Skip to main content
SpringerLink
Log in

Edge states and distributions of edge currents in semi-infinite graphene

  • Mesoscopic and Nanoscale Systems
  • Regular Article
  • Published:
The European Physical Journal B Aims and scope Submit manuscript

Abstract

The distributions of edge currents in semi-infinite graphene under a uniform perpendicular magnetic field are investigated. We show unambiguously that the edge current is finite at the armchair edge but vanishes at the zigzag edge. It is shown that the current density oscillates with the distance away from the boundary and tends to zero deep inside the graphene. The study shows that the total current is independent of edge configurations. The interplay of the bulk and edge contributions to the total current is presented. The quantized plateaus of Hall conductivity at (4e 2/h)(n+1/2) provide a direct evidence of the connection between the edge states and topological properties of relativistic fermions in a magnetic field.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. K.S. Novoselov, A.K. Geim, S.V. Morozov, D. Jiang, Y. Zhang, S.V. Dubonos, I.V. Grigorieva, A.A. Firsov, Science 306, 666 (2004)

    Article  ADS  Google Scholar 

  2. A.H. Castro Neto et al., Rev. Mod. Phys. 81, 109 (2009)

    Article  ADS  Google Scholar 

  3. K.S. Novoselov, A.K. Geim, S.V. Morozov, D. Jiang, M.I. Katsnelson, I.V. Grigorieva, S.V. Dubonos, A.A. Firsov, Nature (London) 438, 197 (2005)

    Article  ADS  Google Scholar 

  4. Y. Zhang, Y.W. Tan, H.L. Stormer, P. Kim, Nature (London) 438, 201 (2005)

    Article  ADS  Google Scholar 

  5. C. Berger, Z. Song, X. Li, X. Wu, N. Brown, C. Naud, D. Mayou, T. Li, J. Hass, A.N. Marchenkov, E.H. Konrad, P.N. First, W.A. de Heer, Science 312, 1191 (2006)

    Article  ADS  Google Scholar 

  6. H. Suzuura, T. Ando, Phys. Rev. Lett. 89, 266603 (2002)

    Article  ADS  Google Scholar 

  7. H. Suzuura, T. Ando, J. Phys. Soc. Jpn 72, 69 (2003)

    Article  ADS  Google Scholar 

  8. S.V. Morozov, K.S. Novoselov, M.I. Katsnelson, F. Schedin, L.A. Ponomarenko, D. Jiang, A.K. Geim, Phys. Rev. Lett. 97, 016801 (2006)

    Article  ADS  Google Scholar 

  9. D.V. Khveshchenko, Phys. Rev. Lett. 97, 036802 (2006)

    Article  ADS  Google Scholar 

  10. E. McCann et al., Phys. Rev. Lett. 97, 146805 (2006)

    Article  ADS  Google Scholar 

  11. G.W. Semennoff, Phys. Rev. Lett. 53, 2449 (1984)

    Article  ADS  Google Scholar 

  12. F.D.M. Haldane, Phys. Rev. Lett. 61, 2015 (1988)

    Article  ADS  MathSciNet  Google Scholar 

  13. N.M.R. Peres, F. Guinea, A.H. Castro Neto, Phys. Rev. B 73, 125411 (2006)

    Article  ADS  Google Scholar 

  14. K. Ziegler, Phys. Rev. Lett. 97, 266802 (2006)

    Article  ADS  Google Scholar 

  15. I.L. Aleiner, K.B. Efetov, Phys. Rev. Lett. 97, 236801 (2006)

    Article  ADS  Google Scholar 

  16. A. Altland, Phys. Rev. Lett. 97, 236802 (2006)

    Article  ADS  Google Scholar 

  17. J. Cserti, Phys. Rev. B 74, 033405 (2006)

    Article  Google Scholar 

  18. P.M. Ostrovsky, I.V. Gornyi, A.D. Mirlin, Phys. Rev. B 74, 235443 (2006)

    Article  ADS  Google Scholar 

  19. K. Nomura, A.H. MacDonald, Phys. Rev. Lett. 98, 076602 (2007)

    Article  ADS  Google Scholar 

  20. M. Fujita, K. Wakabayashi, K. Nakada, K.K.usakabe, J. Phys. Soc. Jpn 65, 1920 (1996)

    Article  ADS  Google Scholar 

  21. Y. Son, M.L. Cohen, S.G. Louie, Nature 444, 347 (2006)

    Article  ADS  Google Scholar 

  22. L. Pisani, J. Chan, B. Montanari, N. Harrison, Phys. Rev. B 75, 064418 (2007)

    Article  ADS  Google Scholar 

  23. O. Yazyev, L. Helm, Phys. Rev. B 75, 125408 (2007)

    Article  ADS  Google Scholar 

  24. M. Vozmediano, M.P. Lopez-Sancho, T. Stauber, F. Guinea, Phys. Rev. B 72, 155121 (2005)

    Article  ADS  Google Scholar 

  25. Y. Son, M.L. Cohen, S.G. Louie, Nature 444, 347 (2006)

    Article  ADS  Google Scholar 

  26. E.H. Hwang, S. Adam, S. Das Sarma, Phys. Rev. Lett. 98, 186806 (2007)

    Article  ADS  Google Scholar 

  27. X. Du, I. Skachko, F. Duerr, A. Luican, E.Y. Andrei, Nature 462, 192 (2009)

    Article  ADS  Google Scholar 

  28. H.B. Heersche et al., Nature 446, 56 (2007)

    Article  ADS  Google Scholar 

  29. M. Müller, J. Schmalian, L. Fritz, Phys. Rev. Lett. 103, 025301 (2009)

    Article  ADS  Google Scholar 

  30. B. Obradovic, R. Kotlyar, F. Heinz, P. Matagne, T. Rakshit, M.D. Giles, M.A. Stettler, D.E. Nikoniv, Appl. Phys. Lett. 88, 142102 (2006)

    Article  ADS  Google Scholar 

  31. Y. Ouyang, Y. Yoon, J.K. Fodor, J. Guo, Appl. Phys. Lett. 89, 203107 2006

    Google Scholar 

  32. Qimin Yan, Bing Huang, Jie Yu, Fawei Zheng, Ji Zang, Jian Wu, Bing-Lin Gu, Feng Liu, Wenhui Duan, Nano Lett. 7, 1469 (2007)

    Article  ADS  Google Scholar 

  33. R. Rammal, J. Phys. 46, 1345 (1985)

    Article  MathSciNet  Google Scholar 

  34. A. Barelli, J. Bellissard, R. Rammal, J. Phys. 51, 2167 (1990)

    Article  MathSciNet  Google Scholar 

  35. R.R. Gerhardts, D. Pfannkuche, Phys. Rev. B 53, 9591 (1996)

    Article  ADS  Google Scholar 

  36. A.Y. Rom, Phys. Rev. B 55, 11025 (1997)

    Article  ADS  Google Scholar 

  37. Y. Iye, E. Kuramochi, M. Hara, A. Endo, S. Katsumoto, Phys. Rev. B 70, 144524 (2004)

    Article  ADS  Google Scholar 

  38. M.C. Geisler, J.H. Smet, V. Umansky, K. von Klitzing, B. Naundorf, R. Ketzmerick, H. Schweizer, Phys. Rev. Lett. 92, 256801 (2004)

    Article  ADS  Google Scholar 

  39. Y.S. Zheng, T. Ando, Phys. Rev. B 65, 245420 (2002)

    Article  ADS  Google Scholar 

  40. V.P. Gusynin, S.G. Sharapov, Phys. Rev. Lett. 95, 146801 (2005)

    Article  ADS  Google Scholar 

  41. K. Wakabayashi, M. Fujita, H. Ajiki, M. Sigrist, Phys. Rev. B 59, 8271 (1999)

    Article  ADS  Google Scholar 

  42. B.I. Halperin, Phys. Rev. B 25, 2185 (1982)

    Article  ADS  MathSciNet  Google Scholar 

  43. A.H. MacDonald, P. Streda, Phys. Rev. B 29, 1616 (1984)

    Article  ADS  Google Scholar 

  44. M. Buttiker, Phys. Rev. B 38, 9375 (1988)

    Article  ADS  MathSciNet  Google Scholar 

  45. Y. Hasegawa, M. Kohmoto, Phys. Rev. B 74, 155415 (2006)

    Article  ADS  Google Scholar 

  46. D.-K. Ki, S. Jo, H.-J. Lee, Appl. Phys. Lett. 94, 162113 (2009)

    Article  ADS  Google Scholar 

  47. L. Brey, H.A. Fertig, Phys. Rev. B 73, 195408 (2006)

    Article  ADS  Google Scholar 

  48. D.J. Thouless, M. Kohmoto, M.P. Nightingale, M. den Nijs, Phys. Rev. Lett. 49, 405 (1982)

    Article  ADS  Google Scholar 

  49. J. Avron, R. Seiler, B. Simon, Phys. Rev. Lett. 51, 51 (1983)

    Article  ADS  Google Scholar 

  50. B. Simon, Phys. Rev. Lett. 51, 2167 (1983)

    Article  ADS  MathSciNet  Google Scholar 

  51. Q. Niu, D.J. Thouless, Y.S. Wu, Phys. Rev. B 31, 3372 (1985)

    Article  ADS  MathSciNet  Google Scholar 

  52. Q. Niu, D.J. Thouless, J. Phys. A 17, 2453 (1984)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  53. J. Avron, R. Seiler, Phys. Rev. Lett. 54, 2599 (1985)

    Article  MathSciNet  Google Scholar 

  54. Y. Hatsugai, Phys. Rev. Lett. 71, 3697 (1993)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  55. V.P. Gusynin et al., Phys. Rev. B 77, 205409 (2008)

    Article  ADS  Google Scholar 

  56. V.P. Gusynina, V.A. Miranskyb, S.G. Sharapov, I.A. Shovkovyc, Low Temp. Phys. 34, 778 (2008)

    Article  ADS  Google Scholar 

  57. L. Brey, H.A. Fertig, Phys. Rev. B 73, 235411 (2006)

    Article  ADS  Google Scholar 

  58. D.A. Abanin, P.A. Lee, L.S. Levitov, Phys. Rev. Lett. 96, 176803 (2006))

    Article  ADS  Google Scholar 

  59. D.A. Abanin, P.A. Lee, L.S. Levitov, Solid State Commun. 143, 77 (2007)

    Article  ADS  Google Scholar 

  60. P. Delplace, G. Montambaux, Phys. Rev. B 82, 205412 (2010)

    Article  ADS  Google Scholar 

  61. M.V. Entin, L.I. Magarill, M.M. Mahmoodian, JETP 97, 138 (2003)

    Article  ADS  Google Scholar 

  62. M. Arikawa, Y. Hatsugai, H. Aoki, J. Phys. Conf. Ser. 150, 022003 (2009)

    Article  ADS  Google Scholar 

  63. Y. One, T. Ohtsuki, Z. Phys., B, Condens. Matter 68, 445 (1987)

    Article  Google Scholar 

  64. M.R. Geller, G. Vignale, Phys. Rev. B 52, 14137 (1995)

    Article  ADS  Google Scholar 

  65. K. Shizuya, Phys. Rev. Lett. 73, 2907 (1994)

    Article  ADS  Google Scholar 

  66. A.M.J. Schakel, G.W. Semennoff, Phys. Rev. Lett. 66, 2653 (1991)

    Article  ADS  Google Scholar 

  67. M. Heuser, J. Hajdu, Z. Phys. 270, 289 (1974)

    Article  ADS  Google Scholar 

  68. L. Smrčka, J. Phys. C Solid State Phys. 17, L63 (1984)

    Article  Google Scholar 

  69. P. Středa, L. Smrčka, J. Phys. C Solid State Phys. 16, L895 (1983)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Physics, Peking University, Beijing, 100871, P.R. China

    W. Wang & Z. S. Ma

Authors
  1. W. Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Z. S. Ma
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Z. S. Ma.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Wang, W., Ma, Z.S. Edge states and distributions of edge currents in semi-infinite graphene. Eur. Phys. J. B 81, 431–439 (2011). https://doi.org/10.1140/epjb/e2011-20242-7

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1140/epjb/e2011-20242-7

Keywords

Search

Navigation