The low-temperature properties of dilute Ruderman-Kittel-Kasuya-Yosida (RKKY)-coupled spin systems have been investigated by computer simulation. Our results are based upon classical "ground states" for finite systems of spins distributed randomly in a cubic volume of fcc lattice, generated by an algorithm which lowers the system energy at each step. The (Heisenberg) RKKY exchange interaction is taken to be of the form appropriate to local moments in noble-metal hosts. In studying the number of such states which occur for a single system of 172 spins, we found only seven independent equilibrium configurations (EC's) among 70 which were generated from different random starting orientations. EC's are found to possess short-range angular correlation, such that a suitably defined correlation volume contains an average of ∼ two neighbor spins. The latter result is found to be independent of concentration, as expected for power-law range functions. The distribution $P_0\mathrm{}\left(H\right)\mathrm{}$ of exchange fields for EC's is compared with an analytic expression derived for randomly oriented spins in the dilute limit. The effect of ordering is to broaden the distribution by ∼ 50% and to diminish the number of spins with very small fields. Quantum equations of motion for small oscillations about equilibrium are derived in a spin-wave approximation and are shown to be equivalent to their classical counterpart. Other features of the spin-wave picture are discussed. Low-lying modes related to rotational symmetry are discussed in detail in the Appendix. Calculated spectra are exhibited for systems of 96 and 172 spins and are found to scale correctly with concentration. Both local and dipolar anisotropy are found to introduce a zero-frequency gap in the excitation spectrum; the classical dipolar term is, however, too small to give an appreciable effect. A similar zero-frequency gap results from the application of a uniform field. Treating the excitations as bosons leads to a successful interpretation of specific-heat data for $\mathrm{Cu}\mathrm{Mn}$. For the case of $\mathrm{Au}\mathrm{Fe}$ general agreement is also found, but with a discrepancy in form attributable to the Kondo effect. Damping of the RKKY interaction is found to cause a simple scaling of the excitation spectrum to lower frequencies. These results are used to interpret the concentration variation of the specific heat and the spin-glass ordering temperature for $\mathrm{Cu}\mathrm{Mn}$. We also employ EC's of 96 and 188 spins to calculate the zero-temperature reversible susceptibility, yielding good agreement with $\mathrm{Cu}\mathrm{Mn}$ data (with $S=\frac{5}{2}$) and $\mathrm{Au}\mathrm{Fe}$ data (with $S=\frac{3}{2}$).

• Received 21 April 1980

DOI:https://doi.org/10.1103/PhysRevB.22.3816