Spin waves and the anomalous Hall effect in ferromagnetic (Ga,Mn)As

The (Ga,Mn)As systems near the insulator-metal transition, where themagnetic order starts to set in, are sometimes regarded as an example ofthe double-exchange ferromagnetism in the (III,Mn)V literature [88]. Inthis regime, the Mn acceptor states form an impurity band with mixed sp–dcharacter. Hopping with the impurity band allows the electric conductionand the exchange coupling of Mn ions.4.2.4 RKKY interactionIndirect exchange interaction can act over large distances when mediated byfree carriers. The interactions of local moments embedded into the carriergas are then described by the Ruderman–Kittel–Kasuya–Yosida (RKKY)theory [42]. It considers the perturbation of the carriers by the magneticmoments and treats the system by second-order perturbation theory. Thelocal moments create wave-like perturbations, similar to a body droppedinto water (Fig. 4.3), leading to an interaction oscillating with the distancebetween them.J(r)rFigure 4.3: The RKKY interaction: mechanical analogy (a diving duck making acircular wave on the water) and distance dependence [107].In the simplest case of magnetic ions in metals coupled by conductionelectrons, the strength of the RKKY interaction can be expressed asJ(r) = − ρ(E F)k 3 F J2 02πsin(2k F r)−2k F rcos(2k F r)(2k F r) 4 , (4.1)where ρ(E F ) and k F are the density of states at the Fermi level and theFermi wavevector of the carriers, J 0 is the exchange coupling between thecarriers’ and ions’ spins and r is the average distance between the magneticions. As follows from the above equation, the RKKY coupling can be eitherferromagnetic or antiferromagnetic, depending on the separation betweenthe magnetic moments, and tends to vary in space on the length scale of thecarriers’ Fermi wavelength (Fig. 4.3) [107].38