Quantum transport through a double Aharonov-Bohm interferometer in the presence of Andreev reflection

Quantum transport through a double Aharonov-Bohm interferometer in the presence of Andreev reflection is investigated in terms of the nonequilibrium Green function method with which the reflection current is obtained. Tunable Andreev reflection probabilities depending on the interdot coupling strength and magnetic flux as well are analysized in detail. It is found that the oscillation period of the reflection probability with respect to the magnetic flux for the double interferometer depends linearly on the ratio of two parts magnetic fluxes $n$, i.e., $2(n+1)\pi$, while that of a single interferometer is $2\pi$. The coupling strength not only affects the height and the linewidth of Andreev reflection current peaks vs gate voltage but also shifts the peak positions. It is furthermore demonstrated that the Andreev reflection current peaks can be tuned by the magnetic fluxes.

Figure 1

Schematic diagram for the double AB interferometer connected with normal metal and the superconductor leads, respectively.

Figure 2

The probability of the Andreev reflection $T_{AR}$ vs $\epsilon$ in the unit of the energy gap $\Delta$ for (a) the symmetry case $({\Gamma }_1^L={\Gamma }_1^R={\Gamma }_2^L={\Gamma }_2^R=0.02)$ and asymmetry case (b) ${\Gamma }_1^L={\Gamma }_2^L=0.08$, ${\Gamma }_1^R={\Gamma }_2^R=0.02$, (c) ${\Gamma }_1^L={\Gamma }_1^R=0.08$, ${\Gamma }_2^L={\Gamma }_2^R=0.02$. The solid, dotted, dot-dashed, and dashed lines correspond to $\Omega =0$, 0.02, 0.04, and 0.08, respectively.

Figure 3

The magnetic flux $\Phi$ dependence of the Andreev reflection probability for ${\Gamma }_1^L={\Gamma }_1^R={\Gamma }_2^L={\Gamma }_2^R=0.02$, ${\epsilon }_1={\epsilon }_2=0$, $\Omega =0.2$, and ${\Phi }_{12}=0$; solid line: $\Phi =\pi ∕3$; dotted line: $\Phi =2\pi ∕3$.

Figure 4

The periodic oscillation of the Andreev reflection probability $T_{AR}$ with the magnetic flux $\Phi$ (${\Gamma }_1^L={\Gamma }_1^R={\Gamma }_2^L={\Gamma }_2^R=0.02$, ${\epsilon }_1=-0.1$, ${\epsilon }_2=0.1$), (a) ${\alpha }_1∕{\alpha }_2=1$. Solid line: $\Omega =0$, $\epsilon =0.1$; dotted line: $\Omega =0.05$, $\epsilon =0.11$; and dashed line: $\Omega =0.1$, $\epsilon =0.14$. (b) $\Omega =0.05$, $\epsilon =0.11$. Solid line: ${\alpha }_1∕{\alpha }_2=2$, dotted line: ${\alpha }_1∕{\alpha }_2=3$, and $4$ for dashed line.

Figure 5

Paths and corresponding phase shift (a) for the incident electron from left lead to superconductor and (b) for the reflecting hole from superconductor to left lead.

Figure 6

Andreev reflection current $J_A$ vs gate voltage $V_g$ (${\epsilon }_1=0.1$, ${\epsilon }_2=0.3$, ${\Gamma }_1^L={\Gamma }_1^R={\Gamma }_2^L={\Gamma }_2^R=0.02$, and $\Phi ={\Phi }_{12}=0$) for bias voltage $V=0.4$, $\Omega =0$ (solid line), $\Omega =0.05$ (dotted line), and $\Omega =0.1$ (dashed line).

Figure 7

Andreev reflection current vs gate voltage with the fixed total magnetic flux that $\Phi =\pi ∕2$ for $\Omega =0.05$. ${\Phi }_{12}=0$ (solid line), $\pi ∕2$ (dotted line), $3\pi ∕4$ (dot-dashed line), $\pi$ (dashed line).

Figure 8

Andreev reflection current vs gate voltage for $\Phi =3\pi ∕2$. ${\Phi }_{12}=0$ (solid line), $\pi ∕4$ (short-dashed line), $\pi ∕2$ (dotted line), $\pi$ (long-dashed line).

Figure 9

Andreev reflection current vs gate voltage with ${\Phi }_{12}=\pi ∕5$ for $\Phi =0$ (solid line), $2\pi ∕3$ (dotted line), $5\pi ∕3$ (dashed line).