On the magnetic ordering implemented by the s–d exchange interaction

The RKKY perturbation scheme for the spin 12 s–d exchange model is reexamined in the case of a finite number of electrons per impurity. The distribution of impurities is assumed to be random. The free electron gas representing the unperturbed system in the RKKY approach is replaced by a mean-field system hr asymptotically equivalent to the reduced s–d model Hr which includes part of the s–d exchange. Below the transition temperature Tc of hr the lowest energy levels of the s–d exchange Hamiltonian HK resulting from second-order perturbation theory prove to be ordered in the same manner as those of hr in the limit of weak s–d exchange, favouring the same definite parallel alignment of any pair of impurity spins. It follows therefore that in the low temperature, weak coupling regime, the s–d model exhibits ferromagnetic ordering of impurities. At temperatures above Tc there is no ordering and impurity–impurity exchange has the RKKY form. These observations are consistent with the presence of a low-temperature ferromagnetic phase in numerous compounds with s–d exchange interaction.