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A Cayley-Sudoku table of a finite group G is a Cayley table for G subdivided into uniformly sized rectangular blocks, in such a way that each group element appears once in each block. They were introduced by J. Carmichael, K. Schloeman, and M. B. Ward [CSW], who also gave three ways to construct them. This paper has four aims. First, we review Constructions 1 and 2 of [CSW] and uncover their unexpected heritage in the work of R. Baer and J. Denes. Next we turn to some new instances of Construction 2 inspired by Baer, which answer an open question in [CSW]. Third, we provide a very brief outline of recent reinventions of special cases of Construction 1 in the literature. We conclude with an invitation to seek out the heritage of Construction 3. Portions of this paper appear in: Kady Hossner Boden and Michael B. Ward (2019) A New Class of Cayley-Sudoku Tables, Mathematics Magazine, 92:4, 243-251, DOI: 10.1080/0025570X.2019.1613949

A Cayley-sudoku table of a finite group G is a Cayley table for G, the body of which is partitioned into uniformly sized rectangular blocks in such a way that each group element appears exactly once in each block. A Cayley-sudoku table is pandiagonal magic provided the blocks are square and the "sum" (using the group operation) of elements in every row, column, broken diagonal, and broken antidiagonal in each block gives the group identity. We provide sufficient conditions for a group to have a pandiagonal magic Cayley-sudoku table and give some examples.

Chiral domain walls in ultrathin perpendicularly magnetised layers have a N\'{e}el structure stabilised by a Dzyaloshinskii-Moriya interaction (DMI) that is generated at the interface between the ferromagnet and a heavy metal. Different heavy metals are required above and below a ferromagnetic film in order to generate the structural inversion asymmetry needed to ensure that the DMI arising at the two interfaces does not cancel. Here we report on the magnetic properties of epitaxial Pt/Co/Au$_x$Pt$_{1-x}$ trilayers grown by sputtering onto sapphire substrates with 0.6 nm thick Co. As $x$ rises from 0 to 1 a structural inversion asymmetry is generated. We characterise the epilayer structure with x-ray diffraction and cross-sectional transmission electron microscopy, revealing (111) stacking. The saturation magnetization falls as the proximity magnetisation in Pt is reduced, whilst the perpendicular magnetic anisotropy $K_\mathrm{u}$ rises. The micromagnetic DMI strength $D$ was determined using the bubble expansion technique and also rises from a negligible value when $x=0$ to $\sim 1$ mJ/m$^2$ for $x = 1$. The depinning field at which field-driven domain wall motion crosses from the creep to the depinning regime rises from $\sim 40$ to $\sim 70$ mT, attributed to greater spatial fluctuations of the domain wall energy with increasing Au concentration. Meanwhile, the increase in DMI causes the Walker field to rise from $\sim 10$ to $\sim 280$ mT, meaning that only in the $x = 1$ sample is the steady flow regime accessible. The full dependence of domain wall velocity on driving field bears little resemblance to the prediction of a simple one-dimensional model, but can be described very well using micromagnetic simulations with a realistic model of disorder. These reveal a rise in Gilbert damping as $x$ increases.

The coupling of electron spin to real-space magnetic textures leads to a variety of interesting magnetotransport effects. The skyrmionic spin textures often found in chiral B20-lattice magnets give rise, via real-space Berry phases, to the topological Hall effect, but it is typically rather small. Here, B20-ordered Fe$_{0.7}$Co$_{0.3}$Si epilayers display a giant topological Hall effect due to the combination of three favourable properties: they have a high spin-polarisation, a large ordinary Hall coefficient, and dense chiral spin textures. The topological Hall resistivity is as large as 820 n$\Omega$cm at helium temperatures. Moreover, we observed a drop in the longitudinal resistivity of 100 n$\Omega$cm at low temperatures in the same field range, suggesting that it is also of topological origin. That such strong effects can be found in material grown in thin film form on commercial silicon wafer bodes well for skyrmion-based spintronics.