Indirect Exchange in Many-Valley Semiconductors

The effect of intervalley transitions on the indirect exchange interaction in magnetic semiconductors is considered. Using second-order perturbation theory, we derive an expression for the spin-spin coupling energy in terms of the susceptibility χ(R₁₂) for one valley. When the spins are located on lattice sites, and a local s-f exchange interaction is used, the coupling energy reduces to a especially simple form. Calculations are based on a two-valley model, for simplicity. The resulting coupling is used to study the paramagnetic Curie temperature and ferromagnetic spin-wave spectrum for a lattice of spins. The latter quantity is shown to be especially sensitive to the ratio of Fermi wavenumber to the separation of valley minima. The paramagnetic Curie temperature θ is shown to be independent of the direction and of the valley separation. Using the number of valleys as a parameter, we have fit the variation of θ with concentration in Eu_xGd_1-xSe and Eu_xGd_1-xTe. In this way we obtain four for the valley number in Se and sixteen for the case of Te.