Quantum charge pumping through a superconducting double barrier structure in graphene

We consider the phenomenon of quantum charge pumping of electrons across a superconducting double barrier structure in graphene in the adiabatic limit. In this geometry, quantum charge pumping can be achieved by modulating the amplitudes (${\Delta }_1$ and ${\Delta }_2$) of the gaps associated with the two superconducting strips. We show that the superconducting gaps give rise to a transmission resonance in the ${\Delta }_1$-${\Delta }_2$ plane, resulting in a large value of pumped charge, when the pumping contour encloses the resonance. This is in sharp contrast to the case of charge pumping in a normal double barrier structure in graphene, where the pumped charge is very small, due to the phenomenon of Klein tunneling. We analyze the behavior of the pumped charge through the superconducting double barrier geometry as a function of the pumping strength and the phase difference between the two pumping parameters, for various angles of the incident electron.

Figure 1

Cartoon of the SDB structure where a graphene sheet is connected to two reservoirs labeled 1 and 2. Superconducting material is deposited on top of the two patches labeled S and connected to contacts labeled 3 and 4. The schematic of the potential profile seen by an incident electron is shown below.

Figure 2

Contours of the transmission probability $\left|t_c\right|{}^2$ in the ${\Delta }_1-{\Delta }_2$ plane for three different values of $k_y$. The steps between the maxima and minima of $\left|t_c\right|{}^2$ range approximately from 1.0 to 0.1, 0.6 to 0.05, and 0.3 to 0.01 for the three cases $k_y=0.0,\hspace{0.5em}0.3,$ and 0.75 respectively. In (a), (b), and (c) $a/L=0.017,{\Delta }_0=40.0,U_0=10.0,{\varphi }_1={\varphi }_2=0.0$, and $\epsilon =0.001\hspace{0.5em}42,0.001\hspace{0.5em}21,0.000\hspace{0.5em}82$ for the three contour plots, respectively. The black circle represents the pumping contour for the parameter values $\omega =1.0,P=30.0$, and $\eta =4\pi /9$.

Figure 3

The value of the pumped charge $\mathcal{Q}$ in units of electron charge $e$, for pumping in the ${\Delta }_1-{\Delta }_2$ plane, is shown as a function of the pumping strength $P$ for three different values of $k_y$. Here $a/L=0.017,\omega =1.0,\eta =4\pi /9,{\Delta }_0=40.0\hspace{0.5em}$, $U_0=10.0$, ${\varphi }_1={\varphi }_2=0.0$, and $\epsilon =0.001\hspace{0.5em}42,0.001\hspace{0.5em}21,0.000\hspace{0.5em}82$ for the three plots, respectively.

Figure 4

The value of the pumped charge $\mathcal{Q}$ in units of electron charge $e$, for pumping in the ${\Delta }_1-{\Delta }_2$ plane, is shown as a function of the phase difference between the two pumping parameters $\eta$ for three different values of $k_y$. Here $a/L=0.017,\omega =1.0,P=30.0,{\Delta }_0=40.0$, $U_0=10.0$, ${\varphi }_1={\varphi }_2=0.0$, and $\epsilon =0.001\hspace{0.5em}42,0.001\hspace{0.5em}21,0.000\hspace{0.5em}82$ for the three plots, respectively.

Figure 5

The maximum value of the pumped charge ${\mathcal{Q}}_{\max }$ in units of electron charge $e$ through the SDB structure, for pumping in the ${\Delta }_1-{\Delta }_2$ plane, is shown as a function of the incident angle $\alpha$.