Toroidal Moments in Confined Nanomagnets and their Impact on Magnonics

The nonreciprocity created by dipolar coupling, electric currents, and Dzyaloshinskii-Moriya interactions is discussed in cases where the magnon propagation direction has a component parallel to the toroidal moment. A criterion for calculating the toroidal moments is established, addressing the issue of correct origin selection by considering compensated and uncompensated magnetization distributions. This criterion is then applied to various nonreciprocal magnetic systems, with the calculations consistent with those reported in the literature and predicting the existence of nonreciprocity in a more general manner. These results broaden the physical significance of the toroidal moment and facilitate the identification and estimation of nonreciprocity in magnonic systems. This work also clarifies the interrelations between different definitions of the toroidal moment for confined structures, where a surface term arising from surface-bound currents connects these definitions without the need for time-averaging. Comparing these definitions of the toroidal moment applied to different magnetic textures demonstrates that they are always parallel but may differ in magnitude and sign. The discrepancy in the different definitions is deemed irrelevant since its direction, rather than its magnitude, primarily predicts the existence of magnon nonreciprocity.