Negative quantum capacitance in graphene nanoribbons with lateral gates

R. Reiter, U. Derra, S. Birner, B. Terrés, F. Libisch, J. Burgdörfer, and C. Stampfer

Phys. Rev. B 89, 115406 – Published 10 March 2014

Abstract

We present numerical simulations of the capacitive coupling between graphene nanoribbons of various widths and gate electrodes in different configurations. We compare the influence of lateral metallic or graphene side gate structures on the overall back gate capacitive coupling. Most interestingly, we find a complex interplay between quantum capacitance effects in the graphene nanoribbon and the lateral graphene side gates, giving rise to an unconventional negative quantum capacitance. The emerging nonlinear capacitive couplings are investigated in detail. The experimentally relevant relative lever arm, the ratio between the coupling of the different gate structures, is discussed.

Received 19 December 2013
Revised 20 February 2014

DOI:https://doi.org/10.1103/PhysRevB.89.115406

Authors & Affiliations

R. Reiter¹, U. Derra², S. Birner³, B. Terrés², F. Libisch^1,4,*, J. Burgdörfer¹, and C. Stampfer²

¹Institute for Theoretical Physics, Vienna University of Technology, AT-1040 Wien, Austria, EU
²JARA-FIT and II. Institute of Physics, RWTH Aachen University, D-52074 Aachen, Germany, EU
³Walter Schottky Institute, Physik-Department, Technische Universität München, Am Coulombwall 4, D-85748 Garching, Germany, EU
⁴Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, New Jersey 08544, USA

^*florian.libisch@tuwien.ac.at

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Vol. 89, Iss. 11 — 15 March 2014

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Images

Figure 1
(a) Schematic illustration of a typical model system: a graphene nanoribbon (GNR) on an isolating substrate (dark gray), capacitively coupled to a back gate (BG, light red) and side gates (SG1, SG2). $t$ = 30 nm, $d$ = 300 nm, and varying $w$ . (b) Electrostatic potential and field lines on a two-dimensional plane perpendicular to the nanoribbon direction. Voltages are $V_{BG} = 2$ V and $V_{SG} = - 3$ V; nanoribbon width $w = 80$ nm.
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Figure 2
(a) Capacitance of an infinite graphene sheet with respect to a back gate with $d = 300$ nm, $ɛ = 3.9$ , and $T = 300$ K. The analytic solution [34] (gray line) coincides exactly with our numerical calculation (black line). (b) Capacitance of a graphene nanoribbon as a function of back gate voltage for different nanoribbon widths $w$ (see insets). $w = \infty$ denotes the bulk limit shown in (a). For narrower ribbons, the quantum capacitance dip at $V_{BG} = 0$ V becomes increasingly prominent. (c) Width-dependent capacitance of graphene nanoribbon: analytical model (from electrostatics, blue dashed line) and at $V_{BG} = 0$ V (quantum capacitance, red trace) and $V_{BG} = 60$ V (classical limit, green trace).
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Figure 3
Voltage-dependent relative change in capacitance (relative to respective classical capacitance $C_{cl}$ ) of graphene nanoribbons featuring (a) grounded metal side gates and (b) grounded graphene side gates, for different nanoribbon widths [Same parameters as in Fig. 2]. Width dependence of (c) the total capacitance (at $V_{BG} = 0$ ) and (d) the ratio of classical and quantum capacitance [see Eq. (2.8)] for a nanoribbon with no side gates (analytical model from Ref. [34], dashed black line), graphene side gates (blue solid line), and metal side gates (green solid line).
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Figure 4
Capacitance of a graphene nanoribbon with metallic side gates at symmetric [ $V_{SG 1} = V_{SG 2}$ , top row panels (a) and (b)] and antisymmetric [ $V_{SG 1} = - V_{SG 2}$ , bottom row panels (c) and (d)] finite side gate voltages (see insets). (b) and (d) Contour plots of the total capacitance as a function of $V_{SG}$ and $V_{BG}$ . Dashed white lines mark the voltage combinations where charge accumulated at one of the edges of the nanoribbon is zero.
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Figure 5
Same as Fig. 4 for graphene side gates. Note the regions of increased capacitance due to quantum-capacitance effects in the side gates (blue areas in color maps). These regions are delimited by voltage combinations where charge vanishes at the ribbon edges (white dashed lines). Black triangles on the color scale denote the classical capacitance value.
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Figure 6
Relative lever arm $α_{rel}$ [see Eq. (3.1)], i.e., relative capacitance of the nanoribbon towards back gate and side gate, in symmetric gate voltage configuration for (a) metallic and (b) graphene side gates. The graphene nanoribbon has a width of 80 nm and the spacing to the side gates is 30 nm.
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