Gauge field formulation of adiabatic spin torques

Journal of the Physical Society of Japan

Volumn 76, Issue 6, 2007, Pages

  Kohno, Hiroshi ^a   Shibata, Junya ^b,c  

^a OSAKA UNIVERSITY   (Japan)

^b RIKEN   (Japan)

^c KANAGAWA INSTITUTE OF TECHNOLOGY   (Japan)

Author keywords

Adiabatic spin frame; Gauge field; Gilbert damping; Landau Lifshitz Gilbert equation; Magnetic impurity; s d exchange interaction; Spin current; Spin relaxation; Spin torque

Indexed keywords

EID: 34547466825     PISSN: 00319015     EISSN: 13474073     Source Type: Journal    
DOI: 10.1143/JPSJ.76.063710     Document Type: Article

References (58)

1. J. C. Slonczewski: J. Magn. Magn. Mater. 159 (1996) L1.
2. L. Berger: Phys. Rev. B 54 (1996) 9353.
3. Y. Tserkovnyak et al.: Rev. Mod. Phys. 77 (2005) 1375.
4. L. Berger: J. Appl. Phys. 55 (1984) 1954;
5. L. Berger: J. Appl. Phys. 71 (1992) 2721.
6. A. Yamaguchi et al.: Phys. Rev. Lett. 92 (2004) 077205.
7. M. Yamanouchi et al.: Nature 428 (2004) 539;
8. M. Yamanouchi et al.: Phys. Rev. Lett. 96 (2006) 096601.
9. E. Saitoh et al.: Nature 432 (2004) 203.
10. M. Kläui et al.: Phys. Rev. Lett. 94 (2005) 106601;
11. M. Kläui et al.: Phys. Rev. Lett. 95 (2005) 026601.
12. M. Hayashi et al.: Phys. Rev. Lett. 96 (2006) 197207;
13. M. Hayashi et al.: Phys. Rev. Lett. 97 (2006) 207205;
14. M. Hayashi et al.: Nat. Phys. 3 (2007) 21.
15. L. Thomas et al.: Nature 443 (2006) 197.
16. G. Tatara and H. Kohno: Phys. Rev. Lett. 92 (2004) 086601;
17. G. Tatara and H. Kohno: Phys. Rev. Lett. 96 (2006) 189702.
18. Z. Li and S. Zhang: Phys. Rev. Lett. 92 (2004) 207203.
19. S. Zhang and Z. Li: Phys. Rev. Lett. 93 (2004) 127204.
20. A. Thiaville et al.: Europhys. Lett. 69 (2005) 990.
21. S. E. Barnes and S. Maekawa: Phys. Rev. Lett. 95 (2005) 107204;
22. S. E. Barnes: Phys. Rev. Lett. 96 (2006) 189701.
23. G. Tatara et al.: J. Phys. Soc. Jpn. 75 (2006) 064708.
24. A. K. Nguyen et al.: Phys. Rev. Lett. 97 (2006) 136603;
25. A. K. Nguyen et al.: Phys. Rev. Lett. 98 (2007) 146602.
26. J. Shibata et al.: Phys. Rev. B 73 (2006) 020403.
27. T. Ishida, T. Kimura, and Y. Otani: Phys. Rev. B 74 (2006) 014424.
28. S. Kasai et al.: Phys. Rev. Lett. 97 (2006) 107204.
29. K. Yamada et al.: Nat. Mater. 6 (2007) 270.
30. We define n to be a unit vector in the direction of spin, which is opposite to the direction of magnetization.
31. Ya. B. Bazaliy, B. A. Jones, and S.-C. Zhang: Phys. Rev. B 57 (1998) R3213.
32. J.-Ph. Ansermet: IEEE Trans. Magn. 40 (2004) 358.
33. J. Fernández-Rossier et al.: Phys. Rev. B 69 (2004) 174412.
34. J. Shibata, G. Tatara, and H. Kohno: Phys. Rev. Lett. 94 (2005) 076601.
35. J. Xiao, A. Zangwill, and M. D. Stiles: Phys. Rev. B 73 (2006) 054428.
36. Y. Tserkovnyak et al.: Phys. Rev. B 74 (2006) 144405.
37. H. Kohno, G. Tatara, and J. Shibata: J. Phys. Soc. Jpn. 75 (2006) 113706. Referred to as I in the text.
38. F. Piéchon and A. Thiaville: Phys. Rev. B 75 (2007) 174414.
39. R. A. Duine et al.: cond-mat/0703414.
40. tot to all terms on the r.h.s. of eq. (1).
41. Please note that, in the literature (except for, e.g., ref. 17), the β-term is regarded as non-adiabatic. In this letter, we take a different view that "adiabatic following" is a consequence of the adiabatic "theorem", rather than the definition of adiabaticity. We are grateful to Y. Tserkovnyak for sharing this viewpoint, to H. J. Skadsem for informing us of ref. 17, and to M. D. Stiles for his thoughtful comments on this issue.
42. na, due to electron reflection, which is spatially nonlocal and oscillatory. We do not study this torque in this letter; please refer to refs. 27, 35, and 36.
43. X. Waintal and M. Viret: Europhys. Lett. 65 (2004) 427.
44. G. Tatara et al.: J. Phys. Soc. Jpn. 76 (2007) 054707.
45. M. D. Stiles et al.: cond-mat/0702020.
46. Y. Tserkovnyak, G. A. Fiete, and B. I. Halperin: Appl. Phys. Lett. 84 (2004) 5234.
47. A. Sakuma: cond-mat/0602075.
48. H. J. Skadsem et al.: cond-mat/0611588.
49. V. Korenman, J. L. Murray, and R. E. Prange: Phys. Rev. B 16 (1977) 4032.
50. G. Tatara and H. Fukuyama: Phys. Rev. Lett. 72 (1994) 772;
51. G. Tatara and H. Fukuyama: Phys. Rev. Lett. 78 (1997) 3773;
52. G. Tatara and H. Fukuyama: J. Phys. Soc. Jpn. 63 (1994) 2538.
53. We use the vector (bold italic) notation for the spin component. The space-time components are indicated by subscripts such as μ, ν (= 0, 1, 2, 3) or i; j (= 1, 2, 3).
54. The subscript σ =↑, ↓ corresponds, respectively, to σ = +1, -1 in the formula (and to σ = ↑, ↓ or -1; +1), as in I.
55. ⊥ already contains a time derivative.
56. R. Kubo: J. Phys. Soc. Jpn. 12 (1957) 570.
57. sreproduces eq. (2).
58. el.