A Depinning Model for Josephson Junction Tuning

Abstract

Building more powerful quantum computers requires manufacturing processes with tight tolerances. To improve the tolerances on Josephson junctions, techniques to fine tune their properties after fabrication have been developed. Understanding how tuning techniques may physically modify the tunnel barrier of a Josephson junction is important and will enable these techniques to be optimised. We develop a model of junction tuning based on depinning theory to interpret a phase diagram of tuning rate. We extract the dependence on temperature, time-varying voltages and oscillation frequency. Using depinning theory we are able to show both why time-varying annealing potentials result in controlled junction tuning and how such protocols can be optimised. We examine how tuning changes the electrical breakdown of barriers and discrepancies between modeled and measured higher energy levels of transmon qubits.

Figures (3)

• Figure 1: (a) A schematic of the experimental setup showing a lock-in amplifier applying a voltage across a load resistor and a JJ in series. A heater controls the temperature of the JJ. (b) Resistance as a function of the frequency of the lock-in amplifier. Above $\sim$8 kHz the nominal resistance of the junction increases rapidly with frequency, a region we omit from study. (c) Three example tuning curves where the fractional resistance of the JJ (i.e. resistance normalised to that at the start of the protocol) against the elapsed time in the protocol performed at 80 $^\circ$C and a tuning voltage of 0.95 V. Dashed lines are a fit of the data to Eq. \ref{['eq:tuning']}.
• Figure 2: (a) Generic phase diagram for depinning theory, showing three operating regimes, where the creep-regime is where tuning occurs. (b) Threshold voltage phase diagram as a function of tuning voltage and temperature. Measurements of the fraction of junctions failing during processing show this boundary. Overlaid are DC measurements of breakdown which follow the same boundary separating the creep and linear regimes. These data are fit by Eq. \ref{['eq:vp']} (c) 1 kHz and (d) 103 Hz resistance tuning speed phase diagram. The parameter $a$ from Eq. \ref{['eq:tuning']} is shown using a colour scale. The boundary between the creep and linear regime found by fitting DC breakdown in (b) is shown in both phase diagrams, as is a straight line fit to points where the interpolated value of a is 0.01 showing the stable/creep boundary. In (d) we show the boundary for both 1000 Hz and 103 Hz showing that with decreasing frequency the boundary has moved to lower voltages and temperatures.
• Figure 3: Investigation of the effect of tuning on DC IVs, electric breakdown and Josephson harmonics. (a) An example IV from an untuned JJ where voltage is ramped up from 0 V. It is fit to the Simmons model to determine model nominal barrier thickness and height. It is also fit to Ohm's law to determine junction resistance. Breakdown voltage is extracted where the current suddenly jumps and is indicated with a star. (b) The resistances of the junctions which have been tuned vs. those that have only been subject to the thermal history associated with tuning. The junctions are tuned an average of $\sim$35% with the resistance prior to tuning shown in transparent blue markers. (c) The thickness extracted from the Simmons model, (d) the barrier height extracted from the Simmons model and (e) the breakdown voltage are all shown comparing tuned and untuned junctions. (f) The results from Josephson harmonics experiments performed on a tuned and an untuned qudit. The difference between the frequencies predicted by the transmon Hamiltonian and those found experimentally are plot for the different transitions.