Influence of a uniform current on collective magnetization dynamics in a ferromagnetic metal

Physical Review B - Condensed Matter and Materials Physics

Volumn 69, Issue 17, 2004, Pages

  Fernandez Rossier J ^a,b   Braun, M ^a,c   Nunez A S ^a   MacDonald, A H ^a  

^a UNIVERSITY OF TEXAS AT AUSTIN   (United States)

^b UNIVERSITY OF ALICANTE   (Spain)

^c UNIVERSITY OF KARLSRUHE   (Germany)

Author keywords

[No Author keywords available]

Indexed keywords

FERROMAGNETIC MATERIAL; METAL; TRANSITION ELEMENT;

ARTICLE; ATOMIC PARTICLE; CALCULATION; DYNAMICS; ENERGY; MAGNETISM; MAGNON; MATHEMATICAL ANALYSIS; MODEL; PREDICTION; ROTATION; TORQUE;

EID: 42749098421     PISSN: 01631829     EISSN: None     Source Type: Journal    
DOI: 10.1103/PhysRevB.69.174412     Document Type: Article

References (60)

1. Spin Dependent Transport in Magnetic Nanostructures, edited by S. Maekawa and T. Shinjo (Taylor and Francis, London, 2002).
2. J.C. Slonczewski, J. Magn. Magn. Mater. 159, L1 (1996).
3. L. Berger, Phys. Rev. B 54, 9353 (1996).
4. M. Tsoi, A.G.M. Jansen, J. Bass, W.-C. Chiang, M. Seck, V. Tsoi, and P. Wyder, Phys. Rev. Lett. 80, 4281 (1998); M. Tsoi, A.G.M. Jansen, J. Bass, W.-C. Chiang, V. Tsoi, and P. Wyder, Nature (London) 406, 46 (2000).
5. M. Tsoi, A.G.M. Jansen, J. Bass, W.-C. Chiang, M. Seck, V. Tsoi, and P. Wyder, Phys. Rev. Lett. 80, 4281 (1998); M. Tsoi, A.G.M. Jansen, J. Bass, W.-C. Chiang, V. Tsoi, and P. Wyder, Nature (London) 406, 46 (2000).
6. M. Tsoi, V. Tsoi, J. Bass, A.G.M. Jansen, and P. Wyder, Phys. Rev. Lett. 89, 246803 (2002).
7. J.Z. Sun, J. Magn. Magn. Mater. 202, 157 (1999).
8. E.B. Myers, D.C. Ralph, J.A. Katine, R.N. Louie, and R.A. Buhrman, Science 285, 867 (1999); J.A. Katine, F.J. Albert, R.A. Buhrman, E.B. Myers, and D.C. Ralph, Phys. Rev. Lett. 84, 4212 (2000); E.B. Myers, F.J. Albert, J.C. Sankey, E. Bonet, R.A. Buhrman, and D.C. Ralph, ibid. 89, 196801 (2002); S.I. Kiselev, J.C. Sankey, I.N. Krivorotov, N.C. Emley, R.J. Schoelkopf, R.A. Buhrman, and D.C. Ralph, Nature (London) 425, 380 (2003); W.H. Rippard, M.R. Pufall, and T.J. Silva, Appl. Phys. Lett. 82, 1260 (2003); F.B. Mancoff, R.W. Dave, N.D. Rizzo, T.C. Eschrich, B.N. Engel, and S. Tehrani, ibid. 83, 1596 (2003).
9. E.B. Myers, D.C. Ralph, J.A. Katine, R.N. Louie, and R.A. Buhrman, Science 285, 867 (1999); J.A. Katine, F.J. Albert, R.A. Buhrman, E.B. Myers, and D.C. Ralph, Phys. Rev. Lett. 84, 4212 (2000); E.B. Myers, F.J. Albert, J.C. Sankey, E. Bonet, R.A. Buhrman, and D.C. Ralph, ibid. 89, 196801 (2002); S.I. Kiselev, J.C. Sankey, I.N. Krivorotov, N.C. Emley, R.J. Schoelkopf, R.A. Buhrman, and D.C. Ralph, Nature (London) 425, 380 (2003); W.H. Rippard, M.R. Pufall, and T.J. Silva, Appl. Phys. Lett. 82, 1260 (2003); F.B. Mancoff, R.W. Dave, N.D. Rizzo, T.C. Eschrich, B.N. Engel, and S. Tehrani, ibid. 83, 1596 (2003).
10. E.B. Myers, D.C. Ralph, J.A. Katine, R.N. Louie, and R.A. Buhrman, Science 285, 867 (1999); J.A. Katine, F.J. Albert, R.A. Buhrman, E.B. Myers, and D.C. Ralph, Phys. Rev. Lett. 84, 4212 (2000); E.B. Myers, F.J. Albert, J.C. Sankey, E. Bonet, R.A. Buhrman, and D.C. Ralph, ibid. 89, 196801 (2002); S.I. Kiselev, J.C. Sankey, I.N. Krivorotov, N.C. Emley, R.J. Schoelkopf, R.A. Buhrman, and D.C. Ralph, Nature (London) 425, 380 (2003); W.H. Rippard, M.R. Pufall, and T.J. Silva, Appl. Phys. Lett. 82, 1260 (2003); F.B. Mancoff, R.W. Dave, N.D. Rizzo, T.C. Eschrich, B.N. Engel, and S. Tehrani, ibid. 83, 1596 (2003).
11. E.B. Myers, D.C. Ralph, J.A. Katine, R.N. Louie, and R.A. Buhrman, Science 285, 867 (1999); J.A. Katine, F.J. Albert, R.A. Buhrman, E.B. Myers, and D.C. Ralph, Phys. Rev. Lett. 84, 4212 (2000); E.B. Myers, F.J. Albert, J.C. Sankey, E. Bonet, R.A. Buhrman, and D.C. Ralph, ibid. 89, 196801 (2002); S.I. Kiselev, J.C. Sankey, I.N. Krivorotov, N.C. Emley, R.J. Schoelkopf, R.A. Buhrman, and D.C. Ralph, Nature (London) 425, 380 (2003); W.H. Rippard, M.R. Pufall, and T.J. Silva, Appl. Phys. Lett. 82, 1260 (2003); F.B. Mancoff, R.W. Dave, N.D. Rizzo, T.C. Eschrich, B.N. Engel, and S. Tehrani, ibid. 83, 1596 (2003).
12. E.B. Myers, D.C. Ralph, J.A. Katine, R.N. Louie, and R.A. Buhrman, Science 285, 867 (1999); J.A. Katine, F.J. Albert, R.A. Buhrman, E.B. Myers, and D.C. Ralph, Phys. Rev. Lett. 84, 4212 (2000); E.B. Myers, F.J. Albert, J.C. Sankey, E. Bonet, R.A. Buhrman, and D.C. Ralph, ibid. 89, 196801 (2002); S.I. Kiselev, J.C. Sankey, I.N. Krivorotov, N.C. Emley, R.J. Schoelkopf, R.A. Buhrman, and D.C. Ralph, Nature (London) 425, 380 (2003); W.H. Rippard, M.R. Pufall, and T.J. Silva, Appl. Phys. Lett. 82, 1260 (2003); F.B. Mancoff, R.W. Dave, N.D. Rizzo, T.C. Eschrich, B.N. Engel, and S. Tehrani, ibid. 83, 1596 (2003).
13. E.B. Myers, D.C. Ralph, J.A. Katine, R.N. Louie, and R.A. Buhrman, Science 285, 867 (1999); J.A. Katine, F.J. Albert, R.A. Buhrman, E.B. Myers, and D.C. Ralph, Phys. Rev. Lett. 84, 4212 (2000); E.B. Myers, F.J. Albert, J.C. Sankey, E. Bonet, R.A. Buhrman, and D.C. Ralph, ibid. 89, 196801 (2002); S.I. Kiselev, J.C. Sankey, I.N. Krivorotov, N.C. Emley, R.J. Schoelkopf, R.A. Buhrman, and D.C. Ralph, Nature (London) 425, 380 (2003); W.H. Rippard, M.R. Pufall, and T.J. Silva, Appl. Phys. Lett. 82, 1260 (2003); F.B. Mancoff, R.W. Dave, N.D. Rizzo, T.C. Eschrich, B.N. Engel, and S. Tehrani, ibid. 83, 1596 (2003).
14. Y. Ji and C.L. Chien and M.D. Stiles, Phys. Rev. Lett. 90, 106601 (2003).
15. S. Urazhdin, N.O. Birge, W.P. Pratt, Jr., and J. Bass, Phys. Rev. Lett. 91, 146803 (2003).
16. J.Z. Sun, Phys. Rev. B 62, 570 (2000); A. Brataas, Y.V. Nazarov, and G.E.W. Bauer, Phys. Rev. Lett. 84, 2481 (2000); X. Waintal and P.W. Brouwer, Phys. Rev. B 63, 220407 (2001); C. Heide, ibid. 65, 054401 (2002); M. Stiles and A. Zangwill, ibid. 65, 014407 (2002); J.-E. Wegrowe, Appl. Phys. Lett. 80, 3775 (2002); S. Zhang, P.M. Levy, and A. Fert, Phys. Rev. Lett. 88, 236601 (2002); G.E.W. Bauer, Y. Tserkovnyak, D. Huertas-Hernando, and A. Brataas, Phys. Rev. B 67, 094421 (2003); M.L. Polianski and P.W. Brouwer, Phys. Rev. Lett. 92, 026602 (2004); A. Shpiro, P.M. Levy, and S. Zhang, Phys. Rev. B 67, 104430 (2003); A. Fert, V. Cros, J.-M. George, J. Grollier, H. Jaffres, A. Hamzic, A. Vaures, G. Faini, J. Ben Youssef, and H. Le Gall, cond-mat/0310737 (unpublished).
17. J.Z. Sun, Phys. Rev. B 62, 570 (2000); A. Brataas, Y.V. Nazarov, and G.E.W. Bauer, Phys. Rev. Lett. 84, 2481 (2000); X. Waintal and P.W. Brouwer, Phys. Rev. B 63, 220407 (2001); C. Heide, ibid. 65, 054401 (2002); M. Stiles and A. Zangwill, ibid. 65, 014407 (2002); J.-E. Wegrowe, Appl. Phys. Lett. 80, 3775 (2002); S. Zhang, P.M. Levy, and A. Fert, Phys. Rev. Lett. 88, 236601 (2002); G.E.W. Bauer, Y. Tserkovnyak, D. Huertas-Hernando, and A. Brataas, Phys. Rev. B 67, 094421 (2003); M.L. Polianski and P.W. Brouwer, Phys. Rev. Lett. 92, 026602 (2004); A. Shpiro, P.M. Levy, and S. Zhang, Phys. Rev. B 67, 104430 (2003); A. Fert, V. Cros, J.-M. George, J. Grollier, H. Jaffres, A. Hamzic, A. Vaures, G. Faini, J. Ben Youssef, and H. Le Gall, cond-mat/0310737 (unpublished).
18. J.Z. Sun, Phys. Rev. B 62, 570 (2000); A. Brataas, Y.V. Nazarov, and G.E.W. Bauer, Phys. Rev. Lett. 84, 2481 (2000); X. Waintal and P.W. Brouwer, Phys. Rev. B 63, 220407 (2001); C. Heide, ibid. 65, 054401 (2002); M. Stiles and A. Zangwill, ibid. 65, 014407 (2002); J.-E. Wegrowe, Appl. Phys. Lett. 80, 3775 (2002); S. Zhang, P.M. Levy, and A. Fert, Phys. Rev. Lett. 88, 236601 (2002); G.E.W. Bauer, Y. Tserkovnyak, D. Huertas-Hernando, and A. Brataas, Phys. Rev. B 67, 094421 (2003); M.L. Polianski and P.W. Brouwer, Phys. Rev. Lett. 92, 026602 (2004); A. Shpiro, P.M. Levy, and S. Zhang, Phys. Rev. B 67, 104430 (2003); A. Fert, V. Cros, J.-M. George, J. Grollier, H. Jaffres, A. Hamzic, A. Vaures, G. Faini, J. Ben Youssef, and H. Le Gall, cond-mat/0310737 (unpublished).
19. J.Z. Sun, Phys. Rev. B 62, 570 (2000); A. Brataas, Y.V. Nazarov, and G.E.W. Bauer, Phys. Rev. Lett. 84, 2481 (2000); X. Waintal and P.W. Brouwer, Phys. Rev. B 63, 220407 (2001); C. Heide, ibid. 65, 054401 (2002); M. Stiles and A. Zangwill, ibid. 65, 014407 (2002); J.-E. Wegrowe, Appl. Phys. Lett. 80, 3775 (2002); S. Zhang, P.M. Levy, and A. Fert, Phys. Rev. Lett. 88, 236601 (2002); G.E.W. Bauer, Y. Tserkovnyak, D. Huertas-Hernando, and A. Brataas, Phys. Rev. B 67, 094421 (2003); M.L. Polianski and P.W. Brouwer, Phys. Rev. Lett. 92, 026602 (2004); A. Shpiro, P.M. Levy, and S. Zhang, Phys. Rev. B 67, 104430 (2003); A. Fert, V. Cros, J.-M. George, J. Grollier, H. Jaffres, A. Hamzic, A. Vaures, G. Faini, J. Ben Youssef, and H. Le Gall, cond-mat/0310737 (unpublished).
20. J.Z. Sun, Phys. Rev. B 62, 570 (2000); A. Brataas, Y.V. Nazarov, and G.E.W. Bauer, Phys. Rev. Lett. 84, 2481 (2000); X. Waintal and P.W. Brouwer, Phys. Rev. B 63, 220407 (2001); C. Heide, ibid. 65, 054401 (2002); M. Stiles and A. Zangwill, ibid. 65, 014407 (2002); J.-E. Wegrowe, Appl. Phys. Lett. 80, 3775 (2002); S. Zhang, P.M. Levy, and A. Fert, Phys. Rev. Lett. 88, 236601 (2002); G.E.W. Bauer, Y. Tserkovnyak, D. Huertas-Hernando, and A. Brataas, Phys. Rev. B 67, 094421 (2003); M.L. Polianski and P.W. Brouwer, Phys. Rev. Lett. 92, 026602 (2004); A. Shpiro, P.M. Levy, and S. Zhang, Phys. Rev. B 67, 104430 (2003); A. Fert, V. Cros, J.-M. George, J. Grollier, H. Jaffres, A. Hamzic, A. Vaures, G. Faini, J. Ben Youssef, and H. Le Gall, cond-mat/0310737 (unpublished).
21. J.Z. Sun, Phys. Rev. B 62, 570 (2000); A. Brataas, Y.V. Nazarov, and G.E.W. Bauer, Phys. Rev. Lett. 84, 2481 (2000); X. Waintal and P.W. Brouwer, Phys. Rev. B 63, 220407 (2001); C. Heide, ibid. 65, 054401 (2002); M. Stiles and A. Zangwill, ibid. 65, 014407 (2002); J.-E. Wegrowe, Appl. Phys. Lett. 80, 3775 (2002); S. Zhang, P.M. Levy, and A. Fert, Phys. Rev. Lett. 88, 236601 (2002); G.E.W. Bauer, Y. Tserkovnyak, D. Huertas-Hernando, and A. Brataas, Phys. Rev. B 67, 094421 (2003); M.L. Polianski and P.W. Brouwer, Phys. Rev. Lett. 92, 026602 (2004); A. Shpiro, P.M. Levy, and S. Zhang, Phys. Rev. B 67, 104430 (2003); A. Fert, V. Cros, J.-M. George, J. Grollier, H. Jaffres, A. Hamzic, A. Vaures, G. Faini, J. Ben Youssef, and H. Le Gall, cond-mat/0310737 (unpublished).
22. J.Z. Sun, Phys. Rev. B 62, 570 (2000); A. Brataas, Y.V. Nazarov, and G.E.W. Bauer, Phys. Rev. Lett. 84, 2481 (2000); X. Waintal and P.W. Brouwer, Phys. Rev. B 63, 220407 (2001); C. Heide, ibid. 65, 054401 (2002); M. Stiles and A. Zangwill, ibid. 65, 014407 (2002); J.-E. Wegrowe, Appl. Phys. Lett. 80, 3775 (2002); S. Zhang, P.M. Levy, and A. Fert, Phys. Rev. Lett. 88, 236601 (2002); G.E.W. Bauer, Y. Tserkovnyak, D. Huertas-Hernando, and A. Brataas, Phys. Rev. B 67, 094421 (2003); M.L. Polianski and P.W. Brouwer, Phys. Rev. Lett. 92, 026602 (2004); A. Shpiro, P.M. Levy, and S. Zhang, Phys. Rev. B 67, 104430 (2003); A. Fert, V. Cros, J.-M. George, J. Grollier, H. Jaffres, A. Hamzic, A. Vaures, G. Faini, J. Ben Youssef, and H. Le Gall, cond-mat/0310737 (unpublished).
23. J.Z. Sun, Phys. Rev. B 62, 570 (2000); A. Brataas, Y.V. Nazarov, and G.E.W. Bauer, Phys. Rev. Lett. 84, 2481 (2000); X. Waintal and P.W. Brouwer, Phys. Rev. B 63, 220407 (2001); C. Heide, ibid. 65, 054401 (2002); M. Stiles and A. Zangwill, ibid. 65, 014407 (2002); J.-E. Wegrowe, Appl. Phys. Lett. 80, 3775 (2002); S. Zhang, P.M. Levy, and A. Fert, Phys. Rev. Lett. 88, 236601 (2002); G.E.W. Bauer, Y. Tserkovnyak, D. Huertas-Hernando, and A. Brataas, Phys. Rev. B 67, 094421 (2003); M.L. Polianski and P.W. Brouwer, Phys. Rev. Lett. 92, 026602 (2004); A. Shpiro, P.M. Levy, and S. Zhang, Phys. Rev. B 67, 104430 (2003); A. Fert, V. Cros, J.-M. George, J. Grollier, H. Jaffres, A. Hamzic, A. Vaures, G. Faini, J. Ben Youssef, and H. Le Gall, cond-mat/0310737 (unpublished).
24. J.Z. Sun, Phys. Rev. B 62, 570 (2000); A. Brataas, Y.V. Nazarov, and G.E.W. Bauer, Phys. Rev. Lett. 84, 2481 (2000); X. Waintal and P.W. Brouwer, Phys. Rev. B 63, 220407 (2001); C. Heide, ibid. 65, 054401 (2002); M. Stiles and A. Zangwill, ibid. 65, 014407 (2002); J.-E. Wegrowe, Appl. Phys. Lett. 80, 3775 (2002); S. Zhang, P.M. Levy, and A. Fert, Phys. Rev. Lett. 88, 236601 (2002); G.E.W. Bauer, Y. Tserkovnyak, D. Huertas-Hernando, and A. Brataas, Phys. Rev. B 67, 094421 (2003); M.L. Polianski and P.W. Brouwer, Phys. Rev. Lett. 92, 026602 (2004); A. Shpiro, P.M. Levy, and S. Zhang, Phys. Rev. B 67, 104430 (2003); A. Fert, V. Cros, J.-M. George, J. Grollier, H. Jaffres, A. Hamzic, A. Vaures, G. Faini, J. Ben Youssef, and H. Le Gall, cond-mat/0310737 (unpublished).
25. J.Z. Sun, Phys. Rev. B 62, 570 (2000); A. Brataas, Y.V. Nazarov, and G.E.W. Bauer, Phys. Rev. Lett. 84, 2481 (2000); X. Waintal and P.W. Brouwer, Phys. Rev. B 63, 220407 (2001); C. Heide, ibid. 65, 054401 (2002); M. Stiles and A. Zangwill, ibid. 65, 014407 (2002); J.-E. Wegrowe, Appl. Phys. Lett. 80, 3775 (2002); S. Zhang, P.M. Levy, and A. Fert, Phys. Rev. Lett. 88, 236601 (2002); G.E.W. Bauer, Y. Tserkovnyak, D. Huertas-Hernando, and A. Brataas, Phys. Rev. B 67, 094421 (2003); M.L. Polianski and P.W. Brouwer, Phys. Rev. Lett. 92, 026602 (2004); A. Shpiro, P.M. Levy, and S. Zhang, Phys. Rev. B 67, 104430 (2003); A. Fert, V. Cros, J.-M. George, J. Grollier, H. Jaffres, A. Hamzic, A. Vaures, G. Faini, J. Ben Youssef, and H. Le Gall, cond-mat/0310737 (unpublished).
26. J.Z. Sun, Phys. Rev. B 62, 570 (2000); A. Brataas, Y.V. Nazarov, and G.E.W. Bauer, Phys. Rev. Lett. 84, 2481 (2000); X. Waintal and P.W. Brouwer, Phys. Rev. B 63, 220407 (2001); C. Heide, ibid. 65, 054401 (2002); M. Stiles and A. Zangwill, ibid. 65, 014407 (2002); J.-E. Wegrowe, Appl. Phys. Lett. 80, 3775 (2002); S. Zhang, P.M. Levy, and A. Fert, Phys. Rev. Lett. 88, 236601 (2002); G.E.W. Bauer, Y. Tserkovnyak, D. Huertas-Hernando, and A. Brataas, Phys. Rev. B 67, 094421 (2003); M.L. Polianski and P.W. Brouwer, Phys. Rev. Lett. 92, 026602 (2004); A. Shpiro, P.M. Levy, and S. Zhang, Phys. Rev. B 67, 104430 (2003); A. Fert, V. Cros, J.-M. George, J. Grollier, H. Jaffres, A. Hamzic, A. Vaures, G. Faini, J. Ben Youssef, and H. Le Gall, cond-mat/0310737 (unpublished).
27. Y.B. Bazaliy, B.A. Jones and S.-C. Zhang, Phys. Rev. B 57, R3213 (1998).
28. →↑], where N is the number of sites in the lattice, and ↑ and ↓ are defined in the axis of the average magnetization.
29. P. Lederer and D.L. Mills, Phys. Rev. 148, 542 (1966).
30. C. Herring and C. Kittel, Phys. Rev. 81, 869 (1951).
31. E.P. Wolfarth, Rev. Mod. Phys. 25, 211 (1953);
32. D.A. Papaconstantopoulos, Handbook of the Band Structure of Elemental Solids (Plenum Press, New York, 1986).
33. T. Moriya, Spin Fluctuations in Itinerant Electron Magnetism (Springer-Verlag, Berlin, 1985).
34. R.E. Prange and V. Korenman, Phys. Rev. B 19, 4691 (1979).
35. J. Hubbard, Phys. Rev. B 19, 2626 (1979).
36. H.J. Schulz, Phys. Rev. Lett. 65, 2462 (1990).
37. E. Fradkin, Field Theories of Condensed Matter Systems (Addison-Wesley, Reading, MA, 1991), Secs 3.3 and 3.4.
38. S. Doniach and E.H. Sondheimer, Green's Functions for Solid State Physicists (Benjamin, New York, 1974).
39. J.W. Negele and H. Orland, Quantum Many-Particle Systems (Addison-Wesley, Reading, MA, 1988).
40. R.L. Stratonovitch, Dok. Akad. Nauk. SSSR 115, 1097 (1957) [Sov. Phys. Dokl. 2, 416 (1958)]; J. Hubbard, Phys. Rev. Lett. 3, 77 (1959).
41. R.L. Stratonovitch, Dok. Akad. Nauk. SSSR 115, 1097 (1957) [Sov. Phys. Dokl. 2, 416 (1958)]; J. Hubbard, Phys. Rev. Lett. 3, 77 (1959).
42. R.L. Stratonovitch, Dok. Akad. Nauk. SSSR 115, 1097 (1957) [Sov. Phys. Dokl. 2, 416 (1958)]; J. Hubbard, Phys. Rev. Lett. 3, 77 (1959).
43. Assa Auerbach, Interacting Electrons and Quantum Magnetism (Springer-Verlag, New York, 1994).
44. R. Skomski and J.M.D. Coey, Permanent Magnetism (Institute of Physics, Bristol, 1999).
45. Claude Cohen-Tannoudji, Gilbert Grynberg, Jacques Dupont-Roc, Atom-Photon Interactions: Basic Processes and Applications (Wiley-Interscience, New York, 1998).
46. A. Fert, J. Phys. C 2, 1784 (1969), and references therein.
47. J.H. Van Vleck, Phys. Rev. 52, 1178 (1937); H. Brooks, Phys. Rev. 58, 909 (1940).
48. J.H. Van Vleck, Phys. Rev. 52, 1178 (1937); H. Brooks, Phys. Rev. 58, 909 (1940).
49. M.D. Stiles, S.V. Halilov, R.A. Hyman, and A. Zangwill, Phys. Rev. B 64, 104430 (2001).
50. V. Korenman, J.L. Murray, and R.E. Prange, Phys. Rev. B 16, 4032 (1977).
51. A.J. Millis, P.B. Littlewood, and B.I. Shraiman, Phys. Rev. Lett. 74, 5144 (1995).
52. P.M. Levy and S. Zhang, Phys. Rev. Lett. 79, 5110 (1997).
53. G. Tatara and H. Fukuyama, Phys. Rev. Lett. 78, 3773 (1997).
54. V. Kambersky, Can. J. Phys. 48, 2906 (1970).
55. V. Kambersky and C.E. Patton, Phys. Rev. B 11, 2668 (1975).
56. V. Korenman, J.L. Murray, and R.E. Prange, Phys. Rev. B 16, 4048 (1977).
57. L. Berger, J. Polym. Sci., Polym. Lett. Ed. 38, 1321 (1977).
58. N.W. Ashcroft and N.D. Mermin, Solid State Physics (Saunders, Philadelphia, 1976)
59. R. Arias and D. Mills, Phys. Rev. B 60, 7395 (1999).
60. J. Fernádez-Rossier, A.S. Núñez, M. Abolfath, and A.H. MacDonald, cond-mat/0304492 (unpublished).