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Landau Levels and Edge States in Graphene with Strong Spin-Orbit Coupling

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Low-Dimensional Functional Materials

Abstract

We investigate the electronic properties of graphene in a magnetic and a strain-induced pseudo-magnetic field in the presence of strong spin-orbit interactions (SOI). For a homogeneous field we provide analytical results for the Landau level eigenstates for arbitrary intrinsic and Rashba SOI, including also the effect of a Zeeman field. We then study the edge states in a semi-infinite geometry in the absence of the Rashba term. We find that, for a critical value of the magnetic field, a quantum phase transition occurs, which separates two phases both with spin-filtered helical edge states but with opposite direction of the spin current. Finally,we discuss magnetic waveguides with inhomogeneous field profiles that allow for chiral snake orbits. Such waveguides are practically immune to disorder-induced backscattering, and the SOI provides non-trivial spin texture to these modes.

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Notes

  1. 1.

    A different representation for \(\mathcal{T}\) has been given in Ref. [2] because of a different arrangement of the sublattice components in the spinor (Eq. 8.1).

  2. 2.

    For notational convenience, we shift p + 1 → p for s = ↓ in the discussion of the purely intrinsic SOI.

  3. 3.

    See footnote 2.

  4. 4.

    We note that we made a wrong statement in Ref. [32] in that direction: For the case j < 0 on page 3 therein, we stated that for a′ = 0 there are two normalizable states with E =  ± | M | which for M → 0 coalesce into a single zero-energy Landau level. However, only the state \(E = -\vert M\vert \) is allowed, and the other one is not normalizable.

  5. 5.

    See footnote 2.

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Acknowledgements

We acknowledge financial support by the DFG programs SPP 1459 and SFB TR 12.

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Authors and Affiliations

  1. Department of Mathematical Science, City University London, Northampton Square, London, EC1V 0HB, UK

    Alessandro De Martino

  2. Institut für Theoretische Physik, Heinrich-Heine-Universit, D-40225, Düsseldorf, Germany

    Artur Hütten & Reinhold Egger

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  1. Alessandro De Martino
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  2. Artur Hütten
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  3. Reinhold Egger
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Correspondence to Alessandro De Martino .

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Editors and Affiliations

  1. , Institut für Theoretische Physik IV, Heinrich-Heine-Universität, Universitätsstrasse 1, Düsseldorf, 40225, Germany

    Reinhold Egger

  2. Laboratory for Advanced Studies, Department of Energy, Turin Polytechnic University in Tashkent, Niyazov St. 17, Tashkent, 100174, Uzbekistan

    Davron Matrasulov

  3. , Department of Physics, National University of Uzbekistan, Niyazov St. 17, Tashkent, 100174, Uzbekistan

    Khamdam Rakhimov

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De Martino, A., Hütten, A., Egger, R. (2013). Landau Levels and Edge States in Graphene with Strong Spin-Orbit Coupling. In: Egger, R., Matrasulov, D., Rakhimov, K. (eds) Low-Dimensional Functional Materials. NATO Science for Peace and Security Series B: Physics and Biophysics. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-6618-1_8

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