Effective capacitance of a single-electron transistor

Starting from the Kubo formula for conductance, we calculate the frequency-dependent response of a single-electron transistor (SET) driven by an ac signal. Treating tunneling processes within the lowest order approximation valid for a wide range of parameters, we discover a finite reactive part even under Coulomb blockade due to virtual processes. At low frequencies this can be described by an effective capacitance. This effect can be probed with microwave reflection measurements in radio-frequency (rf) SET provided that the capacitance of the surroundings does not completely mask that of the SET.

Figure 1

(Color online) Schematic drawing of a single-electron transistor considered in this Brief Report. Left electrode is connected to an ac voltage source of angular frequency $\omega$. Central island is coupled capacitively to a gate electrode, and the gate charge can be adjusted with the gate voltage $V_G$.

Figure 2

(Color online) Real (a) and imaginary (b) parts of the zero-temperature admittance as a function of the frequency of the driving signal and the gate voltage. If the SET is assumed to stay in the ground state at all times, the patterns repeat periodically as functions of the gate voltage.

Figure 3

(Color online) The total capacitance of the SET as a function of $R_T/R_Q$. The different curves correspond to different values of gate voltage (from bottom to top): $e{V}_G/E_C=0$ (blue), $e{V}_G/E_C=0.2$ (cyan), $e{V}_G/E_C=0.4$ (green), $e{V}_G/E_C=0.6$ (magenta) and $e{V}_G/E_C=0.8$ (dark green). The dashed red line corresponds to the geometric capacitance without the quantum correction. The first-order approximation is no longer valid when $R_T/R_Q$ approaches unity. Inset shows the gate dependence of the total capacitance for $R_T/R_Q=2$.

Figure 4

Proposed measurement setup for the measurement of gate charge with the reactive impedance of a SET.

Figure 5

(Color online) Phase of the reflection coefficient in a setup shown in Fig. 4. Phase changes sign at the resonance frequency, which depends on the gate voltage. The parameters that were used are $R_T/R_Q=2$, $E_{C}L/\hslash R_Q=0.6$, $C_{\parallel }/C=250$, and $Z_0/R_Q=0.002$. For $E_C/k_B=1\text{K}$, these correspond to $Z_0\approx 50\Omega$, $L\approx 160\text{nH}$, and $C_{\parallel }\approx 250\text{fF}$. Same values were used in the measurement of Ref. 12. The resonance frequency at zero gate is around ${\omega }_0\approx 6\times {10}^9{\text{s}}^{-1}$.