Bullets and droplets: Two-dimensional spin-wave solitons in modern magnonics (Review Article)

In this review we consider theoretical and experimental results related to the properties of two-dimensional spin-wave (SW) solitons, so-called SW bullets and SW droplets. Such nonlinear self-localized SW modes possess very interesting physical properties, and could have practical applications in modern and future magnonics and spintronics. The experimental and theoretical results presented in this review have undeniably proven the existence of SW bullets in magnetic films and confined magnetic nanostructures (magnetic nanocontacts), and have elucidated the essential distinctions between the properties of the one-dimensional nonlinear SW solitons, two-dimensional nonlinear SW bullets and linear spin wave packets, as well as the possibility of self-generation, parametrical excitation, and phase-conjugation of the SW bullets, similar to the case of the conventional linear spin waves. Also, in this review we presented experimental results demonstrating the nucleation, dynamics, and annihilation of two-dimensional strongly nonlinear SW “droplets” in spin-torque-driven magnetic nanocontacts. The properties of these exotic nonlinear objects are analyzed using recently developed theoretical models and illustrated by micromagnetic numerical simulations.

PACS: 75.30.Ds Spin waves;
PACS: 85.75.–d Magnetoelectronics; spintronics: devices exploiting spin polarized transport or integrated magnetic fields;
PACS: 72.25.–b Spin-polarized transport;
PACS: 73.50.Jt Galvanomagnetic and other magnetotransport effects (including thermomagnetic effects).

References

1. L. Landau and E. Lifshitz, Phys. Z. Sowjetunion 8, 153 (1935). Google Scholar

2. E. Zavoisky, J. Phys. USSR 9, 245 (1945). Google Scholar

3. J. H. E. Griffiths, Nature 158, 670 (1946). CrossRef Google Scholar

4. Ferromagnetic Resonance. The Phenomenon of Resonant Absorption of a High-Frequency Magnetic Field in Ferro-magnetic Substances, edited by S. V. Vonsovskii (Elsevier Ltd., Amsterdam, 1966). Google Scholar

5. A. G. Gurevich, Magnetic Resonance in Ferrites and Antiferromagnets (Nauka, Moscow, 1978) (in Russian). Google Scholar

6. A. G. Gurevich and G. A. Melkov, Magnetization Oscillations and Waves (CRC Press, New York, 1996). Google Scholar

7. A. I. Akhiezer, V. G. Bar'yakhtar, and S. V. Peletminskii, Spin Waves (Interscience (Wiley), New York, 1968). Google Scholar

8. V. S. L'vov, Nonlinear Spin Waves (Nauka, Moscow, 1987) (in Russian); V. S. L'vov, Wave Turbulence Under Parametric Excitation (Springer-Verlag, Berlin, 1994). Google Scholar

9. Spin Dynamics in Confined Magnetic Structures I–III, edited by B. Hillebrands (Springer-Verlag, Berlin, 2002), Vol. I; (2003), Vol. II; (2006), Vol. III. Google Scholar

10. D. D. Stancil and A. Prabhakar, Spin Waves: Theory and Applications (Springer-Verlag, New York, 2009). Google Scholar

11. Magnetism: Materials and Applications, edited by É. du Trémolet de Lacheisserie , D. Gignoux , and M. Schlenker (Springer-Verlag, Boston, 2005). Google Scholar

12. Magnetoelectronics of Microwaves and Extremely High Frequencies in Ferrite Films, edited by A. A. Ignatiev (Springer-Verlag, New York, 2009). Google Scholar

13. J. Scott Russell, in Report of Fourteenth Meeting of the British Association for the Advancement of Science (1844), p. 311. Google Scholar

14. N. J. Zabusky and M. D. Kruskal, Phys. Rev. Lett. 15, 240 (1965). CrossRef Google Scholar

15. C. S. Gardner, J. M. Greene, M. D. Kruskal, and R. M. Miura, Phys. Rev. Lett. 19, 1095 (1967). CrossRef Google Scholar

16. V. E. Zakharov and A. B. Shabat, Funkts. Anal. Prilozh. 13, 166 (1979). CrossRef Google Scholar

17. B. A. Ivanov and A. M. Kosevich, Sov. Phys. JETP 45, 1050 (1977). Google Scholar

18. A. M. Kosevich, B. A. Ivanov, and A. S. Kovalev, Fiz. Nizk. Temp. 3, 906 (1977)A. M. Kosevich , B. A. Ivanov , and A. S. Kovalev, [Sov. J. Low Temp. Phys. 3, 440 (1977)]. Google Scholar

19. A. M. Kosevich, B. A. Ivanov, and A. S. Kovalev, Physica D 1/2, 363 (1981). CrossRef Google Scholar

20. A. M. Kosevich, Fiz. Met. Metalloved. 53, 420 (1982). Google Scholar

21. A. M. Kosevich, B. A. lvanov, and A. S. Kovalev, Sov. Sci. Rev. Sect. A 6, 161 (1985). Google Scholar

22. A. M. Kosevich, B. A. Ivanov, and A. S. Kovalev, Phys. Rep. 194, 117 (1990). CrossRef Google Scholar

23. A. M. Kosevich, B. A. Ivanov, and A. S. Kovalev, in Nonlinear Waves (Nauka, Moscow, 1979), p. 45 (in Russian). Google Scholar

24. A. M. Kosevich, B. A. Ivanov, and A. S. Kovalev, in Nonlinear Waves of Magnetization. Dynamical and Topological Solitons (Naukova dumka, Kiev, 1983) (in Russian). Google Scholar

25. A. M. Kosevich, in Solitons, edited by S. E. Trullinger , V. E. Zakharov , and V. L. Pokrovsky (Elsevier, Amsterdam, 1986), p. 555. Google Scholar

26. A. Slavin and V. Tiberkevich, IEEE Trans. Magn. 45, 1875 (2009). CrossRef Google Scholar

27. A. N. Slavin, B. A. Kalinikos, and N. G. Kovshikov, in Nonlinear Phenomena and Chaos in Magnetic Materials, edited by P. E. Wigen (World Scientific, Singapore, 1994), Chap. 9. Google Scholar

28. N. N. Akhmediev and A. Ankiewicz, Solitons, Nonlinear Pulses and Beams (Chapman and Hall, London, 1997). Google Scholar

29. A. Slavin and V. Tiberkevich, Phys. Rev. Lett. 95, 237201 (2005). CrossRef Google Scholar

30. C. Boone, J. A. Katine, J. R. Childress, J. Zhu, X. Cheng, and I. N. Krivorotov, Phys. Rev. B 79, 140404 (2009). CrossRef Google Scholar

31. F. Sanches, V. Tiberkevich, K. Y. Guslienko, J. Sinha, M. Hayashi, O. Prokopenko, and A. N. Slavin, Phys. Rev. B 89, 140410 (2014). CrossRef Google Scholar

32. E. Grimaldi, A. Dussaux, P. Bortolotti, J. Grollier, G. Pillet, A. Fukushima, H. Kubota, K. Yakushiji, S. Yuasa, and V. Cros, Phys. Rev. B 89, 104404 (2014). CrossRef Google Scholar

33. S. Bonetti, V. Tiberkevich, G. Consolo, G. Finocchio, P. Muduli, F. Mancoff, A. Slavin, and J. Åkerman, Phys. Rev. Lett. 105, 217204 (2010). CrossRef Google Scholar

34. V. E. Demidov, S. Urazhdin, and S. O. Demokritov, Nat. Mater. 9, 984 (2010). CrossRef Google Scholar

35. O. Prokopenko, G. Melkov, E. Bankowski, T. Meitzler, V. Tiberkevich, and A. Slavin, Appl. Phys. Lett. 99, 032507 (2011). CrossRef Google Scholar

36. S. Bonetti, V. Puliafito, G. Consolo, V. S. Tiberkevich, A. N. Slavin, and J. Åkerman, Phys. Rev. B 85, 174427 (2012). CrossRef Google Scholar

37. O. V. Prokopenko, I. N. Krivorotov, E. Bankowski, T. Meitzler, S. Jaroch, V. S. Tiberkevich, and A. N. Slavin, J. Appl. Phys. 111, 123904 (2012). CrossRef Google Scholar

38. O. V. Prokopenko, I. N. Krivorotov, E. N. Bankowski, T. J. Meitzler, V. S. Tiberkevich, and A. N. Slavin, J. Appl. Phys. 114, 173904 (2013). CrossRef Google Scholar

39. O. V. Prokopenko, I. N. Krivorotov, T. J. Meitzler, E. Bankowski, V. S. Tiberkevich, and A. N. Slavin, in Magnonics: From Fundamentals to Applications. Topics in Applied Physics (Springer-Verlag, Berlin, 2013), Chap. 11, Vol. 125, p. 143. CrossRef Google Scholar

40. V. S. Tiberkevich, R. S. Khymyn, H. X. Tang, and A. N. Slavin, Sci. Rep. 4, 3873 (2014). CrossRef Google Scholar

41. L. Yang, R. Verba, V. Tiberkevich, T. Schneider, A. Smith, Z. Duan, B. Youngblood, K. Lenz, J. Lindner, A. N. Slavin, and I. N. Krivorotov, Sci. Rep. 5, 16942 (2015). CrossRef Google Scholar

42. O. V. Prokopenko, Ukr. J. Phys. 60, 104 (2015). CrossRef Google Scholar

43. O. V. Prokopenko and A. N. Slavin, Fiz. Nizk. Temp. 41, 457 (2015)O. V. Prokopenko and A. N. Slavin, [Low Temp. Phys. 41, 353 (2015)]. CrossRef Google Scholar

44. A. Houshang, E. Iacocca, P. Dürrenfeld1, S. R. Sani, J. Åkerman, and R. K. Dumas, Nat. Nanotechnol. 11, 280 (2016). CrossRef Google Scholar

45. O. R. Sulymenko and O. V. Prokopenko, in Nanophysics, Nanomaterials, Interface Studies, and Applications, Springer Proceedings in Physics (Springer-Verlag, Berlin, 2017), Chap. 12, Vol. 195, p. 157; O. R. Sulymenko and O. V. Prokopenko, J. Phys. Stud. 21, 4801 (2017). , Google Scholar Google Scholar

46. A. A. Awad, P. Dürrenfeld, A. Houshang, M. Dvornik, E. Iacocca, R. K. Dumas, and J. Åkerman, Nat. Phys. 13, 292 (2017). CrossRef Google Scholar

47. J. C. Slonczewski, J. Magn. Magn. Mater. 159, L1 (1996). CrossRef Google Scholar

48. L. Berger, Phys. Rev. B 54, 9353 (1996). CrossRef Google Scholar

49. M. J. Lighthill, IMA J. Appl. Math. 1, 269 (1965). CrossRef Google Scholar

50. V. P. Lukomski, Ukr. Fiz. Zh. 23, 134 (1978). Google Scholar

51. B. A. Kalinikos, N. G. Kovshikov, and A. N. Slavin, JETP Lett. 38, 413 (1983).CAS Google Scholar

52. A. K. Zvezdin and A. F. Popkov, Sov. Phys. JETP 57, 350 (1983).CAS Google Scholar

53. A. A. Serga, A. V. Chumak, and B. Hillebrands, J. Phys. D 43, 264002 (2010). CrossRef Google Scholar

54. B. A. Kalinikos, N. G. Kovshikov, and A. N. Slavin, Sov. Phys. JETP 67, 303 (1988). Google Scholar

55. B. A. Kalinikos, N. G. Kovshikov, and A. N. Slavin, Phys. Rev. B 42, 8658 (1990). CrossRef Google Scholar

56. B. A. Kalinikos, N. G. Kovshikov, and A. N. Slavin, IEEE Trans. Magn. 26, 1477 (1990). CrossRef Google Scholar

57. B. A. Kalinikos, N. G. Kovshikov, and A. N. Slavin, IEEE Trans. Magn. 28, 3207 (1992). CrossRef Google Scholar

58. M. Chen, M. A. Tsankov, J. M. Nash, and C. E. Patton, Phys. Rev. B 49, 12773 (1994). CrossRef Google Scholar

59. B. A. Kalinikos, N. G. Kovshikov, and A. N. Slavin, J. Appl. Phys. 67, 5633 (1990). CrossRef Google Scholar

60. A. N. Slavin and I. V. Rojdestvenski, IEEE Trans. Magn. 30, 37 (1994). CrossRef Google Scholar

61. A. D. Boardman, S. A. Nikitov, K. Xie, and H. Mehta, J. Magn. Magn. Mater. 145, 357 (1995). CrossRef Google Scholar

62. A. N. Slavin, Phys. Rev. Lett. 77, 4644 (1996). CrossRef Google Scholar

63. A. N. Slavin, Y. S. Kivshar, E. A. Ostrovskaya, and H. Benner, Phys. Rev. Lett. 82, 2583 (1999). CrossRef Google Scholar

64. A. V. Bagada, G. A. Melkov, A. A. Serga, and A. N. Slavin, Phys. Rev. Lett. 79, 2137 (1997). CrossRef Google Scholar

65. A. L. Gordon, G. A. Melkov, A. A. Serga, V. S. Tiberkevich, A. V. Bagada, and A. N. Slavin, JETP Lett. 67, 913 (1998). CrossRef Google Scholar

66. B. A. Kalinikos, N. G. Kovshikov, and C. E. Patton, Phys. Rev. Lett. 80, 4301 (1998). CrossRef Google Scholar

67. M. Wu, B. A. Kalinikos, and C. E. Patton, Phys. Rev. Lett. 93, 157207 (2004). CrossRef Google Scholar

68. B. A. Kalinikos, N. G. Kovshikov, M. P. Kostylev, P. Kabos, and C. E. Patton, JETP Lett. 66, 371 (1997). CrossRef Google Scholar

69. G. A. Melkov, Yu. V. Kobljanskij, A. A. Serga, V. S. Tiberkevich, and A. N. Slavin, J. Appl. Phys. 89, 6689 (2001). CrossRef Google Scholar

70. A. B. Ustinov, N. Yu. Grigor'eva, and B. A. Kalinikos, JETP Lett. 88, 31 (2008). CrossRef Google Scholar

71. S. O. Demokritov, B. Hillebrands, and A. N. Slavin, Phys. Rep. 348, 441 (2001). CrossRef Google Scholar

72. M. Remoissenet, Waves Called Solitons: Concepts and Experiments (Springer-Verlag, Berlin, 1999). CrossRef Google Scholar

73. A. Slavin and V. Tiberkevich, in Spin Wave Confinement, edited by S. O. Demokritov (Pan Stanford Publishing, Singapore, 2009), Chap. 6. Google Scholar

74. B. A. Kalinikos and A. B. Ustinov, in Recent Advances in Magnetic Insulators—From Spintronics to Microwave Applications, edited by M. Wu , A. Hoffmann , R. E. Camley , and R. L. Stamps (Elsevier, Amsterdam, 2013), Chap. 7. Google Scholar

75. Y. Silberberg, Opt. Lett. 15, 1282 (1990). CrossRef Google Scholar

76. L. Bergé, Phys. Rep. 303, 259 (1998). CrossRef Google Scholar

77. M. Bauer, O. Büttner, S. O. Demokritov, B. Hillebrands, V. Grimalsky, Yu. Rapoport, and A. N. Slavin, Phys. Rev. Lett. 81, 3769 (1998). CrossRef Google Scholar

78. N. Akhmediev and A. Ankiewicz, Dissipative Solitons: From Optics to Biology and Medicine (Berlin, Springer, 2008). Google Scholar

79. S. M. Mohseni, S. R. Sani, J. Persson, T. N. Anh Nguyen, S. Chung, Ye. Pogoryelov, P. K. Muduli, E. Iacocca, A. Eklund, R. K. Dumas, S. Bonetti, A. Deac, M. A. Hoefer, and J. Åkerman, Science 339, 1295 (2013). CrossRef Google Scholar

80. M. A. Hoefer, T. J. Silva, and M. W. Keller, Phys. Rev. B 82, 054432 (2010). CrossRef Google Scholar

81. M. A. Hoefer, M. Sommacal, and T. J. Silva, Phys. Rev. B 85, 214433 (2012). CrossRef Google Scholar

82. M. A. Hoefer and M. Sommacal, Physica D 241, 890 (2012). CrossRef Google Scholar

83. L. D. Bookman and M. A. Hoefer, Phys. Rev. B 88, 184401 (2013). CrossRef Google Scholar

84. M. D. Maiden, L. D. Bookman, and M. A. Hoefer, Phys. Rev. B 89, 180409 (2014). CrossRef Google Scholar

85. L. D. Bookman and M. A. Hoefer, Proc. R. Soc. A 471, 20150042 (2015). CrossRef Google Scholar

86. J. J. Rasmussen and K. Rypdal, Phys. Scr. 33, 481 (1986). CrossRef Google Scholar

87. M. V. Goldman, K. Rypdal, and B. Hafizi, Phys. Fluids 23, 945 (1980). CrossRef Google Scholar

88. O. Büttner, M. Bauer, S. O. Demokritov, B. Hillebrands, M. P. Kostylev, B. A. Kalinikos, and A. N. Slavin, Phys. Rev. Lett. 82, 4320 (1999). CrossRef Google Scholar

89. A. N. Slavin, O. Büttner, M. Bauer, S. O. Demokritov, B. Hillebrands, M. M. Kostylev, B. A. Kalinikos, V. V. Grimalsky, and Y. Rapoport, Chaos 13, 693 (2003). CrossRef Google Scholar

90. A. A. Serga, S. O. Demokritov, B. Hillebrands, and A. N. Slavin, Phys. Rev. Lett. 92, 117203 (2004). CrossRef Google Scholar

91. A. A. Serga, B. Hillebrands, S. O. Demokritov, A. N. Slavin, P. Wierzbicki, V. Vasyuchka, O. Dzyapko, and A. Chumak, Phys. Rev. Lett. 94, 167202 (2005). CrossRef Google Scholar

92. G. A. Melkov, A. A. Serga, V. S. Tiberkevich, A. N. Oliynyk, and A. N. Slavin, Phys. Rev. Lett. 84, 3438 (2000). CrossRef Google Scholar

93. G. A. Melkov, A. A. Serga, A. N. Slavin, V. S. Tiberkevich, A. N. Oleinik, and A. V. Bagada, JETP 89, 1189 (1999). CrossRef Google Scholar

94. V. I. Vasyuchka, A. A. Serga, C. W. Sandweg, D. V. Slobodianiuk, G. A. Melkov, and B. Hillebrands, Phys. Rev. Lett. 111, 187206 (2013). CrossRef Google Scholar

95. A. A. Serga, M. P. Kostylev, and B. Hillebrands, Phys. Rev. Lett. 101, 137204 (2008). CrossRef Google Scholar

96. M. P. Kostylev, A. A. Serga, and B. Hillebrands, Phys. Rev. Lett. 106, 134101 (2011). CrossRef Google Scholar

97. V. Veerakumar and R. E. Camley, Phys. Rev. B 74, 214401 (2006). CrossRef Google Scholar

98. V. E. Demidov, S. O. Demokritov, D. Birt, B. O'Gorman, M. Tsoi, and X. Li, Phys. Rev. B 80, 014429 (2009). CrossRef Google Scholar

99. T. Schneider, A. A. Serga, A. V. Chumak, C. W. Sandweg, S. Trudel, S. Wolff, M. P. Kostylev, V. S. Tiberkevich, A. N. Slavin, and B. Hillebrands, Phys. Rev. Lett. 104, 197203 (2010). CrossRef Google Scholar

100. T. Sebastian, T. Brächer, P. Pirro, A. A. Serga, B. Hillebrands, T. Kubota, H. Naganuma, M. Oogane, and Y. Ando, Phys. Rev. Lett. 110, 067201 (2013). CrossRef Google Scholar

101. F. Heussner, A. A. Serga, T. Brächer, B. Hillebrands, and P. Pirro, Appl. Phys. Lett. 111, 122401 (2017). CrossRef Google Scholar

102. M. Tsoi, A. G. M. Jansen, J. Bass, W.-C. Chiang, M. Seck, V. Tsoi, and P. Wyder, Phys. Rev. Lett. 80, 4281 (1998). CrossRef Google Scholar

103. M. Tsoi, A. G. M. Jansen, J. Bass, W.-C. Chiang, V. Tsoi, and P. Wyder, Nature 406, 46 (2000). CrossRef Google Scholar

104. S. I. Kiselev, J. C. Sankey, I. N. Krivorotov, N. C. Emley, R. J. Schoelkopf, R. A. Buhrman, and D. C. Ralph, Nature 425, 380 (2003). CrossRef Google Scholar

105. W. H. Rippard, M. R. Pufall, and T. J. Silva, Appl. Phys. Lett. 82, 1260 (2003). CrossRef Google Scholar

106. K. J. Lee, A. Deac, O. Redon, J. P. Nozières, and B. Dieny, Nat. Mater. 3, 877 (2004). CrossRef Google Scholar

107. W. H. Rippard, M. R. Pufall, S. Kaka, T. J. Silva, and S. E. Russek, Phys. Rev. B 70, 100406(R) (2004). CrossRef Google Scholar

108. W. H. Rippard, M. R. Pufall, S. Kaka, S. E. Russek, and T. J. Silva, Phys. Rev. Lett. 92, 027201 (2004). CrossRef Google Scholar

109. I. N. Krivorotov, N. C. Emley, J. C. Sankey, S. I. Kiselev, D. C. Ralph, and R. A. Buhrman, Science 307, 228 (2005). CrossRef Google Scholar

110. J. C. Slonczewski, J. Magn. Magn. Mater. 195, 261 (1999). CrossRef Google Scholar

111. S. M. Rezende, F. M. de Aguiar, and A. Azevedo, Phys. Rev. Lett. 94, 037202 (2005). CrossRef Google Scholar

112. A. N. Slavin and P. Kabos, IEEE Trans. Magn. 41, 1264 (2005). CrossRef Google Scholar

113. G. Bertotti, C. Serpico, I. D. Mayergoyz, A. Magni, M. d'Aquino, and R. Bonin, Phys. Rev. Lett. 94, 127206 (2005). CrossRef Google Scholar

114. G. Consolo, B. Azzerboni, G. Gerhart, G. A. Melkov, V. Tiberkevich, and A. N. Slavin, Phys. Rev. B 76, 144410 (2007). CrossRef Google Scholar

115. G. Gerhart, E. Bankowski, G. A. Melkov, V. S. Tiberkevich, and A. N. Slavin, Phys. Rev. B 76, 024437 (2007). CrossRef Google Scholar

116. G. Consolo, B. Azzerboni, L. Lopez-Diaz, G. Gerhart, E. Bankowski, V. Tiberkevich, and A. N. Slavin, Phys. Rev. B 78, 014420 (2008). CrossRef Google Scholar

117. V. Puliafito, G. Siracusano, B. Azzerboni, and G. Finocchio, IEEE Magn. Lett. 5, 3000104 (2014). CrossRef Google Scholar

118. D. Backes, F. Maciá, S. Bonetti, R. Kukreja, H. Ohldag, and A. D. Kent, Phys. Rev. Lett. 115, 127205 (2015). CrossRef Google Scholar

119. V. E. Demidov, H. Ulrichs, S. V. Gurevich, S. O. Demokritov, V. S. Tiberkevich, A. N. Slavin, A. Zholud, and S. Urazhdin, Nat. Commun. 5, 3179 (2014). CrossRef Google Scholar

120. M. B. Jungfleisch, W. Zhang, J. Sklenar, J. Ding, W. Jiang, H. Chang, F. Y. Fradin, J. E. Pearson, J. B. Ketterson, V. Novosad, M. Wu, and A. Hoffmann, Phys. Rev. Lett. 116, 057601 (2016). CrossRef Google Scholar

121. S. Urazhdin, V. E. Demidov, R. Cao, B. Divinskiy, V. Tyberkevych, A. Slavin, A. B. Rinkevich, and S. O. Demokritov, Appl. Phys. Lett. 109, 162402 (2016) CrossRef Google Scholar

122. B. Georges, J. Grollier, M. Darques, V. Cros, C. Deranlot, B. Marcilhac, G. Faini, and A. Fert, Phys. Rev. Lett. 101, 017201 (2008). CrossRef Google Scholar

123. P. Wills, E. E. Iacocca, and H. A. Hoefer, Phys. Rev. B 93, 144408 (2016). CrossRef Google Scholar

124. D. V. Slobodianiuk, O. V. Prokopenko, and G. A. Melkov, J. Magn. Magn. Mater. 439, 144 (2017) CrossRef Google Scholar

125. E. Iacocca, R. K. Dumas, L. Bookman, M. Mohseni, S. Chung, M. A. Hoefer, and J. Åkerman, Phys. Rev. Lett. 112, 047201 (2014). CrossRef Google Scholar

126. S. Chung, S. M. Mohseni, S. R. Sani, E. Iacocca, R. K. Dumas, T. N. Anh Nguyen, Ye. Pogoryelov, P. K. Muduli, A. Eklund, M. Hoefer, and J. Åkerman, J. Appl. Phys. 115, 172612 (2014). CrossRef Google Scholar

127. D. Xiao, V. Tiberkevich, Y. H. Liu, Y. W. Liu, S. M. Mohseni, S. Chung, M. Ahlberg, A. N. Slavin, J. Åkerman, and Y. Zhou, Phys. Rev. B 95, 024106 (2017). CrossRef Google Scholar

128. S. Lendínez, N. Statuto, D. Backes, A. D. Kent, and F. Macià, Phys. Rev. B 92, 174426 (2015). CrossRef Google Scholar

129. F. Macià, D. Backes, and A. D. Kent, Nat. Nanotechnol. 9, 992 (2014). CrossRef Google Scholar

130. S. Chung, S. M. Mohseni, A. Eklund, P. Dürrenfeld, M. Ranjbar, S. R. Sani, T. N. Anh Nguyen, R. K. Dumas, and J. Åkerman, Fiz. Nizk. Temp. 41, 1063 (2015)S. Chung , S. M. Mohseni , A. Eklund , P. Dürrenfeld , M. Ranjbar , S. R. Sani , T. N. Anh Nguyen , R. K. Dumas , and J. Åkerman, [Low Temp. Phys. 41, 833 (2015)].Scitation, ISI CrossRef Google Scholar

131. S. Chung, Q. T. Le, M. Ahlberg, M. Weigand, I. Bykova, A. A. Awad, H. Mazraati, A. Houshang, S. Jiang, T. N. Nguyen, E. Goering, G. Schütz, J. Gräfe, and J. Åkerman, e-print arXiv:1707.01595 cond-mat.mes-hall. CrossRef Google Scholar

132. M. A. Hoefer, T. J. Silva, and M. D. Stiles, Phys. Rev. B 77, 144401 (2008). CrossRef Google Scholar