Pair-breaking effect on mesoscopic persistent currents

We consider the contribution of superconducting fluctuations in the mesoscopic persistent current (PC) of an ensemble of normal metallic rings, made of a superconducting material whose low bare transition temperature $T_c^0$ is much smaller than the Thouless energy $E_c$. The effect of pair breaking is introduced via the example of magnetic impurities. We find that over a rather broad range of pair-breaking strength $\hslash /{\tau }_s$, such that $T_c^0\lesssim \hslash /{\tau }_s\lesssim E_c$, the superconducting transition temperature is normalized down to minute values or zero while the PC is hardly affected. This may provide an explanation for the magnitude of the average PCs in copper and gold as well as a way to determine their $T_c^0$’s. The dependence of the current and the dominant superconducting fluctuations on $E_c{\tau }_s$ and on the ratio between $E_c$ and the temperature is analyzed. The measured PCs in copper (gold) correspond to $T_c^0$ of a few (a fraction of) mK.

Figure 1

The $h/2e$ harmonic (full line) and $T_c/T_c^0$ (dashed line) as functions of the pair-breaking strength displayed on a logarithmic scale. The current in units of $I\left(s=0\right)$ is plotted for $T=E_c$ and $T_c^0=0.1E_c$. The PC reduction at $s=10$ corresponds to $1/{\tau }_s=\pi E_c$.

Figure 2

The amplitude of the $h/2e$ harmonic in units of $I^{\ast }=-e{E}_c$ as a function of the temperature for two values of $T_c^0/E_c$ and several values of $s$. Note that the $s=0$ curve in the upper panel is valid only for $T/T_c\ge 1+Gi$, where $Gi$ is the Ginzburg parameter.

Figure 3

The digamma function (solid line) and $\text{ln}\left(T_c^0/4{\gamma }_{E}T\right)$ for $T_c^0/T=0.6$ (dashed line). The first three solutions $F_{\text{zero}}^{\ell }$ of Eq. (32) are marked on the $x$ axis with their indices indicated below it. The first values of the set $F_{\text{pole}}^{\ell }$, Eq. (33), are marked by arrows.

Figure 4

The PC as computed from Eq. (24) with the summation over $n$ cut at 1000, 3, and 1 (solid, dashed, and dash-dotted curves, respectively). The plots are for $T=T_c^0=0.1E_c$ and $s=1$.

Figure 5

The current in units of $e{E}_c$ as a function of the flux $\varphi$ for $T_c^0/E_c=0.1$; for several temperatures, $T/E_c=5$, 1, and 0.15 in the solid, dashed, and dash-dotted curves, respectively. In the lower panel the dash-dotted curve corresponds to $T/E_c=0.1$. For $s=0$ the current attains the sawtooth form (upper panel) which is lost for $s=1$ (lower panel).

Figure 6

The amplitude of the $h/2e$ harmonic is plotted in units of $I^{\ast }=-e{E}_c$ as a function of the temperature for $T_c^0/E_c=0.1$ and $s=1$. The exact results can be approximated by Eq. (38) for $T⪢E_c$.

Figure 7

The current in units of $e{E}_c$ as a function of the magnetic flux $\varphi$ plotted for $T=0.1T_c^0=0.01E_c$ and $s=1$. The low-temperature approximation Eq. (41) is compared with the exact result, Eq. (34). We take $\alpha =3$ in the logarithm of Eq. (41).

Figure 8

The first-order approximation for the $h/2e$ harmonic of the current Eq. (45) (bold lines) is compared with the exact result (thin lines). Here $T_c^0=0.1E_c$. In drawing the former, we have used the simplest expression for the cutoff $\omega =T+E_c+1/{\tau }_s$.

Figure 9

The bare transition temperatures corresponding to the measured PC as a function of $L/L_s$. The dashed and dash-dotted curves correspond to the PC measured in copper and gold, respectively. The solid curve gives the maximal possible $T_c^0$ satisfying $T_c=0$.