Electronic transmission through $AB$-$BA$ domain boundary in bilayer graphene

We study the electron transmission through the domain boundary on bilayer graphene separating $\mathit{AB}$ and $\mathit{BA}$ stacking regions. Using the effective continuum model, we calculate the electron transmission probability as a function of the electron energy and the incident angle, for several specific boundary structures. The transmission strongly depends on the crystallographic direction of the boundary and also on the atomic configuration inside. At low energy, the boundary is either insulating or highly transparent depending on the structure. In insulating cases, the transmission sharply rises when the Fermi energy is increased to a certain level, suggesting that the electric current through the boundary can be controlled by the field effect. The boundary parallel to the zigzag direction generally has different transmission properties between the two different valleys, and this enables one to generate the valley polarized current in a certain configuration. We show that those characteristic features can be qualitatively explained by the transverse momentum conservation in the position-dependent band structure in the intermediate region.

Figure 1

Various stacking structures of bilayer graphene. (a) $\mathit{AB}$, (b) $\mathit{BA}$, (c) $\mathit{AA}$, and (d) $\mathit{SP}$ stacking.

Figure 2

Schematic structures of several types of $\mathit{AB}$-$\mathit{BA}$ domain boundary: (a) Armchair $\mathit{AA}$, (b) armchair $\mathit{SP}$, (c) zigzag $\mathit{AA}$, and (d) zigzag $\mathit{SP}$. In (c) and (d), a narrow strip in the upper part represents the uniform $\mathit{AB}$ stacking bilayer before the distortion.

Figure 3

The interlayer displacement $\delta \left(x\right)$ along the armchair direction in the $\mathit{AA}$-type boundary and $\mathit{SP}$-type boundary. (Inset) Bilayer graphene with a displacement $\mathbf{\delta }$.

Figure 4

(a) Interlayer matrix elements $U_{A_2{A}_1}=U_{B_2{B}_1}$, $U_{A_2{B}_1}$, and $U_{B_2{A}_1}$ in Eq. (12) as functions of the interlayer displacement $\delta$ along the armchair direction ($y^{\prime }$). (b)–(d) Energy band structures of bilayer graphene with several values of $\delta$ along the armchair direction. In each figure, the left panel shows the surface plot of the energy band in $k_{y^{\prime }}>0$, and the right panel shows the contour plot at several energies. Energy and wave vector are scaled in units of ${\gamma }_1$ and ${\gamma }_1/\left(\hslash v\right)$, respectively.

Figure 5

Electron transmission probability $P\left(\theta \right)$ through the domain boundaries in Fig. 2, calculated for the $K_{-}$ electron with several different energies. Lower panel shows the local electronic band structure in the intermediate regions as a function of $\delta$.

Figure 6

Electronic channel diagonally crossing the $\mathit{AB}$-$\mathit{BA}$ boundary: (a) single boundary and (b) multiple boundary case.

Figure 7

$\mathit{AB}$-$\mathit{BA}$ island structures separated by (a) $\mathit{AA}$-type boundary and (b) $\mathit{SP}$-type boundary, where we translate the layers 1 and 2 inside the inner circle by $\Delta y^{\prime }$ and $-\Delta y^{\prime }$ [defined in Eq. (1)], respectively.