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

the electrons localised on neighbouring magnetic atoms are coupled by theexchange interaction. As its name suggests, it proceeds directly withoutthe need for an intermediary. It is caused by the Coulomb energy differencebetween the singlet and triplet states of two 1 2spins. However, thereis usually not enough direct overlapping between electronic orbitals of thedistant spins to make the direct exchange the dominant mechanism in controllingthe magnetic properties; this is also the case in (Ga,Mn)As. Thisforces us to consider some kind of indirect exchange interactions, which cancouple magnetic moments over large distances. They come in many complicatedvariants, some of them being the Ruderman–Kittel–Kasuya–Yosida(RKKY) exchange, Stoner itinerant exchange, superexchange, double exchangeor Zener kinetic exchange and anisotropic exchange. In the followingsections, I will describe the types of indirect exchange most commonlyassociated with (Ga,Mn)As.4.2.1 SuperexchangeKramer’s superexchange occurs between magnetic ions separated by a nonmagneticatom [111]. In a crystal environment, an electron of the nonmagneticatom can be transferred to the empty shell of a magnetic ion and interact,through the direct exchange interaction, with electrons forming its localmoment. During this process, the nonmagnetic atom becomes polarised andcouples in the same way also with its other magnetic neighbours.3d 5 3d 5pFigure 4.1: Two Mn moments in Ga positions coupled by superexchange interactionvia the nonmagnetic As anion shell. The indirect interaction is a result of theexchange interaction between the electrons from the 3d 5 levels of both Mn ions andthe p electrons of a fully filled As shell. The virtual interatomic transitions of theelectrons (enhanced by the sp–d hybridisation) are marked by dotted lines.Figure 4.1 illustrates this situation in the (Ga,Mn)As lattice. The magneticMn ions, which substitute cation sites, are separated by nonmagneticAs anions. The spin-up electron from the As p orbital can jump to the Mnd orbital on the left. In this process the first Hund rule has to be fulfilled,which in the case of singly occupied Mn orbitals means that the electronhas to have the opposite spin to that of the d spins. At the same timethe remaining p electron with the opposite spin can jump to the right dorbital only if the Mn ion is oppositely polarised, i.e. anti-parallel to theleft Mn moment. As a result, the superexchange favours antiferromagnetic36