Magnetic flux pinning in superconductors with hyperbolic-tessellation arrays of pinning sites

We study magnetic flux interacting with arrays of pinning sites (APSs) placed on vertices of hyperbolic tessellations (HTs). We show that, due to the gradient in the density of pinning sites, HT APSs are capable of trapping vortices for a broad range of applied magnetic fluxes. Thus, the penetration of magnetic field in HT APSs is essentially different from the usual scenario predicted by the Bean model. We demonstrate that, due to the enhanced asymmetry of the surface barrier for vortex entry and exit, this HT APS could be used as a “capacitor” to store magnetic flux.

Figure 1

Hyperbolic tessellations in a Poincaré disk. The Shchläfli symbols $\{p,q\}$ describe tessellations where $q$ regular $p$-gons meet at each vertex: (a) $\{3,7\}$ and (b) $\{4,5\}$ (shown by dark green/black lines). The corresponding dual tessellations (a) $\{7,3\}$ and (b) $\{5,4\}$ are shown by orange (gray in black and white) lines.

Figure 2

Vortex configurations in a $\{3,7\}$ hyperbolic-tessellation (HT) APS for varying number of vortices per simulation cell (inside the HT): (a) $N_v=49$ ($N_v^{\mathrm{HT}}=29$), (b) $N_v=81$ ($N_v^{\mathrm{HT}}=49$), and (c) $N_v=144$ ($N_v^{\mathrm{HT}}=83$). Pinning sites are shown by dark blue (black) open circles, vortices by green (gray) dots. Red (gray) dashed lines show the boundaries of concentric rings [numbered 1 to 4 (a)] of the same area containing 1, 7, 21, and 56 pins. (d) Number of pinned (p) and unpinned (up) vortices in the rings, corresponding to $N_v^{\mathrm{HT}}$ in (a)–(c), shown by symbols (lines connecting the symbols are guides for the eye).

Figure 3

Penetration of magnetic flux in a HT APS: The ground-state vortex configurations for varying number of vortices, $N_v=28$ (a), 63 (b), 81 (c), and 125 (d). For low and moderate applied fluxes [(a), (b)], the penetration of the magnetic flux in the HT is shielded by the outer row (i.e., at the edge) of the HT, while for higher applied fluxes [(c), (d)], vortices jump into the inner pinning sites, i.e., at the second pinning “row” and the pins at the central region, although vortices are still pinned by the outer pins. This scenario results in a very inhomogeneous magnetic flux penetration in the HT APS. The corresponding radial distribution function (RDF) of the magnetic flux penetrating in the HT APS is shown on the right-hand side panel for each $N_v$.

Figure 4

Accumulation of the magnetic flux in a HT APS and its expulsion: Vortex patterns for $N_v=81$ (a) and 93 (b). The right-hand side panels show the corresponding RDF for the magnetic flux inside ($r/\lambda \lesssim 11$) and outside ($r/\lambda \gtrsim 11$) the HT [solid dark blue (black) line] and the average magnetic flux inside the HT [dashed red (gray) line].

Figure 5

Magnetization curves for the HT APS with varying maximum pinning strength: $f_p/f_0=1.2$ (squares), 1 (circles), and 0.8 (open triangles).