Spin waves in disordered FeCo magnonic crystal at finite temperatures

We study theoretically the influence of the temperature and disorder on the spin-wave spectrum of the magnonic crystal Fe$_x$Co$_{1-x}$. Our formalism is based on the analysis of Heisenberg Hamiltonian with the exchange integrals obtained from the \textit{ab initio} magnetic force theorem by means of vector and frequency dependent transverse magnetic susceptibility. The coherent potential approximation is employed to treat the disorder, and random phase approximation in order to account for the softening of the magnon spectrum at finite temperatures. The alloy turns out to exhibit many advantageous properties for spintronic applications. Apart from high Curie temperature, its magnonic bandgap remains stable at elevated temperatures and is largely unaffected by the disorder. We pay particular attention to the attenuation of magnons introduced by the alloying. The damping turns out to be a non-monotonic function of the impurity concentration due to the non-trivial evolution of exchange integrals with the Co concentration. The disorder induced damping of magnons is estimated to be much smaller than their Landau damping.