Understanding magnetotransport signatures in networks of connected permalloy nanowires

The change in electrical resistance associated with the application of an external magnetic field is known as the magnetoresistance (MR). The measured MR is quite complex in the class of connected networks of single-domain ferromagnetic nanowires, known as “artificial spin ice,” due to the geometrically induced collective behavior of the nanowire moments. We have conducted a thorough experimental study of the MR of a connected honeycomb artificial spin ice, and we present a simulation methodology for understanding the detailed behavior of this complex correlated magnetic system. Our results demonstrate that the behavior, even at low magnetic fields, can be well described only by including significant contributions from the vertices at which the legs meet, opening the door to new geometrically induced MR phenomena.

Figure 1

(a) Scanning electron microscope image of an armchair orientation connected kagome artificial spin ice lattice. An external magnetic field, $H$, could be applied along any in-plane direction, denoted by the angle $\theta$ between the field direction and the nominal current direction, $I$. (b) Corresponding magnetic force microscope image. The black and white dots at the vertices are domain walls, indicative of the Ising-like behavior of the individual nanowires. (c) A schematic illustration of the lead arrangement used for transport studies. Large connective pads on each end supply an excitation current, while the thin nanowire leads along the long axis were used for voltage measurement. (d) A schematic illustration of the nanowire leg and vertex regions.

Figure 2

Experimental longitudinal (a) and transverse (c) magnetoresistance data, and corresponding simulated longitudinal (b) and transverse (d) magnetoresistance data. The down field sweeps (red) and up field sweeps (blue) are symmetric under field inversion. For viewing ease, all data except for $\theta =0^{\circ }$ have been vertically offset. Simulated data are for applied field angles at $90.2^{\circ }, {60}^{\circ }, {30}^{\circ }$, and $0.2^{\circ }$.

Figure 3

Pseudoangular tracking of transverse resistance values for identical field strengths at varying $\theta$. For each curve, the connected data were taken from sweeps of the magnetic field magnitude at different angles. The high-field data reveal the expected symmetry of the planar Hall effect, with extrema at ${45}^{\circ }$ and ${135}^{\circ }$. Inset: Zero-field transverse resistance values as a function of $\theta$. The remanent zero-field resistance value depends on the angle at which the field was applied, reflecting the effect of the vertex regions.

Figure 4

Transverse magnetoresistance during upwards field sweeps $(-10\mathrm{kOe}\to 10\mathrm{kOe})$ with small angular variations around $\theta ={90}^{\circ }$ showing both experimental (a) and simulation data (b). Note that the features are inverted above and below $\theta ={90}^{\circ }$, and that the results are well reproduced by the simulations, as described in the text. Deconstructing the simulation data into contributions from the nanowire legs (green dashed line) and vertices (green dotted line) reveals the critical importance of the vertices to the overall magnetoresistance.

Figure 5

Simulated magnetization and $y$-component electric field maps of a 17-hexagon armchair network. (a) The micromagnetic state at 800 Oe and $\theta =90.2^{\circ }$ after applying a $-4000\mathrm{Oe}$ saturating field at the same orientation. At this field, only a fraction of the nanowire leg moments have undergone magnetization reversal. (b) Electric field maps of the same state, generated as described in the text. (c) Expanded section of the magnetization map, showing that the vertex regions (circled) have different magnetization profiles that depend on the adjacent nanowire leg moments. (d) Expanded section of the electric field map. Note that the electric field profile of the vertex regions changes with the magnetization profile, while there is no change for the nanowire legs.