Resonator Heat Map Measurements

Explanation of the heat map measurements shown in "A Resonant Metamaterial Clock Distribution Network for Superconducting Logic". This post explains how the measurements were made, and their role in illuminating how the resonators couple to the rest of the circuitry.
Resonator Heat Map Measurements
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In our paper “A Resonant Metamaterial Clock Distribution Network for Superconducting Logic”, figure 2c is a heat map showing the magnitude of transmission through a resonant clock network as a function of RF clock power and DC bias current applied to an RQL circuit.  When the clock power is high enough to cause the Josephson junctions to trigger, they draw power from the resonator.  This causes a shift in the transmitted amplitude.  The measurement is a valuable tool that may provide insight into the health of our circuits before any logic tests are performed.

We used a computer controlled four-port network analyzer and a current source (implemented by a voltage source and a 1Kohm resistor in series) to make the measurement.  The analyzer is set up to sweep over a small set of frequencies centered on the resonance frequency. There are two traces, one for |S21|  (the I resonator) and one for |S43| (the Q resonator).  A marker is placed on each trace to follow the minimum of the trace on each network analyzer frequency sweep. 

The control software employs a double loop.  First the RF power is incremented one step, then the bias current is stepped from minimum current to maximum.  The network analyzer frequency is then swept for each current step.

At the end of each sweep, the software reads the data back from the markers (for the minimum of each of the two S parameters, the marker provides frequency of the minimum, amplitude of |S21|   (or of |S43| ) at the minimum, and the angle of the measured S parameter at the minimum.  In principle, it is possible to create six heat maps from the measurement; the magnitude heat maps (one for I and one for Q), frequency (one for I and one for Q), and angle (one for I and one for Q).  But in practice, the only significant heat maps are the amplitude heat maps, because frequency changes are often (but not always) imperceptible.  Usually, the heat maps for I and Q resonator amplitude are nearly identical.  If they are not, there is a (often unintentional) difference in the design of the two resonators. 

Figure 1 A typical heat map for the DynaZOR resonator
Figure 1.  A typical heat map for the DynaZor resonator

For a densely populated DynaZOR the amplitude at resonance may shift by as much as a few dB to as little few tenths of a dB, depending on design parameters.  If the resonator is sparsely populated with junctions, the shift may be much smaller, and may be undetectable.  A typical heat map shows the cumulative effect of many thousands of junctions drawing power from a DynaZOR clock network.

Figure 2.  This DynaZOR is sparsely populated and the periodic signature of switching junctions is barely perceptible.  Note the span of the insertion loss change on the color bar to the right.
Figure 2.  This DynaZOR is sparsely populated and the periodic signature of switching junctions is barely perceptible.  Note the span of the insertion loss change on the color bar to the right.

The heat map measurement can be a useful diagnostic tool and can be used to identify chips with manufacturing defects, such as wires with low critical current.  Figure 3 shows a tulip-shaped heat map that resulted from a bias line with low critical current.

Figure 3 This chip had a bias line with an abnormally low critical current.  As the current (x-axis) exceeded +/- 4mA, the resonance shifted due to heating, which resulted in this tulip-shaped heat map.

Figure 3 This chip had a bias line with an abnormally low critical current.  As the current (x-axis) exceeded +/- 4mA, the resonance shifted due to heating, which resulted in this tulip-shaped heat map.

Figure 4.  Frequency Heat Map for chip shown in Figure 3.
Figure 4 This is the frequency heat map for the chip shown in Figure 3.  Note how the frequency gets lower as the bias current (and power) increase.  This is typical for a chip with a bias line having abnormally low critical current.

The heat map measurement is also interesting because it allows us to explore circuit dynamics in regimes that are not accessed in typical logic operation.  For example, in Figure 5, we see interesting structure at higher powers.  We observe diamond-shaped regimes where the junctions switch once per cycle, and higher diamonds where the junctions switch twice per cycle.  They are also useful for detecting low critical current in parts of the bias circuitry (see Figures 3 and 4).

 

Figure 5.  Interesting junction dynamics at high power
Figure 5 This heat map shows interesting junction dynamics at higher powers; lower diamond regimes where the junctions switch once per cycle and higher diamonds where the junctions switch twice per cycle.

We used the “jet” color mapping originally as it is the default for Matplotlib, the Python package used to create these plots.  Because of the way color is perceived, Jet emphasizes some small differences in the values being plotted.  I tried using perceptually uniform colormaps, in particular Viridis.  The structures that we were looking at were very muted in perceptually uniform color mappings.  In this case, the perceptual non-uniformity of jet actually helps by emphasizing structure.  (See Figures 6 and 7 below.)

Figure 6 Heat map using jet colormapping.
Figure 6.  A heat map for a chip using Jet colormapping.Caption
Figure 7.  Positive half of previous heatmap, using Viridis colormapping
Figure 7.  Half map (from 0 to 6 mA only) for the same chip using Viridis colormapping.  The structures in the heat map are not as obvious as they were using the Jet colormapping.

 

 

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