Measuring the local quantum capacitance of graphene using a strongly coupled graphene nanoribbon

We present electrical transport measurements of a van-der-Waals heterostructure consisting of a graphene nanoribbon separated by a thin boron nitride layer from a micron-sized graphene sheet. The interplay between the two layers is discussed in terms of screening or, alternatively, quantum capacitance. The ribbon can be tuned into the transport gap by applying gate voltages. Multiple sites of localized charge leading to Coulomb blockade are observed, in agreement with previous experiments. Due to the strong capacitive coupling between the ribbon and the graphene top layer sheet, the evolution of the Coulomb blockade peaks in gate voltages can be used to obtain the local density of states and therefore the quantum capacitance of the graphene top layer. Spatially varying density and doping are found, which are attributed to a spatial variation of the dielectric due to fabrication imperfections.

Figure 1

(a) Schematic of the device: a graphene ribbon is fabricated on a ${\mathrm{SiO}}_2$ substrate. Source, drain, and two side gates are contacted by gold contacts (ribbon layer). A hBN flake is deposited on top of the device. On the hBN flake, an additional graphene flake is deposited and contacted by gold contacts (top layer). (b) False-color SFM image of the final device: the graphene top layer is colored in white and the corresponding metal contacts in red. The ribbon layer contacts as well as the overlay of the ribbon (not visible in the SFM scan) are marked in blue. The orange background is the hBN flake. (c) Zoom into (b): the top layer is not flat on a large scale, but the area directly above the ribbon is wrinkle-free. The wrinkles and bubbles are highlighted with red dashed lines. (d) SFM image of the graphene nanoribbon before transfer of the hBN flake.

Figure 2

(a) Two-terminal resistance of the graphene top layer as a function of $V_{\mathrm{TL}}$ and $V_{\mathrm{BG}}$. The orange box marks the range of the following plots. (b) Derivative of (a) along $V_{\mathrm{BG}}$. The change in sign (purple line) marks the position of the charge neutrality point of the top layer that is not above the ribbon layer. (c) Derivative of (a) along $V_{\mathrm{TL}}$. The yellow line marks the position of the charge neutrality point in the top layer that is located above the ribbon layer. The black dashed lines mark the range where charge puddles are expected (disorder density), and the arrow marks the position where the kink of the line is the strongest. (d) Current flowing through the ribbon as a function of $V_{\mathrm{TL}}$ and $V_{\mathrm{BG}}$ voltage at a fixed bias voltage. The green lines mark the position of the region of suppressed conductance. (e) and (f) show the absolute value of the calculated charge carrier density for the same range as in (c) and (d).

Figure 3

(a) and (b) show the current flowing through the ribbon as a function of applied bias and $V_{\mathrm{BG}}$ for the right region of suppressed conductance. (c) and (d) show the current flowing through the ribbon as a function of applied side gate voltages at a fixed $V_{\mathrm{BG}}$ and $V_{\mathrm{TL}}$. For better visibility, the features of enhanced current are extracted and shown in the middle. Parts (a) and (c) are recorded at low top layer charge carrier density, and (b) and (d) are recorded at high top layer charge carrier density.

Figure 4

(a) Current flowing through the ribbon as a function of $V_{\mathrm{TL}}$ and $V_{\mathrm{BG}}$ at a fixed bias. To improve the electrical stability of the measurements, several rectangular regions in $V_{\mathrm{BG}}$ and $V_{\mathrm{TL}}$ were measured consecutively. For each of these measurements, the axes were transformed (rotation followed by horizontal squeeze—similar to the n-D transformation in bilayer graphene [30]) in order to obtain a better visibility of the features. Finally, the different measurements were put together in one plot. Lines of constant $V_{\mathrm{TL}}$ and $V_{\mathrm{BG}}$ are marked in green and brown. The direction of the axes $V_{\mathrm{TL}}$ and $V_{\mathrm{BG}}$ is marked with black arrows. (b) Same data as (a) where a number of different Coulomb resonances are marked. (c) Slopes extracted from the lines in (b) as a function of the applied gate voltages. (d) Local quantum capacitance of the top layer calculated from the slopes in (c) using different thicknesses for the hBN layer. (e) Local charge carrier density of the top layer using the data from (d) and the same hBN thicknesses as before. Magenta curves are shifted to the left. (f) Calculated “capacitance disorder” $n_{c,\mathrm{dis}}$ with a locally varying dielectric thickness of $2\mathrm{nm}$ and a mean thickness of $14.5\mathrm{nm}$. The brown lines mark the position at which this “capacitance disorder” starts to dominate over the intrinsic fabrication disorder of around ${10}^{11}{\mathrm{cm}}^{-2}$.