Defect Dynamics in Artificial Colloidal Ice: Real-Time Observation, Manipulation, and Logic Gate

We study the defect dynamics in a colloidal spin ice system realized by filling a square lattice of topographic double well islands with repulsively interacting magnetic colloids. We focus on the contraction of defects in the ground state, and contraction or expansion in a metastable biased state. Combining real-time experiments with simulations, we prove that these defects behave like emergent topological monopoles obeying a Coulomb law with an additional line tension. We further show how to realize a completely resettable “nor” gate, which provides guidelines for fabrication of nanoscale logic devices based on the motion of topological magnetic monopoles.

Figure 1

(a) Schematic showing the colloidal spin ice composed by interacting colloids in a square lattice of double wells. The red line shows a defect line separating two $q=\pm 2$ defects in the GS. (b) Microscope image of an experimental defect line in a square lattice of lithographic double wells filled with paramagnetic colloids. Blue arrows denote spin directions, red arrows highlight the defect line. Scale bar is $15\mu \mathrm{m}$. (c) Vertex configurations for the colloidal square ice. Vertex energy increases from left to right. (d) Optical profilometer image of the lithographic square lattice. (e) Cross section of a typical double well characterized by a central hill of height $h=0.73\mu \mathrm{m}$. (f) Distribution of hill height $h$ fitted with a Gaussian function (continuous line).

Figure 2

(a) Color map showing the net vertex charges in the experiments for a defect line connecting two $q=\pm 2$ defects under a field $B_z=25.7\mathrm{mT}$ (VideoS1 in Ref. [34]). The line consists of high energy $S_{IV}$ vertices with a zero charge but a net dipole, which give raise to the additional line-tension term. (b) Average line length $⟨l⟩$ versus time for three different magnetic fields. Closed (empty) symbols denote experiments (numerical simulation), continuous lines are fit from Eq. (1) in the text. Inset: normalized interaction potential $V(l)/\omega$ between two topological defects minus its linear contribution ($\alpha l+\beta$). Red line is a fit using the potential described in the text.

Figure 3

(a),(b) Experimental vertex charges for a closing line (a) and for a single $q=-2$ propagating defect (b) in the biased state. Black (blue) arrows are spins flipped by the motion of the original (spontaneously emerged) defects. Corresponding movies (VideoS2, VideoS3) are in Ref. [34]. (c) Numerical simulation of case (a) (with one end fixed) and (b) showing the evolution of the line length $⟨l⟩$ for an applied field $B_z=18.8\mathrm{mT}$. Continuous lines are fits from Eq. (1). Bottom inset: difference between the two curves in the main panel (empty squares) versus line length plotted with Eq. (1) with $\kappa =0$ (continuous line).

Figure 4

Realization of a nor gate via numerical simulation. (a),(b) Images showing the system preparation: (a) a force $F_1$ applied along the diagonal biases the system, except for one fixed spin; (b) a smaller force $F_2$ shifts two rows of particles (magenta) with high magnetic susceptibilities. (c),(d) Images showing a 1 output obtained from (0,0) inputs. In (c) an external field $B_z$ is applied perpendicular to the plane to start defect propagation, while a small force $F_3$ along the diagonal prevents motion of other defects from the upper left corner. The final result shown in (d) is the 1 output. (e),(f) Images showing a 0 output obtained from (1,0) input. In (e) an input current causes the whole line of magenta spin to flip (1 input). The 0 output results from the changed trajectory of the propagating defect (f). All the corresponding videos can be found in Ref. [34].