Long-range order in arrays of composite and monolithic magneto-toroidal moments

Magneto-toroidal order, also called ferrotoroidicity, is the most recently established type of ferroic state. It is based on a spontaneous and uniform alignment of unit-cell-sized magnetic whirls, called magneto-toroidal moments, associated with a macroscopic toroidization. Because of its intrinsic magnetoelectric coupling, this new ferroic state could be useful in the development of spintronic devices. We exploit two-dimensional periodic arrays of magnetostatically coupled nanomagnets as model systems for the investigation of long-range magneto-toroidal order. We present two pathways promoting this order, namely (i), structures comprising a ring of uniformly magnetized sub-micrometer-sized bar magnets and (ii), structures in which each magnetic building block itself hosts a magnetic vortex. For both cases calculations of the magnetic-dipole interaction and micromagnetic simulations reveal the conditions for the formation of spontaneous magneto-toroidal order. We confirm this order and the formation of magneto-toroidal domains in our arrays by magnetic force microscopy. We identify the presence of two types of domain-wall states emerging from the competition of two intrinsic microscopic couplings. Our work not only identifies the microscopic conditions promoting spontaneous magneto-toroidal order but also highlights the possibility tailor mesoscale magnetic arrays towards elusive types of ferroic order.