa M., Liu X., Zhao W., Zhao G., Woo S. Skyrmion-electronics: writing, deleting, reading and processing magnetic skyrmions toward spintronic applications. J. Phys.: Cond. Mat., 2020, 14, P. 143001.Belavin A.A. , Polyakov A.M. Metastable states of two-dimensional isotropic ferromagnets. JETP Lett., 1975 22, P. 503–506. [JETP Lett. 22, 245 (1975)].Belavin A.A. , Polyakov A.M. Metastable states of two-dimensional isotropic ferromagnets. JETP Lett., 1975 22, P. 503–506. [JETP Lett. 22, 245 (1975)].Potkina, M.N., Lobanov, I.S., J´ onsson, H., Uzdin, V.M. Lifetime of skyrmions in discrete systems with infinitesimal lattice constant. J. Magn. Magn. Mater., 2022, 549, P. 168974.Potkina, M.N., Lobanov, I.S., J´ onsson, H., Uzdin, V.M. Lifetime of skyrmions in discrete systems with infinitesimal lattice constant. J. Magn. Magn. Mater., 2022, 549, P. 168974.von Malottki S., Bessarab P.F., Haldar S., Delin A. and Heinze S. Skyrmion lifetime in ultrathin films. Phys. Rev. B, 2019, 99, P. 060409.von Malottki S., Bessarab P.F., Haldar S., Delin A. and Heinze S. Skyrmion lifetime in ultrathin films. Phys. Rev. B, 2019, 99, P. 060409.Capic D., Garanin D.A., Chudnovsky E.M., Skyrmions in an oblique field. J. Magn. Magn. Mater., 2021, 537, P. 168215.Capic D., Garanin D.A., Chudnovsky E.M., Skyrmions in an oblique field. J. Magn. Magn. Mater., 2021, 537, P. 168215.Peng L., Takagi R., Koshibae W., Shibata K., Nakajima K., Arima T.H., Nagaosa N., Seki S., Yu X. Tokura, Y., Controlled transformation of skyrmions and antiskyrmions in a non-centrosymmetric magnet. Nat. nanotech., 2020, 15, P. 181–186.Peng L., Takagi R., Koshibae W., Shibata K., Nakajima K., Arima T.H., Nagaosa N., Seki S., Yu X. Tokura, Y., Controlled transformation of skyrmions and antiskyrmions in a non-centrosymmetric magnet. Nat. nanotech., 2020, 15, P. 181–186.Jena J., G¨obel B., Ma T., Kumar V., Saha R., Mertig I., Felser C., Parkin, S.S.P. Elliptical Bloch skyrmion chiral twins in an antiskyrmion system. Nat. Commun., 2020, 11, P. 1115.Jena J., G¨obel B., Ma T., Kumar V., Saha R., Mertig I., Felser C., Parkin, S.S.P. Elliptical Bloch skyrmion chiral twins in an antiskyrmion system. Nat. Commun., 2020, 11, P. 1115.Hagemeister J., Romming N., von Bergmann K., Vedmedenko E.Y., Wiesendanger R. Stability of single skyrmionic bits. Nat. Commun., 2015, 6, P. 8455.Hagemeister J., Romming N., von Bergmann K., Vedmedenko E.Y., Wiesendanger R. Stability of single skyrmionic bits. Nat. Commun., 2015, 6, P. 8455.Nayak A.K., Kumar V., Ma T., Werner P., Pippel E., Sahoo R., Damay F., R¨ oßler U.K., Felser C., Parkin, S.S.P. Magnetic antiskyrmions above room temperature in tetragonal Heusler materials. Nature, 2017, 548(7669), P. 561–566.Nayak A.K., Kumar V., Ma T., Werner P., Pippel E., Sahoo R., Damay F., R¨ oßler U.K., Felser C., Parkin, S.S.P. Magnetic antiskyrmions above room temperature in tetragonal Heusler materials. Nature, 2017, 548(7669), P. 561–566.Potkina M.N., Lobanov I.S., Tretiakov O.A., J´ onsson H., Uzdin, V.M. Stability of long-lived antiskyrmions in the Mn-Pt-Sn tetragonal Heusler material. Phys. Rev. B, 2020, 102, P. 134430.Potkina M.N., Lobanov I.S., Tretiakov O.A., J´ onsson H., Uzdin, V.M. Stability of long-lived antiskyrmions in the Mn-Pt-Sn tetragonal Heusler material. Phys. Rev. B, 2020, 102, P. 134430.Bessarab P. F., Uzdin V. M., 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.Bessarab P. F., Uzdin V. M., 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.Lobanov, I. S., Uzdin, V. M. The lifetime of micron scale topological chiral magnetic states with atomic resolution. Comp. Phys. Commun., 2021, 269, P. 108136.Lobanov, I. S., Uzdin, V. M. The lifetime of micron scale topological chiral magnetic states with atomic resolution. Comp. Phys. Commun., 2021, 269, P. 108136.R´ ozsa L., Simon E., Palot´ as K., Udvardi L., Szunyogh L. Complex magnetic phase diagram and skyrmion lifetime in an ultrathin film from atomistic simulations. Phys. Rev. B, 2016, 93, P. 024417.R´ ozsa L., Simon E., Palot´ as K., Udvardi L., Szunyogh L. Complex magnetic phase diagram and skyrmion lifetime in an ultrathin film from atomistic simulations. Phys. Rev. B, 2016, 93, P. 024417.Wang C., Du H., Zhao X., Jin C., Tian M., Zhang Y., Che R. Enhanced stability of the magnetic skyrmion lattice phase under a tilted magnetic f ield in a two-dimensional chiral magnet. Nano lett., 2017 17, P. 2921.Wang C., Du H., Zhao X., Jin C., Tian M., Zhang Y., Che R. Enhanced stability of the magnetic skyrmion lattice phase under a tilted magnetic f ield in a two-dimensional chiral magnet. Nano lett., 2017 17, P. 2921.

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