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The polarity of magnetic vortex core can be switched by current or magnetic field through a vortex-antivortex pair creation and annihilation process, in which the significant change of the exchange energy during the switching takes an important role. To further unveil the energetic origin of magnetic vortex switching, we investigate the evolution of the maximum exchange energy density of the sample by using micromagnetic finite-element simulations based on the Landau-Lifshitz-Gilbert equation including the adiabatic and the nonadiabatic spin torque terms. Our micromagnetic calculations indicate that maximum exchange energy density for the considered sample must exceed a critical value of ~3.0106 J/m3 in order to achieve the magnetic vortex switching. The threshold value corresponds to the maximum exchange energy density at the time of creation of new vortex-antivortex pair. Following the nucleation of antivortex, the maximum exchange energy density increases rapidly with the antivortex approaching the original vortex. The maximum exchange energy density can become large at the time of annihilation of two vortexes. To explain well the critical value of the local maximum exchange energy density, we use the rigid vortex model（in which the spin distribution is unchangeable while vortex is displaced) to develop an analytical model. For a magnetic vortex confined in a thin ferromagnetic nanodisk, the magnetization distribution is unchanged along the thickness and can be seen as a two-dimensional model when the thickness is less than or on the order of the exchange length. The components of vortex magnetization vector in a ferromagnetic dot can be expressed by using a complex function w(,). Corresponding to the trivortex state appearing in vortex core reversal process, the local exchange energy density Wex around the vortexes cores is obtained. Simultaneously, we obtain the maximum exchange energy density:Wex2.3106 J/m3. In a realistic system, the shape of vortexes will deform during the vortex core reversal, which leads to the analytical result lower than the simulation value. Based on this reason, the analytical result matches well with our simulation value.
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Keywords:
- micromagnetic simulation /
- magnetic vortex /
- spin-transfer torque
[1] Kikuchi N, Okamoto S, Kitakami O, Shimada Y, Kim S G, Otani Y, Fukamichi K 2001 J. Appl. Phys. 90 6548
[2] Van-Waeyenberge B, Puzic A, Stoll H, Chou K W, Tyliszczak T, Hertel R, Fahnle M, Bruckl H, Rott K, Reiss G, Neudecker I, Weiss D, Back C H, Schutz G 2006 Nature 444 461
[3] Liu Y W, Gliga S, Hertel R, Schneider C M 2007 Appl Phys. Lett. 91 112501
[4] Hertel R, Gliga S, Fahnle M, Schneider C M 2007 Phys. Rev. Lett. 98 117201
[5] Weigand M, Van-Waeyenberge B, Vansteenkiste A, Curcic M, Sackmann V, Stoll H, Tyliszczak T, Kaznatcheev K, Bertwistle D, Woltersdorf G, Back C H, Schutz G 2009 Phys. Rev. Lett. 102 077201
[6] Vansteenkiste A, Chou K W, Weigand M, Curcic M, Sackmann V, Stoll H, Tyliszczak T, Woltersdorf G, Back C H, Schutz G, Van-Waeyenberge B 2009 Nat. Phys. 5 332
[7] Yamada K, Kasai S, Nakatani Y, Kobayashi K, Kohno H, Thiaville A, Ono T 2007 Nat. Mater. 6 270
[8] Sheka D D, Gaididei Y, Mertens F G 2007 Appl. Phys. Lett. 91 082509
[9] Liu Y W, He H, Zhang Z Z 2007 Appl. Phys. Lett. 91 242501
[10] Guslienko K Y, Lee K S, Kim S K 2008 Phys. Rev. Lett. 100 027203
[11] Noske M, Stoll H, Föhnle M, Gangwar A, Woltersdorf G, Slavin A, Weigand M, Dieterle G, Förster J, Back H C, Schtz G 2016 J. Appl. Phys. 119 173901
[12] Agramunt-Puig S, Del-Valle N, Navau C, Sanchez A 2014 Appl. Phys. Lett. 104 012407
[13] Jenkins A S, Grimaldi E, Bortolotti P, Lebrun R, Kubota H, Yakushiji K, Fukushima A, de Loubens G, Klein O, Yuasa S, Cros V 2014 Appl. Phys. Lett. 105 172403
[14] Sun M J, Liu Y W 2015 Acta Phys. Sin. 64 247505(in Chinese)[孙明娟, 刘要稳2015物理学报64 247505]
[15] Xiao Q F, Rudge J, Choi B C, Hong Y K, Donohoe G 2006 Appl. Phys. Lett. 89 262507
[16] Lee K S, Guslienko K Y, Lee J Y, Kim S K 2007 Phys. Rev. B 76 174410
[17] Kim S K, Choi Y S, Lee K S, Guslienko K Y, Jeong D E 2007 Appl. Phys. Lett. 91 082506
[18] Hertel R, Schneider C M 2006 Phys. Rev. Lett. 97 177202
[19] Zhang H, Liu Y W 2012 J. Nanosci. Nanotechnol. 12 1063
[20] L G, Cao X C, Qin Y F, Wang L H, Li G H, Gao F, Sun F W, Zhang H 2015 Acta Phys. Sin. 64 217501(in Chinese)[吕刚, 曹学成, 秦羽丰, 王林辉, 厉桂华, 高峰, 孙丰伟, 张红2015物理学报64 217501]
[21] Papanicolaou N, Zakrzewski W J 1995 Physica D 80 225
[22] Guslienko K Y, Novosad V, Otani Y, Shima Y, Fukamichi K 2001 Phys. Rev. B 65 024414
[23] Jubert P O, Allenspach R 2004 Phys. Rev. B 70 144402
[24] Lee K S, Choi Y S, Kim S K 2005 Appl. Phys. Lett. 87 192502
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[1] Kikuchi N, Okamoto S, Kitakami O, Shimada Y, Kim S G, Otani Y, Fukamichi K 2001 J. Appl. Phys. 90 6548
[2] Van-Waeyenberge B, Puzic A, Stoll H, Chou K W, Tyliszczak T, Hertel R, Fahnle M, Bruckl H, Rott K, Reiss G, Neudecker I, Weiss D, Back C H, Schutz G 2006 Nature 444 461
[3] Liu Y W, Gliga S, Hertel R, Schneider C M 2007 Appl Phys. Lett. 91 112501
[4] Hertel R, Gliga S, Fahnle M, Schneider C M 2007 Phys. Rev. Lett. 98 117201
[5] Weigand M, Van-Waeyenberge B, Vansteenkiste A, Curcic M, Sackmann V, Stoll H, Tyliszczak T, Kaznatcheev K, Bertwistle D, Woltersdorf G, Back C H, Schutz G 2009 Phys. Rev. Lett. 102 077201
[6] Vansteenkiste A, Chou K W, Weigand M, Curcic M, Sackmann V, Stoll H, Tyliszczak T, Woltersdorf G, Back C H, Schutz G, Van-Waeyenberge B 2009 Nat. Phys. 5 332
[7] Yamada K, Kasai S, Nakatani Y, Kobayashi K, Kohno H, Thiaville A, Ono T 2007 Nat. Mater. 6 270
[8] Sheka D D, Gaididei Y, Mertens F G 2007 Appl. Phys. Lett. 91 082509
[9] Liu Y W, He H, Zhang Z Z 2007 Appl. Phys. Lett. 91 242501
[10] Guslienko K Y, Lee K S, Kim S K 2008 Phys. Rev. Lett. 100 027203
[11] Noske M, Stoll H, Föhnle M, Gangwar A, Woltersdorf G, Slavin A, Weigand M, Dieterle G, Förster J, Back H C, Schtz G 2016 J. Appl. Phys. 119 173901
[12] Agramunt-Puig S, Del-Valle N, Navau C, Sanchez A 2014 Appl. Phys. Lett. 104 012407
[13] Jenkins A S, Grimaldi E, Bortolotti P, Lebrun R, Kubota H, Yakushiji K, Fukushima A, de Loubens G, Klein O, Yuasa S, Cros V 2014 Appl. Phys. Lett. 105 172403
[14] Sun M J, Liu Y W 2015 Acta Phys. Sin. 64 247505(in Chinese)[孙明娟, 刘要稳2015物理学报64 247505]
[15] Xiao Q F, Rudge J, Choi B C, Hong Y K, Donohoe G 2006 Appl. Phys. Lett. 89 262507
[16] Lee K S, Guslienko K Y, Lee J Y, Kim S K 2007 Phys. Rev. B 76 174410
[17] Kim S K, Choi Y S, Lee K S, Guslienko K Y, Jeong D E 2007 Appl. Phys. Lett. 91 082506
[18] Hertel R, Schneider C M 2006 Phys. Rev. Lett. 97 177202
[19] Zhang H, Liu Y W 2012 J. Nanosci. Nanotechnol. 12 1063
[20] L G, Cao X C, Qin Y F, Wang L H, Li G H, Gao F, Sun F W, Zhang H 2015 Acta Phys. Sin. 64 217501(in Chinese)[吕刚, 曹学成, 秦羽丰, 王林辉, 厉桂华, 高峰, 孙丰伟, 张红2015物理学报64 217501]
[21] Papanicolaou N, Zakrzewski W J 1995 Physica D 80 225
[22] Guslienko K Y, Novosad V, Otani Y, Shima Y, Fukamichi K 2001 Phys. Rev. B 65 024414
[23] Jubert P O, Allenspach R 2004 Phys. Rev. B 70 144402
[24] Lee K S, Choi Y S, Kim S K 2005 Appl. Phys. Lett. 87 192502
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