  Artificial square spin ices are structures composed of magnetic elements
arranged on a geometrically frustrated lattice and located on the sites of a
two-dimensional square lattice, such that there are four interacting magnetic
elements at each vertex. Using a semi-analytical approach, we show that square
spin ices exhibit a rich spin wave band structure that is tunable both by
external magnetic fields and the configuration of individual elements. Internal
degrees of freedom can give rise to equilibrium states with bent magnetization
at the edges leading to characteristic excitations; in the presence of
magnetostatic interactions these form separate bands analogous to impurity
bands in semiconductors. Full-scale micromagnetic simulations corroborate our
semi-analytical approach. Our results show that artificial square spin ices can
be viewed as reconfigurable and tunable magnonic crystals that can be used as
metamaterials for spin-wave-based applications at the nanoscale.
