Magnetic response of brickwork artificial spin ice

We have investigated the response of brickwork artificial spin ice to an applied in-plane magnetic field through magnetic force microscopy, magnetotransport measurements, and micromagnetic simulations. We find that, by sweeping an in-plane applied field from saturation to zero in a narrow range of angles near one of the principal axes of the lattice, the moments of the system fall into an antiferromagnetic ground state in both connected and disconnected structures. Magnetotransport measurements of the connected lattice exhibit unique signatures of this ground state. Also, modeling of the magnetotransport demonstrates that the signal arises at vertex regions in the structure, confirming behavior that was previously seen in transport studies of kagome artificial spin ice.

Figure 1

A schematic of (a) disconnected and (b) connected brickwork artificial spin ice. The moment of each island and leg can be treated as a giant uniaxial spin, as indicated by the arrows, where colors correspond to directions. (c) Each junction of the connected lattice can be divided into a “leg” region (grey) and a “vertex” region (blue), the definition of which is used in understanding our magnetotransport simulations.

Figure 2

(a) Scanning electron microscopy (SEM) image of a portion of disconnected brickwork artificial spin ice. The angle between the applied field and the $x$ axis is defined as θ. (b) SEM image of connected brickwork artificial spin ice. The angle between the applied field and the excitation current direction is defined as θ. (c) A schematic of the leads for magnetotransport measurement.

Figure 3

MFM images of (a) disconnected and (b) connected brickwork artificial spin ice in their ground states in zero magnetic field. For the disconnected islands, a black dot represents a magnetic north pole and a white dot represents a magnetic south pole for the islands. In the image for the connected lattice, two-north/one-south vertices appear as black dots, and one-south/two-north vertices appear as white dots.

Figure 4

The geometry of the structure used for micromagnetic simulation. For the simulated longitudinal signal, local electric field vector was integrated along the red path (1 to 2) to obtain the simulated voltage. For the simulated transverse signal, the same calculation was performed along the blue path (3 to 4). The line integral from point 5 to point 6, which included the horizontal elements, did not give a qualitatively different result than the one from point 3 to point 4.

Figure 5

The comparison of the simulated transverse signal for two different paths. The black curve was obtained by doing the line integral from point 5 to 6 in Fig. 4, whereas the red curve was obtained by doing the integral from point 3 to 4.

Figure 6

MFM images of (a) disconnected and (b) connected brickwork artificial spin ice after applying the field of 3500 Oe at $2^{\circ }$. The images were taken in zero-field. (c) A schematic that shows the magnetization configuration for the MFM image.

Figure 7

The field dependence of the simulated energy difference between the states obtained by micromagnetic relaxation from the ground state and from the polarized state. ΔE is defined as the energy obtained after relaxation from the ground state subtracting the energy obtained after relaxation from the polarized state, i.e., $\mathrm{\Delta }\mathrm{E}={\mathrm{E}}_{\mathrm{GS}}-{\mathrm{E}}_{\mathrm{PS}}$. (a) Disconnected lattice, and (b) Connected lattice.

Figure 8

(a) Longitudinal magnetoresistance (MR) plots of brickwork artificial spin ice at selected angles. (b) Corresponding simulation data for longitudinal MR. From the top, the MR plots are for $\theta =1.5^{\circ },\theta ={30}^{\circ },\theta ={60}^{\circ }$, and $\theta =91.5^{\circ }$, respectively. Plots except for $\theta =91.5^{\circ }$ have been shifted upwards for viewing ease. The insets show schematics of the magnetization configurations at different points in the field sweeps.

Figure 9

(a) Transverse MR plots of brickwork artificial spin ice at selected angles. (b) Corresponding simulation data for transverse MR. From the top, the MR plots are for $\theta =1.5^{\circ },\theta ={30}^{\circ },\theta ={60}^{\circ }$, and $\theta =91.5^{\circ }$, respectively. Plots except for $\theta =91.5^{\circ }$ have been shifted upward for viewing ease. The insets show schematics of the magnetization configurations at different points in the field sweeps.

Figure 10

Nanowire leg and vertex contributions to the simulated transverse resistance, obtained by integrating only over portions of the path. Note the significant contributions from the vertex regions at every angle.

Figure 11

(a) Longitudinal MR plots at closely spaced angles around $\theta =0^{\circ }$. (b) Corresponding simulation data for longitudinal MR. Plots except for $\theta =-1.5^{\circ }$ have been shifted upward for viewing ease. The insets show schematics of the magnetization configurations at different points in the field sweeps.

Figure 12

(a) Experimental and (b) simulated transverse MR during down sweep in small angular steps around $\theta =0^{\circ }$. The shape of the plot abruptly changes as the angle approaches closer to $0^{\circ }$. There is also an inversion symmetry between the plots above and below $0^{\circ }$, associated with the reversal of polarization of the moments aligned with $\theta ={90}^{\circ }$.