Tunable quantum Hall edge conduction in bilayer graphene through spin-orbit interaction

Jun Yong Khoo and Leonid Levitov

Phys. Rev. B 98, 115307 – Published 26 September 2018

Abstract

Bilayer graphene, in the presence of a one-sided spin-orbit interaction (SOI) induced by a suitably chosen substrate, is predicted to exhibit unconventional quantum Hall states. The new states arise due to strong SOI-induced splittings of the eight zeroth Landau levels, which are strongly layer polarized, residing fully or partially on one of the two graphene layers. In particular, an Ising SOI on the meV scale is sufficient to invert the Landau level order between the $n = 0$ and $n = 1$ orbital levels under moderately weak magnetic fields $B ≲ 10$ T. Furthermore, when the Ising field opposes the $B$ field, the order of the spin-polarized levels can also be inverted. We show that, under these conditions, three different compensated electron-hole phases, with equal concentrations of electrons and holes, can occur at $ν = 0$ filling. The three phases have distinct edge conductivity values. One of the phases is especially interesting, since its edge conduction can be turned on and off by switching the sign of the interlayer bias.

Received 19 May 2018

DOI:https://doi.org/10.1103/PhysRevB.98.115307

Physics Subject Headings (PhySH)

Edge states Landau levels Quantum Hall effect Quantum anomalous Hall effect Spin-orbit coupling Valleytronics

Bilayer films Graphene

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Jun Yong Khoo and Leonid Levitov

Department of Physics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139, USA

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Issue

Vol. 98, Iss. 11 — 15 September 2018

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Images

Figure 1
Edge-state configurations for (a) para-phase, (b) ortho/para-phase, and (c) ortho-phase. Edge conduction is reduced due to backscattering between the counterpropagating modes with equal spin polarization. Backscattering between the modes with opposite spins cannot occur because of the orthogonality of the spin wave functions, leading to quantized edge conduction. In phase (b), spin polarization of one of the edge modes can be reversed by a transverse electric field, giving rise to a switchable edge conduction.
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Figure 2
Single-particle zeroth LL spectrum (including Zeeman splitting) as a function of interlayer bias $u$ for $λ = λ_{R} = 5$ meV at (a) $B = 5$ T, (b) $B = 10$ T, (c) $B = 20$ T, and (d) $B = 31$ T. Different colors and line styles are used to differentiate different LLs in each panel. The levels are labeled in panel (a) by $| ξ n s_{z} 〉$ , the notation defined in Eq. (4). A reversal of the order of the levels $| + 0 ↑ 〉$ (black solid line) and $| + 1 ↓ 〉$ (blue solid line) occurs with increasing magnetic field, with the transition taking place at $B \approx 10$ T shown in panel (b).
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Figure 3
Illustration of the extreme smallness of the level shifts induced by the Rashba SOI $λ_{R}$ in the zeroth LL, as compared with those induced by the Ising SOI $λ$ illustrated in Fig. 2. The spectrum found in the absence of SOI interaction remains essentially unchanged after adding a relatively large Rashba SOI: (a) $λ = λ_{R} = 0$ meV and (b) $λ = 0$ , $λ_{R} = 15$ meV. The magnetic field is $B = 5$ T in both cases. Labeling of Landau levels (color and line style) is the same as that in Fig. 2.
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Figure 4
Phase diagrams in the $B - u$ plane showing how the regions of normal and inverted orbital ordering change with the Ising SOI $λ$ magnitude and sign. Panels (a)–(c) correspond to $λ > 0$ and normal or inverted ordering refers to the ordering between the levels $| + 1 ↓ 〉$ and $| + 0 ↑ 〉$ : (a) $λ = 5$ meV, (b) $λ = 3$ meV, and (c) $λ = 1$ meV. Panels (d)–(f) correspond to $λ < 0$ and normal or inverted ordering refers to the ordering between the levels $| + 1 ↑ 〉$ and $| + 0 ↓ 〉$ [see Eq. (4) for notation]: (d) $λ = - 5$ meV, (e) $λ = - 3$ meV, and (f) $λ = - 1$ meV. Dashed lines indicate the phase boundary along which (a)–(c) $δ E_{λ Δ}^{+} = 0$ and (d)–(f) $δ E_{λ Δ}^{-} = 0$ . The energy difference $δ E_{λ Δ}^{+}$ (in meV) is indicated in Fig. 2.
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Figure 5
Phase diagram in the $λ - B$ plane showing the different regions in which the system is found in one of the three different compensated electron-hole phases: the para-phase, the ortho-phase, and the ortho/para-phase. The notation for the energy differences $δ E_{λ B}^{\pm}$ is given in Eq. (10).
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