Thermal decay of Coulomb blockade oscillations

We study transport properties and the charge quantization phenomenon in a small metallic island connected to the leads through two quantum point contacts (QPCs). The linear conductance is calculated perturbatively with respect to weak tunneling and weak backscattering at QPCs as a function of the temperature $T$ and gate voltage. The conductance shows Coulomb blockade (CB) oscillations as a function of the gate voltage that decay with the temperature as a result of thermally activated fluctuations of the charge in the island. The regimes of quantum $T\ll E_C$ and thermal $T\gg E_C$ fluctuations are considered, where $E_C$ is the charging energy of an isolated island. Our predictions for CB oscillations in the quantum regime coincide with previous findings by Furusaki and Matveev [Phys. Rev. B 52, 16676 (1995)]. In the thermal regime the visibility of Coulomb blockade oscillations decays with the temperature as $\sqrt{T/E_C}exp(-{\pi }^{2}T/E_C)$, where the exponential dependence originates from the thermal averaging over the instant charge fluctuations, while the prefactor has a quantum origin. This dependence does not depend on the strength of couplings to the leads. The differential capacitance, calculated in the case of a single tunnel junction, shows the same exponential decay, however the prefactor is linear in the temperature. This difference can be attributed to the nonlocality of the quantum effects. Our results agree with the recent experiment [Nature (London) 536, 58 (2016)] in the whole range of the parameter $T/E_C$.

Figure 1

The experimental system in Ref. [9] is schematically shown. In the integer quantum Hall regime, where only one spinless edge mode contributes to the transport, a metallic island (containing the charge $Q$) is connected to the leads by two QPCs, characterized by the backscattering amplitudes ${\tau }_L$ and ${\tau }_R$. The edge states are described by four bosonic operators, labeled by $\alpha =\text{in,}\text{out}$ and $j=1,2$. The average current $\langle I\rangle$ is calculated through the cross section immediately to the right of the right QPC.

Figure 2

The asymmetric setup, where the left QPC is almost fully open, while the right QPC is weakly transmitting, is schematically shown. We investigate CB oscillations to leading order in the backscattering amplitude ${\tau }_L$ and tunneling amplitude ${\tau }_R$.

Figure 3

The exact temperature dependence of the visibility (green line) is shown for the case of the symmetric setup together with the asymptotic low- and high-temperature solutions (dashed lines) and experimental results (black squares). The visibility is normalized to the value $V_0=|{\tau }_L\left|\right|{\tau }_R|/v_F^2$ in order to get rid of the nonuniversal backscattering amplitudes. The experimental results are obtained for various transmissions at the left QPC [9]: $G_L/G_q=0.983,0.974,0.974,0.75,0.75,0.75$ for $T/E_C=0.055,0.108,0.157,0.273,0.397,0.553$, respectively. The transmission at the right QPC, $G_R/G_q$, is kept very close to 1.

Figure 4

The exact temperature dependence of the visibility (green line) is shown for the case of the asymmetric setup together with the asymptotic low- and high-temperature solutions (dashed lines) and experimental results (black squares). The visibility is normalized to the value $V_0=\left|{\tau }_L\right|/v_F$ in order to get rid of the nonuniversal backscattering amplitude. The experimental results are obtained for transmission at the left QPC [9]: $G_L/G_q=0.075$ for $T/E_C=0.055,0.108,0.157,0.273,0.397,0.553$ and the transmission at the right QPC, $G_R/G_q$, is kept very close to 1. The green shadowed region shows the error bars of the numerical evaluation.