Penetration of nonintegral magnetic flux through a domain-wall bend in time-reversal symmetry broken superconductors

Physical Review B - Condensed Matter and Materials Physics

Volumn 84, Issue 1, 2011, Pages

  Ferguson, David G ^a   Goldbart, Paul M ^a  

^a United States of America   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 79961240109     PISSN: 10980121     EISSN: 1550235X     Source Type: Journal    
DOI: 10.1103/PhysRevB.84.014523     Document Type: Article

References (72)

1. A. P. Mackenzie and Y. Maeno, Rev. Mod. Phys. RMPHAT 0034-6861 10.1103/RevModPhys.75.657 75, 657 (2003). (Pubitemid 36737415)
2. T. M. Rice and M. Sigrist, J. Phys. Condens. Matter JCOMEL 0953-8984 10.1088/0953-8984/7/47/002 7, L643 (1995).
3. K. Ishida, H. Mukuda, Y. Kitaoka, K. Asayama, Z. Q. Mao, Y. Mori, and Y. Maeno, Nature (London) NATUAS 0028-0836 10.1038/25315 396, 658 (1998). (Pubitemid 29003940)
4. K. D. Nelson, Z. Q. Mao, Y. Maeno, and Y. Liu, Science SCIEAS 0036-8075 10.1126/science.1103881 306, 1151 (2004). (Pubitemid 39488987)
5. J. Jang, D. G. Ferguson, V. Vakaryuk, R. Budakian, S. B. Chung, P. M. Goldbart, and Y. Maeno, Science SCIEAS 0036-8075 10.1126/science.1193839 331, 186 (2011).
6. N. B. Kopnin and M. M. Salomaa, Phys. Rev. B PLRBAQ 1098-0121 10.1103/PhysRevB.44.9667 44, 9667 (1991).
7. N. Read and D. Green, Phys. Rev. B PLRBAQ 1098-0121 10.1103/PhysRevB.61. 10267 61, 10267 (2000).
8. A. Y. Kitaev, Ann. Phys. (NY) APNYA6 0003-4916 10.1016/S0003-4916(02) 00018-0 303, 2 (2003). (Pubitemid 36177417)
9. C. Nayak, S. H. Simon, A. Stern, M. Freedman, and S. D. Sarma, Rev. Mod. Phys. RMPHAT 0034-6861 10.1103/RevModPhys.80.1083 80, 1083 (2008).
10. Y. Yoshioka and K. Miyake, J. Phys. Soc. Jpn. JUPSAU 0031-9015 10.1143/JPSJ.78.074701 78, 074701 (2009).
11. S. Raghu, A. Kapitulnik, and S. A. Kivelson, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.105.136401 105, 136401 (2010).
12. G. M. Luke, Y. Fudamoto, K. M. Kojima, M. I. Larkin, J. Merrin, B. Nachumi, Y. J. Uemura, Y. Maeno, Z. Q. Mao, Y. Mori, H. Nakamura, and M. Sigrist, Nature (London) NATUAS 0028-0836 10.1038/29038 394, 558 (1998). (Pubitemid 28366825)
13. J. Xia, Y. Maeno, P. T. Beyersdorf, M. M. Fejer, and A. Kapitulnik, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.97.167002 97, 167002 (2006). (Pubitemid 44611085)
14. C. Kallin and A. J. Berlinsky, J. Phys. Condens. Matter JCOMEL 0953-8984 10.1088/0953-8984/21/16/164210 21, 164210 (2009).
15. M. Matsumoto and M. Sigrist, J. Phys. Soc. Jpn. JUPSAU 0031-9015 10.1143/JPSJ.68.994 68, 994 (1999).
16. P. G. Bjornsson, Y. Maeno, M. E. Huber, and K. A. Moler, Phys. Rev. B PLRBAQ 1098-0121 10.1103/PhysRevB.72.012504 72, 012504 (2005).
17. J. R. Kirtley, C. Kallin, C. W. Hicks, E. A. Kim, Y. Liu, K. A. Moler, Y. Maeno, and K. D. Nelson, Phys. Rev. B PLRBAQ 1098-0121 10.1103/PhysRevB.76. 014526 76, 014526 (2007).
18. C. W. Hicks, J. R. Kirtley, T. M. Lippman, N. C. Koshnick, M. E. Huber, Y. Maeno, W. M. Yuhasz, M. B. Maple, and K. A. Moler, Phys. Rev. B PLRBAQ 1098-0121 10.1103/PhysRevB.81.214501 81, 214501 (2010).
19. G. Volovik and L. Gor'kov, Zh. Eksp. Teor. Fiz. 88, 1412 (1985)
20. G. Volovik and L. Gor'kov, [Sov. Phys. JETP 61, 843 (1985)].
21. A. J. Leggett, Quantum Liquids: Bose Condensation and Cooper Pairing in Condensed-Matter Systems (Oxford University Press, New York, 2006).
22. P. E. C. Ashby and C. Kallin, Phys. Rev. B PLRBAQ 1098-0121 10.1103/PhysRevB.79.224509 79, 224509 (2009).
23. F. London and H. London, Proc. R. Soc. London, Ser. A JUPSAU 1364-5021 10.1098/rspa.1935.0048 149, 71 (1935).
24. V. Ginzburg and L. Landau, Zh. Eksp. Teor. Fiz. 20, 1064 (1950);
25. L. D. Landau, edited by, D. ter Haar, (Pergamon, New York, 1965), Vol. 1, pp. 138-167.
26. J. Bardeen, L. N. Cooper, and J. R. Schrieffer, Phys. Rev. PHRVAO 0031-899X 10.1103/PhysRev.108.1175 108, 1175 (1957).
27. M. Sigrist and K. Ueda, Rev. Mod. Phys. RMPHAT 0034-6861 10.1103/RevModPhys.63.239 63, 239 (1991).
28. V. P. Mineev and K. V. Samokhin, Introduction to Unconventional Superconductivity (Gordon and Breach, Amsterdam, 1999).
29. M. Sigrist, T. M. Rice, and K. Ueda, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.63.1727 63, 1727 (1989).
30. M. Sigrist and D. F. Agterberg, Prog. Theor. Phys. PTPKAV 0033-068X 10.1143/PTP.102.965 102, 965 (1999).
31. E. Babaev, A. Sudbo, and N. W. Ashcroft, Nature (London) NATUAS 0028-0836 10.1038/nature02910 431, 666 (2004). (Pubitemid 39382608)
32. G. E. Volovik and L. P. Gor'kov, Pis'ma Zh. Eksp. Teor. Fiz. 39, 550 (1984)
33. G. E. Volovik and L. P. Gor'kov, [JETP Lett. 39, 674 (1984)].
34. G. E. Volovik, Proc. Natl. Acad. Sci. USA PNASA6 0027-8424 10.1073/pnas.97.6.2431 97, 2431 (2000). (Pubitemid 30159188)
35. M. Sigrist, D. B. Bailey, and R. B. Laughlin, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.74.3249 74, 3249 (1995).
36. F. London, Superfluids, 2nd ed. (Dover, New York, 1961).
37. C. Bergemann, S. R. Julian, A. P. Mackenzie, S. NishiZaki, and Y. Maeno, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.84.2662 84, 2662 (2000).
38. M. W. Haverkort, I. S. Elfimov, L. H. Tjeng, G. A. Sawatzky, and A. Damascelli, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.101.026406 101, 026406 (2008).
39. R. Heeb and D. F. Agterberg, Phys. Rev. B PLRBAQ 1098-0121 10.1103/PhysRevB.59.7076 59, 7076 (1999).
40. Z. Q. Mao, Y. Maeno, S. NishiZaki, T. Akima, and T. Ishiguro, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.84.991 84, 991 (2000).
41. K. Deguchi, Z. Q. Mao, H. Yaguchi, and Y. Maeno, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.92.047002 92, 047002 (2004).
42. S. Kittaka, H. Taniguchi, S. Yonezawa, H. Yaguchi, and Y. Maeno, Phys. Rev. B PLRBAQ 1098-0121 10.1103/PhysRevB.81.180510 81, 180510 (2010).
43. We do not need to specify whether the representation is Γ 5 + or Γ 5 -, for which the basis functions transform respectively as { X Z, Y Z } or { X, Y }. The results of the present work apply to both cases.
44. J.-X. Zhu, C. S. Ting, J. L. Shen, and Z. D. Wang, Phys. Rev. B PLRBAQ 1098-0121 10.1103/PhysRevB.56.14093 56, 14093 (1997).
45. N. D. Mermin, Rev. Mod. Phys. RMPHAT 0034-6861 10.1103/RevModPhys.51.591 51, 591 (1979).
46. R. Jackiw and C. Rebbi, Phys. Rev. D PRVDAQ 1550-7998 10.1103/PhysRevD.13.3398 13, 3398 (1976).
47. W. P. Su, J. R. Schrieffer, and A. J. Heeger, Phys. Rev. B PLRBAQ 1098-0121 10.1103/PhysRevB.22.2099 22, 2099 (1980).
48. Y. Ran, Y. Zhang, and A. Vishwanath, Nat. Phys. PRVDAQ 1745-2473 10.1038/nphys1220 5, 298 (2009).
49. / 2 • (cos (χ + π / 4), i sin (χ + π / 4)).
50. The global structure of this order parameter space is that of a twisted U (1) bundle (see, e.g., Ref. 67).
51. We note that, for weak coupling and specular reflection, the condition at the sample boundary requires β = 0 (see, e.g., Ref. 25), and thus, for a finite sample, all regions can be considered to be surrounded by domain-wall loops. However, the coupling between Γ and the normal direction of either the surface or the domain wall is generically different, which can result in distinct equilibrium orientations of Γ for each of these cases (see, e.g., Ref. 15).
52. Y. A. Izyumov and V. M. Laptev, Phase Transitions PHTRDP 0141-1594 10.1080/01411599008206870 20, 95 (1990).
53. T. A. Tokuyasu, D. W. Hess, and J. A. Sauls, Phys. Rev. B PLRBAQ 1098-0121 10.1103/PhysRevB.41.8891 41, 8891 (1990).
54. J. A. Sauls and M. Eschrig, New J. Phys. NJOPFM 1367-2630 10.1088/1367-2630/11/7/075008 11, 075008 (2009).
55. M. Ichioka, Y. Matsunaga, and K. Machida, Phys. Rev. B PLRBAQ 1098-0121 10.1103/PhysRevB.71.172510 71, 172510 (2005). (Pubitemid 43285296)
56. A. M. Polyakov, Gauge Fields and Strings (Harwood Academic, New York, 1987)
57. A. Zee, Quantum Field Theory in a Nutshell (Princeton University Press, Princeton, 2003).
58. Here and elsewhere in this paper, we take into account the core energies of domain walls but not the core energies of vortices. Our justification for doing this is that, in the standard London limit, the energy cost of a vortex core is negligibly small compared with the kinetic energy of the supercurrents and magnetic fields, whereas the core energy of a domain wall is not.
59. In the case that the equilibrium value of Γ - φ is π / 2, the dipole part of the vorticity is given by d = [- (π τ / 4) - μ] I. Thus, as is shown in the referring paragraph, the part of the domain-wall current proportional to τ can flow in either direction, for a given pattern of chirality, depending on the value of Γ - φ.
60. H.-J. Kwon, V. M. Yakovenko, and K. Sengupta, Synth. Met. SYMEDZ 0379-6779 10.1016/S0379-6779(02)00421-6 133-134, 27 (2003).
61. 4 (Ref. 68) is the absence of translational invariance (i.e., for the presence of multiple domains) along sample boundaries, which can be regarded in certain respects as analogous to domain walls. However, alternate explanations exist that are based on assumed line nodes in the superconducting order parameter (Ref. 69).
62. V. B. Geshkenbein, A. I. Larkin, and A. Barone, Phys. Rev. B PLRBAQ 1098-0121 10.1103/PhysRevB.36.235 36, 235 (1987).
63. dw on Γ(s)-φ(s) should have multiple minima per π period, then there could be topologically stable solitons in Γ(s) along a straight domain wall, each having ΔΓ?π and connected with nonquantized amounts of flux (see Refs. 27, 28, and 66). In this case, in addition to the bend flux of Eq., the flux associated with a bent domain wall may have a further contribution.
64. L. D. Landau and E. M. Lifshitz, Quantum Mechanics: Non-Relativistic Theory, 3rd ed. (Pergamon, New York, 1977).
65. H. Bluhm, Phys. Rev. B PLRBAQ 1098-0121 10.1103/PhysRevB.76.144507 76, 144507 (2007).
66. V. O. Dolocan, C. Veauvy, F. Servant, P. Lejay, K. Hasselbach, Y. Liu, and D. Mailly, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.95.097004 95, 097004 (2005). (Pubitemid 41505846)
67. V. O. Dolocan, P. Lejay, D. Mailly, and K. Hasselbach, Phys. Rev. B PLRBAQ 1098-0121 10.1103/PhysRevB.74.144505 74, 144505 (2006). (Pubitemid 44527858)
68. N. A. Logoboy and E. B. Sonin, Phys. Rev. B PLRBAQ 1098-0121 10.1103/PhysRevB.79.094511 79, 094511 (2009).
69. A. Bouhon and M. Sigrist, New J. Phys. NJOPFM 1367-2630 10.1088/1367-2630/12/4/043031 12, 043031 (2010).
70. M. Stone and P. M. Goldbart, Mathematics for Physics: A Guided Tour for Graduate Students (Cambridge University Press, New York, 2009).
71. F. Kidwingira, J. D. Strand, D. J. V. Harlingen, and Y. Maeno, Science SCIEAS 0036-8075 10.1126/science.1133239 314, 1267 (2006). (Pubitemid 44846394)
72. K. Sengupta, H.-J. Kwon, and V. M. Yakovenko, Phys. Rev. B PLRBAQ 1098-0121 10.1103/PhysRevB.65.104504 65, 104504 (2002)