Note on the Coulomb blockade of a weak tunnel junction with Nyquist noise: Conductance formula for a broad temperature range

TL;DR

This paper derives an analytical formula for the conductance of a weak tunnel junction influenced by Nyquist noise, valid over a broad temperature range, and confirms it with numerical and experimental data.

Contribution

It presents a new analytical expression for the temperature-dependent conductance of a Coulomb blockade junction considering Nyquist noise, extending validity beyond previous limits.

Findings

Analytical conductance formula valid for broad temperature range

Good agreement with numerical simulations without semiclassical assumptions

Consistent with experimental data for resistances near the quantum resistance

Abstract

We revisit the Coulomb blockade of the tunnel junction with conductance much smaller than $e^2/\hbar$. We study the junction with capacitance $C$, embedded in an Ohmic electromagnetic environment modelled by a series resistance $R$ which produces the Nyquist noise. In the semiclassical limit the Nyquist noise charges the junction by a random charge with a Gaussian distribution. Assuming the Gaussian distribution, we derive analytically the temperature-dependent junction conductance $G(T)$ valid for temperatures $k_BT \gtrsim (R_K/2\pi R)E_c$ and resistances $R \gtrsim R_K$, where $R_K = h/e^2$ and $E_c=e^2/2C \ \text{is}$ the single-electron charging energy. Our analytical result shows the leading dependence $G(T) \propto e^{-E_c/4k_BT}$, so far believed to exist only if $(R_K/\pi R)E_c \ll k_BT \ll E_c$ and $R \gg R_K$. The validity of our result for $k_BT \gtrsim (R_K/2\pi R)E_c$ and $R \gtrsim R_K$ is confirmed by a good agreement with the numerical studies which do not assume the semiclassical limit, and by a reasonable agreement with experimental data for $R$ as low as $R_K$. Our result also reproduces various asymptotic formulae derived in the past. The factor of $1/4$ in the activation energy $E_c/4$ is due to the semiclassical Nyquist noise.