Heisenberg pseudo-exchange and emergent anisotropies in field-driven pinwheel artificial spin ice

Rotating all islands in square artificial spin ice (ASI) uniformly about their centers gives rise to the recently reported pinwheel ASI. At angles around ${45}^{\circ }$, the antiferromagnetic ordering changes to ferromagnetic and the magnetic configurations of the system exhibit near degeneracy, making it particularly sensitive to small perturbations. We investigate through micromagnetic modeling the influence of dipolar fields produced by physically extended islands in field-driven magnetization processes in pinwheel arrays and compare the results to hysteresis experiments performed in situ using Lorentz transmission electron microscopy. We find that magnetization end states induce a Heisenberg pseudoexchange interaction that governs both the interisland coupling and the resultant array reversal process. Symmetry reduction gives rise to anisotropies and array-corner mediated avalanche reversals through a cascade of nearest-neighbor (NN) islands. The symmetries of the anisotropy axes are related to those of the geometrical array but are misaligned to the array axes as a result of the correlated interactions between neighboring islands. The NN dipolar coupling is reduced by decreasing the island size and, using this property, we track the transition from the strongly coupled regime towards the pure point dipole one and observe modification of the ferromagnetic array reversal process. Our results shed light on important aspects of the interactions in pinwheel ASI and demonstrate a mechanism by which their properties may be tuned for use in a range of fundamental research and spintronic applications.

Figure 1

ASI geometries considered. (a) A single island, (b) two islands forming a ‘T shape,’ (c) a pinwheel ‘unit,’ and larger (d) ‘asymmetric’ and (e) ‘symmetric’ pinwheel arrays. Square ASI (f) ‘vertex’ and (g) ‘unit.’ All ‘units’ and ‘vertices’ are referred to as the latter in the main text. In (a)–(e) the red islands are those added to the previous structure to generate the current one. The solid and dashed crosses in (d) and (e) show the centers of the subarrays with islands outlined in the corresponding lines. These sets of islands form subarrays of $n\times n, (n-1)\times n$, or $n\times (n-1)$ islands, as annotated. The blue ‘$C_x$’ fonts show the 2D rotational symmetry of the geometry. The same parameter is displayed in brackets when the field process increases the symmetry of the switching field. The inset to (g) shows the applied field orientation definition and the simulation grid axes. The overlays in the bottom island in (f) show the regions used to track the end states of each island. The annotations in (d) and (e) refer to the subarray repeat shapes for the two edge symmetry types.

Figure 2

Single island magnetic properties with $H$ applied at ${45}^{\circ }$. (a) $M-H$ loop, showing magnetization configurations at key points (i)–(iii) and the switching field by the annotation. The solid (dashed) lines show the increasing (decreasing) field segments, as indicated by the arrows. (b) Time resolved end-state mediated magnetization reversal. The last panel depicts the steady state configuration. The stray field is plotted on a logarithmic scale. The color wheel in (a) represents the orientations and relative magnitudes of the vectors. The white arrows within the islands are sketches to aid the eye and are not quantitative.

Figure 3

Magnetic property angular dependence of a single island. (a) Applied field angle dependence of $H_s$ for a vertical and horizontal island. (b) Vertical island end-state phase diagram with $H_s\left(\theta \right)$ superimposed (line). For all pure states, increasing lightness marks a stronger component perpendicular to the long axis of the island, i.e., a stronger end state. The color bars are normalized for each end state between zero and the maximum value. (c) Example end-state schematics. The arrows in (a) and (b) represent the field directions at positive (top) and negative (bottom) fields.

Figure 4

Magnetic properties of the T-shaped geometry. (a) $H_s\left(\theta \right)$ astroids. The thin lines in the inset to (a) are the equivalent lines for the isolated islands of Fig. 3. (b) Vertical and (c) horizontal island end-state phase diagrams. The color maps in (b) and (c) are the same as those presented in Fig. 3. The solid lines in the phase diagrams are the superimposed $H_s\left(\theta \right)$ profiles of the highlighted island. The dashed lines are for the other island highlighted in (a).

Figure 5

Time evolution of the magnetization and stray field of the two-island T-shaped geometry during reversal at the anisotropy axis of $38.21\pm 0.{01}^{\mathrm{o}}$, showing the synchronized reversal. The field was increased (by $25\mu \mathrm{T}$) to just above the switching field. The stray field magnitude is presented on a logarithmic scale. The last panel depicts the equilibrium configuration.

Figure 6

(a) $H_s$ astroids and (b) end-state phase diagram for selected islands in a pinwheel vertex. The color maps in the phase diagram are the same as those presented in Fig. 3. The solid line in the phase diagram is the superimposed $H_s\left(\theta \right)$ profile of the highlighted island. The dashed line is for the other island highlighted in (a). (c) Pinwheel magnetization and stray field during reversal at the anisotropy axis after increasing the field (by $25\mu \mathrm{T}$) to just above the switching field. The stray field magnitude is presented on a logarithmic scale.

Figure 7

Magnetization and stray field maps just before reversal of (a) and (b) the full T-shaped array and (d)–(g) the full pinwheel vertex, drawn with all configurations of one island (gray) being removed. The end-state configurations are sketched in the top panels of (c) and (h), while the average induction from all islands (excluding the removed one) in the area of the removed island is shown by the blue arrows in the bottom of the same panels. The applied field angle is ${45}^{\circ }$. The red arrows indicate the direction of the anisotropy axes, with the angles drawn to scale. The stray field is plotted on a logarithmic scale and uses the same color wheel as the magnetization, shown in the inset to (b).

Figure 8

(a) The angle of the anisotropy axis of pinwheel geometries with symmetric and asymmetric edges as a function of the number of islands. For clarity, only the first few island shapes are drawn. (b) Dependence of switching field on array size. The resolution of the simulations is $0.{25}^{\circ }$ and $25\mu \mathrm{T}$. Periodic boundary conditions (PBC) were employed for the infinite array. Five repeats on each side in $x$ and $y$ of a 64 island pattern was used, with the split down the centers of the islands at the array edges. (c) $H_s\left(\theta \right)$ astroids of the $4\times 4$ array, uncoupled islands, and the PBC array in a small angular range centered on $\theta ={45}^{\circ }$. The yellow islands in (a) are the the first to reverse. The arrows in (a) and (b) are for (green) an uncoupled array (identical to single islands) and (magenta) an infinite array.

Figure 9

Time evolution of the magnetization and stray field during reversal of the smallest (a) asymmetric $(4\times 4)\times (4\times 4)$ and (b) symmetric $(5\times 4)\times (4\times 5)$ arrays to show asymptotic characteristics at their respective anisotropy axes. The field was increased (by $25\mu \mathrm{T}$) to just above the switching field. The stray field magnitude is presented on a logarithmic scale. The last panels in each row depict the equilibrium configuration after reversal. The arrows show the applied field direction.

Figure 10

(a) Experimental reversal pattern for a large asymmetric array at the anisotropy axis during a decreasing field sweep. (b) Island switching time map for simulations of the asymmetric $(4\times 4)\times (4\times 4)$ array in Fig. 9, during an increasing field sweep. (c) Average time between row reversals and (d) standard deviation in switching time along rows of the array in (b). The data in (c) and (d) are from islands in the third column to the last column of (b). The array is rotated clockwise by ${45}^{\circ }$ for display.

Figure 11

Magnetization and stray field for the $(4\times 4)\times (4\times 4)$ (left column) and $(5\times 4)\times (4\times 5)$ (right column) arrays just before reversal for different models (in rows): (a) and (d) micromagnetic, (b) and (e) uniformly magnetized extended island, and (c) and (f) point dipole, showing the strong influence of the end states on the stray field distribution. The magnetization and stray field configurations in (a) and (d) are identical to those in the first columns of Figs. 9 and 9, respectively. The stray field magnitude is presented on a logarithmic scale. The color map is shown in the inset to (f).

Figure 12

$H_s\left(\theta \right)$ astroids for periodic arrays of islands of different in-plane sizes (a) original, (b) half, (c) quarter, all on lattices of the original spacing, showing reduction in the angular range of the strongly coupled regime. The angular resolution was $0.{25}^{\circ }$. The insets show example islands on a common scale. The in-plane grid size of the simulations was 2.5 nm in (c) and 5 nm otherwise.

Figure 13

Island switching time during time evolution of the asymmetric array formed by two interleaved $4\times 4$ subarrays of quarter sized islands $(42\mathrm{nm}\times 117\mathrm{nm}\times 10\mathrm{nm})$, at applied field angles of (a) ${45}^{\circ }$ and (b) ${44}^{\circ }$. The field was increased (by $25\mu \mathrm{T}$) to just above the switching field in (a), and was increased at a rate of $1\mu \mathrm{T}/10\mathrm{ps}$ in (b) from two different points in the field sweep. The arrays are rotated ${45}^{\circ }$ clockwise for display. Each island is drawn $3\times$ the real size for clarity. Real-scale islands are shown in Supplemental Material Fig. S16 [32].