Spin-Wave Impurity States in Ferromagnets

The effect of dilute impurities on the spin-wave spectrum of ferromagnetic insulators has been studied. The theory of spin-wave impurity states is developed from the Heisenberg exchange Hamiltonian using Green's function techniques. The low-lying, $s$-like, impurity states for cubic crystals are discussed and shown to depend only upon the unperturbed spin-wave density of states. Numerical results are obtained for simple cubic crystals. It is shown that localized states lying outside of the spin-wave band as well as virtual states which decay into the continuum can exist. The energy of the $s$-, $p$-, and $d$-like states is obtained as a function of the ratios of the impurity spin and effective exchange to those of the host atoms. The nature of the virtual states is discussed and a general expression for the width is obtained by considering the change in the density of states due to the impurities. Low-lying, $s$-like, virtual states with long lifetimes are found to exist, and it is suggested that these states may cause significant effects in the spin-wave specific heat and thermal conductivity of impure ferroinsulators.