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Chinese Physics Letters

A Stochastic Theory of dc Superconducting Quantum Interference Device Responsitivity in the Presence of Thermal Fluctuations

Ya Jia, Wen‐Rong Sun, Jiarong Li

Published May 28, 2002• 3 citations
Publisher

Abstract

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.

DOI
10.1088/0256-307x/19/6/305

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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.
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 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.
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.
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.
Direct current superconducting quantum interference devices (SQUIDs) made from high transition temperature superconductors typically exhibit a still unexplained higher white noise and smaller transfer function than numerical simulations based on the coupled Langevin equations predict. We present a systematic study that compares experimental results with both numerical simulations and a recently developed analytical approach based on a Fokker–Planck equation. To cover a wide range of SQUID parameters, two samples of different inductances were measured at variable temperature. While several predictions of the analytical theory could be verified and a good qualitative agreement with both theories is observed, only one sample showed good quantitative agreement. Furthermore, our simulations performed in the limit of large thermal fluctuations show excellent agreement with the analytic theory.
We present a theoretical model for two-dimensional (2D) superconducting quantum interference device (SQUID) and superconducting quantum interference filter (SQIF) arrays with overdamped Josephson junctions for uniform bias-current injection at 77 K. Our simulations demonstrate the importance of including Johnson thermal noise and reveal that only the SQUID-loops self-inductance contributions are of importance. Our numerical results establish the validity of a scaling behavior between the voltages of one-dimensional (1D) and 2D SQUID arrays and show that the same scaling behavior applies to the maximum transfer functions. The maximum transfer function of a 2D SQUID array can be further optimized by applying the optimal bias current, which depends on the SQUID-loop self-inductance and the junction critical current. Our investigation further reveals that a scaling behavior exists between the maximum transfer function of a 2D SQUID array and that of a single dc SQUID. Finally, we investigate the voltage response of 1D and 2D SQIF arrays and illustrate the effects of adding spreads in the heights and widths of SQUID loops.