Electron flow in circular $n\text{−}p$ junctions of bilayer graphene

We present a theoretical study of electron wave functions in ballistic circular $n\text{−}p$ junctions of bilayer graphene. Similarly to the case of a circular $n\text{−}p$ junction of monolayer graphene, we find that (i) the wave functions form caustics inside the circular region and (ii) the shape of these caustics are well described by a geometrical optics model using the concept of a negative refractive index. In contrast to the monolayer case, we show that the strong focusing effect is absent in the bilayer. We explain these findings in terms of the angular dependence of Klein tunneling at a planar $n\text{−}p$ junction.

Figure 1

(Color online) An incident plane wave of electrons in bilayer graphene is scattered by a circular $n\text{−}p$ junction created by a gate-induced circular potential barrier $V\left(r\right)$. In the $n$ $\left(p\right)$ region the Fermi energy lies in the conduction (valence) band.

Figure 2

(Color online) The spatial dependence of the intensity of the wave function ${\left|\psi \left(\mathit{r}\right)\right|}^2$ is plotted in the scattering area. Here $k_{n}R=300$ and $k_{p}R=300$ corresponding to $n=-1$. The solid (dashed) line corresponds to the caustic for $p=1$ $\left(p=2\right)$, where $p$ denotes the number of chords inside the NPJ (Ref. 11).

Figure 3

(Color online) The same as in Fig. 2 with $k_{n}R=200$ and $k_{p}R=300$ corresponding to $n=-1.5$.

Figure 4

(Color online) Angular dependence of transmission probability through a planar NPJ of bilayer graphene. The corresponding values of the refractive index $n$ are shown in the figure.

Figure 5

(Color online) Refraction of electron rays at the interface of the $n\text{−}p$ junction. The darkness of the refracted rays reflects the transmission probability: a white ray corresponds to $T=0$ and a black ray would correspond to $T=1$. The dashed curve shows the shape of the caustic derived from Snell’s law (Ref. 11). Only the rays with no internal reflections are displayed. Here the refractive index is $n=-1$.