Analytical results of the Fokker–Planck equation derived for one superconducting nanowire quantum interference device and for DC SQUID: symmetric devices

Abstract

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.

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

“…(13)) also is changed. Therefore except for special cases [28,29], a definite flux shift D/ x in the voltage-flux curves hV ð/ x Þi are appears.…”

“…Bias current I also will be assumed to be small (Section 3 of Ref. [28]). Nanowires and edges of the leads define a closed geometrical contour, which will be referred to as the Aharonov-Bohm (AB) contour.…”

“…[2]), the resistance of the two-junction device is invariant under changing the direction of applied magnetic field or direction of bias current [25,28,29] or transposition junction indexes (1 M 2). According to our formulation, these symmetries exist only in especial cases, e.g., for symmetric device [28].…”

“…Recently we extended [28] Chesca formulation [3] by considering a Sin Fourier series for CHR, and allowing for conditions of a symmetric NQUID. In this paper, by considering:…”

The influences of the second harmonics together with the asymmetries of junctions on the voltage–current characteristics of high TC DC SQUIDs

“…For optimization the applicable SQUID characteristics or for developing the research methods based on SQUID characteristics, the exact analytic formulations for VCC and VFC which incorporate the mentioned facts are necessary. To arrive these needs extensive works have been done [4][5][6][7][8][9][10]. Usually one assumes the non-equilibrium phenomena are localized and takes a set of simple expressions for the superconducting current, the quasiparticle current and the fluctuating current and makes two coupled Langevin type equations for the DC SQUID.…”