Crossover from Coulomb Blockade to Quantum Hall Effect in Suspended Graphene Nanoribbons

Suspended graphene nanoribbons formed during current annealing of suspended graphene flakes have been investigated experimentally. Transport measurements show the opening of a transport gap around charge neutrality due to the formation of “Coulomb islands”, coexisting with quantum Hall conductance plateaus appearing at moderate values of the magnetic field $B$. Upon increasing $B$, the transport gap is rapidly suppressed, and is taken over by a much larger energy gap due to electronic correlations. Our observations show that suspended nanoribbons allow the investigation of phenomena that could not so far be accessed in ribbons on ${\mathrm{SiO}}_2$ substrates.

Figure 1

(a) Two-terminal resistance ($R$) versus $V_g$ of the wide suspended graphene device shown in the inset, measured at $T=0.24\mathrm{K}$ (blue solid line) and 2.0 K (green dashed line). (b) Semilog plot of the conductance $G$ measured at 0.24 K, as a function of carrier density $n$ for both electrons (blue solid line) and holes (red dashed line). The resistance peak width is approximately ${10}^{10}\mathrm{carriers}/{\mathrm{cm}}^2$. (c) Conductance $G$ as a function of carrier density $n$ and magnetic field $B$, showing well defined quantum Hall states with $G=1$ and $2e^2/h$, and an insulating state around charge neutrality. (d) $G$($n$) at different values of $B$ selected from (c), zooming in on quantum Hall plateaus; the dashed lines correspond to $G=1$ and $2e^2/h$.

Figure 2

(a) Scanning electron microscope image of a GNR formed while annealing a suspended graphene layer (the bar is 100 nm long). (b) Gate dependence of the differential conductance $G=\frac{dI}{dV}(V_g)$ measured after a first annealing step of a suspended graphene device (green line on top) and after a second step (blue line at the bottom), in which a suspended GNR is formed. Panel (c) zooms in on the transport gap region of (b), and shows resonances due to Coulomb blockade. (d) $\log (G)$ as a function of bias $V$ and gate $V_g$ voltages, showing the Coulomb diamonds characteristically seen in GNRs. All data are taken at 4.2 K.

Figure 3

(a) Color plot of $G$($V_g$,$B$), with quantized conductance values indicated. The broken line is a guide to the eye, separating the conducting and insulating regions as a function of $V_g$ and $B$. (b) $G$($V_g$) at $B=2.4$, 3.4, 4.4, 5.4, 6.5, and 8.0 T (from the right to the left, respectively), showing conductance plateaus for hole accumulation. (c) Schematic view of the procedure followed to estimate $V_{\mathrm{CNP}}$ and $\alpha (\equiv \Delta n/\Delta V_g)$ described in the text. (d) $G$ as a function of the filling factor $\nu (\equiv n{\varphi }_0/B)$ in the magnetic field range between 5.0 and 8.0 T: all different measurements tend to collapse on a single curve. The presence of $2e^2/h$ conductance plateaus for electron accumulation becomes apparent in this plot. All data are taken at 4.2 K.

Figure 4

(a)–(c) From left to right, logarithm of the differential conductance $G=\frac{dI}{dV}$ versus $V$ and $V_g$, for $B=0$, 0.5, and 1 T: increasing $B$ causes a decrease of the $V_g$ interval in which Coulomb diamonds dominate low-bias transport. (d) Conductance averaged in $V_g$’s between 0 and 20 V, $⟨G⟩$, as a function of $B$. At $B>2\mathrm{T}$, $⟨G⟩$ decreases due to the large, correlation-induced gap around the charge neutrality, which becomes larger as $B$ increases. (e) $\log (G)$ versus $V$ and $V_g$ at $B=5.0\mathrm{T}$, showing the opening of a large gap around charge neutrality due to electronic correlations (the color scale in the legend is the same for all panels). The broken line at $V_g=8.5\mathrm{V}$ corresponds to $V_{\mathrm{CNP}}$ extracted from the quantum Hall effect analysis [see Fig. 3c]. All data are taken at 4.2 K.

Figure 5

(a) $I\mathrm{\text{−}}V$ characteristics of the device at $V_g=1.4\mathrm{V}$ (at $B=0$, 0.6 T and in the range between 1.4 and 2.0 T, as indicated). (b) Logarithm of the differential conductance $G=\frac{dI}{dV}$ versus $V$ and $B$ for $V_g=1.4\mathrm{V}$ away from the charge neutrality, showing the closing of the transport gap with increasing $B$ (the low-bias conductance suppression above 4 T is due to the insulating state around the charge neutrality). All data are taken at 4.2 K.