Influence of parity on the persistent currents of superconducting nanorings

We argue that at sufficiently low temperatures $T$ superconducting parity effect may strongly influence equilibrium persistent currents (PCs) in isolated superconducting nanorings containing a weak link with few conducting modes. An odd electron, being added to the ring, occupies the lowest available Andreev state and produces a countercurrent circulating inside the ring. For a single channel quantum point contact at $T=0$ this countercurrent exactly compensates the supercurrent $I_e$ produced by all other electrons and, hence, yields complete blocking of PCs for any value of the external magnetic flux. In superconducting nanorings with embedded normal metal the odd electron countercurrent can “overcompensate” $I_e$ and a novel “$\pi ∕N$-junction” state occurs in the system. Changing the electron parity number from even to odd results in spontaneous supercurrent in the ground state of such rings without any externally applied magnetic flux.

Figure 1

Andreev levels inside QPC and their occupation at $T=0$ for even (a) and odd (b) ensembles.

Figure 2

Superconducting ring with embedded SNS junction of length $d$.

Figure 3

The ratio between canonical and grand canonical values of PCs $I_{e∕o}∕I_{\mathit{g.c.}}$ [represented by the term in the square brackets in Eq. (20)] versus $\phi$ in a single mode QPC at different temperatures for even (three upper curves) and odd (three lower curves) ensembles. Here we have chosen $T_S^{*}=0.1\Delta$ and $\mathcal{T}=0.99$.

Figure 4

Andreev levels in a single mode SNS junction with $d=6\hslash v_F∕\Delta$ and their occupation at $T=0$ for even (a) and odd (b) ensembles.

Figure 5

The zero temperature current-phase dependence (24) for SNS rings with $N=1$: $I_e\left(\phi \right)$ (dashed) line and $I_o\left(\phi \right)$ (solid) line.

Figure 6

The same as in Fig. 5 only for the odd ensembles [second Eq. (24)] and for $N=2$, 3, and 4.

Figure 7

The zero temperature current-phase relation $I_o\left(\phi \right)$ $(-\pi <\phi <\pi )$ for $N=1$ and different values of the parameter $y=d\Delta ∕\hslash v_F$.

Figure 8

The spontaneous current amplitude $I_{\mathit{sp}}$ as a function of the parameter $y$ at $T=0$. In the inset, the same function is shown on the log-log scale. Dashed lines indicate the asymptotic behavior of $I_{\mathit{sp}}\left(y\right)$ in the limits of small and large $y$.