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

We revisit the Coulomb blockade of the tunnel junction with conductance much smaller than e2/ℏ. We study the junction with capacitance C, embedded in an Ohmic electromagnetic environment modeled 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 kBT≳(RK/2πR)Ec and resistances R≳RK, where RK=h/e2 and Ec=e2/2C is the single-electron charging energy. Our analytical result shows the leading dependence G(T)∝e−Ec/4kBT, so far believed to exist only if (RK/πR)Ec≪kBT≪Ec and R≫RK. The validity of our result for kBT≳(RK/2πR)Ec and R≳RK 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 RK. Our result also reproduces various asymptotic formulae derived in the past. The factor of 1/4 in the activation energy Ec/4 is due to the semiclassical Nyquist noise.