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Electrical Conduction of Edge States

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A Short Course on Topological Insulators

Part of the book series: Lecture Notes in Physics ((LNP,volume 919))

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Abstract

Electrical conduction in clean, impurity-free nanostructures at low temperatures qualitatively deviate from the behavior of Ohmic conductors. We demonstrate such deviations using a simple zero-temperature model of a clean and phase-coherent metallic wire, leading us toward the Landauer-Büttiker description of phase-coherent electrical conduction. We also discuss how scattering at static impurities affects electrical conduction in general. As the main subject of the chapter, we show how the presence of edge states in two-dimensional topological insulators can have an easily measurable physical consequence: a nonvanishing, quantized conductance, even in the presence of disorder.

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References

  1. A.A. Aligia, G. Ortiz, Quantum mechanical position operator and localization in extended systems. Phys. Rev. Lett. 82, 2560–2563 (1999)

    Article  ADS  Google Scholar 

  2. Y. Ando, Topological insulator materials. J. Phys. Soc. Jpn. 82(10), 102001 (2013)

    Google Scholar 

  3. G. Bastard, Wave Mechanics Applied to Semiconductor Heterostructures (Les Editions de Physique, Les Ulis, 1988)

    Google Scholar 

  4. B.A. Bernevig, Topological Insulators and Topological Superconductors (Princeton University Press, Princeton, 2013)

    MATH  Google Scholar 

  5. B.A. Bernevig, T.L. Hughes, S.-C. Zhang, Quantum spin hall effect and topological phase transition in HgTe quantum wells. Science 314, 1757 (2006)

    Article  ADS  Google Scholar 

  6. M.V. Berry, Quantal phase factors accompanying adiabatic changes. Proc. R. Soc. Lond. A 392, 45–57 (1984)

    Article  ADS  MATH  Google Scholar 

  7. J.C. Budich, B. Trauzettel, From the adiabatic theorem of quantum mechanics to topological states of matter. Physica Status Solidi RRL 7(1–2), 109–129 (2013)

    Article  ADS  Google Scholar 

  8. C.-Z. Chang, J. Zhang, X. Feng, J. Shen, Z. Zhang, M. Guo, K. Li, Y. Ou, P. Wei, L.-L. Wang, et al., Experimental observation of the quantum anomalous hall effect in a magnetic topological insulator. Science 340(6129), 167–170 (2013)

    Article  ADS  Google Scholar 

  9. L. Du, I. Knez, G. Sullivan, R.-R. Du, Robust helical edge transport in gated InAs∕GaSb bilayers. Phys. Rev. Lett. 114, 096802 (2015)

    Article  ADS  Google Scholar 

  10. M. Franz, L. Molenkamp, Topological Insulators, vol. 6 (Elsevier, Oxford, 2013)

    Google Scholar 

  11. L. Fu, C.L. Kane, Time reversal polarization and a Z 2 adiabatic spin pump. Phys. Rev. B 74, 195312 (2006)

    Article  ADS  Google Scholar 

  12. T. Fukui, Y. Hatsugai, H. Suzuki, Chern numbers in discretized brillouin zone: efficient method of computing (spin) hall conductances. J. Phys. Soc. Jpn. 74(6), 1674–1677 (2005)

    Article  ADS  Google Scholar 

  13. I.C. Fulga, F. Hassler, A.R. Akhmerov, Scattering theory of topological insulators and superconductors. Phys. Rev. B 85, 165409 (2012)

    Article  ADS  Google Scholar 

  14. A. Garg, Berry phases near degeneracies: Beyond the simplest case. Am. J. Phys. 78(7), 661–670 (2010)

    Article  ADS  Google Scholar 

  15. D.J. Griffiths, Introduction to Quantum Mechanics (Pearson Education Limited, Harlow, 2014)

    Google Scholar 

  16. F. Grusdt, D. Abanin, E. Demler, Measuring Z 2 topological invariants in optical lattices using interferometry. Phys. Rev. A 89, 043621 (2014)

    Article  ADS  Google Scholar 

  17. M.Z. Hasan, C.L. Kane, Colloquium: topological insulators. Rev. Mod. Phys. 82, 3045 (2010)

    Article  ADS  Google Scholar 

  18. B.R. Holstein, The adiabatic theorem and Berry’s phase. Am. J. Phys. 57(12), 1079–1084 (1989)

    Article  ADS  MathSciNet  Google Scholar 

  19. C.L. Kane, E.J. Mele, Z 2 topological order and the quantum spin hall effect. Phys. Rev. Lett. 95, 146802 (2005)

    Article  ADS  Google Scholar 

  20. M. König, H. Buhmann, L.W. Molenkamp, T. Hughes, C.-X. Liu, X.-L. Qi, S.-C. Zhang, The quantum spin hall effect: theory and experiment. J. Phys. Soc. Jpn. 77(3), 031007 (2008)

    Google Scholar 

  21. M. König, S. Wiedmann, C. Brüne, A. Roth, H. Buhmann, L.W. Molenkamp, X.-L. Xi, S.-C. Zhang, Quantum spin hall insulator state in HgTe quantum wells. Science 318(6), 766–770 (2007)

    Article  ADS  Google Scholar 

  22. C. Liu, T.L. Hughes, X.-L. Qi, K. Wang, S.-C. Zhang, Quantum spin hall effect in inverted type-ii semiconductors. Phys. Rev. Lett. 100, 236601 (2008)

    Article  ADS  Google Scholar 

  23. N. Marzari, A.A. Mostofi, J.R. Yates, I. Souza, D. Vanderbilt, Maximally localized wannier functions: Theory and applications. Rev. Mod. Phys. 84, 1419–1475 (2012)

    Article  ADS  Google Scholar 

  24. X.-L. Qi, Y.-S. Wu, S.-C. Zhang, Topological quantization of the spin hall effect in two-dimensional paramagnetic semiconductors. Phys. Rev. B 74, 085308 (2006)

    Article  ADS  Google Scholar 

  25. X.-L. Qi, S.-C. Zhang, Topological insulators and superconductors. Rev. Mod. Phys. 83, 1057–1110 (2011)

    Article  ADS  Google Scholar 

  26. R. Resta, Berry Phase in Electronic Wavefunctions. Troisieme Cycle de la Physique en Suisse Romande (1996)

    Google Scholar 

  27. R. Resta, Macroscopic polarization from electronic wavefunctions. arXiv preprint cond-mat/9903216 (1999)

    Google Scholar 

  28. R. Resta, What makes an insulator different from a metal? arXiv preprint cond-mat/0003014 (2000)

    Google Scholar 

  29. S. Ryu, A.P. Schnyder, A. Furusaki, A.W.W. Ludwig, Topological insulators and superconductors: tenfold way and dimensional hierarchy. New J. Phys. 12(6), 065010 (2010)

    Google Scholar 

  30. S.-Q. Shen, Topological insulators: Dirac equation in condensed matter. Springer Ser. Solid-State Sci. 174 (2012)

    Google Scholar 

  31. A.A. Soluyanov, D. Vanderbilt, Smooth gauge for topological insulators. Phys. Rev. B 85, 115415 (2012)

    Article  ADS  Google Scholar 

  32. J. Sólyom, Fundamentals of the Physics of Solids: Volume III: Normal, Broken-Symmetry, and Correlated Systems, vol. 3 (Springer Science & Business Media, Berlin, 2008)

    Google Scholar 

  33. D.J. Thouless, Quantization of particle transport. Phys. Rev. B 27, 6083–6087 (1983)

    Article  ADS  MathSciNet  Google Scholar 

  34. G.E. Volovik, The Universe in a Helium Droplet (Oxford University Press, New York, 2009)

    Book  Google Scholar 

  35. F. Wilczek, A. Zee, Appearance of gauge structure in simple dynamical systems. Phys. Rev. Lett. 52, 2111–2114 (1984)

    Article  ADS  MathSciNet  Google Scholar 

  36. D. Xiao, M.-C. Chang, Q. Niu, Berry phase effects on electronic properties. Rev. Mod. Phys. 82, 1959–2007 (2010)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  37. R. Yu, X.L. Qi, A. Bernevig, Z. Fang, X. Dai, Equivalent expression of Z 2 topological invariant for band insulators using the non-abelian Berry connection. Phys. Rev. B 84, 075119 (2011)

    Article  ADS  Google Scholar 

  38. J. Zak, Berry’s phase for energy bands in solids. Phys. Rev. Lett. 62, 2747–2750 (1989)

    Article  ADS  Google Scholar 

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Authors and Affiliations

  1. Wigner Research Centre for Physics, Hungarian Academy of Sciences, Budapest, Hungary

    János K. Asbóth

  2. Department of Physics of Complex Systems, Eötvös Loránd University, Budapest, Hungary

    László Oroszlány

  3. Department of Materials Physics, Eötvös Loránd University, Budapest, Hungary

    András Pályi

  4. Department of Physics, Budapest University of Technology and Economics, Budapest, Hungary

    András Pályi

Authors
  1. János K. Asbóth
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  2. László Oroszlány
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  3. András Pályi
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Asbóth, J.K., Oroszlány, L., Pályi, A. (2016). Electrical Conduction of Edge States. In: A Short Course on Topological Insulators. Lecture Notes in Physics, vol 919. Springer, Cham. https://doi.org/10.1007/978-3-319-25607-8_10

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