The intrinsic nonlinearity of Josephson junctions converts Gaussian current noise in the input into non-Gaussian voltage noise in the output. For a resistively shunted Josephson junction with white input noise we determine numerically exactly the properties of the few lowest cumulants of the voltage fluctuations, and we derive analytical expressions for these cumulants in several important limits. The statistics of the voltage fluctuations is found to be Gaussian at bias currents well above the Josephson critical current, but Poissonian at currents below the critical value. In the transition region close to the critical current the higher-order cumulants oscillate and the voltage noise is strongly non-Gaussian. For coloured input noise we determine the third cumulant of the voltage.
We investigate the dynamics of a resistively shunted Josephson junction. We compute the Josephson frequency and the generalized impedances for a variety of the parameters, particularly with relevance to predicting the measurable effects of zero-temperature current noise in the resistor.
This is a detailed derivation of the equation for the voltage noise of a resistively shunted Josephson junction with non-sinusoidal current-phase relation, which was presented in Sov. Phys. JETP {\bf 67}, 579 (1988).
We have studied the Josephson dynamics of resistively-shunted ballistic superconducting quantum point contacts at finite temperatures and arbitrary number of conducting modes. Compared to the classical Josephson dynamics of tunnel junctions, dynamics of quantum point contacts exhibits several new features associated with temporal fluctuations of the Josephson potential caused by fluctuations in the occupation of the current-carrying Andreev levels.
We present a cluster algorithm for resistively shunted Josephson junctions and similar physical systems, which dramatically improves sampling efficiency. The algorithm combines local updates in Fourier space with rejection-free cluster updates which exploit the symmetries of the Josephson coupling energy. As an application, we consider the localization transition of a single junction at intermediate Josephson coupling and determine the temperature dependence of the zero bias resistance as a function of dissipation strength.
The dynamic critical exponent $z$ is determined numerically for the $d$-dimensional XY model ($d=2, 3$, and 4) subject to relaxational dynamics and resistively shunted junction dynamics. We investigate both the equilibrium fluctuation and the relaxation behavior from nonequilibrium towards equilibrium, using the finite-size scaling method. The resulting values of $z$ are shown to depend on the boundary conditions used, the periodic boundary condition, and fluctuating twist boundary condition (FTBC), which implies that the different treatments of the boundary in some cases give rise to differen…
We theoretically investigate the phase and voltage correlation dynamics, which includes both the deterministic contribution and stochastic fluctuations, under a current noise generated by a resistor including thermal and quantum fluctuations in a resistively and capacitively shunted Josephson junction. An external current is found to shift and intensify the deterministic contributions in phase and voltage. In addition to effects of external current, we observe the relaxation of autocorrelation functions of phase and voltage, which includes the variances due to the current noise, to finite valu…
We investigated current-voltage characteristics of unshunted and externally shunted Josephson junctions (JJs) with high critical current densities, Jc, in order to extract their basic parameters and statistical characteristics for JJ modeling in superconducting integrated circuits and to assess their potential for future technology nodes. Nb/AlOx-Al/Nb JJs with diameters from 0.5 μm to 6 μm were fabricated using a fully planarized process with Mo or MoNx thin-film shunt resistors with sheet resistance Rsq = 2 Ω/sq and Rsq = 6 Ω/sq, respectively. We used our standard MIT LL process SFQ5ee to fa…
Using a new cluster Monte Carlo algorithm, we study the phase diagram and critical properties of an interacting pair of resistively shunted Josephson junctions. This system models tunneling between two electrodes through a small superconducting grain, and is described by a double sine-Gordon model. In accordance with theoretical predictions, we observe three different phases and crossover effects arising from an intermediate coupling fixed point. On the superconductor-to-metal phase boundary, the observed critical behavior is within error-bars the same as in a single junction, with identical v…
We have fabricated the Bloch transistor shunted on-chip by a small-sized Cr resistor with Rs about 1 kOhm. The Bloch transistor normally consists of two small Josephson junctions connected in series, which in our case have been replaced by two superconducting interferometer loops, each with two junctions in parallel. A capacitively coupled gate is supplied to control the induced charge of the small intermediate electrode (island) of the transistor. The measured I-V curves show no hysteresis and correspond to the operation of a effective Josephson junction at the high-damping and strong-noise l…
We have studied both theoretically and experimentally how an LC series circuit connected in parallel to a Josephson junction influences the Josephson dynamics. The presence of the shell circuit introduces two energy scales, which in specific cases can strongly differ from the plasma frequency of the isolated junction. Josephson junctions were manufactured using Nb/Al-AlOx/Nb fabrication technology with various on-chip LC shunt circuits. Spectroscopic measurements in the quantum limit show an excellent agreement with theory taking into account the shunt inductance and capacitance in the Resisti…
In this Chapter, we present our latest experimental results (along with their theoretical interpretation) related to some unusual magnetic properties of artificially prepared ordered two-dimensional Josephson junction arrays (of both shunted and unshunted Nb-AlO_x-Nb tunnel junctions). Namely, we address the following topics: (i) influence of non-uniform critical current density profile on magnetic field behavior of AC susceptibility, (ii) origin of the so-called dynamic reentrance phenomenon, and (iii) manifestation of novel geometric quantization effects in temperature behavior of AC magneti…
We have investigated the phase dynamics and IV-characteristics of shunted Josephson junctions coupled to an $LC$ circuit. When the Josephson frequency $ω_J$ is close to the eigen frequency $ω_{rc}$ of the coupled resonance circuit, the IV-characteristic demonstrates an additional $rc$-branch. We have investigated the features of the $rc$-branch and of the superconducting current component for different values of the resonance frequency. It has been found that the maximal value of the superconducting current depends on the resonance frequency and that it increases when the end point of the $rc$…
We are developing a superconductor electronics fabrication process with up to nine planarized superconducting layers, stackable stud vias, self-shunted Nb/AlOx-Al/Nb Josephson junctions, and one layer of MoNx kinetic inductors. The minimum feature size of resistors and inductors in the process is 250 nm. We present data on the mutual inductance of Nb stripline and microstrip inductors with linewidth and spacing from 250 nm to 1 μm made on the same or adjacent Nb layers, as well as the data on the linewidth and resistance uniformity.
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 inductan…
Zero-bias resistances of a single resistance-shunted Josephson junction are calculated as a function of the temperature by means of the path-integral Monte Carlo method in case a charging energy $E_{\rm C}$ is comparable with a Josephson energy $E_{\rm J}$. The low-temperature behavior of the zero-bias resistance changes around $α=R_{\rm Q}/R_{\rm S}=1$, where $R_{\rm S}$ is a shunt resistance and $R_{\rm Q}=h/(2e)^2$. The temperature dependence of the zero-bias resistance shows a power-law-like behavior whose exponent depends on $E_{\rm J}/E_{\rm C}$. These results are compared with the exper…
Within the framework of the microscopic model of tunneling, we modelled the behavior of the Josephson junction shunted by the Superconductor-Insulator-Normal metal (SIN) tunnel junction. We found that the electromagnetic impedance of the SIN junction yields both the frequency-dependent damping and dynamic reactance which leads to an increase in the effective capacitance of the circuit. We calculated the dc I-V curves and transient characteristics of these circuits and explained their quantitative differences to the curves obtained within the resistively shunted junction model. The correct oper…
Self-shunted aluminum Josephson junctions with high-transparency barriers were fabricated using the shadow-evaporation technique and measured at low temperatures, T=25 mK. Due to high junction transparency, the IV-characteristics were found to be of only small hysteresis with retrapping-to-switching current ratio of up to 80%. The observed critical currents were close to the Ambegaokar-Baratoff values (up to 80-100%). Good barrier quality was confirmed by the low subgap leakage currents in the quasiparticle branches, which makes the self-shunted Al junctions promising for application in integr…
Recent progress in superconductor electronics fabrication has enabled single-flux-quantum (SFQ) digital circuits with close to one million Josephson junctions (JJs) on 1-cm$^2$ chips. Increasing the integration scale further is challenging because of the large area of SFQ logic cells, mainly determined by the area of resistively shunted Nb/AlO$_x$-Al/Nb JJs and geometrical inductors utilizing multiple layers of Nb. To overcome these challenges, we are developing a fabrication process with self-shunted high-J$_c$ JJs and compact thin-film MoN$_x$ kinetic inductors instead of geometrical inducto…
Thermodynamics and transport properties of resistance-shunted Josephson junctions are studied theoretically in the tight-binding limit E_C/E_J<<1, where E_C and E_J are a charging energy and a Josephson coupling energy respectively. Based on a phenomenological harmonic-oscillator model, weak coupling region K=R_Q/R<<1 is analytically studied, where R and R_Q=h/(2e)^2 are a shunted resistance and the quantum resistance. In addition to the effective bandwidth, we find that this multi-level system genuinely has a novel crossover at lower energy below which the density of states becomes strongly d…
The choice of impedance used to shunt a Josephson junction determines if the charge transferred through the circuit is quantized: a capacitive shunt renders the charge discrete, whereas an inductive shunt leads to continuous charge. This discrepancy leads to a paradox in the limit of large inductances L. We show that while the energy spectra of the capacitively and inductively shunted junction are vastly different, their high-frequency responses become identical for large L. Inductive shunting thus opens the possibility to observe charging effects unimpeded by charge noise.
A theoretical study is presented on a number N of resistively shunted Josephson junctions connected in parallel as a disordered 1D array by superconducting wiring in such a manner that there are N-1 individual SQUID loops with arbitrary shape formed.
Superconducting weak-link (WL), behaving like a Josephson junction (JJ), is fundamental to many superconducting devices such as nanoSQUIDs, single-photon detectors, and bolometers. The interplay between unique nonlinear dynamics and inevitable Joule heating in a JJ leads to new characteristics. Here, we report a time-dependent model incorporating thermal effect in the AC Josephson regime for a Josephson WL shunted by a resistor together with an inductor to investigate the dynamics as well as the resulting current-voltage characteristics. We find that the dynamic regime where phase and temperat…
We present an electronic circuit whose dynamical properties emulate those of a resistively and capacitively shunted Josephson junction (RCSJ). We show how it reproduces the switching properties of a shunted junction and its dependence on the quality factor. A thermal noise source is then used to characterize the temperature dependence of the phase dynamics. In the presence of an AC drive, phase locking is observed at integer and rational multiples of the drive frequency, and it competes with chaotic behavior when the quality factor of the junction exceeds unity. We characterize the stability o…
We report on anomalous conductance in a resistively and capacitively shunted Josephson junction which is simultaneously driven by ac and dc currents. The dependence of the voltage across the junction on the frequency of the ac current shows windows of absolute negative conductance regimes, i.e. for a positive (negative) dc current, the voltage is negative (positive).
Recently transport measurements have been carried out in resistively shunted long superconducting nanowires [M. W. Brenner {\it et. al.}, Phys. Rev. B {\bf{85}}, 224507 (2012)]. The measured voltage-current ($V$-$I$) characteristics was explained by the appearance of the phase slip centers in the shunted wire, and the whole wire was modeled as a single Josephson junction. The kinetic inductance of the long nanowires used in experiments is generally large. Here we argue that the shunted superconducting nanowire acts as a Josephson junction in series with an inductor. The inductance depends on t…
We present the first study of superconducting nanowires shunted with an external resistor, geared towards understanding and controlling coherence and dissipation in nanowires. The dynamics is probed by measuring the evolution of the V-I characteristics and the distributions of switching and retrapping currents upon varying the shunt resistor and temperature. Theoretical analysis of the experiments indicates that as the value of the shunt resistance is decreased, the dynamics turns more coherent presumably due to stabilization of phase-slip centers in the wire and furthermore the switching curr…
We investigate theoretical aspects of the detection of Majorana bound states in Josephson junctions using the semiclassical RCSJ model of junction dynamics. The influence of a 4$π$-periodic supercurrent contribution can be detected through its effect on the width of the Shapiro steps and the Fourier spectrum of the voltage signal. We explain how the inclusion of a capacitance term results in a strong quenching of the odd steps when the junction is underdamped, and hence may be used to effectively detect Majorana bound states. Furthermore, in presence of capacitance the first and third steps ar…
The form of the fluctuation-dissipation theorem for a resistively shunted Josephson juction array is derived with the help of the method which explicitely takes into acoount screening effects. This result is used to express the flux noise power spectrum in terms of frequency dependent sheet impedance of the array. The relation between noise amplitude and parameters of the detection coil is analysed for the simplest case of a single-loop coil.
Josephson radiation is a useful signature for detecting Majorana zero modes in topological superconductors. We study the Josephson radiation of the underdamped topological Josephson junction within the quantum resistively and capacitively shunted junction model. We show that the quantum dynamics of the Majorana two-level system induce oscillatory patterns in the Josephson emission spectra. With the Floquet method, we obtain analytical results for these oscillatory patterns and find that they are well described by Bessel functions. We perform numerical simulations to verify the analytical resul…
The RCSJ (Resistor-Capacitor Shunted Junction) model provides us an analytical formula for the conversion efficiency for a resonant Josephson effect mixer. An external rf signal is applied to a long Josephson junction operating in the fluxon oscillating regime. The sine-Gordon equation is used to investigate the dynamics of the phase-locking between the fluxon oscillations and the external signal. We obtain that it is possible to have conversion efficiency greater than unity.
The dynamical vortex response of a two-dimensional array of the resistively shunted Josephson junctions in a perpendicular magnetic field is inferred from simulations. It is found that, as the magnetic field is increased at a fixed temperature, the response crosses over from normal to anomalous, and that this crossover can be characterized by a single dimensionless parameter. It is described how this crossover should be reflected in measurements of the complex impedance for Josephson junction arrays and superconducting films.
We report on an experimental and theoretical study of the high-frequency mixing properties of ion-irradiated YBa2Cu3O7 Josephson junctions embedded in THz antennas. We investigated the influence of the local oscillator power and frequency on the device performances. The experimental data are compared with theoretical predictions of the general three-port model for mixers, in which the junction is described by the resistively shunted junction model. A good agreement is obtained for the conversion efficiency in different frequency ranges, spanning above and below the characteristic frequencies f…
In this paper we present a method for processing a hybrid Josephson junction on Co-doped BaFe2As2(Ba-122) single crystals with a thin film Pb-counter electrode and a barrier layer of TiOx. This includes the leveling and polishing of the crystals and structuring them with thin film techniques such as photo lithography, sputtering and ion beam etching (IBE). The junctions show hysteretical resistively and capacitively shunted junction (RCSJ)-like I-V characteristics with an IcRn-product of about 800uV.
In this paper, we report on different phenomena related to the magnetic properties of artificially prepared highly ordered (periodic) two-dimensional Josephson junction arrays (2D-JJA) of both shunted and unshunted Nb-AlOx-Nb tunnel junctions. By employing mutual-inductance measurements and using a high-sensitive bridge, we have thoroughly investigated (both experimentally and theoretically) the temperature and magnetic field dependence of complex AC susceptibility of 2D-JJA. We also demonstrate the use of the scanning SQUID microscope for imaging the local flux distribution within our unshunt…
Josephson junctions were fabricated in epitaxial films of cobalt-doped BaFe2As2 on [001]-tilt (La,Sr)(Al,Ta)O3 bicrystal substrates. 10m-wide microbridges spanning a 30-degrees-tilted bicrystal grain boundary (BGB bridge) exhibited resistively-shunted-junction (RSJ)-like current-voltage characteristics up to 17 K, and the critical current was suppressed remarkably by a magnetic field. Microbridges without a BGB did not show the RSJ-like behavior, and their critical current densities were 20 times larger than those of BGB bridges, confirming BGB bridges display a Josephson effect originating fr…
The effect of external radiation on the phase dynamics of Josephson junctions shunted by an LC circuit is examined. It is shown that additional resonant circuit (rc) branches appear in the current voltage characteristics of the junctions when the Josephson frequency is equal to the frequency of the formed resonance circuit. We show that the amplitude dependence of the Shapiro step width crucially changes when the Shapiro step is on the rc-branch in comparison to the case of Josephson junction without shunt. The experimental implementation of these effects might give very important advantages f…
Current-voltage characteristics and the linear resistance of the two-dimensional XY model with and without external uniform current driving are studied by Monte Carlo simulations. We apply the standard finite-size scaling analysis to get the dynamic critical exponent $z$ at various temperatures. From the comparison with the resistively-shunted junction dynamics, it is concluded that $z$ is universal in the sense that it does not depend on details of dynamics. This comparison also leads to the quantification of the time in the Monte Carlo dynamic simulation.
We demonstrate that shunting of Superconductor-Insulator-Superconductor Josephson junctions by Superconductor-Insulator-Normal metal (S-I-N) structures having pronounced non-linear I-V characteristics can remarkably modify the Josephson dynamics. In the regime of Josephson generation the phase behaves as an overdamped coordinate, while in the superconducting state the damping and current noise are strikingly small, that is vitally important for application of such junctions for readout and control of Josephson qubits. Superconducting Nb/AlO${_x}$/Nb junction shunted by Nb/AlO${_x}$/AuPd juncti…
A comparative study of the magnetic properties of shunted and unshunted two-dimensional Josephson junction arrays (2D-JJA) is presented. Using a single-plaquette approximation of the 2D-JJA model, we were able to successfully fit all our experimental data (for the temperature, AC and DC field dependencies of susceptibility) and demonstrate that the dynamic reentrance of AC susceptibility is directly linked to the value of the Stewart-McCumber parameter β_C. Based on extensive numerical simulations, a phase diagram β_C vs β_L is plotted which demarcates the border between the reentrant and non-…
We investigate classical anomalous electrical transport in a driven, resistively and capacitively shunted Josephson junction device. Novel transport phenomena are identified in chaotic regimes when the junction is subjected to both, a time periodic (ac) and a constant, biasing (dc) current. The dependence of the voltage across the junction on the dc-current exhibits a rich diversity of anomalous transport characteristics: In particular, depending on the chosen parameter regime we can identify so termed absolute negative conductance around zero dc-bias, the occurrence of negative differential c…
Point contact junctions made from two pieces of MgB2 can be adjusted to exhibit either superconductor-insulator-superconductor (SIS) or superconductor-normal metal-superconductor (SNS) current-voltage characteristics. The SIS characteristics are in good agreement with the standard tunneling model for s-wave superconductors, and yield an energy gap of (2.02 +/- 0.08) meV. The SNS characteristics are in good agreement with the predictions of the resistively-shunted junction model. DC Superconducting QUantum Interference Devices made from two SNS junctions yield magnetic field noise as low as 35 …
We have observed the Josephson effect in junctions formed between single crystals of SrFe1.74Co0.26As2 and Ba0.23K0.77Fe2As2. I-V curves showed resistively-shunted junction characteristics, and the ac Josephson effect was observed under microwave irradiation. By applying an in-plane magnetic field, the critical current is completely modulated and shows a relatively symmetric diffraction pattern, consistent with the intermediate junction limit. The observation of the Josephson effect in the p-n bicrystal structure not only has significant implications for designing phase-sensitive junctions to …
Using a point (superconductor-constriction-superconductor, ScS) contact in a single-Josephson-junction superconducting quantum interference device (RF SQUID) provides stochastic resonance conditions at any arbitrary small value of loop inductance and contact critical current, unlike SQUIDs with more traditional tunnel (superconductor-insulator-superconductor, SIS) junctions. This is due to the unusual potential energy of the ScS RF SQUID which always has a barrier between two wells thus making the device bistable. This paper presents the results of a numerical simulation of the stochastic dyna…
We investigate numerically the critical current of two-dimensional fully frustrated arrays of resistively shunted Josephson junctions at zero temperature. It is shown that a domino-type mechanism is responsible for the existence of a critical current lower than the one predicted from the translationally invariant flux lattice. This domino mechanism is demonstrated for uniform-current injection as well as for various busbar conditions. It is also found that inhomogeneities close to the contacts makes it harder for the domino propagation to start, which increases the critical current towards the…
We have measured the low frequency critical current noise in Nb/AlO$_{\mathrm{x}}$/Nb Josephson junctions. Unshunted junctions biased above the gap voltage and resistively shunted junctions biased near the critical current, $I_{c}$, have been measured. For both, the spectral density of $δI_{c}/I_{c}$, $S_{i_{c}}(f)$, is proportional to $1/f$, scales inversely as the area, $A$, and is independent of $J_{c} \equiv I_{c}/A$ over a factor of nearly 20 in $J_{c}$. For all devices measured at 4.2 K, $S_{i_{c}}$(1 Hz)$= 2.0 \pm 0.4 \cdot 10^{-12}$/Hz when scaled to A=1 $μ$m$^{2}$. We find that, from …
The boundary effects on the current-voltage characteristics in two-dimensional arrays of resistively shunted Josephson junctions are examined. In particular, we consider both the conventional boundary conditions (CBC) and the fluctuating twist boundary conditions (FTBC), and make comparison of the obtained results. It is observed that the CBC, which have been widely adopted in existing simulations, may give a problem in scaling, arising from rather large boundary effects; the FTBC in general turn out to be effective in reducing the finite-size effects, yielding results with good scaling behavi…
In this paper we demonstrate that, if it exists, coherent quantum phase slip is the exact dual to Josephson tunneling. We use the duality to predict kinetic capacitance and a sharp resonance in narrow wires. Biased resistively and driven at high frequency, quantum phase slip junctions should exhibit current plateaus of interest for a fundamental standard.
A non-band theory is developed to describe the superconductor-insulator (SI) transtition in resistively shunted, single Josephson junctions. The $I-V$ characteristic is formulated by a Landauer-like formula and evaluated by the path-integral transfer-matrix method. The result is consistent with the recent experiments at around 80 $mK$. However, the insulator phase shrinks with decreasing temperature indicating that the single Josephson junction becomes all superconducting at absolute zero temperature, as long as dissipation is present.
The lattice effects on the current-voltage characteristics of two-dimensional arrays of resistively shunted Josephson junctions are investigated. The lattice potential energies due to the discrete lattice structure are calculated for several geometries and directions of current injection. We compare the energy barrier for vortex-pair unbinding with the lattice pinning potential, which shows that lattice effects are negligible in the low-current limit as well as in the high-current limit. At intermediate currents, on the other hand, the lattice potential becomes comparable to the barrier height…
We calculate the current-voltage characteristics and critical current I_c^{array} of an array of Josephson-coupled superconducting wires. The array has two layers, each consisting of a set of parallel wires, arranged at right angles, such that an overdamped resistively-shunted junction forms wherever two wires cross. A uniform magnetic field equal to f flux quanta per plaquette is applied perpendicular to the layers. If f = p/q, where p and q are mutually prime integers, I_c^{array}(f) is found to have sharp peaks when q is a small integer. To an excellent approximation, it is found in a squar…
A Lagrangian formalism is used to derive the Hamiltonian for a $λ$/4 resonator shunted by a current-biased Josephson junction. The eigenstates and the quantum dynamics of the system are analyzed numerically, and we show that the system can function as an efficient detector of weak incident microwave fields.
By employing mutual-inductance technique and using a high-sensitive bridge, we have thoroughly investigated (both experimentally and theoretically) the temperature and magnetic field dependence of complex AC susceptibility of artificially prepared highly ordered (periodic) two-dimensional Josephson junction arrays (2D-JJA) of both shunted and unshunted Nb-based tunnel junctions as well as disordered three-dimensional arrays (3D-JJA). This paper reviews some of our latest results regarding the influence of non-uniform critical current density profile on magnetic field behavior of AC susceptibil…
