Energy symmetry and interlayer wave function ratio of tunneling electrons in partially overlapped graphene

Abstract

While the exponential decay of tunneling probability with barrier thickness is well known, the accompanying oscillations with thickness have been comparatively less explored. Using a tight binding model, we investigate an AB-stacked bilayer graphene region acting as an energy barrier between two monolayer graphene leads, under a vertical electric field. We discuss the case where the energy gap induced by the vertical electric field is comparable to the interlayer transfer integral. In the up (down) junction, the left and right monolayer leads are connected to different layers (a common layer) of the central bilayer, while the remaining, unconnected layers form armchair-type open edges. We reveal a characteristic relation between the tunneling probability and the wave function structure. Among the valley-resolved transmission probabilities, only the valley-reversed transmission in the up junction exhibits even symmetry with respect to energy $E$. This result is counterintuitive. The interlayer wave function ratio $\beta$ is asymmetric in $E$, i.e., $\beta(-E) \neq \beta(E)$, and electrons cannot bypass the interlayer path in the up junction, whereas they can in the down junction. We attribute this unexpected symmetry to a self-cancellation effect of $\beta$, which arises from chiral and rotational symmetry operations combined with the conservation of probability. Our results demonstrate that the energy dependence of conductance in double junction structures serves as evidence of this effect.