Analysis of Josephson Junction Barrier Variation: A Combined Electron Microscopy, Breakdown and Monte-Carlo Approach

TL;DR

The problem addressed is the variability of the Al/AlOx/Al barrier in Josephson junctions and its impact on device metrics such as the current–phase relation. The authors combine electrical IV measurements, Monte-Carlo simulations of multi-valued barrier thickness, breakdown statistics, and STEM-EDS analysis to infer barrier thickness distributions and their influence on conduction. They find that a skewed, log-normal thickness distribution best reproduces both IV characteristics and breakdown voltages, with the tails driving conductance and revealing sub-ensembles; STEM measurements provide a robust average thickness around 1.9–2.4 nm but are not definitive for the distribution due to projection and edge-detection uncertainties. Overall, the multi-modal approach offers actionable feedback for barrier optimization in JJ fabrication, while highlighting limitations in reconstructing full thickness distributions from STEM alone and pointing to the need for direct barrier-height measurements and more realistic barrier models.

Abstract

Josephson junctions manufactured to tight tolerances are necessary components for superconducting quantum computing. Developing precise manufacturing techniques for Josephson junctions requires an understanding of their make-up and robust feedback metrics against which to optimise. Here we consider complementary techniques assessing what conclusions they allow us to draw about the barriers in junctions. Monte-Carlo simulations of barriers show that standard deviations of 15-20% of the total barrier thickness are compatible with our experimental data. Electrical breakdown allows us to probe the weakest points in barriers. Narrowing the distribution of this breakdown provides a promising feedback mechanism for barrier optimisation. Grouping junctions by breakdown voltage allows us to identify sub-ensembles of junctions with different median resistance. Transmission electron microscopy can be used to find average barrier thickness, although we highlight challenges forming robust conclusions on the distribution of thicknesses in a barrier from these experiments.

Figures (5)

• Figure 1: (a) An AFM measurement of a JJ from the ensemble of JJs (b) An IV of a typical junction showing the extraction of the breakdown voltage, resistance from a linear fit to the low voltage regime and a fit to the full Simmons model. (c,d) Results from analysing an ensemble of 598 junctions fabricated on a 3" wafer. (c) A histogram showing the outputs from fitting Eq. \ref{['eq:iv']} to measured IVs with thickness, barrier height and nominal area shown on the same x axis. (d) A histogram of the resistance deviation from the median resistance of the junction ensemble with a median resistance value of 7122 $\Omega$.
• Figure 2: Results from Monte-Carlo simulations of barriers with normally (a - e) and log-normally (f - j) distributed thicknesses. Heatmaps of (a, f) resistance, (b, g) resistance spread (c, h) barrier height ($\phi$), (d, i) fitted thickness. Resistance and its spread are found by fitting a straight line to the low voltage region of the simulated IV and other parameters are from found from fitting Eq. \ref{['eq:iv']} to the full voltage range of the simulated IV. Any regions within the experimental variation (or 10% instead for resistance) are shown using a purple to orange heatmap, whereas values outside this range are shown in gray-scale (e,j) For each pixel we sum the number of experimental criteria matched by the simulation. Dark hues indicate regions where the experiment and simulation coincide, indicating that there is a parameter regime for each distribution where this matching is achieved.
• Figure 3: Comparison of (a) experimentally measured breakdown voltages with simulated breakdown voltages for (b) log-normally and (c) normally distributed barriers created with distributions in regions where all experimental criteria are matched in Fig. \ref{['fig:mciv']}. (a) Shows a double-Gaussian fit indicating a bimodal breakdown distribution. Simulated distributions are found by taking the thinnest points in randomly generated barriers with a mesh size of 0.2 nm and rescaling this by a dielectric strength ($E_{\rm ds}$) to keep the mean breakdown voltage equal to the experimental values. (d) The resistance of junctions distinguishing junctions with breakdown above/below the midpoint of the bimodal distribution 1.3 V, shown by the black dashed line in (a). (e) A histogram of the full thickness distribution of example junctions generated with the parameters from (b, c). (f) Cumulative conductance of junctions from 10 different junctions generated with parameters from (e) as a fraction of the total junction area.
• Figure 4: Experimental STEM-EDS imaging of an Al/AlO$_{\rm x}$/Al barrier of the oxygen peak. (a-c) show heatmaps indicating the oxygen peak with the edges of the barrier as detected by Kernel integration overlaid to the maps. Different values of $\delta$ are used to define the kernel for the three maps (0, 0.2, 0.4 respectively). (d-f) show histograms of thicknesses as inferred from the different kernel integrations.
• Figure 5: (a, b) two high resolution AFM scans of representative bottom leads (region indicated in Fig. \ref{['fig:dc_meas']} a). The route mean squared roughness for the areas shown are 0.7 and 0.5 nm respectively. In (a, b) five 30 $\times$ 100 nm areas are marked by dashed lines. These regions are used to simulate STEM profiles shown in (c-g) assuming a 2 nm oxide barrier coating the AFM topography with the labels matching the AFM region to the STEM profile inserted into the boxes. (c-g) Edges of simulated oxygen content are detected using a symmetric kernels and shown overlaid, as well as measurements of average thickness and thickness standard deviations being noted in each figure. The labels (c - g) are in the same side of the scan region in the simulated STEM profile and the AFM regions.