  In this paper, we present a theoretical formulation of magnetization dynamics
in disordered binary alloys based on Kubo linear response theory interfaced
with the combination of seamlessly three approaches; density functional based
tight-binding linear muffin-tin orbitals, generalized recursion and Augmented
space formalism. We apply this method to study the magnetization dynamics in
chemically disordered Fe$_x$Co$_{1-x}$ ($x$ = 0.2, 0.5, 0.8) alloys. We
reported that the magnon energies decrease with an increase in Co
concentration. Significant magnon softening has been observed in
Fe$_{20}$Co$_{80}$ at the Brillouin zone boundary. The magnon-electron
scattering increases with increasing Co content which in turn modifies the
hybridization between the Fe and Co atoms. This reduces the exchange energy
between the atoms and soften down the magnon energy. The lowest magnon lifetime
in found in Fe$_{50}$Co$_{50}$, where disorder is maximum. This clearly
indicates that the damping of magnon energies in Fe$_x$Co$_{1-x}$ is governed
by the hybridization between Fe and Co whereas the magnon lifetime is
controlled by disorder configuration. Our atomistic spin dynamics simulations
show a reasonable agreement with our theoretical approach in magnon dispersion
for different alloy compositions.
