Parity effect in Josephson junction arrays

We study the parity effect and transport due to quasiparticles in circuits comprised of many superconducting islands. We develop a general approach and show that it is equivalent to previous methods for describing the parity effect in their more limited regimes of validity. As an example we study transport through linear arrays of Josephson junctions in the limit of negligible Josephson energy and observe the emergence of the parity effect with decreasing number of nonequilibrium quasiparticles. Due to the exponential increase in the number of relevant charge states with increasing length, in multijunction arrays the parity effect manifests in qualitatively different ways to the two-junction case. The role of charge disorder is also studied as this hides much of the parity physics that would otherwise be observed. Nonetheless, we see that the current through a multijunction array at low bias is limited by the formation of metastable even-parity states.

Figure 1

(a) Quasiparticle tunneling rate [Eq. (5)] as a function of energy difference between charge configurations $\left(\delta E\right)$ for four different parity configurations, i.e., odd or even charge states on the source and destination islands respectively. The rates are evaluated at $T_e=222\mathrm{mK}$, which is just below the parity temperature $T^{*}=269\mathrm{mK}$ for these parameters, see text for details. (b) Parity rates evaluated in the middle of the subgap region $\left(\delta E=\Delta \right)$ as a function of $T_e$. The scaling behavior for the four rates as a function of $\Delta /k_B{T}_e$ can be clearly seen above and below the parity temperature.

Figure 2

(a) JJA circuit under consideration, consisting of a linear chain of Josephson junctions with Josephson energy $E_J$ and capacitance $C_J$. The circuit is driven by a symmetrically applied voltage source $V$ and each junction sees an effective capacitance to ground $C_G$. (b) Current through the array (logarithmic scale) as a function of applied voltage. Effective electron temperature is varied from below to above the parity crossover temperature, which is $T^{*}=269\mathrm{mK}$ for these parameters. The lines in the figure are the average of 10 KMC runs, each of which consists of ${10}^6$ events. The maximum and minimum of these 10 runs are indicated by the pale shading.

Figure 3

(a) Current as a function of voltage and electron temperature for maximal disorder. Below the parity temperature, the current is strongly suppressed at low voltages. The oscillations in the current due to parity effects are also lost due to disorder. (b) The current at $T_e=170\mathrm{mK}$ as a function of disorder strength. In (b) we ensemble averaging over 50 different disorder realizations, the variance of which is shown via pale shading. Although there is considerable variation with disorder realization, qualitatively we see that using $\eta =1$ insures numerical convergence for a given realization.

Figure 4

(a) Current plotted as a function of $T_e$ for various values of $V$. Here we see at low voltages a characteristic crossover from $2\Delta /k_B{T}_e$ (dashed-dotted line) to $\Delta /k_B{T}_e$ (dashed line) scaling, indicating parity limited conduction. At higher voltages, the increased state space means that the current is no longer limited by the even-even rate but is instead dominated by the odd parity processes, showing an initial constant scaling with temperature. Irrespective of voltage (while still in the subgap region), at temperatures above $T^{*}$ the current scales with the parity independent rate. (b) The inset shows the same data plotted as a function of inverse temperature, which clearly shows the two regimes.

Figure 5

(a) Fourier transform of the charge-charge correlation function as a function of temperature. As the temperature is increased, the mean transition rate and therefore the current also increases. As the temperature is increased, correlated conduction sets in soon after conduction itself begins. The ratio of correlation frequency to current $f_{\mathrm{peak}}/I=e$ is consistent with the charge carriers being single electrons. (b) The width of the correlation peaks normalised by the position (the reduced line width) shows a particularly clear signature of the parity effect. As a function of temperature we see the reduced line width plateau above the parity temperature (indicated with triangles for different values of the superconducting gap).