Advertisement

Anomalous Hall Effect

  • Yuriy MokrousovEmail author
Chapter
Part of the Springer Series in Solid-State Sciences book series (SSSOL, volume 192)

Abstract

The anomalous Hall effect (AHE) is one of the most fundamental, practically important and for a long time most enigmatic phenomena exhibited by magnetic materials. Here, we briefly outline the relation of the anomalous Hall effect to the geometric properties of the electronic states as given by the Berry phase. The Berry phase origins of the AHE lead to its topological manifestations in insulators, which we review in detail based on key examples. In addition to the intrinsic AHE and its anisotropy in solids, we draw a deep correlation of this effect with orbital magnetism and magnetoelectric response, and discuss its emergence in non-collinear magnets.

Notes

Acknowledgements

I would like to thank Frank Freimuth, Hongbin Zhang, Jürgen Weischenberg, Jan-Philipp Hanke, Patrick Buhl, Chengwang Niu, Gustav Bihlmayer, Klaus Seemann, Christopher Marrows, Martin Gradhand, Diemo Ködderitzsch, Hubert Ebert, Jairo Sinova, Stefan Blügel, Jakub Železný, Bernd Zimmermann and Ivo Souza for numerous discussions on the anomalous Hall effect.

References

  1. 1.
    E.H. Hall, Am. J. Math. 2, 287 (1879)CrossRefGoogle Scholar
  2. 2.
    E.H. Hall, Philos. Mag. 10, 301 (1880)CrossRefGoogle Scholar
  3. 3.
    C.M. Hurd, (Plenum, New York, 1972)Google Scholar
  4. 4.
    C.L. Chien, C.R. Westgate, (Plenum, New York, 1980)Google Scholar
  5. 5.
    J.E. Hirsch, Phys. Rev. Lett. 83, 1834 (1999)ADSCrossRefGoogle Scholar
  6. 6.
    J. Sinova et al., Phys. Rev. Lett. 92, 126603 (2004)ADSCrossRefGoogle Scholar
  7. 7.
    N.A. Sinitsyn, J. Phys.: Condens. Matter 20, 023201 (2008)ADSGoogle Scholar
  8. 8.
    D. Xiao, M.-C. Chang, Q. Niu, Rev. Mod. Phys. 82, 1959 (2010)ADSCrossRefGoogle Scholar
  9. 9.
    N. Nagaosa, J. Sinova, S. Onoda, A.H. MacDonald, N.P. Ong, Rev. Mod. Phys. 82, 1539 (2010)ADSCrossRefGoogle Scholar
  10. 10.
    M.V. Berry, Proc. R. Soc. Lond. A 392, 45 (1984)ADSCrossRefGoogle Scholar
  11. 11.
    A. Bohm, A. Mostafazadeh, H. Koizumi, Q. Niu, J. Zwanziger, The Geometric Phase in Quantum Systems (Springer, Berlin, 2003)zbMATHCrossRefGoogle Scholar
  12. 12.
    Y. Yao et al., Phys. Rev. Lett. 92, 037204 (2004)ADSCrossRefGoogle Scholar
  13. 13.
    Y. Mokrousov et al., J. Phys.: Cond. Mat. 25, 163201 (2013)Google Scholar
  14. 14.
    T. Jungwirth, Q. Niu, A.H. MacDonald, Phys. Rev. Lett. 88, 207208 (2002)ADSCrossRefGoogle Scholar
  15. 15.
    T. Jungwirth et al., Appl. Phys. Lett. 83, 320 (2003)ADSCrossRefGoogle Scholar
  16. 16.
    Z. Fang et al., Science 302, 92 (2003)ADSCrossRefGoogle Scholar
  17. 17.
    P. Czaja et al., Phys. Rev. B 89, 014411 (2014)ADSCrossRefGoogle Scholar
  18. 18.
    J. Weischenberg et al., Phys. Rev. Lett. 89, 014411 (2014)Google Scholar
  19. 19.
    B. Zimmermann et al., Phys. Rev. B 90, 220403 (2014)ADSCrossRefGoogle Scholar
  20. 20.
    P.N. Dheer, Phys. Rev. 156, 637 (1967)ADSCrossRefGoogle Scholar
  21. 21.
    J. Kötzler, W. Gil, Phys. Rev. B 72, 060412 (2005)ADSCrossRefGoogle Scholar
  22. 22.
    J. Smit, Physica 21, 877 (1955)ADSCrossRefGoogle Scholar
  23. 23.
    Y. Tian et al., Phys. Rev. Lett. 103, 087206 (2009)ADSCrossRefGoogle Scholar
  24. 24.
    S. Lowitzer et al., Phys. Rev. Lett. 105, 266604 (2010)ADSCrossRefGoogle Scholar
  25. 25.
    H. Ebert et al., Phys. Rev. B 91, 165132 (2015)ADSCrossRefGoogle Scholar
  26. 26.
    X. Wang, D. Vanderbilt, J.R. Yates, I. Souza, Phys. Rev. B 76, 195109 (2007)ADSCrossRefGoogle Scholar
  27. 27.
    M. Bode et al., Phys. Rev. Lett. 89, 237205 (2002)ADSCrossRefGoogle Scholar
  28. 28.
    C. Gould et al., Phys. Rev. Lett. 93, 117203 (2004)ADSCrossRefGoogle Scholar
  29. 29.
    T. McGuire, R. Potter, IEEE Trans. Magn. 11, 1018 (1975)ADSCrossRefGoogle Scholar
  30. 30.
    J. Velev et al., Phys. Rev. Lett. 94, 127203 (2005)ADSCrossRefGoogle Scholar
  31. 31.
    K.M. Seemann et al., Phys. Rev. Lett. 107, 086603 (2011)ADSCrossRefGoogle Scholar
  32. 32.
    B.G. Park et al., Phys. Rev. Lett. 100, 087204 (2008)ADSCrossRefGoogle Scholar
  33. 33.
    R.C. Fivaz, Phys. Rev. 183, 586 (1969)ADSCrossRefGoogle Scholar
  34. 34.
    H. Zhang, S. Blügel, Y. Mokrousov, Phys. Rev. B 84, 024401 (2011)ADSCrossRefGoogle Scholar
  35. 35.
    E. Roman et al., Phys. Rev. Lett. 103, 097203 (2009)ADSCrossRefGoogle Scholar
  36. 36.
    A. Hirsch, Y. Weissman, Phys. Lett. A 44, 239 (1973)ADSCrossRefGoogle Scholar
  37. 37.
    N. Volkenshtein et al., Fiz. Metal. Metalloved. 11, 152 (1961)Google Scholar
  38. 38.
    T. Hiraoka, J. Sci. Hiroshima Univ., Ser. A-2 32, 153 (1968)Google Scholar
  39. 39.
    R.S. Lee, S. Legvold, Phys. Rev. 162, 431 (1967)ADSCrossRefGoogle Scholar
  40. 40.
    K. Ohgushi et al., J. Phys. Soc. Jpn. 75, 013710 (2006)ADSCrossRefGoogle Scholar
  41. 41.
    B.C. Sales et al., Phys. Rev. B 77, 024409 (2008)ADSCrossRefGoogle Scholar
  42. 42.
    J. Stankiewicz, K.P. Skokov, Phys. Rev. B 78, 214435 (2008)ADSCrossRefGoogle Scholar
  43. 43.
    J. Stankiewicz et al., Phys. Rev. B 83, 014419 (2011)ADSCrossRefGoogle Scholar
  44. 44.
    N.H. Long et al., Phys. Rev. B 90, 064406 (2014)ADSCrossRefGoogle Scholar
  45. 45.
    H. Zhang et al., Phys. Rev. B 87, 205132 (2013)ADSCrossRefGoogle Scholar
  46. 46.
    J. Bellissard, arXiv:cond-mat/9504030
  47. 47.
    D.J. Thouless, M. Kohmoto, M.P. Nightingale, M. den Nijs, Phys. Rev. Lett. 49, 405 (1982)ADSCrossRefGoogle Scholar
  48. 48.
    C. Chang et al., Science 340, 167 (2013)ADSCrossRefGoogle Scholar
  49. 49.
    F.D.M. Haldane, Phys. Rev. Lett. 61, 2015 (1988)ADSCrossRefGoogle Scholar
  50. 50.
    Y. Ren et al., Rep. Prog. Phys. 79, 066501 (2016)ADSCrossRefGoogle Scholar
  51. 51.
    C.L. Kane, E.J. Mele, Phys. Rev. Lett. 95, 226801 (2005)ADSCrossRefGoogle Scholar
  52. 52.
    C. Chang et al., Nat. Mater. 14, 473 (2015)ADSCrossRefGoogle Scholar
  53. 53.
    C. Liu et al., Phys. Rev. Lett. 101, 146802 (2008)ADSCrossRefGoogle Scholar
  54. 54.
    Q. Wang et al., Phys. Rev. Lett. 113, 147201 (2014)ADSCrossRefGoogle Scholar
  55. 55.
    Z. Qiao et al., Phys. Rev. B 82, 161414 (2010)ADSCrossRefGoogle Scholar
  56. 56.
    J. Ding et al., Phys. Rev. B 84, 195444 (2011)ADSCrossRefGoogle Scholar
  57. 57.
    H. Zhang et al., Phys. Rev. Lett. 108, 056802 (2012)ADSCrossRefGoogle Scholar
  58. 58.
    Z. Qiao et al., Phys. Rev. Lett. 112, 116404 (2014)ADSCrossRefGoogle Scholar
  59. 59.
    S. Wu et al., Phys. Rev. Lett. 113, 256401 (2014)ADSCrossRefGoogle Scholar
  60. 60.
    C. Niu et al., Phys. Rev. B 91, 041303 (2015)ADSCrossRefGoogle Scholar
  61. 61.
    K. Garrity et al., Phys. Rev. Lett. 110, 116802 (2013)ADSCrossRefGoogle Scholar
  62. 62.
    K. Garrity et al., Phys. Rev. B 90, 041109 (2014)CrossRefGoogle Scholar
  63. 63.
    H. Zhang et al., Phys. Rev. Lett. 112, 096804 (2014)ADSCrossRefGoogle Scholar
  64. 64.
    X. Sheng, B. Nikolić, Phys. Rev. B 95, 201402 (2017)ADSCrossRefGoogle Scholar
  65. 65.
    Y. Chen et al., Phys. Rev. B 90, 195145 (2014)ADSCrossRefGoogle Scholar
  66. 66.
    H. Zhang et al., Phys. Rev. B 90, 156143 (2014)Google Scholar
  67. 67.
    J.-P. Hanke et al., Nat. Comm. 8, 1479 (2017)ADSCrossRefGoogle Scholar
  68. 68.
    B.A. Bernevig, T.L. Hughes, Topological Insulators and Topological Superconductors (Princeton University Press, 2013)Google Scholar
  69. 69.
    A. Chernyshov et al., Nat. Phys. 5, 656 (2009)CrossRefGoogle Scholar
  70. 70.
    I.M. Miron et al., Nat. Mater. 9, 230 (2010)ADSCrossRefGoogle Scholar
  71. 71.
    K. Garello et al., Nat. Nanotech. 8, 587 (2013)ADSCrossRefGoogle Scholar
  72. 72.
    I.M. Miron et al., Nature 476, 189 (2011)ADSCrossRefGoogle Scholar
  73. 73.
    L. Liu et al., Phys. Rev. Lett. 109, 096602 (2012)ADSCrossRefGoogle Scholar
  74. 74.
    P. Wadley et al., Science 351, 6273 (2016)CrossRefGoogle Scholar
  75. 75.
    F. Freimuth et al., Phys. Rev. B 90, 174423 (2014)ADSCrossRefGoogle Scholar
  76. 76.
    I. Dzyaloshinskii, J. Phys. Chem. Sol. 4, 241 (1958)ADSCrossRefGoogle Scholar
  77. 77.
    T. Moriya, Phys. Rev. 120, 91 (1960)ADSCrossRefGoogle Scholar
  78. 78.
    N. Kanazawa et al., Phys. Rev. Lett. 106, 156603 (2011)ADSCrossRefGoogle Scholar
  79. 79.
    A. Neubauer et al., Phys. Rev. Lett. 102, 186602 (2009)ADSCrossRefGoogle Scholar
  80. 80.
    C. Franz et al., Phys. Rev. Lett. 112, 186601 (2014)ADSCrossRefGoogle Scholar
  81. 81.
    J. Gayles et al., Phys. Rev. Lett. 115, 036602 (2015)ADSCrossRefGoogle Scholar
  82. 82.
    F. Freimuth et al., J. Phys.: Cond. Mat. 26, 104202 (2014)Google Scholar
  83. 83.
    J. Kübler, C. Felser, Europ. Phys. Lett. 108, 67001 (2014)ADSCrossRefGoogle Scholar
  84. 84.
    M. Hoffmann et al., Phys. Rev. B 92, 020401(R) (2015)ADSCrossRefGoogle Scholar
  85. 85.
    C.O. Avci et al., Nat. Mater. 16, 309 (2017)ADSCrossRefGoogle Scholar
  86. 86.
    Y.-H. Chu et al., Nat. Mater. 6, 478 (2008)ADSCrossRefGoogle Scholar
  87. 87.
    W. Zhang et al., Phys. Rev. Lett. 113, 196602 (2014)ADSCrossRefGoogle Scholar
  88. 88.
    H. Chen et al., Phys. Rev. Lett. 112, 017205 (2014)ADSCrossRefGoogle Scholar
  89. 89.
    S. Nakatsuji et al., Nature 527, 212 (2015)ADSCrossRefGoogle Scholar
  90. 90.
    A.K. Nayak et al., Sci. Adv. 2, 150187 (2016)CrossRefGoogle Scholar
  91. 91.
    T. Tomizawa, H. Kontani, Phys. Rev. B 80, 100401(R) (2009)ADSCrossRefGoogle Scholar
  92. 92.
    T. Tomizawa, H. Kontani, Phys. Rev. B 82, 104412 (2010)ADSCrossRefGoogle Scholar
  93. 93.
    Y. Zhang et al., Phys. Rev. B 95, 075128 (2017)ADSCrossRefGoogle Scholar
  94. 94.
    R. Shindou, N. Nagaosa, Phys. Rev. Lett. 87, 116801 (2001)ADSCrossRefGoogle Scholar
  95. 95.
    P. Bruno et al., Phys. Rev. Lett. 93, 096806 (2004)ADSCrossRefGoogle Scholar
  96. 96.
    P.M. Buhl et al., Phys. Status Solidi RRL 11, 1700007 (2017)CrossRefGoogle Scholar
  97. 97.
    J. Železný et al., Phys. Rev. Lett. 119, 187204 (2017)ADSCrossRefGoogle Scholar
  98. 98.
    Y. Machida et al., Phys. Rev. Lett. 98, 057203 (2007)ADSCrossRefGoogle Scholar
  99. 99.
    Y. Taguchi et al., Science 291, 2573 (2001)ADSCrossRefGoogle Scholar
  100. 100.
    Y. Machida et al., Nat. (Lond.) 463, 210 (2010)ADSCrossRefGoogle Scholar
  101. 101.
    C. Sürgers et al., Nat. Comm. 5, 3400 (2014)CrossRefGoogle Scholar
  102. 102.
    J.-P. Hanke et al., Sci. Rep. 7, 41078 (2017)ADSCrossRefGoogle Scholar
  103. 103.
    S. Heinze et al., Nat. Phys. 7, 713 (2011)CrossRefGoogle Scholar
  104. 104.
    J.-P. Hanke et al., Phys. Rev. B 94, 121114(R) (2016)ADSCrossRefGoogle Scholar
  105. 105.
    J. Zhou et al., Phys. Rev. Lett. 116, 256601 (2016)ADSCrossRefGoogle Scholar
  106. 106.
    M.d. Santos Dias et al., Nat. Comm. 7, 13613 (2016)Google Scholar
  107. 107.
    T. Thonhauser, Int. J. Mod. Phys. B 25, 1429 (2011)ADSCrossRefGoogle Scholar
  108. 108.
    D. Go et al., Sci. Rep. 7, 46742 (2017)ADSCrossRefGoogle Scholar
  109. 109.
    T. Yoda et al., Sci. Rep. 5, 12024 (2015)ADSCrossRefGoogle Scholar
  110. 110.
    S. Zhong et al., Phys. Rev. Lett. 116, 077201 (2016)ADSCrossRefGoogle Scholar
  111. 111.
    F. Lux et al., arXiv:1807.05040 (2018)
  112. 112.
    M. Menzel et al., Phys. Rev. Lett. 108, 197204 (2012)ADSCrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institute for Advanced SimulationForschungszentrum Jülich GmbHJülichGermany
  2. 2.Institute of PhysicsJohannes Gutenberg-University MainzMainzGermany

Personalised recommendations