Charge relaxation resistance in the Coulomb blockade problem

We study the dissipation in a system consisting of a small metallic island coupled to a gate electrode and to a massive reservoir via single tunneling junction. The dissipation of energy is caused by a slowly oscillating gate voltage. We compute it in the regimes of weak and strong Coulomb blockade. We focus on the regime of not very low temperatures when electron coherence can be neglected but quantum fluctuations of charge are strong due to Coulomb interaction. The answers assume a particularly transparent form while expressed in terms of specially chosen physical observables. We discovered that the dissipation rate is given by a universal expression in both limiting cases.

Figure 1

Measurement of resistance $R_q$. The SEB is subjected to a constant gate voltage $U_0$ The dissipative current through the tunneling contact is caused by a weak ac voltage $U\left(t\right)$.

Figure 2

Measurement of conductance. The SET is subjected to a constant gate voltage $U_g$ and constant bias $U$.

Figure 3

(Color online) Charging energy $E_{\text{ch}}=E_c{\left(n-q\right)}^2$ as a function of gate charge $q$.

Figure 4

Feynman rules for pseudofermion action; ${\xi }_{\sigma }=-\eta +\frac{\sigma \Delta }{2}$.

Figure 5

Feynman diagrams defining the polarization operator in the lowest order.

Figure 6

Dyson equation for polarization operator ${\Pi }_{s,\text{pf}}\left(i{\omega }_n\right)$.

Figure 7

Dyson equation for the vertex ${\mathbf{\Gamma }}_{\sigma }$.

Figure 8

The dissipative part of admittance of the SEB at fixed $\omega$ as a function of $\Delta$. Three plots using three different formulae are presented. $G_r$ is given by Eq. (107), $G_0$ is given by Eq. (109), and $G_f$ is given by Eq. (110). We use $g=0.5$, $E_c=10T$, and $\omega =0.8T$.

Figure 9

Schematic for comparison of quantum and classical mechanisms of the energy dissipation. The quantum dissipation dominates in the filled region.

Figure 10

Keldysh contour.

Figure 11

Contour for polarization operator $\Pi \left(\omega \right)$.

Figure 12

Contour for the vertex function ${\Gamma }^{\text{ARR}}$.