Rogue wave solution in ferromagnetic nanowires

Li Zai-Dong; Guo Qi-Qi

doi:10.7498/aps.69.20191352

Abstract

In this paper, we introduce some new excited states of magnetization in ferromagnetic nanowires, including Akhmediev breathers, Kuznetsov-Ma soliton and rogue wave in isotropic ferromagnetic nanowires, and rogue wave in anisotropic ferromagnetic nanowires driven by spin-polarized current. The isotropic case demonstrates a spatial periodic process of a magnetic soliton forming the petal with four pieces and a localized process of the spin-wave background. In a limit case, we get rogue waves and clarify its formation mechanism. In the case of anisotropy, it is found that the generation of rogue waves mainly comes from the accumulation of energy and rapid dispersion in the center. In addition, rogue waves are unstable, the spin-polarized current can control the exchange rate of magnons between the envelope soliton and the background. These results can be useful for the exploration of nonlinear excitation in Bosonic and fermionic ferromagnet.

Keywords:

Authors and contacts

Li Zai-Dong^1,2,
Guo Qi-Qi¹

1.
Department of Applied Physics, Hebei University of Technology, Tianjin 300401, China
2.
School of Science, Tianjin University of Technology, Tianjin 300384, China

Corresponding author: Li Zai-Dong, lizd@hebut.edu.cn

Funds: Project supported by the National Natural Science Foundation of China (Grant No. 61774001) and the Natural Science Foundation of Hebei Province, China (Grant No. F2019202141)

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References

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[3]	Yu X Z, Kanazawa N, Onose Y, Kimoto K, Zhang W Z, Ishiwata S, Matsui Y, Tokura Y 2011 Nat. Mater. 10 106 Google Scholar
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[7]	Parkin S S P, Hayashi M, Thomas L 2008 Science 320 190 Google Scholar
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[9]	He P B, Xie X C, Liu W M 2005 Phys. Rev. B 72 172411 Google Scholar
[10]	Tretiakov O A, Liu Y, Abanov Ar 2012 Phys. Rev. Lett. 108 247201 Google Scholar
[11]	Li Q Y, Zhao F, He P B, Li Z D 2015 Chin. Phys. B 24 037508 Google Scholar
[12]	Kosevich A M, Ivanov B A, Kovalev A S 1990 Phys. Rep. 194 117 Google Scholar
[13]	Mikeska H J, Steiner M 1991 Adv. Phys. 40 191 Google Scholar
[14]	Haazen P P J, Mure E, Franken J H, Lavrijsen R, H. Swagten J M, Koopmans B 2013 Nat. Mater. 12 299 Google Scholar
[15]	Miron I M, Gaudin G, Auffret S, Rodmacq B, Schuhl A, Pizzini S, Vogel J, Gambardella P 2010 Nat. Mater. 9 230 Google Scholar
[16]	Di K, Zhang V L, Lim H S, Ng S C, Kuok M H, Yu J, Yoon J, Qiu X, Yang H 2015 Phys. Rev. Lett. 114 047201 Google Scholar
[17]	Wang W W, Albert M, Beg M, Bisotti M A, Chernyshenko D 2015 Phys. Rev. Lett. 114 087203 Google Scholar
[18]	Li Z D, Cui H, Li Q Y, He P B 2018 Ann. Phys. 388 390 Google Scholar
[19]	Chesi S, Coish W A 2015 Phys. Rev. B 91 245306 Google Scholar
[20]	Li Z D, Liu F, Li Q Y, He P B 2015 J. Appl. Phys. 117 173906 Google Scholar
[21]	Pfleiderer C, Julian S R, Lonzarich G G 2001 Nature 414 427 Google Scholar
[22]	Uchida M, Onose Y, Matsui Y, Tokura Y 2006 Science 311 359 Google Scholar
[23]	Meckler S, Mikuszeit N, Preßler A, Vedmedenko E Y, Pietzsch O, Wiesendanger R 2009 Phys. Rev. Lett. 103 157201 Google Scholar
[24]	Moriya T 1960 Phys.Rev. 120 91 Google Scholar
[25]	Zakharov D V, Deisenhofer J 2006 Phys. Rev. B 73 094452 Google Scholar
[26]	Gangadharaiah S, Sun J M, Starykh O A 2008 Phys. Rev. B 78 054436 Google Scholar
[27]	Albert F J, Emley N C, Myers E B, Ralph D C, Buhrman R A 2002 Phys. Rev. Lett. 89 226802 Google Scholar
[28]	Mucciolo E R, Chamon C, Marcus C M 2002 Phys. Rev. Lett. 89 146802 Google Scholar
[29]	Beach G S D, Knutson C, Nistor C, Tsoi M, Erskine J L 2006 Phys. Rev. Lett. 97 057203 Google Scholar
[30]	Bertotti G, Serpico C, Mayergoyz I D, Magni A, Aquino M, Bonin R 2005 Phys. Rev. Lett. 94 127206 Google Scholar
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[33]	Tsoi M, Jansen A G M, Bass J, Chiang W C, Seck M, Tsoi V, Wyder P 1998 Phys. Rev. Lett. 80 4281 Google Scholar
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[35]	Li Z D, He P B, Liu W M 2014 Chin. Phys. B 23 117502 Google Scholar
[36]	Tsoi M, Tsoi V, Bass J, Jansen A G M, Wyder P 2002 Phys. Rev. Lett. 89 246803 Google Scholar
[37]	Tserkovnyak Y, Brataas A, Bauer G E W 2002 Phys. Rev. Lett. 88 117601 Google Scholar
[38]	Li Z D, Li Q Y, He P B, Liang J Q, Liu W M, Fu G S 2010 Phys. Rev. A 81 015602 Google Scholar
[39]	Freimuth F, Blugel S, Mokrousov Y 2014 Phys. Rev. B 90 174423 Google Scholar
[40]	Santos T S, Lee J S, Migdal P, Lekshmi I C, Satpati B, Moodera J S 2007 Phys. Rev. Lett. 98 016601 Google Scholar
[41]	Li Z, Zhang S 2004 Phys. Rev. Lett. 92 207203 Google Scholar
[42]	Kasai S, Nakatani Y, Kobayashi K, Kohno H, Ono T 2006 Phys. Rev. Lett. 97 107204 Google Scholar
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[46]	Liu C, Yang Z Y, Zhao L C, Yang W L 2015 Phys. Rev. E 91 022904 Google Scholar
[47]	Duan L, Zhao L C, Xu W H 2017 Phys. Rev. E 95 042212 Google Scholar
[48]	Wang L, X Wuan, Zhang H Y 2018 Phys. Lett. A 382 2650 Google Scholar
[49]	Wang L, Liu C, Wu X, Wang X, Sun W R 2018 Nonlinear Dyn. 94 977 Google Scholar
[50]	Zakharov V E, Gelash A A 2013 Phys. Rev. Lett. 111 054101 Google Scholar
[51]	Gelash A A, Zakharov V E 2014 Nonlinearity. 27 R1 Google Scholar
[52]	Yan P, Wang X S, Wang X R 2011 Phys. Rev. Lett. 107 177207 Google Scholar
[53]	Zhao F, Li Z D, Li Q Y, Wen L, Fu G S, Liu W M 2012 Ann. Phys. 327 2085 Google Scholar
[54]	Li Z D, Li Q Y, Xu T F, He P B 2016 Phys. Rev. E 94 042220 Google Scholar
[55]	Li Q Y, Li Z D, He P B, Song W W, Fu G S 2010 Can. J. Phys. 88 9
[56]	Hasegawa A 1984 Opt. Lett. 9 288; Tai K, Tomita A, Jewell J L, Hasegawa A 1986 Appl. Phys. Lett. 49 236
[57]	Akhmediev N N 2001 Nature. 413 267; Van Simaeys G, Emplit P, Haelterman M 2001 Phys. Rev. Lett. 87 033902; Mussot A, Kudlinski A, Droques M, Szriftgiser P, Akhmediev N 2014 Phys. Rev. X 4 011054
[58]	Zhang H Q, Tian B, Xing L, Meng X H 2010 Physica A 389 367 Google Scholar
[59]	Matveev V B, Salli M A 1991 Darboux Transformations and Solitons, Vol. 5 (Berlin: Springer) pp7−15
[60]	谷超豪, 胡和生, 周子翔 2005 孤立子理论中的达布变换及其几何应用(第二版)(上海: 上海科学技术出版社)第18−24页 Gu C H, Hu H S, Zhou Z X 2005 Darboux Transformation in Soliton Theory and Its Geometric Applications (Shanghai: Scientific and Technical Publishers) pp18−24 (in Chinese).
[61]	Li Z D, Wu X, Li Q Y, He P B 2016 Chin. Phys. B 25 010507 Google Scholar
[62]	Li Z D, Huo C Z, Li Q Y, He P B, Xu T F 2018 Chin. Phys. B 27 040505 Google Scholar
[63]	Ho T L 1998 Phys. Rev. Lett. 81 742; Law C K, Ohmi T, Machida K, 1998 J. Phys. Soc. Jpn. 67 1822; Law C K, Pu H, Bigelow N P 1998 Phys. Rev. Lett. 81 5257
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Cited By

图 1 方程(3)在$\mu \to {A_{\rm{s}}}{k_{\rm{s}}}, \nu = {k_{\rm{s}}}\sqrt {1 - A_{\rm{s}}^2} $的极限条件下, 磁化分量${m_3}$的渐近过程, 参数如下: A_s = 0.9, k_s = 1　(a) μ = 0.82; (b) μ = 0.89; (c) μ = 0.89999; (d) μ = 1.1; (e) μ = 0.96; (f) μ = 0.9001

Figure 1. The asymptotic processes of the magnetic component ${m_3}$ in the limit processes $\mu \to {A_{\rm{s}}}{k_{\rm{s}}}$ and $\nu = {k_{\rm{s}}}\sqrt {1 - A_{\rm{s}}^2} $ in Eq. (3), where the parameters are as follows: A_s = 0.9, k_s = 1: (a) μ = 0.82; (b) μ = 0.89; (c) μ = 0.89999; (d) μ = 1.1; (e) μ = 0.96; (f) μ = 0.9001, respectively.

DownLoad: Full-Size Img PowerPoint

图 2 方程(3)和(7)中磁化强度${{m}} = \left( {{m_1}, {m_2}, {m_3}} \right)$的怪波演化图, 即(a)−(c)为亮怪波, (d)−(f)为暗怪波. 参数: ${A_{\rm{s}}} = \sqrt 3 /2, {k_{\rm{s}}} = 1.5, \nu = {k_{\rm{s}}}\sqrt {1 - A_{\rm{s}}^2}, \mu = \pm 0.75 \sqrt 3 $, ±分别表示亮怪波和暗怪波^[54]

Figure 2. The graphical evolution of rogue waves for the magnetization ${{m}} = \left( {{m_1}, {m_2}, {m_3}} \right)$ in Eq. (3) and (7), i.e., bright rogue waves (a)−(c) and dark rogue waves (d)−(f). The parameters are as follows: ${A_{\rm{s}}} = \sqrt 3 /2, {k_{\rm{s}}} = 1.5, \nu = {k_{\rm{s}}}\sqrt {1 - A_{\rm{s}}^2} $, and $\mu = \pm 0.75 \sqrt 3 $ with the sign $ \pm $ corresponding to the bright and dark rogue waves, respectively^[54].

DownLoad: Full-Size Img PowerPoint

图 3 方程(3)在A_s = 1, $\mu \to {A_{\rm{s}}}{k_{\rm{s}}}, \nu = {k_{\rm{s}}}\sqrt {1 - A_{\rm{s}}^2} $的条件下, 磁化分量${{m_3}}$四片花瓣结构的演化图. 参数如下: A_s = 1, $ {k_{\rm{s}}} = 0.9, \mu = 0.8999$

Figure 3. The formation of magnetic petal in the component ${{m_3}}$ of Eq.(3) under the special condition of A_s = 1, $ \mu \to {A_{\rm{s}}}{k_{\rm{s}}}, \nu = {k_{\rm{s}}}\sqrt {1 - A_{\rm{s}}^2}$. The parameters are as follows: ${A_{\rm{s}}} = 1, {k_{\rm{s}}} = 0.9, \mu = 0.8999$.

DownLoad: Full-Size Img PowerPoint

图 4 不同的参数${\mu _1}$下的磁振子密度分布图^[53], 范围从0.09到0.29间隔0.05. 插图为怪波形成时的磁振子密度. 其余参数为${A_{\rm{c}}} = 0.2, {A_{\rm{J}}} = {k_c} = 0.1$

Figure 4. The magnon density distribution against the background for the different parameter ${\mu _1}$, which ranges from 0.09 to 0.29 in 0.05 steps^[53]. The inset figure is the magnon density distribution against the background for the excited formation of magnetic rogue wave. Other parameters are ${A_{\rm{c}}} = 0.2, {A_{\rm{J}}} = {k_{\rm{c}}} = 0.1$.

DownLoad: Full-Size Img PowerPoint

图 5 (a)−(e)不同电流值激发怪波在区域$\left( {x, t} \right)$的分布图^[53], 电流从0到0.8, 间隔为0.2; (f)不同电流怪波形成时的图形, 插图为磁振子积聚的最大时情况. 临界电流${A_{\rm{J}}} = 2{k_{\rm{c}}}$, 其它参数${A_{\rm{c}}} = {k_{\rm{c}}} = 0.2$

Figure 5. (a)−(e) The formation region in space $\left( {x, t} \right)$ for magnetic rogue wave with different current^[53]. The parameter ${A_J}$ ranges form 0 to 0.8 in 0.2 steps; (f) The nonuniform exchange of magnons between rogue wave and background for the different spin current. The inset figure in (f) denotes the maximal accumulation (or dissipation) process for the critical current value${A_{\rm{J}}} = 2{k_{\rm{c}}}$. Other parameters are ${A_{\rm{c}}} = {k_{\rm{c}}} = 0.2$.

DownLoad: Full-Size Img PowerPoint

map

[1]	Bu K M, Kwon H Y, Kang S P, Kim H J, Won C 2013 J. Magn. Magn. Mater. 343 32 Google Scholar
[2]	Moon J H, Seo S M, Lee K J, Kim K W, Ryu J, Lee H W, McMichael R D, Stiles M D 2013 Phys. Rev. B 88 184404 Google Scholar
[3]	Yu X Z, Kanazawa N, Onose Y, Kimoto K, Zhang W Z, Ishiwata S, Matsui Y, Tokura Y 2011 Nat. Mater. 10 106 Google Scholar
[4]	Rohart S, Thiaville A 2013 Phys. Rev. B 88 184422 Google Scholar
[5]	Robler U K, Bogdanov A N, Pfleiderer C 2006 Nature 442 797 Google Scholar
[6]	Allwood D A, Xiong G, Faulkner C C, Atkinson D, Petit D, Cowburn R P 2005 Science 309 1688 Google Scholar
[7]	Parkin S S P, Hayashi M, Thomas L 2008 Science 320 190 Google Scholar
[8]	Li Z D, Hu Y C, He P B, Sun L L 2018 Chin. Phys. B 27 077505 Google Scholar
[9]	He P B, Xie X C, Liu W M 2005 Phys. Rev. B 72 172411 Google Scholar
[10]	Tretiakov O A, Liu Y, Abanov Ar 2012 Phys. Rev. Lett. 108 247201 Google Scholar
[11]	Li Q Y, Zhao F, He P B, Li Z D 2015 Chin. Phys. B 24 037508 Google Scholar
[12]	Kosevich A M, Ivanov B A, Kovalev A S 1990 Phys. Rep. 194 117 Google Scholar
[13]	Mikeska H J, Steiner M 1991 Adv. Phys. 40 191 Google Scholar
[14]	Haazen P P J, Mure E, Franken J H, Lavrijsen R, H. Swagten J M, Koopmans B 2013 Nat. Mater. 12 299 Google Scholar
[15]	Miron I M, Gaudin G, Auffret S, Rodmacq B, Schuhl A, Pizzini S, Vogel J, Gambardella P 2010 Nat. Mater. 9 230 Google Scholar
[16]	Di K, Zhang V L, Lim H S, Ng S C, Kuok M H, Yu J, Yoon J, Qiu X, Yang H 2015 Phys. Rev. Lett. 114 047201 Google Scholar
[17]	Wang W W, Albert M, Beg M, Bisotti M A, Chernyshenko D 2015 Phys. Rev. Lett. 114 087203 Google Scholar
[18]	Li Z D, Cui H, Li Q Y, He P B 2018 Ann. Phys. 388 390 Google Scholar
[19]	Chesi S, Coish W A 2015 Phys. Rev. B 91 245306 Google Scholar
[20]	Li Z D, Liu F, Li Q Y, He P B 2015 J. Appl. Phys. 117 173906 Google Scholar
[21]	Pfleiderer C, Julian S R, Lonzarich G G 2001 Nature 414 427 Google Scholar
[22]	Uchida M, Onose Y, Matsui Y, Tokura Y 2006 Science 311 359 Google Scholar
[23]	Meckler S, Mikuszeit N, Preßler A, Vedmedenko E Y, Pietzsch O, Wiesendanger R 2009 Phys. Rev. Lett. 103 157201 Google Scholar
[24]	Moriya T 1960 Phys.Rev. 120 91 Google Scholar
[25]	Zakharov D V, Deisenhofer J 2006 Phys. Rev. B 73 094452 Google Scholar
[26]	Gangadharaiah S, Sun J M, Starykh O A 2008 Phys. Rev. B 78 054436 Google Scholar
[27]	Albert F J, Emley N C, Myers E B, Ralph D C, Buhrman R A 2002 Phys. Rev. Lett. 89 226802 Google Scholar
[28]	Mucciolo E R, Chamon C, Marcus C M 2002 Phys. Rev. Lett. 89 146802 Google Scholar
[29]	Beach G S D, Knutson C, Nistor C, Tsoi M, Erskine J L 2006 Phys. Rev. Lett. 97 057203 Google Scholar
[30]	Bertotti G, Serpico C, Mayergoyz I D, Magni A, Aquino M, Bonin R 2005 Phys. Rev. Lett. 94 127206 Google Scholar
[31]	Garcia-Sanchez F, Borys P, Soucaille R, Adam J P, Stamps R L, Kim J V 2015 Phys. Rev. Lett. 114 247206 Google Scholar
[32]	Katine J A, Albert F J, Buhrman R A, Myers E B, Ralph D C 2000 Phys. Rev. Lett. 84 3149 Google Scholar
[33]	Tsoi M, Jansen A G M, Bass J, Chiang W C, Seck M, Tsoi V, Wyder P 1998 Phys. Rev. Lett. 80 4281 Google Scholar
[34]	He P B, Liu W M 2005 Phys. Rev. B 72 064410 Google Scholar
[35]	Li Z D, He P B, Liu W M 2014 Chin. Phys. B 23 117502 Google Scholar
[36]	Tsoi M, Tsoi V, Bass J, Jansen A G M, Wyder P 2002 Phys. Rev. Lett. 89 246803 Google Scholar
[37]	Tserkovnyak Y, Brataas A, Bauer G E W 2002 Phys. Rev. Lett. 88 117601 Google Scholar
[38]	Li Z D, Li Q Y, He P B, Liang J Q, Liu W M, Fu G S 2010 Phys. Rev. A 81 015602 Google Scholar
[39]	Freimuth F, Blugel S, Mokrousov Y 2014 Phys. Rev. B 90 174423 Google Scholar
[40]	Santos T S, Lee J S, Migdal P, Lekshmi I C, Satpati B, Moodera J S 2007 Phys. Rev. Lett. 98 016601 Google Scholar
[41]	Li Z, Zhang S 2004 Phys. Rev. Lett. 92 207203 Google Scholar
[42]	Kasai S, Nakatani Y, Kobayashi K, Kohno H, Ono T 2006 Phys. Rev. Lett. 97 107204 Google Scholar
[43]	Kasai S, Fischer P, Im M Y, Yamada K, Nakatani Y, Kobayashi K, Kohno H, Ono T 2008 Phys. Rev. Lett. 101 237203; Sugimoto S, Fukuma Y, Otani Y C 2011 IEEI. T. Magn. 47 2951
[44]	Yamada K, Kasai S, Nakatani Y, Kobayashi K, Kohno H, Thiaville A, Ono T 2007 Nat. Mater. 6 270; Moriya R, Thomas L, Hayashi M, Bazaliy Y B, Rettner C, Parkin S S P 2008 Nat. Phys. 4 368
[45]	Zhao L C, Ling L 2016 J. Opt. Soc. Am. B 33 850 Google Scholar
[46]	Liu C, Yang Z Y, Zhao L C, Yang W L 2015 Phys. Rev. E 91 022904 Google Scholar
[47]	Duan L, Zhao L C, Xu W H 2017 Phys. Rev. E 95 042212 Google Scholar
[48]	Wang L, X Wuan, Zhang H Y 2018 Phys. Lett. A 382 2650 Google Scholar
[49]	Wang L, Liu C, Wu X, Wang X, Sun W R 2018 Nonlinear Dyn. 94 977 Google Scholar
[50]	Zakharov V E, Gelash A A 2013 Phys. Rev. Lett. 111 054101 Google Scholar
[51]	Gelash A A, Zakharov V E 2014 Nonlinearity. 27 R1 Google Scholar
[52]	Yan P, Wang X S, Wang X R 2011 Phys. Rev. Lett. 107 177207 Google Scholar
[53]	Zhao F, Li Z D, Li Q Y, Wen L, Fu G S, Liu W M 2012 Ann. Phys. 327 2085 Google Scholar
[54]	Li Z D, Li Q Y, Xu T F, He P B 2016 Phys. Rev. E 94 042220 Google Scholar
[55]	Li Q Y, Li Z D, He P B, Song W W, Fu G S 2010 Can. J. Phys. 88 9
[56]	Hasegawa A 1984 Opt. Lett. 9 288; Tai K, Tomita A, Jewell J L, Hasegawa A 1986 Appl. Phys. Lett. 49 236
[57]	Akhmediev N N 2001 Nature. 413 267; Van Simaeys G, Emplit P, Haelterman M 2001 Phys. Rev. Lett. 87 033902; Mussot A, Kudlinski A, Droques M, Szriftgiser P, Akhmediev N 2014 Phys. Rev. X 4 011054
[58]	Zhang H Q, Tian B, Xing L, Meng X H 2010 Physica A 389 367 Google Scholar
[59]	Matveev V B, Salli M A 1991 Darboux Transformations and Solitons, Vol. 5 (Berlin: Springer) pp7−15
[60]	谷超豪, 胡和生, 周子翔 2005 孤立子理论中的达布变换及其几何应用(第二版)(上海: 上海科学技术出版社)第18−24页 Gu C H, Hu H S, Zhou Z X 2005 Darboux Transformation in Soliton Theory and Its Geometric Applications (Shanghai: Scientific and Technical Publishers) pp18−24 (in Chinese).
[61]	Li Z D, Wu X, Li Q Y, He P B 2016 Chin. Phys. B 25 010507 Google Scholar
[62]	Li Z D, Huo C Z, Li Q Y, He P B, Xu T F 2018 Chin. Phys. B 27 040505 Google Scholar
[63]	Ho T L 1998 Phys. Rev. Lett. 81 742; Law C K, Ohmi T, Machida K, 1998 J. Phys. Soc. Jpn. 67 1822; Law C K, Pu H, Bigelow N P 1998 Phys. Rev. Lett. 81 5257
[64]	Pu H, Zhang W P, Meystre P 2001 Phys. Rev. Lett. 87 140405 Google Scholar
[65]	Li Z D, He P B, Li L, Liang J Q, Liu W M 2005 Phys. Rev. A 71 053611 Google Scholar

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Vol. 69, Issue 1 - 2020-01-05

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