Thermal Properties of Spin-Wave Impurity States

The effect of dilute magnetic impurities on the thermal properties of an ideal simple cubic spin-½ Heisenberg ferromagnet has been investigated using the thermal-Green's-function procedure with a simple randomphase decoupling scheme. It is shown that for a small ratio $\epsilon$ of impurity-host to host-host exchange, low-lying "$s$-type" virtual spin-wave states result which cause a large density of states to occur at low energies. These low-energy states lead to an accumulation of spin disorder at and near the impurity site. Consequently the impurity magnetization decreases far more rapidly than that of the host. This effect is accompanied by a large increase in the low-temperature spin-wave specific heat. Analytic solutions to the Green's-function equations are calculated for temperatures near 0 and near $T_c$ the Curie temperature. Self-consistent numerical solutions are presented for both the magnetization and the spin-wave specific heat as a function of temperature. For small $\epsilon$ the impurity magnetization is approximated by the Brillouin function $m_I={\mu }_{\beta }tanh\{{\mu }_{\beta }H+\frac{6J^{\prime }〈{S_1}^z〉}{\mathrm{kT}}\},$ where $J^{\prime }$ is host-impurity exchange and $〈{S_1}^z〉$ is the thermal average for the impurity nearest-neighbor spins. $〈{S_1}^z〉$ is found to be depressed from the bulk value by an amount which increases with temperature and is about 0.84 of the bulk value as $T\to T_c$.