Cone-dependent retro- and specular Andreev reflections in AA-stacked bilayer graphene

We theoretically study the Andreev reflection (AR) in AA-stacked bilayer graphene-superconductor junction. AABG has a linear gapless energy band with two shifted Dirac cones and the electronic states are described by the cone indices. The results indicate that the property of AR strongly depends on the cone degree of freedom. In the absence of the interlayer potential difference, only intracone AR and normal reflection (NR) could occur, and the intercone process is forbidden. By adjusting the potential, the intracone AR can be specular AR (SAR) in one cone and it is retro-AR (RAR) in the other cone. The existence of the interlayer potential difference would lead to the intercone scattering. As a result, double ARs and double NRs can take place between the two cones. The intercone SAR could happen in a broad potential region. Furthermore, the intercone retro-NR (RNR) could happen as well. The switch between SAR and RAR, and the switch between specular NR (SNR) and RNR can be achieved by regulating the potential. Therefore, different cone carriers can be separated spatially based on the RAR and SAR. The cone-dependent Andreev conductance may be separately measured near the critical values where RAR crosses over to SAR.

Figure 1

(a) Band structures in the AABG region. An incident electron from the $A_{\pm }$ points can be Andreev reflected as a hole from $B_{\pm }$ points and normally reflected as an electron from $C_{\pm }$ points. The solid and dashed curves are the energy bands for electron and hole. The black and red curves are the energy bands at upper and lower cones, respectively. (b) and (c) The projection of the Andreev scattering and normal scattering processes (b) on the $k_x\text{−}{k}_y$ plane and (c) on the $x\text{−}y$ real space. The black and red arrows denote the group velocities at the upper and lower cones. The arrows with filled circles and empty circles are for the electrons and holes, respectively. The potential satisfies $U_{c1}<U<U_{c2}$ in (a)–(c). (d) The Andreev scattering and normal scattering processes on the $k_x\text{−}{k}_y$ plane when $U_{c3}<U<U_{c4}$.

Figure 2

AR probabilities $R_{\text{A}}^{++}$ and $R_{\text{A}}^{--}$ in the $(U,k_y)$ plane for the incident electron from the (a) upper cone and (b) lower cone at $E=\mathrm{\Delta }$ and $\delta =0$. The white dashed curves are energy bands in the AABG region, which are also the boundaries of the zero and nonzero probabilities.

Figure 3

Andreev conductance $G_{\text{AR}}^{-}$ of the AABG-SC interface for the lower cone at $\delta =0$. The insets show the AR probability $R_{\text{A}}^{--}$ in the $(E,\theta )$ plane at $U=99.6\mathrm{\Delta }$ and $100.0\mathrm{\Delta }$. The cyan curve are the Andreev conductance $G_{\text{AR}}^{+}$ for the upper cone at $\delta =0$ and $U=99.4\mathrm{\Delta }$.

Figure 4

Andreev conductances (a) $G_{\text{AR}}^{+}$, (b) $G_{\text{AR}}^{-}$, and (c) $G_{\text{AR}}^{\text{total}}$ of the AABG-SC interface at $\delta =0$. The insets in (a) and (b) show the AR probabilities $R_{\text{A}}^{++}$ and $R_{\text{A}}^{--}$ in the $(E,\theta )$ plane at $U=60.0\mathrm{\Delta }$ and $\delta =0$, respectively.

Figure 5

AR and NR probabilities $R_{\text{N,A}}^{\pm \pm }$ versus incident angle $\theta$ for the electron from the (a)–(c) upper cone and (d)–(f) lower cone at $E=0.8\mathrm{\Delta }$. The parameters $U=0.0$ and $\delta =0.0$ in (a) and (d), $U=0.0$ and $\delta =120.0\mathrm{\Delta }$ in (b) and (e), and $U=60.0\mathrm{\Delta }$ and $\delta =120.0\mathrm{\Delta }$ in (c) and (f).

Figure 6

The projection of the Andreev scattering and normal scattering processes on the $k_x\text{−}{k}_y$ plane when (a) $U<U_{c1}$, (b) $U_{c2}<U<U_{c3}$, and (c) $U>U_{c4}$. The black and red arrows denote the group velocity at upper and lower cones. The arrows with filled circles and empty circles are for the electrons and holes, respectively.

Figure 7

AR and NR probabilities (a) $R_{\text{N}}^{++}$, (b) $R_{\text{N}}^{+-}$, (c) $R_{\text{A}}^{++}$, and (d) $R_{\text{A}}^{+-}$ in the $(U,k_y)$ plane for the electron from the upper cone at $E=0.0$ and $\delta =120.0\mathrm{\Delta }$. The white dashed curves are energy bands in the AABG region, which are also the boundaries of the zero and nonzero probabilities.

Figure 8

Andreev conductances (a) $G_{\text{AR}}^{+}$, (b) $G_{\text{AR}}^{-}$, and (c) $G_{\text{AR}}^{\mathrm{total}}$ of the AABG-SC interface at $\delta =120.0\mathrm{\Delta }$.