Steady-state thermodynamics of nonequilibrium quasiparticles in a Cooper-pair box

A superconducting Coulomb-blockade electrometer was used to measure the Coulomb staircase of an $\mathrm{Al}∕\mathrm{Al}{\mathrm{O}}_x∕\mathrm{Al}$ Cooper-pair box from a temperature of $30\text{to}300\mathrm{mK}$. At the lowest temperature, the Coulomb staircase displays effects from nonequilibrium quasiparticles. As the temperature is increased, an initial decrease is found in the width of the odd steps in the staircase, which corresponds to a reduction in the probability of having a quasiparticle on the island of the box. Above $180\mathrm{mK}$, the width of the odd steps increases, eventually producing a staircase with $1e$ features. We develop a steady-state model of the system and find that the presence of quasiparticles at low temperature is consistent with the assumption of Aumentado et al. [Phys. Rev. Lett. 92, 066802 (2004)] that nonequilibrium quasiparticles are generated in the leads. Above $180\mathrm{mK}$, our results are consistent with the quasiparticle states of the island being thermally populated.

Figure 1

(a) Schematic of the experimental setup used to measure the charge staircases of the Cooper-pair box (CPB). A superconducting Coulomb-blockade electrometer [labeled SET in (b) and (c)] is capacitively coupled to a CPB to measure the charge on the CPB. A dc voltage is placed across the electrometer (labeled $V_{ds}$), and the current through the electrometer is measured. To keep the gain of the electrometer constant while adjusting the voltage on the gate of the CPB $\left(V_g\right)$, the electrometer is placed in a proportional-integral feedback circuit. (b) Scanning-electron micrograph of sample JPL1. (c) Scanning-electron micrograph of sample LPS1.

Figure 2

(Color online) Coulomb staircases at different temperatures for sample JPL1. Black curves are data (500 averages) taken at temperatures of 30, 40, 50, 60, 70, 80, 90, 100, 200, 250, and $300\mathrm{mK}$ (from bottom to top). The smooth (red online) curves are corresponding model predictions. The left axis is the shift in the reduced gate voltage on the electrometer, and the right axis is the change in the charge on the CPB. Curves have been displaced upward successively by $e$ on the CPB. To take each temperature trace, a drain-source voltage of $575\mu \mathrm{V}$ and a current of $55\mathrm{pA}$ were used for the electrometer.

Figure 3

(Color online) Coulomb staircases at different temperatures for sample LPS1. Black curves are data (500 averages) taken at temperatures of 35, 50, 60, 70, 80, 90, 100, 110, 120, 130, 140, 150, and $160\mathrm{mK}$ (from bottom to top). The smooth (red online) curves are corresponding model predictions. The left axis is the shift in the reduced gate voltage on the electrometer, and the right axis is the change in the charge on the CPB. Curves have been displaced upward successively by $e$ on the CPB. To take each temperature trace, a drain-source voltage of $470\mu \mathrm{V}$ and a current of $36\mathrm{pA}$ were used for the electrometer.

Figure 4

(Color online) Measured width of the odd plateau versus temperature for samples JPL1 (filled triangles) and LPS1 (open squares). The solid (red online) curve is based on the full model using the parameters found for a best fit for sample JPL1, and the straight solid line is based on Eq. (19). The dashed-dotted (red online) curve and straight dashed-dotted line are the corresponding fit to the model and result from Eq. (19) for sample LPS1. The inset shows the theoretical plots of the shift in the chemical potential used in the Owen-Scalapino distribution versus temperature.