Crossed Andreev effects in two-dimensional quantum Hall systems

We study the crossed Andreev effects in two-dimensional conductor/superconductor hybrid systems under a perpendicular magnetic field. Both a graphene/superconductor hybrid system and an electron gas/superconductor one are considered. It is shown that an exclusive crossed Andreev reflection, with other Andreev reflections being completely suppressed, is obtained in a high magnetic field because of the chiral edge states in the quantum Hall regime. Importantly, the exclusive crossed Andreev reflection not only holds for a wide range of system parameters, e.g., the size of system, the width of central superconductor, and the quality of coupling between the graphene and the superconductor, but also is very robust against disorder. When the applied bias is within the superconductor gap, a robust Cooper-pair splitting process with high-efficiency can be realized in this system.

Figure 1

Schematic diagram for three-terminal hybrid system of (a) graphene/superconductor and (b) two-dimensional electron gas/superconductor. The red lines denote the edge states due to the strong magnetic field $B$. With the aid of the unidirectional chiral edge states, the Cooper pairs in the superconductor lead can be split into two separated terminals.

Figure 2

Andreev reflection coefficients $T_A$ versus incident energy $\omega$ for the graphene/superconductor hybrid system at $\varphi =0.003$. Due to the chiral edge states, the exclusive CAR coefficient $T_{13A}$ is very large, with the other Andreev reflection coefficients being prohibited. The parameters are listed as follows: the Dirac point energy $E_0=-5\mathrm{\Delta }$, the width of the graphene nanoribbon $N=50$, the length of the central region $L=30$, the size of the covered area of the superconductor $M=15$, the coupling strength between the graphene and the superconductor $t_c=t$, and the disorder strength $W=0$.

Figure 3

Andreev reflection coefficients $T_A$ and normal transmission coefficients $T_N$ versus incident energy $\omega$ for the graphene/superconductor hybrid system under different magnetic fields $\varphi$. The other parameters are the same as those of Fig. 2.

Figure 4

(a) Andreev reflection coefficients $T_A$ and (b) transmission coefficient $T_{13N}$ versus incident energy $\omega$ in the regime of strong magnetic fields $\varphi$. In panel (a), only $T_{13A}$ has large value; $T_{11A},T_{31A}$, and $T_{33A}$ are almost zero, and their curves overlap together and are shown by the black bold lines. The other parameters are the same as those of Fig. 2.

Figure 5

Energy-dependent CAR coefficient $T_{13A}$ by considering different $L$ (a), $M$ (b), $N$ (c), and $t_c$ (d) in the graphene/superconductor hybrid system. If the parameters are not shown in the legend, they are as follows: $L=30,N=50,M=15,t_c=t,E_0=-5\mathrm{\Delta },\varphi =0.01$, and $W=0$. Here $T_{11A},T_{31A}$, and $T_{33A}$ are not shown in the figure because they are all zero.

Figure 6

$T_A$ and $T_N$ versus $\omega$ for different disorder strengths $W$. The parameters are $E_0=-5\mathrm{\Delta },N=50,L=30,M=15,t_c=t$, and $\varphi =0.01$.

Figure 7

Andreev reflection coefficients $T_A$ and normal transmission coefficients $T_N$ versus energy $\omega$ for the graphene/superconductor hybrid system under different magnetic fields, with the Dirac point energy $E_0=-0.5\mathrm{\Delta }$. The other parameters are $N=50,L=30,M=15,t_c=t$, and $W=0$.

Figure 8

(a) Schematic diagram of positive chiral edge states (red lines) for the conduction band and negative chiral edge states (blue lines) for the valence band in the three-terminal graphene/superconductor system. (b),(c) Schematic view of the specular reflection and the retroreflection in the energy space. While in the presence of the magnetic field and the chiral edge states, the specular reflection and the retroreflection are changed into the LAR and the CAR, respectively.

Figure 9

Andreev reflection coefficients $T_A$ and normal transmission coefficients $T_N$ versus incident energy $\omega$ for a 2D electron gas/superconductor system under different magnetic fields. The parameters are $E_b=-0.02t,N=100,L=50,M=20,t_c=0.03t$, and $W=0$.

Figure 10

$T_{13A}$ versus $\omega$ for different $\varphi$ (a), $L$ (b), $t_c$ (c), and $W$ (d) in the electron gas/superconductor hybrid system, with $E_b=-0.02t,N=100$, and $M=20$. The other parameters are (a) $L=50,t_c=0.03t$, and $W=0$; (b) $t_c=0.03t,\varphi =0.005$, and $W=0$; (c) $L=50,\varphi =0.005$, and $W=0$; and (d) $L=50,t_c=0.03t$, and $\varphi =0.01$. $T_{11A},T_{31A}$, and $T_{33A}$ are almost zero, so they are not shown in the figure.