We consider a symmetric two-junction superconducting quantum interference device, whose junctions are assumed to be overdamped, and consider the 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 the distribution function and obtain relations for current–voltage and the circulating current. We consider the measured resistance of the superconducting nanowire quantum interference device with mesoscopic leads that Hopkins et al reported in Hopkins et al (2005 Science 308 1762) and Pekker et al (2005 Phys. Rev. B 72 104517), by defining the loop inductance, and by considering appropriate relations for the resistance of nanowires. In fact, we extend the Chesca formulation (Chesca 1998 J. Low Temp. Phys. 112 165) and give a unification formulation for symmetric nanowire two-junction devices, low and high Tc DC superconducting quantum interference devices (SQUIDs) in restricted conditions.

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The Shapiro effect is known, with regular steps in the voltage-current characteristic of a Josephson junction in the presence of microwave radiation, and so are its applications. In asymmetric two-junction SQUIDs (superconducting quantum interference devices), in particular the interactions between various current contributions in the two junctions lead to more varied and richer conditions. In this paper, we discuss the possibility and the impacts of a kind of internal Shapiro effect in some portions of the parameter space of the asymmetric high-T C SQUID with overdamped junctions and large loop inductance. In this predicted phenomenon, the required high-frequency radiation is produced spontaneously in the interior of the SQUID. The main message is the importance of achieving a set of nine parameters for the device whose voltage-current characteristics show no indication of spike-step-like structures.

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