To validate how much fully simulating an array could improve pixel yield over interpolation, we studied an example group of 57 resonators between 6501.2MHz and 6388.9MHz with the length of their last capacitor fingers between 2µm and 286µm respectively. For one data set we simulated both edge resonators fully and linearly extrapolated the capacitor finger length for the 55 resonators in between for a desired 2 MHz spacing in frequency (Fig. 4, green points). According to this linear interpolation, the step size between capacitor lengths should be approximately 5µm. The same group of 57 resonators was then constructed with AEM to automate finding the optimal capacitor lengths for each exact resonant frequency (Fig. 4, red points). We compare both methods by the distance in frequency space from the intended designed value (Fig. 4, Left) and if pixels would fall into the same FFT bin before fabrication (Fig. 4, Right). Resonators are classed as “clashing” if they have less than 1.5 MHz distance to their nearest neighbour as they then risk to fall within the same 1 MHz FFT window. For this definition we assume an additional 0.5 MHz movement of the resonances caused by fabrication inaccuracies and increased by the typical 0.2–0.3 MHz resonance width.
It can be seen in Fig. 4 that the interpolated values lead to systematic deviations of above 6 MHz from the intended design, while the automated geometries only deviated by less than 0.3 MHz. Deviations in the automated values, specifically around the 6400 MHz and 6470 MHz range are caused by the used cell size of 1µm chosen in Sonnet. AEM is programmed to design a resonant structure as close to the desired resonant frequency as possible and thus the limiting factor in resonant frequency accuracy is purely down to the simulation’s cell size and thus the available fabrication capabilities.
The important deviation in the interpolated and automated geometries can be seen much better when looking at clashing resonators that risk to fall within the same 1.0 MHz FFT bin as these resonators would be lost in readout and hence would reduce the overall pixel yield The pixel yield for the interpolated and automated groups in our example in Fig. 4 are 86% and 98% respectively, showing a clear improvement with automated geometries by AEM. It should be noted that the one resonator lost with AEM sits exactly at 1.5 MHz to its nearest neighbour and may still be usable. Even though a single clashing resonator always makes two pixels unusable we calculated the above yield with only one lost resonator per clash as our definition of clashing at 1.5 MHz distance in frequency is rather strict.
For a better estimate on the required simulation times we performed AEM simulations for a small scale prototype with a 100 pixels array. 100 resonators would be time-consuming to construct by hand, and interpolation would as discussed reduce the expected pixel yield of the mask. To test AEM’s array building performance, we went for 100 MKIDs with 5pH/sq for the superconducting film, designed to be equidistant (approx. 40 MHz) within the 4–8 GHz octave. The initial parameters given for this run are shown in Table 1.
Table 1
Initial starting dimensions for the simulation test; details see text.
Capacitor Leg Thickness | Capacitor Leg Spacing | Initial Coupling Bar Thickness | Number of MKIDs & Resonances | Qc Range |
2 µm | 2 µm | 4 µm | 100 MKIDs, 4 GHz – 8 GHz | 20,000–30,000 |
The automation had finished the 100 MKIDs (some examples shown in Fig. 5) with a runtime of 10 hours and 45 minutes and a total of 1457 simulations performed. All resonators lay between Qc-20,000 and 30,000 (see Fig. 3) with a mean resonant frequency accuracy of ± 0.188 MHz. This run was performed on a not especially strong PC using a 12 thread CPU and 16 GB of RAM. To date, Sonnet allows up to 64 threads[7] for calculations on a single machine and thus this simulation trial can be expeditated to be much faster on a more specialized computer.
The amount of simulations performed is the result of two main causes:
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More sophisticated optimization method then binary-search could allow to further decrease the number of simulations performed.
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Crashes & non-physical data produced within Sonnet occur with a rough rate of one in about 20 simulations. Further updates of the Sonnet lab toolbox by the manufacturer could offer further improvements.
Using the results above, 2,000 MKIDs simulated lying within the 4–8 GHz octave can be estimated to have a runtime of 215 hours or roughly 9 days with similar accuracies for f0 and Qc. Assuming linear dependency on utilized CPU cores this could likely be reduced to about 40 hours on a modern CPU. Further improvements by providing more RAM are expected to be less significant.