Quantized charge pumping in a superconducting double-barrier structure: Nontrivial correlations due to proximity effect

We consider quantum charge pumping of electrons across a superconducting double-barrier structure in the adiabatic limit. The superconducting barriers are assumed to be reflectionless so that an incident electron on the barrier can either tunnel through it or Andreev reflect from it. In this structure, quantum charge pumping can be achieved (a) by modulating the amplitudes, ${\Delta }_1$ and ${\Delta }_2$, of the gaps associated with the two superconductors or alternatively, (b) by a periodic modulation of the order-parameter phases, ${\varphi }_1$ and ${\varphi }_2$, of the superconducting barriers. In the former case, we show that the superconducting gap gives rise to a very sharp resonance in the transmission, resulting in quantization of pumped charge, when the pumping contour encloses the resonance. On the other hand, we find that quantization is hard to achieve in the latter case. We show that inclusion of weak electron-electron interaction in the quantum wire leads to renormalization-group evolution of the transmission amplitude toward the perfectly transmitting limit due to interplay of electron-electron interaction and proximity effects in the wire. Hence as we approach the zero-temperature limit due to renormalization-group flow of transmission amplitude, we get destruction of quantized pumped charge. This is in sharp contrast to the case of charge pumping in a double barrier through a Luttinger liquid where quantized charge pumping is actually achieved in the zero-temperature limit.

Figure 1

(Color online) A one-dimensional quantum wire (e.g., carbon nanotube) connected to two reservoirs, labeled by 1 and 2. The two thin strips on the wire depict layers of superconducting material deposited on top of the wire. The superconducting strips are connected to contacts labeled 3 and 4.

Figure 2

(Color online) Contours of transmission probability ${\left|T_{ee}\right|}^2$ in the ${\Delta }_1-{\Delta }_2$ plane at two different values of length scale, $L_P=1$ and $L_P=10$, at which the RG flow is cut off for values of $V\left(0\right)=0.1$ and $V\left(2k_F\right)=0.1$. The red ellipse $C_1$ represents the pumping contour.

Figure 3

(Color online) Contours of transmission probability ${\left|T_{ee}\right|}^2$ in the ${\varphi }_1-{\varphi }_2$ plane at two different values of length scale, $L_P=1$ and $L_P=10$, at which the RG flow is cut off for values of $V\left(0\right)=0.1$ and $V\left(2k_F\right)=0.1$. The red circle $C_2$ represents the pumping contour.

Figure 4

(Color online) Pumped charge $\mathcal{Q}$, for pumping in ${\Delta }_1-{\Delta }_2$ plane, is shown as a function of the dimensionless parameter $l$ where $l=\text{ln}\left(L_P/d\right)$, $L_P$ is either $L_T=\hslash v_F/k_{B}T$ at zero bias or $L_V=\hslash v_F/eV$ at zero temperature, and $d$ is the short-distance cutoff for the RG flow. The three curves correspond to three different values of $V\left(0\right)$ and $V\left(2k_F\right)$.

Figure 5

(Color online) The plot shows the variation of the AR phase $\varphi$ with time $t$ along the pumping contour $C_1$ in the plane of ${\Delta }_1-{\Delta }_2$ and the inset shows the variation of the same along the pumping contour $C_2$ in the plane of ${\varphi }_1-{\varphi }_2$.

Figure 6

(Color online) Pumped charge $\mathcal{Q}$, for pumping in ${\varphi }_1-{\varphi }_2$ plane, is shown as a function of the dimensionless parameter $l$ where $l=\text{ln}\left(L_P/d\right)$, $L_P$ is either $L_T=\hslash v_F/k_{B}T$ at zero bias or $L_V=\hslash v_F/eV$ at zero temperature, and $d$ is the short-distance cutoff for the RG flow. The three curves correspond to three different values of $V\left(0\right)$ and $V\left(2k_F\right)$.