Diagrammatic real-time approach to adiabatic pumping through metallic single-electron devices

We present a real-time diagrammatic formalism to study adiabatic pumping in chains of tunnel-coupled metallic islands. This approach is based on an expansion to linear order in the frequency of the time-dependent parameters and on a systematic perturbation expansion in the tunnel-coupling strength. We apply our formalism to single-island and double-island systems. In the single-island setup, we find that the first-order contribution in the tunnel-coupling strength is purely due to the renormalization of the charging-energy gap. In the double-island system, we investigate the transition between weak and strong pumping.

Figure 1

Example of diagrams for the single-island system in first order in the tunnel coupling which are needed to calculate the kernels ${\mathbf{W}}_t^{\left(i/a,1\right)}\left(z\right)$ and $\partial {\mathbf{W}}_t^{\left(i,1\right)}\left(z\right)$. The additional frequency lines needed for the adiabatic corrections to the kernels are shown as dotted lines.

Figure 2

Sketch of the single-island system. The possible pumping parameters are the charging-energy gap $\Delta$ of the island and the tunnel-coupling strengths ${\alpha }_0^r$.

Figure 3

(Color online) Pumped charge through the single-island system up to first order in ${\alpha }_0$ in units of $e\beta {\eta }_1{\overline{\alpha }}_0^{\text{R}}/{\overline{\alpha }}_0^2$ as a function of the time average of the charging-energy gap for different values of $E_{\text{C}}$. The pumping parameters are ${\alpha }_0^{\text{L}}$ and $\Delta$. The tunnel-coupling strength is ${\overline{\alpha }}_0=0.01$.

Figure 4

(Color online) Pumped charge through the single-island system up to first order in ${\alpha }_0$ in units of $e{\eta }_2/{\overline{\alpha }}_0$ as a function of the time average of the charging-energy gap for different values of $E_{\text{C}}$. The pumping parameters are ${\alpha }_0^{\text{L}}$ and ${\alpha }_0^{\text{R}}$.

Figure 5

Sketch of the double-island system. It consists of two metallic islands tunnel coupled in series to noninteracting leads which are kept at the same chemical potential. We consider as pumping parameters the two charging-energy gaps ${\Delta }_r$.

Figure 6

(Color online) Pumped charge through the double-island system in the weak-pumping limit in zeroth order in ${\alpha }_0$ in units of $e{\beta }^2{\eta }_3$ as a function of the time averages of the pumping parameters ${\overline{\Delta }}_{\text{L}}$ and ${\overline{\Delta }}_{\text{R}}$ in units of the inverse temperature ${\beta }^{-1}=k_{\text{B}}T$. The white square marks the area in parameter space over which we integrate to obtain the pumped charge in the strong-pumping limit.

Figure 7

Pumped charge through the double-island system in the weak-pumping limit in zeroth order in ${\alpha }_0$ in units of $e{\beta }^2{\eta }_3$ for ${\Delta }_{\text{L}}$ equal to ${\Delta }_{\text{R}}$ (dashed line) and ${\Delta }_{\text{L}}$ equal to $-{\Delta }_{\text{R}}$ (solid line).

Figure 8

Pumped charge per cycle as a function of the side of the square cycle in parameter space obtained by means of a numerical method.

Figure 9

(a) ${\left({\mathbf{W}}_t^{\left(i/a,1\right)}\right)}_{0,0}$, (b) ${\left({\mathbf{W}}_t^{\left(i/a,1\right)}\right)}_{0,1}$, (c) ${\left({\mathbf{W}}_t^{\left(i/a,1\right)}\right)}_{1,0}$, and (d) ${\left({\mathbf{W}}_t^{\left(i/a,1\right)}\right)}_{1,1}$. Diagrams contributing to the instantaneous/adiabatic matrix elements in first order in ${\alpha }_0$. The dotted frequency lines are only needed for the adiabatic corrections ${\left({\mathbf{W}}_t^{\left(a,1\right)}\right)}_{i,j}$.

Figure 10

Diagrams contributing to the instantaneous matrix element ${\left({\mathbf{W}}_t^{\left(i,2\right)}\right)}_{0,1}$ in second order in ${\alpha }_0$.

Figure 11

(a) ${\left({\mathbf{W}}_t^{\left(i,1\right)}\right)}_{00,01}$, (b) ${\left({\mathbf{W}}_t^{\left(i,1\right)}\right)}_{10,01}$, and (c) ${\left({\mathbf{W}}_t^{\left(i,1\right)}\right)}_{01,01}$. Diagrams contributing to the instantaneous matrix elements in first order in ${\alpha }_0$.