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Journal of Applied Physics

Modeling a voltage-locked Josephson junction array amplifier: Gain, input impedance, and bandwidth

D. G. McDonald

Published Nov 1, 1986• 5 citations
Publisher

Abstract

Previously published experimental results are used to deduce an equivalent circuit for this amplifier, based on the shunted junction model. It is found that the very small inductances of the junction shunt resistors play an important role in determining the shape of the current-voltage curves in the voltage-locked region. Once the circuit is determined using the shunted junction model a combination of an approximation method (the method of slowly varying amplitudes) and the shunted junction model is used to approximately maximize the power gain of the circuit. The maximum gain achieved in these simulations of the two-junction amplifier is 11.3. This gain occurred with a negative resistance imput impedance of −0.41R and a signal source impedance of 0.167R, where R is the junction shunt resistance. It is estimated that the bandwidth of this amplifier is about 1 GHz.

DOI
10.1063/1.337746

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Journal of Applied Physics•Jun 1, 1980•3 citations
Previously published experimental results are used to deduce an equivalent circuit for this amplifier, based on the shunted junction model. It is found that the very small inductances of the junction shunt resistors play an important role in determining the shape of the current-voltage curves in the voltage-locked region. Once the circuit is determined using the shunted junction model a combination of an approximation method (the method of slowly varying amplitudes) and the shunted junction model is used to approximately maximize the power gain of the circuit. The maximum gain achieved in these simulations of the two-junction amplifier is 11.3. This gain occurred with a negative resistance imput impedance of −0.41R and a signal source impedance of 0.167R, where R is the junction shunt resistance. It is estimated that the bandwidth of this amplifier is about 1 GHz.
We have studied a new type of Josephson junction amplifier which is based on a two-junction array; the junctions are mutually phase locked at the Josephson self-oscillation frequency. With this frequency at 82 GHz, the voltages of the junctions remain equal (locked) for a bias current range as large as 60% of the critical current. Over a much smaller bias range, with an applied signal frequency of 1 kHz, a small-signal power gain of 19 dB was measured, accompanied by a negative resistance input impedance. This performance is consistent with a quasistatic theory of the amplifier.
The high-frequency amplification properties of transformer coupled, resistively shunted Josephson tunnel junctions have been investigated. The importance of the shunt loop inductance is stressed. It allows a high cutoff frequency, of significance for good high-frequency performance. The self-pumped parametric amplifier showed none of the excessive noise rise, which has hitherto plagued the development of externally pumped Josephson junction amplifiers. Around 10 GHz, we estimated a noise temperature less than 30 K for an amplifier pumped by a Josephson oscillation with a frequency well above twice the signal frequency. The corresponding gain of 5 dB may be increased in a better impedance matched circuit. The gain was very stable against variations in the bias conditions. A gain-bandwidth product as high as 0.3 was registered. The experimental results agreed well with the established theory for self-pumped parametric Josephson amplifiers. It should be possible to extend the low noise amplification by this device to mm wave frequencies. A relaxation oscillation occurred at a subharmonic of the Josephson frequency when the shunt loop inductance became large. The amplification in this mode followed closely the predictions of a simple model, where the signal modulated the switching of the sawtooth-like (relatively low frequency) relaxation current. Gains of about 15 dB were measured around 10 GHz, but the amplification was sensitive to bias conditions and noisy in this case where the relaxation frequency fell well below the signal frequency. Much improved properties were registered when the inductance was decreased so that the relaxation frequency approached the Josephson frequency and exceeded twice the signal frequency. The behavior then resembled that of a Josephson mode parametric amplifier, but the high content of harmonics of a relaxation oscillation meant that the amplifier became noisier due to converted noise from the many idler frequencies.
The use of unshunted junctions to close the superconducting loop of two shunted junctions in series is shown to improve the voltage locking range compared to the more common use of passive elements such as resistors or capacitors. Stacking pairs of junctions in a ladder-like form with one junction common for each two steps makes it possible to phase lock more than two junctions. Simulations show that four junctions with 5% matching can be locked from 0 to 55μV. The device can also be used as a voltage amplifier by driving one of the junctions. Simulations show that an array of four series junctions operates as an amplifier between 12μV to 33μV input. Input impendance was 6 ohms and output impedance was 12 ohms.
It is shown that inductively resistively shunted junctions are good candidates for Josephson-junction oscillator arrays. The shunt resonates out the junction intrinsic capacitance, allowing a much higher effective impedance at the resonance frequence, but still with a small shunt resistance giving a narrow linewidth. Compared to resistively shunted junctions they are advantageous for frequencies greater than about 100 GHz, when fewer junctions and lumped arrays can be used, giving larger bandwidths. Design rules that avoid chaotic solutions and optimize output power are given. A design example of an integrated circuit consisting of an SIS-mixer pumped by an array of as few as four junctions at 350 GHz is given. A bandwidth of 160 GHz and a linewidth of 400 kHz should be feasible in a practical situation.>
Frequency locking effects in two interacting Josephson junctions are investigated. The junctions are described in the usual resistively-shunted-junction (RSJ) model and coupling between junctions is provided by a resonant circuit. The system can be used to simulate interactions between superconducting weak links in a resonant cavity or via a general impedance shunt. Various types of v-i curves in the voltage locking region are obtained in this calculation by using a first harmonic approximation in the high-frequency regime. Stability of order parameter phase oscillations for two identical junctions in the voltage-locked region is examined. We also show that dissimilar junctions can be synchronized in a resonant cavity using a common series bias current.
Some types of Josephson junctions such as long microbridges, S-N-S junctions, and junctions with a series palastic inductance are known to have nonsinusoidal current-phase relations which can be approximated by linear-periodic functions. In this paper the equivalent circuit equation of the resistively shunted junction model with linear-periodic current-phase relation is solved. The heights of zeroth and first constant-voltage steps in the I-V characteristics and the impedance at these steps are obtained analytically. This analysis shows that the impedance at the first step lies on a semicircle in the impedance plane with its center at the impedance of the zeroth step. The calculated impedance is compared with the experimental results by Claridge et al. The theory presented here can explain some features of experimental results which differ considerably from predictions based on a sinusoidal current-phase relation. Among these are the value of bias current corresponding to the center of the first step, and the appearance of ’’shoulders’’ in the imaginary part of the impedance at the first step.