Electric-Field-Tunable Edge Transport in Bernal-Stacked Trilayer Graphene

Saurabh Kumar Srivastav, Adithi Udupa, K. Watanabe, T. Taniguchi, Diptiman Sen, and Anindya Das

Phys. Rev. Lett. 132, 096301 – Published 27 February 2024

Abstract

This Letter presents a nonlocal study on the electric-field-tunable edge transport in $h$ -BN-encapsulated dual-gated Bernal-stacked ( $A B A$ ) trilayer graphene across various displacement fields ( $D$ ) and temperatures ( $T$ ). Our measurements revealed that the nonlocal resistance ( $R_{N L}$ ) surpassed the expected classical Ohmic contribution by a factor of at least 2 orders of magnitude. Through scaling analysis, we found that the nonlocal resistance scales linearly with the local resistance ( $R_{L}$ ) only when the $D$ exceeds a critical value of $\sim 0.2 V / nm$ . Additionally, we observed that the scaling exponent remains constant at unity for temperatures below the bulk-band gap energy threshold ( $T < 25 K$ ). Further, the value of $R_{N L}$ decreases in a linear fashion as the channel length ( $L$ ) increases. These experimental findings provide evidence for edge-mediated charge transport in $A B A$ trilayer graphene under the influence of a finite displacement field. Furthermore, our theoretical calculations support these results by demonstrating the emergence of dispersive edge modes within the bulk-band gap energy range when a sufficient displacement field is applied.

Received 5 June 2023
Revised 29 August 2023
Accepted 29 January 2024

DOI:https://doi.org/10.1103/PhysRevLett.132.096301

Physics Subject Headings (PhySH)

Edge states Mesoscopics Transport phenomena

Graphene

Hall bar Resistivity measurements

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Saurabh Kumar Srivastav ^1,*, Adithi Udupa ², K. Watanabe ³, T. Taniguchi³, Diptiman Sen ², and Anindya Das ^1,†

¹Department of Physics, Indian Institute of Science, Bangalore 560012, India
²Centre for High Energy Physics, Indian Institute of Science, Bangalore 560012, India
³National Institute of Material Science, 1-1 Namiki, Tsukuba 305-0044, Japan

^*Corresponding author: ssaurabh@iisc.ac.in
^†Corresponding author: anindya@iisc.ac.in

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Issue

Vol. 132, Iss. 9 — 1 March 2024

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Images

Figure 1
(a) Schematic of the device configuration: Bernal-stacked trilayer graphene (TLG) is encapsulated between two $h$ -BN substrates and is gated by a graphite back gate ( $V_{BG}$ ) and a metal top gate ( $V_{TG}$ ). (b),(c) Band dispersions of Bernal-stacked TLG at zero and finite ( $Δ = 200 meV$ ) displacement field, respectively. (d) $R_{x y}$ response of the device as a function of $V_{TG}$ for several values of the magnetic field. (e) $(d R_{x y} / d V_{TG})$ is plotted as a function of $V_{TG}$ and magnetic field. The observation of the crossing points between the different Landau levels, depicted as discontinuities in the QH plateau structure (blue strips) along the white dashed lines, confirms the $A B A$ character of the TLG.
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Figure 2
Color map of $R_{L}$ (a) and $R_{N L}$ (b) as a function of total carrier density $n$ and the displacement field $D$ . (c) The line cuts of $R_{L}$ (red) and $R_{N L}$ (black) are plotted with density at $D = - 0.4 V / nm$ . The $R_{N L}$ is multiplied by a factor of 20. The magenta and green curves (both multiplied by 20) represent the theoretically expected nonlocal contributions from the charge accumulation (near the edge) and classical Ohmic one, respectively. (d),(e) Log-log plots of $R_{N L}$ with $R_{L}$ . Open circles are extracted from Figs. 2 and 2 for different values of $n$ (along vertical black arrows) near the Dirac point, (d) and (e), for different values of $D$ (along horizontal yellow arrows). The solid lines correspond to the linear fitting of the data points with slope $\sim 1$ . (f) The activation gap extracted from $R_{N L}$ is plotted versus the gap extracted from $R_{L}$ . The red line is the linear fit to these data with slope $\sim 1.1$ . Different filled circles correspond to the gap extracted at $D$ ranging from $- 0.29$ to $- 0.50 V / nm$ . Error bars correspond to the standard deviation associated with the slope of the linear fit.
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Figure 3
(a) The scaling exponent $α$ plotted as a function of $D$ at $T = 5.2 K$ . For $| D | < 0.2 V / nm$ , $α$ deviates significantly from unity. (b) $α$ plotted as a function of $T$ for $D = - 0.45 V / nm$ . For $T ≳ 25 K$ $α$ starts deviating from unity. (c) $R_{N L}$ is plotted with density for channel lengths of $L = 4 μ m$ (black), $8 μ m$ (blue), and $12 μ m$ (red). Inset: The peak value of $R_{N L}$ (blue circles) plotted for different $L$ . The red line is a linear fit. (d) The dispersion of TLG, along with the zigzag edge, is plotted for $Δ = 20 meV$ (top) and $Δ = 40 meV$ (bottom). The green (red) curves correspond to modes on the right (left) edge of the system.
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