# Thermal decay of the Coulomb blockade oscillations

## TL;DR

This paper analyzes how Coulomb blockade oscillations in a small metallic island decay with temperature, providing a unified description of quantum and thermal regimes and matching recent experimental observations.

## Contribution

It offers a theoretical prediction for the thermal decay of Coulomb blockade oscillations across different regimes, including the quantum and thermal limits, with explicit temperature dependence.

## Key findings

- Coulomb blockade oscillations decay as ^{-rac{1}{2}} ext{exp}(-rac{ ext{} ext{pi}^2 T}{E_C}) in the thermal regime.
- The decay is independent of coupling strength to leads in the thermal regime.
- Results agree with recent experimental data across all temperature ranges.

## Abstract

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 in [A. Furusaki and K. A. Matveev, Phys. Rev. B {\bf 52}, 16676 (1995)]. In the thermal regime the visibility of Coulomb blockade oscillations decays with the temperature as $\sqrt{T/E_C}\exp(-\pi^2T/E_C)$, 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 non-locality of the quantum effects. Our results agree with the recent experiment [S. Jezouin {\em et al}., Nature {\bf 536}, 58 (2016)] in the whole range of the parameter $T/E_C$.