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Superconductor Science and Technology

An analysis theory of symmetric dc SQUID driven by thermal noises

Ya Jia, W R Sun, J R Li

Published Feb 7, 2003• 2 citations
Publisher

Abstract

An analysis expression for the stationary probability distribution of the symmetric superconducting quantum interference device with two Josephson junctions (dc SQUID) driven by thermal noise is derived from two-dimensional Fokker–Planck equations where the potential condition is satisfied. The analytical formulae for SQUID characteristics, e.g. the circulating current, the current–voltage relationship and the transfer function, are obtained by using the probability distribution. The ‘ripple’ phenomenon of the circulating current versus the applied flux, the effects of thermal fluctuations on current–voltage relationship and the optimum operating condition of dc SQUID system are represented.

DOI
10.1088/0953-2048/16/4/303

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An analysis expression for the stationary probability distribution of the symmetric superconducting quantum interference device with two Josephson junctions (dc SQUID) driven by thermal noise is derived from two-dimensional Fokker–Planck equations where the potential condition is satisfied. The analytical formulae for SQUID characteristics, e.g. the circulating current, the current–voltage relationship and the transfer function, are obtained by using the probability distribution. The ‘ripple’ phenomenon of the circulating current versus the applied flux, the effects of thermal fluctuations on current–voltage relationship and the optimum operating condition of dc SQUID system are represented.
An analysis expression for the stationary probability distribution of asymmetric superconducting quantum interference device with two Josephson junctions (dc SQUID) driven by thermal noise is derived from two-dimensional Fokker–Planck equation, where the potential condition is satisfied. Two of the three asymmetric parameters, inductance asymmetric parameter η and critical current asymmetric parameter α, can be changed in this condition, but resistance asymmetric parameter ρ cannot be changed. The "ripple" phenomenon of circulating current can disappear with the change of coefficient α. The effects of asymmetric parameters on current-voltage relationship and transfer function of dc SQUID system are also represented.
An analytical expression for the stationary probability distribution of the DC superconducting quantum interference device (SQUID) with a resistively shunted inductance driven by thermal noise is derived from the two-dimensional Fokker–Planck equation. The effects on the SQUID characteristics subject to a large thermal fluctuation with a noise parameter Γ>0.20 are discussed by taking into account the thermal noise in the accuracy of numerical simulation. This theory is valid for a reduced inductance β⩽1. The analytical formulae for the SQUID characteristics, e.g. the circulating current, the average voltage and the voltage modulation, are obtained and discussed. The theory shows that the voltage modulation increases with the shunted inductance more efficiently for a large inductance parameter β and small fluctuation parameter Γ.
A symmetric superconducting quantum interference device (SQUID) with two Josephson junctions in the presence of thermal fluctuations is theoretically studied through two-dimensional Fokker-Planck equations. An analytical formula of the SQUID transfer function (or responsitivity) is derived by using the potential condition in the case of the steady state. It is found that the relationship between the optimum bias current I opt and the thermal fluctuation parameter Γ is I opt /Γ>1. The optimal reduced dc SQUID inductance β opt is in the range of 0.80<β opt <1.25, which is remarkably in agreement with the result of numerical simulation β opt ≈1.
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.
Effects of thermal noise on the characteristics of the dc superconducting quantum interference device (SQUID) have been studied. Numerical simulation on the SQUID characteristics operating at T=77 K has been performed by taking into account the thermal noise. It is shown that the voltage versus flux relation of the dc SQUID is degraded considerably with the thermal noise. The degradation becomes significant when the inductance of the SQUID increases. Due to this degradation, there exists significant limitation for the range of the inductance available at T=77 K, unlike the case at T=4.2 K. The maximum inductance should be around 200 pH in order to avoid significant degradation of the transfer function. This limited value of the inductance must be taken into account when we realize the SQUID coupled to an input coil. The analytical expression for the degradation of the transfer function due to the thermal noise is also obtained. The theoretical result explains experimental results reported recently.