Improving the design of superconducting quantum interference devices (SQUIDs) for quantum sensing applications can involve a compromise between competing device parameters. For SQUIDs made from high-temperature superconductors (HTS), accurate modelling for optimization should include the device geometry, material properties and their temperature dependences, as well as thermal noise. We outline here a lumped element model that calculates the voltage response to magnetic flux of a hairpin SQUID and use it to explore the effect of these parameters on optimizing the SQUID voltage output. We first validate our model by showing close agreement with current-voltage and voltage-flux measurements performed on Yttrium Barium Copper Oxide hairpin SQUIDs with different geometries by including structures such as the bias leads. We show the effect of track width on the kinetic to geometric inductance ratios ( $\kappa$ ) for a range of film thicknesses and temperatures, to aid researchers how to optimize SQUID designs. Then, through numerical simulations, we show on how (i) the voltage modulation depth decreases exponentially with the total inductance, independent of the asymmetry ratio; (ii) narrower superconducting tracks lead to a broader temperature operation range, $\Delta T \sim 30$ K, while wider tracks operate in a narrower temperature range, $\Delta T \sim 10$ K, but are more sensitive to temperature changes; and (iv) the optimal bias current, which maximises the device performance, depends on the operating temperature.

In this study, we investigated the performance of two 2D superconducting quantum interference filter (SQIF) arrays fabricated from YBCO thin films at a temperature of 77 K. Each array consisted of 6 Josephson junctions (JJs) in parallel and 167 in series. We conducted both experimental and theoretical analyses, measuring the arrays' voltage responses to an applied magnetic field and their voltage versus bias-current characteristics. To properly model the planar array layouts, our theoretical model used the stream function approach and also included the Johnson noise in the JJs. The model further divides the superconducting current density of the arrays into its Meissner current, circulating current, and bias current parts for practicality. Since the fabrication process of YBCO thin films cannot produce identical JJ critical currents, we assumed a log-normal distribution to model the JJ critical current disorder. Our model predictions, with a JJ critical current spread of 50%, agreed well with our experimental data. Using our model, we were able to study the dependence of the voltage modulation depth on critical current disorder and London penetration depth. We also analyzed the observed reflection asymmetries of the voltage versus magnetic field characteristics, which might provide insight into the degree of critical current disorder. Overall, our findings suggest that the use of YBCO thin films in SQIF arrays is promising, despite the critical current disorder inherent in their fabrication process. Our study highlights the importance of theoretical modeling in understanding the performance of superconducting devices and provides insights that could inform the design of future SQIF arrays.

In this letter, we demonstrate for the first time the creation of Josephson-like superconducting nanojunctions using a thermal scanning probe to directly inscribe weak links into microstrips of YBa2Cu3O7-x (YBCO). Our method effectively reduces the critical current (Ic) over an order of magnitude. The resulting nanobridges exhibit clear evidence of Josephson effects, of SNS-type junctions, as shown by both the DC and AC Josephson effects. This approach provides a novel and flexible method for scaling up quantum mechanical circuits that operate at liquid nitrogen temperatures. Additionally, it offers a promising pathway for modifying properties of the junctions in-situ and post fabrication.

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.

Superconducting quantum interference device arrays have been extensively studied for their high magnetic field sensitivity. The performance of these devices strongly depends on the characteristic parameters of their Josephson junctions, i.e. their critical currents and shunt resistances. Using a resistively shunted junction model and including thermal noise, we perform a numerical investigation of the effects of asymmetric Josephson junctions by independently studying variations in the critical currents and junction resistances. We compare the voltage response of a dc-SQUID with a 1D parallel SQUID array and study the maximum transfer function dependence on the number of junctions in parallel, the screening parameter and thermal noise strength. Our results show that the maximum transfer function and linearity increase with the number of junctions in parallel for arrays with different junction resistances, in contrast to SQUID arrays with identical junctions or with spreads in the critical currents.

Inductance plays a crucial role in the design and optimization of superconducting quantum interference devices (SQUIDs) for quantum sensing applications, since it dictates the sensitivity and coupling ratio with other circuit elements. In high-temperature superconductors the kinetic inductance, which depends on both geometry and temperature, becomes a dominant part of the device's total self-inductance, since their London penetration depth is considerably larger compared to low-temperature superconductors. In this work, we use an asymmetric SQUID to investigate the kinetic self-inductance ratio and voltage modulation depth at different operating temperatures, device geometries and bias currents. We first validate our approach by comparing our modelled data with experimental measurements. Then, through numerical simulations, we show: (i) kinetic inductance dominates for thin superconducting films, while for thicker films the inductance is less sensitive to temperature changes; (ii) the voltage modulation depth decreases exponentially with the total inductance independent of the asymmetry ratio; (iii) narrower superconducting tracks lead to a broader temperature operation range, $\Delta T \sim 30 K$, while wider tracks operate in a smaller temperature range, $\Delta T \sim 10 K$, but are more sensitive to temperature changes; and (iv) the device performance versus temperature strongly depends on the bias current used.

We investigate theoretically the effect of the coupling radius on the transfer function in 1D parallel and 2D SQUID arrays with different number of Josephson junctions in parallel and series at 77 K. Our results show a plateauing of the array maximum transfer function with the number of junctions in parallel. The plateauing defines the array coupling radius which we show increases with decreasing the normalised inductive reactance of the SQUID loop. The coupling radius is found to be independent of the number of junctions in series. Finally, we investigate the voltage versus magnetic field response and maximum transfer function of one 1D and two 2D SQIF arrays with different SQUID loop area distributions.

An important parameter of Josephson junctions (JJs) is the product of normal state resistance ( R n ) and critical current ( I c ) for designing superconductor analogue devices or digital circuits. Determination of R n and I c from voltage – current ( U–I ) characteristic curves often faces difficulties; in particular I c is considerably reduced by intrinsic thermal or extrinsic electrical noises. Here, we propose a standard measurement method of R n and intrinsic critical current ( I ci ) for high- T c superconductor (HTS) grain boundary JJs operated in liquid nitrogen and low- T c superconductor (LTS) multilayer superconductor/normal-conductor/superconductor (SNS) JJs in liquid helium. The applicable condition of this method is that both HTS and LTS JJs have U–I curves compatible with resistively-shunted junction (RSJ) model. Both R n and I ci values are extracted by combining a geometric mean criterion to select a data set and a least-squares fitting method with the RSJ model, eliminating two distortion effects on U–I curves: noise-rounding and self-heating. The combined method ensures relative standard uncertainty values of 1.9% for R n and 8% for I ci or better, when the users follow the standard protocol. It is demonstrated that the combined method is valid for d -wave HTS JJs near 77 K, regardless of peculiarities such as 0– π junction transition in d -wave superconductors at lower temperatures, and s -wave LTS SNS JJs with a wide range of junction parameters. This work is the first step to facilitate standardization for superconductor electronics with JJs.

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.

We have developed a comprehensive theoretical model for predicting the magnetic field response of a parallel superconducting quantum interference device (SQUID) array in the voltage state. The model predictions are compared with our experimental data from a parallel SQUID array made of a yttrium barium copper oxide thin film patterned into wide tracks, busbars, and leads, with 11 step-edge Josephson junctions. Our theoretical model uses the Josephson equations for resistively shunted junctions as well as the second Ginzburg-Landau equation to derive a system of coupled first-order nonlinear differential equations to describe the time evolution of the Josephson junction phase differences which includes Johnson noise. Employing the second London equation and Biot-Savart's law, the supercurrent density distribution is calculated, using the stream function approach, which leads to a two-dimensional second-order linear Fredholm integro-differential equation for the stream function with time-dependent boundary conditions. The model calculates the stream function everywhere in the thin-film structure to determine during the time evolution the fluxoids for each SQUID array hole. Our numerical model calculations are compared with our experimental data and predict the bias-current-versus-voltage and the voltageversus-magnetic-field response with accuracy. The model elucidates the importance of fully taking Meissner shielding and current crowding into account in order to properly describe fluxoid focusing and bias-current injection. Furthermore, our model illustrates the failure of the simple lumped-element approach to describe a parallel SQUID array with a wide thin-film structure.

The optimum design of high-sensitivity Superconducting Quantum Interference Devices (SQUIDs) and other devices based on thin high-temperature superconductor (HTS) films requires accurate inductance modeling. This needs the London penetration depth λ to be well defined, not only at 77 K, but also for any operating temperature, given the increasingly widespread use of miniature low-noise single-stage cryocoolers. Temperature significantly affects all inductances in any active superconducting device, and cooling below 77 K can greatly improve device performance; however, accurate data for the temperature dependence of inductance and λ(T) for HTS devices are largely missing in the literature. We report here inductance measurements on a set of 20 different thin-film YBa2Cu3O7−x SQUIDs at 77 K with thickness t = 220 or 113 nm. By combining experimental data and inductance modeling, we find an average penetration depth λ(77)=391 nm at 77 K, which was independent of t. Using the same methods, we derive an empirical expression for λ(T) for a further three SQUIDs measured on a cryocooler from 50 to 79 K. Our measured value of λ(77) and our inductance extraction procedures were then used to estimate the inductances and the effective areas of directly coupled SQUID magnetometers with large washer-style pickup loops. The latter agrees better than 7% with experimentally measured values, validating our measured value of λ(77) and our inductance extraction methods.

We investigate theoretically the maximum transfer function of 1D and 2D SQUID arrays with different number of Josephson junctions in parallel and series at 77 K. Our results show a plateauing of the array maximum transfer function with the number of junctions in parallel. The plateauing defines the array coupling radius which we show increases with decreasing the normalised impedance of the SQUID loop inductance. The coupling radius is found to be independent of the number of junctions in series. Finally, we investigate the voltage versus magnetic field response and maximum transfer function of one 1D and two 2D SQIF arrays with different SQUID loop area distributions.

CSIRO is the Australian national science agency and has been working in superconducting electronics since the 1970s. With the discovery of high-temperature superconductivity in 1986, CSIRO supported a team of researchers to work on YBa2Cu3O7-x (YBCO) step-edge Josephson junctions. This junction technology is the basis of many different HTS devices and sensors. The CSIRO team used these devices in a range of systems for many different applications. This paper briefly reports on the development of the HTS YBCO step-edge junction formed on MgO. It also presents several systems developed with a focus on the most successful commercial system so far called LANDTEM™. This system is used for mineral exploration and has been successful in supporting the delineation and discovery of many billions of dollars of ore bodies. Over a 30-year period, the Josephson junction has been the basis of research for the CSIRO superconductivity team and provided all the team members with rich and fulfilling careers.

The quantum interference effects of one-dimensional (1D) parallel arrays of high-temperature superconducting (HTS) SQUIDs were investigated experimentally and theoretically via the voltage–magnetic field responses for 4–81 Josephson junctions. The sensitivity of the arrays generally decreased as the number of junctions (and SQUIDs) in parallel increased, contrary to the predictions of models in the low (zero) inductance limit. A full theoretical description was developed to describe 1D parallel HTS SQUID arrays with finite inductances in an applied magnetic field, by extending the model for a single DC SQUID to multiple loops in parallel and including the flux generated by currents circulating through all loops in the array. Calculations were extended from SQUID arrays with equal loop areas to arrays with a distribution of loop areas, otherwise known as superconducting quantum interference filters. The model uses parameters relevant to HTS arrays, including typical variations (up to 30%) in HTS Josephson junction parameters, such as critical current and normal resistance. The effect of the location of the current biasing leads was also explored through the calculations. This model shows good agreement with experimentally measured 1D arrays of different lengths and highlights the importance of the geometry of the current biasing leads to the arrays when optimizing the array response.

Conventional magnetometry yields a low temperature bulk magnetic moment of about 4 mu(B)/Dy3+ in an applied field of mu H-0 = 9 T for thin and thick dysprosium nitride ( DyN) films. This is significantly lower than the maximum possible value of 10 mu(B)/Dy3+. Ion-assisted deposition was used to grow 5.7-mu m-thick rare earth nitride DyN films on organic Kapton (R) substrates. Dy-161 Mossbauer spectroscopy (with its time scale on the order of nanoseconds) indicates thermal relaxation between fully stretched +/- 10 mu(B) levels of a low-lying Kramers doublet, which is inconsistent with the Dy3+ site's ideal cubic symmetry. However, a small tetragonal distortion [is an element of approximate to -0.024(10)] observed using x-ray powder diffraction is compatible with an additional rank 2 crystal field term, B-2(0) approximate to -1.0(4) K, approaching the magnitude estimated to bring this about. The observed magnetic behavior can then be described using a two-level, molecular field model with theta(C) set to approximate to 6-8K, which is substantially smaller than the accepted ordering temperature of T-C approximate to 17-26 K.

With the recent development of multijunction superconducting quantum interference device (SQUID) arrays and superconducting quantum interference filters (SQIFs) using high-temperature superconducting (HTS) materials, it has become increasingly desirable to accurately model HTS devices to enable the optimization of the design for enhanced performance before fabrication. Initially, to develop our simulation capability for HTS devices, we have been working with a commercial software package to fit the data obtained from the characterization of two HTS devices. The models were iteratively adjusted by increasing the number of parameters taken into account, including the aspect ratio of the SQUID loop and the bias injection point. These models were developed for devices fabricated using YBCO step-edge junctions formed on MgO substrates. It was found that the biasing mechanism affects the SQIF voltage response to applied magnetic fields. This has implications for the design of larger arrays.

Superconducting quantum interference filters (SQIFs) have been created using two dimensional arrays of YBCO step-edge Josephson junctions connected together in series and parallel configurations via superconducting loops with a range of loop areas and loop inductances. A SQIF response, as evidenced by a single large anti-peak at zero applied flux, is reported at 77 K for step-edge junction arrays with the junction number N = 1 000 up to 20 000. The SQIF sensitivity (slope of peak) increased linearly with N up to a maximum of 1530 V T−1. Array parameters related to geometry and average junction characteristics are investigated in order to understand and improve the SQIF performance in high temperature superconducting arrays. Initial investigations also focus on the effect of the SQUID inductance factor on the SQIF sensitivity by varying both the mean critical current and the mean inductance of the loops in the array. The RF response to a 30 MHz signal is demonstrated.

We explore bi-superconducting quantum interference device (bi-SQUID) designs suitable for the fabrication process of high-temperature superconducting step-edge Josephson junctions. The bi-SQUID offers increased linearity and improved signal detection performance over a SQUID. The realization of high-temperature superconducting (HTS) bi-SQUIDs opens up new potential applications given the reduced constraints of the cryogenic package. In this paper, we explore the effects of Josephson junction and temperature variations through modeling a bi-SQUID system of equations for different designs based on step-edge Josephson junctions.

Fabrication and characterisation of YBCO step-edge Josephson junction arrays in a series configuration are reported. The junction arrays were fabricated using CSIRO YBCO step-edge junction technology, which, owing to the flexibility of locating the junctions anywhere on the chip, makes it very attractive for implementation of multiple HTS devices and circuits. Arrays of 50 junctions in series were fabricated, and the DC current-voltage (I-V) characterisations were studied at temperatures ranging from 15 to 77 K; temperatures were obtained via either placing the samples in the vent gas of a liquid He cryostat or use of a commercial cryocooler. Eight fabricated and tested arrays, totalling 400 junctions, showed no electrically open or shorted junctions (i.e., a 100% yield of working Josephson junctions), demonstrating good fabrication reliability. Large junction normal resistances (several hundred Omega s) were obtained from the series junction arrays, making them suitable for a range of high-frequency applications. A study of the junction critical current spread within an array, magnetic field dependence of the critical current, and microwave responses of the arrays is also described.