Chiral tunneling through generic one-dimensional potential barriers in bilayer graphene

We study tunneling of charge carriers in single- and bilayer graphene. We propose an explanation for nonzero “magic angles” with 100% transmission for the case of symmetric potential barrier, as well as for their almost-survival for slightly asymmetric barrier in the bilayer graphene known previously from numerical simulations. Most importantly, we demonstrate that these magic angles are not protected in the case of bilayer and give an explicit example of a barrier with very small electron transmission probability for any angles. This means that one can lock charge carriers by a p-n-p (or n-p-n) junction without opening energy gap. This creates new opportunities for the construction of graphene transistors.

Figure 1

Numerical simulation of transmission probability for n-p-n junction in single-layer graphene. Energy of a particle 80 meV, height of the junction 200 meV. (a) Transmission probability for a symmetric potential of barrier width $l_2=250$ nm, for which n-p and p-n regions have characteristic widths $l_1=l_3=100$ nm. (b) Transmission probability for an asymmetric potential with $l_1=150$ nm and $l_3=50$ nm. (c), (d) The corresponding potentials; see Ref. [7, Fig. 6].

Figure 2

Numerical simulations of a transmission through an n-p-n junction in bilayer graphene. Energy of a particle 17 meV, height of the junction 50 meV. (a) Transmission probability for a symmetric potential of barrier width $l_2=100$ nm, with n-p and p-n regions of characteristic width $l_1=l_3=10$ nm. (b) Transmission probability for an asymmetric potential with $l_1=20$ nm and $l_3=40$ nm. (c), (d) The corresponding potentials.

Figure 3

Blue line: the transmission probability for the symmetric potential from Figs. 1 and 1. Green line: real-valued analytic function $f\left(\theta \right)$ such that $\left|r\right(\theta \left)\right|=\left|f\right(\theta \left)\right|$. Its zeros correspond to magic angles.

Figure 4

Blue line: the transmission probability for the symmetric potential from Figs. 2 and 2. Green line: real-valued analytic function $f\left(\theta \right)$ such that $\left|r\right(\theta \left)\right|=\left|f\right(\theta \left)\right|$. Again, its zeros correspond to magic angles.

Figure 5

Fast-oscillating potential $U\left(x\right)$.

Figure 6

(a) Green line: maximal transmission probability $p_{\text{max}}={max}_{\theta }p\left(\theta \right)$ for the potential Eq. (17) as a function of the given energy $E$ of the wave. Note that in the band $E=20...30$ meV the transmission probability does not exceed $2\times {10}^{-8}$. Blue line: maximal transmission probability for the cosine-oscillating potential Eq. (18). Note that in the same energy band the transmission probability is still very small (it does not exceed ${10}^{-6}$). (b) Transmission probability for the potential Eq. (17) as a function of the angle $\theta$ for energies $E=20,22.5,25,27.5,30$ meV.

Figure 7

Zoom in: absolute value of the reflection amplitude $\left|r\right(\theta \left)\right|$ for n-p-n junction in bilayer graphene near magic angles. Settings are the same as in Figs. 2 and 2.

Figure 8

Transmission probability for n-p-n junction in bilayer graphene for asymmetric 20-100-40 nm (red lines), symmetric 20-110-20 nm (green lines), and 30-100-30 nm (blue lines) barriers.

Figure 9

The function $log\mu (\theta ,l)$ for a wave of energy $E=17$ meV, crossing on bilayer graphene an n-p barrier of height $V=50$ meV (same energy and height as for Fig. 2). (a) The dependence on the width $l$ for (almost) magic angles $\theta \approx 30.{64}^{\circ }$ (blue line), $\theta \approx 65.{34}^{\circ }$ (red), and for $\theta ={45}^{\circ }$ (green). (b) The dependence on the angle of incidence $\theta$, for widths $l=20$ nm (blue line), 30 nm (green), and 40 nm (red).

Figure 10

Blue (solid) line: exact transmission probability for the symmetric n-p-n junction. Red (dashed bold) line: Lorenz-Breit-Wigner approximation $p\left(\theta \right)\approx \frac{1}{1+c{(\theta -{\theta }_0)}^2}$.