Minigap, parity effect, and persistent currents in SNS nanorings

We have evaluated a proximity-induced minigap in the density of states (DOS) of SNS junctions and SNS nanorings at an arbitrary concentration of nonmagnetic impurities. We have demonstrated that an isotropic energy minigap in the electron spectrum opens up already at arbitrarily weak disorder, while the angle-resolved DOS at higher energies can remain strongly anisotropic. The minigap value ${\epsilon }_g$ can be tuned by passing a supercurrent through an SNS junction or by applying a magnetic flux $\Phi$ to an SNS ring. A nonmonotonous dependence of ${\epsilon }_g$ on $\Phi$ has been found at weak disorder. We have also studied persistent currents in isolated SNS nanorings. For an odd number of electrons in the ring we have found a nontrivial current-phase (current-flux) relation which, at relatively high disorder, may lead to a $\pi$-junction state and spontaneous currents in the ground state of the system.

Figure 1

A superconducting ring with embedded normal metal of length $d$.

Figure 2

(Color online) Angle-averaged DOS in the middle of the normal layer. Here and below we choose $d=10{\xi }_0$.

Figure 3

(Color online) The angle-resolved DOS in the middle of the normal layer (thin lines) together with the angle-averaged DOS (bold solid line). The angle between quasiparticle momentum and the $x$ direction (normal to NS interfaces) is denoted as $\theta$.

Figure 4

(Color online) The minigap ${\epsilon }_g\left(0\right)$ and the zero-temperature Josephson critical current of an SNS junction as a function of $d∕\ell$. For sufficiently short values of $\ell$ both quantities approach the corresponding results derived in the diffusive limit from the Usadel equation analysis.

Figure 5

(Color online) Phase dependence of the minigap ${\epsilon }_g\left(\phi \right)$ for an SNS structure at different values of the electron mean free path $\ell$.

Figure 6

(Color online) Phase dependence for the zero-temperature Josephson current $I\left(\phi \right)$ at different values of the mean free path $\ell$.

Figure 7

(Color online) Phase dependence of the Josephson current at $T=0$ for an odd and even number of electrons in the ring.

Figure 8

(Color online) Josephson energy $E\left(\phi \right)$ of an SNS ring as a function of the phase difference $\phi$ for the even and odd ensembles. The solid curve corresponds to a $\pi$-junction state.