Josephson Junction Arrays

This project was done at UIUC in the Bezryadin Condensed Matter Lab. This project can be split into two parts. (1) Modeling and (2) Disorder and Symmetry Analysis.

Modeling

I first developed a critical current model of a many Josephson Junction Array. In order to do this, I used the current-phase-relationship (CPR) of a traditional tunnel junction of

\[\begin{equation} I(\varphi) = \sin(\varphi) \end{equation}\]

Here, $\varphi$ is the phase difference in the superconducting wavefunction across the tunnel junction. Then, lining many tunnel junctions nanowires in a parallel fashion and applying a magnetic field $b$ would generate Meissner currents. Such currents arise because the cumulative phase difference traced out by any closed loop must be equal to $2\pi n$ where $n$ is an integer. I used this boundary condition to then derive the following Generalized Meissner Phase correlation:

\[\begin{equation} \oint \vec{\nabla}\phi \cdot \vec{dl} = 2\pi n \iff \phi_j - \phi_i - 2\pi b = 2\pi n \end{equation}\]

I modeled the critical current, or the maximum current that the device can support, using Equations 1 and 2. Then, I verified the model against analytically known solutions. The code can be found here.

Disorder and Symmetry Analysis

After developing the model, I led an extensive analysis into the effects of fabrication disorder. In this analysis, I ended up developing a python package which allows the user to inverse generate data points that correspond to a standard deviation and mean. I used this python package to study the effects of increase the standard deviation of the junction widths (increasing fabrication disorder) on the resulting critical current. I also used this python package to solve a 10 year node-lifting problem posed by Professor Alexey Bezryadin.

I also led a symmetry analysis of the device. In this analysis, I discovered several rotational symmetries as well as a current-magnetic symmetry that was always preserved. These results are currently being published.