Tunable artificial vortex ice in nanostructured superconductors with a frustrated kagome lattice of paired antidots

Theoretical proposals for spin-ice analogs based on nanostructured superconductors have suggested larger flexibility for probing the effects of fluctuations and disorder than in the magnetic systems. In this paper, we unveil the particularities of a vortex ice system by direct observation of the vortex distribution in a kagome lattice of paired antidots using scanning Hall probe microscopy. The theoretically suggested vortex ice distribution, lacking long-range order, is observed at half matching field ($H_1/2$). Moreover, the vortex ice state formed by the pinned vortices is still preserved at $2H_1/3$. This unexpected result is attributed to the introduction of interstitial vortices at these magnetic-field values. Although the interstitial vortices increase the number of possible vortex configurations, it is clearly shown that the vortex ice state observed at $2H_1/3$ is less prone to defects than at $H_1/2$. In addition, the nonmonotonic variations of the vortex ice quality on the lattice spacing indicates that a highly ordered vortex ice state cannot be attained by simply reducing the lattice spacing. The optimal design to observe defect-free vortex ice is discussed based on the experimental statistics. The direct observations of a tunable vortex ice state provides new opportunities to explore the order-disorder transition in artificial ice systems.

Figure 1

Schematic representation of the nanostructured superconducting sample patterned with a kagome lattice consisting of paired antidots (white dashed rounded ellipses) on ${\mathrm{Si}/\mathrm{SiO}}_2$ substrate. The paired vortex-vortex (V-V) distance and the nearest neighbor V-V distance at each vertex of the kagome lattice are denoted by $L_1$ and $L_2$, respectively. The kagome lattice spacing is $L_s$. The magnetic field $H_a$ is perpendicular to the plane of the film.

Figure 2

(a) Optical microscope image (16 $\times 16\mu {\text{m}}^2$) of Sample III with lattice parameters: $L_1=2\mu \text{m}$ and $L_2=2\mu \text{m}$. (b) Vortex configurations in Sample III at $H_a=H_1/3$, $5H_1/6$, and $1.53H_1$. The positions of the antidots are marked by white circles. (c) The observed vortex ice states (upper panels) and their schematic representation (bottom panels) at four different locations of Sample III for $H_a=H_1/2$, where the vortices obey $n_{\text{in}}=1$ or $n_{\text{in}}=2$ at each vertex. The interstitial vortices are marked by red circles. (d) Sketch of vortex occupations in paired antidots: empty pair (no vortex in paired antidots, surrounded by white dashed ellipse), ice pair (only one vortex in paired antidots, black dashed ellipse), and saturated pair (two vortices in paired antidots, orange dashed ellipse). Six possible vortex ice configurations for each vertex, which is analogous to the ice rules, i.e., one-in/two-out (upper) and two-in/one-out (bottom). The ice defects with $n_{\text{in}}=0$ and $n_{\text{in}}=3$ (rightmost sketches).

Figure 3

(a) The occurrence (in percentage) of ice pairs (red diamond), empty pairs (blue triangle), and saturated pairs (cyan circle) vs external magnetic field based on the statistics of vortex occupations in paired antidots (see Ref. [44]: original SHPM images of vortex states at different locations of Sample III under various magnetic fields). (b) The occurrence (in percentage) of the different occupation numbers for each vertex as a function of the applied magnetic field. Purple triangle: $n_{\text{in}}=1$; green star: $n_{\text{in}}=2$; black cross: interstitial vortex. (c) Vortex configurations obtained after field cooling at different magnetic fields as indicated by the capital letters in (a). (d) The vortex ice states with additional interstitial vortices observed at $H_a=2H_1/3$ in four different locations of Sample III (also see the results of the tdGL simulations in Ref. [44]).

Figure 4

Statistics of vortex ice states in Sample I with $L_1=2\mu \text{m}$, $L_2=1.1\mu \text{m}$ (a), Sample II with $L_1=2.2\mu \text{m}$, $L_2=1.5\mu \text{m}$ (b), and Sample IV with $L_1=3\mu \text{m}$, $L_2=1.5\mu \text{m}$ (c). Upper left: schematic representations of the sample with same size as scanned area (16 $\times 16\mu {\text{m}}^2$). Upper middle: The vortex configurations observed in three samples at $H_a=H_1/2$ (points $A_1$, $A_2$, and $A_3$ in bottom panels). Upper right: The vortex ice states at a magnetic field value $H_a^{*}$ defined by the constrained: $P_{n_{\text{in}}=1}\approx P_{n_{\text{in}}=2}$ (points $H_a^{*}$ in bottom panels). Bottom panels: The occurrence (in percentage) of the different pair distributions [ice pairs (red diamond), empty pairs (blue triangle), and saturated pairs (cyan circle)] and the different vertex distributions [$n_{\text{in}}=1$ (purple triangle) and $n_{\text{in}}=2$ (green stars)]. The black crosses mark averaged number of interstitial vortices per hexagon.

Figure 5

The variation of vortex ice with the kagome lattice spacing $L_s$. Black circle: the magnetic field value ($H_a^{*}$) at which $P_{(n_{\text{in}}=1)}=P_{(n_{\text{in}}=2)}$; red triangle: the difference between the percentage of $n_{\text{in}}=1$ and $n_{\text{in}}=2$ at $H_1/2$; blue triangle and green square: percentage of ice pairs at $H_1/2$ and $H_a^{*}$, respectively.