Crossover from $hc/e$ to $hc/2e$ current oscillations in rings of $s$-wave superconductors

We analyze the crossover from an $hc/e$ periodicity of the persistent current in flux-threaded clean metallic rings toward an $hc/2e$-flux periodicity of the supercurrent upon entering the superconducting state. On the basis of a model calculation for a one-dimensional ring we identify the underlying mechanism, which balances the $hc/e$ versus the $hc/2e$ periodic components of the current density. When the ring circumference exceeds the coherence length of the superconductor, the flux dependence is strictly $hc/2e$ periodic. Further, we develop a multichannel model which reduces the Bogoliubov–de Gennes equations to a one-dimensional differential equation for the radial component of the wave function. The discretization of this differential equation introduces transverse channels whose number scales with the thickness of the ring. The periodicity crossover is analyzed close the critical temperature.

Figure 1

(Color online) Energy dispersion of a ring with an order parameter $\Delta =0.22t$ for $\varphi =0$ (dashed line) and $\varphi ={\varphi }_c\approx 0.24t$ (solid line), where the indirect energy gap closes. The filled (empty) circles represent occupied (unoccupied) $k$ states for a ring with $N=18$. The asymmetry for $\pm k$ scales with $1/R$.

Figure 2

(Color online) Eigenenergies $E_{\pm }\left(k,\varphi \right)$ [Eq. (10)] as a function of flux $\varphi$ for $N=26$ and a self-consistently calculated order parameter $\Delta$; lines below $E=0$: $E_{-}\left(k,\varphi \right)$ and lines above $E=0$: $E_{+}\left(k,\varphi \right)$. Upper panel: large-gap regime ($V=1.9t$ and ${\Delta }_{1/2}\approx 0.30t$). Lower panel: small-gap regime ($V=1.1t$ and ${\Delta }_{1/2}\approx 0.08t$). Superconductivity occurs only in the odd-$q$ sectors for $V=1.1t$ (see Fig. 3). The bold line marks the highest occupied state for all $\varphi$. For the definition of ${\Delta }_{1/2}$ see text after Eq. (11).

Figure 3

(Color online) Solution of the self-consistency equation [Eq. (13)] for different values of the pairing energy $V$ at $T=0$. From top to bottom: $V=1.9t,1.6t,1.35t,1.1t$.

Figure 4

Crossover from the $hc/e$-periodic normal persistent current to the $hc/2e$-periodic supercurrent in a ring with $N=26$ at $T=0$. For this ring size ${\Delta }_c\approx 0.24t$. The discontinuities occur where the $\varphi$ derivative of the highest occupied state energy changes sign.

Figure 5

Annulus with inner radius $R_1$ and outer radius $R_2$. For a magnetic flux threading the interior of the annulus, the radial part of the Bogoliubov–de Gennes equations is solved numerically with a discretized radial coordinate.

Figure 6

(Color) Non-self-consistent calculation of current and energy at $T=0$. The circulating current (upper panel) in an annulus with an inner radius $R_1=100a$ and an outer radius $R_2=150a$ is shown for fixed, $\varphi$ independent $\Delta =0$ (light blue line), $\Delta =0.002t$ (blue line), $\Delta =0.004t$ (dark blue line), and $\Delta =0.006t$ (black line). The lower panel shows the difference between the total energy of the annulus as a function of $\varphi$ and the total energy at zero flux for the same values for $\Delta$ as above.

Figure 7

(Color) Self-consistent calculations for the same annulus as in Fig. 6. In addition, the top panel displays the self-consistent order parameter $\Delta$ as a function of $\varphi$. The lines correspond to the pairing interaction $V=0$ (light blue line), $V=0.28t$ (blue line), $V=0.32t$ (dark blue line), and $V=0.38t$ (black line). The black arrows mark the positions of the $q$ jump for $V=0.38t$ and $V=0.32t$.

Figure 8

(Color online) The order parameter $\Delta$ and the persistent current for the temperature-driven transition from the normal to the superconducting state in an annulus with inner radius $R_1=30a$ and outer radius $R_2=36a$. The pairing intercation is $V=0.7t$, with a critical temperature of $T_c\approx 0.0523t$ for zero flux. For these parameters $\Delta \left(T=0\right)\approx 0.1t$. The lines (from top to bottom) correspond to the temperatures $T=0.0513t$, $T=0.0520t$, and $T=0.0522t$. Notice that $\Delta$ is slightly different for the flux values $\varphi =0$ and $\pm 1/2$.