Klein tunneling and cone transport in AA-stacked bilayer graphene

We investigate the quantum tunneling of electrons in an AA-stacked bilayer graphene (BLG) $n$-$p$ junction and $n$-$p$-$n$ junction. We show that Klein tunneling of an electron can occur in this system. The quasiparticles are not only chiral but are additionally described by a “cone index.” Due to the orthogonality of states with different cone indexes, electron transport across a potential barrier must strictly conserve the cone index, and this leads to the protected cone transport which is unique in AA-stacked BLG. Together with the negative refraction of electrons, electrons residing in different cones can be spatially separated according to their cone index when transmitted across an $n$-$p$ junction. This suggests the possibility of “cone-tronic” devices based on AA-stacked BLG. Finally, we calculate the junction conductance of the system.

Figure 1

Lattice structure of AA-stacked bilayer graphene. The $\mathcal{A}1$ sublattice is stacked directly above the $\mathcal{A}2$ sublattice. The interlayer and intralayer hopping energies are $\gamma$ and $t$, respectively.

Figure 2

(a) Band structures of electrons (left) and holes (right). The band structure is made up of two up-down shifted Dirac cones. Electrons and holes reside in one of the Dirac cones which is labeled by a cone index $c$. (b) The chirality of the quasiparticles are labeled by $\chi$. For $\chi =+1$($\chi =-1$), the wave vector is (anti)parallel to the group velocity. (c) and (d) show the tunneling of an electron through an $n$-$p$ junction and an $n$-$p$-$n$ junction, respectively.

Figure 3

Transmission probabilities for the (a) upper and (b) lower cones across an $n$-$p$ junction. The transmission angle is shown in (c) for the upper cone and in (d) for the lower cone. Cross-hatched pattern represents an evanescent transmitted mode.

Figure 4

Schematic drawing of a cone polarizer based on AA-stacked BLG $n$-$p$ junction: (a) Incident energy potential height to achieve cone polarization. The incident energy has to be in the range of $\gamma <E<V_0-\gamma$ and the potential height has to be greater than $2\gamma$. (b) Spatial separation of electrons according to their cone index. Electrons injected at point X are refracted through different transmission angles according to their cone index. $c=+1$ electrons are focused at Point B while $c=-1$ electrons are focused at Point A.

Figure 5

Transmission probabilities for (a) upper and (b) lower cones across an $n$-$p$-$n$ junction.

Figure 6

Conductance for (a) lower, (b) upper cones, and (c) the conductance sum of upper and lower cones. The incident energy $E$ is in units of $V$ and the potential height $V$ is in units of $\gamma$.

Figure 7

The total integrated conductance of an $n$-$p$ junction. The integrated conductances are plotted for three Fermi levels, $E_F/V=0.25$ (solid line), $E_F/V=0.50$ (dashed line), and $E_F/V=0.75$ (dash-dotted line).