Theory of weak localization in ferromagnetic (Ga,Mn)As

Physical Review B - Condensed Matter and Materials Physics

Volumn 79, Issue 15, 2009, Pages

  Garate, Ion ^a   Sinova, Jairo ^b,c   Jungwirth, T ^c,d   MacDonald, A H ^a  

^a UNIVERSITY OF TEXAS AT AUSTIN   (United States)

^b TEXAS A AND M UNIVERSITY   (United States)

^c INSTITUTE OF PHYSICS   (Czech Republic)

^d UNIVERSITY OF NOTTINGHAM   (United Kingdom)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 66049093467     PISSN: 10980121     EISSN: 1550235X     Source Type: Journal    
DOI: 10.1103/PhysRevB.79.155207     Document Type: Article

References (51)

1. For reviews see, e.g., B. L. Altshuler, A. G. Aronov, D. E. Khmelnitskii, and A. I. Larkin, in Quantum Theory of Solids, edited by, I. M. Lifshits, (Mir, Moscow, 1982);
2. G. Bergmann, Phys. Rep. 107, 1 (1984) 10.1016/0370-1573(84)90103-0
3. E. Akkermans and G. Montambaux, Mesoscopic Physics of Electrons (Cambridge University Press, London, 2007).
4. G. Bergmann, Phys. Rep. 107, 1 (1984). 10.1016/0370-1573(84)90103-0
5. Y. Lyanda-Geller, Phys. Rev. Lett. 80, 4273 (1998). 10.1103/PhysRevLett. 80.4273
6. W. Knap, C. Skierbiszewski, A. Zduniak, E. Litwin-Staszewska, D. Bertho, F. Kobbi, J. L. Robert, G. E. Pikus, F. G. Pikus, S. V. Iordanskii, V. Mosser, K. Zekentes, and Y. B. Lyanda-Geller, Phys. Rev. B 53, 3912 (1996). 10.1103/PhysRevB.53.3912
7. S. Pedersen, C. B. Sorensen, A. Kristensen, P. E. Lindelof, L. E. Golub, and N. S. Averkiev, Phys. Rev. B 60, 4880 (1999). 10.1103/PhysRevB.60.4880
8. E. McCann, K. Kechedzhi, V. I. Falko, H. Suzuura, T. Ando, and B. L. Altshuler, Phys. Rev. Lett. 97, 146805 (2006) 10.1103/PhysRevLett.97.146805
9. D. V. Khveshchenko, Phys. Rev. Lett. 97, 036802 (2006) 10.1103/PhysRevLett.97.036802
10. A. F. Morpurgo and F. Guinea, Phys. Rev. Lett. 97, 196804 (2006). 10.1103/PhysRevLett.97.196804
11. For a more rigorous statement, see T. Ando, T. Nakanishi, and R. Saito, J. Phys. Soc. Jpn. 67, 2857 (1998) 10.1143/JPSJ.67.2857
12. and also H. Suzuura and T. Ando, Phys. Rev. Lett. 89, 266603 (2002). 10.1103/PhysRevLett.89.266603
13. I. P. Smorchkova, N. Samarth, J. M. Kikkawa, and D. D. Awschalom, Phys. Rev. Lett. 78, 3571 (1997) 10.1103/PhysRevLett.78.3571
14. F. G. Aliev, E. Kunnen, K. Temst, K. Mae, G. Verbanck, J. Barnas, V. V. Moshchalkov, and Y. Bruynseraede, Phys. Rev. Lett. 78, 134 (1997) 10.1103/PhysRevLett.78.134
15. M. Rubinstein, F. J. Rachford, W. W. Fuller, and G. A. Prinz, Phys. Rev. B 37, 8689 (1988) 10.1103/PhysRevB.37.8689
16. S. Kobayashi, Y. Ootuka, F. Komori, and W. Sasaki, J. Phys. Soc. Jpn. 51, 689 (1982) 10.1143/JPSJ.51.689
17. M. Brands, A. Carl, and G. Dumpich, Europhys. Lett. 68, 268 (2004). 10.1209/epl/i2004-10191-8
18. K. Wagner, D. Neumaier, M. Reinwald, W. Wegscheider, and D. Weiss, Phys. Rev. Lett. 97, 056803 (2006). 10.1103/PhysRevLett.97.056803
19. D. Neumaier, K. Wagner, S. Geiszler, U. Wurstbauer, J. Sadowski, W. Wegscheider, and D. Weiss, Phys. Rev. Lett. 99, 116803 (2007) 10.1103/PhysRevLett.99.116803
20. D. Neumaier, K. Wagner, U. Wurstbauer, M. Reinwald, W. Wegscheider, and D. Weiss, New J. Phys. 10, 055016 (2008). 10.1088/1367-2630/10/5/055016
21. L. Vila, R. Giraud, L. Thevenard, A. Lemaitre, F. Pierre, J. Dufouleur, D. Mailly, B. Barbara, and G. Faini, Phys. Rev. Lett. 98, 027204 (2007). 10.1103/PhysRevLett.98.027204
22. F. Matsukura, M. Sawicki, T. Dietl, D. Chiba, and H. Ohno, Physica E 21, 1032 (2004). 10.1016/j.physe.2003.11.165
23. B. Kramer and A. MacKinnon, Rep. Prog. Phys. 56, 1469 (1993). 10.1088/0034-4885/56/12/001
24. L. P. Rokhinson, Y. Lyanda-Geller, Z. Ge, S. Shen, X. Liu, M. Dobrowolska, and J. K. Furdyna, Phys. Rev. B 76, 161201 (R) (2007). 10.1103/PhysRevB.76.161201
25. V. K. Dugaev, P. Bruno, and J. Barnas, Phys. Rev. B 64, 144423 (2001). 10.1103/PhysRevB.64.144423
26. S. Sil, P. Entel, G. Dumpich, and M. Brands, Phys. Rev. B 72, 174401 (2005). 10.1103/PhysRevB.72.174401
27. V. K. Dugaev, P. Bruno, and J. Barnas, Phys. Rev. Lett. 101, 129701 (2008). 10.1103/PhysRevLett.101.129701
28. D. Neumaier, K. Wagner, S. Geiszler, U. Wurstbauer, J. Sadowski, W. Wegscheider, and D. Weiss, Phys. Rev. Lett. 101, 129702 (2008). 10.1103/PhysRevLett.101.129702
29. T. Jungwirth, Jairo Sinova, J. Masek, J. Kucera, and A. H. MacDonald, Rev. Mod. Phys. 78, 809 (2006). 10.1103/RevModPhys.78.809
30. The AMR contribution to the conductance G in Ref. is ∼0.006 e2 /h per wire. The authors assume that the AMR contribution scales with the high-field conductance and therefore that it is reduced by about 10% between 65 and 20 mK. This ∼0.0006 e2 /h reduction in AMR G is a significant (30%) fraction of the conductance peak interpreted as due to WAL.
31. Another possible culprit for the early saturation is dimensionality crossover induced by an external magnetic field. However, one difficulty of this explanation is that even at zero fields the samples are not clearly in the one-dimensional (1D) limit.
32. K. S. Burch, D. B. Shrekenhamer, E. J. Singley, J. Stephens, B. L. Sheu, R. K. Kawakami, P. Schiffer, N. Samarth, D. D. Awschalom, and D. N. Basov, Phys. Rev. Lett. 97, 087208 (2006). 10.1103/PhysRevLett.97.087208
33. K. S. Burch, D. D. Awschalom, and D. N. Basov, J. Magn. Magn. Mater. 320, 3207 (2008). 10.1016/j.jmmm.2008.08.060
34. T. Jungwirth, J. Sinova, A. H. MacDonald, B. L. Gallagher, V. Novak, K. W. Edmonds, A. W. Rushforth, R. P. Campion, C. T. Foxon, L. Eaves, E. Olejnik, J. Masek, S. R. Eric Yang, J. Wunderlich, C. Gould, L. W. Molenkamp, T. Dietl, and H. Ohno, Phys. Rev. B 76, 125206 (2007). 10.1103/PhysRevB.76.125206
35. S. R. Eric Yang and A. H. MacDonald, Phys. Rev. B 67, 155202 (2003). 10.1103/PhysRevB.67.155202
36. A. P. Dmitriev, V. Y. Kachorovskii, and I. V. Gornyi, Phys. Rev. B 56, 9910 (1997). 10.1103/PhysRevB.56.9910
37. L. E. Golub, Phys. Rev. B 71, 235310 (2005). 10.1103/PhysRevB.71.235310
38. The Jz basis representation choice is motivated by convenience-it diagonalizes the impurity potential we consider. Choosing Jx or Jy representations would eventually lead to the same answer.
39. I. Garate and A. H. MacDonald, Phys. Rev. B 79, 064404 (2009). 10.1103/PhysRevB.79.064404
40. However, it will turn out that there are qualitative differences between the M2DEG model and more realistic models when it comes to evaluating quantum corrections to conductivity in magnetic semiconductors (see Sec. 4).
41. Our assumption neglects additional Q -dependent matrix elements in U and thereby in the Cooperon C. However, these extra terms are proportional to either Qx, Qy, or Qx Qy, which do not survive the angular part of the Q integral in Eq. 1 provided that the eigenstates in Eq. 5 are taken to be independent of Q. Neglecting the Q dependence of the eigenstates carries an error that goes like Q/k, where k is the momentum of the quasiparticle. Since only quasiparticles near the Fermi energy contribute to transport, the characteristic error of our approximation is Q/ kF ≤1/ (l kF), where l is the quasiparticle mean-free path. Hence our assumption is quantitatively accurate in conducting materials.
42. The precise values of F τ and τφ in the figures of Sec. 4 are somewhat arbitrary since M2DEGs bear little resemblance to ferromagnets studied experimentally. We choose F τ1 and τφ τ because (i) our calculation of quantum interference is most accurate in highly conducting materials and (ii) a large dephasing time makes quantum effects more robust and hence easier to analyze theoretically.
43. We have verified, for example, that when the chirality of the Rashba SO is doubled (i.e. when spinors rotate twice as fast along the Fermi circle), the singular mode would correspond to | 1,0 (triplet) rather than | 0,0 (singlet). Consequently, WAL is replaced by WL.
44. We note that in general spin-flip impurities dephase both WAL and WL. However, we are restricting our attention to impurities whose spins commute with the exchange field; these do not affect the equal spin correlations that contribute to WL as can be verified from Eq. 13.
45. For a review on (Ga,Mn)As, see, e.g., T. Jungwirth, J. Sinova, J. Masek, J. Kucera, and A. H. MacDonald, Rev. Mod. Phys. 78, 809 (2006). 10.1103/RevModPhys.78.809
46. N. Averkiev, L. E. Golub, and G. E. Pikus, Zh. Eksp. Teor. Fiz. 113, 1429 (1998) 10.1134/1.558539
47. N. Averkiev, L. E. Golub, and G. E. Pikus, [JETP 86, 780 (1998)].
48. While light-hole states are not spin coherent states, they still satisfy LH1,k | J | LH1,k = LH1,-k | J | LH1,-k and LH2,k | J | LH2,k = LH2,-k | J | LH2,-k.
49. The characteristic Hext at which quantum interference decays in Fig. 7 is much larger than that of Ref., where the WL suppression is evident at as low a field as 0.03 T. Our results are more in tune with Refs., where the decay of quantum corrections is observed at the level of several Tesla.
50. A. W. Rushforth, K. Vyborny, C. S. King, K. W. Edmonds, R. P. Campion, C. T. Foxon, J. Wunderlich, A. C. Irvine, P. Vasek, V. Novak, K. Olejnik, J. Sinova, T. Jungwirth, and B. L. Gallagher, Phys. Rev. Lett. 99, 147207 (2007). 10.1103/PhysRevLett.99.147207
51. R. N. Bhatt, P. Wolfle, and T. V. Ramakrishnan, Phys. Rev. B 32, 569 (1985). 10.1103/PhysRevB.32.569