Mechanically driven domain wall movement in magnetoelastic nanomagnets

  • Théo Mathurin
  • Stefano Giordano
  • Yannick Dusch
  • Nicolas Tiercelin
  • Philippe Pernod
  • Vladimir Preobrazhensky
Regular Article

Abstract

Magnetic domain walls are fundamental objects arising in ferromagnetic materials, largely investigated both through micromagnetic simulations and experiments. While current- and field-based techniques for inducing domain wall propagation have been widely studied for fundamental understanding and application-oriented purposes, the possibility to manipulate domain walls using mechanical stress in magnetoelastic materials has only recently drawn interest. Here, a complete analytical model describing stress-induced transverse domain wall movement in ferromagnetic nanostripe with variable cross-section is presented. This approach yields a nonlinear integro-differential equation describing the magnetization field. Its numerical implementation, based on the nonlinear relaxation method, demonstrates the possibility to precisely control the position of a domain wall through mechanical action.

Keywords

Solid State and Materials 

References

  1. 1.
    D.A. Allwood, G. Xiong, C. Faulkner, D. Atkinson, D. Petit, R.P. Cowburn, Science, 309, 1688 (2005)ADSCrossRefGoogle Scholar
  2. 2.
    S.S. Parkin, M. Hayashi, L. Thomas, Science 320, 190 (2008)ADSCrossRefGoogle Scholar
  3. 3.
    T. Ono, H. Miyajima, K. Shigeto, K. Mibu, N. Hosoito, T. Shinjo, Science 284, 468 (1999)ADSCrossRefGoogle Scholar
  4. 4.
    D. Atkinson, D.A. Allwood, G. Xiong, M.D. Cooke, C.C. Faulkner, R.P. Cowburn, Nat. Mater. 2, 85 (2003)ADSCrossRefGoogle Scholar
  5. 5.
    G.S.D. Beach, C. Nistor, C. Knutson, M. Tsoi, J.L. Erskine, Nat. Mater. 4, 741 (2005)ADSCrossRefGoogle Scholar
  6. 6.
    M. Hayashi, L. Thomas, C. Rettner, R. Moriya, Y.B. Bazaliy, S.S.P. Parkin, Phys. Rev. Lett. 98, 037204 (2007)ADSCrossRefGoogle Scholar
  7. 7.
    D. Ravelosona, S. Mangin, J.A. Katine, E.E. Fullerton, B.D. Terris, Appl. Phys. Lett. 90, 072508 (2007)ADSCrossRefGoogle Scholar
  8. 8.
    A.V. Khvalkovskiy, V. Cros, D. Apalkov, V. Nikitin, M. Krounbi, K.A. Zvezdin, A. Anane, J. Grollier, A. Fert, Phys. Rev. B 87, 020402 (2013)ADSCrossRefGoogle Scholar
  9. 9.
    P.N. Skirdkov, K.A. Zvezdin, A.D. Belanovsky, J. Grollier, V. Cros, C.A. Ross, A.K. Zvezdin, Appl. Phys. Lett. 104, 242401 (2014)ADSCrossRefGoogle Scholar
  10. 10.
    J. Dean, M.T. Bryan, T. Schrefl, D.A. Allwood, J. Appl. Phys. 109, 023915 (2011)ADSCrossRefGoogle Scholar
  11. 11.
    M.T. Bryan, J. Dean, D.A. Allwood, Phys. Rev. B 85, 144411 (2012)ADSCrossRefGoogle Scholar
  12. 12.
    N. Lei, T. Devolder, G. Agnus, P. Aubert, L. Daniel, J.-V. Kim, W. Zhao, T. Trypiniotis, R.P. Cowburn, C. Chappert, D. Ravelosona, P. Lecoeur, Nat. Commun. 4, 1378 (2013)ADSCrossRefGoogle Scholar
  13. 13.
    B. Van de Wiele, L. Laurson, K.J.A. Franke, S. van Dijken, Appl. Phys. Lett. 104, 012401 (2014)ADSCrossRefGoogle Scholar
  14. 14.
    K.J.A. Franke, B. Van de Wiele, Y. Shirahata, S.J. Hämäläinen, T. Taniyama, S. van Dijken, Phys. Rev. X 5, 011010 (2015)Google Scholar
  15. 15.
    R. Tolley, T. Liu, Y. Xu, S. Le Gall, M. Gottwald, T. Hauet, M. Hehn, F. Montaigne, E.E. Fullerton, S. Mangin, Appl. Phys. Lett. 106, 242403 (2015)ADSCrossRefGoogle Scholar
  16. 16.
    E. De Ranieri, P.E. Roy, D. Fang, E.K. Vehsthedt, A.C. Irvine, D. Heiss, A. Casiraghi, R.P. Campion, B.L. Gallagher, T. Jungwirth, J. Wunderlich, Nat. Mater. 12, 808 (2013)ADSCrossRefGoogle Scholar
  17. 17.
    G. Catalan, J. Seidel, R. Ramesh, J.F. Scott, Rev. Mod. Phys. 84, 119 (2012)ADSCrossRefGoogle Scholar
  18. 18.
    M. Sharad, C. Augustine, G. Panagopoulos, K. Roy, IEEE Trans. Nanotechnol. 11, 843 (2012)ADSCrossRefGoogle Scholar
  19. 19.
    B. Behin-Aein, D. Datta, S. Salahuddin, S. Datta, Nat. Nanotechnol. 5, 266 (2010)ADSCrossRefGoogle Scholar
  20. 20.
    S. Parkin, S.-H. Yang, Nat. Nanotechnol. 10, 195 (2015)ADSCrossRefGoogle Scholar
  21. 21.
    N. Locatelli, V. Cros, J. Grollier, Nat. Mater. 13, 11 (2014)ADSCrossRefGoogle Scholar
  22. 22.
    H. Sohn, M.E. Nowakowski, C.Y. Liang, J.L. Hockel, K. Wetzlar, S. Keller, B.M. McLellan, M.A. Marcus, A. Doran, A. Young, M. Kläui, G.P. Carman, J. Bokor, R.N. Candler, ACS Nano 9, 4814 (2015)CrossRefGoogle Scholar
  23. 23.
    M. Kruzík, A. Prohl, SIAM Rev. 48, 439 (2006)ADSMathSciNetCrossRefGoogle Scholar
  24. 24.
    T. Mathurin, S. Giordano, Y. Dusch, N. Tiercelin, P. Pernod, V. Preobrazhensky, Appl. Phys. Lett. 108, 082401 (2016)ADSCrossRefGoogle Scholar
  25. 25.
    C.W. Nan, M.I. Bichurin, S. Dong, D. Viehland, G. Srinivasan, J. Appl. Phys. 103, 031101 (2008)ADSCrossRefGoogle Scholar
  26. 26.
    S. Giordano, M. Goueygou, N. Tiercelin, A. Talbi, P. Pernod, V. Preobrazhensky, Int. J. Eng. Sci. 78, 134 (2014)MathSciNetCrossRefGoogle Scholar
  27. 27.
    S.-T. Gu, Q.-C. He, Philos. Mag. 95, 2793, (2015)ADSCrossRefGoogle Scholar
  28. 28.
    S. Giordano, Mech. Res. Comm. 55, 18 (2014)CrossRefGoogle Scholar
  29. 29.
    N. Tiercelin, Y. Dusch, V. Preobrazhensky, P. Pernod, J. Appl. Phys. 109, 07D726 (2011)CrossRefGoogle Scholar
  30. 30.
    N. Tiercelin, Y. Dusch, A. Klimov, S. Giordano, V. Preobrazhensky, P. Pernod, Appl. Phys. Lett. 99, 192507 (2011)ADSCrossRefGoogle Scholar
  31. 31.
    Y. Dusch, N. Tiercelin, A. Klimov, S. Giordano, V. Preobrazhensky, P. Pernod, J. Appl. Phys. 113, 17C719 (2013)CrossRefGoogle Scholar
  32. 32.
    Y. Dusch, V. Rudenko, N. Tiercelin, S. Giordano, V. Preobrazhensky, P. Pernod, Nanomater. Nanostruct. 2, 44 (2012)Google Scholar
  33. 33.
    A.K. Biswas, S. Bandyopadhyay, J. Atulasimha, Appl. Phys. Lett. 104, 232403 (2014)ADSCrossRefGoogle Scholar
  34. 34.
    S. Giordano, Y. Dusch, N. Tiercelin, P. Pernod, V. Preobrazhensky, Phys. Rev. B 85, 155321 (2012)ADSCrossRefGoogle Scholar
  35. 35.
    S. Giordano, Y. Dusch, N. Tiercelin, P. Pernod, V. Preobrazhensky, J. Phys. D 46, 325002 (2013)CrossRefGoogle Scholar
  36. 36.
    N. Tiercelin, Y. Dusch, S. Giordano, A. Klimov, V. Preobrazhensky, P. Pernod, Strain Mediated Magnetoelectric Memory, in Nanomagnetic and Spintronic Devices for Energy-Efficient Memory and Computing, edited by S. Bandyopadhyay, J. Atulasimha (John Wiley & Sons Ltd., 2016)Google Scholar
  37. 37.
    K. Roy, S. Bandyopadhyay, J. Atulasimha, J. Appl. Phys. 112, 023914 (2012)ADSCrossRefGoogle Scholar
  38. 38.
    H. Ahmad, J. Atulasimha, S. Bandyopadhyay, Sci. Rep. 5, 18264 (2015)ADSCrossRefGoogle Scholar
  39. 39.
    L.D. Landau, E.M. Lifshitz, Electrodynamics of Continuous Media (Pergamon Press, London, 1984)Google Scholar
  40. 40.
    W.F. Brown, Micromagnetics (Interscience Publisher, New York, 1963)Google Scholar
  41. 41.
    W.F. Brown, Magnetoelastic Interactions (Springer-Verlag, Berlin, 1966)Google Scholar
  42. 42.
    J.A. Stratton, Electromagnetic theory (Mc Graw Hill, New York, 1941)Google Scholar
  43. 43.
    C. Miehe, G. Ethiraj, Comput. Methods Appl. Mech. Eng. 245, 331 (2012)ADSMathSciNetCrossRefGoogle Scholar
  44. 44.
    G. Bertotti, I. Mayergoyz, C. Serpico, Nonlinear Magnetization Dynamic in Nanosystems (Elsevier, Oxford, 2000)Google Scholar
  45. 45.
    N.L. Schryer, L.R. Walker, J. Appl. Phys. 45, 5406 (1974)ADSCrossRefGoogle Scholar
  46. 46.
    Y. Dusch, N. Tiercelin, A. Klimov, V. Rudenko, Y. Ignatov, S. Hage-Ali, P. Pernod, V. Preobrazhensky, J. Appl. Phys. 109, 07A720 (2011)CrossRefGoogle Scholar
  47. 47.
    P. Gaunt, Philos. Mag. 48, 261 (1983)ADSCrossRefGoogle Scholar
  48. 48.
    T. Rojac, M. Kosec, B. Budic, N. Setter, D. Damjanovic, J. Appl. Phys. 108, 074107 (2010)ADSCrossRefGoogle Scholar
  49. 49.
    D.I. Paul, J. Appl. Phys. 53, 1649 (1982)ADSCrossRefGoogle Scholar
  50. 50.
    J.P. Attané, Y. Samson, A. Marty, D. Halley, C. Beigné, Appl. Phys. Lett. 79, 794 (2001)ADSCrossRefGoogle Scholar
  51. 51.
    M. Kläui, C.A.F. Vaz, J. Rothman, J.A.C. Bland, W. Wernsdorfer, G. Faini, E. Cambril, Phys. Rev. Lett. 90, 097202 (2003)ADSCrossRefGoogle Scholar
  52. 52.
    D. Petit, A.-V. Jausovec, D. Read, R.P. Cowburn, J. Appl. Phys. 103, 114307 (2008)ADSCrossRefGoogle Scholar
  53. 53.
    M. Kläui, H. Ehrke, U. Rüdiger, T. Kasama, R.E. Dunin-Borkowski, D. Backes, L.J. Heyderman, C.A.F. Vaz, J.A.C. Bland, G. Faini, E. Cambril, W. Wernsdorfer, Appl. Phys. Lett. 87, 102509 (2005)ADSCrossRefGoogle Scholar
  54. 54.
    A.N. Kolmogorov, S.V. Fomin, Elements of the Theory of Functions and Functional Analysis (Dover, New York, 1999)Google Scholar
  55. 55.
    D.B. Gopman, J.W. Lau, K.P. Mohanchandra, K. Wetzlar, G.P. Carman, Phys. Rev. B 93, 064425 (2016)ADSCrossRefGoogle Scholar
  56. 56.
    Y.Y. Huang, Y.M. Jin, Appl. Phys. Lett. 93, 142504 (2008)ADSCrossRefGoogle Scholar
  57. 57.
    H.B. Keller, Numerical Solution of Two Point Boundary Value Problems (SIAM, Philadelphia, 1976)Google Scholar
  58. 58.
    P. Xu, K. Xia, C. Gu, L. Tang, H. Yang, J. Li, Nat. Nanotechnol. 3, 97 (2008)ADSCrossRefGoogle Scholar
  59. 59.
    L. Landau, E. Lifshitz, Phys. Zeitsch. der Sow. 8, 153 (1935)Google Scholar
  60. 60.
    T.L. Gilbert, Phys. Rev. 100, 1243 (1955) (abstract only)Google Scholar
  61. 61.
    T.L. Gilbert, IEEE Trans. Mag. 40, 3443 (2004)ADSCrossRefGoogle Scholar
  62. 62.
    S. Giordano, Y. Dusch, N. Tiercelin, P. Pernod, V. Preobrazhensky, Eur. Phys. J. B 86, 249 (2013)ADSCrossRefGoogle Scholar
  63. 63.
    R. Ravaud, G. Lemarquand, Prog. Electromagn. Res. B 98, 207 (2009)CrossRefGoogle Scholar
  64. 64.
    I.S. Gradshteyn, I.M. Ryzhik, Table of Integrals, Series and Products (Academic Press, San Diego, 1965)Google Scholar
  65. 65.
    M. Abramowitz, I.A. Stegun, Handbook of Mathematical Functions (Dover Publication, New York, 1970)Google Scholar
  66. 66.
    F.W.J. Olver, D.W. Lozier, R.F. Boisvert, C.W. Clark, NIST Handbook of Mathematical Functions (National Institute of Standards and Technology and Cambridge University Press, New York, 2010)Google Scholar

Copyright information

© EDP Sciences, SIF, Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  • Théo Mathurin
    • 1
  • Stefano Giordano
    • 1
  • Yannick Dusch
    • 1
  • Nicolas Tiercelin
    • 1
  • Philippe Pernod
    • 1
  • Vladimir Preobrazhensky
    • 1
    • 2
  1. 1.International Associated Laboratory LEMAC/LICS: IEMN, UMR CNRS 8520, ComUE Lille Nord de France, ECLilleVilleneuve d’AscqFrance
  2. 2.Wave Research Center, Prokhorov General Physics Institute, Russian Academy of ScienceMoscowRussia

Personalised recommendations