Duality breaking of vortex configuration in a hierarchical honeycomb network

We report measurements of Little-Parks oscillation on the hierarchical honeycomb-superconducting network for investigating possible effects of hierarchical structure in terms of spatial symmetry, parity, and duality. We observed an asymmetric Little-Parks oscillation about $\Phi /{\Phi }_0=1/2$, although spatial symmetry was kept in the network. In comparison with a regular honeycomb network, the asymmetric oscillation is attributed to hierarchy which induces mixture of commensurate and incommensurate regions. The asymmetric oscillation is found to indicate breaking of the duality of vortex configuration.

Figure 1

SEM image of the samples. (a) The regular honeycomb network, which has about 5000 cells with lattice constant of $2\mu \text{m}$, line width of $0.6\mu \text{m}$. (b) The hierarchical honeycomb network. The elementary hexagon side length is $2\mu \text{m}$ with line width of $0.2\mu \text{m}$ and has five classes of hierarchy.

Figure 2

The magnetic-flux dependence of the sample resistance normalized by the normal-state resistance $R_N$. We found periodic dips indicated by the arrows. The inset shows the index number of dip positions as a function of the magnetic-flux. The slope shows the period of oscillation as 2.22 G.

Figure 3

(Color online) (a) The sample resistance as a function of the filling ratio $\Phi /{\Phi }_0$ in range from 0 to 1. The arrows indicate dips with the fundamental filling ratio of 1/4, 1/3, 2/5, 1/2, 3/5, 2/3, and 3/4. (b) Error from Farey sequence versus the dip positions. Shape of symbols denotes correspondence relation in terms of the symmetry about 1/2. Solid and open symbols denote $0<\Phi /{\Phi }_0<1/2$ and $1/2<\Phi /{\Phi }_0<1$, respectively. Solid line in the center is a guide to the eye indicating $\Phi /{\Phi }_0=1/2$. Every dips correspond to $F_5$ within 1.3% accuracy. The inset is vortex configuration on the regular honeycomb network. The unit cells occupied with vortices are shown shaded. Left side is at $\Phi /{\Phi }_0=1/3$ and right side is $\Phi /{\Phi }_0=2/3$. Spatial vortex configuration of 2/3, is essentially identical to that of 1/3.

Figure 4

The magnetic-flux dependence of the sample resistance normalized by $R_N$. (a) In range from $-10$ to 0 G. (b) In range from 0 to 10 G. The arrows indicate periodic dips. (c) The index number of dip positions as a function of the magnetic flux. The slope of the line is 2.00 G.

Figure 5

(Color online) (a) The sample resistance as a function of the filling ratio $\Phi /{\Phi }_0$ in range from 0 to 1. The arrows indicate fundamental dips. (b) Error from Farey sequence versus the dip positions. Same notation as in Fig. 3 is used. The dashed arrows are guides to the eye indicating correspondence in terms of the symmetry about 1/2. Dips of 1/5, 1/3, 1/2, and 3/4 correspond to $F_5$ within 1.3% accuracy. The other dips deviate from $F_5$ with accuracy up to 6%.

Figure 6

(Color online) The model of vortex configuration on the hierarchical honeycomb network. Blue and red denote commensurate and incommensurate region where vortices are allocated, respectively. Color depth denotes density of vortices. (a) At $\Phi /{\Phi }_0=1/3$. In some regions vortices are commensurately allocated with base structure and other regions are not. (b) At $\Phi /{\Phi }_0=2/3$. In spite of duality operation, the configuration of 2/3 is not dual for that of 1/3.