Mesoscopic fluctuations of the supercurrent in diffusive Josephson junctions

We study mesoscopic fluctuations and weak-localization correction to the supercurrent in Josephson junctions with coherent diffusive electron dynamics in the normal part. Two kinds of junctions are considered: a chaotic dot coupled to superconductors by tunnel barriers and a diffusive junction with transparent normal-superconducting interfaces. The amplitude of current fluctuations and the weak-localization correction to the average current are calculated as functions of the ratio between the superconducting gap and the electron dwell energy, temperature, and superconducting phase difference across the junction. Technically, fluctuations on top of the spatially inhomogeneous proximity effect in the normal region are described by the replicated version of the $\sigma$ model. For the case of diffusive junctions with transparent interfaces, the magnitude of mesoscopic fluctuations of the critical current appears to be nearly three times larger than the prediction of the previous theory, which did not take the proximity effect into account.

Figure 1

The minigap $E_{*}$ vs $\Delta ∕E_g$ in a superconductor–quantum dot–superconductor Josephson junction with symmetric tunnel barriers. The two curves correspond to $E_{*}$ in units of $\Delta$ (solid line) and in units of $E_g$ (dashed line).

Figure 2

Quasiclassical results for a superconductor–quantum dot–superconductor Josephson junction with symmetric tunnel barriers: (a) the critical phase ${\chi }_c$ (in units of $\pi ∕2$) vs $\Delta ∕E_g$ at zero temperature; (b) the critical current $I_c$ (solid line: in units of $G\Delta ∕e$, dashed line: in units of $G{E}_g∕e$, dotted line: in units of $G{E}_{*}∕e$) vs $\Delta ∕E_g$ at zero temperature.

Figure 3

(a) Mesoscopic fluctuations of the critical current, $\delta I_c={\left(\mathrm{var}{I}_c\right)}^{1∕2}$ (solid line: in units of $e\Delta ∕\hslash$, dashed line: in units of $e{E}_g∕\hslash$, dotted line: in units of $e{E}_{*}∕\hslash$) vs $\Delta ∕E_g$ at zero temperature. (b) The ratio $G\delta I_c∕G_Q{I}_c$ vs $\Delta ∕E_g$ at zero temperature.

Figure 4

Mesoscopic fluctuations of the Josephson current $\delta I\left(\chi \right)$ vs phase $\chi$ at zero temperature (solid line) and at temperature $kT=E_{*}∕10$ (dashed line). The curves are plotted for different ratios $\Delta ∕E_g$: 0 (top), 0.2, 0.5, 1, 2, 5 (bottom).

Figure 5

The minigap $E_{*}$ induced in the normal wire at $\chi =0$ (solid line: in units of $\Delta$; dashed line: in units of $\pi E_T$) vs $\Delta ∕E_T$.

Figure 6

Josephson relation $I\left(\chi \right)$ (in units of $G_N\Delta ∕e$) for a short wire at zero temperature.

Figure 7

(a) The critical phase ${\chi }_c$ (in units of $\pi ∕2$) vs $\Delta ∕E_T$ at zero temperature. (b) The critical current $I_c$ (solid line: in units of $G_N\Delta ∕e$, dashed line: in units of $\pi G_N{E}_T∕e$, and dotted line: in units of $G_N{E}_{*}∕e$) vs $\Delta ∕E_T$ at zero temperature.

Figure 8

Josephson relation $I\left(\chi \right)$ (in units of $G_N{E}_T∕e$) for a long wire at zero temperature.

Figure 9

Mesoscopic fluctuations $\delta I\left(\chi \right)$ (in units of $e\Delta ∕\hslash$) for a short wire at $T=0$.

Figure 10

Weak-localization correction $-{\Delta }_{\mathrm{WL}}I\left(\chi \right)$ (in units of $e\Delta ∕\hslash$) for a short wire at $T=0$.

Figure 11

(a) Weak-localization correction to the critical current $-{\Delta }_{\mathrm{WL}}{I}_c$ (solid line: in units of $e\Delta ∕\hslash$, dashed line: in units of $\pi e{E}_g∕\hslash$, and dotted line: in units of $e{E}_{*}∕\hslash$) vs $\Delta ∕E_T$ at zero temperature. (b) The ratio $-{\Delta }_{\mathrm{WL}}{I}_c∕I_c$ (in units of $G_Q∕G_N$) vs $\Delta ∕E_T$ at zero temperature.

Figure 12

(a) The amplitude of mesoscopic fluctuations of the critical current $\delta I_c$ (solid line: in units of $e\Delta ∕\hslash$, dashed line: in units of $\pi e{E}_T∕\hslash$, and dotted line: in units of $e{E}_{*}∕\hslash$) vs $\Delta ∕E_T$ at zero temperature. (b) The ratio $\delta I_c∕I_c$ (in units of $G_Q∕G_N$) vs $\Delta ∕E_T$ at zero temperature.

Figure 13

Weak-localization correction $-{\Delta }_{\mathrm{WL}}I\left(\chi \right)$ (in units of $e{E}_T∕\hslash$) vs $\chi$ for long wires at zero temperature.

Figure 14

Mesoscopic fluctuations $\delta I\left(\chi \right)$ (in units of $e{E}_T∕\hslash$) vs $\chi$ for long wires at zero temperature.