Equilibration of quantum Hall edges in symmetry-broken bilayer graphene

Chandan Kumar, Saurabh Kumar Srivastav, and Anindya Das

Phys. Rev. B 98, 155421 – Published 16 October 2018

Abstract

Equilibration of quantum Hall edges is studied in a high quality dual gated bilayer graphene device in both unipolar and bipolar regimes when all the degeneracies of the zero energy Landau level are completely lifted. We find that in the unipolar regime when the filling factor under the top gate region is higher than the back gate filling factor, the equilibration is partial based on their spin polarization. However, the mixing of the edge states in the bipolar regime is insensitive to the spin configurations of the Landau levels and the values are very close to the full equilibration prediction. This has been explained by Landau level collapsing at the sharp $p - n$ junction in our thin hBN ( $\sim 15$ nm) encapsulated device, in consistent with the existing theory.

Received 17 July 2018

DOI:https://doi.org/10.1103/PhysRevB.98.155421

Physics Subject Headings (PhySH)

Edge states Integer quantum Hall effect Landau levels Quantum transport

Graphene

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Chandan Kumar, Saurabh Kumar Srivastav, and Anindya Das^*

Department of Physics, Indian Institute of Science, Bangalore 560 012, India

^*anindya@iisc.ac.in

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Issue

Vol. 98, Iss. 15 — 15 October 2018

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Images

Figure 1
(a) The schematic of an encapsulated hBN/bilayer graphene/hBN device. The 300-nm-thick ${SiO}_{2}$ acts as a back gate while the top thin hBN acts as a top gate and controls density only in the middle portion of BLG. (b) 2D color plot of resistance as a function of back gate and top gate voltage at $B = 0$ T. The inset shows the resistance at $V_{TG} = - 0.315$ V. The red line is fit to resistance to extract mobility. (c) 2D color plot of transconductance $d G / d V_{BG}$ as a function of $V_{BG}$ and perpendicular magnetic field $B$ . Clear Landau levels can be seen emitting from $V_{BG} = - 1.3$ V at $B = 0$ T. (d) Two probe resistance as a function of back gate voltage at zero top gate voltage measured at 40 mK and 10 T magnetic field. All the degeneracy of the zero energy Landau level is lifted, leading to the observation of QH plateaus at an integer multiple of $e^{2} / h$ .
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Figure 2
Schematic of the chiral QH edge state. (a) For $ν_{BG} = ν_{TG} = - 1$ , here, since $| ν_{BG} | = | ν_{TG} |$ , all the $ν_{BG}$ edge states are completely transmitted through the top gate region. (b) For $| ν_{BG} | > | ν_{TG} |$ , in this scenario only $| ν_{TG} |$ edge states are completely transmitted and the rest of the ( $| ν_{BG} | - | ν_{TG} |$ ) edge states gets reflected back. (c) For $| ν_{BG} | < | ν_{TG} |$ , all the $| ν_{BG} |$ edge states completely transmitted through the top gate region while the $| ν_{TG} | - | ν_{BG} |$ edge states keep circulating in the top gate region. These inner circulating edge states can now interact with the transmitting edge states and the conductance gets modified depending on the equilibration. (d) Since electron and hole edge states have opposite chirality, the electron and hole edge states move copropagating along the $p - n$ junction and the full equilibration of these edge states leads to a value of conductance given by Eq. (3).
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Figure 3
(a) 2D plot of conductance as a function of $ν_{BG}$ and $ν_{TG}$ at 10 T magnetic field. The horizontal and vertical strips in the unipolar regime corresponds to back gate and top gate filling factor, respectively. Quantized conductance of $2 e^{2} / h$ is observed for $ν_{BG} = - 2$ from $ν_{TG} = - 2$ to $ν_{TG} = - 5$ . Similarly for $ν_{BG} = - 4$ , quantized conductance of $4 e^{2} / h$ is observed from $ν_{TG} = - 4$ to $ν_{TG} = - 5$ . The checkerboard pattern in the bipolar regime is represented by a red dashed line. (b) 2D plot of displacement field as a function of electron density. The LL crossing point is at $D \sim 0.09$ V/nm.
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Figure 4
Line trace of G as a function of $ν_{TG}$ at different set of $ν_{BG}$ at 10 T and 40 mK. (a) Cut across $ν_{BG} = - 1$ . (b) Cut across $ν_{BG} = - 2$ . (c) Cut across $ν_{BG} = - 3$ . (d) Cut across $ν_{BG} = - 4$ . For $ν_{BG} = - 2$ ( $ν_{BG} = - 4$ ) conductance remains quantized at $2 e^{2} / h$ ( $4 e^{2} / h$ ) from $ν_{TG} = - 2$ to $ν_{TG} = - 5$ ( $ν_{TG} = - 4$ to $ν_{TG} = - 5$ ), suggesting that these edge states do not equilibrate in our device. Each figure in the lower panel represents the different spin configuration associated with each of the LL. The spin states are obtained from Fig. 5 below “b.” The up and down spins are represented by blue and red color, respectively. BG and TG in the lower panel represents back gate and top gate region, respectively.
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Figure 5
Schematic representation of LL evolution with magnetic field and the perpendicular electric field. The solid (dashed) line represents the up (down) spin states while the orbital index and valley is represented by (0,1) and (+, $-$ ) respectively. (a) Model adapted from Refs. [31, 32]. (b) Model adapted from Refs. [27, 28, 29, 30]. The filling factors in (a) are indicated for point below “b.”
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