Microscopic dipole-exchange theory for planar nanostriped magnonic crystals

Microscopic (or Hamiltonian-based) calculations are reported for the collective spin-wave bands and gaps in one-dimensional magnonic crystals consisting of a periodic array of ferromagnetic stripes separated by nonmagnetic spacers. This is achieved by generalizing a previous approach for the dipole-exchange spin waves in individual (or non-interacting) stripes and small finite arrays of nanoelements to cases where there is an effectively infinite periodic array of striped elements. This involves introducing a Bloch wavenumber associated with the periodicity property and reformulating the microscopic dipole-dipole and exchange sums for a periodic structure to include the inter-stripe and intra-stripe contributions. The theory gives good agreement when compared with Brillouin light scattering data for Permalloy magnonic structures. Applications of the microscopic theory are also made to cases where the applied magnetic field has a component transverse to the easy axis of the stripes, favouring the formation of edge modes.