McCumber Parameter Research Articles

Time-Domain Simulator of Josephson Junctions Based on the BCS Theory

We realized a time-domain simulator based on the electrodynamics of Cooper pairs and quasiparticles in Josephson junctions. The tool, based on the charge carriers’ densities of states described by the Werthamer and Harris formalisms of Bardeen-Cooper-Schrieffer (BCS) theory, allows to analyze the behavior of current- or voltage-controlled Josephson junction-based circuits for any signal waveform, at a physical temperature comprised between the absolute temperature and the critical temperature. The simulator can account for <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">I-V</i> curve hysteresis depending on the McCumber parameter, or Shapiro and photo-assisted steps in the presence of a monochromatic microwave signal. We used the simulator to assess the behavior of MgB <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> -based Josephson junctions at THz frequencies by taking into account the presence of the two anisotropic gaps of MgB2.

Characterization of Shapiro steps in the presence of a 4π-periodic Josephson current

The Majorana zero-energy modes (MZMs) residing at the boundary of topological superconductors have attracted a great deal of interest recently, as they provide a platform to explore fundamental physics such as non-Abelian statistics, as well as fault-tolerant quantum computation. Period doubling of Shapiro steps in a Josephson junction under microwave irradiation has been regarded as strong evidence for the emergence of the MZMs at the junction edges. However, questions remain as to how the Shapiro steps respond to the presence of a 4{\pi}-periodic Josephson current. In this study, we investigated the characteristic features of Shapiro steps with respect to the ratio ({\alpha}) of the 4{\pi}-periodic current to the topologically trivial 2{\pi}-periodic one, as well as the reduced microwave frequency ({\Omega}) and McCumber parameter ({\beta}) of the junction. Our analysis reproduced Shapiro steps similar to those observed experimentally for specific parameter sets of {\alpha},{\Omega} ({\lesssim 0.1}), and {\beta} ({gtrsim 1.0}). Full suppression of the first lobe of the n=1 step guarantees the presence of a 4{\pi}-periodic Josephson current.In addition, we discuss the range of {\Omega} and {\beta} needed for full suppression of the first lobe of the {n=1} step, even for small {\alpha} ({<0.1}). To observe "period-doubled Shapiro steps", even with a small {\alpha}, the junction should have a large {I_c}{R_N} product and sufficiently large junction capacitance.

CURRENT-VOLTAGE CHARACTERISTIC OF JOSEPHSON JUNCTION WITH FRACTIONAL TERM IN CURRENT-PHASE RELATION

In this study, we carried out the analysis of the influence of unconventional current-phase relation of Josephson junction with additional fractional term sin\phi/2 to first harmonic sin\phion current-voltage characteristics. The current-voltage characteristic of the junction was calculated for the different amplitudes of fractional term m and McCumber parameter . It was shown increasing the amplitude of fractional terms leads to increasing in the hysretesis on the current-voltage characteristics.

Possible dynamics of Josephson junction arrays connected to high-Q tank circuit

Serial Josephson junction arrays connected to high-Q tank circuit are analysed and discussed. Numerical simulation of the systems shows two possible oscillation modes above resonance frequency when the tank impedance is capacitive. These are the inphase oscillation mode and the collectively antiphase oscillation mode. This fact is responsible for complicate switching dynamics, which can block parametric resonance excitation. Stability domains of the modes depend on McCumber parameter of the junctions. Increase in this parameter is favourable to the inphase mode. When the inphased oscillations are feasible, the resonance peak-like peculiarities on IV curve and Shapiro steps can be described with the analytic theory derived earlier for one-junction case and extended over the array systems.

The Influence of the McCumber Parameter on Parametric Amplification of High-Frequency Radiation by Josephson Junctions

The influence of the internal capacity of a Josephson junction on the parametric amplification of external electromagnetic radiation was studied in terms of a resistively and capacitively shunted junction model. The influence of the McCumber parameter on parametric amplification was clarified. It is shown that the additional regions of amplification can occur near subharmonic Shapiro steps in the case of Josephson junctions with internal capacity.

Study on Noise Matching Between SQUID Sensor and Its Readout Electronics

Noise matching between superconducting quantum interference device (SQUID) sensor and readout electronics has always been an important problem. Conventionally, in order to avoid the hysteresis, SQUIDs should be operated in the range of $\beta _{{{c}}}\, , where $\beta _{{{c}}}$ is the Stewart–McCumber parameter. Recently, Liu et al. extended the SQUID operating area to $\beta _{{{c}}}\,> \,1$ due to a large noise parameter Γ*. When $\beta _{{{c}}}\,> \,1$ , SQUID exhibits a large ∂ V /∂Φ, which is beneficial in reducing the noise contribution of the readout electronics, $\delta \Phi _{{{e}}}$ . In this work, we compare the SQUID system noise $\delta \Phi _{{{s}}}$ measured with four different readout electronics. Here, the four readout electronics employed are as follows: 1) the flux modulation scheme (FMS); 2) the direct readout scheme (DRS), which uses six parallel connected bipolar transistor as its preamplifier (DRS-PCBT); 3) the DRS, which uses a single preamplifier AD797 (DRS-AD797); and 4) the two-stage scheme. The preamplifier voltage noise of DRS-PCBT and DRS-AD797 are about 0.4 nV/√Hz and 0.9 nV/√Hz, respectively. For a two-stage scheme, the equivalent flux noise reaches about 0.25 μΦ0/√Hz, the measured SQUID intrinsic noise, δΦ $ _{{{i}}}$ , will dominate $\delta \Phi _{{{s}}}$ . In other readout schemes, however, $\delta \Phi _{{{s}}}$ depends on the value of $\beta _{{{c}}}$ . Finally, we found that FMS is suitable to match SQUIDs with $\beta _{{{c}}}\, . DRS-PCBT is a good choice for SQUIDs with $1\, whereas DRS-AD797 suffices for SQUIDs with $\beta _{{{c}}}\,> \,3$ .

Dependence of SQUID Intrinsic Flux Noise on Stewart–McCumber Parameter &lt;inline-formula&gt; &lt;tex-math notation="LaTeX"&gt;$\beta_{c}$&lt;/tex-math&gt; &lt;/inline-formula&gt; of Josephson Junction

It is known that three superconducting quantum interference device (SQUID) parameters, namely, the intrinsic flux noise δΦ s , the flux-to-voltage transfer coefficient ∂V/∂Φ, and the dynamic resistance R d , increase with increasing the Stewart- McCumber parameter β c of Josephson junction. It is very difficult to obtain the low values of δΦ s for a strongly damped SQUID due to its small ∂V/∂Φ. In this paper, we employ a two-stage scheme (TSS) to measure δΦ s directly. Here, we use single chip readout electronics and a weakly damped SQUID with a β c of flux noise of the readout-SQUID system is about 4.8 μΦ 0 /√Hz 1.2, both acting as a readout-SQUID system (a preamplifier). The with a corner frequency of about 10 Hz. The front-end SQUID is coupled to the readout SQUID via the input coil with a mutual inductance of M i . In our experiment, the flux gain for a readout SQUID, i.e., G Φ = M i × (∂I/∂Φ) front-end is adjusted to be 20. Using the TSS, we measured δΦ s with different β c changing from 0.3 to 13.5 and found δΦ s increasing from 1.0 to 5.0 μΦ 0 /√Hz. We also discussed the relations between L s , β c , and δΦ s quantitatively. Finally, we proposed designing a suitable β c to match the preamplifier in a direct readout scheme to simplify a SQUID measurement system.

Characterization of an On-Chip Magnetic Shielding Technique for Improving SFQ Circuit Performance

An on-chip magnetic shielding technique was characterized on several DC-SQUIDs to prepare the next generation of complex digital RSFQ circuits designed by the FLUXONICS foundry. The maximal frequency of operation of a toggle flip-flop cell was used as a criteria to validate the concept. It goes from a simulated value of 52 GHz (46 $\pm$ 2 GHz experimentally), for a McCumber parameter $\beta_{c}=1$ with $J_{C}=1 \text{kA}/\text{cm}^{2}$ in the absence of external magnetic field, down to 29 GHz in the presence of an external field of 88 $\mu\text{T}$ . The shielding technique consists of surrounding a circuit by one or several superconducting niobium loops with the same layers as for the SFQ circuit. The loops are connected to the ground plane with vias. It is shown experimentally that the local magnetic field is reduced by a factor of 4 for the best shielded configuration, which limits the reduction of the maximal frequency of operation in presence of external magnetic field.

Low-Tc direct current superconducting quantum interference device magnetometer-based 36-channel magnetocardiography system in a magnetically shielded room**Project supported by “One Hundred Persons Project” of the Chinese Academy of Sciences and the Strategic Priority Research Program (B) of the Chinese Academy of Sciences (Grant No. XDB04020200).

We constructed a 36-channel magnetocardiography (MCG) system based on low-Tc direct current (DC) superconducting quantum interference device (SQUID) magnetometers operated inside a magnetically shielded room (MSR). Weakly damped SQUID magnetometers with large Steward–McCumber parameter βc (βc ≈ 5), which could directly connect to the operational amplifier without any additional feedback circuit, were used to simplify the readout electronics. With a flux-to-voltage transfer coefficient ∂ V/∂ Φ larger than 420 μV/Φ 0, the SQUID magnetometers had a white noise level of about 5.5 fT·Hz−1/2 when operated in MSR. 36 sensing magnetometers and 15 reference magnetometers were employed to realize software gradiometer configurations. The coverage area of the 36 sensing magnetometers is 210×210 mm2. MCG measurements with a high signal-to-noise ratio of 40 dB were done successfully using the developed system.

Superconducting quantum interference devices with different damped junctions operated in directly coupled current- and voltage-bias modes

We investigate niobium thin film superconducting quantum interference devices (SQUIDs) with different Steward—McCumber parameters βc operated in both current- and voltage-bias modes. We experimentally prove that there is no difference between the two bias modes with respect to the SQUID intrinsic noise and the noise contribution from the preamplifier. Furthermore, the relationships of the SQUID dynamic parameters, (Rd)current bias ≈ (Rd)voltage bias and (∂V/∂Φ)current bias ≈ [(∂i/∂Φ)Rd]voltage bias, are always satisfied. For a strongly damped SQUID with βc ≈ 0.25, additional positive feedback (APF) and noise cancellation (NC) were employed to enhance ∂V/∂Φ, the former showing a degradation in the linear flux range but otherwise the same with NC. For a weakly damped SQUID with βc ≈ 3, it is directly connected to the preamplifier without APF or NC, and a low SQUID system noise of about 4 μΦ0/√Hz is measured, which is close to its intrinsic noise.

Statistical characterization of voltage-biased SQUIDs with weakly damped junctions

Recently, it has been shown that voltage-biased readout of SQUIDs with weakly damped junctions (large Stewart–McCumber parameter βc, due to high shunt resistance) is useful for suppression of preamplifier noise. We experimentally studied the characteristics of 53 planar niobium–SQUID magnetometers with junction shunt resistors RJ nominally of 30 Ω fabricated on 5 × 5 mm2 chips. The field-to-flux transfer coefficient ∂B/∂Φ of the magnetometers was 1.5 nT/Φ0, with a SQUID loop inductance Ls of about 350 pH. The distributions of important SQUID parameters, such as the current swing Iswing, the dynamic resistance Rd, and the flux-to-voltage transfer coefficient ∂V/∂Φ, are given. Nearly all the SQUIDs could be stably operated in the voltage bias mode and their ∂V/∂Φ reached a large mean value of 380 μV/Φ0. In this case, the SQUIDs can be read out directly by a commercial operational amplifier without any additional means to suppress preamplifier noise. The mean flux noise of the SQUIDs was found to be 4.5 μΦ0 Hz−1/2, corresponding to a field resolution of 7 fT Hz−1/2. To demonstrate the applicability of these SQUIDs in the direct readout scheme, a simple four-channel SQUID gradiometer system was set up to perform magnetocardiography and magnetoencephalography measurements in a magnetically shielded room.

An insight into voltage-biased superconducting quantum interference devices

We experimentally studied two important parameters of helium-cooled superconducting quantum interference devices (SQUIDs) in the voltage bias mode: the dynamic resistance Rd and the flux-to-current transfer coefficient ∂i/∂Φ, with different junction shunt resistors RJ. We investigated a voltage-biased SQUID using the direct readout current-to-voltage converter scheme involving an operational amplifier. At higher RJ, the flux-to-voltage conversion coefficient ∂V/∂Φ becomes sufficiently large to effectively suppress the room-temperature amplifier's noise without any need for additional feedback circuits. The McCumber parameter limits the rise of ∂V/∂Φ. We discuss the performance of voltage-biased SQUIDs at different effective McCumber parameters.

Planar SQUID magnetometer integrated with bootstrap circuitry under different bias modes

A planar superconducting quantum interference device (SQUID) magnetometer consisting of a parallel gradiometer SQUID with integrated input coils connected to an on-chip pickup loop was designed and fabricated in conventional niobium technology. SQUID bootstrap circuitry (SBC) incorporating suitable current and voltage feedbacks was also integrated into the design. For a SQUID inductance of Ls = 350 pH and a chip size of 5 × 5 mm2, the field resolution of the voltage-biased SQUID magnetometer reached <5 fT Hz−1/2 with the bootstrap circuit and an ordinary preamplifier. We also observed that the effective McCumber parameter βc of the junctions is influenced by the bias mode. Indeed, when the nominal junction βc was larger than unity, our SQUID magnetometer operated stably in the voltage bias mode. The device exhibited low noise even without SBC.

Charging of superconducting layers and resonance-related hysteresis in the current-voltage characteristics of coupled Josephson junctions

A manifestation of a resonance-type hysteresis related to the parametric resonance in the system of coupled Josephson junctions is demonstrated. In contrast with the McCumber and Steward hysteresis, we find that the width of this hysteresis is inversely proportional to the McCumber parameter and it also depends on the coupling between junctions and the boundary conditions. Investigation of the time dependence of the electric charge in superconducting layers allows us to explain the origin of this hysteresis by different charge dynamics for increasing and decreasing bias current processes. The effect of the wavelength of the longitudinal plasma wave created at the resonance on the charging of superconducting layers is demonstrated. We find a strong effect of the dissipation in the system on the amplitude of the charge oscillations at the resonance.

Electrical parameters of niobium-based overdamped superconductor-normal metal-insulator-superconductor Josephson junctions for digital applications

We have demonstrated superconductor-normal metal-insulator-superconductor Nb/Al–AlOX/Nb Josephson junctions developed with Al layer thicknesses between 30 and 100 nm, resulting in nonhysteretic current-voltage characteristics for critical current densities between 20 and 60 kA/cm2. Specific capacitances ranged from 180 to 480 fF/μm2, depending on temperature and of barrier deposition conditions. It is shown that the high capacitance seen by the junction depends on the Josephson junction barrier but only to a least extent on the parasitic environment. Although the specific capacitance is unusually high, Stewart–McCumber parameters from 1 to 3, at 4.2 K, optimum for digital single-flux-quantum applications were obtained without external shunting.

Josephson effects in weakly coupled MgB2 intergrain nanobridges prepared by focused ion beam

We have fabricated weakly coupled intergrain nanobridges from MgB2 films by a focused ion beam (FIB) patterning technique and studied their transition properties. The bridges were nominally 300 nm wide and 100 nm long, and crossed a single grain boundary perpendicularly. The temperature-dependent resistance data showed a two-step transition after FIB pattern with more-than-two-decade increase in the resistivity. Current-voltage curves showed the characteristics of an ideal Josephson junction. The measured data were perfectly matched with the theory of the resistively shunted junction model with thermal fluctuations at all measured temperatures. At 4.2 K, the measured data showed the effect of hysteresis in agreement with the estimated McCumber parameter βc&amp;gt;0.3. The hysteresis effect disappeared above 6 K. The critical currents obtained from fitting to the RSJ model were linearly dependent on temperature, implying that the grain boundary played an insulating barrier.

Design of single flux quantum cells for a 10-Nb-layer process

We present design of single flux quantum (SFQ) cells for a 10-Nb-layer process which has been developed to fabricate SFQ VLSI circuits. The device fabricated by the process has a structure of an active layer on the top, two passive transmission line (PTL) layers in the middle, and a DC power (DCP) layer at the bottom. We have determined a unit cell size of 30 μm × 30 μm and a unit cell structure by taking accounts of the design rules and the experimental data on the PTLs. This cell size enables us to draw two PTLs of each PTL layer. PTL driver and receiver cells have a unit cell size, whereas a half unit cell with a size of 15 μm × 15 μm is used for PTL segments and vias. On the active layer, circuit parameters in analog simulation level are based on those of CONNECT cells, except for junction parameters with McCumber parameter β c of 2.0. Major cells including logic cells and PTL driver/receiver cells have been developed. We have designed a 2 × 2 switch on-chip test circuit using the cells and successfully tested them at high speed.

Experimental manifestation of the breakpoint region in the current-voltage characteristics of intrinsic Josephson junctions

The experimental evidence of the breakpoint on the current-voltage characteristics (IVCs) of the stacks of intrinsic Josephson junctions (IJJs) is presented. The influence of the capacitive coupling on the IVCs of Bi2Sr2CaCu2Oy IJJs has been investigated. At 4.2K, clear breakpoint region is observed on the branches in the IVCs. It is found that due to the coupling between junctions, the hysteresis observed on the IVC is small compared to that expected from the McCumber parameter. Measurements agree well with the results predicted by the capacitively coupled Josephson junction model including the diffusion current.

Current–voltage characteristics of intrinsic Josephson junctions with charge-imbalance effect

The current–voltage characteristics (IVC) of intrinsic Josephson junctions are numerically calculated taking into account the quasiparticle charge-imbalance effect. We solve numerically the full set of the equations including second order differential equations for phase differences, kinetic equations and generalized Josephson relations for a stack of Josephson junctions. The boundary conditions due to the proximity effect are used. We obtain the branch structure of IVC and investigate it as a function of disequilibrium parameter at different values of coupling constant and McCumber parameter. An increase in the disequilibrium parameter essentially changes the character of IVC at large values of McCumber parameter.

High quality ferromagnetic 0 and π Josephson tunnel junctions

The authors fabricated high quality Nb∕Al2O3∕Ni0.6Cu0.4∕Nb superconductor-insulatorferromagnet-superconductor Josephson tunnel junctions. Depending on the thickness of the ferromagnetic Ni0.6Cu0.4 layer and on the ambient temperature, the junctions were in the 0 or π ground state. All junctions have homogeneous interfaces showing almost perfect Fraunhofer patterns. The Al2O3 tunnel barrier allows one to achieve rather low damping, which is desired for many experiments especially in the quantum domain. The McCumber parameter βc increases exponentially with decreasing temperature and reaches βc≈700 at T=2.11K. The critical current density in the π state was up to 5A∕cm2 at T=2.11K, resulting in a Josephson penetration depth λJ as low as 160μm. Experimentally determined junction parameters are well described by theory taking into account spin-flip scattering in the Ni0.6Cu0.4 layer and different transparencies of the interfaces.