Fig. 3: Control of magnetism via modulation of the substrate dielectric constant ϵ_s.

a Schematic view of a hexagonal array of cylindrical nanoparticles placed at a distance d from the substrate. The semiconducting substrate can be tuned from being an insulator to a conductor via an applied voltage. The corresponding change in its static dielectric constant alters the polariton modes that mediate the spin–spin interaction. b Dependence of the critical temperature T_c of the unconventional ferromagnetic state on d when the substrate is an electrical insulator (ϵ_s = 2.5; orange line, ϵ_s = 10; blue line) or a conductor (black line).