Chiral Tunneling in a Twisted Graphene Bilayer

The perfect transmission in a graphene monolayer and the perfect reflection in a Bernal graphene bilayer for electrons incident in the normal direction of a potential barrier are viewed as two incarnations of the Klein paradox. Here we show a new and unique incarnation of the Klein paradox. Owing to the different chiralities of the quasiparticles involved, the chiral fermions in a twisted graphene bilayer show an adjustable probability of chiral tunneling for normal incidence: they can be changed from perfect tunneling to partial or perfect reflection, or vice versa, by controlling either the height of the barrier or the incident energy. As well as addressing basic physics about how the chiral fermions with different chiralities tunnel through a barrier, our results provide a facile route to tune the electronic properties of the twisted graphene bilayer.

Figure 1

Tunneling through a barrier in a twisted graphene bilayer. (a) Schematic diagram of an electron coming to a potential-energy barrier of height $E+\Delta U$ and width D. $E$ is the Fermi energy of the twisted graphene bilayer and the one-dimensional barrier is infinite along the $y$ direction. (b) Electronic spectrum of the quasiparticles in a twisted graphene bilayer with a finite interlayer coupling in the proximity of one of the two valleys. Two saddle points form between the two Dirac cones, $K$ and $K_{\theta }$. (c) Density plot of the energy dispersion of the twisted graphene bilayer around $K$ and $K_{\theta }$. $k_x$ is the direction perpendicular to the barrier.

Figure 2

Quantum tunneling in a twisted graphene bilayer for low incident energy. Transmission probability $T$ through a 100-nm wide barrier as a function of the incident angle $\phi$ for (a),(c) a twisted graphene bilayer and (b),(d) the system described by the Hamiltonian $(2v_F^2\Delta K/15{\tilde{t}}_{\perp })\overrightarrow{\sigma }·\overrightarrow{q}$. The remaining parameters are the twist angle of the graphene bilayer $\theta =3.89°$, $t_{\perp }=0.12\mathrm{eV}$, and $E_V=0.15\mathrm{eV}$. We only plotted the transmission of the states in a particular cone; the transmission of the corresponding states in the other cone is related by mirror symmetry. The angular behavior of $T(\phi )$ in (b),(d) is similar to that of a graphene monolayer, and the chiral fermions are always perfectly tunneling for normal incidence irrespective of the parameters of the barrier. The $T(\phi )$ of the twisted graphene bilayer shows both similarities and differences with respect to that of the graphene monolayer. For twisted graphene bilayers, the $T(\phi )$ is asymmetric about $\phi =0$ and the asymmetry increases with increasing height of the barrier.

Figure 3

Quantum tunneling in a twisted graphene bilayer for normally incident electrons. (a) Transmission probability for normally incident electrons as a function of the incident energy. The curves with different colors correspond to different values of the $\Delta U$. (b) Transmission probability for normally incident electrons with different incident energy as a function of the value of $\Delta U$. Here the height of potential barrier $U(x)$ increases from zero; i.e., the value of $\Delta U$ increases from $-E$. In the middle and lower panels, the transmission is completely suppressed when $|\Delta U|<E_g/2$ [$E_g$ is defined in panel (c)]. For a certain value of $\Delta U$ (here $\Delta U>E_g/2$), the amplitude of the oscillations increases with the incident energy. For a fixed incident energy, both the periodicity and the amplitude of the oscillations increase with the positive value of $\Delta U$ (here $\Delta U>E_g/2)$. (c) The right panel is a density plot of the energy dispersion of the twisted graphene bilayer around $K$ and $K_{\theta }$. The dark dots are the positions of normal incident electrons in the superlattice Brillouin zone. The energy spectra in the left panel are a section view of band structures through serifs that cross normal incident modes. The dashed curves represent the electrons that are moving away from the barrier in the perpendicular direction.

Figure 4

Quantum tunneling in a twisted graphene bilayer with different width of the barrier. (a) Transmission probability as a function of the values of the $\Delta U$. The curves with different colors correspond to different widths of the barriers. (b) The average periodicity of the oscillations for $0.3\mathrm{eV}<\Delta U<0.4\mathrm{eV}$ as a function of $D^{-1}$. Here $D$ is the width of the barrier. The periodicity almost increases linearly with $D^{-1}$.