najoNanosystems: Physics, Chemistry, MathematicsНаносистемы: физика, химия, математика2220-80542305-7971Университет ИТМО10.17586/2220-8054-2018-9-3-356-363najo-793Research ArticlePHYSICSФИЗИКАInterplay between size and stability of magnetic skyrmionsVarentsovaA. S.

197101 St. Petersburg

varentsova.a@mail.ruPotkinaM. N.

198504 St. Petersburg

107 Reykjav´ık, Iceland

MalottkiS. von

24098 Kiel

HeinzeS.

24098 Kiel

BessarabP. F.

197101 St. Petersburg

bessarab@hi.isTMO UniversityRussian FederationSt. Petersburg State University; Science Institute, University of IcelandRussian FederationInstitute of Theoretical Physics and Astrophysics, University of KielGermany20181408202593356363Copyright © Varentsova A.S., Potkina M.N., Malottki S., Heinze S., Bessarab P.F., 20252025Varentsova A.S., Potkina M.N., Malottki S., Heinze S., Bessarab P.F.Varentsova A.S., Potkina M.N., Malottki S., Heinze S., Bessarab P.F.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/793

The relationship between the size and stability of isolated skyrmions in a magnetic monolayer is analyzed based on minimum energy path calculations and atomistic spin Hamiltonian. It is demonstrated that the energy barrier protecting the skyrmion from collapse to the ferromagnetic state is not uniquely defined by the skyrmion size, although these two properties as functions of relevant material parameters follow similar trends. Stability of nanoscale skyrmions can be enhanced by a concerted adjustment of material parameters. The proposed parameter transformation conserves the skyrmion size, but does not conserve the skyrmion shape which changes from an arrow-like pattern to a profile that resembles magnetic bubbles. This transformation of the skyrmion shape is accompanied by an increase in the collapse energy barrier and thus enhancement of skyrmion stability.

magnetic skyrmionthermal stabilityenergy barrierThe authors would like to thank H. J´onsson and V.M. Uzdin for helpful discussions and useful comments. This work is supported by the Russian Science Foundation (Grant No. 17-72-10195).ReferencesBogdanov A. N., Yablonsky D. A. Thermodynamically stable “vortices” in magnetically ordered crystals. The mixed state of magnets. Sov. Phys. JETP., 1989, 68(1), P. 101–103.Bogdanov A. N., Yablonsky D. A. Thermodynamically stable “vortices” in magnetically ordered crystals. The mixed state of magnets. Sov. Phys. JETP., 1989, 68(1), P. 101–103.Bogdanov A., Hubert A. Thermodynamically stable magnetic vortex states in magnetic crystals. J. Magn. Magn. Mater., 1994, 138(3), P. 255–269.Bogdanov A., Hubert A. Thermodynamically stable magnetic vortex states in magnetic crystals. J. Magn. Magn. Mater., 1994, 138(3), P. 255–269.Bogdanov A., Hubert A. The properties of isolated magnetic vortices. Phys. Stat. Sol. (b), 1994, 186(2), P. 527–543.Bogdanov A., Hubert A. The properties of isolated magnetic vortices. Phys. Stat. Sol. (b), 1994, 186(2), P. 527–543.Kiselev N. S., Bogdanov A. N., Sch¨afer R., R¨oßler U. K. Chiral skyrmions in thin magnetic films: new objects for magnetic storage technologies? J. Phys. D: Appl. Phys., 2011, 44, P. 392001.Kiselev N. S., Bogdanov A. N., Sch¨afer R., R¨oßler U. K. Chiral skyrmions in thin magnetic films: new objects for magnetic storage technologies? J. Phys. D: Appl. Phys., 2011, 44, P. 392001.Fert A., Cros V., Sampaio J. Skyrmions on the track. Nat. Nanotechnol., 2013, 8, P. 152–156.Fert A., Cros V., Sampaio J. Skyrmions on the track. Nat. Nanotechnol., 2013, 8, P. 152–156.M¨uller J. Magnetic skyrmions on a two-lane racetrack. New J. Phys., 2017, 19, P. 025002(2017).M¨uller J. Magnetic skyrmions on a two-lane racetrack. New J. Phys., 2017, 19, P. 025002(2017).Nagaosa N., Tokura Y. Topological properties and dynamics of magnetic skyrmions. Nat. Nanotechnol., 2013, 8, P. 899–911.Nagaosa N., Tokura Y. Topological properties and dynamics of magnetic skyrmions. Nat. Nanotechnol., 2013, 8, P. 899–911.Woo S., Litzius K., Kr¨uger B., Im M.-Y., Caretta L., Richter K., Mann M., Krone A., Reeve R. N., Weigand M., Agrawal P., Lemesh I., Mawass M.-A., Fischer P., Kl¨aui M., Beach G. S. D. Observation of room-temperature magnetic skyrmions and their current-driven dynamics in ultrathin metallic ferromagnets. Nat. Mater., 2016, 15, P. 501–506.Woo S., Litzius K., Kr¨uger B., Im M.-Y., Caretta L., Richter K., Mann M., Krone A., Reeve R. N., Weigand M., Agrawal P., Lemesh I., Mawass M.-A., Fischer P., Kl¨aui M., Beach G. S. D. Observation of room-temperature magnetic skyrmions and their current-driven dynamics in ultrathin metallic ferromagnets. Nat. Mater., 2016, 15, P. 501–506.Yu X. Z., Kanazawa N., Zhang W. Z., Nagai T., Hara T., Kimoto K., Matsui Y., Onose Y., Tokura Y. Skyrmion flow near room temperature in an ultralow current density. Nat. Commun., 2012, 3, P. 988.Yu X. Z., Kanazawa N., Zhang W. Z., Nagai T., Hara T., Kimoto K., Matsui Y., Onose Y., Tokura Y. Skyrmion flow near room temperature in an ultralow current density. Nat. Commun., 2012, 3, P. 988.Sampaio J., Cros V., Rohart S., Thiaville A., Fert A. Nucleation, stability and current-induced motion of isolated magnetic skyrmions in nanostructures. Nat. Nanotechnol., 2013, 8, P. 839–844.Sampaio J., Cros V., Rohart S., Thiaville A., Fert A. Nucleation, stability and current-induced motion of isolated magnetic skyrmions in nanostructures. Nat. Nanotechnol., 2013, 8, P. 839–844.Moreau-Luchaire C., Moutas C., Reyren N., Sampaio J., Vaz C. A. F., Van Horne N., Bouzehouane K., Garcia K., Deranlot C., Warnicke P., Wohlh¨uter P., George J.-M., Weigand M., Raabe J., Cros V., Fert A. Additive interfacial chiral interaction in multilayers for stabilization of small individual skyrmions at room temperature. Nat. Nanotechnol., 2016, 11, P. 444–448.Moreau-Luchaire C., Moutas C., Reyren N., Sampaio J., Vaz C. A. F., Van Horne N., Bouzehouane K., Garcia K., Deranlot C., Warnicke P., Wohlh¨uter P., George J.-M., Weigand M., Raabe J., Cros V., Fert A. Additive interfacial chiral interaction in multilayers for stabilization of small individual skyrmions at room temperature. Nat. Nanotechnol., 2016, 11, P. 444–448.Boulle O., Vogel J., Yang H., Pizzini S., de Souza Chaves D., Locatelli A., Mentes¸ T. O., Sala A., Buda-Prejbeanu L. D., Klein O., Belmeguenai M., Roussign´e Y., Stashkevich A., Ch´erif S. M., Aballe L., Foerster M., Chshiev M., Auffret S., Miron I. M., Gaudin G. Room-temperature chiral magnetic skyrmions in ultrathin magnetic nanostructures. Nat. Nanotechnol., 2016, 11, P. 449–454.Boulle O., Vogel J., Yang H., Pizzini S., de Souza Chaves D., Locatelli A., Mentes¸ T. O., Sala A., Buda-Prejbeanu L. D., Klein O., Belmeguenai M., Roussign´e Y., Stashkevich A., Ch´erif S. M., Aballe L., Foerster M., Chshiev M., Auffret S., Miron I. M., Gaudin G. Room-temperature chiral magnetic skyrmions in ultrathin magnetic nanostructures. Nat. Nanotechnol., 2016, 11, P. 449–454.Yu G., Upadhyaya P., Li X., Li W., Kim S. K., Fan Y., Wong K. L., Tserkovnyak Y., Amiri P. K., Wang K. L. Room-temperature creation and spinorbit torque manipulation of skyrmions in thin films with engineered asymmetry. Nano Lett., 2016, 16(3), P. 1981–1988.Yu G., Upadhyaya P., Li X., Li W., Kim S. K., Fan Y., Wong K. L., Tserkovnyak Y., Amiri P. K., Wang K. L. Room-temperature creation and spinorbit torque manipulation of skyrmions in thin films with engineered asymmetry. Nano Lett., 2016, 16(3), P. 1981–1988.Jiang W., Upadhyaya P., Zhang W., Yu G., Jungfleisch M. B., Fradin F. Y., Pearson J. E., Tserkovnyak Y., Wang K. L., Heinonen O., te Velthuis S. G. E., Hoffmann A. Blowing magnetic skyrmion bubbles. Science, 2015, 17(6245), P. 283–286.Jiang W., Upadhyaya P., Zhang W., Yu G., Jungfleisch M. B., Fradin F. Y., Pearson J. E., Tserkovnyak Y., Wang K. L., Heinonen O., te Velthuis S. G. E., Hoffmann A. Blowing magnetic skyrmion bubbles. Science, 2015, 17(6245), P. 283–286.Legrand W., Maccariello D., Reyren N., Garcia K., Moutafis C., Moreau-Luchaire C., Collin S., Bouzehouane K., Cros V., Fert A. Room-temperature current-induced generation and motion of sub-100 nm skyrmions. Nano Lett., 2017, 17(4), P. 2703–2712.Legrand W., Maccariello D., Reyren N., Garcia K., Moutafis C., Moreau-Luchaire C., Collin S., Bouzehouane K., Cros V., Fert A. Room-temperature current-induced generation and motion of sub-100 nm skyrmions. Nano Lett., 2017, 17(4), P. 2703–2712.Jiang W., Zhang X., Yu G., Zhang W., Wang X., Jungfleisch M. B., Pearson J. E., Cheng X., Heinonen O., Wang K. L., Zhou Y., Hoffmann A., te Velthuis S. G. E. Direct observation of the skyrmion Hall effect. Nat. Phys., 2016, 11, P. 449–454.Jiang W., Zhang X., Yu G., Zhang W., Wang X., Jungfleisch M. B., Pearson J. E., Cheng X., Heinonen O., Wang K. L., Zhou Y., Hoffmann A., te Velthuis S. G. E. Direct observation of the skyrmion Hall effect. Nat. Phys., 2016, 11, P. 449–454.Hrabec A., Sampaio J., Belmeguenai M., Gross I., Weil R., Ch´erif S. M., Stashkevich A., Jacques V., Thiaville A., Rohart S. Currentinduced skyrmion generation and dynamics in symmetric bilayers. Nat. Commun., 2017, 8, P. 15765.Hrabec A., Sampaio J., Belmeguenai M., Gross I., Weil R., Ch´erif S. M., Stashkevich A., Jacques V., Thiaville A., Rohart S. Currentinduced skyrmion generation and dynamics in symmetric bilayers. Nat. Commun., 2017, 8, P. 15765.Litzius K., Lemesh I., Kr¨uger B., Bassirian P., Caretta L., Richter K., B¨uttner F., Sato K., Tretiakov O. A., F¨orster J., Reeve R. M., Weigand M., Bykova I., Stoll H., Sch¨utz G., Beach G. S. D., Kl¨aui M. Skyrmion Hall effect revealed by direct time-resolved X-ray microscopy. Nat. Phys., 2017, 13, P. 170–175.Litzius K., Lemesh I., Kr¨uger B., Bassirian P., Caretta L., Richter K., B¨uttner F., Sato K., Tretiakov O. A., F¨orster J., Reeve R. M., Weigand M., Bykova I., Stoll H., Sch¨utz G., Beach G. S. D., Kl¨aui M. Skyrmion Hall effect revealed by direct time-resolved X-ray microscopy. Nat. Phys., 2017, 13, P. 170–175.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.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.Romming N., Kubetzka A., Hanneken C., von Bergmann K., Wiesendanger R. Field-dependent size and shape of single magnetic skyrmions. Phys. Rev. Lett., 2015, 114(17), P. 177203.Romming N., Kubetzka A., Hanneken C., von Bergmann K., Wiesendanger R. Field-dependent size and shape of single magnetic skyrmions. Phys. Rev. Lett., 2015, 114(17), P. 177203.Hanneken C., Otte F., Kubetzka A., Dup´e B., Romming N., von Bergmann K., Wiesendanger R., Heinze S. Electrical detection of magnetic skyrmions by tunnelling non-collinear magnetoresistance. Nat. Nanotechnol., 2015, 10, P. 1039–1042.Hanneken C., Otte F., Kubetzka A., Dup´e B., Romming N., von Bergmann K., Wiesendanger R., Heinze S. Electrical detection of magnetic skyrmions by tunnelling non-collinear magnetoresistance. Nat. Nanotechnol., 2015, 10, P. 1039–1042.R¨oßler U. K., Leonov A. A., Bogdanov A. N. Chiral skyrmionic matter in non-centrosymmetric magnets. J. Phys. Conf. Ser., 2011, 303, P. 012105.R¨oßler U. K., Leonov A. A., Bogdanov A. N. Chiral skyrmionic matter in non-centrosymmetric magnets. J. Phys. Conf. Ser., 2011, 303, P. 012105.R¨oßler U. K., Bogdanov A. N., Pfleiderer C. Spontaneous skyrmion ground states in magnetic metals. Nature, 2006, 442, P. 797–801.R¨oßler U. K., Bogdanov A. N., Pfleiderer C. Spontaneous skyrmion ground states in magnetic metals. Nature, 2006, 442, P. 797–801.Wilson M. N., Butenko A. B., Bogdanov A. N., Monchesky T. L. Chiral skyrmions in cubic helimagnet films: The role of uniaxial anisotropy. Phys. Rev. B, 2014, 89, P. 094411.Wilson M. N., Butenko A. B., Bogdanov A. N., Monchesky T. L. Chiral skyrmions in cubic helimagnet films: The role of uniaxial anisotropy. Phys. Rev. B, 2014, 89, P. 094411.Leonov A. O., Monchesky T. L., Romming N., Kubetzka A., Bogdanov A. N., Wiesendanger R. The properties of isolated chiral skyrmions in thin magnetic films. New J. Phys., 2016, 18, P. 065003.Leonov A. O., Monchesky T. L., Romming N., Kubetzka A., Bogdanov A. N., Wiesendanger R. The properties of isolated chiral skyrmions in thin magnetic films. New J. Phys., 2016, 18, P. 065003.Bogdanov A., Hubert A. The stability of vortex-like structures in uniaxial ferromagnets. J. Magn. Magn. Mater., 1999, 195(1), P. 182–192.Bogdanov A., Hubert A. The stability of vortex-like structures in uniaxial ferromagnets. J. Magn. Magn. Mater., 1999, 195(1), P. 182–192.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. Comput. Phys. Commun., 2015, 196, P. 335–347.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. Comput. Phys. Commun., 2015, 196, P. 335–347.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.Rohart S., Miltat J., Thiaville A. Path to collapse for an isolated N´eel skyrmion. Phys. Rev. B, 2016, 93, P. 214412.Rohart S., Miltat J., Thiaville A. Path to collapse for an isolated N´eel skyrmion. Phys. Rev. B, 2016, 93, P. 214412.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.Lobanov I. S., J´onsson H., Uzdin V. M. Mechanism and activation energy of magnetic skyrmion annihilation obtained from minimum energy path calculations. Phys. Rev. B, 2016, 94, P. 174418.Lobanov I. S., J´onsson H., Uzdin V. M. Mechanism and activation energy of magnetic skyrmion annihilation obtained from minimum energy path calculations. Phys. Rev. B, 2016, 94, P. 174418.Stosic D., Mulkers J., Van Waeyenberge B., Ludermir T., MiloȈsevi´c M. V. Paths to collapse for isolated skyrmions in few-monolayer ferromagnetic films. Phys. Rev. B, 2017, 95, P. 214418.Stosic D., Mulkers J., Van Waeyenberge B., Ludermir T., MiloȈsevi´c M. V. Paths to collapse for isolated skyrmions in few-monolayer ferromagnetic films. Phys. Rev. B, 2017, 95, P. 214418.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. 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: Condens. Matter. In press. DOI: 10.1016/j.physb.2017.09.040.Cort´es-Ortu~no D., Wang W., Beg M., Pepper R. A., Bisotti M.-A., Carey R., Vousden M., Kluyver T., Hovorka O., Fangohr H. Thermal stability and topological protection of skyrmions in nanotracks. Sci. Rep., 2017, 7, P. 4060.Cort´es-Ortu~no D., Wang W., Beg M., Pepper R. A., Bisotti M.-A., Carey R., Vousden M., Kluyver T., Hovorka O., Fangohr H. Thermal stability and topological protection of skyrmions in nanotracks. Sci. Rep., 2017, 7, P. 4060.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.Stosic D., Ludermir T. B., MiloȈsevi´c M.V. Pinning of magnetic skyrmions in a monolayer Co film on Pt(111): Theoretical characterization and exemplified utilization. Phys. Rev. B, 2017, 96, P 214403.Stosic D., Ludermir T. B., MiloȈsevi´c M.V. Pinning of magnetic skyrmions in a monolayer Co film on Pt(111): Theoretical characterization and exemplified utilization. Phys. Rev. B, 2017, 96, P 214403.von Malottki S., Dup´e B., Bessarab P. F., Delin A., Heinze S. Enhanced skyrmion stability due to exchange frustration. Sci. Rep., 2017, 7, P. 12299.von Malottki S., Dup´e B., Bessarab P. F., Delin A., Heinze S. Enhanced skyrmion stability due to exchange frustration. Sci. Rep., 2017, 7, P. 12299.Bessarab P. F., M¨uller G. P., Lobanov I. S., Rybakov F. N., Kiselev N. S., J´onsson H., Uzdin V. M., Bl¨ugel S., Bergqvist L., Delin A. Lifetime of racetrack skyrmions. Sci. Rep., 2018, 8, P. 3433.Bessarab P. F., M¨uller G. P., Lobanov I. S., Rybakov F. N., Kiselev N. S., J´onsson H., Uzdin V. M., Bl¨ugel S., Bergqvist L., Delin A. Lifetime of racetrack skyrmions. Sci. Rep., 2018, 8, P. 3433.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.Draaisma H. J. G., de Jonge W. J. M. Surface and volume anisotropy from dipole-dipole interactions in ultrathin ferromagnetic films. J. Appl. Phys., 1988, 64(7), P. 3610–3613.Draaisma H. J. G., de Jonge W. J. M. Surface and volume anisotropy from dipole-dipole interactions in ultrathin ferromagnetic films. J. Appl. Phys., 1988, 64(7), P. 3610–3613.Bessarab P. F. Comment on “Path to collapse for an isolated N´eel skyrmion”. Phys. Rev. B, 2017, 95, P. 136401.Bessarab P. F. Comment on “Path to collapse for an isolated N´eel skyrmion”. Phys. Rev. B, 2017, 95, P. 136401.Butenko A. B., Leonov A. A., Bogdanov A. N., R¨oßler U. K. Theory of vortex states in magnetic nanodisks with induced Dzyaloshinskii-Moriya interactions. Phys. Rev. B, 2009, 80, P. 134410.Butenko A. B., Leonov A. A., Bogdanov A. N., R¨oßler U. K. Theory of vortex states in magnetic nanodisks with induced Dzyaloshinskii-Moriya interactions. Phys. Rev. B, 2009, 80, P. 134410.Siemens A., Zhang Y., Hagemeister J., Vedmedenko E. Y., Wiesendanger R. Minimal radius of magnetic skyrmions: statics and dynamics. New J. Phys., 2016, 18, P. 045021.Siemens A., Zhang Y., Hagemeister J., Vedmedenko E. Y., Wiesendanger R. Minimal radius of magnetic skyrmions: statics and dynamics. New J. Phys., 2016, 18, P. 045021.B¨uttner F., Lemesh I., Beach G. S. D. Theory of isolated magnetic skyrmions: From fundamentals to room temperature applications. Sci. Rep. 2018. 8, P. 4464.B¨uttner F., Lemesh I., Beach G. S. D. Theory of isolated magnetic skyrmions: From fundamentals to room temperature applications. Sci. Rep. 2018. 8, P. 4464.Rohart S., Thiaville A. Skyrmion confinement in ultrathin film nanostructures in the presence of Dzyaloshinskii-Moriya interaction. Phys. Rev. B, 2013, 88, P. 184422.Rohart S., Thiaville A. Skyrmion confinement in ultrathin film nanostructures in the presence of Dzyaloshinskii-Moriya interaction. Phys. Rev. B, 2013, 88, P. 184422.Bedanta S., Kleemann W. Supermagnetism. J. Phys. D, 2008, 42, P. 013001.Bedanta S., Kleemann W. Supermagnetism. J. Phys. D, 2008, 42, P. 013001.Kapaklis V., Arnalds U. B., Farhan A., Chopdekar R. V., Balan A., Scholl A., Heyderman L. J., Hj¨orvarsson B. Thermal fluctuations in artificial spin ice. Nat. Nanotechnol., 2014, 9, P. 514–519.Kapaklis V., Arnalds U. B., Farhan A., Chopdekar R. V., Balan A., Scholl A., Heyderman L. J., Hj¨orvarsson B. Thermal fluctuations in artificial spin ice. Nat. Nanotechnol., 2014, 9, P. 514–519.Krause S., Herzog G., Stapelfeldt T., Berbil-Bautista L., Bode M., Vedmedenko E. Y., Wiesendanger R. Magnetization reversal of nanoscale islands: How size and shape affect the Arrhenius prefactor. Phys. Rev. Lett., 2009, 103, P. 127202.Krause S., Herzog G., Stapelfeldt T., Berbil-Bautista L., Bode M., Vedmedenko E. Y., Wiesendanger R. Magnetization reversal of nanoscale islands: How size and shape affect the Arrhenius prefactor. Phys. Rev. Lett., 2009, 103, P. 127202.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.Wild J., Meier T. N. G., P¨ollath S., Kronseder M., Bauer A., Chacon A., Halder M., Schowalter M., Rosenauer A., Zweck J., M¨uller J., Rosch A., Pfleiderer C., Back C. H. Entropy-limited topological protection of skyrmions. Sci. Adv., 2017, 3(9), P. e1701704.Wild J., Meier T. N. G., P¨ollath S., Kronseder M., Bauer A., Chacon A., Halder M., Schowalter M., Rosenauer A., Zweck J., M¨uller J., Rosch A., Pfleiderer C., Back C. H. Entropy-limited topological protection of skyrmions. Sci. Adv., 2017, 3(9), P. e1701704.

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