Realization of Artificial Ice Systems for Magnetic Vortices in a Superconducting MoGe Thin Film with Patterned Nanostructures

We report an anomalous matching effect in MoGe thin films containing pairs of circular holes arranged in such a way that four of those pairs meet at each vertex point of a square lattice. A remarkably pronounced fractional matching was observed in the magnetic field dependences of both the resistance and the critical current. At the half matching field the critical current can be even higher than that at zero field. This has never been observed before for vortices in superconductors with pinning arrays. Numerical simulations within the nonlinear Ginzburg-Landau theory reveal a square vortex ice configuration in the ground state at the half matching field and demonstrate similar characteristic features in the field dependence of the critical current, confirming the experimental realization of an artificial ice system for vortices for the first time.

Figure 1

Measured resistance vs magnetic field curves for a 20-nm-thick MoGe film with a square ice arrangement of holes (diameter $a=102\mathrm{nm}$ and interhole spacing $d=300\mathrm{nm}$) at different temperatures for the applied dc current $I=500\mu \mathrm{A}$. Inset shows a scanning electron microscopy image of the sample. Dashed ellipses highlight the pair of holes (which are highlighted in white) that creates an effective double-well pinning potential for the vortices.

Figure 2

The critical current of the sample in Fig. 1 [normalized to zero magnetic field critical current $I_c(0)$] as a function of the magnetic field $H$ at $T=4.8\mathrm{K}$ (red circles) and $T=4.9\mathrm{K}$ (black squares). Top inset shows the results obtained at $T=4.72\mathrm{K}$. Zero magnetic field critical currents are: $I_c(0)=802\mu \mathrm{A}$ at $T=4.72\mathrm{K}$, $I_c(0)=613\mu \mathrm{A}$ at $T=4.8\mathrm{K}$, and $I_c(0)=335\mu \mathrm{A}$ at $T=4.9\mathrm{K}$. Lower inset shows contour plots of the simulated local magnetic field $h$ for the half matching field, which shows the ground state vortex configuration at $T=4.8\mathrm{K}$. White circles indicate the positions of vortex-free holes and the dashed red square shows the unit cell of the simulation area.

Figure 3

Simulation results. (a),(b) Current-voltage characteristics of the sample with $a=102\mathrm{nm}$ and $d=300\mathrm{nm}$ at $T=4.8\mathrm{K}$ (a) and $T=4.9\mathrm{K}$ (b) for applied field values: $H=0$ (circles), $H=0.5H_1$ (squares), and $H=H_1$ (triangles). Inset in (b) shows the model system: a superconducting strip (width $w$ and with periodic boundary condition in the $x$ direction) with a square ice arrangement of holes (size $a$ and separation $d$) under applied dc current $I$ and a perpendicular magnetic field $H$. (c) The critical current of the sample (normalized to the one at zero magnetic field) as a function of $H$ at two temperatures.

Figure 4

(a),(b) Simulated voltage vs time response of the sample of Fig. 3 at $T=4.8\mathrm{K}$ for zero (a) and half matching field (b). Panels 1–5 show snapshots of $|\psi {|}^2$ at times indicated on the $V(t)$ curve. White (blue) arrow in panel 1 shows the direction of motion of vortices (antivortices) and dashed white ellipses highlight the annihilation point of vortex-antivortex pairs. Blue ellipses in panel 3 show the square ice arrangement of vortices in neighboring unit cells. White circles in panels 4 and 5 show positions of holes with a pinned vortex.