Probing barrier transmission in ballistic graphene

We derive the local density of states from itinerant and boundary states around transport barriers and edges in graphene and show that the itinerant states lead to mesoscale undulations that could be used to probe their scattering properties in equilibrium without the need for lateral transport measurements. This finding will facilitate vetting of extended structural defects, such as grain boundaries or line defects as transport barriers for switchable graphene resonant tunneling transistors. We also show that barriers could exhibit double minima and that the charge density away from highly reflective barriers and edges scales as $x^{-2}$.

Figure 1

Schematics of the relationship between the barrier transmissivity and the undulations in the local density of states (LDOS) itinerant from: (a) both sides and (b) the left side, representing equilibrium and nonequilibrium, respectively. Because carriers originating from the left and right do not interfere, the undulations are the same in (a) and (b), except for the transmitted side in (b) where they are absent. The transmission probability through the barrier is related to the undulations, which could be probed in equilibrium, even though no net current flows through the barrier.

Figure 2

Contours to determine the LDOS around a graphene barrier. (a) The unit circle and real axis contours account for the itinerant and boundary states in graphene, respectively. (b) These contributions are evaluated for a barrier with an effective coupling $\lambda$ and effective potential $\varepsilon$.

Figure 3

Local densities of states around neutral barriers ($\varepsilon =0$) at the carrier energy $E=50\mathrm{meV}$ that include: (a) all contributions and (b) contributions from states originating from the left. Contributions from transmitted carriers in (b) are proportional to the average transmission probability and do not exhibit undulations.

Figure 4

Local densities of states around decoupled barriers ($\lambda =0$) at the carrier energy $E=50\mathrm{meV}$. The effective potential shifts the undulation peaks in (a), and if positive, leads to states localized around the barrier. The boundary states take their measure from the itinerant states as shown in (b) for $\varepsilon =0.8$.

Figure 5

Properties in parameter space. The derivative of the density undulations (a) at the barrier $\Delta {\rho }_E^{\prime }\left(0^{+}\right)$ correlates with the number of boundary states (b), a result of the boundary states taking measure from the itinerant states. The boundary states are maximally localized at the bold cross. The average transmission probability (c) ranges from 0 for decoupled barriers (or edges) to 1 for nonexistent barriers.

Figure 6

LDOS comparison between the graphene barrier model and the nearest-neighbor tight-binding (TB) model at the carrier energy of $E=50\mathrm{meV}$. There are three sets of TB curves for the three barrier structures shown as insets, each with a different barrier orientation with respect to the graphene lattice. The highlighted barrier sites have a potential of $-1.2\mathrm{eV}$. The barrier model is computed for $(\lambda ,\varepsilon )=(-1,-0.5)$.