Symmetry-broken magnetotoroidal artificial spin ices: Magnetization states and dynamics

Magnetotoroidal artificial spin ices (MT-ASIs) are arrangements of nanomagnets that exhibit spontaneous toroidization. A ferrotoroidic order could have implications on the propagation of spin waves through this artificial spin ice, including the development of topological edge modes. Here, we numerically investigate the magnetization dynamics of an MT-ASI with and without spatial symmetry breaking. Through micromagnetic simulations, we compute the energies and ferromagnetic resonance spectra of the four lowest-order states, which exhibit ferrotoroidicity, antiferrotoroidicity, and no toroidicity. As expected, we find that the resonant modes split when spatial symmetry is broken. To determine whether our system exhibits topologically protected edge modes, we perform semianalytical calculations to first estimate the ferromagnetic resonance and then compute the band structure. Our results show that symmetry-broken MT-ASIs are reconfigurable by magnetic field protocols, and that their band structures depend on magnetization state. Calculation of the Chern number indicates that the bands are topologically trivial in all cases, suggesting that the dynamic magnetic coupling is weak. The absence of a nonzero Chern number suggests that the dynamic coupling in ASIs is too weak to unlock their full potential in magnonics applications.