Nanoscale structure of the magnetic induction at monopole defects in artificial spin-ice lattices

Artificially frustrated spin-ice systems are of considerable interest since they simulate the spin frustration and concomitant rich behavior exhibited by atoms on a crystal lattice in naturally occurring spin-ice systems such as pyrochlores. As a result of the magnetic frustration, these systems can exhibit “magnetic monopole” type defects, which are an example of an exotic emergent quasiparticle. The local magnetization structure of such monopole defects determines their stability and thus is critical to understanding their behavior. In this paper, we report on the direct observation at room temperature of the nanoscale magnetic structure of individual magnetic monopoles in an artificially frustrated two-dimensional square spin-ice lattice, using high-resolution aberration-corrected Lorentz transmission electron microscopy. By combining the high-resolution microscopy with micromagnetic simulation, we demonstrate how nucleation of defect strings, reminiscent of Dirac strings, connecting monopole defects controls the demagnetization process in these spin-ice lattices.

Figure 1

(a) Schematic showing the different types of nodes possible in a square spin-ice lattice; (b) schematic showing the uniformly magnetized square lattice and the generation of a monopole pair ($\oplus$ and $\ominus$) after flipping the magnetization in two elements and the corresponding vortex state formed indicated by a magenta dot and a curved arrow; (c) experimental TEM image of the patterned square lattice; (d) magnetic induction color map from the section highlighted in (c) with the cosine of the phase overlaid to show the magnetic interaction between the islands; and (e) similar color induction and phase map computed from micromagnetic simulations.

Figure 2

(a) Magnetic induction color map reconstructed after the first demagnetization cycle, showing the formation of defect strings and two magnetic monopole defects with negative charge indicated by $\ominus$ in the field of view; (b) magnetic induction color map reconstructed after the second demagnetization cycle, showing the presence of many defect strings and three monopole defects with negative charge ($\ominus$) and two with positive charge ($\oplus$) in the field of view.

Figure 3

(a) A plot of normalized magnetization and monopole density (inset) as a function of applied field along the diagonal of the square lattice; (b) and (c) show the color-coded magnetic induction map from micromagnetic simulations at applied field values of $520$ and $580$ Oe.

Figure 4

The top row shows phase maps with color-coded magnetic induction for a neutral node surrounded by four vortices, and the bottom row shows the same for a node with a magnetic monopole defect of positive charge. (a) and (d) show simulated phase maps using the approximation of uniform magnetization for each element, (b) and (e) show the experimentally obtained high-resolution phase maps from the two regions circled in Fig. 2b, and (c) and (f) show simulated phase maps using the micromagnetic simulation data.