Band structure and Wannier-Stark ladders in the spin wave spectrum

Two models of Hamiltonians are proposed to describe the propagation of magnetic disturbances within materials whose form is of long and narrow film strips. Both models are defined in terms of special Heisenberg Hamiltonians where it is assumed that the interaction of the spins of the electrons with the structure of the material can be replaced in first approximation by a Hamiltonian of that type. The first one is a one-dimensional periodic Heisenberg Hamiltonian applicable to describe the transport of spin waves in magnonic crystals and it is shown that the associated Schrodinger equation supports analytical solutions and closed expressions are obtained for the dispersion relation and the density of states. This Hamiltonian of course leads to a structure of bands in the spectrum of energies. However, several essential differences are found with respect to the band structure of the well-known Kronig-Penney model. The second model is a Hamiltonian, in principle periodic, but to which a gradient has been added that alters the interaction between the spins. This Hamiltonian only supports numerical solution. This model is applicable to the transport of spin waves in magnetic crystals in the presence of some external agent. For example a temperature gradient, exposure to some non-uniform luminous signal, a non-uniform chemical attack, etc., which modifies the constant interactions from spin to spin. It is shown that in this case the original band structure is completely destroyed and under certain circumstances it appears a series of resonances which are similar to the Wannier-Stark ladder resonances.