Normal persistent currents in proximity-effect bilayers

We calculate the contribution of superconducting fluctuations to the mesoscopic persistent current of an ensemble of rings, each made of a superconducting layer in contact with a normal one, in the Cooper limit. The superconducting transition temperature of the bilayer decays very quickly with the increase of the relative width of the normal layer. In contrast, when the Thouless energy is larger than the temperature, then the suppression of the persistent current with the increase of this relative width is much slower than that of the transition temperature. This effect is similar to that predicted for magnetic impurities, although the proximity effect considered here results in pair weakening as opposed to pair breaking.

Figure 1

Proximity-effect ring in which the width of the strong (weak) superconductor is $d_S$ ($d_N$), threaded by a magnetic flux $\Phi$ (measured in units of the flux quantum).

Figure 2

The transition temperature of a proximity-effect sandwich in the Cooper limit [Eq. (7)] as a function of the weighed width of the normal layer for ${\lambda }_N=0$, 0.05, and $-0.05$ (solid, dashed, and dotted curves, respectively). In all curves ${\lambda }_S=0.17$. The inset depicts the Abrikosov-Gorkov expression [Eq. (8)] for the reduction of the transition temperature by pair breakers.

Figure 3

The transition temperature of the bilayer $T_{c0}^{NS}$, normalized by $T_{c0}^S$ (blue solid line) and the first harmonic of the current, divided by its value at $p_N^{}=0$, from Eq. (36) (red dashed line). The green dash-dotted line shows the approximation (41) for the current. The parameters used for all graphs are $T_{c0}^S=1.27$ K (equivalent to ${\lambda }_S^{}=0.17$), ${\omega }_D=400$ K, $T_{c0}^N=0$ (i.e., ${\lambda }_N^{}=0$) and $1/{\tau }_s^{}=0$. The current is plotted for $T=4T_{c0}^S$ and $E_c^{}=10T_{c0}^S$.

Figure 4

The first harmonic of the current, in units of the Thouless energy $E_c$, versus $T/E_c$, for $T_{c0}^S=1.27$ K, $E_c=0.015$ K, ${\omega }_D=400$ K, $T_{c0}^N=1/{\tau }_s=0$, and ${\tilde{\mathcal{N}}}_{\text{eff}}{T}_{c0}^S={10}^5$. The current is plotted for $p_N^{}=0.45$, $0.5$, and $0.7$ and for $T>T_{c0}^{NS}\left(p_N\right)$.

Figure 5

Same as Fig. 4, but with $T_{c0}^N=0.001$ K (i.e., ${\lambda }_N^{}=0.077$). The current is plotted for $p_N^{}=0.7$ (blue dashed line) and $p_N^{}=0.8$ (red dash-dotted line). Thick (thin) lines represent $1/{\tau }_s=0$ ($1/{\tau }_s=E_c$). The graphs are plotted for $T>T_c^{NS}$.

Figure 6

$T_{Gi}/T_{c0}^S$ and $T_{c0}^{NS}/T_{c0}^S$ versus $p_N$ for ${\tilde{\mathcal{N}}}_{\text{eff}}{T}_{c0}^S={10}^5$. Note the factor of ${10}^{-5}$ on the $y$ axis.