The Landau–Lifshitz equation in atomistic models

Физика Низких Температур: Том 41, Выпуск 9 (Сентябрь 2015), c. 908-916 ( к оглавлению , назад )

The Landau–Lifshitz equation in atomistic models

M.O.A. Ellis¹, R.F.L. Evans¹, T.A. Ostler¹, J. Barker², U. Atxitia³, O. Chubykalo-Fesenko⁴, and R.W. Chantrell¹

¹Department of Physics, University of York, York, YO10 5DD, United Kingdom

²Institute for Materials Research, Tohoku University, Sendai, Miyagi 980-8577, Japan

³ Fachbereich Physik and Zukunftskolleg, Universität Konstanz, Konstanz D-78457, Germany

⁴Instituto de Ciencia de Materiales de Madrid, CSIC, Cantoblanco, Madrid, 28049 Spain
E-mail: me521@york.ac.uk

Received May 9, 2015

Abstract

The Landau–Lifshitz (LL) equation, originally proposed at the macrospin level, is increasingly used in Atomistic Spin Dynamic (ASD) models. These models are based on a spin Hamiltonian featuring atomic spins of fixed length, with the exchange introduced using the Heisenberg formalism. ASD models are proving a powerful approach to the fundamental understanding of ultrafast magnetization dynamics, including the prediction of the thermally induced magnetization switching phenomenon in which the magnetization is reversed using an ultra-fast laser pulse in the absence of an externally applied field. This paper outlines the ASD model approach and considers the role and limitations of the LL equation in this context.

PACS: 75. Magnetic properties and materials;
PACS: 75.10.Hk Classical spin models;
PACS: 75.78.–n Magnetization dynamics.

Key words: Landau–Lifshitz equation, atomistic spin models.

Published online: July 24, 2015

References

1. Monte Carlo Methods in Statistical Physics, edited by K. Binder (Springer-Verlag, Berlin, 1979), p. 204. Google Scholar

2. U. Nowak , R. W. Chantrell , and E. C. Kennedy , Phys. Rev. Lett. 84, 163 (2000). CrossRef Google Scholar

3. R. Koctema and A. Berkowitz , Phys. Rev. B 59, 6321 (1999). CrossRef Google Scholar

4. C. Mitsumata , A. Sakuma , and K. Fukamichi , Phys. Rev. B 68, 014437 (2003). CrossRef Google Scholar

5. U. Nowak , R. Wieser , O. N. Mryasov , K. Guslienko , and R. W. Chantrell , Phys. Rev. B 72, 172410 (2005). CrossRef Google Scholar

6. E. Boerner , O. Chubykalo-Fesenko , R. Chantrell , O. Heinonen , and O. Mryasov , IEEE Trans. Magn. 41, 936 (2005). CrossRef Google Scholar

7. B. Skubic , J. Hellsvik , L. Nordstrom , and O. Eriksson , J. Phys.: Condens. Matter 20, 315203 (2008). CrossRef Google Scholar

8. W. F. Brown, Jr. , Micromagnetics Interscience (New York, 1963). Google Scholar

9. G. Grinstein and R. H. Koch , Phys. Rev. Lett. 90, 207201 (2003). CrossRef Google Scholar

10. E. Martinez , L. Lopez-Diaz , L. Torres , and O. Alejos , IEEE Trans. Magn. 39, 2522 (2003). CrossRef Google Scholar

11. T. W. McDaniel , J. Phys.: Condens. Matter 17, R315 (2005). CrossRef Google Scholar

12. E. Rottmayer , S. Batra , D. Buechel , W. A. Challenger , J. Hohlfeld , Y. Kubota , L. Li , B. Lu , C. Mihalcea , K. Mountfield , K. Pelhos , C. Peng , T. Rausch , M. A. Siegler , D. Weller , and X. M. Yang , IEEE Trans. Magn. 42, 2417 (2006). CrossRef Google Scholar

13. K. Schwarz , P. Mohn , P. Blaha , and J. Kobler , J. Phys. F: Met. Phys. 14, 2659 (1984). CrossRef Google Scholar

14. R. F. L. Evans , W. J. Fan , P. Chureemart , T. A. Ostler , M. O. A. Ellis , and R. W. Chantrell , J. Phys.: Condens. Matter 26, 103202 (1984). Google Scholar

15. O. N. Mryasov , U. Nowak , K. Y. Guslienko , and R. W. Chantrell , Europhys. Lett. 69, 805 (2005). CrossRef Google Scholar

16. L. Szunyogh , L. Udvardi , J. Jackson , U. Nowak , and R. W. Chantrell , Phys. Rev. B 83, 024401 (2011). CrossRef Google Scholar

17. T. A. Ostler , R. Evans , R. W. Chantrell , U. Atxitia , O. Chubykalo-Fesenko , I. Radu , R. Abrudan , F. Radu , A. Tsukamoto , A. Itoh , A. Kirilyuk , Th. Rasing , and A. Kimel , Phys. Rev. B 84, 024407 (2011). CrossRef Google Scholar

18. R. F. L. Evans , D. Givord , R. Cuadrado , M. I. J. Probert , T. Shoji , A. Manabe , M. Yano , G. Hrkac , T. Schrefl , and R. W. Chantrell (submitted). Google Scholar

19. G. Kresse and J. Furthmuller , Phys. Rev. B 54, 11169 (1996). CrossRef Google Scholar

20. J. M. Soler , E. Artacho , J. D. Gale , A. GarcÃa , J. Junquera , P. OrdejÃ³n , and D. SÃ¡nchez Portal , J. Phys.: Condens. Matter 14, 2745 (2002). CrossRef Google Scholar

21. J. Zabloudil , R. Hammerling , L. Szunyogh , and P. Weinberger , Electron Scattering in Solid Matter (Springer, Berlin, 2005). Google Scholar

22. H. Ebert , D. KÃ¶dderitzsch , and D. MinÃ¡r , Rep. Prog. Phys. 74, 096501 (2011). CrossRef Google Scholar

23. M. I. Katsnelson and A. I. Lichtenstein , Phys. Rev. B 61, 8906 (2000). CrossRef Google Scholar

24. M. Pajda , J. Kudrnovsky , I. Turek , V. Drchal , and P. Bruno , Phys. Rev. B 64, 174402 (2001). CrossRef Google Scholar

25. A. Liecntenstein , M. Katsnelson , V. Antropov , and V. Gubanov , J. Magn. Magn. Mater. 67, 65 (1987). CrossRef Google Scholar

26. N. Kazantsev , D. Hinzke , R. W. Chantrell , U. Atxitia , and O. Chubykalo-Fesenko , Phys. Rev. B 77, 184428 (2008). CrossRef Google Scholar

27. O. N. Mryasov , U. Nowak , K. Y. Guslienko , and R. W. Chantrell , Europhys. Lett. 69, 805 (2005). CrossRef Google Scholar

28. M. O. A. Ellis and R. W. Chantrell , Appl. Phys. Lett. 106, 162407 (2015). CrossRef Google Scholar

29. O. Hovorka , S. Devos , Q. Coopman , W. J. Fan , C. J. Aas , R. F. L. Evans , X. Chen , G. Ju , and R. W. Chantrell , Appl. Phys. Lett. 101, 052406 (2012). CrossRef Google Scholar

30. A. Lyberatos , D. Weller , and G. J. Parker , J. Appl. Phys. 114, 233904 (2013). CrossRef Google Scholar

31. A. Lyberatos , D. Weller , G. J. Parker , and B. C. Stipe , J. Appl. Phys. 112, 113915 (2012). CrossRef Google Scholar

32. E. R. Callen and H. B. Callen , J. Phys. Chem. Solids 27, 1271 (1966). CrossRef Google Scholar

33. J.-U. Thiele , K. R. Coffey , M. F. Toney , J. A. Hedstrom , and A. J. Kellock , J. Appl. Phys. 91, 6595 (2002). CrossRef Google Scholar

34. S. Okamoto , N. Kikuchi , O. Kitacami , T. Miyazaki , Y. Shi-Mada , and K. Fukamichi , Phys. Rev. B 66, 24413 (2002). CrossRef Google Scholar

35. A. Lyberatos , D. V. Berkov , and R. W. Chantrell , J. Phys.: Condens. Matter 5, 8911 (1993). CrossRef Google Scholar

36. W. F. Brown, Jr. , Phys. Rev. 130, 1677 (1963). CrossRef Google Scholar

37. J. L. GarciÃ¡-Palacios and F. J. Lazaro , Phys. Rev. B 58, 14937 (1998). CrossRef Google Scholar

38. D. V. Berkov , IEEE Trans. Magn. 38, 2489 (2002). CrossRef Google Scholar

39. E. Beaurepaire , J.-C. Merle , A. Daunois , and J.-Y. Bigot , Phys. Rev. Lett. 76, 4250 (1996). CrossRef Google Scholar

40. C. D. Stanciu , F. Hansteen , A. V. Kimel , A. Kirilyuk , A. Tsukamoto , A. Itoh , and Th. Rasing , Phys. Rev. Lett. 99, 047601 (2007). CrossRef Google Scholar

41. K. Vahaplar , A. M. Kalashnikova , A. V. Kimel , D. Hinzke , U. Nowak , R. Chantrell , A. Tsukamoto , A. Itoh , A. Kirilyuk , and Th. Rasing , Phys. Rev. Lett. 103, 117201 (2009). CrossRef Google Scholar

42. P. Nieves , U. Atxitia , R. W. Chantrell , and O. Chubykalo-Fesenko , Fiz. Nizk. Temp. 41, 949 (2015) P. Nieves , U. Atxitia , R. W. Chantrell , and O. Chubykalo-Fesenko , [Low Temp. Phys. 41, 9 (2015)]. CrossRef Google Scholar

43. N. Kazantseva , D. Hinzke , R. W. Chantrell , and U. Nowak , Europhys. Lett. 86, 27006 (2009). CrossRef Google Scholar

44. C.-H. Lambert , S. Mangin , B. S. D. C. S. Varaprasad , Y. K. Takahashi , M. Hehn , M. Cinchetti , G. Malinowski , K. Hono , Y. Fainman , M. Aeschlimann , and E. E. Fullerton , Science 345, 1337 (2014). CrossRef Google Scholar

45. J. Chen , D. Tzou , and J. Beraun , Int. J. Heat Mass Transfer 49, 307â316 (2006). CrossRef Google Scholar

46. I. Radu , K. Vahaplar , C. Stamm , T. Kachel , N. Pontius , H. A. Drr , T. A. Ostler , J. Barker , R. F. L. Evans , R. W. Chantrell , A. Tsukamoto , A. Itoh , A. Kirilyuk , Th. Rasing , and A. V. Kimel , Nature 472, 205 (2011). CrossRef Google Scholar

47. T. A. Ostler , J. Barker , R. F. L. Evans , R. Chantrell , U. Atxitia , O. Chubykalo-Fesenko , S. El Moussaoui , L. Le Guyader , E. Mengotti , L. J. Heyderman , F. Nolting , A. Tsukamoto , A. Itoh , D. Afanasiev , B. A. Ivanov , A. M. Kalashnikova , K. Vahaplar , J. Mentink , A. Kirilyuk , Th. Rasing , and A. V. Kimel , Nat. Commun. 3, 666 (2012). CrossRef Google Scholar

48. A. R. Khorsand , M. Savoini , A. Kirilyuk , A. V. Kimel , A. Tsukamoto , A. Itoh , and Th. Rasing , Phys. Rev. Lett. 108, 127205 (2012). CrossRef Google Scholar

49. J. H. Mentinck , J. Hellsvik , D. V. Afanasiev , B. A. Ivanov , A. Kirilyuk , A. V. Kimel , O. Eriksson , M. I. Katsnelson , and Th. Rasing , Phys. Rev. Lett. 108, 057202 (2012). CrossRef Google Scholar

50. U. Atxitia , T. Ostler , R. F. L. Evans , R. W. Chantrell , and O. Chubykalo-Fesenko , Phys Rev. B 87, 224417 (2013). CrossRef Google Scholar

51. J. Barker , U. Atxitia , T. A. Ostler , O. Hovorka , O. Chubykalo-Fesenko , and R. W. Chantrell , Sci. Rep. 3, 3262 (2013). CrossRef Google Scholar

52. R. F. L. Evans , T. A. Ostler , R. W. Chantrell , I. Radu , and Th. Rasing , Appl. Phys. Lett. 104, 082410 (2014). CrossRef Google Scholar

53. R. F. L. Evans , R. W. Chantrell , U. Nowak , A. Lyberatos , and H.-J. Richter , Appl. Phys. Lett. 100, 102402 (2012). CrossRef Google Scholar

54. K. Capelle and B. L. Gyorffy , Europhys. Lett. 61, 354 (2003). CrossRef Google Scholar

55. K. Blum , Density Matrix Theory and Applications (Plenum Press, New York, 1981). Google Scholar

56. R. Wieser , Eur. Phys. J. B 88, 77 (2015). CrossRef Google Scholar

57. U. Atxitia , O. Chubykalo-Fesenko , R. W. Chantrell , U. Nowak , and A. Rebei , Phys. Rev. Lett. 102, 057203 (2009). CrossRef Google Scholar

58. K. Miyazaki and K. Seki , J. Chem. Phys. 108, 7052 (1998). CrossRef Google Scholar

59. J. M. Sancho and M. San Miguel , Phys. Rev. A 26, 1589 (1982). CrossRef Google Scholar

60. P.-W. Ma , C. Woo , and S. Dudarev , Phys. Rev. B 78, 024434 (2008). CrossRef Google Scholar

61. P.-W. Ma , S. Dudarev , and C. Woo , Phys. Rev. B 85, 184301 (2012). CrossRef Google Scholar

62. S. Karakurt , R. Chantrell , and U. Nowak , J. Magn. Magn. Mater. 316, 280 (2007). CrossRef Google Scholar

63. D. Beaujouan , P. Thibaudeau , and C. Barreteau , Phys. Rev. B 86, 174409 (2012). CrossRef Google Scholar

64. M. D. Kuzmin , Phys. Rev. Lett. 94, 107204 (2005). CrossRef Google Scholar

65. A. W. Sandvik and J. Kurkijarvi , Phys. Rev. B 43, 5950 (1991). CrossRef Google Scholar

66. R. F. L. Evans , U. Atxitia , and R. W. Chantrell , Phys. Rev. B 91, 144425 (2015). CrossRef Google Scholar