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Journal of Computational Electronics

A new behavioral-level model of superconducting Josephson junctions with Simulink

Yalin Zhong, Peng Chen

Published Aug 3, 2024• 1 citation
Publisher

Abstract

Abstract Josephson junctions based on superconducting materials are fundamental components for quantum detection, quantum communication and quantum computers. An accurate behavioral model of Josephson junctions is the prerequisite for predicting various superconducting circuits. In this study, We present a resistively and capacitively shunted junction (RCSJ) model-based behavioral-level model for the current-voltage characteristics of Josephson junctions. This model accurately predicts the current-voltage characteristics and their temperature dependencies of Josephson junctions made of different materials under three typical working modes: underdamped voltage-driven, overdamped current-driven, and underdamped current-driven. Additionally, it forecasts the critical current and superconducting energy gap characteristics with respect to temperature, as well as the constraint relationship between the shunt resistance, superconducting energy gap, and critical current. Comparing the measured data with the simulation predictions, the model has an average accuracy of 85.9%, which illustrates the reliability of the model.

DOI
10.1007/s10825-024-02206-0

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Physical review. B, Condensed matter•Jul 1, 1982•117 citations
Abstract Josephson junctions based on superconducting materials are fundamental components for quantum detection, quantum communication and quantum computers. An accurate behavioral model of Josephson junctions is the prerequisite for predicting various superconducting circuits. In this study, We present a resistively and capacitively shunted junction (RCSJ) model-based behavioral-level model for the current-voltage characteristics of Josephson junctions. This model accurately predicts the current-voltage characteristics and their temperature dependencies of Josephson junctions made of different materials under three typical working modes: underdamped voltage-driven, overdamped current-driven, and underdamped current-driven. Additionally, it forecasts the critical current and superconducting energy gap characteristics with respect to temperature, as well as the constraint relationship between the shunt resistance, superconducting energy gap, and critical current. Comparing the measured data with the simulation predictions, the model has an average accuracy of 85.9%, which illustrates the reliability of the model.
Josephson junction, as the core nonlinear element underpinning superconducting electronics, is characterized by its current-phase relation (CPR), which fundamentally determines the dynamical properties and functional capabilities of superconducting quantum devices. Traditional Josephson junctions typically exhibit a traditional sinusoidal CPR; however, the junctions characterized by non-sinusoidal CPR have recently attracted considerable attention due to their distinctive physical properties and promising quantum device applications. In this work, a numerical model tailored specifically for junctions exhibiting non-sinusoidal CPR is developed by integrating experimentally measured current-voltage (I-V ) characteristics from Nb/Al-AlOx/Nb junctions into a resistively and capacitively shunted junction (RCSJ) framework. By leveraging this refined model, the influence of CPR skewness on Josephson junction dynamics is systematically investigated. Our results indicate that in underdamped junctions, the critical current significantly diminishes with the increase of CPR skewness, a behavior reminiscent of the adjustable critical currents typically observed in DC superconducting quantum interference devices (SQUIDs). Conversely, in overdamped junctions, the influence of CPR skewness on the I-V characteristics is found to be negligible. However, our numerical simulations under microwave irradiation indicate that nonsinusoidal CPRs readily promote the emergence of half-integer Shapiro steps in overdamped junctions, thereby establishing CPR skewness as a plausible microscopic origin for this phenomenon. In addition, the advanced design system (ADS) simulations is employed to model nonlinear resonators and DC SQUID circuits, offering a detailed investigation into how nonsinusoidal CPRs modulate the Josephson inductance and magnetic flux response. Our findings reveal that engineering the CPR of Josephson junctions provides substantial flexibility in the design of superconducting qubits, parametric amplifiers, and non-magnetic nonreciprocal devices. This tunability underscores significant opportunities for developing next-generation superconducting electronic components. The Josephson junctions with engineered CPR offer expanded functionality for superconducting quantum technologies. This study suggests that customized CPR can enhance control over the dynamical behavior of junctions, and promote the optimized designs of superconducting qubits, parametric amplifiers, and nonmagnetic nonreciprocal devices.
Abstract The pursuit of high-speed response and precise reconstruction of phase reversal within Josephson junction constitutes the principal research objectives for the device model in the superconducting single flux quantum (SFQ) circuits. The proposing of the resistive-capacitive-inductive-shunted junction (RCLSJ) model by incorporating a shunted inductance on the classical resistive-capacitive-shunted junction (RCSJ) model, indeed achieves a more accurate description of the Current–Voltage ( I – V ) characteristics of Josephson junctions. However, the effectiveness of the RCLSJ model in dynamic simulations of SFQ circuits has not yet been verified. We proposed a simple and convenient test structure for D flip-flop (DFF) circuits to obtain the variation of its bias margin over a wide frequency sweep range. Post-simulations using the RCLSJ model were performed for DFF circuits. It was found that there are still matching problems between the simulations and tests, especially in the higher frequency region. Based on the junction area structure and layout application under the SIMIT-Nb03 process, a corrected RCLSJ model, designated as the RCLSJv model, was developed. This model mainly reassigns the parasitic inductance of the junction area and introduces correction coefficients by fitting the test data. The validity of the model has been verified through the comprehensive fitting of multiple sets of DFFs’ test data. In comparation with the RCSJ model, the deviation between the test and simulation results was successfully confined within 3%. Notably, in the frequency range above 60 GHz, the matching degree exhibits a remarkable enhancement, with an improvement factor ranging from 2 to 5 times.
We have measured Nb–AlOx–Nb Josephson tunnel junctions which have resistive shunts with different parasitic inductances. Numerical simulations reveal that specific features in the experimental current–voltage (I–V) characteristics of these devices are dc signatures of complex ac behavior. Depending on the inductance of the shunt loop and the capacitance of the junction, these features may either appear or disappear as the temperature of the device is increased. Examination of the simulated voltage waveforms allows us to map regions of the parameter space which exhibit complicated behavior. These regions should be avoided when a nearly sinusoidal voltage waveform is desired, as is the case for Josephson junction-based oscillators. The agreement of the experimental and simulated I–V curves also enables us to accurately determine the inductance of the shunts and the capacitance of the junctions.
Dynamical simulations and scaling arguments are used to study the current-voltage $(I\ensuremath{-}V)$ characteristics of a two-dimensional model of resistively shunted Josephson-junction arrays in presence of percolative disorder, at zero external field. Two different limits of the Josephson-coupling concentration ppp are considered, where pc{p}_{c}pc​ is the percolation threshold. For p>pcp>{p}_{c}p>pc​ and zero temperature, the $I\ensuremath{-}V$ curves show power-law behavior above a disorder-dependent critical current. The power-law behavior and critical exponents are consistent with a simple scaling analysis. At pc{p}_{c}pc​ and finite temperature T,T,T, the results show the scaling behavior of a T=0T=0T=0 superconducting transition. The resistance is linear but vanishes for decreasing TTT with an apparent exponential behavior. Crossover to nonlinearity appears at currents proportional to ${T}^{1+{\ensuremath{\nu}}_{T}},$ with a thermal-correlation length exponent ${\ensuremath{\nu}}_{T}$ consistent with the corresponding value for the diluted XY\mathrm{XY}XY model at pc.{p}_{c}.pc​.
Conventional Josephson metal-insulator-metal devices are inherently underdamped and exhibit hysteretic current-voltage response due to a very high subgap resistance compared to that in the normal state. At the same time, overdamped junctions with single-valued characteristics are needed for most superconducting digital applications. The usual way to overcome the hysteretic behavior is to place an external low-resistance normal-metal shunt in parallel with each junction. Unfortunately, such solution results in a considerable complication of the circuitry design and introduces parasitic inductance through the junction. This paper provides a concise overview of some generic approaches that have been proposed in order to realize internal shunting in Josephson heterostructures with a barrier that itself contains the desired resistive component. The main attention is paid to self-shunted devices with local weak-link transmission probabilities that are so strongly disordered in the interface plane that transmission probabilities are tiny for the main part of the transition region between two super-conducting electrodes, while a small part of the interface is well transparent. We discuss the possibility of realizing a universal bimodal distribution function and emphasize advantages of such junctions that can be considered as a new class of self-shunted Josephson devices promising for practical applications in superconducting electronics operating at 4.2 K.
Measurements have been made of the low-frequency spectral density of the voltage noise in current-biased resistively shunted Josephson tunnel junctions under conditions in which the noise mixed-down from frequencies near the Josephson frequency (${\ensuremath{\nu}}_{J}$) to the measurement frequency ($\ensuremath{\ll}{\ensuremath{\nu}}_{J}$) is in the regime $h{\ensuremath{\nu}}_{J}>{k}_{B}T$. In this limit, quantum corrections to the mixed-down noise are important. The spectral densities measured on junctions with current-voltage characteristics close to the Stewart-McCumber model were in excellent agreement with the predicted values, with no fitted parameters. The mixed-down noise for a wide range of bias voltages was used to infer the spectral density of the current noise in the shunt resistor at frequency $\ensuremath{\nu}$. With no fitted parameters, this spectral density at frequencies up to 500 GHz was in excellent agreement with the prediction $(\frac{2h\ensuremath{\nu}}{R})coth(\frac{h\ensuremath{\nu}}{2{k}_{B}T})$. The presence of the zero-point term, $\frac{2h\ensuremath{\nu}}{R}$, at frequencies $h\ensuremath{\nu}>{k}_{B}T$ was clearly demonstrated. The current-voltage characteristics of a junction with ${\ensuremath{\beta}}_{L}\ensuremath{\equiv}\frac{2\ensuremath{\pi}{L}_{s}{I}_{0}}{{\ensuremath{\Phi}}_{0}}\ensuremath{\sim}1$ and ${\ensuremath{\beta}}_{C}\ensuremath{\equiv}\frac{2\ensuremath{\pi}{I}_{0}{R}^{2}C}{{\ensuremath{\Phi}}_{0}}\ensuremath{\ll}1$, where I0{I}_{0}I0​ is the critical current, CCC is the junction capacitance, and Ls{L}_{s}Ls​ is the shunt inductance, showed structure at voltages where the Josephson frequency was near a subharmonic of the LsC{L}_{s}CLs​C resonant frequency. The additional nonlinearity of the $I\ensuremath{-}V$ characteristic caused mixing down of noise near higher harmonics of the Josephson frequency, thereby greatly enhancing the voltage noise. The measured noise was in good agreement with that predicted by computer simulations.