Analytical results of the Fokker-Planck equation derived for one superconducting nanowire quantum interference device and for DC SQUIDs-asymmetric devices

We consider an asymmetric two-junction superconducting quantum interference device, whose junctions are assumed to be overdamped, and regard Sin Fourier series for their current-phase relations. We take into account the effects of thermal fluctuations by forming a two-dimensional Fokker-Planck equation for the distribution function. We judge a series expansion of first order with respect to the components of the reduced inductance for distribution function and obtain current-voltage relation. We consider the measured resistance of the superconducting nanowire quantum interference device with mesoscopic leads that Hopkins et al. reported in Hopkins et al. [D.S. Hopkins, D. Pekker, P.M. Goldbart, A. Bezryadin, Science 308 (2005) 1762] and analyzed in Pekker et al. [D. Pekker, A. Bezryadin, D.S. Hopkins, P.M. Goldbart, Phys. Rev. B 72 (2005) 104517], by defining loop inductance, and by considering appropriate relations for resistance of nanowires. In fact we extend Chesca formulation [B. Chesca, J. Low Temp. Phys. 112 (1998) 165] simultaneously in three aspects and give a unified theory for nanowire two-junction devices, low Tc and high Tc DC SQUIDs, in restricted conditions.