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arXiv:2106.11439 · Physical Review Applied

Modeling the Transfer Function of Two-Dimensional SQUID and SQIF Arrays with Thermal Noise

M. A. Galí Labarias, K.‐H. Müller, E.E. Mitchell

Published Jun 3, 2022• 9 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevapplied.17.064009

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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.
Abstract Designing superconducting electronic devices involves a careful study of all the elements in the circuit, including the superconducting bias leads. In this work, we introduce a theoretical model for two-dimensional (2D) superconducting quantum interference device (SQUID) arrays capable of simulating the voltage-to-magnetic flux response of devices with different bias current configurations. First, we compare uniformly biased and centre biased SQUID arrays by investigating the voltage versus magnetic flux response, maximum transfer function and voltage modulation depth. Then, we calculate the time-averaged fluxoid distributions for one-dimensional (1D) and 2D centre biased arrays. Finally, we study the performance of the two bias current configurations depending on array size, screening parameter, thermal noise strength and kinetic self-inductance fraction. Our calculations reveal: (i) centre biased 1D parallel SQUID arrays present an unusual voltage response caused by the asymmetric fluxoid distribution; (ii) the optimal transfer function of centre biased arrays strongly depends on the number of junctions in parallel; (iii) the performance of centre biased arrays approaches the uniform biased ones when the number of junctions in series exceeds those in parallel; (iv) while the screening parameter and the thermal noise strength clearly affect the device performance, the kinetic to self-inductance fraction seems to play only a secondary role.
We have successfully developed a novel superconducting quantum interferometer based on Josephson junction networks with unconventional loop size distribution. For distinct theoretically determined distributions, the magnetic field B dependent dc voltage V(B) of the interferometer possesses a unique delta-peak like characteristics around B=0. Such devices are called Superconducting Quantum Interference Filters (SQIFs). The unique voltage response of SQIFs allows novel applications, e.g., absolute magnetic field sensors, high speed logical switches and non hysteretic low noise amplifiers which can be directly connected to standard room temperature electronics. In this paper we present new experimental and theoretical results on high performance two dimensional Superconducting Quantum Interference Filters (2D SQIFs). Such 2D SQIFs can be used as absolute magnetic field sensors. Our results indicate that due to the scaling behavior of the flux to voltage transfer function and the scaling of the white output noise, a highly sensitive absolute field sensor based on 2D SQIFs can be very small in size.
Superconducting Quantum Interference Devices (SQUIDs), formed by incorporating Josephson junctions into loops of superconducting material, are the backbone of many modern quantum sensing systems. It has been demonstrated that, by combining multiple SQUID loops into a two-dimensional (2D) array, it is possible to fabricate ultra-high-performing radio-frequency sensors. However, to function as absolute magnetometers, current-in-use arrays require the area of each SQUID loop in the array to be incommensurate. Doing so forbids the achievement of their full potential of performance, limited only by the standard quantum limit. This is because imposing incommensurability in the areas contrasts with optimised performance in each single SQUID loop. In this work, we report that by selectively inserting bare sections of a superconducting circuit with no Josephson junctions, 2D SQUID arrays can operate as an absolute magnetometer even when no physical area spread is applied. Based on a generalisation of currently available theories, a complete analytical formulation for the one-to-one correspondence between the distribution of these bare loops and what we call a synthetic areas spread is unveiled. This synthetic spread represents the equivalent physical spread of incommensurate SQUID loops that you would use to obtain the absolute Voltage-Magnetic Flux response if no bare loops were in use. Our work opens the way to a broader use of this technology for the fabrication of ultra-high-performance absolute quantum sensors. Our approach is also experimentally verified by fabricating several 2D Superconducting Quantum Interference Filter (SQIF) arrays incorporating bare superconducting loops and by demonstrating that they behave in alignment with what is suggested by our theory.
The superconducting quantum interference filter (SQIF) is a new type of superconducting device which has been recently proposed for highly sensitive magnetometers for absolute magnetic field measurements. It benefits of very high voltage-to-field response, which is, in contrast conventional dc SQUIDs, not periodical. The SQIF can also be used as a radiofrequency amplifier in a similar way as the dc SQUID that can operate in a gigahertz frequency range. We designed a series type of SQIF amplifier that is compatible to conventional YBa2Cu3O7(YBCO) technology on bicrystal substrates. We present analytical, numerical and scale modeling as well as first electrical measurements results at frequencies up to 10 GHz. The SQIF array consists of 50 loops with randomly distributed areas from 0.5 to 1.5 times of . We also compared it to the regular array of conventional SQUIDs with the same loop areas. We have additional dc contacts to each 5-th SQUIDs and the SQIFs for control and comparison. Devices are fabricated using Josephson junctions with 3 mum width formed in YBCO over 24/24 and 12/12 degrees grain boundaries in yttrium-stabilized zirconia (YSZ) bicrystal substrates.
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.
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.