Guiding Dirac Fermions in Graphene with a Carbon Nanotube

Relativistic massless charged particles in a two-dimensional conductor can be guided by a one-dimensional electrostatic potential, in an analogous manner to light guided by an optical fiber. We use a carbon nanotube to generate such a guiding potential in graphene and create a single mode electronic waveguide. The nanotube and graphene are separated by a few nanometers and can be controlled and measured independently. As we charge the nanotube, we observe the formation of a single guided mode in graphene that we detect using the same nanotube as a probe. This single electronic guided mode in graphene is sufficiently isolated from other electronic states of linear Dirac spectrum continuum, allowing the transmission of information with minimal distortion.

Figure 1

Electron waveguide in graphene. (a) Schematic of graphene with a potential well represented by the blue region which confines electrons along the $y$ direction. Below, electrostatic potential along the $x$ direction generated by a charged carbon nanotube (CNT). (b) Schematic of the band structure as a function of momentum $k_y$. The gray lines correspond to the bulk states, and the blue lines correspond to the guided modes. On the right: global density of states (DOS) and local density of states (LDOS) as a function of energy. (c) Diagram showing the condition (blue line) for a waveguide to host a Dirac single mode and nonrelativistic single mode. (d) Optical image of one of the devices. EFM picture of a CNT on top of a h-BN encapsulated graphene device with metallic electrodes and a schematic of the device structure.

Figure 2

Graphene density of states measured with the carbon nanotube (CNT). (a) Schematic of the measurement setup (top) and CNT conductance measured as a function of the backgate showing the Coulomb blockade behavior. All measurements presented in this Letter are performed at 1.6 K. (b) Operational principle of the CNT sensor. (c) CNT conductance $G_{\mathrm{NT}}$ vs $V_G$ and $V_{\mathrm{bg}}$. The wide dark brown area around $V_G=0$ corresponds to the semiconducting gap of the CNT, in which the latter is not charged. Inset shows $d{G}_{\mathrm{NT}}/d{V}_G$ over a small region in order to highlight a double kink corresponding to the Dirac point followed by a guided mode resonance (blue arrow). (d) Local quantum capacitance measured as a function of global charge carrier density $n_G$ for two different voltage differences $V_G$ between the CNT and graphene.

Figure 3

Potential depth dependence. (a) Evolution of the graphene local density of states with the potential well depth $U_0$ controlled by $V_G$. (b) Comparison with theoretical calculations obtained from a tight-binding modeling. (c) Theoretical evolution of the band structure for increasing $U_0$. (d) Positions of local Dirac point and first guided mode as a function of $V_G$ (experiment: continuous lines) and $U_0$ (theory: dashed lines). Here, the scales of $U_0$ and $V_G$ are adjusted to show the apparent linear relation between $U_0$ and $V_G$ in the accessible gate voltage range where the guided mode forms.