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IEEE Transactions on Instrumentation and Measurement

Calculation of the Shapiro step amplitudes in two-dimensional Josephson tunnel junction

U. Klein, S. Schroder, Johann Heyen Hinken

Published Apr 1, 1991• 4 citations
Publisher

Abstract

We have investigated theoretically the amplitudes of RF- induced constant voltage steps in two-dimensional Josephson tunnel junctions, using both a first-order approximation and a numerical it- eration procedure to calculate the current density distribution in the junction. We use a planar resonator model to analyze the electromag- netic field in the microwave-driven junction. Integrating the local Jo- sephson current density over the junction area yields the total dc current on a chosen Shapiro step. Numerical calculations of the Sha- piro step amplitudes in two-dimensional resonant junctions show a qualitatively different behavior from that in one-dimensional resonant junctions. Depending on the dominant resonant mode, the microwave power dependence of the step amplitude gives a smooth curve without zeros. This behavior is advantageous for biasing the Josephson junc- tion in its maximum step amplitude. The main difference between the results of the first-order approximation and the full theory is that- using the iteration procedure-the Shapiro steps in resonant junctions become asymmetric with respect to the quasi-particle curve. This is due to consideration of the losses and the internal Josephson currents in the numerical iteration procedure of the full theory.

DOI
10.1109/tim.1990.1032941

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We have investigated theoretically the amplitudes of RF- induced constant voltage steps in two-dimensional Josephson tunnel junctions, using both a first-order approximation and a numerical it- eration procedure to calculate the current density distribution in the junction. We use a planar resonator model to analyze the electromag- netic field in the microwave-driven junction. Integrating the local Jo- sephson current density over the junction area yields the total dc current on a chosen Shapiro step. Numerical calculations of the Sha- piro step amplitudes in two-dimensional resonant junctions show a qualitatively different behavior from that in one-dimensional resonant junctions. Depending on the dominant resonant mode, the microwave power dependence of the step amplitude gives a smooth curve without zeros. This behavior is advantageous for biasing the Josephson junc- tion in its maximum step amplitude. The main difference between the results of the first-order approximation and the full theory is that- using the iteration procedure-the Shapiro steps in resonant junctions become asymmetric with respect to the quasi-particle curve. This is due to consideration of the losses and the internal Josephson currents in the numerical iteration procedure of the full theory.
The amplitudes of the current steps in the I-V characteristics of mutually coupled two-dimensional distributed Josephson tunnel junctions driven by microwaves are investigated. For this purpose a numerical computation algorithm that is based on a planar resonator model for the individual Josephson tunnel junctions is used to calculate the DC current density distribution. In addition to the fundamental microwave frequency, harmonic contents of the tunneling current are also considered. The lateral dimensions of the individual junctions are small compared to the microwave wavelength and the Josephson penetration depth, giving an almost constant current density distribution. Therefore, the coupled junctions can give much greater step amplitudes than a single junction with an equal tunneling area, because of their nonuniform current density distribution. The calculation show a strong dependence of the Shapiro step amplitudes with respect to the coupling of the tunnel junctions. These studies imply that the choice of proper coupling conditions is important to achieve maximum step amplitudes. These results are helpful in designing tunnel junctions with optimized step amplitudes.
Experimental I-V curves of microwave-driven Josephson tunnel junctions with resistive shunts are reported. The results are in very good agreement with numerical calculations using the resistively shunted junction model. In the low-frequency regime there are three distinct regions in the I-V curves and two different types of Shapiro steps. It is demonstrated that the overall shape of the I-V curves can be explained by using an adiabatic interpretation of the junction response. The two different types of Shapiro steps are related to a compensation effect between dc and rf bias currents.
The microwave response of the dc quasiparticle tunneling current in Josephson tunnel junctions, where the Josephson current is suppressed by an external magnetic field, has been studied quantitatively in order to clarify its characteristics as a probe for the measurement of the junction capacitance. Extensive experiments for both small and long junctions are carried out for distinguishing between microwave behaviors of lumped and distributed constant junctions. It is shown that the observed voltage dependence of the dc quasiparticle tunneling current modified by an applied rf field is in good agreement with a theoretical result which takes into account the influence of the microwave circuit connected to the junction. The comparison between theory and experiment gives the magnitude of the internal rf field in the junction. Together with the applied rf field, this internal rf field leads to the junction rf impedance which is dominated by the junction capacitance in our experimental condition. In the case of lumped junctions, this experimental rf impedance is in reasonable agreement with the theoretical one with the junction capacitance estimated from the Fiske step of the distributed junction fabricated on the same substrate; the obtained ratio of the experimental impedance to the theoretical one is approximately 0.6–1.7. In the case of distributed junctions, however, experimental values of their characteristic impedances are approximately 0.2–0.3 of theoretical values calculated by assuming the one-dimensional junction model and taking account of the standing-wave effect in the junction.
Both resonant and microwave-induced steps in the V-I characteristics of four-Josephson-junction loop with a small junction-parameter spread have been investigated. An rf current was superimposed onto a dc current and the radio frequency was tuned to the resonance frequency of the loop. In the high-Q region, the rf-power dependence of the step amplitude was similar to that of a single junction. However, we obtained V-I characteristics which indicated that the behavior of the junction in the loop was coherent in the low-Q region.
Using Takanaka's theory of the zero-field dc-current steps of a Josephson tunnel junction, we have studied the effect of the junction QQQ factor on the shape of the current step near the first even-order resonant voltage $V=h\frac{{\ensuremath{\nu}}_{c}}{e}$, where ${\ensuremath{\nu}}_{c}$ is the frequency of the fundamental cavity mode. In addition, we have found that the effect of a magnetic field on the step height can be qualitatively explained by modifying Takanaka's theory to include a magnetic-field-dependent term.
High Tc Josephson junctions have IcRn products as large as a few mV at liquid helium temperature, and hence, the frequency normalized by IcRn∕Φ0 becomes on the order of 10−3 for a gigahertz (GHz) microwave signal. (Here, Ic, Rn, and Φ0 are the critical current, the normal resistance, and a flux quantum, respectively.) It is well known that a microwave-current-driven Josephson junction does not exhibit clear Shapiro steps under such low frequency conditions. Instead of Shapiro steps, however, we often observe a current-step-like structure in the current–voltage curve of YBa2Cu3O7−δ grain boundary Josephson junctions coupled to a GHz signal. In this article, we present a model analysis for such current-step-like structures. Two types of device models are employed: a simple resistively and capacitively shunted junction model and a rf-field-driven two-junction superconducting quantum interference device model. Numerical results of the latter model agree with the experimental results, which indicates that the envelope of small Shapiro steps, including the effects of microwave-induced motion of flux quanta, appears as a current-step-like structure in a high Tc Josephson device coupled to a GHz microwave.