Synaptic weighting in single flux quantum neuromorphic computing

Josephson junctions act as a natural spiking neuron-like device for neuromorphic computing. By leveraging the advances recently demonstrated in digital single flux quantum (SFQ) circuits and using recently demonstrated magnetic Josephson junction (MJJ) synaptic circuits, there is potential to make rapid progress in SFQ-based neuromorphic computing. Here we demonstrate the basic functionality of a synaptic circuit design that takes advantage of the adjustable critical current demonstrated in MJJs and implement a synaptic weighting element. The devices were fabricated with a restively shunted Nb/AlOx-Al/Nb process that did not include MJJs. Instead, the MJJ functionality was tested by making multiple circuits and varying the critical current, but not the external shunt resistance, of the oxide Josephson junction that represents the MJJ. Experimental measurements and simulations of the fabricated circuits are in good agreement.


Details of timing in the synaptic signal modulation circuit
The synaptic signal modulation is envisioned to operate at high speeds with SFQ pulses. As such, it is important to understand the timing conditions of the circuit as a function of the Ic value of the JJsyn. Such timing considerations can be seen in fig 1c of the main text for the JJsyn with Ic = 200 µA, where a peak occurs in the coupled current around 10 ps. This peak is in part a reflection of how the data were normalized. Figure S1a shows the coupled current as a function of time for a 100 ps input pulse. The current value for normalization was taken at the middle of the pulse time. Thus the 10 ps center pulse time current value was ~ 7 % higher than the 10 ns center pulse time current value, though they had the same peak value. Figure S1b shows the same trace for the simulation of the JJsyn Ic = 100 µA. Note that in this case there is no overshoot in the time trace and therefore no peak in the corresponding data in fig 1c of the main text.
When changing Ic of the JJsyn, we also change the screening current, characteristic time constant, and damping parameter. Below we test each of these changes to see which effect dominates the short time scale response in our circuit. Figure S1c shows the time trace with the βL of JJsyn Ic = 200 µA adjusted to match that of JJsyn Ic = 100 µA in the original simulations. Note, that the screening current enhances the short time scale peak, and therefore does not explain the effect.
The slowest characteristic time for the overdamped JJsyn is given by which is about 20 ps for the 200 µA JJsyn. However, if the characteristic time were the dominate effect, we would expect the 100 µA JJsyn to have an even larger effect at 10 ps, since its is 10 ps.
Finally, we are left with the Stewart-McCumber parameter 1,2 βc , which represents the damping in the junction with βc < 1 being in the overdamped regime, βc = 1 being critically damped and βc > 1 being underdamped, where where C is the junction capacitance. βc for the JJsyn values tested here was between 6.7×10 -5 and 2.6×10 -3 corresponding to highly overdamped junctions in all cases. In fig S1d we adjust the shunt resistor in the simulation to increase the damping of the JJsyn with Ic = 200 µA again to match that of the case where JJsyn has Ic = 100 µA. In this case, we see that the short time scale peak is eliminated implying that change in damping caused the short time scale overshoot in coupled current for the JJsyn with Ic = 200 µA.
It is worth noting that a large Ic value JJsyn is intended to couple the least amount of current into the output SQUID. As such, the slightly higher peak current could reduce the dynamic range of the signal modulation if the output SQUID circuit is fast enough to respond to the shot time scale current. In future design considerations, it is important to take this into account by designing the circuit to have consistent dynamics across the intended range of Ic values that a JJsyn would potentially access. For example, if the damping was increased, then all the Ic values could be damped out. If a faster circuit is desired, then the damping would need to be decreased enough for all the Ic values to have a fast response. While, either of these cases would lead to the largest possible dynamic range of coupled current, a slight reduction, e.g. 7 %, should still yield synaptic signal modulation that is quite acceptable.

Schmatic layout of the test circuits:
The circuits were designed for ease of testing and were not optimized for minimizing the space. As such the circuits are rather large, with the largest amount of space being taken up by the inductors in the circuit. A wire width of 2.5 µm was used for the inductors, and that could easily be reduced to sub-micron width. If one works with the narrower width inductor, it will have a higher inductance per micron, which will lead to a more compact layout. The smallest junction that we tested was roughly 0.7 µm in diameter. This was the 32 µA synaptic junction and we did not observe any issues with yeild at this device size, though we only tested three chips. If we were to redesign the circuit to minimize the area, our nominal 7 pH inductor would need to be be about 7.8 µm in total length assume that we designed the wire width to have 0.9 pH/µm. The junctions would remain the same size, which is around 0.7-1.5 µm in diameter depending on the desired Ic. Below we show both the schematic layout of the test circuits ( fig. S2), and an optical image of one of the actual circuits tested. Both the schematic layout and optical image are for the circuit that has a 100 µA synaptic junction. The SQUID neurons in all circuits are the same. Figure S2. (color online) Schmatic layout of the synaptic test circuit. The Josephson junctions are the three red circles. The two on the bottom compose the SQUID neuron and the one on the top is a 100 uA synaptic junction. The five larger yellow rectangles are resistors that isolate the circuit from the incoming wiring used to test the circuit. The three smaller yellow rectangles are shunt resistors to the ground plane. The center "U" shape is the inductive coupling from the synaptic JJ to the SQUID neuron. A ground plan is present under circuit, but is not shown for clarity. The scale bar on the lower left side represents 10 µm.