Probabilistic Analysis of Tunability of Step-Edge Josephson Junction Arrays’ Inductance in HTS Microwave Metamaterials

Abstract

Josephson junction in superconductor circuits and metamaterials is modeled as a tunable inductance. The Josephson inductance is tunable by the bias current passing through it and can be tuned from an initial value to a very large value as much as one can push bias current near the critical current but not exceeding it. Tunability by bias current allows design of programmable metamaterials with flexible functions in microwave regime. In High-Temperature Superconductivity (HTS) tunable metamaterials, Josephson junctions should be used in series configuration to give usable tunable inductance. Each Step-Edge Josephson junction (SEJs) in HTS has a critical current that is dependent on its geometrical dimensions such as the width of the junction. In a practical fabrication process, the widths and critical currents will be distributed with a Gaussian Probability Distribution Function (PDF). This will decrease the tunability of the Josephson inductance of the array. We have calculated the tunability of an array with defined critical current distribution, in terms of probabilistic analysis.

Appendix

In reality, the probability distribution function must vanish for negative values of i_cns. But the nature of Gaussian distribution does not allow this. Nonetheless, modeling of positive random variables like mass and size using Gaussian distribution is quite common in the literature. This is because the probability significantly drops for two or three σ_i around μ_i and one should not worry much about the incorrectness of the model. But since we deal with the product of many probabilities in Eq. 4, there is a concern about the validity of i_cmin in Fig. 2 for large values of σ_i. Thus, we have tried to introduce a criterion to evaluate the error in our model. From Eq. 4, the probability of all currents being positive is evaluated from:

$$ P\left({i}_{c1},{i}_{c2},\dots, {i}_{cN}>0\right)=\frac{1}{2^N}{\left(1-\mathit{\operatorname{erf}}\left(\frac{-{\mu}_i}{\sqrt{2}{\sigma}_{i\max }}\right)\right)}^N=1-{P}_e $$

In a physically correct model, this probability should be exactly equal to unity. But it is smaller than unity, and the deviation from unity P_e is a measure for evaluating the error caused by the Gaussian approximation. For a given P_e, one can calculate σ_imax (maximum tolerated standard deviation) below which the resulting error is smaller than P_e:

$$ {\sigma}_{i\max }=-\frac{\mu_i}{\sqrt{2}{\operatorname{erf}}^{-1}\left(1-2\sqrt[N]{1-{P}_e}\right)}={\alpha}_i{\mu}_i $$

in which ${\alpha}_i=\frac{\sigma_{i\max }}{\mu_i}$ is the ratio of the maximum tolerated standard deviation to the mean value. Figure 7 shows α_i versus P_e. One can see that for α_i < 0.3 (as in Fig. 2), we will expect P_e < 0.05 which is a good approximation. Since P_e has a small value, we neglect its effect on the results obtained in sections 4 and 5.

Fig. 7

Error probability versus the ratio of the maximum tolerated standard deviation to the mean value