Charge transport gap in graphene antidot lattices

A. J. M. Giesbers, E. C. Peters, M. Burghard, and K. Kern

Phys. Rev. B 86, 045445 – Published 26 July 2012

Abstract

Graphene antidot lattices (GALs) offer an attractive approach to band-gap engineering in graphene. Theoretical studies indicate that the size of the opened gap is sensitive to the shape, size, and architecture of the nanoholes introduced into the graphene sheet. We have investigated the temperature-dependent electrical conductivity of GALs comprising 50-nm-diameter nanoholes with a pitch of 80, 100, and 200 nm, respectively. The data reveal the presence of localized states within a transport gap, whose interactions lead to a soft Coulomb gap and associated Efros–Shklovskii variable range hopping (ES-VRH). This conduction type is preserved upon application of magnetic fields up to 1 Tesla, above which a transition to Mott variable range hopping occurs. Such a crossover can alternatively be introduced at zero magnetic fields by increasing either the nanohole spacing or the gate-controlled carrier concentration. Furthermore, at intermediate magnetic fields, the hopping exponent assumes a value of 2/3, as predicted by percolation theory for ES-VRH under this condition.

Received 13 April 2012

DOI:https://doi.org/10.1103/PhysRevB.86.045445

Authors & Affiliations

A. J. M. Giesbers^1,*, E. C. Peters¹, M. Burghard¹, and K. Kern^1,2

¹Max Planck Institute for Solid State Research, Heisenbergstrasse 1, D-70569 Stuttgart, Germany
²Institute de Physique de la Matière Condensée, Ecole Polytechnique de Lausanne, CH-1015 Lausanne, Switzerland

^*Corresponding author: a.j.m.giesbers@tue.nl; current address: Molecular Materials and Nanosystems, Eindhoven University of Technology, NL-5600 MB Eindhoven, The Netherlands.

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Issue

Vol. 86, Iss. 4 — 15 July 2012

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Figure 1
Analysis of the temperature dependence of conductivity and gate dependence of the Coulomb gap. (a) Conductivity of sample no. 4, comprising 50-nm-sized nanoholes with 100-nm spacing, at the charge neutrality point ( $V_{G}$ $=$ 10.6 V) as a function of temperature. The solid line is a fit of the experimental data using the general thermal activation dependency valid for hopping conduction (exponent ν $=$ 0.49 ± 0.03). The inset shows a scanning electron micrograph of the patterned graphene between two gold contacts serving as voltage probes. (b) The conductance through sample no. 4 as a function of back-gate voltage for various temperatures $T$ $=$ 1.4, 2, 4, 6, 10, 15, 20, 30, 50, and 70 K. (c) Coulomb gap as a function of charge carrier concentration. The background shows a schematic illustration of the Coulomb gap in the localized density of states (black shaded area). The red and blue areas, respectively, represent the electron and hole density of states, which overlap around the CNP.[30]Reuse & Permissions
Figure 2
Concentration dependence of the Coulomb gap. (a) Close to the CNP the conduction is dominated by ES-VRH (sample no. 4). The extracted Coulomb gap decreases with increasing charge carrier concentration, in both the hole and electron regime. In the regimes $- 2.3$ × 10 $^{11}$ cm $^{- 2}$ < $n$ < $- 3$ × 10 $^{11}$ cm $^{- 2}$ and 1.6 × 10 $^{11}$ cm $^{- 2}$ < $n$ < 2.7 × 10 $^{11}$ cm $^{- 2}$ , the exponential fits (1) give an exponent close to ν $=$ 1/3, indicating transport is governed by Mott-VRH. For $| n |$ > 4 × 10 $^{11}$ cm $^{- 2}$ , transport follows a weak localization (WL) behavior. (b) Temperature-dependent WL fits of sample no. 4 at $n$ $=$ $- 4.0$ × 10 $^{11}$ cm $^{- 2}$ (blue triangles), $- 5.0$ × 10 $^{11}$ cm $^{- 2}$ (green triangles), $- 6.1$ × 10 $^{11}$ cm $^{- 2}$ (red circles), and −7.2 × 10 $^{11}$ cm $^{- 2}$ (black squares). Similar curves are obtained for the electrons. (c) Annealing sample no. 2 leads to a strong increase of the maximum resistance at the CNP and a shift closer to $V_{g}$ $=$ 0. The inset shows the increase of the Coulomb gap due to annealing. Measurements were performed at $T$ $=$ 5.1 K.Reuse & Permissions
Figure 3
Aharonov–Bohm oscillations in GALs. (a) Aharonov–Bohm oscillations in sample no. 6 at $T$ $=$ 1.4 K and $V_{g} = V_{CNP} + 0.3$ V, where $G_{0}$ is the smoothly varying background subtracted from the raw conductance data ( $G$ ) to visualize the oscillations. (b) Corresponding FFT of the oscillations, displaying clear maxima at $e A$ / $h$ $=$ 2.1 and 2 $e A$ / $h$ ≈ 4.2 and a third, weak maximum at 2 $e A$ / $h$ ≈ 6.3.Reuse & Permissions
Figure 4
Magnetoresistance of several GAL devices. (a) Resistance at the CNP as a function of magnetic field for the bare graphene sample (sample no. 8, black squares), the 200-nm spacing antidot sample (sample no. 7, red triangles), and a 100-nm spacing antidot sample (sample no. 6, blue circles). All curves were measured at $T$ $=$ 1.4 K. The dotted line shows a linear fit to the low field dependency of sample no. 6. The inset shows a zoom of the main panel. (b) Magnetic field dependence of the conductivity correction in sample no. 5 in the weak localization regime ( $n$ $=$ $- 4.0$ × 10 $^{11}$ cm $^{- 2}$ ), the red dashed line is a weak localization fit.[10] (c) Plot of the exponent ν, extracted from Eq. (1), for all the samples as a function of the B field, sample 1 (black squares), sample 3 (red circles), sample 4 (green up triangles), sample 5 (blue down triangles), and sample 6 (cyan diamonds). The dashed lines are guides to the eye.Reuse & Permissions

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