Multiple energy scales in mesospin systems: The vertex-frustrated Saint George lattice

The interplay between topology and energy hierarchy plays a vital role in the collective magnetic order in artificial ferroic systems. Here we investigate, experimentally, the effect of having one or two activation energies of interacting Ising-like magnetic islands—mesospins—in thermalized, vertex-frustrated lattices. The thermally arrested magnetic states of the elements were determined using synchrotron-based magnetic microscopy after cooling the samples from temperatures above the Curie temperature of the material. Statistical analysis of the correlations between mesospins across several length scales reveals changes in the magnetic order, reflecting the amount of ground state plaquettes realized for a vertex-frustrated lattice. We show that the latter depends on the presence, or not, of different activation energies.

Figure 1

Illustration of the Saint George lattices. (a) The Saint George (SG) lattice with its two different island (mesospin) sizes. (b) The modified Saint George (mSG) lattice has only one island size. In the Saint George lattice geometry there are two different horizontal chains, colored green and yellow. These horizontal chains are connected by vertical islands. The green island chains are referred to as long-mesospin chains in the main text, as they are consisting of the long islands in the SG lattice and have double the amount of islands in the mSG. The yellow island chains are equal in the SG and mSG lattice.

Figure 2

Photoemission electron microscopy images using the x-ray magnetic circular dichroism effect to visualize the magnetization direction. Combined measurements of (a) the SG lattice and (b) of the mSG lattice. The magnetization direction of each individual island can be determined by its color. Black illustrated mesospins pointing left or up while white mesospins have their magnetization pointing right or down.

Figure 3

Magnetic order of long-mesospin chains in the SG and mSG lattice. (a) Correlation between neighboring long mesospins in the SG lattice shows an almost random alignment for the first mesospin neighbor, the slight negative correlation value indicates a preference towards antiferromagnetic order. (b) The same mesospin chains are also investigated in the mSG, where two short mesospins resemble one long mesospin. Depending on the starting element ($★$ or $+$), the first mesospin neighbor has a positive value ($★$) and therefore shows a preference for ferromagnetic alignment, or a negative value ($+$) and therefore a slight preference for antiferromagnetic alignment.

Figure 4

Vertex statistics for the SG and mSG lattice. (a) The degeneracy corrected vertex abundance for the fourfold coordinated vertices in the SG (blue) and mSG (red) lattice. (b) The degeneracy corrected vertex abundance for the threefold coordinated vertices in the SG (blue) and mSG (red) lattice. The gray bars illustrate the vertex abundance for a random distribution as it would be received if the mesospins would not interact. The vertex type energies are corresponding to their notation, namely type ${\mathrm{I}}_{4\left(3\right)}$ being the lowest energy and therefore the ground state for the fourfold (threefold) coordinated vertex.

Figure 5

Mesospin circulation in the SG and mSG lattice. The circulation for loops of five mesospin in the SG lattice (blue), compared to the circulation for six-mesospin loops in the mSG lattice (red) are shown in the left graph. The circulation of the loops was normalized to $+1$ ($-1$) for a total clockwise (anticlockwise) alignment of all mesospins of a loop. This normalization results in a change of the loop circulation by $\pm 0.4$ for each mesospin flipped in the SG lattice and by $\pm \frac{1}{3}$ in the mSG lattice. The right graph illustrates the degeneracy corrected abundance of flux closure loops and higher energy states for both the SG and mSG.

Figure 6

Proposed ground state configuration in the SG (left) and mSG lattice (right). This ground state configuration takes into account only the vertex energies, while ignoring the different activation energies for short and long mesospins. All fourfold-coordinated vertices are in their lowest energy state. This configuration leads to a frustration among the threefold-coordinated vertices, as 50% of them are in their lowest energy state but the other 50% are in their first excited state, equivalent to the Shakti lattice [12]. The proposed ground state creates flux closure loops, as unit cells of the ground state manifold. The four different flux closure loops are colored orange or green, depending on their sense of flux circulation (sense of rotation). The border mesospins, eight in the SG and ten in the mSG lattice, are head to tail aligned, with an either clockwise or anticlockwise sense of rotation. The nonborder mesospin is frustrated and located in the center of the flux closure loop. The magnetic orientation of this island has no impact on the overall energy of the entire lattice, but it defines the position of the excited threefold-coordinated vertex.

Figure 7

Mapping of ground state unit cells in the SG and mSG lattice. The ground state unit cells introduced in Fig. 6 are mapped out experimentally in our arrays. There is a clear difference in the quantity of ground state unit cells found in the SG and the mSG lattice. A 22% ground state realization is observed in the SG lattice, with a 10% realization in the mSG lattice.

Figure 8

Conditional arrangement of the ground state unit cells in the SG and mSG lattice. The two long mesospins from subsequent long-mesospin chains are antiferromagnetically aligned to each other. In this conditional mesospin arrangement we compare the abundance of the ground state unit cells in the SG and mSG lattice and see a clear evidence of a stronger ordering in the SG lattice.