In this paper, we analyze nonequilibrium effects within a simple heating approximation for the case of metallic Josephson weak links which have a favorable three-dimensional cooling geometry. Our principal conclusions are the following: (1) The temperature in the center of the junction with voltage V applied is Tm=[Tb2+3 (eV/2πk)2]1/2, where Tb is the bath temperature, e is the electronic charge, and k is Boltzmann’s constant. This Tm can be as high as 70 K in Nb point contacts on a high Josephson step; thus the device noise temperature TN? (1/2)(Tb+Tm) can greatly exceed Tb. (2) The critical current falls approximately as exp(−P/Po), where P is the power dissipated in the junction and Po is typically 10 μW for Al, Sn, Pb, and Nb, but much smaller for Nb3Sn. This leads to a decrease in the amplitude of Josephson steps at high voltages which is in good agreement with data on the best Sn variable-thickness microbridges and Nb point contacts. In a junction of optimized resistance level, the overall maximum voltage at which microwave-induced steps can still exceed the noise is predicted to be proportional to Tc[Ωξ (0)/ρo]2/7× (ν1/Tc)1/7, where Ω is the solid angle of the three-dimensional cooling, ξ (0) is the extrapolated coherence length at T=0, ρo is the residual resistivity of the metal, and ν1 is the microwave frequency. (3) The thermal response should be fast enough (∼τGL) to allow detection (heterodyne or square law) with bandwidth as large as the energy gap frequency; it is estimated that this bolometric detection effect will dominate true Josephson detection for carrier frequencies above ∼3 THz (i.e., λ≲100 μm) in junctions with resistance 10 Ω or less.

Tinkham, M., Octavio, M., & Skocpol, W. J. (1977). Heating effects in high-frequency metallic Josephson devices: Voltage limit, bolometric mixing, and noise. Journal of Applied Physics, 48(3), 1311–1320.