Direct visualization of vortex ice in a nanostructured superconductor

Artificial ice systems have unique physical properties that are promising for potential applications. One of the most challenging issues in this field is to find novel ice systems that allow precise control over the geometries and many-body interactions. Superconducting vortex matter has been proposed as a very suitable candidate to study artificial ice, mainly due to the availability of tunable vortex-vortex interactions and the possibility to fabricate a variety of nanoscale pinning potential geometries. So far, a detailed imaging of the local configurations in a vortex-based artificial ice system is still lacking. Here we present a direct visualization of the vortex-ice state in a nanostructured superconductor. By using scanning Hall probe microscopy, a large area with the vortex-ice ground-state configuration has been detected, which confirms the recent theoretical predictions for this ice system. Besides the defects analogous to artificial spin-ice systems, other types of defects have been visualized and identified. We also demonstrate the possibility to realize different types of defects by varying the magnetic field.

Figure 1

Vortex ice in a nanostructured superconductor. (a) Atomic-force microscope image of the sample. The dashed square indicates one unit cell. (b) Schematic presentation of the ground-state ordering of the vortex lattice. Two antidots are paired to imitate an elongated double-well pinning site. The arrows in each pair, pointing from the empty antidot to the occupied one, represent the corresponding ground-state arrangement of artificial spin ice. (c) Scanning Hall probe microscope image, showing the magnetic field distribution of the ground-state vortex ice, observed after field cooling at half matching field. The squares indicate the positions of antidots. (d) Possible vortex configurations corresponding to the magnetic-moment configurations of an artificial square spin ice, grouped by increasing energy (type-1 to type-4). The scale bars in (a) and (c) equal $4\mu \mathrm{m}$.

Figure 2

Vortex pattern evolution with magnetic field. Vortex images taken at (a) $H=H_1/4=0.45$, (b) $H=0.6$, (c) $H=0.72$, (d) $H=H_1/2=0.9$, (e) $H=1.02$, (f) $H=1.2$, (g) $H=3H_1/4=1.35$, (h) $H=1.65$, (i) $H=1.7$, and (j) $H=H_1=1.8$ Oe. The white squares indicate the positions of antidots. At $H=H_1/4$, there is only one vortex in each vertex. Above $H_1/4$, two-vortex vertices appear and their number increases until a perfect ground state of vortex ice is formed at $H=H_1/2$. With increasing field, three-vortex vertices appear so that at $H=3H_1/4$, each vertex carry three vortices. At higher fields, four-vortex vertices are formed and eventually all the antidots are occupied by vortices at $H_1$. The images with the same frame color correspond to the maximum number of vortices per vertex equal to 1 (black), 2 (blue), 3 (red), and 4 (green). The scale bar equals $4\mu \mathrm{m}$.

Figure 3

SHPM images showing the magnetic field distribution of the vortex-ice pattern. Vortex patterns observed at (a) $H_1/2-0.03$, (b) $H_1/2$, and (c) $H_1/2+0.045$ Oe. The white squares indicate the positions of antidots. (d)–(f) Vertex configurations corresponding to the SHPM images of (a)–(c). The blue open circles represent vertices where the ice rule for the ground state is obeyed. The green open circles correspond to ice defects of type-2 to type-4 shown in Fig. 1. The solid circles indicate the non-ASI-like defects shown in (g).

Figure 4

Statistics of vertex configurations in vortex ice. (a) Field dependence of the fraction of the spinlike pinning pairs (circles) and the vertices (squares) corresponding to the four types of vortex configurations in Fig. 1. Both the fraction of spinlike pinning pairs and ASI-like vertices reach maximum at half matching field. The dashed lines are a guide to the eye. (b) Gibbs free-energy density (circles) for differ- ent vertex configurations defined in Fig. 1. The fraction of different ASI-like vertex configurations (squares) at half matching field. The type-4 vertex is not detected in the scanned area. Solid lines are a guide to the eye.