Single-loop-like energy oscillations and staircase vortex occupation in superconducting double networks

The magnetic-field dependence of the energy and vortex occupation is calculated for the recently realized superconducting double network consisting of two interlaced subnetworks of small and large loops. Two different approaches are employed, both based on the $J^2$ model: Mean-field analysis that minimizes the network energy assuming random-vortex configurations and numerical simulations in which energy is minimized avoiding this assumption. In the mean-field analysis the vortex population in both subnetworks increases linearly with the applied field. In contrast the simulations show that while the population of the large loops increases linearly with field, the occupation of the small loops grows in steps, resembling the behavior of an ensemble of decoupled loops. This decoupling is also reflected in the waveform of the energy versus applied field. A modified mean-field analysis, which introduces decoupling between the small loops, yields results in excellent agreement with the simulations. These findings suggest that the behavior of a single loop is reflected in the double network and thus constitute it as a favorable system for the experimental study of quantization effects in superconducting loops.

Figure 1

Schematic diagram of (a) the double network and (b) the simple square network.

Figure 2

Normalized energy per loop obtained from the mean-field analysis [Eq. (9)] and from the simulations (solid line and open circles, respectively) plotted versus the normalized field. The bold circles indicate the theoretical value of the energy corresponding to the checkerboard configuration of vortices in the square network $E=N_T{\varphi }_0^2/(32L)$.

Figure 3

(a) Mean-field calculations [Eq. (25)] of the normalized energy per unit-cell of the double network, and (b) the occupation $N_v=N+F$ of large and small loops $n_v=n+f$ for $L/\ell =5$. Note that the ratio of the slopes of the two lines is 24, which corresponds to the ratio between the areas of the large and small loops.

Figure 4

(a) Normalized energy per unit-cell obtained theoretically after minimizing Eq. (33) (bold solid lines) and from simulations (squares connected by thin lines as guide for the eye). (b) Average number of vortices per large loop$\langle N_v\rangle$ and per small loop $\langle n_v\rangle$ obtained theoretically [Eq. (33)] (dashed and solid lines, respectively) and from simulations (diamonds and circles, respectively). The different panels relate to double networks with size ratios $L/\ell =10$, $5$, and $3$ (from top to bottom). Both numerical and analytical solutions show breaks around the middle of the steps resulting from competition in occupation of large and small loops. This competition occurs in the field range where the energy cost to insert a vortex into a small loop or a large loop is similar. The field increment (in units of $H{\ell }^2/{\varphi }_0$) in the simulations is $0.02$ for $L/\ell =3$ and $0.01$ for $L/\ell =5$ and $10$. The step in the $\langle n_v\rangle$ plot for $L/\ell =10$ is relatively sharp ($<0.01$) and hence points on this step are absent.

Figure 5

Vortex configuration in the double network at low normalized fields, $H{\ell }^2/{\varphi }_0=0.01,0.02$ and $0.03$ for $L/\ell =5$. The large loops are continuously occupied in the same way as in a simple square network, while the small loops remain empty. Note the checkerboard distribution at $H{\ell }^2/{\varphi }_0=0.02$ corresponding to half filling of the large loops $H(L^2-{\ell }^2)/{\varphi }_0\approx 0.5$.

Figure 6

Vortex configuration in the double network at relatively high fields, $H{\ell }^2/{\varphi }_0=0.48,0.5$, and $0.52$ for $L/\ell =5$. Note that in this narrow field range the number of vortices in the small loops increases sharply from zero to one.