Depinning of domain walls with an internal degree of freedom

Physical Review B - Condensed Matter and Materials Physics

Volumn 80, Issue 5, 2009, Pages

  Lecomte, V ^a   Barnes, S E ^a,b   Eckmann, J P ^a   Giamarchi, T ^a  

^a UNIVERSITY OF GENEVA   (Switzerland)

^b UNIVERSITY OF MIAMI   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 70249111813     PISSN: 10980121     EISSN: 1550235X     Source Type: Journal    
DOI: 10.1103/PhysRevB.80.054413     Document Type: Article

References (46)

1. S. Lemerle, J. Ferré, C. Chappert, V. Mathet, T. Giamarchi, and P. Le Doussal, Phys. Rev. Lett. 80, 849 (1998). 10.1103/PhysRevLett.80.849
2. M. Bauer, A. Mougin, J. P. Jamet, V. Repain, J. Ferré, R. L. Stamps, H. Bernas, and C. Chappert, Phys. Rev. Lett. 94, 207211 (2005). 10.1103/PhysRevLett.94.207211
3. M. Yamanouchi, D. Chiba, F. Matsukura, T. Dietl, and H. Ohno, Phys. Rev. Lett. 96, 096601 (2006). 10.1103/PhysRevLett.96.096601
4. P. J. Metaxas, J. P. Jamet, A. Mougin, M. Cormier, J. Ferré, V. Baltz, B. Rodmacq, B. Dieny, and R. L. Stamps, Phys. Rev. Lett. 99, 217208 (2007). 10.1103/PhysRevLett.99.217208
5. P. Paruch, T. Giamarchi, and J.-M. Triscone, Phys. Rev. Lett. 94, 197601 (2005). 10.1103/PhysRevLett.94.197601
6. P. Paruch and J.-M. Triscone, Appl. Phys. Lett. 88, 162907 (2006). 10.1063/1.2196482
7. A.-L. Barabasi and H. E. Stanley, Fractal Concepts in Surface Growth (Cambridge University Press, Cambridge, 1995).
8. J. Krim and G. Palasantzas, Int. J. Mod. Phys. B 9, 599 (1995). 10.1142/S0217979295000239
9. S. Moulinet, A. Rosso, W. Krauth, and E. Rolley, Phys. Rev. E 69, 035103 (R) (2004). 10.1103/PhysRevE.69.035103
10. M. Kardar, Phys. Rep. 301, 85 (1998). 10.1016/S0370-1573(98)00007-6
11. T. Giamarchi and P. Le Doussal, Statics and Dynamics of Disordered Elastic Systems (World Scientific, Singapore, 1998), p. 321
12. Spin Glasses and Random Fields, edited by, A. P. Young, (World Scientific, Singapore, 1998).
13. M. Yamanouchi, J. Ieda, F. Matsukura, S. E. Barnes, S. Maekawa, and H. Ohno, Science 317, 1726 (2007). 10.1126/science.1145516
14. S. S. P. Parkin, M. Hayashi, and L. Thomas, Science 320, 190 (2008). 10.1126/science.1145799
15. A. Yamaguchi, T. Ono, S. Nasu, K. Miyake, K. Mibu, and T. Shinjo, Phys. Rev. Lett. 92, 077205 (2004). 10.1103/PhysRevLett.92.077205
16. L. Thomas, M. Hayashi, X. Jiang, R. Moriya, C. Rettner, and S. S. P. Parkin, Nature (London) 443, 197 (2006). 10.1038/nature05093
17. G. S. D. Beach, Corneliu Nistor, Carl Knutson, Maxim Tsoi, and James L. Erskine, Nature Mater. 4, 741 (2005). 10.1038/nmat1477
18. N. L. Schryer and L. R. Walker, J. Appl. Phys. 45, 5406 (1974). 10.1063/1.1663252
19. L. D. Landau, E. M. Lifshitz, and L. Pitaevskii, Electrodynamics of Continuous Media (Pergamon, Oxford, 1960).
20. A. P. Malozemoff and J. C. Slonczewski, Magnetic Domain Walls in Bubble Materials (Academic, New York, 1979).
21. J. C. Slonczewski, J. Magn. Magn. Mater. 159, L1 (1996). 10.1016/0304-8853(96)00062-5
22. G. Tatara and H. Kohno, Phys. Rev. Lett. 92, 086601 (2004). 10.1103/PhysRevLett.92.086601
23. S. E. Barnes and S. Maekawa, Phys. Rev. Lett. 95, 107204 (2005). 10.1103/PhysRevLett.95.107204
24. R. A. Duine, A. S. Núñez, and A. H. MacDonald, Phys. Rev. Lett. 98, 056605 (2007). 10.1103/PhysRevLett.98.056605
25. R. A. Duine and C. M. Smith, Phys. Rev. B 77, 094434 (2008). 10.1103/PhysRevB.77.094434
26. V. Lecomte, S. E. Barnes, J.-P. Eckmann, and T. Giamarchi (unpublished).
27. P. H. J. Guckenheimer, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields (Springer-Verlag, New York, 2002).
28. P. Le Doussal, M. C. Marchetti, and K. J. Wiese, Phys. Rev. B 78, 224201 (2008). 10.1103/PhysRevB.78.224201
29. H. D. Vollmer and H. Risken, Z. Phys. B 37, 343 (1980). 10.1007/BF01352745
30. H. Risken, The Fokker-Planck equation (Springer-Verlag, Berlin, 1996).
31. H. Freidlin and D. Wentzell, Random Perturbations of Dynamical Systems (Springer-Verlag, Berlin, 1984).
32. J.-P. Eckmann, L. Thomas, and P. Wittwer, J. Phys. A 14, 3153 (1981). 10.1088/0305-4470/14/12/013
33. S. E. Barnes and S. Maekawa, Phys. Rev. Lett. 98, 246601 (2007). 10.1103/PhysRevLett.98.246601
34. W. F. Brown, Phys. Rev. 130, 1677 (1963). 10.1103/PhysRev.130.1677
35. G. Tatara, H. Kohno, and J. Shibata, J. Phys. Soc. Jpn. 77, 031003 (2008). 10.1143/JPSJ.77.031003
36. B. Derrida, J. Stat. Phys. 31, 433 (1983). 10.1007/BF01019492
37. P. Le Doussal and V. M. Vinokur, Physica C 254, 63 (1995). 10.1016/0921-4534(95)00545-5
38. Although this coupling is well known in the magnetic DW community (Ref.), it has to our knowledge always been discarded in interface physics.
39. vs =P js / (e ρs) with P as the current polarization, e as the carrier charge, and ρs as the spin density.
40. It has zero mean and variance (Ref.): η i (x,t) η j (x′, t′) =2 -1 α kB Tδ (x′ -x) δ (t′ -t) δij.
41. Bloch DWs are treated similarly (Ref.).
42. We also translated φ→φ+ π 2 for convenience.
43. For T=0 and js =0, and using the scheme proposed in Ref., we have solved numerically (Ref.) the bulk Eq. 1 with V (r) = 1 κ sinκr, and we checked as in Ref. that the ansatz gives a good description of solution although it is no longer exact.
44. For an overdamped particle in an arbitrary potential (Refs.), taking V (r) random yields r (t) ∼ tγ and v (f) is not finite in general, while taking V (r) periodic, while being a crude model, leads to the standard depinning of the elastic model, with different exponents β= 1 2 and ψ= 1 3.
45. This is at variance with the thermal rounding of the v∼ (f- fc) β law, which develops no crossings for T>0.
46. In simulations we discretize Eqs. 2 3 in time (> 106 steps with dt=5× 10-3) and average over 2048 realizations. In both cases, the DW width was taken to λ=15nm (Ref.) and the temperature to T=300K.