Indirect Exchange in 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 particularly simple form. Calculations are based on a two-valley model, for simplicity. The resulting coupling is used to study the ferromagnetic spin wave spectrum for a lattice of spins. The latter quantity is shown to be especially sensitive to the ratio of the Fermi wave number to the separation of valley minima.