Nonequilibrium Quasiparticles and $2e$ Periodicity in Single-Cooper-Pair Transistors

We have fabricated single-Cooper-pair transistors in which the spatial profile of the superconducting gap energy was controlled by oxygen doping. The profile dramatically affects the switching current vs gate voltage curve of the transistor, changing its period from $1e$ to $2e$. A model based on nonequilibrium quasiparticles in the leads explains our results, including the observation that even devices with a clean $2e$ period are “poisoned” by small numbers of these quasiparticles.

Figure 1

(a) Schematic representation of $I_{sw}^{\mathrm{e}\mathrm{v}\mathrm{e}\mathrm{n}\mathrm{/}\mathrm{o}\mathrm{d}\mathrm{d}}$. Single-Cooper-pair transistor circuit shown at right. (b),(c) gray scale plots of $I_{sw}(n_g)$ for codeposited type $L$ and type $H$ SCPTs at $T=30\mathrm{m}\mathrm{K}$, measured with $\dot{I}=1\mu \mathrm{A}/\mathrm{s}$. The corresponding gap profiles are shown on the right.

Figure 2

Three-state model of poisoning. (a) Schematics of the “0,” “$\ell$,” and “$i$” states (see text). (b) Transitions from 0 to $\ell$ are driven by a nonequilibrium QP source in the leads. When $\Delta E_{\ell i}<0$ these QPs see the island as a barrier and parity is predominantly even. When $\Delta E_{\ell i}>0$ QPs see the island as a trap and parity is predominantly odd at low temperatures, but QPs can be thermally activated out of the island at higher temperatures.

Figure 3

(a),(b) $\Delta E_{\ell i}$ for the $L$ and $H$ devices. The range of $\Delta E_{\ell i}$ for all values of $\phi$ is indicated by the dark gray region. (c),(d) Predicted crossover in $I_{sw}$ between the even parity $\ell$ state and the odd parity $i$ state (see Fig. 2). The dotted (dashed) curves are the even (odd) curves from Fig. 1a while the thick lines follow the curve corresponding to the lowest energy QP state. The island traps QPs for $n_g>n_{g,\mathrm{c}\mathrm{r}}$ (light gray regions).

Figure 4

(a) $I_{sw}(n_g)$ for the $L$ device ramped at $\dot{I}=100\mu \mathrm{A}/\mathrm{s}$ and $T=30\mathrm{m}\mathrm{K}$. The gray scale is saturated at half the maximum count number to emphasize rare events. (b) Same as (a) but with all switching events shown equally. The gray region marks the predicted range for QP trapping. (c) $I_{sw}(n_g)$ for the $H$ device ramped at $\dot{I}=1\mu \mathrm{A}/\mathrm{s}$ with all counts shown equally. (We show data for the slower ramp rate to emphasize events that occur below the dominant $2e$ curve.)

Figure 5

(a) Thermal activation of QPs out of the island of the $L$ device in the trapping regime. (b) $I_{sw}(n_g)$ for the $L$ device at 200 mK. (c) Histograms of $I_{sw}$ at $n_g=0.7$ for the $L$ device as a function of temperature. The histograms span $T=30\mathrm{m}\mathrm{K}$ to 350 mK in 20 mK increments (from bottom to top) and are offset vertically for clarity. Inset: Peak height ratio and fit to thermal activation model (see text).